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IN MEMORIAM 
L. P. SHIDY 




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r 



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FORMULAS AND TABLES 



ARCHITECTS AND ENGINEERS 



CALCULATING THE STRAINS AND CAPACITY 



STRUCTURES IN IRON AND WOOD, 



BY 



F. SCHUMANN, C. E. 



ILLUSTRATXD WITH MOU THAW THBKB HVMDBBD DUGKAKB, SIKONKL Alii> 
BNQBATID KgPKOIALLT FOB THIS WOBK BT J. 0. LTOXS. 



WASHINGTON CITY: 

WARREN CHOATE A CO. 

1873. 

Digitized by CjOOQIC 



11 






Entered aooordtng to Act of Congress, in the year 18T3, by 

F. SCHUMANN. 
In the Offiee of the Librarian of Congress, at Washington. 



IN MEMORIAM 



a 



8TBEK0TTPKD BT 
H*OZLL * WRHBItOW, 



(li) Digitized by Google 






THIS VOLUME 

n 

RESPECTFULLY DEDIt)ATED 



A. B. MULLETT, 

(TOPZKTIBINO ABCHmCT OF THC V. 8. TBIASUBt DBPABTMillt, 



BY THE AUTHOR. 



922910 

Cm) 



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Google 



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PRE5FAOE. 



The following collection of Formnlas and Tables is intended as 
a ready reference for Architects and Engineers in compnting the 
Strains and Strength of Stmctnres in Iron or Wood. The anthor 
has attempted to give concisely all the necessary data required 
for the calculations. The object is, to give a compilation and 
transformation into simple formulas and tables, adapted to practice, 
of matter contained in the works of WeUbach, Bankine, BebJuihn, 
Bitter, Brer/man, Gordon, Brandt, Moll & Beuleaux, Lai8$le dt 
Schvihler, Fairbaim, and others. 

Among other matter may be particularly mentioned the tables 
giving the capacity of rolled and cast-iron beams, and the exten- 
sive table of formulas for the Moment of Inertia and Resistance 
of various sections. The formulas and constants for ascertaining 
the Strains in Roof and other Trusses will be found quite complete, 
extending to all the different forms or systems in general use. 
Under "Miscellaneous" is given necessary formulas for the cal- 
culation of Lines, Areas, and Solids; also tables of Natural Sines, 
Cosines, Tangents, Cotangents, Secants, and Cosecants, Area and 
Circumference of Circles, Weight of Materials, <fcc. 

WAsmroTOM, D. C, Mcweh, 1878. 



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I 



CONTENTS. 



ERBATA. 



30 



On page 4, 10th line from bottom, read — ~- instead of 30. 

100 

On page 4, 10th line from bottom, read 10.0036 instead ot 
10.036. 

On page 4, 14th, 15th, and 16th lines from bottom, read -— 

instead of a. 

On page 32, Fig. 70, insert I = distance between supports. 

On page 34, Fig. 72, insert I = distance between supports. 

On page 34, Fig. 74, insert I = length of beam. 

On pages 38 and 39 ti;^ total weight of beam between supports. 

On page 39, 5th line from top, read 1099000 instead of 1000000. 

On page 39, 5th line from top, read 1754 instead of 1757. 

On pages 144 and 145, in formulas for -H„, change places of last 
minus sign with foregoing plus sign. (See 13th line from top.) 

Page 145, lines 1 to 7 from bottom, ] Chanee places of Cand T 

Page 146, lines 1 to 3 from top, [ under strains in Figs. 

Page 146, lines 13 to 22 from top, J 225, 226, 227, and 228. 

^ Iw D 

On page 149, 1st line from bottom, read — -^- -= — ^ instead 

JS £1 — D 

- Iw 

On page 197, 7th line from bottom, read 3.14159 instead of 
1.14159. 
On page 204, 1st line from bottom, read A+ Ay instead of AAy. 

^ A^ I^J fVt it-, liu ^i'vvvivt/*^*^*'* ,V 

YTmgui* vi iuwiU{^ xvwuo.... • ...^^ 

Static and moving loads on bridges of wrought iron...... 192, 193 

(vii) 



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CONTENTS. 



PAOU. 

Strmmary of defini1*ofld and gWietfel priffdp^.. 1-5 

]4onients of inertia and resistanxj^ of vftriOtfd BfeCtloni. . . . 5-25 

SbBfngth of materials, &c v 2S-20 

Besistance to cross-breaking and shearing 29 

Capacity and strength of beams 2^99 

Presenre on supports 100-102 

Compressive strains and -pus^toe on snppo^ 102 

Sloping beams, raftetti, Ac 102-103' 

B^tanee to cmsfaing 103 

Strength of cohmrtft. pillkw, and sttnts 103-110 

Parallelogram of forces Ill 

Strains in frames 112-114 

Strains in boom derricks 114-115 

Strains in tmsses 115-121 

Strains in trussed beams 122-125 

Strains in trasses with parallel booms 126-146 

Strains in parabolic curved tmsses 147 

"Bow-string girders" 147-153 

Capacity and strength of parabolic arched beams or ribs 

originally curved 153, 154 

Strains in a polygonal frame 154, 155 

Strains in roof trusses 156-178 

Pressure of wind on roofs 178,179 

Pressure of snow on roofe 180 

Tie rods and bars 181, 183 

Joints or connections in iron constructions 184-186 

Dimensions of bolts and nuts 187, 188 

Compound strain in horizontal and sloping beams 188-190 

Weight of moving loads 191 

Static and moving loads on bridges of wrought iron 192, 193 

(vii) 



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via CONTEHTS. 

MISCELLANEOUS. 

PA0I8. 

Geometry 197-201 

Center of gravity of planes 202-204 

Trigonometrical formulas 205 

Trigonometrical functions 206-217 

Circumference, area, and cubic contents of circles 218-223 

Specific gravities of materials 224-226 

Weight of a superficial inch of wrought and cast iron... 227 

Weight per square foot of metals 228 

Weight of a lineal foot of flat and square, bar iron 229-233 

Weight of a lineal foot of rolled round iron 234 

Weight of bolts, nuts, and heads 235-237 

Weight of materials used in building 238' 

Divisions of a foot expressed in equivalent decimals 239 

Table for comparing measures and weights of different 

countries 240,241 

To cut the strongest and stiffest beam from a log 242 



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FORMULAS AND TABLlfig" ' " " ' " 



ARCHITECTS AND ENaiNEERS. 



Summary of Definitions and General Principles. 

External forces are those forces (loads, &c.) which caose or 
tend to cause the rupture of a structure. 

Internal torces are those forces which resist the external 
forces; when one balances the other, the structure is said to pos- 
8688 Stahiliiy. 

BXTKBITAL FOKOIfl. IHTBEHAl. FOKOW. 

ComprespiFe atraio. Resistance to CompressioiL 

Tensional strain. Resistance to Tension. 

Shearing strain. Resistance to Shearing. 

Gross-breaking strain. Resistance to Cross-breaking. 

Compression causes the material to fail by crushing or })uch' 
ling^ or both. 

Besistanoe to direct Crushing: In case pillars, blocks, struts, 
or rods, along which the strains act, are not so long in propor- 
tion to their diameter as to have a tendency to give way by 
bending sideways. This includes — 
Stone and brick pillars and blocks, of ordinary proportions ; 
Pillars, struts, and rods of cast iron, in which the length is 
not more than five times the diameter, approximatelv ; 

Pillars, struts, and rods of wrought iron, in which the length 
is not more than ten times the diameter, approximately ; 

Pillars, struts, and rods of dry timber, m which the length is 
not more than twenty times the diameter. 
Let W= Crushing load in lbs. 

C = Ultimate resistance of material to crashing in 

lbs. per square inch. 
A = Sectional area of pillar, Ac., in square inches. 

W 
Then will W=: A x C; and il = -7^ 

Tbvsioh, causes the material to be torn asunder. 
0) 



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2 .: ^^ '* DJUVStiij^'JifD OENBBAL PBIETOIFLES. 

" * ]V. . 

/, j?W^att(fe<ff;KaS-s, &4.,/« teaiing: the ultimate strength of a 
:b%C(*«> JeAHDg>i8> whw ' *• '•' 

T= Ultimate resistance of the material to tearing, in 
lbs. per square inch. 

Tr= Tearing load in lbs. 

A = Sectional area of bar, in square inches. 

W 
Then will W= A X T; 9,ndA = -^ 

Sheasing causes the fibres of the material to be shorn by each 
other ; when a bolt pulls out of its nut, the threads of the screw 
are said to be sheared. 

Resistance of bars, bolts, Ac, when sheared at one place, is: 
when 

8 = Ultimate resistance of material to shearing, in 

lbs. per square inch. 
pr= Shearing load in lbs. 
A = Sectional area of bar, Ac, in square inches. 

Then will W=^ A X S; and il = -^ 

Cboss-bbeaeiitg a beam, Ac, supported at one or both ends, 
with a force at right angles to its length, sufficient to cause rup- 
ture, is said to be broken across. 

Besktance to cross-breaking is the resistance of the material 
to compression, tension, and shearing combined ; — — . 

The flanges or booms, in beams or trusses, resist the bending 
moment, or moment of rupture. 

Tlie web or braces, in oeams or trusses, resist the shearing 
forces. 

Capaoitt means the load or pressure a structure is capable of 
sustaining with safety. 

Deflection is the displacement of a beam from a horizontal, 
caused by its own weight or the applied load, or both. 

Cambee is given a beam to counter balance the deflection, so 
that the beam may be horizontal when loaded ; the camber should 
be equal to the computed deflection. 

To find the effect of combining several loads on one beam, whose 
separate actions are known: lor each cross section, the shearing 
force is the sum of the shearing forces, and the bending moment 
the sum of the bending moments, which the loads would produce 
separately. 

When a load on a structure is partly static and partly moving, 
multiply each part of the load by its proper factor of safety, and 

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DEFISXTIOSS AND GENERAL FRIN0IFLE3. 



3 



add together the products : the sum will be the load to which the 
structure is to be adapted. 

For a Bridge with two platforms, one carrying a road and the 
other a railway, those two loads are to be combined. 

Of Iron Ties, Struts, and Beams. 

In designing ordinary structures of wrought iron, it saves time 
and expense to use iron bars of such forms of cross section as are 
usually to be met with in the market. An engineer should 
avoid introducing new sections for bars into his designs, except 
when, by so doing, some important purpose is to be served, or 
some decided advantage to be gained. The most common forms 
of rolled bars is shown by the following enumerated figures : 



No. of 
figure. 


Name of Form. 


Applicable for— 


4 


flniiArA iron....... 


Ties. 


13 

2 


Round iron 

Flat iron 


Ties, bolts, and rivets. 
Ties. 


29 


1 or double T-iron 


Beams, rafters, and stmts. 
Rafters and struts. 
Rafters and struts. 


30 
37 


Channel iron '. 

T-iron 


47 


L or anele iron 


Various purposes. 
Beams and rafters. 


1 


Deck Beam 









Avoid the use of cast iron for ties, also trussed cast-iron beams. 

When a member of a structure acts alternately as a strut and 
as a tie, it must have sufficient total sectional area, and sufficient 
stiffness, to resist the ^eatest compressive strain that can act, and 
sufficient effective sectional area to resist the greatest tensional 
strain which can act along it. 

Let all pins, bolts, rivets, <&c., exposed to a shearing strain, 
fit tight in its hole or socket. 

Roof trusses, the divisions of a rafter, and also the struts, may 
be considered as hinged at the ends. 

In members under a compound strain, as for instance a trussed 
beam with a distributed load, be careful to take into account the 
additional compression, caused^by the bending moment. 

The best distribution of the material in a section of a cast-iron 
beam, for cross-breaking, is that — = — ; or — ^ = —p=- 

8 8/ 8 jL 

When T=z Ultimate resistance of the material to tension. 

0= Ultimate resistance of the material to compression. 

8 = Distance from neutral axis to most extended fibres. 

8/ = Distance from neutral axis to most compressed fibres. 

That is, the fibres most in tension should be nearest the neutral 

axis of beam. 



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DBFIHITI0H8 AWD aUNEEAL PBIH0IPLE8. 



In wrongbt-iron beams, ibe section may be made alike above 
and below the neutral axis. 

The Modulus op Rupture should be applied to beams with 
full section, or beams with continuous web only ; for all open web 
beams, or beams with very thin web, the resistance of the mate- 
rial to crushing or tearing, respectively, must be used instead. 

Expansion and Contraction of long beams, which arise from 
the changes of atmospheric temperature, are usually provided for 
by supporting one end of each oeam on rollers of steel or hard- 
ened cast iron. The following table shows the proportions in 
which the length of a bar of certain materials is increased by an 
elevation of temperature from the melting point of ice (32° Fahr., 
or 0° Centigrade) to the boiling point of water under the mean 
atmospheric pressure, (212® Fahr., or 100° Cent. ;) that is, by an 
elevation of 180° Fahr., or 100° Cent.: 



METALS. 

Brass 0.00216 

Bronze 0.00181 

Copper 0.00184 

Cast iron aooiU 

Wrought iron 0.00120 

Tin 0.00225 

Zinr 0.00294 

Lead 0.00290 



EARTHY MATERIALS. 

Brick, common 0.00366 

Brick, fire 0.00050 

Cement 0.00140 

Glass, average 0.00090 

Granite 0.00086 

Marble- 0.00087 

Sandstone 0.00106 

Slate 0.00104 



Beference. 
Let u = Value given in above table, for a certain material. 
I = Length of a bar at 0° Centigrade, 
and ly= its length at a given number of degrees Centigrade. 
a = Given number of degrees, at which l^ is required. 
A = Superficial area of a plate ; 
and Ay= its area at a given number of 0° C. 

B = Cubic contents of a body, 
and By= its contents at a given number of 0° C. 
Then will ly= I (l-^au); 
Ay^zAh 4-2a w); 
By=z Bil-^Sau). 
Example : A bar of wrought iron 2 inches square, is 10 feet 
long at a temperature of 0° Centigrade ; what is its length at an 
increased temperature of 30° ? 
Ans : Z^ = 10 (1 + 30 X 0.00120) = 10.036 feet. 

The Neutral Axis, in a cross section of a beam, is that layer of 
fibres which are neither in compression or tension, by the action 
of a load on the beam ; it always passes through the centre of 
gravity of the section: provided the limits of elasticity of the 
material is not exceeded. A beam supported at both ends, with 
a load acting perpendicular between the supports, will cause the 
fibres above the nentral axis to be compressed, and those below, 
extended: the farther from the fibres to the neutral axis, the 
greater the stress. 

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XOHBHTS OF IHEBTIA AKB BESISTANGE. 5 

Under Moment of Inertia of a cross section, may be under • 
stood : an internal force at rest ; a static force resisting an exter- 
nal force ; it means the snm of all the area elements, multiplied 
by the square of their perpendicular heights from the neutral 
axis of tne section. The moment of inertia, which we have 
denoted with I, depends on the form and dimensions of the cross 
section, and is expressed in square inches. 

Moment of Resistance of a cross section is that static force 
resisting an external force of compression or tension ; it is equal to 
the moment of Inertia divided by the distance of the most ex- 
tended or compressed fibres, respectively, from the neutral axia. 



MOMENTS OF INERTIA AND RESISTANCE OF VARIOUS 
SECTIONS. 

To find the moment of inertia of any given cross section — 
First. Divide the section into as many simple figures as possi- 
ble. (See diagram, fig. 1.) 

Second. Find the moment of inertia of each of the simple figures 
about its own neutral axis, and insert the value in the following 
formula: 

Reference. 
Letters A, -4/, -4//, = area of simple figure, respectively ; and 
d, c//, c///, = its distance from its centre of gravity 
to that of the whole section. 
ti Vi V/i = moment of inertia of simple figures, re- 
spectively. 
For neutral axis see centre of gravity. 



Fi0.U 




Formula. 

(t^/ + d^J^A^/j + cfec, = moment 
of inertia of whole section. 



Moments of Inertia I and Moments of Resistance — 

Reference, 
m — n = neutral axis of section, 
r = radius. 

s = distance from neutral axis to most compreflsed or 
extended fibres. 
6, ^, &c. = dimensions. 
A = area. 

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16 



UOHBirrS OF iitebtia akd besistanob. 



No. of Section. 



No. of Figure. 



Form of Section. 



XXVI. 



31 




XXVII. 



32 



XVIII. 



33 



XXIX. 



34 



XXX. 



35 




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MOMENTS OP INEBXIA AND BESISTANCB. 17 



Moment of Inertia /. 


Moment of Resistance -£ 

8 


12 




ij[bh*-hh,*-{h-\)h,»^ 


^[hh^^b,h/-(h^h,)h,,^] 


^b[h»-K>] 


b {¥ - A/) 
6h 


A[M'-6V+6A'] 


-l-[&/t»~6V+M/] 



Vi[(6W-5A»)-(6/>»)] 



6h 



db, Google 



i2 



MOMENTS OF INEBTIA AND BESISTAJSCOK. 



No. of Section, 



No. of Figure. 



Form of Section. 



XLI. 



49, 50, and 51 




— ^-H 1^— i~^ ,kr-i ^ 



XLII. 



XLIII. 



XLIV. 



52 



53 



54 





XLV. 



56 




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MOMENTS OF nTEBTIA ANB SESISTANOB. 



23 



Moment of Inertia /. 


Moment of Resistance -/ 
» 


(bh^ - bX^) ' - ^bJib.hy ( Wv) 2 


(6A»- 6,V)«- 4W>, (A-M' 


12 (bn — b,h,) 


b {bh^ + byh,^^ 2b,hh,) 


3^ r*%/3=: 0.5413 r* 


I 


^+1^^ H^ 0.6381 r* 


I 
ih 


= 0.5413 (r<—r/) 


I 
ih 


1 + 2\/2"(r* — r/) 
= 6381 (»^—r/) 


I 

ih 



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24 



MOMENTS OF INEBTIA A.HD BESISTANOE. 



No. of Section. No. of Figure. 



XLVI. 



XLVII. 



66 



57 



Form of Section. 




;<■— ■^— 




XLVIII. 



58 



J^-^\ 




XLIX. 



59 



60 




-^ 




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MOMENTS OF INERTIA AHD BESHSTAKOB. 



Moment of Inertia /. 



n, = number of sides. 
^ n^r^ ain.v (2 + cos.v) 



n y ss number of sides. 
b = length of a side. 



l2 









Moment of Resistance JL 



^n^r^ sin.v (2 + cos,v) 



A-~-(3A»+i6«) 



6ii 



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26 



BTEENGTH OF MATEEIALS. 



STRENGTH OF MATERIALS, &a, 
In IWs;, ftve4rd«pois, per square inch of «ross-8ectk>n. 







Ultimate Resistance to— 


li 


Materials. 


TeariDg. 


Cruahinc. 


BiMuteK 


Cws-brt 
Modulus of 




Metals. 


505.7 

533 

o24 

537 
549 

445 
434 
to 
456 

481 
490* 


18000 
49000 
3UO0O 

19000 
:^>(K)00 
ZAiOO 
GOOOO 
16500 
13400 

to 
29000 

65000 

*5ioiK> 

35700 

28600 

6«»000 

to 
70000 
64(KM) 
7(M)00 

to 
100000 
«K)000 


10300 

117000 

112006 

80000 

to 

145000 

36006 
to 

mm 


......... 




9170000 


" wire. 

Bronze or gun metal, (cop- 
per 8, tin 1) 


14230000 
9900000 







" slio**t. 




bolts 


17000000 




27700 


noooiKX) 




14000000 


" beams, average... 
" " op*»n work. 
" solid rect. bars, 
various quallities. 

Iron, wrouglit, average 


28800 
17000 

a-jooo 

to 
43500 

38000 

*8o6bo 


to 
22900000 


50000 




" plates 

joints, d'ble 
riveted. 
Iron, wrought, joints, single 

riveted. 
Iron, wrought, bars and 
bolts. 

" hoop, best best 







29000000 
25300000 


" wire ropes.... 
flteel av<*rftflre . . 







15000000 


*' bars 


100000 

to 
130000 
80000 
3:100 
4000 
7000 

to 
8oOO 

17000 

6300 
11500 


12000 

7730 
15500 


15000 


29000000 


" plates 






to 
42000000 


Lead, sheet 

Tin ca<>t 


712 

402 

43a 

47 

25 
43 


720000 





4000000 




13000000 


Timber, (well sea-soned and 

dry.) 
j^sh 


9000 
9360 


1400 


12000 

to 
14000 

9000 

to 

2000U 


1600000 


Bamboo 




Beech 


1350000 









Digitized by VjUU^ It: 



8TSEK0TH OF MATERIALS. 



27 





IS 


Ultimate resistance to— 




Materials. 


T««tog. 


GfMbi>t. 


SlMWiH- 


CroM-br'k 
Modulus ui 
Rupture. 


II 


Timber— Continued. 
Birch 


44 

60 
33.4 

34 

74.6 
37 

37 

33 

52 

47 

62.6 

44 

62 

35 

49 
52.6 

47.4 

42 

54 

346 

29 

37 

48 

62.6 
40 
26 
60 

126 

136 
137.6 

luo 

89 


151 KM) 
2CKMK) 
KHMHJ 

to 

131 KHJ 
140(10 

12000 

to 
14(MI0 
12400 

9(KiO 
to 

26000 
2(MM)(J 
2:M4K) 
16000 
IIWX) 
80<J0 
to 
21800 

lor,ot) 

10000 

o 
19800 

iSoiio 

10250 
11500 

150(M> 
130(10 
15000 

"'soTio 

14000 
8000 

280 

• to 

300 

280 

to 
300 


6400 
io:ioo 
6300 

10300 

mm 

6;i75 
to 
62(K) 
6U00 

6670 

11000 
7300 

"9000 
9900 
8200 

6500 
10000 

7700 
6100 
6000 
6300 
5400 
12000 
120U0 

11000 
6600 
4000 

650 
to 
800 
lino 
1700 
417 
to 
612 




11700 

1066 

600(> 

to 
9. Ovi 
2700 . 
7100 

to 
954U 
990.. 
to 
ll;yOt> 
60.0 

to 
lOOOC 

1735I'- 
11.00 
1200 
10000 

10000 

to 
IStOo 

10609 

9600 
12«»00 

to 
19000 
174tK) 

6600 


1645000 


Box 






11-10000 


Elm 


1400 


700000 


Tlhonv Went Indian 


to 
1340000 


Fir Red Pine 


506 
to 
800 
600 

970 

to 

1700 


146 000 


*• Spruce 


to 
1900000 
1400 00 


** Larch 


to 
1800000 
90000Q 


Hickory 


to 
1360000 
1.040000 


Lance wood 




Locust ~ 

Lifrnum vitffi.. 








Mahogany 

Maple... 


1256000 


2300 




Oalc British..*.*.* -*.' 


1200000 


" Dantzic 


to 
1750000 


" American white 






«* " red 


•^150000 


Pine, American, white 

" yellow 

Svcamore .. . 






Teak. Indian 




2400000 


Water ffum— •••••.•••»•••••••••••«•• 







Walnut 




Yew 






Stokks, (natural and arti- 
ficial.) 
Briok. weak red... 




" strong red 








** work 




Cement .•••• 









d by Google 



28 



STRENGTH OF MATERIALS. 





'fl 


Ultimate resistance to— 




Materials. 


T«rii«. 


Cuhbig. 


ShMriag. 


tfodohuof 
Rupture. 


11 


Stonbs— Continued. 

Chalk — 

Gla8«» 


146.6 

173 

168 

172 
197 

109 
116 

144* 

ifs" 


118 
9400 


330 

"6500 

to 

11000 

6500 

4000 

to 
4600 

About 
4-10 cut 
stone. 
5600 
3300 
to 4400 




2860 

1100 
6000 


8000000 


Granite 






., 




" granular 




Mortar, hydraulic 


100 
to 

170 
60 




" ordinary. 




Sandsit-one stronir. ^ 






ordinary « 

** weak 

Slate « 






""wm 

to 
12800 

26000 
14000 
6300 
4200 
6200 
7700 


13000000 


Miscellaneous. 

Flaxen yarn 

Hempen ropes 






to 
16000000 


Hide« ox, undressed 

Leather, ox « 

Silk fibre 







Whalebone 








Modulus of Rupture R. 
According to Professor Rankine, the modulus of rupture is 
eighteen times the weight that is recjuired to break a bar of a 
given material one inch scfuare (section) and one foot between 
supports, the weight concentrated at the middle. 

Modulus op Elasticity £ 
Is that power (in lbs. generally) through which a prismatic body 
of a given material, of section := 1, is assumed to be extended 
double its length, or compressed to 0. 

Let A = Sectional area of a rod of the material. 

Wss Weight or power in lbs., which causes the eztextsion 

or compression of the rod. 
I s: Length in inches of rod before W is applied. 
Y as The extension or compression of the rod in inches, 
caused by W. 

Thenwill^ = -^^-;r = -2^./- 



y Google 



BESISTANOE TO CROSS -BBEAEING AITD 9HEABINQ. 29 

Factors of Safety k. 

The ultimate resistance of material should be divided by — 

^ Wrouglftl^in!"** ^°^ P'*^^^ Strength. For Working Stress. 

Steady load *. 2 3 

Moving load 4 to 6 

Cast Iron. 

Steady load , 2 to 3 3 to 4 

Moving load , 6 to 8 

Timber. 

Average 3 8 to 10 ' 



RESISTANCE TO CROSS-BREAKING AND SHEARING. 

Capacity and Strength of Beams. 

Reference, 
A = Area of cross-section of beam. 
D = Deflection of beam from a horizontal. 
E = Modulus of elasticity. 
/= Moment of inertia of cross section. 
Jlf = Maximum moment of rupture, or bending moment. 
B = Modulus of rupture. 
S = Vertical shearing force. 
V = Pressure on supports. 
IF = Capacity or weight of load. 
c, d, l^=i Dimensions in units of length. 
k = Factor of safety. 
w = Weight of load per unit of length. 

= Moment of resistance of cross-section. 

8 

It I 

For the stability of a beam: ifss jr=s — r- . . 

k 8 

The web of a metal beam must have sufficient area to resist the 

Sk 



shearing force 8; that is, A -. 



Ultimate resistance to shearing. 



The weight of the beam must be added to TT, except in small 
beams, under 60 lbs. per lineal foot, when it may be disregarded. 

[Note. — Always oso the same units of dimensions or weight.] 

Digitized by LnOOQlC 



36 EESISTAUOE TO CROSS- BREAKING AND SHEARING. 



77 



78 



Manner of loading and feistening beams. 



.-d^A 



-c — M 



'm 






Maximum moment of 
rupture M. 



Whencf <(i — c); 



Whenc?>(i — c); 
W 

—d-d)' 



-i — 




3^ 



=^p 



w- 



,V(3;- ;,)(;-;,) 



2P 



w 



hHi-h) 






d by Google 



EESISTANCE TO CROSS- BREAKING AND SHEARING. 



37 



Capacity W of any 
section. 


MAximum 
deflection D. 


foci 

•-.is 
gal 


Shearing 
force & 


Pressure 
on sup- 
ports V. 


^ K 










2K 
{l-d)' 


... .%.. . 




W 








2^ rr 
ii'i.si-h){l-h) 


SB.I • 

























gitizedby^jUUyH:: 



3d BESISTAirOE TO OBOSS-BBEAEIKG AND SHEABUTO. 

Example. — Capacity of wrought-iron l-shaped beams ; top and 
bottom flange alike ; load equally distributed ; ends not fixed. 

Dimensiona of OroBt-section, 

h =s Height = 10 inches. 
b = Width of flange = 4 inches. 
t s= Thickness of nange =s 0.8 inches. 
ty =s Thickness of web = 0.5 inches. 

hy=sh — 2t: by==b — ty. 

Distance between supports -= 20 feet =3 240 inches. Factor of 
safety = 3. 

Moment of Resistakce. 

I b¥-^b,h,^ 4X10»-3.5X8.4» _^. 

— = 6A elTio =^^-^^- 

Capacity W. 
ti; =r (4 X 0.8 X 2 + 8.4 X 0.5) X 240 X 0.28 = 712.32 lbs. 

. 32.09 = 406473.33. 



712.32 = 12836.72 lbs. 



Example. — Capacity of cast-iron X-shaped beams; load equally 
distributed; ends not fixed; flange down. 

Dimermons of Cross-section. 

Let h = Height = 18 inches. 

b = Width of flange = 9 inches. 
t = Thickness of flange = 1.25 inches. 
ty = Thickness of web = 1 inch. 

hy ^ h t; b/ ^=S b ty. 

Area = 28 square inches. Distance between supports = 20 
feet = 240 inches. Factor of safety ^ =: 4. 

Moment of Resistance. 
/ _ r (W — M/)' ^bhbyky (h — h,y -1 

~8 * L W — 2byhh, + byhy^ bh^ — 2b,hhj. + byh,^ J 

(9 X 18^ — 8 X 16.752)« 




-'[ 



9 X 18» — 2 X 8 X 18 X 16.75 + 8 X 16.752 
4X9X18X8X 16.75 (18 -- 16.75)» n 
9 X 18« — 2 X 8 X 18 X 16.75+ 8 X 16.75«J 

Digitized by V^OOQIC 



RESISTANCE TO CROSS- BBEA KING AND SHEARINO. 39 

452256.25 135675.00 



r 452256 .2 
' * L 336.5 



336 5 J-'^'- 



Capacity W. 
«; = 28 X 240 X 0.261 = 1754. lbs. 

E I 28000 

K= 4^ . _ = 7^ . 157 = 1099000. 
k 8 4 

rF=84-«,= 8.-i20^_1757 = 34879 1bs. 

For light beams no attention need be paid to weight of beam w. 

Capacity IT of Rolled I -shaped Beams. 
Load equally distributed. 

The calculations are based upon the patterns or section^ used 
by the Phoenixville Iron Company. Practically this applies to 
all similar beams rolled in the united States, the difference in the 
profile of section being slight. 

In the following table the factor of safety k = 2.53: 

Beference. 

W= Load in tons of 2,000 lbs., equally distributed. 
w =s Weight of beam in tons of 2,000 lbs. 
L = Distance between supports in feet. 
I =^Distance between supports in inches. 
w^= Weight per square foot of floor. 

TF^= Capacity of coupled or trebled beams in tons of 2,000 lbs. 
D = Deflection in inches at centre, between supports. 
d = Distance between centres of beams, when spacing for 
floors, in feet. 

.^ ^ K ^, R I E 38000 ,.^^„ 

IF= 8 . w, K-ss. —— . — , -r-= -^r-— - = 15000 lbs. = 

I k s k 2.53 

7.5 tons. d=: — , orc/=:-_ '^-, i)=J — J . — -. 

L.Wy L.Wy "^ E.I 4& 

K^ = Constant, computed by formulas. (See under examples.) 

Digitized by LnOOQlC 



40 KESISTA5CE TO CBOSS BBEAKIKG AZTD SHEARING. 

Tlie rivets for coupled or trebled beams sbould be about J inch 
in diameter, and 8 inches apart. 

Trebled Beams. • 

Coupled Beamsr. 



Tr,= W^X5.33. Mp Wy = Wx2.*7. 



Fig 80. 

Fig. 79. 

Examples explanatory of the following Table. 

Example. — What is the capacity of a 15-inch light beam, load 
equally distributed, distance between supports = 20 feet? 

A'l = — — — , and TF= —=r- ; for 15-inch light beam —y- =s 

— '-■ — = 17.2 ioTis. This is also found at the intersection of 
20 feet and column under capacity W. 

Example. — What distance apart should 9-inch medium beams 
be placed, the distance between supports being 20 feet, and to 
carry a total load of 140 lbs. per square foot of floor surface? 

Ans. 4.4 feet; being found at the intersection of the horizontal 
line from 20 feet and the vertical column under 140 lbs. 

Example. — Wliat is the capacity of 12 inch light beams trebled, 
load equallv distributed, distance between supports = 25 feet? 

Ans. W for 12-inch light beam = 9.19 and Wy=Wx 6.33 = 
9.19 X 5.33 = 48.98 tons. 



y Google 



BE3ISTANCE TO CROSS-BREAKING AND 8HEAEIN0. 41 



Capacity of Rolled Beams. 

Explanation of Tables for I Beams. 

The first column gives the distance between sapports in feet. 

The second column gives the capacity iu tons of 2,000 lbs., 
equally distributed. 

The third column gives the deflection in inches at centre of 
beam. 

The fourth column gives the weight of beam in lbs. for length 
between supports. 

The fifth to fifteenth column (inclusive) gives the distance in 
feet that the beams should be spaced from centre to centre, for 
weight in lbs., per sq. ft. of surface for floors. 

Pounds in decimals of a ton. 

lbs. tons. 

60 = 0.03 

70 = 0.035 

80 = 0.04 

90 = 0.045 
100 = 0.05 
140 = 0.07 
160 = 0.08 
180 = 0.085 
200 = 1 
250 = 0.125 
300 = 0.15 

In using these beams for floors, with brick arching, the ends 
resting on supports should have a bearing of about 8 inches, 
resting on a cast-iron plate, 8 X 12 in. sq,, by 1 in. thick. 

Tie rods should be used where floors are subject to heavy con- 
centrated moving loads, (as trucks with merchandise, &c. ;) these 
rods should be about 8 times the depth of beam apart, fastened 
about J from the bottom of beam. 

When beams are used to support walls, or as girders to carry 
floor beams, and put side by siae (II,) they should be fastened to- 
gether with cast-iron blocks, fitting between the flanges, so as to 
securely combine the two beams. The blocks may be put about 
the same distance apart as the tie-rods. 



y Google 



RESISTANCE TO CROSS BBEAEINO AND SHEARING. 



16" 



Fig. 81. 

I 

6.31" 



W^ "Heavy " Beam. Weight per If. == 66.66 lbs. 



Sectional area -* 20.0" 

Moment of inertia / -* 662.42 

a7" Ck>n8tantir' —434.95 

K' 

W . 

L 



I 


a 


1 




Distance d bet. centres of beams in feet, for 


1 


weight in lbs. per sq. foot of— 


9i 


'^ 


5 


B 




.^ 
























i 


1 


c5 


1 


1 

i 


i 


1 

8 


i 


i 


1 

i 


i 


1 


0? 


1 


1 


6 


72.49 


0.037 


400.0 
























7 


62.13 


0.050 


466.6 
























8 


64.35 


0.066 


6333 
























9 


48 32 


0.084 


600.0 
























10 


43.48 


0.104 


666.6 
























11 


39.54 


0.126 


733.3 
























12 


36.24 
33.45 
31.05 
28.99 


0.150 
0.177 
0.205 
0.236 


800.0 
866.6 
9a3.3 
1000.0 






















20.1 
17.6 


13 























17.7 
15.6 


14 


















22.1 
19.3 


14.7 
12.8 


16 


*""* 




".". 


.!!". 


!!!.". 







22.3 


16 


27.18 


0.270 


1066.6 

















21.2 


18.8 


1C.9 


13.6 


11.3 


17 


25 58 


0305 


1133.3 












...... 


21.4 


19.0 


17.0 


15.0 


12.0 


10.2 


18 


24.16 


0.342 


1200.0 














19.1 


16.7 


14.9 


13,4 


10.7 


8.9 


19 


22.89 


0.383 


1266.6 




"."!. 










17.6 


15.2 


13.4 


12!o 


9.6 


8.0 


20 


21.73 


0.426 


1333.3 










21.7 


15.6 


13.5 


12.0 


10.8 


8.6 


7.2 


21 


20.71 


0471 


1400.0 








22.0 


19.7 


14.7 


12.5 


11.6 


9.8 


7.9 


6.7 


22 


19.58 


0.515 


1466.6 









19.7 


17.8 


12 7 


11.1 


9.8 


8.9 


7.1 


6.9 


23 


18.91 


0.669 


1633.3 






21.0 


18.9 


17.1 


11.8 


10.5 


9.4 


8.2 


6.7 


6.6 


24 


18.12 


0.623 


1600.0 




2*1.5 


18.8 


IC.7 


15.1 


10.7 


9.4 


8.3 


7.5 


6.0 


60 


25 


17.39 


0.677 


1666.6 





19.9 


17.3 


16.5 


14.4 


10.2 


8.6 


7.7 


6.0 


5.6 


4.7 


26 


16.72 


0.735 


1733.3 


21.4 


18.3 


16.7 


15.2 


12.8 


9.2 


8.0 


7.2 


6.4 


6.2 


4.2 


27 


16.10 


0.795 


1800.0 


19,8 


17.1 


14.9 


13.4 


11.9 


8.6 


7.4 


6.7 


5.9 


4.7 


3.9 


28 


15.63 


860 


1866.6 


18.2 


15.8 


13.8 


12.3 


11.0 


7.9 


6.9 


6.2 


5.6 


4.4 


3.6 


29 


14.99 


0.925 


1933.8 


17.2 


14.8 


12.9 


11.5 


10.7 


7.4 


6.8 


5.7 


6.1 


4.1 


3.4 


80 


14.49 


0.994 


2000.0 


16.1 


13.8 


12.0 


10.7 


9.8 


6.9 


6.0 


5.3 


4.8 


3.8 


3.2 


31 


14.03 


1.067 


2066.6 


16.0 


12.9 


11.3 


10.0 


9.0 


6.4 


5.6 


5.0 


4.6 


3.6 


3.0 


82 


13.69 


1.141 


2133.3 


14.0 


12.0 


10.0 


9.4 


8.4 


6.0 


6.3 


4.7 


4.2 


3.3 


2.8 


33 


13.17 


1.219 


22U0.0 


13.3 


11.4 


9.9 


8.8 


7.9 


6.7 


4.9 


4.4 


3.9 


3.1 


2.6 


34 


12.79 


1.304 


2266.6 


12.5 


10.7 


9.4 


8.2 


7.5 


63 


4.7 


4.1 


3.7 


3.0 


2.6 


86 


12.42 


1.384 


2:^33.3 


11.8 


10.1 


8.8 


7.9 


7.1 


6.0 


4.4 


3.9 


3.6 


2.8 


2.3 


86 


12.08 


1.473 


2400.0 


11.1 


9.5 


8.4 


7.4 


6.6 


4.7 


4.1 


37 


3.3 


2.6 


2.2 


37 


11.75 


1.664 


2466.6 


10.8 


9.1 


7.9 


7.0 


e.3 


4.4 


39 


*3.6 


3J 


2.6 


2.1 


38 


11.43 


1.656 


2533.3 10.0 


8.6 


7.6 


6.6 


6.0 


4.2 


3.7 


3.3 


3.0 


2.4 


2.0 


89 


11.16 


1.754 


2600.0 9.6 


8.1 


7.1 


6.3 


6.7 


4.0 


3.6 


3.1 


2.8 


2.2 


1.9 


40 


10.87 


1.854 


2666.6 9.0 7.71 6.7 
1 1 


6.0 


6.4 


3.8 


8.3 2.9 


2.7 


2.1 


1.8 



y Google 



BESISTAKCE TO CROSS BBEAEINQ AND SHEABINQ. 



43 



Pig. 82. 

,* — >, 



W^ "Lighr Beam. Weight per If. = 51.66 lbs. 



Sectional area — 15.5" 

Moment of inertia / -= 517.78 
Constant K' = 346.10 

IF-—. 
L 



4.12 



^ 


s 


i 




Distance d bet. centres of beams in feet, for 


oL 


1 




i 


weight in lbs. per sq. foot of— 




it 


a 


.s 


£> 






k 


a 


fl 
























5_ 


i 

6 


1 




i 
i 


1 


OQ 

i 


1 

i 


1 
1 


i 


i 




1 


1 


1 


6 


57.52 


037 


310.0 
























7 


40.31 


0.050 


.361.0 
























8 


43.13 


0.005 


41.3.3 
























9 


38.35 


0.084 


465.0 
























10 


34.50 
31.38 


0103 
0.124 


516.6 
567.3 






















23.0 


11 












.!..*." 








22.9 


19.0 


12 


28.76 


V 0.150 


620.0 








!!!.*.! 












10.1 


15.9 


18 


26.55 


0.176 


671.6 






.... 










'22.7 


20.*4 


16.4! 13.6 


14 


24.65 


0.205 


723.3 














22.0 


19.5 


17.0' 14.0! 11.7 


16 


23.01 


0.236 


776.0 













21.9 


19.1 


17.0 


16.3 12.3 10.2 


16 


21.57 


0.269 


806.6 




".**' 








19.2 


16.8 


14.9 


13.4' 10.7 


8.9 


17 


20.30 


0..304 


858.3 














17.9 


14.9 


13.2 


11.9 


9.5 


7.9 


18 


19.16 


0.341 


930.0 










21.'3 


16.2 


13.3 


11.8 


10.6 


8.6 


7.1 


19 


18.15 
17.24 


0381 
0.424 


981.6 
1033.3 








2li 
19.1 


19.1 
17.2 


13.6 
12.3 


11.9 

10.7 


10.6 
9.6 


9.5 

8.6 


7.6 
6.9 


6.3 


20 






21.5 


6.7 


21 


16.43 


0.469 


1086.0 






19.6 


17.4 


15.6 


11.1 


9.8 


8.7 


7.8 


6.2 


6.2 


22 


15.68 


0.515 


1136.6 




20.3 


17.8 


15.8 


14.2 


101 


8.9 


7.9 


7.1 


5.7 


4.7 


23 


15.00 


0565 


1187.3 


2iV7 


18.7 


16.3 


14.5 


13.0 


9.3 


8.1 


7.2 


6.5 


5.2 


4.3 


24 


14.38 


0.620 


1240.0 


19.9 


17.1 


14.9 


13JJ 


11.9 


8.6 


7.4 


6.6 


69 


4.8 


3.9 


26 


13.80 


0.674 


1291.6 


18.4 


15.8 


13.8 


12.3 


11.0 


7.8 


6.9 


6.1 


65 


4.4 


3.6 


26 


13.27 


0.732 


1343.3 


17.0 


14.5 


12.8 


11.3 


10.2 


7.2 


6.3 


6.6 


5.1 


4.0 


3.4 


27 


12.78 


0.791 


1395.0 


16.7 


1.3.0 


11.8 


10.6 


9.4 


6.7 


6.9 


6.2 


4.7 


3.7 


3.1 


28 


12.32 


0.855 


1446.6 


14.6 


12.5 


11.0 


9.7 


8.8 


6.2 


6.5 


4.8 


4.4 


3.6 


2.9 


29 


11.93 


0.921 


1498.3 


13.7 


11.8 


10.2 


9.1 


8.2 


6.8 


6.1 


4.6 


4.1 


3.2 


2.7 


30 


11.50 


0.989 


1560.0 


12.7 


10.9 


9.5 


8.5 


7.6 


6.4 


4.7 


4.2 


3.8 


3.0 


2.6 


81 


11.13 


1.060 


1601.6 


11.9 


10.3 


8.9 


8.0 


7.1 


5.1 


4.4 


3.9 


3.5 


2.8 


2.3 


82 


10.78 


1.133 


1653.3 


11.2 


9.6 


§.4 


7.4 


6.7 


4.8 


4.2 


3.7 


3.3 


26 


2.2 


83 


10.46 


1.211 


1706.0 


106 


9.0 


7.9 


7.0 


6.3 


4.6 


39 


3.6 


3.1 


2.5 


2.1 


84 


10.14 


1.292 


1766.6 


9.9 


8.5 


7.4 


6.6 


6.9 


4.2 


3.7 


3.3 


2.9 


2.3 


1.9 


86 


9.86 


1.375 


1808.3 


9.3 


8.0 


7.0 


6.2 


5.6 


4.0 


3.5 


3.1 


2.8 


2.2 


1.8 


86 


9.58 


1.463 


1860.0 


8.8 


7.6 


6.6 


6.9 


6.3 


3.8 


3.3 


2.9 


2.6 


2.1 


1.7 


87 


9.32 


1.553 


1911.6 


8.3 


7.2 


6.2 


6.G 


6.0 


3.5 


3.1 


2.7 


2.5 


2.0 


1.6 


88 


9.08 


1.645 


1963JI 


7.9 


6.8 


5.9 


6.3 


4.7 


3.4 


2.9 


2.6 


2.3 


1.9 


1.6 


89 


8.86 


1.742 


2016.0 


7.6 


6.5 


6.6 


6.0 


4.5 


3.2 


2.8 


2.5 


2.2 


1.8 


1.4 


40 


8.62 


1.841 


2066.6 


7.1 


6.1 


5.3 


4.7 


4-3 


8.0 


2.6 


2.3 


2.1 


1.7 


1.4 



y Google 



44 



EESISTANCE TO CROSS- BREAKING AUD SHEARINQ. 



Fig. 83. 



12^''' ''Heavy'' Beam. Weight per If. = 66.66 lbs. 




Sectional area. — 17.0" 

Moment ol inertia I =- 373.53 
Constant K' — 311.28 

K' 
W . 

L 



6.6" 



1 


s 


in 




Distance d bet, centres of beams 


in feet, for 


3 © 


3 




1 


weight in lbs 


. per sq. foot of— 




























^•2 


1 


c5 

© 


1 


00 


CO 


.o 


1 


OD 
JO 


1 


oS 


1 


1 


1 


1 


.2 


Q • 


3 


i 


i 


i 


i 


1 


o 


i 


1 


1 


1 


1 


6 


51.88 


0.046 


340.0 
























7 


44.54 


0.063 


396.6 
























8 


38.70 


0.082 


453.8 
























9 


34.68 


0.105 


510.0 
























10 
11 
12 
13 


31 12 


0.131 


560.6 






















20.7 
17.1 


28!29 
25 94 


0'.168 
0.188 


62.3!3 
680.0 




















20.6 
17.2 


















21.6 


14.4 


23!94 


0^222 


736!6 




.... 










23.0 


20.4 


18.4 


14.7 


1212 


14 


22.22 


0258 


793.3 












2*2.V 


19.8 


17.6 


15.8 


12.0 


10.6 


15 


20.76 


0.297 


850.0 













19.7 


17.2 


15.3 


13.8 


11.0 


9.2 


16 


19.50 


0.339 


906.6 













17.4 


15.2 


13.5 


12.1 


9.7 


8.1 


' 17 


18.31 


0.383 


963.3 










21J6 


15.3 


13.4 


11.9 


10.7 


8.6 


7.1 


18 


17.29 


0.431 


1020.0 








21.Vi 


19.2 


13.7 


12.0 


10.6 


9.6 


7.6 


6.4 


19 


16.38 


0.481 


1076.6 






21.5 


19.1 


17.2 


12.3 


10.7 


9.6 


8.6 


6.8 


6.7 


20 


15.61 


0.538 


1133.3 






19.5 


17.3 


15.6 


11.1 


9.7 


8.6 


7.8 


6.2 


5.2 


21 


14.82 


0.592 


1190.0 




20.1 


17.6 


15.C 


14.1 


10.0 


8.8 


7.8 


7.0 


5.6 


4.7 


22 


14.14 


0.652 


1246.6 


21.4 


18.3 


16.0 


14 2 


12.8 


9.1 


8.0 


7.1 


6.4 


5.1 


4.2 


23 


13.63 


0.717 


1303.3 


19.6 


16.8 


14.7 


13 


11.7 


8.4 


7.3 


6.6 


5.8 


4.7 


3.9 


24 


12.97 


0.786 


1360.0 


18.0 


15.4 


13.5 


12.0 


10.8 


7.7 


6.7 


6.0 


5.4 


4.3 


ae 


25 


12.45 


0.855 


1416.6 


16.0 


14.2 


12.4 


11.0 


9.9 


7.1 


6.2 


5.5 


4.9 


3.9 


3.3 


26 


11.97 


0.927 


1473.3 


15.3 


13.1 


115 


10.2 


9.2 


6.5 


5.7 


5.1 


4.0 


ac 


3.0 


27 


11.52 


1.003 


1530.0 


14.2 


12.1 


10.0 


9.4 


8.5 


6.0 


5.3 


4.7 


4.2 


a4 


2.8 


28 


11.11 


1.084 


1586.6 


13.2 


11.3 


9.9 


8.8 


7.9 


5.6 


4.9 


4.4 


3.9 


3.1 


2.6 


29 


10.73 


1.170 


1643.8 


12.3 


10.5 


9.2 


8.2 


7.4 


6.2 


4.0 


4.1 


3.7 


2.9 


2.4 


30 


10.37 


1.257 


1700.0 


11.5 


9.8 


8.0 


7.6 


6.9 


4.9 


4.3 


as 


3.4 


2.7 


2.3 


31 


10.04 


1.350 


1756.6 


10. 


9.2 


8.0 


7.1 


6.4 


4.6 


4.0 


3.6 


3.2 


2.5 


2.1 


32 


9.71 


1.443 


1813.3 


10.1 


8.6 


7.5 


67 


6.0 


4.3 


3.7 


3.4 


3.0 


2.4 


2.0 


33 


9.43 


1.546 


1870.0 


0.6 


8.2 


7.1 


6.3 


5.7 


4.0 


3.5 


3.1 


2.8 


2.2 


1.9 


34 


9.1C 


1.650 


1926.6 


8.9 


7.6 


6.7 


5.9 


5.3 


3.8 


3.3 


2.9 


2.6 


2.1 


1.7 


85 


8.89 


1.758 


1983.3 


8.4 


7.2 


6.3 


5.6 


5.0 


3.6 


3.1 


2.8 


2.6 


2JQ 


1.6 


36 


8.64 


1.871 


2040.0 


8.0 


6.8 


6.0 


5.3 


4.8 


3.4 


3.0 


2.6 


2.4 


1.9 


1.6 


37 


8.41 


1.987 


2096.6 


7.6 


6.4 


6.6 


5.0 


4.5 


3.2 


2.8 


•2.5 


2.2 


1.8 


1.5 


38 


8.18 


2.104 


2163.3 


7.1 


6.1 


5.3 


4.7 


4.3 


3.0 


2.0 


2.3 


2.1 


1.7 


1.4 


89 


7.98 


2.234 


2210.0 


6.8 


5.8 


5.1 


4.6 


4.0 


2.9 


2.5 


2.2 


2.0 


1.6 


1.3 


40 


7.78 


2.836 


2266.6 


6.4 


5.5 


4.8 


4.3 


3.8 


2.7 


2.4 


2.1 


1.9 


1.6 


1.2 



y Google 



BESISTANOS TO <:!BOSS BBEAEINO AND SHEABIKG. 



45 



Fig. 84. 
4.75" 



12^^ ''Light" Beam. Weight per If. =41.66 lbs. 



Sectional area -= 12.5" 

Moment of inertia J= 275.92 
0.46" Constant IC = 229.94 

W . 

L 



o © 


s 






Distance d 


bet. centres of beams 


in feet, for 


s 

d 


d 
d 


1 

Ha 


weight in lbs. 


per sq. foot of— 




tt 
























^.2 


i 


i 


§ 


« 


CD 


(B 


00 


i 


i 


s 


i 


1 


1 


1 


m 




vi 


s 


.o 


Xi 


.o 


X> 




s 












P 


s 


& 


^ 


s 


i 


i 


i 


i 


g 


1 


i 


i 


i 


§ 


6 


39.31 


0.047 


250.0 




















7 


32.84 


0.063 


291.6 
























8 


28.74 
25.54 
22.98 
20.90 


0.083 
0.105 
0.131 
0.158 


a33.3 
375.0 
416.6 
458.3 






















24.0 


9 




















23.6 
18.3 
15.2 


IS,Q 


10 


















*22.*6 
19.0 


15.3 


11 










...z 




'^'.6 


21.6 


12.6 


12 


19.16 
17.68 
16.42 
15.32 
14.37 


0.189 
0.222 
0.258 
0.297 
0.339 


600.0 
641.6 
683.3 
625.0 
666.6 




.'.'.'... 








22V6 
19.4 
16.7 
14.6 
12.8 


19.9 
17.0 
14.6 
12.7 
11.2 


17.7 
15.1 
13.0 
11.3 
9.9 


15.9 
13.6 
A.l 
10.2 

8.9 


12.7 
10.9 
9.3 
8.1 
7.1 


10.6 


13 












9^0 


14 












7.8 


16 








'22.0 

19.9 


'20.0 

17.9 


6.7 
6.9 


16 


...... 




22.0 


17 


13.52 


0.383 


708.3 






19.9 


17.7 


16.9 


11.3 


9.9 


8.8 


7.9 


6.3 


6.3 


18 


12.77 


0.431 


750.0 




"20.*6 


17.7 


15.7 


14.1 


10.1 


8.8 


7.8 


7.1 


5.6 


4.7 


19 


12.10 


0.481 


791.6 


21.0 


18.3 


15.9 


14.2 


12.7 


9.1 


7.9 


7.0 


6.3 


5.1 


4.2 


20 


11.48 


0.538 


833.3 


19.1 


16.4 


14.3 


12.7 


11.4 


82 


7.1 


6.3 


6.7 


4.5 


3.8 


21 


10.94 


0.592 


875.0 


17.3 


15.0 


13.0 


ll.f 


10.4 


7.4 


6.6 


6.7 


6.2 


4.1 


3.4 


22 


10.44 


0.652 


916.6 


15.8 


13.5 


11.8 


10.5 


9.6 


6.7 


5.9 


5.2 


4.7 


3.7 


3Jl 


23 


9.99 


0.717 


958.3 


14.4 


12.5 


10.8 


9.7 


8.6 


0.2 


5.4 


4.8 


4.3 


3.4 


2.8 


24 


9.58 


0.786 


1000.0 


13.3 


11.4 


9.9 


8.8 


7.9 


6.7 


4.9 


4.4 


3.9 


3.1 


2.6 


26 


9.19 


0.855 


1041.6 


12.2 


10.6 


9.1 


8.2 


7.3 


6.2 


4.5 


4.0 


3.6 


2.9 


2.4 


26 


8.84 


0.927 


1083.3 


11.3 


9.7 


8.5 


7.5 


6.8 


4.8 


4.2 


3.7 


3.4 


2.7 


2.2 


27 


8.51 


1.003 


1126.0 


10.5 


9.0 


7.8 


7.0 


6.3 


4.6 


3.9 


3.5 


3.1 


2.5 


2.1 


28 


8.21 


1.084 


1166.6 


9.7 


8.3 


7.3 


6.6 


6.8 


4.1 


a6 


3.2 


2.9 


2.3 


1.9 


29 


7.92 


1.170 


1208.3 


9.1 


7.8 


6.8 


6.1 


6.4 


3.8 


3.4 


3.0 


2.7 


2.1 


1.8 


80 


7.66 


1.257 


1250.0 


8.6 


7.2 


6.3 


5.e 


6.1 


3.6 


3.1 


2.8 


2.5 


2.0 


1.7 


81 


7.41 


1.350 


1291.6 


7.9 


6.8 


6.9 


6.3 


4.8 


3.4 


2.9 


2.6 


2.3 


1.9 


1.6 


32 


7.18 


1.443 


1333.3 


7.4 


6.4 


6.6 


4.9 


4.4 


3.2 


2.8 


2.4 


2.2 


1.7 


1.4 


33 


6.96 


1.542 


1375.0 


7.0 


6.0 


6.2 


4.7 


4.2 


3.0 


2.6 


2.3 


2.1 


1.6 


1.4 


84 


'6.75 


1.646 


1416.6 


6.6 


6.6 


4.9 


4.4 


3.9 


2.8 


2.4 


2.2 


2.0 


1.5 


1.3 


86 


6.67 


1.754 


1468.3 


6.2 


6.3 


4.7 


4.1 


3.7 


2.6 


2.3 


2.0 


1.8 


1.5 


1.2 


86 


6.38 


1.871 


1600.0 


6.9 


6.0 


4.4 


8.9 


8.6 


2.6 


2.2 


1.9 


1.7 


1.4 


1.1 


87 


6.21 


1.987 


1641.6 


6.6 


4.8 


4.2 


3.7 


3.3 


2.3 


2.0 


1.8 


1.6 


1.3 


1.1 


38 


6.06 


2.109- 


1683.3 


6.3 


4.6 


3.9 


3.6 


8.1 


2.2 


1.9 


1.7 


1.6 


1.2 


1.0 


89 


6.89 


2.229 


1626.0 


6.0 


4.8 


3.7 


3.8 


8.0 


2.1 


1.8 


1.6 


1.4 


1.1 


1.0 


40 


6.74 


2.366 


1666.6 


4.7 


4.1 


3.6 


3.1 


2.8 


2.0 


1.7 


1.6 


1.3 


1.0 


0.9 



y Google 



46 



RESI8TAKCB TO CROSS BREAKING AND SHEARING. 



Fig. 85. 



10.6' 




— 0.45" 



lO.S^-' Beam. Weight per If. =» 35 lbs. 



Sectional atea — 10.6" 

Moment of inertia / — 179.44 
Constant K' —170.903 

W . 

L 



1 








Distance d bet. centres of beams in feet, tor 


h 


s 

B 

i 


i 
i 


1 

.2 

1 


weight in lbs 


. per sq. 


foot of— 




5-e 


i 


Xi 


1 


i 


i 


.Q 


i 


.s 


1 


i 


i 


Q 


5 


Q 


^ 


i 


s 


o 

00 


i 


1- 


1 


1 


1 


1 


J_ 


J_ 


6 


2U8 


0.053 


210 
























7 


2UI 

21.30 


0.072 
0.095 


245.0 
280.0 






















23,2 


8 










".... 










'21.3 


17.8 


9 


18.08 


0.120 


315.0 

















23.4 


•iiVi 


17.0 


14.0 


10 


17.09 
15.5:; 


0.149 
0.181 


350.0 
385.0 














'21.3 
17.6 


189 
15.G 


17.0 
14.1 


13.6 
11.3 


11.4 


11 













"20.1 


93 


12 


14.21 
1;}.14 


0.216 
0.264 


420.0 
465.0 












16.9 
14.4 


14.8 
12.0 


13.1 
11.1 


11.8 
10,1 


9.4 
8.1 


7 9 


13 








20,6 


*2C).*2 


6;7 


14 


12.2ft 


0.295 


49ao 






iiV? 


19.2 


17.4 


12.4 


10.9 


9.6 


8.7 


6.9 


5.8 


15 


ll.::8 


0.340 


625.0 




21.'9 


18.9 


17.0 


15.1 


10.8 


9.4 


8.4 


7.6 


6.0 


6.0 


16 


10 08 


0.389 


560.0 


22.2 


19.0 


16.6 


14.9 


13.3 


9.6 


8.3 


7.4 


6.6 


5.3 


4.4 


-17 


10.0.1 


0.439 


695.0 


19.7 


17.0 


14.7 


13.2 


11.8 


8.4 


7.3 


6.5 


6.9 


4.7 


3.9 


18 


0.49 


0.494 


6:jo.o 


17.6 


15.0 


131 


11.7 


10.5 


7.6 


6.5 


5.8 


6.2 


4.2 


3.6 


10 


8.00 


0.553 


665.0 


15.7 


13.6 


11.7 


10.6 


94 


6.7 


5.9 


5.2 


4.7 


3.7 


3.1 


23 


8.54 


0.614 


700.0 


14.2 


12w2 


10.6 


9.4 


85 


6.1 


6.3 


4.7 


4.2 


3.4 


2.8 


21 


8.i:i 


0.681 


735.C» 


12.9 


ll.l 


9.6 


8.6 


7.7 


6.5 


4.8 


4.3 


3.8 


3.1 


2A 


22 


7.75 


0.752 


770.0 


11.7 


10.0 


9.1 


7.8 


7,0 


6.0 


4.4 


.3.9 


3.6 


2.8 


2.3 


2J 


7.43 


0.823 


805.0 


10.7 


9.2 


8.0 


72 


6,4 


4.6 


4.0 


3.5 


3.2 


2.5 


2.1 


24 


7.12 


0.903 


840.0 


9.8 


8.4 


7.4 


6.5 


5.9 


4.2 


3.7 


3.2 


2.9 


2.3 


1.9 


25 


6.83 


0.980 


876.0 


9.1 


7.8 


6.8 


6.0 


6.4 


ao 


3.4 


3.0 


2.7 


2.1 


1.8 


21 


6 57 


1.067 


910.0 


8.4 


7.2 


6.3 


6.6 


5.0 


3.6 


3.1 


2.8 


2.6 


20 


1.6 


27 


6.32 


1.154 


945.0 


7.8 


6.7 


6.8 


6.2 


4.6 


3.3 


2.9 


2.6 


2,3 


1.8 


1.6 


28 


6.10 


1.261 


-980.0 


7.2 


6.2 


6.4 


4.8 


4.3 


3.1 


2.7 


2.4 


2.1 


1.7 


1.4 


29 


5.89 


1.346 


1016.0 


6.7 


6.8 


6.0 


4.5 


4.0 


2.9 


2.5 


2.2 


2.0 


1.C 


1.8 


30 


5.00 


1.450 


1050.0 


6.3 


6.4 


4.7 


4.2 


3.7 


2.7 


2.3 


2.1 


1.8 


1.5 


1.2 


31 


5.51 


1.556 


1085.0 


6.9 


6.1 


4.4 


3.9 


3.5 


2.5 


2.2 


1.9 


1.7 


1.4 


1.1 


32 


5.84 


1.672 


1120.0 


6.6 


4.7 


4.1 


3.7 


3.3 


2.3 


2.0 


1.8 


1.6 


1,3 


1.1 


33 


5.17 


1.783 


1155.0 


6.2 


4.4 


3.9 


a4 


3.1 


2.2 


1,9 


1.7 


1.6 


1.2 


1.0 


34 


5.02 


1.906 


1190X) 


4.8 


4.2 


.3.6 


3.2 


2.9 


2.1 


1.8 


l.G 


1.4 


1.1 




35 


4.88 


2.033 


1225.0 


4.6 


4.0 


3.4 


3.1 


2.7 


1.9 


1,7 


1.5 


1,3 


1.1 




36 


4.09 


2.143 


1260.0 


4.3 


3.7 


3.2 


2.8 


2.6 


1.8 


1.6 


1.4 


1.3 


1.0 




37 


4.61 


2.297 


1295.0 


4.1 


3.5 


3.1 


2.7 


2.4 


1,7 


1.6 


1.3 


1.2 






38 


4.50 


2.444 


1.^30.0 


3.9 


8.3 


2.9 


2.6 


2.3 


1.6 


1.4 


1.3 


l.l 






39 


4.38 


2589 


1365.0 


3.6 


3.2 


2.8 


2.5 


2.2 


1.6 


1.4 


1.2 


1.1 






40 


4.20 


2.711 


1400.O 


3.6 


8.0 


2.6 


2.3 


2.1 


1.6 


1.3 


1.1 


1.0 







y Google 



BESISTAKCE TO CBOSS-BBEAEIirO AVD SHEABH^G. 



47 



Fig. 86. 



9^^ " Heavy'* Beam. Weight per If. = 50 lbs. 




Sectional area — 16.0" 

Moment of inertia / — 188.66 
Constant IT' -« 20U.50 

W . 

L 



6.37 



t, 


i 


o 




Distance d bet. centres of beams in feet, for 


a 


1 


of 


weight in lbs 


. per sq. foot of— 




s^ 


.s 


a 


a 












3« 




















fe: 


a 


s 
























ZS 


1 


6 


JO 

to 


i 


a 


i 


i 


1 


i 


i 


i 


i 


1 


i 


.2 
Q 


a 


1 


i 


o 


J- 


g 


8 


i 


i 


1 


1 


i 


i. 


G 


36.91 


0.005 


300.0 
























7 


29.92 


0.084 


3500 
























8 


20.18 
2.3.27 


O.lll 
0.141 


400.0 
450.0 























21.8 


9 






...... 










,"...*! 


..... 


*20.6 


17.2 


10 


20.95 


0.174 


• 500.0 
















*23.2 


20.9 


16.7 


13.9 


11 


19.01 
17.45 


0.211 
0.253 


550.0 
600.0 








...... 






21.0 
18.1 


19.1 
16.1 


17.8 
14.5 


13.7 
11.6 


IIJJ 


12 




...... 








20.7 


9.6 


13 


16.11 
I4.9C 


297 
0.345 


050.0 
700.0 












17.0 
15.2 


15.4 
13.3 


13.8 
11.8 


12.3 
10.0 


9.9 
8.6 


8.2 


14 


...... 








...... 

21.'3 


7.1 


15 


13.90 


0.398 


750.0 








*20.'6 


18.G 


13.2 


11.0 


10.3 


9.3 


7.5 


6.2 


16 


13.09 


0.454 


800.0 






20.4 


19.5 


10.3 


11.0 


10.2 


9.0 


8.1 


6.5 


6.4 


17 


12.32 


0.515 


860,0 




20.7 


18.1 


10.1 


14.5 


10.3 


9.0 


8.0 


7.2 


6.7 


4.8 


18 


11.0.) 


0.680 


900.0 


21.5 


18.4 


10.1 


14.3 


12.9 


9.2 


8.0 


7.1 


6.4 


6.1 


4.8 


19 


11.02 


0.048 


950.0 


i9.;i 


10.5 


14.5 


12.8 


11.0 


8.2 


7.2 


6.4 


5.8 


4.0 


3.8 


20 


10.47 


0.722 


1000.0 


17.4 


14.9 


13.0 


11.4 


10.4 


7.4 


6.5 


6.8 


6.2 


4.1 


3.4 


21 


JI.97 


0.799 


1050.0 


15.8 


13.5 


11.8 


10.5 


9.4 


6.8 


5.9 


6.2 


4.7 


a4 


3.1 


22 


9.52 


0.883 


1100 


14.4 


12.3 


10.3 


9.6 


8.2 


6.1 


6.4 


4.8 


4.3 


3.4 


2.8 


23 


9.10 


0.908 


1150.0 


13.1 


11.3 


9.8 


8.7 


7.9 


6.0 


4.9 


4.3 


3.9 


8.1 


2.6 


24 


8.72 


1.002 


12«J0.0 


12.1 


10.3 


9.0 


8.0 


7.2 


5.1 


4.6 


4.0 


3.0 


2.9 


2.4 


25 


B.M 


1.152 


125(i.O 


11.1 


9.5 


8.3 


7.4 


6.6 


4.7 


4.1 


3.7 


3.3 


2.0 


2.2 


26 


8.05 


1.258 


i:wo.o 


10.3 


8.8 


7.7 


6.8 


6.1 


4.4 


3:8 


3.4 


3.0 


2.4 


2.0 


27 


7.75 


1.303 


i:j50.o 


9.5 


8.0 


7.1 


6.3 


6.7 


4.1 


3.6 


3.1 


28 


2.2 


1.9 


28 


7.48 


1.477 


1400.0 


8.9 


7.0 


6.6 


6.9 


6.3 


3.8 


3.3 


2.9 


2.0 


2.1 


1.7 


29 


7.22 


1.593 


1450.H 


8.3 


7.2 


6.2 


6.5 


4.9 


3.5 


3.1 


2.7 


2.5 


1.9 


1.6 


30 


6.98 


1.718 


1500.0 


7.7 


6.6 


5.8 


6.1 


4.6 


3.3 


2.9 


2.6 


2.3 


1.8 


1.5 


31 


6.75 


1.846 


1650.0 


7-^ 


6.2 


6.4 


4.8 


4.3 


3.1 


2.7 


2.4 


2.1 


1.7 


1.4 


32 


6.64 


1.982 


1000.0 


6.7 


6.8 


5.1 


4.4 


4.0 


2.9 


2.6 


2.2 


2.0 


1.0 


1.8 


83 


6..34 


2.119 


1650.0 


6.4 


5.0 


4.8 


•4.2 


3.8 


2.7 


2.4 


2.1 


1.9 


1.5 


1.2 


34 


6.16 


2.265 


1700.0 


6.0 


6.1 


4.5 


4.0 


3.6 


2.6 


2.2 


2.0 


1.8 


1.4 


1.2 


35 


5.98 


2.416 


1750.0 


6.6 


4.8 


4.2 


3.7 


3.4 


2.4 


2.1 


1.8 


1.7 


1.3 


1.1 


36 


6.81 


2.577 


1800.0 


6.3 


4.0 


4.0 


3.6 


3.3 


2.3 


2.0 


1.7 


1.0 


1.2 


1.0 


37 


6.66 


2.742 


1850.0 


6.0 


4.3 


3.8 


3.3 


3.0 


2.1 


1.9 


1.6 


1.6 


1.2 




38 


6.51 


2.918 


1900.0 


4.8 


4.1 


3.0 


3.2 


2.9 


2.0 


1.8 


1.6 


1.4 


1.1 




39 


6.37 


3.098 


1950.0 


4.5 


3.9 


3.4 


3.0 


2.7 


1.9 


1.7 


16 


1.3 


1.1 




40 


6.27 


3.289 


20004) 


4.3 


3.7 


3.2 


2.9 


2.6 


1.8 


1.6 


1.4 


1.3 


1.0 





y Google 



48 



BESISTANCB TO CROSS BREAKINa AND SHEARIHG. 



Fig. 87. 




9^^ " Medium'' Beam. Weight per If. = 30 lbs. 



Sectional area = 9.0" 

Moment of inertia / -= 111.32 

Constant K' -= 123.69 

0.45" A"' 

W . 

L 



i 


OD 


i 


Distance d bet. centres of beams in feet, for 


a«i 


s 


-g 


OQ 




iveight in lbs 


per sq. foot of— 






1 

a 


a 
a 




















S| 


1 






















1-2 




c? 


s . 1 








OQ 


m 


00 


« 


(K 


00 


m 


fO "^ 






bO 


tn 


n 


s 


2 


jo 


Xi 


.0 


£ 


£ 


£ 


X3 








•M 


^ 


J2 


X3 


.0 
















5 


ft 


^ 


i 


i 


i 


1 


1 


1 


1 


i 


1 


i 


i. 


6 


20.60 
17.C7 
15.46 
13.74 
12.36 


0.062 
0.085 
0.111 
0.141 
0.174 


1800 
210.0 
240.0 
270.0 
300.0 






















22.0 


7 



















'25.0 

19.0 
15.0 
12.0 


20.0 

15.0 
11.0 
9.8 


leio 


8 














24.0 
19.0 
15.0 


21.6 

160 
13.0 


12.0 


9 












2V.O 

17.0 


lO.O 


10 








...... 




8!2 


11 


11.24 

10.30 


0.211 
0.252 


5S0.0 
360.0 








22.0 
19.0 


'20.0 

17.0 


14.0 
12.U 


12.0 
10.0 


11.0 10.0 
9.6 8.5 


8.1 

6.8 


68 


12 






2V.6 


6.7 


13 


9.51 


0.297 


390.0 




2J.6 


18.0 


16.0 


14.0 


10.0 


9.1 


8.1 


7.3 


5.8 


4.8 


14 


8.83 


0.345 


420.0 


21.0 


18.0 


15.0 


14.0 


12.0 


9.0 


7.8 


7.0 


6.3 


5.0 


4.2 


15 


8.24 


0.398 


450.0 


18.0 


15.0 


13.0 


12.0 


10.0 


7.8 


6.8 


6.1 


6.4 


4.3 


3.6 


16 


7.73 


0.455 


480.0 


16.0 


13.0 


12.0 


10.0 


9.6 


6.9 


6.0 


6.3 


4.8 


3.8 


3.2 


17 


7.21 


0.611 


510.0 


14.0 


12.0 


10.0 


9.4 


8.4 


6.0 


5.3 


4.7 


4.2 


3.2 


2.8 


18 


0.87 


0.580 


5400 


12.0 


10.0 


9.5 


8.4 


7.6 


6.4 


4.7 


4.2 


3.8 


3.0 


2.6 


19 


6.51 


0.650 


570.0 


11.0 


9.7 


8.6 


7.6 


6.8 


4.8 


4.2 


3.8 


3.4 


2.7 


2.2 


20 


6.18 


0.722 


600.0 


10.0 


8.8 


7.7 


6.8 


6.1 


4.4 


3.8 


3.4 


3.0 


2.4 


2.0 


21 


5.88 


799 


630.0 


9.3 


8.0 


7.0 


6.2 


6.6 


4.0 


3.5 


3.1 


2.8 


2.2 


1.8 


22 


6.62 


0.884 


660.0 


8.5 


7.2 


6.3 


6.6 


5.1 


3.6 


3.1 


2.8 


2.5 


2.0 


1.7 


23 


6.37 


0.969 


690.0 


7.7 


6.6 


5.8 


5.1 


4.6 


3.3 


2.9 


2.5 


2.3 


1.8 


1.5 


24 


5.16 


1.065 


720.0 


7.1 


6.1 


6.3 


4.7 


4.2 


3.0 


.2.6 


2.3 


2.1 


1.7 


1.4 


25 


4.94 


1.167 


750.0 


6.6 


5.6 


4.9 


4.3 


3.9 


2.8 


2.5 


2.1 


1.9 


1.5 


1.3 


26 


4.83 


1.277 


780.0 


6.1 


5.3 


4.6 


4.1 


3.7 


2.6 


2.3 


2.0 


1.8 


1.4 


1.2 


27 


4.68 


1.365 


810.0 


6.6 


4.8 


4.2 


3.7 


3.3 


2.4 


2.1 


1.8 


1.6 


1.3 


1.1 


28 


4.41 


1.476 


840.0 


5.2 


4.5 


3.9 


3.5 


3.1 


2.2 


1.9 


1.7 


1.5 


1.2 


1.0 


29 


4.26 


1.593 


870.0 


4.8 


4.1 


3.6 


3.2 


2.9 


2.0 


1.8 


1.6 


1.4 


1.1 




30 


4.12 


1.718 


900.0 


4.5 


3.9 


3.4 


3.0 


2.7 


1.9 


1.7 


1.5 


1.3 


1.0 




81 


3.99 


1.846 


930.0 


4.2 


3.6 


3.2 


2.8 


2.5 


1.8 


1.6 


1.4 


1.2 


1.0 




82 


3.86 


1.982 


960.0 


4.0 


3.4 


3.0 


2.6 


2.4 


1.7 


1.6 


1.3 


1.2 






83 


3.74 


2.119 


990 


3.7 


3.2 


2.8 


2.5 


2.2 


1.6 


1.4 


1.2 


1.1 






34 


3.63 


2.235 


1020.0 


3.5 


3.0 


2.6 


2.3 


2.1 


1.5 


1.3 


1.1 


1.0 






85 


3.63 


2.416 


1050.0 


3.3 


2.8 


2.6 


2.2 


2.0 


1.4 


1.2 


1.1 


1.0 






86 


3.43 


2.577 


1080.0 


3.1 


2.7 


2.3 


2.1 


1.9 


1.3 


1.1 


1.0 








87 


3.34 


2.742 


1110.0 


3.0 


2.5 


2.2 


2.0 


1.8 


1.2 


1.1 


1.0 








88 


3.25 


2.918 


1140.0 


28 


2.4 


2.1 


1.9 


1.7 


1.2 


1.0 










89 


3.17 


3.098 


1170.0 


2.7 


2.3 


2.0 


1.7 


1.6 


1.1 


1.0 










40 


3.09 


3.289 


1200.0 


2.5 


2.2 


1.9 


1.6 


1.5 


1.1 













y Google 



BESI8TANCE TO CROSS BEEAKINa AND SHEAEINa. 



49 



Fig. 88. 



9^^ ''Light'' Beam, Weight per If. =e 23.33 ] 




Sectional area — 1,(y 

Moment of inertia /» 91.06 
Constant K' — 101v2 

Tr=— . 
L 



1 


1 


1 




Distance d bet centres of beams in feet, for 


i 


weight in pounds per sq. foot of— 


&a 


a 


a 
a 


a 




z^ 
























i 


1 


6 

£ 
S 


1 


i 

i 


i 
^ 


a 


1 

i 


1 

i 


1 
1 


1 


1 

i 


1 


1 


1 


6 


16.86 


0.062 


140.0 




















22.0 


18.7 


7 


14.45 


00i5 


103.3 










.... 




25.6 


22!6 


2o!6 


16.6 


13.7 


8 


12.65 


0.111 


186.6 













26!6 


19.7 


17.5 


15.8 


12.6 


10.6 


9 


11.24 


0.141 


210 


...... 












17.8 


15.6 


13.8 


12.6 


lO.l 


8.6 


10 


10.12 


0.175 


233.3 








22.6 


ioJo 


14.4 


12.6 


11.2 


10.1 


8.0 


6.7 


11 


9.20 


0.212 


2.36.6 






21.0 


18.7 


16.7 


11.9 


10.4 


9.2 


8.3 


6.7 


5.6 


12 


8.43 


0.253 


280.0 


23.6 


5»*.6 


17.5 


15.6 


14.0 


10.0 


8.7 


7.8 


7.0 


5.6 


4.6 


13 


7.78 


0.297 


303.3 


19.9 


17.9 


14.9 


13.4 


11.9 


8.5 


7.4 


6.6 


5.9 


4.8 


3.9 


14 


7.22 


0.345 


326.6 


17.1 


14.7 


12.8 


11.4 


10.3 


7.3 


6.4 


5.7 


6.1 


4.1 


3.4 


16 


6.74 


0.898 


350.0 


14.9 


12.9 


11.2 


10.0 


8.9 


6.4 


5.0 


6.0 


4.5 


3.6 


2.9 


16 


6.31 


0.453 


373.3 


13.1 


11.2 


9.8 


8.7 


7.8 


5.6 


4.9 


4.3 


3.9 


31 


2.6 


17 


5.95 


0.510 


396,6 


11.6 


lO.O 


8.7 


7.8 


7.0 


5.0 


4.3 


3.8 


3.5 


2.8 


2.3 


18 


6.62 


0.679 


420.0 


10.4 


8.9 


7.8 


G.9 


6.2 


4.4 


3.9 


3.4 


3.1 


2.6 


2.0 


19 


5.32 


0.648 


443.3 


9-3 


8.0 


7.0 


6.2 


5.6 


4.0 


3.5 


3.1 


2.8 


2.2 


1.8 


20 


5.06 


0.721 


466.6 


8.4 


7.2 


63 


6.0 


5.0 


3.6 


3.1 


2.8 


2.5 


2.0 


1.0 


21 


4.81 


0.797 


490.0 


7.6 


6.6 


6.7 


6.1 


4.5 


3.2 


2.8 


2.5 


2.2 


1.8 


1.5 


22 


4.59 


0.879 


513.3 


6.9 


5.9 


5.2 


4.0 


4.1 


2.9 


2.6 


2.3 


2.1 


1.7 


1.3 


23 


4.40 


0.968 


536.6 


6.3 


5.6 


4.7 


4.2 


3.8 


2.7 


2.3 


2.1 


1.9 


1.5 


1.2 


24 


4.21 


1.000 


560.0 


5.8 


5.0 


4.3 


3.8 


3.5 


2.6 


2.1 


1.9 


1.7 


1.4 


1.1 


25 


4.04 


1.151 


583.3 


6.3 


4.6 


4.0 


3.5 


3.2 


2.3 


2.0 


1.7 


1.6 


1.2 


1.0 


26 


3.89 


1.254 


606.6 


4.9 


4.2 


3.7 


3.3 


2.9 


2.1 


1.8 


1.6 


1.4 


1.1 




27 


a74 


1.359 


680.0 


4.6 


3.9 


3.4 


3.0 


2.7 


1.9 


1.7 


1.5 


1.3 


10 




28 


3.G0 


1.4C6 


653.3 


4.2 


3.6 


3.2 


2.8 


2.6 


1.8 


1.6 


1.4 


1.2 


l.O 




29 


a48 


1.682 


676.6 


4.0 


3.4 


3.0 


2.6 


2.4 


1.7 


1.5 


1.3 


1.1 






30 


3.37 


1.711 


700.0 


3.7 


3.2 


2.8 


2.5 


2.2 


1.6 


1.4 


1.2 


l.l 






81 


3.26 


1.837 


723.3 


3.6 


3.0 


2.C 


23 


2.1 


1.5 


1.3 


1.1 


1.0 






32 


3.16 


1.968 


746.6 


3.2 


2.8 


2.4 


2.1 


1.0 


1.4 


12 


1.0 








33 


3.06 


2.104 


770.0 


3.0 


2.6 


2.3 


2.0 


1.8 


1.3 


1.1 










34 


2.97 


2.250 


793.3 


2.9 


2.4 


2.1 


1.9 


1.7 


1.2 


1.0 










36 


2.89 


2.399 


816.6 


2.7 


2.3 


2.0 


1.8 


l.C 


1.1 












36 


2.81 


2.565 


840.0 


2.6 


2.2 


1.9 


1.7 


1.5 


1.0 












37 


2.73 


2.723 


863.3 


2.4 


2.1 


1.8 


1.0 


1.4 














38 


2.66 


2.898 


886.6 


2.3 


2.0 


1.7 


Lf) 


1.4 














39 


2.59 


3.070 


OIO.O 


2.2 


1.9 


1.0 


1.4 


I..'i 














40 


2.52 


3.2o0 


9:J3.3 


2.1 


1.8 


1.6 


1.4 


1.2 















y Google 



m 



BEBIfSAHOE CTa CllOS9'S]tB2.ff IHQ^ AWD 'BfifiABINQ.^ 



Fig. 80. 




S^y Beamy Wdghi per IL\= 21.66 lbs. 



SectienAl nrea. — 6.5" 

Moment of inertia / «= G5.09 
. ConstaDU A'' ~..*... —82.49 

L 



en 

1 


§ 


CO 




Distilneei d bot; cteintres of l^eams in l^et, foS* 
. weiglvt'in pounds -per pq. fooli of^ . 


2 


l.. 


'•i 


fl 


c 


.C " 


1 


~- -'• ' '— . ■ : 
























|£ 




;| 


1 


1 


1 


1 


J3 


X) 


i 


jO 


« 

£• 


Xi 


1- 




i 


;o 


^- 


i 


i 


S 


i 


i 


|_ 


g 


1 


1 


i 


1-: 


- 6 


18.74 


oio70 


13QJ0 


















22.9 


18.3 


Mi.2 


7 


11.78 


0i095 


15U6 


.Z.. 


.!Z. 




.", 






21.V 


is!?; 


10.8 


13.4 


Ilia 


8 


10-30 


0.124 


173.3 


...... 


...... 


...... 




2.5.7 


r8.'3 


1 0.0 


14.3 


12.8 


10.3 


8.6 





9.16 


ai58 


195.0 






.... 


.~. 


io:.\ 


lU. 


12.T 


11.3 


10.1 


8.1 


6.7 


10 


8.23 


0;108 


. 216.0 






:o.5 


18.3 


16.4 


11.7 


10.2 


9.1 


8.2 


6.5 


5.4 


11 


7.49 


0,238 


238^ 


^Izcl 


19.4 


17.0 


iru 


13.0 


9.7 


8.5 


7.6 


6.8 


5.4 


4.5 


12 


0.87 


0,284 


, 260.0 


19.0 


103 


14.3 


12.7 


11,4 


8.1 


7.1 


6.3 


5.7 


4.5 


n 


13 


0.34 


0.335 


281.6 


10.2 


14.0 


12.1 


lO.O 


9 7 


0.9 


6.0 


5.4 


4.8 


3:9 


14 


.5.89 


(1390 


303-3 


14.0 


12.0 


ia5 


0.3 


8.4 


6.0 


6.2 


4.6 


4.2 


3:3 


2.8 


15 


5.49 


0.447 


325.0 
346.6 


12.2 


IQ.6 


9.1 


8.1 


7.B 


;5.2 


4.5 


4.0: 


3.6 


2:9 


2.4 


16 


6.15 


0.511 


10.7 


9.1 


8.0 


7,1 


6.4 


4J) 


4.0 


3.6 


3.2 


2.5 


2.1 


17 


4.85 


0.580 


368.3 


a.^ 


H 


7-1 


6.3 


5.7' 


4;0 


3.7 


3.2 


2.6 


23 


}:? 


18 


4.58 


aG53 


390.0 


8.4 


7.2 


fi.15 


a.6 


6.1 


3.8 


3.1 


2.7 


2.& 


2:0 


19 


4.34 


a731 


. 411.6 


7.T5 


6 '6 


5.7 


5.1 


4.'5 


3.3 


'29 


25 


22 


1.8 


1.5 


20 


4.11 


0.810 


433.3 


6.8 


5,8 


5.1 


4.5 


4.1 


2.4) 


2.5 


2.2 


2.0 


4.6 


i:l 


21 


3.92 


0.898 


455.0 


6.2 


5.3 


4.6 


4.1 


3.7 


26 


2.3 


2.0 


1.8 


1.4 


22 


3.73 


0.989 


470.6 


5.6 


4.8 


4.2 


8.7 


3.4 


24 


2.1 


1*8 


1.6 


1,3 


1.1 


23 


3 58 


1,090 


498.3 


5.1 


4.4 


3.8 


3.4 


3.1 


2.2 


1.9 


1.7 


l.fe 


1.2 


1.0 


24 


3.42 


L192 


520.0 


4.7 


4.0 


3.5 


.3.1 


2.8 


;2i) 


1.7 


1.5 


'.4 


1.1 




25 


3.29 


1.300 


541.6 


4.3 


3.7 


3.2 


2.9 


2.6 


1.8 


mI.5 


1.4 


fl.3 


1.0 




26 


3.17 


1.417 


563.3 


4.0 


3.4 


3.0 


2.7 


2.4 


1.7 


1.4 


1.3 


1.2 






27 


3.05 


1.536 


585.0 


3.7 


3.2 


2£ 


25 


22 


!1.6 


1.4 


1.2 


l.l 






28 


294 


1.602 


606.6 


3.5 


3.0 


2.6 


Z3 


21 


T.6 


1.3 


1.1 


1.0 






29 


2.84 


1.795 


628.3 


3.2 


2.8 


2.4 


2;i 


1.9 


1.4 


1.2 


1.0 








30 


2.73 


1.923 


650.0 


3.0 


2.6 


2.2 


2.0 


1.8 


1.3 


1.1 


J 








31 


2.66 


2.080 


671.6 


2.8 


2.4 


2.1 


l.T) 


1.7 


1.2 


1.0 




r 






32 


2.50 


2.219 


693.3 


2.6 


2.2 


2.0 


1.7 


1.6 


1.1 


r 


' 


? 






33, 


2.49 


2.383 


715.0 


2.5 


2.1 


1.8 


1.6 


1.4 


,1.0 


' 






' ■ 




34 


2.42 


2.550 


. 736.6 


2.3 


2.0 


1.7 


1.5 


1.4 


. ' 






* 




- 


35 


2.35 


2.722 


758.3 


2.2 


1.0 


1.6 


1.4 


1.3 














36 


2.29 


2.907 


780.0 


2.1 


1.8 


1.5 


1.4 


1.2 














37 


2.22 


3.084 


801.6 


2.0 


1.7 


1.5 


1.3 


12 














38 


2.17 


3.290 


823.3 


1.9 


1.6 


1.4 


1.2 


1.1 














39 


2.11 


3.484 


846.0 


1.8 


1.5 


1.3 


1.2 


1.0 














40 


2.06 


3.702 


866.6 


li7 


1.4 


1.21 1.1 


1.0 















y Google 



I^l&SASOE TO CBOSSoBBSAEnre AXD SHXABnTOi 



n 



i^ig.96. 



V^ Beam: Weight per If. = 18.33 lbs. 



' Sectional area.,.; >» 6 5" 

Moment of inertia I — ^.84 
1.3" . Constant JBT '...,, »* 64.06 

. IF—'—. 
" L 



F 




•"""i. 




Distance d bet centres of beams in feet, for 


1 


3 


5 


1 


weight in lbs. per sq. foot of— 





5 


























i-5 

1 


1 

10.67 







1 

J- 


1 


1 
J- 


_s_ 


i 
i 


1 

i 


i 


1 




1 


1 


« 


0.060 


110.0 












25.4 


£22 


19.7 


17.7 


14.2 


11.8 


7 


9.15 


0.109 


128.3 















18.6 


16.3 


14.5 


13,0 


10.6 


8.7 


8 


8.00 


0.143 


146.6 






Sio 


22i2 


20.0 


14.2 


12.5 


ll.l 


lO.O 


8.0 


6.6 


9 


7.11 


0.181 


1G5.0 




22.0 


19.7 


17.5 


15.8 


11.2 


9.8 


8.7 


7.9 


6.3 




10 


6.40 


0,224 


183.3 


Ki 


18.2 


16.0 


14.2 


12.8 


9a 


8.0 


7.1 


6.4 


6.1 


3I 


11 


6.82 


0.278 


201.6 


^7.6 


15.3 


13.2 


11.7 


10.5 


7.5 


6.6 


6.8 


5.2 


4.2 


12 


6.33 


0.326 


220.0 
2S.3 


14.8 


12.6 


ll.l 


94{ 


8.8 


6.3 


6.5 


4.9 


^? 


3.5 


2!9 


13 




0.382 


18.6 


10.9 


9.4 


18.8 


7.5 


5.4 


4.7 


4.1 


3.0 


2.6 


14 


4.66 


0.444 


256w6 


10.8 


9.3 


8.1 


7.2 


6.6 


4:6 


4.0 


8.6 


3.2 


2.6 


2.1 


15 


4!27 


0i613 


2T5.0 


9.4 


8.2 


7.1 


6.3 


!6.7 


4.0 


85 


3.1 


2.^ 


2.2 


1.8 


16 


3.99 


0.686 


298.3 


8.3 


7.1 


6.2 


6.5 


,4.9 


3.5 


3.1 


2.7 


2.4 


1.9 


1.6 


17 


3.76 


0.666 


3U.6 


7.3 


6.5 


5.5 


,4.9 


4.4 


3.1 


2.7 


2.3 


2.1 


1.7 


1.4 


18 


3.55 


0.749 


3».0 


6.6 


5.6 


4.9 


4.3 


3J» 


2.8 


24 


2.1 


1.9 


1.6 


1.3 


19 


3.37, 


0.840 


348.3 


6.9 


5.1 


4.4 


3.9 


3.6 


2:5 


2.2 


1.9 


1.7 


1.4 


1.1 


20 


3.20 


0.936 


366.6 


5.3 


4.5 


4.0 


3.5 


3.2 


2.2 


2.0 


1.7 


1.6 


1.2 


1.0 


21 


3.05 


1.038 


386.0 


4.8 


4.1 


3.6 


3.2 


2.9 


2.0 


1.8 


1.6 


1.4 


1.1 




22 


2.91 


1.146 


^ 


4.4 


3.7 


3.3 


12.9 


2.6 


1.8 


1.6 


1.4 


1.3 


1.0 




23 


2.78 


1J257 


4.0 


3.4 


3.0 


2.7 


2.4 


1.7 


1.5 


1.3 


1.2 






24 


2.66 


1581 


440i) 


3.6 


.3.1 


2.7 


i2.4 


2.2 


1.6 


1.3 


1.2 


1.1 






26 


2.56 


1.504 


458.3 
^6.6 


3.4 


2.9 


2.5 


2.2 


2.0 


1.5 


1.2 


1.1 


1.0 






26 


2.45 


1.630 


3.1 


2.6 


2.3 


2.0 


il.8 


1.4 


1.1 










27 


2.37 


1.775 




2.9 


2.5 


2.1 


1.9 


,1.7 


1.3 


1.0 










28 


2.27 


1.871 


6]^.3 


2.7 


2.3 


2.0 


1.8 


1.6 


1.2 












29 


2.20 


2,075 


631.6 


2.5 


2.1 


1.8 


1.7 


1.5 


11 












30 


2.12 


2.229 


650.0 


2.3 


2.0 


1.7 


1.6 


1.4 


1.0 













y Google 



52 



BESISTAHCE TO CBOSS-BBEAEING ASD SHEABIKG. 



Fig. 91. 



6^^ 5«am. Weight per If. = 13.33 lbs. 



Sectional area — 4.(/' 

Moment of inertia I — 22.5 
Constant K' — 87.64 

IF-—. 
L 



L 




1 




Distance d bet. centres of beams in feet, for 


S 


o 
g 


1 




weight in lbs. per sq. foot of— 


O 9 




d 

1 


d 








OQ 


1 


1 


1 


1 


1 


i 


i 


1 


a 


i 


in 

.22 


5 


^ 


i 


g 


i 


i 


i 


i 


i 


L 


i 


i 


i 


6 


6.27 


0.094 


80.0 








23.2 


20.9 


14.9 


13.0 


11.6 


10.4 


8.3 


6.9 


7 


6.37 


0.128 


93.3 







19.*! 


17.3 


15.3 


10.0 


9.5 


8.4 


7.6 


6.1 


6.1 


8 


4.70 


0.168 


106.6 


lils 


16.8 


14.6 


13.0 


11.7 


8.5 


7.3 


6.5 


5.8 


4.7 


3.9 


9 


4.18 


0.213 


120.0 


15.4 


13.4 


11.6 


10.4 


9.2 


6.6 


5.8 


5.1 


4.6 


3.7 


3.1 


10 


3.75 


0.263 


133.3 


12.5 


10.7 


9.3 


8.3 


7.5 


5.3 


4.7 


4.1 


3.7 


3.0 


2.6 


11 


3.42 


0.320 


146.6 


10.3 


9.0 


7.7 


6.9 


6.2 


4.4 


3.8 


8.4 


3.1 


2.4 


2.0 


12 


3.13 


0.382 


160.0 


8.6 


7.0 


6.5 


5.7 


6.2 


3.7 


3.2 


2.9 


2.6 


2.0 


1.7 


13 


2.89 


0.450 


173.3 


7.4 


6.4 


6.6 


4.9 


4.4 


3.1 


2.7 


2.4 


2.2 


1.7 


1.4 


14 


2.68 


0.624 


186.6 


6.3 


5.4 


4.7 


4.2 


3.8 


2.7 


2.3 


2.1 


1.9 


1.5 


1.2 


15 


2.51 


0.607 


200.0 


6.5 


4.8 


4.2 


3.7 


3.3 


2.3 


2.1 


1.8 


1.6 


1.3 


1.1 


16 


2.34 


0.689 


213.3 


4.8 


41 


3.6 


3.2 


2.9 


2.0 


1.8 


1.6 


1.4 


1.1 




17 


2.21 


0.786 


226.6 


4.8 


3.7 


3.2 


2.9 


2.5 


1.8 


1.6 


1.4 


1..^ 






18 


2.09 


0.888 


240.0 


8.8 


3.3 


2.9 


2.6 


2.3 


1.6 


1.4 


1.2 


1.1 






19 


1.98 


0.995 


253.3 


3.4 


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1.79 


1.231 


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2.4 


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1.9 


1.7 


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22 


1.70 


1.350 


293.3 


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2.2 


1.9 


1.7 


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23 


1.63 


1.493 


306.6 


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1.5 


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1.641 


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1.6 


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1.5 


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346.6 


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28 


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2.286 


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1.29 


2.489 


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y Google 



BESISTAKOE TO CBOSS-BBEAKIHa AKD SHEABIlSra. 



53 



Oast-Ibon Bsams. 

Factor of rupture Cfor cast-iron hearm of various sections. 

The factor Cis based on practical experiments by Hodgkinson- 

Its value alters with the different proportions of the cross-sections 

of beam. 

Beam supported at the ends; load concentrated at the center. 

Beference, 

C = Factor of rupture. 
W= Breaking weight in lbs. 
A = Sectional area of beam in scjuare inches. 

I = Distance between supports in inches. 

h = Height of beam in inches. 

Dimensions in inches, h = Thickness of web at center is the 
unit. 

lig. 92. 0.32 = 0.726 



5.126 = 11.646 




0.44 =6 



0.47 = 1.066 
0.52 = 1,186 



2.27 = 5.156 
il = 3.20 (7=27292 



J%r.e3. 



26 = 0.866 



5.125 = 17.082 



30 = 6 



55 = 1.736 
il = 2 . 73 C= 285iad by Google 



S4 JIBSISTAVCE TO C&OQi'tntAKatQ AVB SHEABIird. 



rO.30=5 0,fl4J 



6.125 = 166 




0.32 = 6 



2^10=r^6^.T6i^ _. -T 
-4 = 2.88 a=i 30330 

lig,95.' ' l!6 = 47216 

10.315 = 0.826 



5.125 = 13.486 




rr-0438s^6 



^0.53 = 1.396 
'4.16 = 10.946 
il=s^.33 0=35262 

JFi^. 96.' 2.3^ = 8.756 



5.125=19.266 




-0.31 = 1.166 



Q.2p6 = 6 



0.66=2.486 



6.67 = 25.076 
4 ==6.23^ (7=441^6 



by Google 



BS?IfiT^A^E TO CSqSS-BR&lKINa AVB 8H&iI(IliO. 



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;^ BE8ISTAFCE TO 0B06B*BBlASIVe A9J) BSBABIVa. 



Load concentrated at center: W= -r-, or ^^ ?= ^ TT. 

Beam fixed at one end; principal flange at top. 
Load equally distributed : W =r —— -, xnK^^2.l.W. 

Load concentrated at free end: If «= -r-r. or ^^ = 4 J. TT. 

4.t 

[Note. — The more the sectional area is contained in coefficient £^, the 
more is the section economical.] 

Example. — Section No. 34. Load equally distributed; beam 
supported at both ends ; thickness of web =5 1 inch ; thickness 
of nange =: 1\ inch; height = 10 inches; width of flange = 5.9 
inches. Distance between supports = 20 feet = 240 inches. 

W = -jT- = -r^ = 5.48 tons capacity. 

I K^ 

The moment of resistance of croBa-section =s -— — 

8 14 



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26 
27 
28 
29 
30 
31 
32 
33 
34 
35 
36 
37 
38 
39 
40 
41 
42 
43 
44 
45 
46 
47 
48 
49 
50 



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7 

8 

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16 

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17 

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4.6 
4.8 
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5.2 
5.4 
5.6 
5.7 
5.9 
6.1 
6.3 
6.5 
6.7 
6.9 
7.1 
7.3 
7.5 
7.7 
7.9 
8.0 
8.2 
8.4 
8.6 
8.8 
9.0 






10.4 
11.1 
11.8 
12.6 
13.2 
13.9 
14.7 
15.4 
16.2 
16.9 
17.6 
18.3 
19.1 
19.8 
20.6 
21.3 
22.1 
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25.8 
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BESISTANOE TO OBOSS-BBEAEING AND SHEABIlfa. 



61 



F%g, 110. 




Fig, 111. 




Fig, 113. 




Id 



51 
52 
53 
54 
55 
56 
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58 
59 
60 
61 
62 
63 
64 
65 
66 
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68 
69 
70 
71 
72 
73 
74 
75 






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4.3 
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5.3 
5.4 
5.6 
5.7 
5.9 
6.0 
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6.6 
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7.3 
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11.4 
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266 

308 

364 

406 

462 

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588 

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5.9 



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64 



BESISTANOE TO OBOSStBBEAEING AKD 8HEABIBG. 




d by Google 



RESlStANOE TO CEOSS-BREAKING AUD SHEARnTG. 66 



Fig. 126. 




Fig. 129. 




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Width B 
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139 


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7 


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16.4 


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66 



BESISTANCE TO CBOSS- BREAKING AND SHEARING. 



Fig. 130. 




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Fig. 132. 




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152 
153 
154 
155 
156 
157 
158 
159 
160 
161 
162 
163 
164 
165 
166 
167 
168 
169 
170 
171 
172 
173 
174 
175 
176 



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6.5 

6.7 

6.9 

7.1 

7.3 

7.5 

7.7 

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8.2 

8.4 

8.6 

8.9 

9.1 

9.3 

9.5 

9.8 

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17.2 
18.3 
19.3 
20.4 
21.5 
22.6 
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25.8 
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30.1 
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EESISTANCE TO OEOSS-BEEAKING AND SHEABING. 



67 



Fig, 134. 







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177 



178 



179 



180 



181 



182 



183 



184 



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187 



188 



189 



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21.6 



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Fig. 140. 



Fig. 141. 



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190 



191 



192 



193 



194 



195 



196 



197 



198 



199 



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BE3I8TAN0E TO OBOSS-BBEAKUTO AHD SHEARING. 



69 









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Width B 
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5 


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203 


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70 BESISTANCE TO CROSS* BBEAEING AND SHEABIKG. 



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Width h 
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2 


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6 


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71 



Fig, 146. 




Fig. 148. 




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26 
27 
28 
29 
30 
31 
32 
33 
34 
35 
36 
37 
38 
39 
40 
41 
42 
43 
44 
45 
46 
47 
48 
49 
50 



■tax: 

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7 
8 
8 
8 
8 
8 
8 
8 
8 
8 
8 
8 
8 
8 
9 
9 
9 
9 
9 
9 
9 
9 
9 
9 
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6 
7 
8 
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10 
11 
12 
13 
14 
15 
16 
17 
18 
7 
8 
9 
10 
11 
12 
13 
14 
15 
16 
17 



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1.0 
1.5 
1.9 
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647 
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54.4 


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59.6 


3,2 


62.4 


3.7 


65.4 


4.1 


68.2 


4.6 


71.2 


2.1 


58.2 


2.5 


61.0 


3.0 


64.0 


3.4 


66.8 


3.9 


69.8 


4.4 


72.8 






4452 
3164 
3444 
3738 
4032 
4296 
4606 
4900 
3742 
4074 
4396 
4718 
5026 
5348 
4060 
4410 
4760 
5110 
5460 
5810 
43^2 
4746 
5124 
5502 
5080 
6258 



d by Google 



98 



B£819TAB(S TO CftOBS'BKSAEIZre AHD 8HBABIHQ. 



Steenqth op Wooden Beams. 

Capacity W in lbs. of American ^hite and yellow pine beams, Joists, 
Ac, from 1" X 1" to 16 X 16 in. 

10000 
The modulus of rupture is taken at — 1250 lbs., or 8 times safety. 



K^ — tabulated coefficient, to be divided by 
I — distance between supports * 
from support to free end of oenm, 



I — distance between supports in inches, or length of beams in inches 



53 


Coefficient 


h 














Height in 


.M C- 
















O C 
















2"^ 


1 


2 


3 


4 


6 


6 


7 


1 


lfi66 


6666 


16000 


26666 


41666 


60000 


81666 


iH 


2500 


10000 


22500 


39999 


62499 


90000 


122499 


2 


3333 


13.333 


30000 


63333 


83333 


120000 


163333 


2}4 


41G6 


16666 


37500 


66666 


104166 


150000 


204166 


3^^ 


5000 


19999 


46000 


80000 


124999 


180000 


244999 


^H 


5833 


23:m 


62700 


93333 


145833 


210000 


285833 


4r 


6666 


26666 


60000 


106666 


166666 


240000 


326666 


^)4 


7499 


29999 


67600 


119999 


187499 


270000 


367499 


5 


8:333 


83333 


75000 


1333a3 


208333 


300000 


4083:33 


by^ 


9166 


36666 


82500 


146666 


229166 


asoooo 


449166 


6 


10000 


39999 


90(K)0 


159999 


249999 


360000 


489999 


. 6H 


10833 


4a333 


.97500 


I733a3 


270833 


390000 


530833 


• 7 


11666 


46666 


105000 


186666 


291666 


420000 


671666 


1% 


12500 


49999 


112600 


199999 


312499 


460000 


612499 


8 


ia333 


53333 


120000 


213333 


333;i33 


' 480000 


663333 


8M 


14166 


56666 


127500 


226666 


854166 


610000 


694166 


9 


14998 


59999 


135000 


230999 


374999 


540000 


734999 


9V$ 


15831 


63333 


142500 


253,333 


395833 


570000 


775*33 


10 


16666 


66666 


150000 


266666 


416666 


600000 


816666 


lOM 


17500 


69999 


157500 


279999 


437499 


630000 


857599 


11 ' 


183:i3 


7:i333 


1(^5000 


293333 


458333 


660000 


898533 


11>^ 


19166 


76666 


172600 


300666 


479166 


690000 


939366 


12 


20000 


79999 


180000 


319999 


499999 


720000 


979999 


12^ 


20833 


8a-W3 


187500 


33:i3;« 


520833 


750000 


1020833 


13 


21606 


86066 


195000 


346666 


641666 


780000 


1061666 


13H 


22500 


80:)99 


202500 


350999 


662499 


810000 


1102499 


U 


2*333 


93:t3:J 


210000 


373333 


683333 


840000 


1143333 


UM 


24166 


96(i66 


S17500 


386666 


604166 


870900 


1184166 


15 


25000 


99099 


225000 


399999 


624999 


900000 


1224999 



y Google 



IUS8I8TANOE TO GB0«8rBBSAKIHe AVD BHEABINa. ^ 



BSAH8 BUPF0BT8D A7 THE BVD8. 
Load equally distributed, F — — or iT" — 2 fT. 1 

Load oeaoentrated at centre, W'^— or K'^TiW, % 

Beams fizsd at one eitb. 

Load equally distributed, IT— — orJP — «ir. 8 

Load concentrated at free end, TT— — oriT — WIT. 4 

82 



inches. 



y Google 



100 



PBESSintE^ OV 81TFP0BTS, 



PRESSURE ON SUPPORTS. 
Reaction of Supports. 

For a continuous beam, horizontal or inclined. Load W/ 
equally distributed, and supports equal distance apart. Appli- 
cable to trussed beams, rafters, or beams supported by three or 
more supports. 

Beferenee. {Fig. 166.) 
Wj ^ Weight of load per unit of length in lbs. 
L = Distance between supports in units of length. 
PtPi, P'i = Pressure on supports in lbs., counting from end 
support to center of beam. 
M, Mif M2 = Moments of rupture over supports, 
w, mj, wig = Moments of rupture between supports. 

I, li, l^ srrThe distance from a support to section where 
moments m, m^, m, occur. 

By this table the pressure upon any support, from 3 to 9 in 
number, can be ascertained; also the moments of rupture. The 
table is used in calculating the strains in roof trusses, &c. 

Fig. 166. 



li 


Number of Supports. 


11 


3 


4 


6 


7 


9 


P 

1 


0.376 WjL 
1.26 W,L 


a4 W,L 
1.1 W,L 


0.3929 ?P,L 
1.1429 ^L 
0.9286 H^L 


0.3942 HJL 
1.1346 HJL 

1.0192 HVL 


0.3W3 TF,2^ 
1.1340 fr,2^ 
0.9C29fr,2/ 
1.0103 W^L 
0.9048 Wjl^ 
















^1 


0.126 W,L2 


0.1 WM 


0.1071 ^La 
0.0714 TF;Z 2 


0.1058^X2 
0.0769 W^l^ 2 
0.0866 Tr,L2 


0.1057 TT.L 2 
0O773iri2 


m\ 






0.0850 TT.i^a 
0.0824 ir,L2 


mI 











d by Google 



^BESSnaB.OH, SU?P0BT3. 



161 



s « 


Number of Supports. 


«s 


3 


4 


6 


7 


9 


m 
mi 


0.0703 Tr,z 2 


0.08 tf:Z2 

0.025 W,L2 


0.0772 WiL2 
0.0364 Tf;i»2 


0.0777 W,L2 
0.0340 Fr,L2 
0.0434 »F,I»2 


0.07771^2.2 
0.0339 H-Il 2 
0.0438 IV 2,2 


Ws 








0.0412 »F,2i 2 










0.375 L 


0.4 2/ 
0.5 L 


0.3928 L 
0.635 If 


0.3942 2; 
0.6288 2/ 
0.4903 .L 


0.3943 £ 
0.6283 If 
4922 L 








06026 L 













Beference. (Figs. 167, 168, and 169.) 
W, TTi. 1^3 = Load in lbs. 

2, ^^,^2 5= Dimensions in units of length. 
P, Fj^, F^ss Pressure on supports in lbs. 



Fig. 16 



Three supports, 
unequal distances 
apart. 



Load equally distributed: 



W,^-fW 



Fig, 168 



Load < 



P^iWi 



= 4ir 






One support, and 
fixed at one end. 



^-'^ 



Digitized by V^OOQIC 



102 . COMPBESSIVE STBflsr AHD PBJKSnmB ON 8XTPPOBT8. 



One sappori, 
and fixed at 
one end. 



Load concentrated at free end: 



P,= P- Tr=(^-i)Tr=^r 



COMPRESSIVE STRAIN AND PRESSIIRE ON SUPPORTS. 

Sloping Beams, Raftebs, Ac. 

Load W equally distributed. 

For the cross- breaking strain, the rafter, Ac, is to be treated 

as a horizontal beam of the length I. (See Compound Strains in 

Beam, &c.) 

Reference. 
C^ Compression in direction of beam. 
JI= Horizontal strain acting on support. 
V^s Pressure on supports. 

Lower end supported vertically and horizontally; npper end 
resting on inclinea support : 



Csss^^ sin.v 

w w 

H=s -—- sin.t; cos.v Fj =-5- (cos.v)* 



FrsTT— Fi = Tr(l-Hco».v)*) 

Google 



BSSlBTAItCE TO CRU&DINa. 103 

Upper end fized; lower end supported Iiorizontally : 
Fig. 171. 

(7=0 
//=0 



Upper end resting against a vertical surface ; lower end sup- 
ported vertically and horizontally : 

Fig. 172. 



(7= 


W 
2sin.t; 


If = 


W 
2 


cotg » 


F= 
^1 = 


W 






RESISTANCE TO CRUSHING. 

Strength of Columns, Pillaes, and Struts. 

Reference. 

A = Area of cross-section in inches. 

Ca= Coefficient, depending on the material. 

I = Least moment of inertia of cross-section. 
W=i Capacity of column, pillar, or strut in lbs. 

a =: Coefficient, depending on the material in respect to flexure. 

c as Coefficient, depending on the material. 

h = The least dimension across the section in inches. 

k = Factor of safety. 

l=s Length of column, Ac, in inches. 

r = Least radius of gyration. 

Digitized by LnOOQlC 



104 BESISTANOE TO CRUSHTira. 

To find the sqaare of the radius of gyration (r«) of a plane 
about a given axis, divide the least moment of inertia by the 

sectional area of the plane ; that is, r' = — — . 



ilues of— 


For Malleable Iron. 


For Cast Iron. 


For Dry Tinaber. 


(7 = 


36.000 lbs. 
36.000 - 
0.000333 


80,000 lbs. 

3.200 " 
0.0025 


7,200 lbs. 
3,000 " 
0.004 



The factor of safety k should be, for wrought iron = 6; for cast 
iron =s 8 ; for timber = 10. This applies to moving loads. 

Case 1. 
Rounded or hinged at both ends, as per — 

For square, rectangular, or circular cross-section : 
TTT 1 OA 



fV — 


■ h 


1 + *" la 


any 


othei 


cross- section 


Tr = 


1 


CA 



Caie2. 

Fixed, or having a flat base at one end, and rounded or hinged 
at the other, as per — 

i For square, rectangular, or circular cross-section : 

For any other cross-section : 
1 CA 



Tr=« 



* 1+ i«-^ 



9.c.r« 



y Google 



BESISTAirOE TO OKUSHING. 



105 



OaseS. 

Fixed, or having flat bases at both ends, as per — 
Fig. 175. 

For square, rectangular, or circular cross-section : 
1 OA 




Tr = - 



1+ a 






For any other cross- section: 



Tr = -, 



OA 



1 + 



c,r^ 



Rounded at both ends : 



Examples. 
Case 1. 



What is the capacity of a urou^ght-iron strut of the annexed 
figure and dimensions? 

I = 10 feet = 120 inches. 
-4 = 4 . 68 inches. 




0.9 X 3.58+5.1x0.38 
12 

3.227 



— 3.227 



Tr=}- 



36000 X 4.68 



1 + 



4 X 120» 
36000x0.689 



= i- 



168480 



57600 
■^ 24804 



I J^!?2_ = 12.679 lbs. 
* 3.322 i-^."'«'*i'»- 

The same as above, in Case 3, fixed at both ends : 
36000 X 4.69 , 168480 



TTs 



1 + 



120« 



= 1- 



36000x0.689 



1 + 



14400 
24804 



,J11L = »,e„,u. 



y Google 



106 



BESISTA.KOE TO C&USHIiTG. 



For the annexed figure and dimensions ; otherwise, same as 
above : 



A = 7 inches. 

Bounded at both ends : 
Fig. 177. 



Case I. 




1 X4»+3X V 
12 

5.6 



:5.6 



ir=t- 



1 + 



36000 X.7_ ^ J J5?92L = 21.000 lbs. 
"36000 X 0.8 



Same as above, in Case 3, fixed at both ends : 
36000 X 7 



Tr=j- 



1 + 



1202 



, 252000 ,c.^r.^^. 

= i — 7-^ — = 42,000 lbs. 
l.o 



Fixed ends : 



36000 X 0.8 

Case 3. 



What is the capacity of a cast-iron pillar of the annexed figure 
and dimensions ? 

I = 10 feet = 120 inches. 
A sss II inches. 

Fig. 178. 

U- ^L -! 




J = 



8 X 4» — 7 X 3« 



12 



:26.9 



W=^i-- 



80000 X 11 



1+0.0025 



120» 
4«"" 



, 880000 ooo.^ii. 
= *— 3-.■25"•"^^•^^^^^^• 



yGoogIe 



BI8ISTANCX TO OSUBHIlFe. 107 

'For the annexed figure and dimensions; otherwise, same as 
above. 



W^i 



W=i 



Fig. 179. 

il =s 28 inches. 

80000 X 28 

1-1.0.0025 -i|^- 

, 2240000 ,^.,«^^„ 

1 5— 179.200 lbs. 

^ 1.6625 A/J7,^vvius. 

For the annexed figure and dimensions; otherwise, same as 
above. 

Fia. 180. 

ii = 22 inches. 

80000 X 22 

1 + 0.0025 -1|^ 

__1!60000__ 
^ 1.5625 

To find the oapacity of a Column, Pillar, or Strut of any 
cross-sectiOA by the following Table : 

Find how many times the least dimension h across the section 

is contained in the length I of column, Ac. — that is, then 

multiply the corresponding number on the same horizontal line, 
under A^^, by the sectional area of cross-section. This gives the 
capacity in tons of 2,000 lbs. 
Let f = Length of column, &c. 

h = Least dimension of cross-section. 
K^^ = Capacity in tons of one scjuare inch of cross-section, to 
be multiplied by sectional area of desired cross- 
section. 
Various sections for which this table is applicable: 
Fig. 181. Fig. 182. 






byt^OOgle 



108 BEBTSTAircE TO CBUSHnTG. 

Fig. 183. Fig. 184. 



Fig. 186. 



Fig. 185. 




Fig. 187. 



i^ipr. 188. 



[Note.— This table is strictly correct, only for columns, Ac, with circular 
or rectangular cross-section. As the error is small, it may be used for 
any cross-section.] 

Example explanatory of the following table. 

What is the capacity of a cast-iron column 10 feet = 120 inches 
long, fixed at both ends, and of the annexed cross-section and 
dimensions? t 



Fig. 189. 




120 



-;-=-—- =40 ^^^for40=1.000ton8. 



l_ 

A ~ 3 
Area= 6 inches. 
Tr= 6 X 1 = 6 tons, 8 times safety. 



y Google 



BE8ISTAN0E TO CBUSHIHG. 

Column^ Ae.^ fixed at both ends. 



109 



Cast Iron—eight times safety. 


Wrought Iron- 


-six times safety. 


1 
h 


IT 


I 
h 


i^' 


I 
h 


K" 


l 
h 


JT' 


I 
h 


K" 


I 
h 


K>' 




Tons. 




Tons. 




Tons. 




Tons. 




Tons. 




Tons. 


1 


4.987 


26 


1.951 


49 


0.714 


1 


2.999 


26 


2.487 


49 


1.674 


2 


4.950 


26 


1.868 


60 


0.689 


2 


2.996 


26 


2.462 


60 


1.644 


8 


4.890 


27 


1.771 


51 


0.666 


3 


2.991 


27 


2.418 


61 


1.616 


4 


4.807 


28 


1.689 


62 


0.644 


4 


2.984 


28 


2.383 


62 


1.686 


5 


4.705 


29 


1.611 


53 


0.623 


6 


2.976 


29 


2.348 


53 


1.657 


6 


4.687 


30 


1.638 


54 


0.608 


6 


2.964 


30 


2.313 


64 


1.529 


7 


4.450 


31 


L469 


56 


0.584 


7 


2.968 


31 


2.277 


65 


1.601 


8 


4.310 


32 


1.404 


66 


0.565 


8 


2.938 


32 


2.242 


56 


1.474 


9 


4.168 


33 


1.343 


57 


0.548 


9 


2.921 


33 


2.206 


57 


1.448 


10 


4.000 


34 


1.285 


68 


0.631 


10 


2.905 


34 


2.172 


68 


1.422 


11 


3.838 


36 


1.230 


69 


0.616 


11 


2.885 


35 


2.136 


59 


1306 


12 


3.676 


36 


L179 


60 


0.500 


12 


2.863 


36 


2.101 


CO 


1.371 


13 


3.514 


37 


1.1.30 


61 


0.485 


13 


2.841 


37 


2.067 


61 


1.347 


14 


3.365 


38 


1.084 


02 


0.471 


14 


2.817 


38 


2.032 


62 


1.323 


15 


3.200 


39 


1.041 


63 


0.457 


15 


2.792 


39 


1.998 


63 


1.299 


16 


3.048 


40 


1.000 


64 


0.446 


16 


2.766 


40 


1.963 


64 


1.276 


17 


2.902 


41 


0.961 


65 


0.432 


17 


2.738 


41 


1.930 


66 


1.263 


18 


2.762 


42 


0.924 


66 


0.420 


18 


2.711 


42 


1.896 


66 


1.228 


19 


2.628 


43 


0.889 


67 


0.409 


19 


2.680 


43 


1.863 


67 


1.209 


20 


2.500 


44 


0.856 


68 


0.398 


20 


2.660 


44 


1.831 


68 


1.187 


21 


2378 


45 


0.824 


69 


0.387 


21 


2.619 


46 


1.798 


69 


1.167 


22 


2.262 


46 


0.794 


70 


0.3T7 


22 


2.686 


46 


1.767 


70 


1.146 


23 


2.152 


47 


0.766 


71 


0.367 


23 


2.664 


47 


1.736 


71 


1.126 


24 


2.049 


48 


0.739 


72 


0.368 


24 


2.620 


48 


1.704 


72 


I.IOT 



y Google 



110 



BESI8TANCE TO CETTSHnTO. 



Strength of Columns, Pillars, or Struts, of seasoned wood, rotm^ <Hr 
square section. 
Fixed at both ends. All dimensions in inches. 
Find how many times the least dimension across the section is 

contained in the length or height of column, Ac, that is, -— - ; 

then multiply the corresponding figures on the same horizontal 

line under K^^ by the sectional area of cross-section. This gives 

the capacity of column, <&c., in tons of 2,000 lbs., 10 times safety. 

Reference, . ,. 

H^ss Length of column, &.c. ^| 

D := Least dimension of cross-section. ' ^f 

K^^ = Capacity in tons of one square inch of cross-section, ^ 

be multiplied by sectional area of desired cross-section. -■ V 

The coefficient C for white and yellow pine in the foUowiiMt 

table is ta^en at ^^ = 600 lbs. for safety : 

For oak at H%^ = 800 lbs. per square inch for safety. i\ 

Example. — What is the capacity of a pillar of oak, sectiotf 
4x6 inches, length = 12 feet = 144 inches ? 

-^ = 4^ = 36, K'' for 36 = 0.064 X 4 X 6 = 1.536 tons. 

Capacity K'^ of one square inch in tons of 2,000 lbs. 



White and Yellow Pine. 


Oak. 


H 




H 




H 




H 






K" 




K" 




K" 




K"' 


"F " 




^" 




'd" 




~D " 


1 


1 


0.299 


26 


0.081 


1 


0.399 


26 


o.ioa-^ 


2 


0.2!)5 


27 


0.076 


2 


0.394 


27 


0.102 f 


8 


0.289 


28 


0.072 


3 


0.386 


28 


0.096 .j; 


4 


0.282 


29 


0.068 




0.376 


29 


0.Q91 p 


6 


0.272 


30 


0.066 




0.363 


30 


0.086 ^ 


6 


0.262 


31 


0.061 




0.349 


31 


0.Q82 


7 


0.261 


32 


0.058 




0.334 


32 


0.078 


8 


239 


33 


0.066 




0.319 


33 


0.074 . 


9 


0.226 


34 


0.053 




0.302 


34 


0.071 


10 


0.214 


85 


0.050 




0.285 


35 


0.067 


11 


0.202 


36 


0.048 




239 


36 


0.064 


12 


0.190 


37 


0.046 




0.254 


37 


0.061 


13 


0.179 


38 


0.044 




0.238 


38 


0.069 


14 


0.168 


39 


0.042 




0.224 


39 


0.056 . 


16 


0.168 


40 


0.040 




0.210 


40 


0.064 


16 


0.148 


41 


0.038 




0.197 


41 


0.051 . 


17 


0.139 


42 


0.037 




0.185 


42 


0.O49 • 


18 


0.130 


43 


0.035 




0.174 


43 


0.047 , 


19 


0.123 


44 


0.034 




0.163 


44 


0.045 


20 


0.116 


46 


0.038 


20 


0.164 


45 


0.044 , 


21 


0.108 


46 


0.031 


21 


0144 


46 


0.042 : 


22 


0.102 


47 


0.030 


22 


0.136 


47 


0040 


23 


0.096 


48 


0.029 


23 


0.128 


48 


0.039 


2i 


0.030 


49 


0.028 


24 


0.121 


49 


0.037 


25 


0.085 


50 


C.027 


25 


0.114 


50 


0.036 



Digitized by VjVJ W V IC 



PABALLELOGBAM OF POBOES. 



Ill 



PARALLELOGRAM OF FORCES. 

CoMPOSITlOir AND RESOLUTION OP FOBCES. 

Heference, 

A, B, (7 = Forces, or strains, acting on a single point, 
v, v\ = angles. 



Fig. 190. 




. Csin.V/ 

"" 8in.(t;+v^)* 

Csin. V 



-i^^'^^"^'"''''"^-^- 



-K^secv; 



when » + », < 90° C'= J^' + £»+ (2 ^ £ cos. {v + v,) \ 



when V 



180° 



-Ft^. 191, 



iy + v,))^ 




Fig. 192. 




v4.v^=90« 
ul = Ccos. V 
B ^ C sin. V ss O COS. Vy 
C = \/A^ + B^ 



v^= 9CP 



COS. V 

B=s tang. V 



y Google 



112 8TBAIHS IK FRAMES. 



STRAINS IN FRAMES. 



Reference. ' . , \ 

C = Compressive strain in units of weight, ^ 

r= Tensile " " ': 

F= Vertical " " .*• 

^= Horizontal " * ' = 

Tr= Load in units of weight. . . ■ ^ 
I = Dimensions in units of length. 
V = Angle between horizontal and inclined member. \ 
For cioss-breaking strain, see "Resistance to cross- breaking.T5 

Fig, 193. 

e_ ^ 

2 sin. V 
C^=i -2~ ^°*g' '^^^ 



Fin. 194. 



C/= H^=\\W cotg.i? = cross-breaking strain at H. 
B,^ -^B^ W'-j- ^cotg. V = tension in E,. 

if— 5". = U. T-^- ^ W cotg. v = comwession in C/ 

Digitized by LnOOQlC 



Mg, 195. 



STRAINS IN FRAMES. 113 



V = — — : = compression. 

l^ Bin. V ^ 

^ ^/ W.l 
Cy = = — = corn- 
cos, y i^^.cos. y 



pression. 

c„=w. 

H— W.l. 



h. 



W.l 
V=^ Hy tang, y = — tang, y 



When I > Z, the portion lyy is in tension == V — W=: 
jrf-i-tang.y— l) 

\ ly/ / 

When i < ^3 the portion lyy is in compression = IT — F = 
Pr(l-.^tang.y) 



Vy = —-I^, W:=z tension. 






\^ — ^/ 



F^ = r- IT = ( ' ^ ^ ) ir, = Ty (tension) = C, (c( 
pression.) 

Digitized by CjOOQIC 



11 i 8TBAIl!rS IN BOOM DEBBICK3. 



Cs= f — ^j — : = (compression) =: T (tension.) 

H = ( — j TFcotg. V = (tension) = Hy (compression.) 



Ends of beams not built into wall or fixed: 
I — I, 



F,= F— TF= \ i' ^ ^= ^/ (compression) = T, 

(tension.) 

C = -J^ = , ^^ = 2 (tension.) 
sin. V ly sm. V 

IpGs: Fcolg. V = — — TFcotg. V = (tension) = ffy (compression.) 

V 



STRAINS IN BOOM DERRICKS. 

Eeference. 
0= Compression in boom. 
C/= Compression in mast. 
T= Tension in tackling. 
Ty= Tension in guy. 

t = Tension in runner from mast head to weight. 

ty= Tension in runner from boom head to weights 
W = Weight or load. 
H = Horizontal strain. 
F = Vertical strain. 
V, Vj, Vg = Angles. (See Figure.) 

Fi(f. 198. 



y Google 



STBAINS nr TBX7BSES. 



115 



t^ 



IT sin. Vi 



sin. (v + Vj) 
V^ty cosin. Vi 
C= Kcosec. Va 
T= Kcosec. Vj 



</=• 



TTsin. V 



sin. (« + Vi) 
JE = Fcotg. Vj 

Cy^ W 

T/= Fcotg. Vg sec. t;^ 



STRAINS IN TRUSSES. 

Load equally distributed. 

Eeference. 
TTss Load eqnally distributed in lbs. 
I := Distance between abutments, 
t; = Angle between liorizontal and diagonal. 
C = Compression in lbs., (denoted by tnick lines.) 
T^ Tension in lbs., (denoted by tbin lines.) 

2 Bays =.-5- 



Fig, 199. 




3Bay8 = — 



Fig. 200. 




C,= 



W 



^1=* 



W 
sin. V 



y Google 



lie 



8IBAIBS IS TVUSSS9. 



4Bay«= — 

Fig. 201. 




3*2= -^ COSfiC. « 



5Bay=_ 

Fig. 202. 



C = r = 3Ca cotg. V 
Ci= ^1= 2(7, cotg. 1? 

^3= 2C, 
T^= O2 cosec. V 
2"«=27'a 



y Google 



STRAINS IH TBUSSES. ' if 7 

6 Bays = — 



(73= ^3= — ^ cotg. t; Te = -^ cosec. » 

TT * 2*4 = 3^, 

Table of Constants, based on fobeooing Fo&kula. 
Load equcUly distributed. 

Table of constants for strains in respective member of trusses, 
from 2 to 6 bays, with diagonals inclined from 5^ to 45^: 

Reference, 

W =K Load in lbs., equally distributed over whole length of 
truss, to be multiplied by constant for strain in re- 
pective member. 
V =^ Angle between horizontal and diagonal. 
C =? Compression in lbs. in respective member. 
!r= Tension in lbs. in respective member. 
Example.— Required, the strain in the various members of a 
truss of 4 bays. Length »*= 40 feet : load W = 80,000 lbs. ; angle 
V = 20°. 

Members. Constants. W. Strains. 

C = r = 1.372 X 80,000 = 109,760 lbs. 
(7i= ^1= 1.029 X 80,000 = 82,320 " 
C7a=0.25 X 80,000= 20,000 " 
a= 0.375 X 80.000= 30,000 " 
2^3=0.365 X 80,000= 29.200 " 
5^= 1.095 X 80,000= 87,600 " 

[NoTs.— When the trusses are inyerted, the strains change in kind, but 
not in amount] 

Digitized by VjUU^ It' 



118 



8TBAIHS IV TBUSSES. 



2 Bays = — 

J?^. 204. 



3 Bays = — 

Fig. 205. 




V 


a 


Oi 


T 


C—T 


Ci 


Ti 


5 


3^72 


0.625 


3.684 


3.810 


0.333 


3.820 


6 


2.972 


** 


2987 


3.170 


M 


ai86 


7 


2.544 


w 


2.662 


2.713 


** 


2.733 


8 


2.226 


<{ 


2.244 


2.370 


«« 


2.393 


9 


1.972 


u 


1.997 


2.103 


M 


2.130 


10 


1.772 


u 


1800 


1.890 


" 


1.920 


11 


1.610 


(( 


1.640 


1.'.10 


*( 


1.747 


12 


1.469 


(( 


1.600 


1.570 


U 


1.603 


13 


1.353 


(( 


1.390 


1.444 


U 


1.483 


14 


1.253 


u 


1.2iK) 


1.333 


U 


1.376 


16 


1.166 


« 


1.210 


1.243 


M 


1.286 


16 


1.087 


M 


1.134 


1.160 


(t 


1.210 


17 


1.022 


*( 


1.070 


1.090 


il 


1.140 


18 


0.969 


i( 


1.013 


1.C23 


« 


1.080 


19 


0.906 


M 


0.959 


0.970 


U 


1.023 


20 


0.859 


U 


0.912 


0.917 


** 


0.973 


81 


0.813 


it 


0.872 


0.866 


{t 


0.930 


22 


0.778 


(1 


0.834 


0.823 


U 


0.890 


23 


0.734 


M 


0.790 


0.783 


(( 


0.853 


24 


0.703 


U 


0.765 


0.750 


M 


0.810 


25 


0.668 


l( 


0.738 


0.713 


if 


0.786 


26 


0.641 


«* 


0.712 


0.G85 


»« 


0.760 


27 


0.613 


M 


0.687 


0.653 


U 


0.730 


28 


0.687 


U 


0.666 


0.626 


«« 


0.701 


29 


0.662 


(( 


0.644 


0.600 


.« 


0.686 


SO 


0.641 


U 


0.625 


0.643 


« 


0.666 


31 


0.619 


u 


0.606 


0.666 


tt 


0.646 


32 


0.600 


u 


0.691 


0.633 


it 


0.630 


83 


0.481 


u 


0.675 


0.513 


u 


0.613 


34 


0.463 


<• 


0.559 


0.493 


{( 


0.690 


35 


0.447 


u 


0.544 


0.476 


(( 


0.680 


36 


0.431 


w 


0.631 


0.460 


{( 


0.566 


37 


0.416 


l< 


0.619 


0.444 


tt 


0.553 


38 


0.400 


*» 


0.506 


0.426 


u 


0.640 


39 


0.384 


(. 


0.497 


0.410 


tt 


0.530 


40 


0.372 


« 


0.487 


0.396 


tt 


0.620 


41 


0.359 


u 


0.475 


0.385 


** 


0.506 


42 


0.347 


ti 


0.466 
0.4^ 


0.370 


tt 


0.496 


43 


0334 


(1 


0JJ67 


M 


0.486 


44 


0»322 


«« 


0.460 


0.343 


<l 


0.480 


45 


0.312 


(( 


0.444 


0.333 


" 


0.473 



y Google 



STBAIKS IN TBUSSE3. 



119 



4 Bays = — 

Fig. 20r>. 




V 


c-r 


Ci'-Ti 


C2 


C3 


T2 


^3 


6 


5.72'D 


4290 


0.2.M) 


0.375 


1.434 


4.032 


6 


4.7uO 


3 570 


" 


" 


1.200 


3.000 


7 


4.0G8 


3.051 


♦« 


»♦ 


1.025 


3.075 


8 


3.5G0 


2.070 




" 


0.897 


2 591 


9 


3.1G4 


2.;i73 


" 


" 


0.799 


2.397 


10 


2.822 


2.124 




'» 


0.720 


2.160 


11 


2.568 


1.926 


•* 


«• 


0.055 


1.905 


12 


2.388 


1.791 


♦• 


'• 


0.(i01 


1.803 


13 


2.104 


1623 


•' 


" 


0.556 


1.668 


14 


2(K)0 


1.500 


•' 


" 


0.516 


1.548 


15 


1.804 


1.308 


«« 




0482 


1.446 


10 


1.740 


1.305 




" 


0.454 


1.362 


17 


1.0.^2 


1.224 


*• 


«• 


0.428 


1.284 


18 


1.5.32 


1.149 


" 


'• 


0.405 


1.215 


19 


1.448 


1.086 




" 


0.384 


1.152 


20 


1.372 


i.oa 


t( 


" 


0.365 


1.095 


21 


1.300 


0.975 


" 


•♦ 


0.349 


1.047 


22 


1.236 


0.927 


« 


*' 


0.3.34 


l.(J02 


23 


^ 1.172 


0.879 


»' 


" 


o.:^J 


0.900 


24 


1.124 


0.843 


" 


*' 


0.306 


0.918 


25 


1.008 


801 


(. 


•« 


0.295 


0.885 


26 


1.024 


0.768 


'■ 


(4 


0.285 


0.855 


27 


0.980 


0.735 


•' 


' 


0.275 


0.825 


2S 


0.940 


0.705 


'• 


•' 


0.266 


0.798 


29 


0.900 


0.675 


« 


'• 


0.258 


0.774 


30 


0.864 


0.648 


•t 


*• 


0.250 


0.750 


31 


0.823 


0.621 


«' 


" 


0.243 


0.729 


32 


0.800 


0.600 


•' 


(4 


236 


0.708 


:ii 


0.708 


0.576 




" 


0.230 


o.ono 


34 


0.740 


0.555 


*• 


•* 


0.224 


0.072 


35 


0.720 


0.540 


" 


♦♦ 


0.218 


0.054 


36 


0.688 


0.516 


•' 


*' 


0.212 


0.636 


37 


0.C64 


0.498 


ti 


♦• 


0.207 


0.621 


38 


0.640 


0.480 


" 


" 


0.203 


0.009 


39 


0.016 


0.462 


«• 


«• 


0.199 


0.597 


40 


0.000 


0.4.50 


<; 


" 


0.195 


0.585 


41 


0.576 


0.432 


«• 


'♦ 


0.190 


0.570 


42 


0.5C0 


0.420 


•* 


" 


0.186 


0.558 


43 


0.5.{6 


0.402 




•' 


0.183 


0.549 


44 


0U>20 


0.390 


•' 


*• 


0.180 


0.540 


45 


0.500 


0.375 


" 


'■ 


0.177 


0.631 



y Google 



120 



STBAI]7S IS TBV8SE8. 



5 Bay8 = — 



V 


<7-r 


Ci-Ti 


C2 


Cs 


T2 


Ts 


6 


6.858 


4.572 


0.200 


0.400 


2.294 


4.588 


6 


5.706 


3.804 


" 


u 


1.912 


3.824 


7 


4.884 


3.266 


•♦ 


" 


1.640 


3.280 


8 


4.272 


2.848 


'• 


" 


1.436 


2.872 


9 


3.786 


2.524 


" 


•• 


1.278 


2.556 


10 


3.402 


2.268 


" 


« 


1.152 


2.304 


11 


3.084 


2.066 


«« 


♦* 


1.048 


2.096 


12 


2.820 


1.880 


»* 


a 


0.962 


1.9-4 


13 


2.698 


l.r32 


" 


*♦ 


a890 


1.780 


14 


2.406 


1.604 


" 


'* 


0.826 


1.G62 


15 


2.2:38 


1.492 


*' 


" 


0.772 


1.544 


16 


2.088 


1.392 


" 


" 


0.726 


1.452 


17 


1.962 


1.308 


" 


♦• 


0.684 


1.3C8 


18 


1.842 


1.228 


*( 


»* 


0.648 


1.296 


19 


1.740 


1.160 


" 


" 


0.614 


1.228 


20 


1.660 


1.100 


" 


" 


0.584 


1.168 


21 


1.560 


1.040 


♦♦ 


»♦ 


0.668 


1.116 


22 


1.482 


0.988 


" 


•* 


0.634 


1.068 


23 


1.410 


0.940 


•* 


" 


0.512 


1.024 


24 


1.360 


0.900 


** 


*' 


0.490 


0.980 


25 


1.284 


0.866 


»♦ 


*♦ 


0.472 


0.944 


26 


1.230 


0.820 


" 


'* 


0.456 


0.912 


27 


1.176 


0.784 


i. 


»• 


0.440 


0.880 


28 


1.128 


0.762 


«' 


*' 


0.426 


0.862 


20 


1.080 


0.720 


u 


C( 


0.412 


0.824 


30 


1.058 


0.692 


«« 


« 


0.400 


0.800 


31 


0.996 


0.064 


" 


{. 


0-388 


0.776 


32 


0.960 


0.640 


" 


*♦ 


0:W8 


0.766 


33 


0.924 


0.616 


(( 


ti 


0.368 


0.736 


34 


0.888 


0.692 


« 


'« 


0.368 


0.716 


35 


0.858 


0.572 


« 


" 


0.348 


0.696 


36 


0.828 


0.552 


" 


« 


0.340 


0.680 


87 


0.798 


0.532 


u 


'• 


0.332 


0.664 


38 


0.-768 


0.512 


«« 


« 


0.324 


0.048 


39 


0.738 


0.492 


** 


" 


0..318 


0.636 


40 


0.714 


0.476 


«• 


(• 


0.312 


0.624 


41 


0.690 


0.460 


«« 


«« 


0.304 


0.008 


42 


0.666 


0.444 


" 


" 


0.298 


0.696 


43 


0.642 


0.428 


«♦ 


" 


0.292 


0.684 


44 


0.618 


0.412 


i( 


♦' 


0.288 


0.676 


46 


0.600 


0.400 


" 


" 


0.284 


0.668 



y Google 



STRAINS IN TRUSSB.S 



V 


C— T 


Cl-Ti 


1 

O2 — T: 


C3 


^4 


C5 


n 


'A 


To 


5 


8.568 


7.616 


4.760 


0.1G6 


0.210 


0.416 


0.952 


2.866 


4.760 


G 


7.123 


6.336 


3.960 


(. 


*« 


** 


0.793 




3.966 


7 


6.102 


6.424 


3.390 


" 


•' 


** 


0.680 


2.041 


3.402 


8 


6.337 


4.744 


2.966 


♦• 


«♦ 


" 


0.696 


1.78^ 


2.980 


9 


4.G2> 


4.200 


2.625 


'• 


" 


" 


0.630 


1.690 


2.660 


10 


4.218 


8.776 


2.360 


(( 


" 


♦• 


0.478 


1.434 


2.390 


11 


3.852 


8.424 


2.140 


*' 


" 


(( 


0.436 


1.305 


2.176 


12 


3.619 


3.128 


1.955 


*' 


" 


*♦ 


0.399 


1.197 


1.996 


i:j 


3.240 


2.880 


1.800 


" 


•' 


•* 


0.369 


1.107 


1.846 


14 


3.00G 


2.G72 


1.670 


" 


«* 


u 


0.343 


li)29 


1.715 


10 


2.799 


2.488 


1,555 


" 


*• 


»i 


0.320 


0.960 


1.600 


IG 


2.610 


2.320 


1.450 


*» 


" 


It 


0.301 


Q.903 


1.505 


17 


2.448 


2.176 


1.360 


" 


*' 


It 


0.284 


0.852 


1.420 


16 


2.:j04 


2X)48 


1.280 


♦• 


" 


ft 


0.269 


0.807 


1.345 


19 


2.1G9 


1.928 


1.205 


" 


*• 


tt 


. 0.255 


0.765 


1.275 


20 


2.0CI 


1.832 


1.146 


•' 


«' 


•♦ 


0.242 


0.726 


1.210 


21 


1.944 


1.728 


1080 


" 


«< 


'* 


0.231 


0.693 


1.165 


22 


1.864 


1.648 


1.030 


it 


{. 


" 


0.221 


0.663 


1.105 


23 


1.7G4 


1.568 


0.980 


" 


" 


it 


0.212 


0.636 


1.060 


21 


l.G8;i 


1.496 


0.935 


« 


(( 


" 


6.203 


0.609 


1.015 


2j 


1.G02 


• 1.424 


0.890 


»« 


«* 


" 


^196 


0.588 


0.980 


2.i 


1.639 


1.368 


0.855 


>( 


. a 


" 


0.189 


0.667 


0.946 


27 


1.467 


1.304 


0.816 


»• 


«• 


" 


0.182 


0.646 


0.910 


28 


1.404 


1.248 


0.780 


" 


f« 


" 


0.177 


0.531 


0.885 


29 


1.350 


1.200 


0.760 


«« 


(( 


•• 


6.171 


0.513 


0.855 


:w 


1.2)6 


1.162 


0.720 


(i 


t« 


t. 


0.166 


0.498 


C.830 


31 


1.242 


1.104 


0.090 


«« 


•' 


" 


0.161 


0.483 


0.805 


32 


1.197 


1.064 


0.666 


(( 


»« 


tt 


0.156 


0.468 


0.780 


33 


1.152 


1.024 


0.640 


" 


" 


t( 


0.162 


0.45G 


0.760 


34 


1.107 


0.984 


0.616 


•» 


« 


" 


0.148 


0.444 


0.740 


86 


1.071 


0.952 


0.696 


'• 


♦♦ 


" 


0.144 


0.432 


0.720 


30 


1.036 


0.9-20 


0.576 


•♦ 


«♦ 


«« 


0.141 


0.423 


0.705 


37 


0.999 


0.888 


0.656 


'« 


« 


" 


0.1.38 


0.414 


0.690 


3S 


0.9.H 


0.848 


0.530 


" 


«» 


« 


0.134 


0.402 


0.670 


39 


0.918 


0.816 


0.510 


•» 


« 


«« 


0.132 


0.396 


0.660 


40 


0.891 


0.792 


0.496 


'• 


•• 


*« 


0.129 


0.387 


0.645 


41 


0.864 


0.768 


0.480 


" 


«' 


•» 


0.126 


0.378 


0.630 


42 


0.82j 


0.736 


0.460 


" 


" 


»t 


0.123 


0.369 


0.616 


43 


0.801 


0.7;2 


0.445 


♦' 


" 


" 


0.121 


0..')63 


0.605 


44 


0.774 


0.688 


0.430 


« 


«4 


tt 


0.119 


0.357 


0.596 


46 


0.747 


0.664 


0.416 


" 


" 


" 


0.118 


0.354 


0.690 



y Google 



122 8TBAINS IN TBUS8ED BEAMS. 



STRAINS IN TRUSSED BEAMS. 

When a beam supported at the ends, is required to carry a 
greater load than its given capacity, and trussing is resorted to, 
it may become necessary to find what portion of the load is borne 
by the different members of the trussea beam. 

Reference. 
Let IF = Load acting on truss at a supported point. (See figure.) 
Wy= That portion of PF acting on (iiagonals. 
Tr^= That portion of TF acting on beam. 
A^=: Sectional area of diagonal. 
A 2= Sectional area of beam. 

JE^j= Modulus of elasticity of material in diagonals. 
J5^2= Modulus of elasticity of material in beam. 

a = Length of diagonal. 

h = Distance between center of beam and point of support. 

c = Distance between abutment and point of support. 

{ = Depth of beam. 
= Depth of truss. 
I = Distance between center of beam and abutment. 

[Note.— Use the same unit of length and weight] 

No. 1. 

Fig. 209 





P 
a' 




^1 


-El' 


w,= 






^1 




w,= 






A, 




A,= 




a* 


PA. 


E, 
■ E, 



d by Google 



STRAINS IN TRUSSED BEAMS. 



123 



_^ __P_ h'^A^ Ei_ 

^^-^ PFi • a» • p ' E, 



W,: 









w 



+ 1 






When load is equally distributed W becomes | W. 
No. 2. 



Fig, 210. 



-Pi^r 211. 






Wi= 



2 ' 



W^=2Wi. 






JL 



Ai 

Ai ' 

Ai 
A^ 

A, 



_Ei_ 
E2 

^1 



ili= 



2J^ 






^ 
^i 



iia= Ji 



Tri=- 



2IFi 



IT 



1^2 



+ 1 






Tr,= 



IT 



_^+l 
Fi ^ 



When load is equally distributed PF becomes J PP. 



y Google 



124 STBAINS IN TRUSSED BEAMS. 

No. 3. 

Fig, 212. 





















When load is equally distributed IT becomes iW> 



y Google 



STRAINS IN TItUSSED BSAHS. 125 

No. 4. 

Figs. 213 and 214. 



TFs "" 2y» • a{a^+bc) ' A^ ' K2 



"^'"~ 2/« • aia*+bc) ' Ai ' Ik 



W-2W f" «(«'+^<') ^» ^ 









'2Tri * p • aia^^bc) ' E^ 






When load is equally distributed TF'becomes \W, 



y Google 



126 STRAINS IN TBUSSES WITH PARALLEL BOOMS. 

STRAINS IN TRUSSES. WITH' PARALLEL BOOMS. 
{Caused hy Static and Moving Loads) 

The strain in the upper boom is always compressive. 
♦►The strain in the lower boom is always tensile. 

All braces inclined down from the nearest abutment are in 
tension. 

All braces inclined up from the nearest abutment are in com 
pression. 

The strains in the verticals and diagonals increase from the 
center of truss to abutment. 

The strains in the booms decrease from the center of truss to 
abutment. 

A moving load, advancing over a truss, &c., causes the maxi- 
mum moment of rupture (which under an ec|ually distributed 
load is at the center of truss) to shift to one side of the center, 
thereby changing the nature and amount of strain in web only. 
This requires either the enlargement of those rnembers consti- 
tuting the web or the addition of so-called counters, (braces, 
struts, or ties.) 

To find the point from center of truss to where the addition oi 
counters must commence, the following formula is used : 

Let d = Distance from center of truss to point where 
maximum moment of rupture occurs, and where 
counter bracing must commence. 

d^= Distance from nearest abutment to ditto. 



These results will be found to agree with formulas for " Counter 
Strains" when Fm becomes negative. 

Eeference. 

iV"^ Total number of bays in a truss. 
Hn = Horizontal strains in booms. 

Vn := Strains in verticals. 

Yn= Strains in diagonals. 

Fa = Vertical strains acting on counters y«. 

y„ = Strains in counters, opposite in kind to Ji. 

Digitized by ^^UUgle 



STEAIN8 IK TRUSSES WITH PARALLEL BOOMS. 127 

W = Weight of static load, equally distributed over whole 

length of truss. 
TF/= Weight of moving load, equally distributed over whole 

length of truss. 

?i ^ Height or depth of truss between the center of gravity of 
booms. 

I = Span or length of truss from abutment to abutments, 
n = Number of member, counting from abutment A. 
■m = Number of member, between center and abutment B. 

r = PlaK the length of a panel* or bay. 

« = Length of a panel or bay. 
w = Weiglit of static load per unit of length I. 
W/= Weight of moving load per unit of length I. 

V = Angle between horizontal and diagonal* 
For other designations, see diagrams and examples. 

The angle v for Howe Truss is generally 45°. 
The angle v for Whipple Truss is generally 45°. 
The angle v for Lattice Truss is generally 45°. 
The angle v for Warren Truss is generally 60°. 

The proportion of height h to span I is from f to -j^, gener- 
ally A. 



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123 STRAINS IK TBUSSESWITH FABAIiLEL BOOKS. 



o 
o 






y Google 



8TBAIKS IK TRUSSES WITH PARALLEL BOOMS. 129 

Howe Truss. (Figs, 215, 216, 217, and 218.) 

Additional Reference, 
x^z=. Distance from abutment A to center of bay. 
y,^= Distance from abutment A to apex of bay. 

Static or Permanent Load^ equally disiribiUed over whole length of 
Truss. 

Strains in Booms, 

Strains in Verticals, 

Strains in Diagonals. 
yi= F, C086C. t;. 



Moving and Static Load^ each eaually distributed per unit of 
length. 

Strains in Booms. 

^'- 2A -^^ — m^-^' 

Strains in Verticals. 
WW W\ 

Strains in Diagonals. 
F,=F,cosec. w. 

Strains in Counters, 

^ Digilized by Google 



130 &TRAIK8 IN TRUSSES WITH PAEALLBL BOOMS. 

Example. {Figs. 215, 216, 217, and 218.) 
Moving Load, (as railway train passing over bridge.) ^ 
We will assume W = 50.000 lbs. ^r 

Wi= 100,000 lbs. * • * 

I =z 100 feet. - 

A = 10 feet. 
i; = 4o°.(co8ec. = 1.414.) 

Ff.^rizontalStrainsin Booms, {compression inupper, iensionin I 
W+ TFi W+ Wx , _ 50000+10 
^^^= 2A •2'- 2hl •^' "" 20 

IL = 7500 . 10 — 75 . 100 = 67,500 lbs. 
// __ 7000.20 — 75.400 = 120,000 lbs. 
H. = 7500.30 — 75.900 = 157,500 lbs. 
Ill = 7500.40.— 75.1600 = 180,000 lbs. 
Jil = 7500.50 — 75.2500 = 187,500 lbs. 

Strains in Verticals. 
„ W W ,Wi ,x,_ 50000 5000^ 

strains in Figa. 215 216 217 
V. = 25000 — 500 . 5 + 5 . \)o^ = 67625 Ten. Ten. Com. 

F2 = 25000 — 500.154- 5.85^ = 53625 " " " ^^ 

F, = 25000 — 500.25+5.702 = 40625 - " " *;- 

F4 = i'5000— 500.35 + 5.652=28625 " " " 5' 

F5 = 25000 — 500.45+ 5.552= 17625 " " " **"' 

Counter Strains (Fm) for Strains in Counters. 
Fg = 25000 — 500.55 + 5.45^ = 7625. 
Fy = 25000 — 500 . 65 + 5 . S52 = 5625. 

Strains in Diagonals. ' ". 

Yn = Fn cosec V. 
Strains in Figs. 215 21G 217 2l» 
7, =67625 . 1.414 = 95,620 lbs. Com. Com. Ten. Tett.t: 
Y2 = 53625 . 1.414 = 75,826 lbs. " " " '* ' 

To = 40625 . 1 414 = 57,441 lbs. " 

Y. = 28625 . 1.414 = 40,476 lbs. " " " " , 

Fg = 17625 . 1.414 = 24.922 lbs. " " " " ': 

Strains in Counters, {doited lines, Fig. 215, /or example.) .. 

Y^ = F„ cosec. V. , 

Slrainy ill Figs. 215 2iG 217 21S ' "■ 

Yq = 7625 . 1.414 = 10,762 lbs. Com. Com. Ten. Ten." 

F, = 5625. 1.414= 7.954 lbs. ,-„e,by^uuyiL^ 



«r 



STEAnrS IN TRUSSES WITH PABALLEL BOOMS. 



131 



Fig, 219; Lattice Truss with Vertical Numbers. 

Fig. 219. Load on either Boom. 

To compute the strains in this truss, the 
I easiest method is to find the values of JEfn, F^, 

V„, y;, and r„ for a Howe Truss. {Figs. 215, 

216, 217, and 218 ) loaded in the same man- 
I ner, (upper or lower boom.) Tljese valuds in 

the following formulas for the above truss will 

give the required strains : 



Strains in Booms, {8.) 



5i= 



m 



* 2 



.ff.+ JSi 



S.= 



S,+ H, 



Qenerally ;6i,i 



^5 = 



2 



Strains in Verticals. {U.) 
Upper boom loaded — compression. 
Lower boom loaded — tension. 

^ = — ?rv? — constant. 
2N 



Strains in End Post ( U^) 

Upper boom loaded. 
?7o= U -\- Si= compression. 

Lower boom loaded. 
C^3= /S^i=r compression. 

Strains in Diagonals. (D.) 



-.=4 
^-.-4- 



D,= 

i>5 = 



21 

2 



^8 = 



2 



Generally 7)^= -^ 



Strains in Counters. 
Generally i>«= — ^ 



d by Google 



132 8TBAJNS IV TBUSSES WITH FABALLEL BOOHS. 

Fig, 220. Warbek Tbuss. 

Fig, 220. Lower Boom Loaded, 

Additional Reference. 
Xn = Distance from abutment A to center 

of diagonal. 
yn = Distance from abutment A to apez 

of bay of upper boom. 
2. == Distance from abutmetnt A to apez 

of bay of lower boom. 

Static or Permanent Load^ equally distributed 

over whole length of Truss. 

Strains in Booms. 

Upper. 

2h ^ 2hl ' 
Lower. 

^•=2A-2'--2W-^-' 
Strains in Verticals. 

W W 

Fi = — J- x^ {V^ acts at the end 

2 I 

of«..) 

Strains in Diagonals. 

Yu=-Vn cosec. V. 

Moving and Static Load, each equally dis- 
tributed per unit of length. 

Strains in Booms. 
Upper. 

W+W, TF+TT, 

■"• 2A~-^ 2W~"^ 

Lower. 

W+Wi W+Wi . 

^• = — 2— -y- 2«-y'' 

Strains in Verticals. 
W W . Wi 



yiagonals. 
cosec V. 

d by Google 



Strains in Diagonals. 
yi = F^ cosec V. 



STBAIH8 Il<r TBUSSES WITH PARALLEL BOOMS. 138 

Strains in Counters. 
F. = -~--y^a;«+ J{^~a^)« F. = F. cosec. v. 



Example. {Fig. 220.) 
Moving Load {as railway train passing over bridge) on lower Bocm. 



We will assume W = 

1 = 
h = 

v = 


50,000 lbs. 
: 100,000 lbs. 

100 feet. 
: 10 feet. 
: 63° 20^ (cosec. = 


= 1.12.) 


Horizontal Strains in Upper Boom. 


{Co7npressio7i.) 




2hl • ^ " 


50000+100000 
2.10 


50000+100000 
^ 2.10 .100 '^ "" 


150000 
20 -^^ 



150000 , ^.^^ ^^ , 

2000 ^' = ^^Q'^~75.2;,» 

J7i = 7500.10 — 75.100 = 67,500 lbs. 
H = 7500 . 20 — 75 . 400 = 120.000 lbs. 
^ = 7500.30 — 75.900 = 157,500 lbs. 
H^ = 7500.40 — 75. 1600 = 180,000 lbs. 
ITj = 7500.50 — 75.2500 = 187,500 lbs. 

Horizontal Strains in Lower Boom. { Tension.) 
W+Wi W+ Wi , 50000 + 100000 



2h ■^' 2/ti •^" ■* 2.10 

50000+100000 ^_ 150000 150000 ^ 



^" 2.10 . 100 ••^' 20 *'^' 2000 

^1 = 7500.5 —75.25 = 37500— 1875= 35,625 lbs. 
ir2 = 7500.15 — 75.225 =112500— 16875= 95,625 lbs. 
IT, = 7500.25 — 75.625 =187500— 46875 = 140,625 lbs. 
E. = 7500.35 — 75. 1225 = 262500 — 91875 = 170,625 lbs. 
jffg =: 7500.45 — 75.2025 = 387500 — 151875= 185,625 lbs. 

Digitized by LnOOQlC 



134 



STBAINS IN TBUSSK WITH PA.BALLEL BOOMS. 



Strains in VerticaU. 
Fn = Fn cos 

^" = -2-~— •""+l^■(^"- 
-o+-^^•(100-x„)«=250O 



2.100» 

= 25000- 

= 25000- 

= 25000- 

= 25000- 

25000- 

25000- 

25000- 

25000- 

25000- 

25000- 



I 

F. =25000 — 
F« = 

r = 



■ 500 . 2.5 
. 500 . 7.5 
. 500 . 12.5 

- 500 . 17.5 
. 500 . 22.5 

■ 500 . 27.5 

- 500 . 32.5 

- 500 . 37.5 
. 500 . 42.5 
. 500 . 47.5 



+ 



•^14 



= 25000- 
: 25000 - 
: 25000 • 
: Null. 






== 71281.25 
= 64031.25 
= 57031.25 
= 50281.25 
= 43781.25 
= 37531.25 
= 31531.25 
= 25781.25 
= 20281.25 
,= 14031.25 



r =10031.25 
^12= 5281.25 
Fi3= 781.25 



Counter Strains, 

- 500 . 52.5 4- 5 . 

- 500 . 57.5 + 5 . 

- 500 . 62.5 + 5 . 

Strains in Diai 

Y^=i FnCose 

1.12 = 79,835 lbs. 
1.12 = 71,715 lbs. 
1.12 = 63,875 lbs. 
1.12 = 56,315 lbs. 
1.12 = 49,035 lbs. 
1.12 = 42.035 lbs. 
1.12 = 35,315 lbs. 
, 1.12 = 28,875 lbs. 
1.12 = 22.715 lbs. 
1.12 = 15,715 lbs. 

Counter Stratno. 

^m=^mCOSeC. V. 

1.12 = 11,235 lbs. Compression in Y^^ and Ti^. 



i.l^= 11, ^ou lus. v/ompreauiuu lu jl iq auu j 

1.12= 5,915 lbs. Tension in Y^ and Y^^. 
1.12^= 875 lbs. Compression m Y^ and ] 



18* 



y Google 



STBAIKS IN TBUSSES WITH PABALLEL BOOMS. 



135 



Fig. 221. Wabren Truss. 

Fig. 221. tipper Boom Loaded, 

Additional Reference. 
Xn = Distance from abutment A to center 

of bay of upper boom, 
yo = Distance from abutment A to apex of 

bay of upper boom. 
2n = Distance from abutment A to apex of 

bay of lower boom. 

[ Static or Permanent Load, equally distributed 
over whole length of Truss. 

Strains in Booms. 
Upper. 






Lower. 



^« = 



w 

2h 



W 
2hl 



'Vn' 



Strains in Verticals. 
^^ W W 

^•=-5 r-'' 

Strains in Diagonals.. 
Y^ =■ V^ ca«ec. v. 

M(mng and Static Load, each equally dis- 
tributed per unit of length. 

Strains in Booms. 

Upper. 



fi.= 



W+Wy 



'-—{ 



r+iT, 



.2.' + 



5,: 





2hl 


; 




Lower. 




w+w, 




W+Wi 


2h 


• y« 


2hl 



^.y.' 



y Google 



136 



STRAUI8 IN 7BU8SES WITH PASALCEL BOOKS. 

I^rains in VerUeah, 

Straim in DiagonaU, 
T^ = Vn cosec. V. 



F„ = 



W 

9 



iSfrains in Counters. 
—-—x. A — (I — xJ\ * 



F«=F« cosec. 



Example, (i^^^. 221.) 
Moving Load {as railway train passing over bridge) on Upper Boom. 

We will assume W= 50.000 lbs. 
JFi= 100,000 lbs. 
Z=r 100 feet. 
^ = 10 feet. 
V = 63° 20^ r = 6 feet. 

Horizontal Strains in Upper Boom. (Compression.) 

^" "" ~2h '^ ■" L 2hi '^ "^ m J "" 



150000 
20 



[ 



150000 



150000.52 



2000 • " ' 2000 
7500.2.— [75. 2„«+ 1875] 



^^]- 






7500.5 — 


[75.25 H 


h 1875] 


7500.15 — 


75.225 H 


h 1875" 


7500.25 — 


75.625 H 


h 1875' 


7500.35 — 


75.1225 H 


h 1875 


7500.45 — 


75.2025 H 


h 1875] 



= 33,750 lbs. 
= 93,750 lbs. 
= 138,750 lbs. 
= 168,750 lbs. 
= 183,750 lbs. 



3.= 



Horizontal ^Strains in Lower Boom 



2h 



.y» 



2hl 



{Tension.) 
.y,« = 7500.y.-75.y.2 



: 7500.10—75.100 = 67,500 lbs. 
'. 7500 . 20 — 75 . 400 ?= 120,000 lbs. 
: 7500.30 — 75.900 = 157,500 lbs. 
; 7500.40 — 75. 1600 = 180,000 lbs. 
: 7500.50 — 75.2500 = 187,500 lbs. 



y Google 



STRAINS IK TRUSSES WITH PAEALLEI, BOOMS. 137 

Strains in VerticaU. 

F.= ^--f.*» + ^(i-.aO«= 25000-500.0^ + 

5.(2~a^)« 

Fi= 25000 — 500.6 + 5.95» = 67,625 lbs. 
F2= 25000 — 500. 15 4- 5 . 85» =5 53,625 lbs. 
Fo= 25000 — 500.25 + 5.75» = 40,625 lbs. 
V.= 25000 — 500.35 + 5.65a = 28,625 lbs. 
F5= 25000 — 500.45 + 5.55« = 17,625 lbs. 

Counter Strains. 
Fe= 25000 — 500.55 + 5.45* = 7,625 lbs. 

StraiJis 171 Diagonals. 
1^ = Fn cosec. 
Fi = 67625 . 1.12 = 75,740 lbs. Tension in Y^ and Fjo; 

compression in I^ and Y^. 
Y^ = 53625 . 1.12 = 60,060 lbs. Tension in Y^ and Y^ ; 

compression in yj, and J^. 
Fg = 40625 . 1.12 = 45,500 lbs. Tension in Fg and F^; 

compression in Y^ and F^. 
F4 = 28625 . 1.12 = 32,060 lbs. Tension in F^ and F, ; 

compression in F^ and F4. 
F5 = 17625 . 1.12 = 19,740 lbs. Tension in F^ and F^; 

compression in F« and F^. 

Cbtm^ Strains. 

Y^ = Vn cosec. v. 
Fe = 7625 .1.12= 8,540 lbs. Compression in Fj and F^; 
tension in F^ and I^. 



y Google 



133 STRAINS IN TUUSSK8 WITU PARALLEL BOOMS. 



y Google 



8T&AIHS IK TRUSSES WITH PARALLEL fiOOtfS. 139 

Lattice Truss. (Figs. 222, 223. and 224.) 

Lower Boom Loaded. 

Additional Reference. 
r= Half the length of a bay of simple truss. (Figs. 222 

and 223.) 
ajB= Distance from abutment A to center of bay of lower boom. 
yii= Distance from abutment A to apex of bay of wpper boom. 
Zj^ss. Distance from abutment A to apex of bay of lower boom. 

The formulas are for the strains in the simple trusses, (Figs. 
222 and 223.) Fig. 224 shows the simple trusses combined, con- 
stituting the Lattice Truss. 

When the upper boom is loaded, treat the strains as acting up- 
ward and the truss inverted: the strains will be of the same 
amount in each member, but different in kind. 

' Static or Permanent Load, equally distributed over whole length oj 
Truss. 

Strains in Booms. 

Upper. 

„ ^ f V ^ \ W f . r \2 Wr^ 

^^=ih • l^+ -ri "" 2Ai • l'»+ -2r) + -m 

Lower. 
TT W f r \ W / r\2 



Lower. 

3Wr* 

m 

Strains in Verticals. 



'^— 4 a • '^ 

Strains in Diagonals* 

1^ = Fn cosec. V. 

Moving and Static Load, each equally distributed per unit of 

length. 

Strains in Booms. 

Upper. 

^-= — 2A— 1'"+ T-) " -2hi-V''+ T) + ~^hi — 

Lower. 

tt-E±Eli !L^_iE±'ii^ r -^ 3(ir+,).-Tr' 

»~ 2A A^" 2 / 2« 'V'' 2 / m 

Digitized by <^00QIC 



140 



STRAINS IK TRUSSES WITH PARALLEL BOOMS. 



Strains in Verticah, 

Strains in Diagonals, 
1^ = T^ cosec v. 



Strains in Counters, 
jr W W , Wi ., ,- 



I^ = Fa cosec. V. 



[Note.— The strains in Fa.b, are equal in araouut, but different 

In kind to f - 



> the strains in Yi,i,s, 



Example. (Figs. 222, 223, and 224.) 

Moving Load (as railway train passing over bridge) on Lower Boom. 

We will assume W = 50.000 Iba. 
Wi= 100,000 lbs. 
/ = 100 feet. 
A = 10 feet. 
V = 63° 20^. (cosec. = 1. 12,) r = 5 feet.. 



Horizontal Strains in Upper Boom. (Compression. Fig. 221.) 



W+Wi 



--^i^(^+x)-^i^(-.+x)'+ 



(W+W,)r^ 



Shi 

J3;=7500 
B,=:7500. 
H.f=^ 7500 . 
i?3s=7500. 
//,= 7500 . 
1(.= 7500. 
-Sc=7500. 
i?,= 7500. 
5«=7500. 
-Hi=7500. 



= 7500(z„+ 2. 5) -.75(2„+ 2.5)2 + 468.75 



( 0+2.5)- 
( 5+2.5)- 
(10+2.6)- 
(15+2.5)- 
(20+2.5)- 
(25 + 2.5)- 
(30+2.5)- 
(35+2.5). 
(40+2.5)- 
^45 4-2.5) - 



-75.(0 + 
-75.( 5 + 
■75.(10 + 
•75 (15 + 
-75.(20 + 
-75.(25 + 
-75.(30 + 
-75.(35 + 
-75.(40 + 
-75.(45 + 



2.5)« + 

2.5)«-- 

2.5)2-- 

2.5)2-- 

2.5)2-- 

2.5)2-- 

2.5)2.- 

2.5)2 

2.5)2 + 

2.5)2 



468.75 = 
468.75 = 
468.75 r 
468.75 = 
468.75 = 
468.75 = 
468.75 = 
468.75 = 

468.75:: 

458.75 = 



: 18,750 Ibe. 
: 52,500 lbs. 
: 82,500 lbs. 
: 108,750 lbs. 
: 131,250 lbs. 
= 150.000 lbs. 
= 165,000 lbs. 
= 176.250 lbs. 
: 183,750 lbs. 
: 187,500 lbs. 



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8TBAINS IK TBUSSES WITH PARALLEL BOOMS. 141 

Horizontal Strains in Lower Boom. {Tension, Fig, 224.) 
W+ Wx I r \ W-\- Wx I r \t 

^'= — 2/r- (y- — 2) — m- • i^- — 2"; "• 

3(PF+ W^r^ - = 7500.(y„— 2.5) — 75. (y„—2.5)«— 1406.25 



Ul 



H. = 7500 . ( 5 — 2 5) — 75 . ( 5 — 2.5)«— 1406.25 = 16,875 lbs 
fl: = 7600 . (10— 2.5)— 75 . (10—2.5)2—1406.25= 50,625 lbs, 
IL = 7500 . (15 — 2.5)— 75 . (15 — 2.5)«— 1406.25= 80,625 lbs, 
R. = 7500 . (20— 2.5)— 75 . (20— 2.5)«— 1406.25 = 106.875 lbs, 
H. = 7500 . (25 — 2.5)— 75 . (25— 2.5)«— 1406.25 = 129,375 lbs, 
IL. = 7500 . (30 — 2.5)— 75 . (30— 2.5)«— 1406.25 = 148.125 lbs, 
fl! = 7500 . (35— 2.5)— 75 . (35— 2.5)«— 1406.25 = 163,1 25 lbs 
Ho = 7500 . (40— 2.5)— 75 . (40— 2.5)«— 1406.25 = 174,375 lbs 
^ = 7500 . (45 — 2.5)— 75 . (45— 2.5)«— 1406.25 =181.875 lbs 
H^^^ 7500 . (50 — 2.5)— 75 . (50— 2.5)«— 1406.25 = 185,625 lbs 



Simple Tbuss. (Fig. 222.) 
Strains in Verticals. (F^.) 

, F,= -^ - ^. x,+ ^. (2 - rrj«= 12500 - 250.a:,+ 
2.5 .(Z — rrj« 

Fi= 12500 — 250 . 4- 2.5 . 100* = 37,250 lbs. Com. in U, 
F2= 12500 — 250 . 10 -f 2.5 . 90* == 30,250 lbs. 

Vf= 12500 — 250 . 20 + 2.5 . 80« = 22,500 lbs. 
F;= 12500 — 250 . 30 + 2.5 . 70* = 17,250 lbs. 

F5= 12500 — 250 . 40 + 2.8 . 60« = 11,500 lbs. 

Counter Strains, ( F„.) 

Fr= 12500 — 250 . 50 + 2.5 . 50« = 6,250 lbs. 
Ft= 12600 — 260 . 60 + 2.5 . 40« = 1,500 lbs. 

Strains in IHagonaU, 
F^= Fn cosec. 

Ti = 37250 . 1.12 =a 41,720 lbs. Tension in Y^ and Fio; 
compression in Y^ and Y^ 

Fa = 30260 . 1.12 = 33,880 lbs. Tension in F, and r» ; 
compression in Y^ and Y^, 

Digitized by LnOOQlC 



142 



STRAlirS IK TRUSSES WITH PARALLEL BOOMS. 



Fg = 22500 . 1.12 = 25,200 lbs. 

compression in Y^ and ¥„. 
Y^ = 17250 . 1.12 = 19,320 lbs. 

compression in Y^ and Y^, 
Fa = 11500 . 1.12 = 12,880 lbs. 

compression in F, and Y^. 

Counter Strai 
^m= ^m cose 
r« = 6250 . 1.12= 7.000 lbs. 

tension in Y and Y . 
Fy = 1500 . 1.12=1,680 lbs. ' 
tension in Y^ and Y^. 



Tension in Fj and Fg; 



Simple Truss. (. 
Strains in Vertical 






12500 — 250 . 
12500 — 250 
12500 — 250 
12500 — 250 , 
12500 — 250 



9t 

SI 



5+ 2.5 
15 + 2.5 
25 + 2.5 . 7t 
35 + 2.5 . et 
45 + 2.5 . 51 



Counter Strains. 

Fe= 12500 — 250 . 55 + 2.5 . 4^ 

Strains in Diac^ 

5^n= K cose 

F,= 33812.5 . 1.12 = 37,870 lbs. 

tension in F^ and F^. 
F,= 26812.5 . 1.12 = 30,030 lbs. 

tension in Y^ and F^,. 
F,= 20312 5 . 1.12 = 22.750 lbs. 

tension in F^ and F^. 
F,r^ 14312.5 . 1.12 = 16,030 lbs. 

tension in F^ and F^. 
F6=8812 5 . 1.12 = 9,870 lbs. 
tension in F. and F^. 



Compression in F, and F^; 
Compression in F^ and F^; 
Compression in Y^ and F^; 



Counter Strains.' 

^m =^ ^m COSeC. V. 

•• 3812.5 . 1.12 = 4,270 lbs. Tension in F, 
pression in F^ and F,. 

Digitized by V^OOQIC 



5 and F^; 



STRAIXS IN TRUSSES WITH PARALLEL BOOMS. 143 



Fi(j. 225. 
Lower boom loaded. 



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144 STBAIKS IN TBUSSES WITH PARALLEL BOOMS. 

Whipple Tbuss. {Figs. 225, 226, 227, and 228.) 

Additional Eefererice* 
*n» yn= Distance from abutment A to end of bay. 

Static or Permanent Load, equally distributed over whole length oj 
Truss. 

Strains in Booms. 

_ TF W ,, sW sW 

•^"- "W • ^«"' IaT* y*^^ "IJiT • ^»"" "IT 

Strains in Verticals, 
^^ W W 

^•=— -- 2r-^ 

Strains in Diagonals. 
Yj^= Fn cosec. V, 

Moving and Static Load, each equally distributed per unit of 
length. 

Strains in Booms. 

W+W, ^_W±W^ ., ^{W+Wy) 
»~ 2h ' ^" 2hl ' ^"^ 2hl ' ^"^ 

s(W+W,) 
4/1 

Strains in Verticals. 

Strains in Diagonals, 
F,=7', cosec. V. 

Strains in Cownters. 



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STRAlirS IN TBHSSES WITH PABALLBL BOQXS. Ii5 

Example. (Figs, 225. 226, 227, and 228.) 

(With 20 Bays.) 

Moving Load^ (as railway train passing over bridge.) 

Let Tr= 50.000 lbs. 
Wi= 100,000 lbs. 
I = 100 feet. 
A = 10 feet, « =x 5 feet. 
V = 45°. (End diagonals v = 26° 30'.) 

Horizontal Strains in Booms. (Compression in upper^ tension in 
lower.) 

W+ W, W+ Wi s(W +W,) 

^- 27t — -y- — ^hf—y"^ — 2hr—'y- 

J!^_EiL = 7500.2/.- 75.y„^ - 375.y,+ 18750 

IT = 7500 . — 75 . 02 — 375 . + 18750 = 18,750 lbs. 
Hy= 7500 . 5 — 75 . 5« — 375 . 5 + 18750 = 52,500 lbs. 
^2= 7500 . 10 — 75 . 102 _ 375 , iq + 18750 = 82,500 lbs. 
£3= 7500 .15 — 75 . 152 — 375 . 15 + 18750 = 108,750 lbs. 
^^= 7500 . 20 — 75 . 202 __ 375 20 + 18750 = 131,250 lbs. 
^5= 7500 . 25 — 75 . 252 __ 375 . 25 + 18750 = 150,000 lbs. 
H^-^ 7500 . 30 — 75 . 302 — 375 . 30 4- 18750 = 165,000 lbs. 
ir.= 7500 . 35 — 75 . 352 _ 375 . 35 ^ 13750 = 176,250 lbs. 
H.^ 7500 . 40 — 75 . 402 _ 375 . 40 4. 18750 = 183.750 lbs. 
J39= 7500 . 45 — 75 . 452 _ 375 . 45 j^ 13750 = 187,500 



"" V 8A I 



lbs. 



Strains in Verticals. 

v.^^^^.x^ + ^.(i-x;i^=.i2mo-^2m.x.+ 

2.5.(Z — aj,)2 

IT 4- IF" 
7;= ^ ^ = 75,000 lbs. 

Strains in Figs. 225 223 227 228 
1^=12500 — 250. 0+2.5. 1002 = 37,500 lbs. C. C. T. T. 
F2= 12500 — 250. 5+25. 952 = 33,812 lbs. 
F,= 12500 — 250 . 10 + 2.5 . 902 ^ 3o,250 lbs. 
F;= 12500 — 250 . 15 + 2.5 . 852 = 26,812 lbs. 
Fe= 12500 — 250 . 20 + 2.5 . 8O2 = 23,500 lbs. 
P>= 12500 — 250 . 25 + 2.2 . 752 ^ 20,312 lbs. 
FJ= 12500 — 250 . 30 + 2.5 . 702 =. 17^250 lbs. 
10 

Digitized by V^OOQIC 



l( tl 




(t II 




CC l( 




t( li 





146 



STBAIHS nr TBXrSSBS WITH PABALLEL BOOKS. 



Strains in Figs. 225 226 2Z7 228 

V. = 12500 — 250 . 35 + 2.5 . 65» = 14.312 lbs. C. C. T. T. 

1^ = 12500 — 250. 40 4- 2.5. 60^=11,500 lbs. " '* - " 

Fio= 12500 — 250. 45 + 2.5. 55^= 8.812 lbs. " " *• " 

Vj^ Acting on Counters. 

Viy= 12500 — 250 . 50 + 2.5 . 50^ = 6,250 lbs. 
P^i2= 12500 — 250 . 55 + 2.5 . 45^ = 3.812 lbs. 
T^3= 12500 — 250 . 60 + 2 5 . 402 = i 500 lbs. 



Ten. 



Com. 



Com. 



Strains in Diagonals. 

F„ = F„ cosec. V. 

Strains in Figs. 225 
F, = 37500 . 1.117 = 41,887 lbs. Ten. 
y; = 33812 . 1.414 = 47,810 lbs. 
F3 = 30250 . 1.414 = 42,773 lbs. 
Y. = 26812 . 1.414 =: 37,913 lbs. 
Y. = 23500 . 1.414 = 33,229 lbs. 
Y = 20312 . 1.414 = 28,722 lbs. ** 
y. = 17250 . 1.414 = 24,391 lbs. '^ 
Fg = 14312 . 1.414 = 20,238 lbs. 
Fg = 11500 . 1.414 = 16.261 lb». 
110= S812 . 1.414 = 12,461 lbs. 

StravrU in Counters. 

F„= 6250 . 1.414 = a^37 lbs. 
^2== 3812 . 1.414 =s 5,391 lbs. 
Fi3= 1500 . 1.414 =a= 2.121 lbs. 

[Note.— If counter twaces are not inserted, Fi 1, Fi 2? and V13, and Ts* 
Y9, and Fi will have an additional strain, opposite m kind and equal 
to Fi 1, Vi 2f and Vi 3, and Fi 1, Fi 2, and Fi 3 ; but if counters are used, 
the strain Fii, Fi2»«nd F13 will not occur in the structure, but will 
be necessary to determine the strain in Fn, Fi2,»nd Fis on!y. Fn, 
yi2,and Fig will then be inclined in the same direction as the dia^ro- 
nals from abutment A to center of truss, the character of strain being the 
same. (Sde also ''Howe Truss.'') 

E^ep in mind that each half truss, as to the character and amount of 
strain in the respective members, is alike.] 



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STBAIKS 15 PABABOLIO OUBVED TRUSSfil. 147 

STRAINS IN PARABOLIC CURVED TRUSSES —" BOW- 
STRING GIRDERS." 

{Figs. 229, 230. 231, 232, 233, and 234.) 

The strains in the lower boom (when horizontal) are the greatest, 
and equal in every bay, when the load is equally distributed over 
the wnole length. 

The strains in the arch or upper boom are also greatest when 
the load is equally distributed over the whole length; the strains 
gradually increasing from the middle to the supports. 

The strains in the diagonals, whether single or double, in a 
bay are, when the load is equally distributed, everywhere null. 
When the load is unequally distriDuted, and one diagonal to each 
bay is used, they will be either in compression or tension. The 
character of the maximum of strains will be as follows: Assume 
the left half of truss to be loaded. All diagonals inclined up from 
left to right abutment are in tension ; if inclined down, in com- 
pression. The character of strains will be vice versa when the 
right half onljr is loaded. 

The strains in verticals are either compression, tension, or null. 
The maximum of compressive strain occurs when the diagonals 
in connection are under the greatest strain; that is, under an 
unequally distributed load. For other explanation, see diagram 
under variously- disposed loads. 

In the following formulas and examples the diagonals (for a 
moving load) resist a tensional strain only, and the verticals a 
compressive. This would not be the case if one diagonal to each 
bay were used. In the latter case the diagonals and verticals 
would have to resist an alternate compressive and tensional strain. 
Wlien the trusses are inverted, the strains are different in kind, 
but not in amount. 

Heference. 
A, B = Reaction of support. 

(7= Compression in arch or upper boom. 
r= Tension in lower boom. 
D and H=i Rise of arch. 
F and / = Vertical forces. 

W=s Weight of moving and static load per unit of span 

or length. 
F= Strain in verticals. 
N = Total number of bays. 
a = Length of a bay. 
c = Length of a diagonal. 
d and h = Ordinates to parabola. 

I = Distance between supports or span. 
k = Total number of verticals = N — 1. 
m = Number of bays between support and P^. 

Digitized by ^^UUgle 



148 BTRAnrs nr pababolto CftmTED TBtrssiiB. 

n = Number of a member, counting from support to 

middle of truss. 
t s= Tension in diagonal. 
V and z = Angle between liorizontal and member of polygon. 
w = Weight of static load per unit of span or lenstn. 
ip^s= Weight of moving load, equally distributea per 
unit of span or length. 
u, x,y ^=: Abscissas. 

In the following diagrams, one-half of truss only is shown, the 
strains being alike in the respective members of each half : 



Lower Boom HorizontaL 
To find the ordinates Ti when ^is given: 

The value of T given, to find h: 

— 2T~ ~T' 



Lowtr Boom Curved^ 
To find the ordinates hoxd when ^or D is given: 

K ^5 ^= ^5 



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RPBAJUS IH PA»ABQLIO CUBVBI) TBUSS?^* Ii9 

The value of 7 given, to find A; 

° 2T *^ IT 

Load equally distrihutedSlaiic Load. (Figs. 231 and 232.) 
W = The weight of construction and applied load. 



Lower Boom Loaded. 
n= J -^ sec. v^ T=zi^ = C F= -^ = tension. 

Upper Boom Loaded. 
^i=*-;g-sec.v„ T=i~— = (7 r=null. 



Fig. 232. 



V=z —— = teneion. 

iv 



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150 STBAhrs nr parabolic cubved teusses. 

Load unequaUy dUtributed— Moving Load. {Figs. 233 and 234.) 

(Strains in Booms, same as for Static Load.) 

Fig. 233. 

yn= K cot- *n; ^11= Vu— *^ 



Lower Boom Loaded, 



<n= -^ Cu ^«i== K—fu= compression. 

r(l + ife — m)(ife— m)-] _, , , ,r(l + m)»»-I 

Upper Boom Loaded, 
r„= -y- = compression. «.= -jj^c^ 

Fig. 234. 



Upper Boom Loaded. 
r.= -^ = compression. «.= 8(5""^rD)"'- 



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STBAIKS IS PABABOLIO OUEVED TBUSSES. 151 

Example. {Fig. 233.) 

Moving Load on Lower Boom. 

Reference. 

I = 64 feet. Ci= 8.7 feet. -w = 125 lbs. 

£•= 8 feet. 0^= c^= 10.0 feet. w^= 625 lbs. 

a= 8 feet. c^= 65= 10.9 feet. 1^= 1^ + «;^= 750 lbs. 

iV= 8, Jk = 7. Ce= 11.3 feel. 

/ts^ftvQ/ftJ. 8^ ^1= 8.0—8= Ofeet. 

4x8xJ(64-8). ^ 35 ^^^ ^^^ 19 2 - 16 = 3.2 feet. 

' 642 ^_ 40.0 — 24 = 16.0 feet. 

^^^ 4X8 X 1^(64-16) _,, ^^ u,= l28.0-32 = 96.0feet. 

^4x8x24(64-24)^^^^^^^^ 

A4=ir= 8.0 feet. 

Tang. ^1= — ^ = — ^ = 23° 37^ 

K—\ 6—3.5 ,^001/ 
Tang. v.f=z -^ — -^ = — 5 — = 17*^ 21^ 

Tang. „,= ^-^=^ = ^-^ = 3» 34' 30". 



a 



V,= 3.5 X 2.28 = 8.0 feet. ^3= 7.5 X 5.37 = 40.0 feet, 

y ~ 6.0 X 3.20 = 19.2 feet. y,= 8.0 X 16.00 = 128.0 feet. 

!!^X^ = 48,000 lbs. 

C sec. Vq. 

;. 0=48000X1.017 =48,816 lbs. 
i. C;=48000xl.0019=48,0911b8. 

625 X 64 8.7 = 5437.5 lbs. 

^8X8 

^ «s= -^1^ X 1^-^ = ^250.0 lbs. 

Digitized by LnOOQlC 



152 STBAIHS Uf FABABOUO OUBYSD TftUMSB. 

625 X 64 
i,= ^= g^^ X 10.9 ^ 6802.5 lbs. 

625 X 64 
^.= 8 X 125 [(i±l=^Mnl) 3^2625 

^^8Xl25r<l+^;i^^^=l^l = 1250 



£.= 8(125 +625)[-^^]= 750 

£i= 8(125+ 625)[-iLtM.] = 2250 

5^ 8(125 J 625) [ ^^3^y ] = 4500 

£,= 8(125 + 625) [iL-t^] = 7500 

p- 750(-^-'->ii-) = . 
» \ 0+1x8 / 



750 



: 6000.0 



/-2625(-^J^-^) = 



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FABABDLXO ABC^SD BEAKS OB BttS. 153 

16 



/-1250(^3-3^)=500.0 



96 



/^= 7S'>(w+-4ir^)=^«2-^ 



Fi=6000 —0 =6,0001bs. 73=9000—500 =8.500 lbs. 
Fjj= 7812.5 — 312.5= 7,500 lbs. F4= 9375— 562.6 = 8,812.5 lbs. 



CAPACITY AND STRENGTH OF PARABOLIC ARCHED 
BEAMS OR RIBS ORIGINALLY CURVED. 

Beference. {All dimensions in inches.) 

A ss Sectional area of beam. 
C= Compressive strain in direction of arch. 
E = Modulus of elasticity. 

jy= Horizontal thrust at abutment, or tension on tie rod. 
Js= Moment of inertia of cross-section of beam. 
M = Resistance of material to crushing, (to be divided by factor 

of safety.) 
W=s Concentrated load at crown of arch. 
a = Vertical deflection at crown. 
b :s Horizontal deflection at abutments. 
h ^ Rise of arch. 

21 = Distance between abutments = span. 
« =r Distance between neutral axis and farthest edge of section. 
w = Load per unit of length, equally distributed horizontally. 
X = Vertical distance from crown to point of arch, intersected 

by y, say at on diagram. 
y = Horizontal distance .from middle of arch to section where 

the amount of strain is desired. 
V = Angle between horizontal and tangent to curve. 

Horizontal Thrust, {resisted either by abutments or tie rod.) 
Fig. 235. {All dimensions to line of pressure.) 




d by Google 



1S4 STEAIUS lU A POLYGONAL FfiAlCE. 

To determine the curve or line of pressure: 

X = i- ir-^T y-Wx ^'^^ir 

Tang. V at any point = = -~ — 

2h 
Tang. V at abutment = -— 

. Load concentrated at crown or middle of arch : 

\ 64A 56^ "^ l^ 32/5 / 

„ 25^ W , SIWU 



il = 



64A ^ ^ 1600J 
25Z X 1600/ 



eih{B 16001— SI Wis) 
Load equally distributed: 



a = o 5 = ^=^ c;=-^+-2e^ 



^ = 



ti; 

2h '_ r» 

i2 



STKAINS IN A POLYGONAL FRAME IN EQUILIBRIUM. 

Load equally distnbuted over members of Frame. 

Reference. 

H = Horizontal strain in units of weight at foot. 

Fn= Vertical strain in units of weight at foot. 

Cn= Compressive strain in units of weight in directiou of 
member. 

TF^= Load in units of weight, equally distributed over a mem- 
ber of the polygon. 

Vn= Angle between horizontal and member. 

Digitized by <^00QIC 



PABABOLIC ARCHED BEAMS OB BIBS. 155 

Fig, 236. 



H:=i TTcotg. v^ (7^= Vj, cosec. v^ 

^«— ^1 ■*■ '2 "~2~ "^ 2 

•^5= ^2 i 2 — "2" "1 2 "^ ? 

J^A— »^8-1 2 2""^ 2 "^ 2 "^ 

-.S±.^ &c. 

2 

For the equilibrium, v^ being given : 

Tang..,= ^ = tang.«.+ iiIl±i^) + M^±^ 

y 
Tang. t;^= -^ = tang. Vi + 

H 

The above can be used to compute the strains in ribs for dome 
construction. 

Digitized by LnOOQlC 



I56r 8TBAIN8 IN BOOF TBUSSES. 

STRAINS IN HOOF TRUSSES. 

Reference. {Figs. 237 to 255.) 

{Weight of construction. 1 
Pressure of wind. >= Load in units of 

Pressure of snow. J 

weight, equally distributed over one rafter. 
(See Fig. 238.) 
C.= Compression of member in units of weight. 
T= Tension of member in units of weight. 
L = Total span, or distance between abutments in 
units of length. 
d, h, I, and S = Dimensions in units of length. (See Figures.) 
v,y = Angles. (See Figures.) 

The diagrams show only one-half of truss, (except Fig. 238,) 
the thick lines indicating compression, and the thin ones tension. 
(See ''Reaction of Supports*' for pressure on joints ; also ''Compound 
Strains in Trussed Beams.'') 

Compression in Rafters. {Trusses Nos. 1, 3, and 4.) 

The compressive strain in the rafter gradually increases from 
ridge to abutments. Let x = Horizontal distance from abutment 

to point where the strain is desired, and I half the span = -— . 

Cfor Truss No. 1 = TTsin. v (l ^) + -^ -^^^^ 

\ 1/2 tg.v 

Cfor Truss No. 3 = PTsin. v (l ^) + ^ , ';^''/ , 

V I J ' 2 tg.{v + t\) 

C for Truss No. 4 = TFsin. v(l ~) + ^ , T' "" , 

V 1^2 tg.(v — Uj) 

In the following examples the maximum of C is given : 
Tniss No. 1. 



- _ . . W COS. v 
C= TTsm. v+ -— 



2 tg. v 
— cotg.v 



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BTBAINS Iir KOOP TRUSSES. 157 

Example. 
Let Tr= 8,000 lbs. 
V = 26° 30^ 

C= 8000 X 0.44619 + i^ ^||- = 10.666 lbs. Com. 

7 8000 

Wben a? = -g- tlien will C = 2 x 44619 ~ ^'^^^ ^^®' ^^°** 

8000 
T= — ^ — 2.00 = 8,000 lbs. Tension. 



Tmas No. 2. 

i^i^. 238. 



W 
(7= -5- sin. v Ci= fr(cos. t;)« 

r = — r- sin. V COS. V = -r- sm. 2v 
2 4 

Example. 
Let Pr= 8,000 lbs. 
t; = 26° 30^ 

8000 
(7= — - — X 0.4462 = 1,786 lbs. Compression. 

Ci= 8000 X 0.895* = 6,668 lbs. Compressioli. 

8000 
T^ —^ X 0.7986 = 1.597 lbs. Tension. 

[NoTB.~'When the rafters are fastened together at the ridge, they are 
under a cross-breaking strain only. Consequently there is no horizontal 
thrust at the abutments ; that is, r— 0, and the oompression IxlCi — TT.] 

Digitized by ^^jUU^ It: 



168 STRAINS IN ROOF TRUSSES. 

Tmss No. 3. 



Fig, 239. 



C=W Bin. V + 



2 tg.(t;+i;i) 



(i^TP COS. y sin, y ^ 
^ sin. (v -j- ^i) 



2'='^ 



2 8in.(v4- Vj) 



Truss No. 4. 

TF COS. V 



(7= 1^810.1; + 



2 tg.(v-t;i) 
TF COS. V 



2 sin.(v — Vj) 



COS. V sin. Vi 



^ Bin.{v — Vj) 

Example. 
Let Tr= 8,000 lbs. 
1; = 26°30^ 
v^= 5° 0^ 

C=8000 X 0.44619+ -^-^51^ = 12,653 lbs. Com. 



8000 0.894 ^n-»Aiu rr • 

r= — — == 9,920 lbs. Tension. 

2 0.366 



Ji= 8000 .M?ixa087 = 1.720 lbs. Tension. 



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nr KOOP TBUSSES. 159 



C7= JfTTcosec.v Ci= J TF cotg. t; T= i(l + ^)w coig.v 

LW 
When there is no tie T, Q is under a tensile strain =fe -tt- i 

^ being the height from C^ to ridge. 

Example. 
Let TF= 8,000 lbs. 
I = 22.36 feet. 
/,= ?_== 11.18 feet. 
V == 26° 30^ 
(7= ^f 8000 X 2.241 = 14,566 lbs. Compression. 
G^=z J 8000 X 2. = 8,000 lbs. Compression. 

2^= J (l + 4h4|-) SOOO X 2. ss 12,000 lbs. Tension. 
\ 22.36 / 



(7= ^S^-miS^-^h) j.=(Tr-ATF)A. 

Digitized by VjUU^ IC 



160 8TBAIKS IK B007 TRUSSES. 



Example. 
Let Tr= 8.000 lbs. 
I = 20 feet. 
?i= 20.6 feet. 
^ = 10 feet. 
hy^ 5 feet. 
/S= 22.36. 

8000X500^15 00(500--10X5)^ t^^^ ^^^ 

5 X 22.36 
Q= 0.625 X .800Of J :==: 10,000 lbs. Compression. 

r= (8000 -- 1500) — ^— = 26,780 lbs. Tension. 
5 

2;= 2(8000 — 1500) ^^""^ = 13,000 lbs. Tension. 



21 sin. V 2 h 2 

C7,= U ^ cosec. V C^= | TT-^^^ 

^ ^® '^ sm. 2v 

A 

Example. 

Let Pr= 8.000 lbs. A = 10 feet. ^; = 26*>30^ 

Z = 20 feet. ^i= X1.18 feet. Vi= 26° 30^ 



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STRAINS IF BOOP TEUSSES. 161 

20 



20 

(7= 8000 -rr z^^r TT-TTTrrjr- = 8,964 lbs. Compression. 

2 X 20 X 0.44619 ^ 

C7i='0 8125 X 8000 X 2.2411 = 14,567 lbs. Compression. 
C,= 0.625 X 8000x1.12 = 5,600 lbs. Compression. 
T= 0.625 X 8000 = 5,000 lb8. Tension. 
^1= 0.8125 X 8000 X 2.0 = 13,000 lbs. 



^^ Ti+jW ^^ I __ W cos.v^ 



(\={iW 



2sin.u 2^1 sin. (v — Vi) 2 sin.(v — vj 

COS. Vi 



8in.(« — Vi) 






'~ L" "sin.rw — »,) "^ ^ sin.fi; — »i+ %) J ~ 



2(3'em.«,+ Gsin.Ct.,-..)) = Jf4^^?^ - ITT 
\ ^ V sm.(v — Vi) 

Example. 
Let W= 8.000 lbs. 
v = 26° 30^ 
Vi= 9° 20^ 
^2= 19° 0^ 
. 11 

Digitized by LnOOQlC 



162 STEA15S rs ROOP TBUSSE8. 

^ 9000 + 0.375 X 8000 , „ ..„ iu n 

0= A Qoo = 13,452 lbs. Compression. 

0.892 

Q8fi 
Ci— 0.812 X 8000 -,^^z^7=- = 21,710 lbs. CompreMion. 

895 
C.y= 0.625 X 8000 ' ' = 7,585 lbs. Compression. 
0.590 

y^ 812 X 8000 — ^ = 19,702 lbs. Tension. 
0.295 

r,= 2 X 8000 [0.8I2 ^'^^Q^J'^^^ + 0.625 X 

0.895x0.168 1 nAAAiu rr 

. 1-^ = 9,000 lbs. Tension. 

0;590 J 

TnuM No. 9. 



C=z U (V-J^ f TFsin. V 

^'^ sin. c 

C,= HW- = H^cosec. V 

^^ sm. ■» 

C.i= iWcos.v 

T=j\Wcotg.v 

T,= If W cotg. v — j\W cotg. V = } TF cotg. v 

7i=||Prcotg.t? 

Example. 
Let Tr= 8.000 lbs. 
V = 26° 30^ 
(7 = 0.812 X 8000 X 2.241 — 0.625 X 8000 X 0.446 = 12,336 lbs. 
Compression. 

Ci= 0.812 X 8000 X 2.241 = 14,560 lbs. Compression. 

a= 0.625 X 8000 X 0.895 = 4,475 lbs. Compression. 

T= 0.312 X 8000 X 2. = 4.992 lbs. Tension. 

2\=0.812 X 8000 X 2— 0.31 2 X 8000 X 2. = 8,000 lbs. Tension. 

2^3= 0.812 X 8000 X 2. = 12,992 lbs. Tension. 



d by Google 



8TEAISS IN ROOP TRUSSES. 163 



^ , - -rr, COS. Vi . ,^ . 

^^ sin.(n; — Vi) ^ 
(7,= if IT- 



sin. (u — Vi) 
^2== ITT COS. v. 



m 1 FfTTT COS. t> sin. Vi . ^ ,„ n 

8in.(2v — i7i) L^^ Bin.(t; — vi) ^^ J 

_ , o TT^ COS. V COS. Vi „ 



~" 2 /t— Ai 



sin.(i; — Vi) 

Example. 
Let Tr= 8,000 lbs. Vi= 9° 20^ A = 10 feet. 

v = 26°30^ Z = 20 feet. A,= 2 feet. 

987 
(7=0.8125 X 8000 ^-^^^ 0.625 x 8000 x 0.446 = 19,517 lbs. 

Compression. 

987 
(7i= 0.8125 X 8000 -^^ = 21,747 lbs. Compression. 

G= 0.625 X 8000 X 0.895 = 4,475 lbs. Compression. 
^== W (°«^2« X «^^ "''Ly' + 0-^25 X 8000 X 
0.895a] = 7,163 lbs. Tension. 

Digitized by LnOOQlC 



1^^ STRAlirS IlSr EOOP TBUSSES. 

m 8000 20 

^'"^ —2— X io_2 =" ^^'^^ ^^^' Tension. 

T,^ 0.8125 X 8000 ^ == 19,720 lbs. Tension. 



n 1 « TTT- COS. Vi 

' sin. (2» - r.) I ^ sin. (»-„,)■ ■*" "/ 



Bin.(t; — vi) 
Example. 
Let Tr= 8,000 lbs. ^==500. 7» = 10fGet Z— 9nf..* 
. = 26=30'. U9»20^ '-ffeef i^limeet. 

C= 0.8125X8000 -2Jl._o.625 x 8000 X 0.446 =19.517 lbs. 
Compression. 

a= 0.8126 X 8000 ^ = 21.747 lbs. Compression. 

ft= 0.366 X 8000 -2^ = 4.070 lbs. Compression. 

Digitized by LnOOQlC 



STEAINS IlSr EOOP TEUSSES. 165 

r=0.125x8000^(6.5 -2^+5. 0.894) = 11.050 lbs. 

Tension. 

r,= 19486 X 0.986 — 7421 x 0.723 — 4930X 0.446 = 10,00011)8. 
Tension. 

2i= 0.812 X 8000 ^^ = 19.486 lbs. Tension. 



43A'+39P j,_i,Wy/?^+W- 
"' 30XAXJ ''-«-2 h 

I Example. 

Let Tr= 8000 lbs. A = 10 feet. 

Z==20feet. iS^ = 22.36 feet. 

43 X 100 X 15600 



^= "sVxVox 22.36^ ^^^ ^ ^^'"^^^ ^^^* Compression. 

20 
Q= 0.366 X 8000 — = 5,856 lbs. Compression. 

8000 22 ^fi 
C^ 0.366 X —2 j^ = 3,280 lbs. Compression. 

r=0.866x-^^J^ = 20,9921bs. Tension. 
!7;= 1.23 X 8000 = 9,840 lbs. Tension. 

Digitized by LnOOQlC 



166 STRAINS IH BOOF TB968S8. 



c= 


..iw^ 






A= 


41 TF 


CV.S. 


Vi 


^^"^ 8in.(». 


-h) 


c,= 


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COS. 


"l 


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Ca= 


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(7,= 


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I, 





y=UPr-.'=o^-'' 



2'x=«ir 



siii.(v — Vi) 

COS. V 

Bin.(v — Vj) 






Example. 
Let Pr== 20,000 lbs. A = 20 feet. v = 2P 40^ 

2 = 50 feet. 1^= 53.8 feet. v = 0<>. 

C = 0.5 X 20000 . = 26,900 lbs. Compression. 
Q= 0.683 X 20000 tt^HS" = ^"^.^IS lbs. Compression. 
(7a= 0.866 X 20000 = 46,937 lbs. Compression. 

C8=0.55 X 20000 -^i^ = ll,7701bs. Compression. 

18 
Q= 0.55 X 20000 — = 9,900 lbs. Compression. 



d by Google 



STBAINS IN BOOF TRUSSES. 167 

r = 0.683 X 20000 X 2.517 = 34,382 lbs. Tension. 
^1= 0.866 X 20000 X 2.517 = 43,594 lbs. Tension. 

!Z>= _20000 X20 5333 33 ^ ^^qqq ^^s. Tension. 

71= 0.183 X 20000 = 3,660 lbs. Tension. 









C,= {T,+ ^W)4- ^l^Ci-T- 



Example. 

Let Tr=r 24,000 lb«. Span = 100 feet /=lj=lr=?r== 1.26 feet. 
A =: 20 feet E=0. /8f= 53.85 feet. 

Digitized by V^OOQ IC 



168 STBAINS IN ROOP TRUSSES. 

C,= 12000 X — 2o~ = ^^'^1^ ^^8- Compression. 

Ci= 49088 — 0.228 X 24000 ^^^ = 41,728 lbs. Com. 
2 X 20 

(73= 58320 — 0.286 X 24000 ^^^ = 49,088 lbs. Com. 
2 X 20 

' 53 85 
Q= 2 1600 — 2Q- = 58,320 lbs. Compression. 

C,= (5801 + 0.286 X 24000) -^^ = 12,493 lbs. Com. 

05= 3432 + 5484 -i|- = 9,282 lbs. Compression. 

C.= 0.286 X 24000 -^!^:^ = 9.245 lbs. Compression. 

50 
7\= (24000 — 0.1 X 24000) — — = 54,000 lbs. Tension. 

7;= 51000 ~ 0.286 X 21000 ^^,^ = 45,420 lbs. Vension. 

^3= 45120 — 9282 ^|;^ 38,170 lbs. Tension. 

In 

7;= 24000 — ^21000 = 19.200 lbs. Tension. 

71= 9282 4?- = 5,801 lbs. Tension . 
lb 

7^6= 0.286 X 24000 ~- = 3,432 lbs. Tension. 



y Google 



STRAIHS IK SOOF TBOSSXS. 169 






sin.(v — Vi) *° * •* sin.(v — v^) 



sin.(v — Vi) * 2 (^ — Ai)cos. Vi 

<^S= U ^ COS. V ^ ^ ^_ COS. V 



C^z= ^ W COS. V 4 6 gjjj ^y ^^ J 

(A— A^) ° o xo 8m.(v — Vi) 

^ COS. Vj 

Example. 

Let IT = 20,000 lbs. h = 20 feet. Vj = . 

Z = 50 feet. v = 2[°W. '^2= 46° 30^ 

C = 0.866 X 20000 -^^ 0.733 X 20000 X 0.369 — 0.183 X 

20000 X 0.929 X 2.517 = 32,959 lbs. Compression. 

Ci= 0.866 X 20000 X -^^ — 0.366 X 20000 x 0.369 

^ 44,236 lbs. Compression. 
C2= 0.866 X 20000 X -tt^ = 46,937 lbs. Compression. 

C,= 0.55 X 20000 X 0.929 = 10,219 lbs. Compression. 
(74= 0.366 X 20000 x 0.929 = 6,800 lbs. Compression. 

40 
T= 20000 X -spT X tang, v = Null. 
20 

2-,= ^^^~y^^ 10,920 lbs. Tension. 

O.DOO 

^2= 0.183 X 20000 X 2.5 = 9,150 lbs. Tension. 

50 
^3= 10000 X -^ r- = 25,000 lbs. Tension. 

7;= 0.683 X 20000 X 2.5 = 34.150 lbs. Tension. 
Ts= a866 X 20000 X 2.5 = 43,300 lbs. Tension. 



)igitized by V^OOQIC 



170 STBAIKS 15 ROOF TRUSSES. 






TTsin. t; 
sin v. 






8in.(v — Vi) 

C^^lWcos.v. 

Q= IjPTcos. V + ^17 COS. V = Jf TTcos. v 

L^° 8in.(v — vi) , ^° J sin.(2v — r>i) 



COS. V ™ ^ 



COS. V 

sin. {v — vi) 
W I 



^'""2 h — h^ 



^ ^° sm.(v — vi) * ^° sin.(« — Vj) 



COS. V 



* ^" sm.(v — Vi) 

COS. V 

* ® ^ ^ sm.(v — Vi) 

Example. 

Let TT = 20,000 lbs. ^ = 20 feet. A,= 0. 

I = 50 feet. V = 21° 40^. vi= 0. 

0= 41885 — 0.286 X 20000 X 0.369 = 39,774 lbs. Compression. 
Ci= 43567 — 0.228 X 20000 X 0.369 = 41,885 lbs. Compression. 

Digitized by VjUU^ It: 



STBAINS lis ROOF TRUSSES. 171 

Cj= 48780 — 5213 = 43,567 lbs. Compression. 

Cj= 0.9 X 20000 , ^■,_ = 48,780 lbs.. Compression. 

C^= 0.286 X 20000 X 0.929 = 5,213 lbs. Compre.spion. 
^5= 0.514 X 20000 X 0.929 = 9,550 lbs. Compression. 

T= (o.9 X 20000 -^1^ — 0.8 X 20000 X SeO-* - 0.1 X 

20000^) -^r^-r = 20,000 lbs. Tension. 
/ 0.686 

Ty= T— ^5= 20000 — 7188 = 12,812 lbs. Tension. 
5-= ^^ X -^ = 25.000 lbs. Tension. 



0.929 
0.369 



^3= T;— ^5= 0.757 X 20000 ^^^^ = 38,1 18 lbs. Tension. 



029 
5^4= 0.9 X 20000 —^ = 45,306 lbs. Tensioa. 

5= T^= T— :7l= 7,188 lbs. Tension. 
7i=r5= 7,188 lbs. Tension. 



When tho. rafter is resting on joint A: 

^ W ^ W cos. V COS. fa — v) 

4 sin. V 8 2 sin. Vj 

Ci= -j-7 — r + } IFsin.t; T= (7, cos. r, + T. 
4 sm.-v 11^1 

^a= J r Ji= a COS. v 

sm. Vi * * 

Bending moment at point B = O^ sin. v, . Z. 

Digitized by LnOOQlC 



172 8TBAI5S IS BOOF TBUSSES. 

When rafter is fixed at joint A: 

4 sin. V ^ sin. v^ 

w w 

Ci=}-^!— !r,= — cotg.i; 

Bending moment at -B = -— . I 



Truss No. 18. 

Fig, 254. 



^__ , TTcos. vi 
* 8in.(v-|-ri) 

/Y— 1 ^cos. Vs 
^8~ 2~^ 



sin.(t?-|- Vi) 
!Zi= Cj cos. V + Cj cos. Vi 



y Google 



STRAINS IN EOOP TBUSSES. 



173 



(7= JPTcosec. v 


Q=JTrtang.t;i 




(\=:ilw C08GC.V 


C^iW 




€2= Jf TTcosec. V 


T=iW 




q;= fPFcotg 


. V 


7\= f W cotg. i; + t ^ tang. 2;, 


Q= iTTcotg 


. V 


T^iW cotg.v 






Example. 




Let W=z 20,000 lbs. 1 


>) = 21° 40^ vi= 


= 56° 30^ 


C= 27.000 lbs. 


Compression 


. Q= 3,533 lbs. 


Compression. 


Cl= 36,900 lbs. 


Compression. 


. C^ 6,666 lbs. 


Compression. 


(72= 46,800 lbs. 


Compression, 


. 2'= 6,666 lbs. 


Tension. 


(7s= 33,466 lbs. 


Compressian, 


. Ty= 37,000 lbs. 


Tension. 


Ci= 6,867 lbs. 


Compression. 


. Tr= 41,831 lbs. 


Tension. 



y Google 



174 



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176 



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STRAIKS IK ROOF TBUSSES. 



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178 PBESSUBE OF TniTD OH BOOFS. 

Example to Table of Cobotakts. (Titm Na. 13.) 

What is the amonnt of strain in the various members of a trass, 
according to Fig. 249, of the following dimensions, viz: Span 60 
feet, distance between trasses 10 feet, height at center 10 feet, 
weight to be carried, including w^eight of coDRtriactioa, 66} lbs. 
per fiquare foot horizontally; hence total weight on one ra/ter 
= 30 X 10 X 66J = 20,000 lbs.? 

Z, = 60feet. r ^a v = 18° 20^ 

A = 10 feet. J^ = ^==e, Tr= 20,000 lbs. 
h 10 

Member. Constnnt. W Btraiiw. 

G, = 2.745 X 20,000 = 54,900 lbs. ) 

Cj = 0.660 X 20.000 = 13.200 lbs. V Compression. 

C^ = 0.567 X 20,000 = 11,340 lbs. J 

T = 1.956 X 20,000 = 39,120 lbs. 

T. «= 2.606 X 20,000 = 52,120 lbs. i r^^-^^ 

± = 0.734 X 20.000 = 14,680 lbs. ^ ^^^^^o^- 

7i « 0.183 X 20,000 = 3,660 lbs. ^ 

[NoTB.-^Iii the foregoing table the proportion of A to Xr is approximate. 
The constants are based on the angles.] 



PRESSURE OF WIND ON ROOFS. 

In the following table the maximum pressure of wind is taken 
at 50 lbs. per square foot: 

The angle between horizontal and direction of wind is generally 
10« 00^ (See diagram.) 



R^jtreact. 

F = Force of wind in lbs. = 50. 

w^ = Pressure at ri^ht angles to surface per square foot in lbs. 

u^/y= Pressure, vertical, per square foot in Ibe. 

W/ = J^Bin.«(t;+ 10) 

W//= 1— 

COS. V 

Digitized by V^OOQIC 



FSBSStTBS OF WIND OH BOOFfif. 



-m 



Proportion of , 
height A to 
span L 


Angle V. 


Pressure w/ ' 
in lbs. 

! 


Pressure w,, 
in lbs. 


-: ■ 


90« 00' 


50.00 


0.00 


-4 


45« 00' 


33.53 1 


47.40 


-s 


33« 41' 50" 


23.80 


28,60 


-4 • 


2B*»33'50'' 


17.64 


19.70 


^-', 


21° 48' 


13.83 


14.80 


-4 


18<> 26' 


11.23 


11.80 


-J 


15« 54' 40" 


9.46 


9.80 


-: 


140 02' 10" 


8.56 


8.80 


-: 


12® 31' 40" 


7.29 


7.40 


-4 


IP 18' 40" 


6.51 


6.60 



y Google 



180 



PBESSUBB OF BVOW OV BOOFS. 



PBESSURE OF SNOW ON ROOFS. 

The average pressure of snow on a level sarface, in the United 
States, is aboot 15 lbs. per sqnare foot. 

The following table gives the pressure per square foot on 
various inclined surfaces : 

Beference, 

F = Pressure per square foot in lbs. = 16. 

Pi= Vertical pressure in lbs. 

j>,= Pressure at right angles to surface in lbs. 

V = Angle between surface and horizontal. 

Pl= P COS. V. 
p^=PlC08. V. 



Proportion of 
height h to 


Angle V. 


Pressure Pi 
in lbs. 


Pressure Pz 
in lbs. 


-1 


45<>00' 


10.60 


7.49 


'-4 


33° 41^ 50^^ 


12.48 


10.38 


»-l 


26« 33^ 50'^ 


13.42 


12.00 


-4 


2P 48^ 


13.93 


12.94 


-- 1- 


18« 26^ 


14.23 


13.50 


-4 


15« 54' 40'^ 


14.41 


13.86 


'-: 


14<> 02' 10'' 


14.52 


. 14.06 


-4 


120 31' 40" 


14.64 


14.29 


-i 


IP 18' 40" 


14.71 


14.43 


*-4 


0« 00' 00" 


15.00 


16.00 






■' bigltlzedby^ 


.UUyiL 



TIB KODS AITD BABS. 



181 



TIE BODS AND BARS. 

Capacity and Proportional Dimennons of Wrought-iron Tie Rodi, 
Tie Bars, and Fins or Bolts. 

Ultimate resistance to tearing = 60,000 lbs. = 30 tons per 
square inch. 

Ultimate resistance to shearing = 50,000 lbs. = 25 tons per 
square inch. (See Fig. 258.) 



6,000 
6,200 
7,400 
8,G00 
10,000 
11,200 
12,400 
13,600 
16,000 

7,400 
9,200 
11,200 
13,000 
15,000 
16,800 
18,600 
20,600 
22,400 

10,000 
12,400 
15,000 
17,400 
20/)00 
22,400 
25,000 
27,400 
90,000 

12,400 
16,600 
18,600 
21,800 
26,000 
28,000 
30»633 



2.50 


3,000 


1.50 


0.26 


0.66 


H 


1 


0.75 


0.62 


3.10 


3,720 


1.86 


a3i 


0.62 


44 


VA 


0.93 


0.69 


3.70 


4,440 


2.22 


0.37 


0.70 


« 


1.12 


0.75 


4.30 


6,100 


2.58 


0.43 


0.74 


U 


1% 


1.31 


0.80 


5.00 


6,000 


3.00 


0.60 


0.79 


«t 


2 


1.50 


0.88 


6.r.o 


6,720 


3.36 


0.66 


0.84 


** 


f4 


1.68 


0.92 


6.23 


7,440 


3.72 


0.62 


0.89 


(t 


1.87 


0.97 


6.80 


8,160 


3.88 


0.68 


0.93 


** 


^* 


2.06 


1.01 


7.50 


9,000 


4.50 


0.76 


0.97 


U 


3 


2.25 


1.08 


3.70 


4,440 


2.22 


0.37 


0.68 


^ 


1 


0.76 


0.75 


4.60 


6,520 


2.76 


0.40 


0.76 


IK 


0.93 


0.83 


5.60 


6,720 


3.36 


0.66 


0.84 


(( 




1.12 


0.92 


6.50 


7,800 


3.90 


0.66 


0.91 


t( 


]a^ 


1.31 


0.99 


7.50 


9,000 


4.50 


0.76 


0.97 


*t 


2 


1.60 


1.08 


8.40 


10,080 


6.04 


0.84 


1.04 


tl 


f6 


1.68 


1.13 


9.30 


11,160 


6.58 


0.93 


1.09 


u 


1.87 


1.19 


10.30 


12,.')r>o 


6.18 


1.03 


1.16 


It 


&^ 


2.06 


1.24 


11.20 


1.3,440 


6.72 


1.12 


1.19 


M 


3 


2.25 


1.29 


5.00 


6,000 


3.00 


0.60 


0.79 


^ 


1 


0.75 


0.88 


0.20 


7,440 


3.72 


0.62 


0.88 


iv^ 


0.93 


0.97 


7.50 


9,000 


4.50 


0.75 


0.97 


** 


lyt 


1.12 


1.08 


8.70 


10,440 


6.02 


0.87 


1.05 


fl 


1% 


1.31 


1.16 


10.00 


12,000 


6.00 


1.00 


1.13 


(( 


2 


1.50 


1.24 


11.20 


13,440 


6.72 


1.12 


1.20 


" 


21^ 


1.68 


1.32 


12.50 


15,000 


7.50 


1.25 


1.26 


tt 


2% 


1.87 


1.39 


13.70 


16,440 


8.22 


1.37 


1.32 


" 


2?4 


2.06 


1.45 


16.00 


18,000 


9.00 


1.60 


1.39 


U 


3 


226 


1.52 


6.20 


7,440 


3.72 


0.62 


0.90 


^ 


1 


0.75 


0.98 


7.80 


9,360 


4.68 


0.78 


1.00 


IV^ 


0.93 


1.09 


9.30 


11,160 


6.58 


0.93 


1.09 


(i 


\\t 


1.12 


1.20 


10.90 


13,080 


6.54 


1.09 


1.18 


'* 


l5i 


1.31 


1.29 


12.60 


15,000 


7.60 


1.25 


1.26 


«< 


2 


1.60 


1.39 


14.00 


16,800 


8.40 


L40 


1.34 


M 


^i 


1.68 


1.47 


16.27 


18,720 


9.36 


1.66 


1.41 


N 


1.87 


1.64 



0.44 
0.48 
0.63 
0.57 
0.62 
0.66 
0.69 
0.72 
0.76 

0.53 
0.68 
0.66 
0.70 
0.76 
0.80 
0.83 
0.83 
0.92 

0.62 
0.69 
0.76 
0.82 
0.88 
0.93 
0.98 
1.03 
1.08 

0.69 
0.77 
0.85 
0.91 
0.98 
1.04 
1.09 



gitizedbyVjUUyit: 



182 



TIE BODS ASD BAX& 



Capacity of tie or bar. 


t 

a 

:n 
1 






afimesBafety. 


6 times 


Btttety. 




Lba. 


Teas. 


LbB. 


Ton* 




94,200 


n.io 


20,620 


10.26 


1.71 


1.48 


'4 


2% 


2.06 


1.62 


1.14 


37,600 


18.75 


22,440 


11.22 


1.87 


1.54 


1i 


2.25 


*1.69 


1.20 


15,000 


7.60 


9,000 


4.60 


0.75 


0.98 


rf 


I 


a76 


1.08 


0.78 


18,600 


9l30 


11,160 


6.58 


0.93 


1.09 




0.93 


1.20 


086 


|g.400 


11.20 


13,440 


6.T2 


1.12 


1.19 


*♦ 


jl? 


1.12 


1.31 


0.93 


S6,200 


13.10 


16,720 


7.86 


1,31 


1.30 


u 


\0 


1.31 


1.41 


1.00 


80,000 


15.00 


18,000 


9.00 


1.60 


1.39 


{( 


2 


1.50 


1.52 


1.08 


93.600 


16.80 


20,160 


10.08 


1.68 


1.46 


" 


2V^ 


1.68 


1.62 


1.14 


37,400 


18.70 


22,440 


11.22 


1.87 


1.64 


t( 


2% 


1.87 


1.69 


1.20 


iI,20O 


20.60 


24,720 


12.38 


2.06 


1.62 


" 


2% 


2.06 


1.77 


1.26 


49»O0O 


22.60 


27,000 


13.60 


2.25 


1.69 


u 


a 


2.26 


1.86 


1.32 


17,400 


8.70 


10,440 


6.22 


tt87 


14)6 


H 


1 


0.76 


1.16 


0.82 


n^oo 


10.90 


13,680 


6.64 


1.09 


1.18 


tr 


11^ 


093 


1.29 


0.91 


86,200 


13.10 


16,720 


T.80 


1.31 


1.29 


« 


I72 


1.12 


1.41 


1.00 


80,600 


15.30 


18,360 


9.18 


1.63 


1.40 


*• 


15^ 


1.31 


1.53 


1.08 


34,800 


n.4o 


20,880 


10.44 


1.74 


1.49 


« 


2 


1.50 


1.63 


1.16 


3!t,2no 


19.60 


23,520 


11.76 


1.96 


L68 


*' 


2^ 


1.68 


1.73 


1.23 


«,600 


21.80 


26,160 


13.08 


2.18 


1.66 


u 


2I2 


1.87 


1.82 


1.29 


48,000 


24.00 


28.800 


14.40 


2.40 


1.76 


it 


274 


2.06 


1.89 


1.84 


52,400 


26.20 


31.440 


15.72 


2.62 


1.83 


M 


3 


2.26 


2.00 


1.42 


20,000 


10.00 


12,000 


6.00 


1.00 


1.13 


1 


1 


0.76 


1.39 


0.80 


25.000 


12.60 


15,000 


7.50 


IJ2& 


1.26 


U 


lii 


0.93 


1.45 


04»8 


30,000 


15.00 


18,000 


9.00 


1.50 


1.39 


u 


IX^ 


1.12 


1.52 


1.08 


35,000 


17.50 


21,000 


10.60 


1.75 


1.49 


*• 


m 


1.31 


1.64 


1.16 


40,000 


20.00 


244)00 


12.00 


2.00 


1.60 


<( 


2 


1.60 


1.75 


1.24 


46,000 


22.50 


27,000 


13 60 


2.25 


1.70 


M 


2W 


1.68 


1.86 


1.32 


50,000 


2».00 


30,000 


15J)0 


2.60 


1.79 


(t 


2V4 


1.87 


1.96 


1.89 


65,000 


27.60 


SSfiOO 


16.60 


2.76 


1.87 


i( 


2% 


2.06 


2.05 


1.45 


eopooo 


30.00 


364)00 


184)0 


3.00 


1.96 


«t 


3 


2.25 


2.15 


1.62 


zz 


14.00 


16,800 


a40 


1.40 


1.84 


^ 


IV 


0.93 


1.47 


1.04 


16.80 


20400 


10.08 


1.68 


1.47 


^2a 


1.12 


1.60 


1.13 


39,0^ 


19.80 


23,520 


11.76 


1.98 


1.68 


t« 


1% 


1.31 


1.73 


1.23 


46,000 


,22.60 


27,000 


1^60 


2.26 


1.69 


M 




1.50 


1.86 


1.32 


60,6«0 


28.30 


30,860 


15.18 


2.53 


1.80 


it 


Jft^ 


1.68 


1.97 


1.39 


66,200 


28.10 


33,720 


16.86 


2.81 


1.89 


ft 


2M 


1.87 


2.09 


L« 


601,800 


30.90 


37,080 


ia64 


3.09 


1.98 


a 


29^ 


2.06 


2.18 


1.04 


ff7,4QO 


33.70 


40440 


20.^ 


3.a7 


2.08 


(C 


3 


2.25 


2:26 


1.60 


73.000 


36.50 


43,800 


21.90 


3.65 


2.16 


M 


^^ 


2.43 


2.36 


1.67 


78,600 


39.30 


«7,160 


23.68 


3.93 


2.24 


l( 


ayL 


2.62 


2.46 


1.74 


81,200 


42.10 


60,520 


25.20 


4.21 


2.32 


K 


35* 


281 


2.63 


1.80 


904)00 


45.00 


64,000 


274)0 


4.50 


2.40 


M 


4 


3.00 


2.63 


1.86 


31,200 


16.60 


18,720 


9.30 


1.66 


1.41 


'Y* 


11^ 


0.93 


1.54 


1.09 


sr,4oo 


18.70 


22,440 


1L22 


1.37 


1.66 


ii2 


1.12 


1.69 


1.20 


43,600 


21.80 


26460 


13.08 


2.18 


1.67 


H 


1^ 


1.81 


1.82 


1.89 


00,000 


26.00 


30,000 


16.00 


2JS0 


1.79 


a 


2 


1.60 


1J96 


:1.89 


66,200 


28.10 


38,720 


16.86 


2.81 


1.80 


o 


^ 


1.68 


24)9 


1.48 


«^«)0 


31.20 


VJ440 


^78 


U2 


IM 


(( 


L87 


249 


U6 














Dig 


tized by 


Goc 


)Qle 





TIB HODS, AND BARS. 



18a 



Capacity of tie or bar. 


t 

a 


1^ 


Dimension of 
flat bars in in., 
uniform thick- 
iless. 


Diameter 
D of pin 
orbolk 








1 


•§1 


5§ 






3 times safety. 


6 times safety. 


U 


ii 


'J 




9. ti\ 










P.» 


Lbs. 


Tons. 


Lbs. 


Tons. 


1 


Q 


S2 


P 


08 






68,600 


3430 


41,160 


20.58 


3.43 


2.10 


1^ 


2% 


2.06 


2.29 


1.62 


76,000 


37.50 


45,000 


22.50 


3.76 


2.19 




3 


2.25 


2.40 


LTft 


81,200 


40.60 


48,720 


24.36 


4.06 


2.27 


(( 


354 


2.43 


2.49 


1.76 


87,400 


4370 


52,440 


26.22 


4.37 


2.36 


it, 


3U 


2.62 


2.60 


1.84 


08,600 


46.80 


56,160 


28.08 


4.68 


2.44 


«* 


3^ 


2.81 


2.68 


1.89 


100,000 


60.00 


60,000 


30.00 


5.00 


2.53 


t( 


4 


aoo 


2.77 


1.96 


41,200 


20.60 


24,720 


12.36 


2.06 


1.62 


^% 


^^ 


1.12 


1.77 


1.26 


48,000 


24.00 


28,800 


14.40 


2.40 


1.75 


«4 


1^ 


1.31 


1.89 


1.34 


55,000 


27..'i0 


33,000 


16.50 


2.76 


1.87 


it 


2 


1.60 


2.06 


1.46 


«1,800 


30.90 


37,080 


18.54 


3.09 


1.98 


u 


f4 


1.68 


2.18 


1.54 


68,600 


3k30 


41,160 


20.58 


3.43 


2.09 


u 


1.87 


2.29 


1.62 


75,600 


37.80 


46,360 


22.68 


3.78 


2.19 


M 


2% 


2.06 


2.41 


1.71 


B2,400 


41.20 


49,440 


24.72 


4.12 


JJ.29 


M 


3 


2.26 


2.51 


1.78 


89,200 


44.60 


63,620 


26.76 


4.46 


2.38 


M 




243 


261 


1.85 


96,200 


48.10 


57,720 


28.86 


4.81 


2.47 


U 


2.62 


2.71 


1.92 


103,000 


51.60 


61,800 


30.90 


5.15 


2.56 


M 


^7* 


2.81 


2.81 


1.99 


110,000 


65.00 


66,000 


33.00 


5.60 


2.65 


** 


4 


3.00 


2.90 


2.05 


45,000 


22.6 


27,000 


13.60 


2.25 


1.70 


'^ 


Ip 


1.12 


1.86 


1.32 


52,400 


20.20 


31.440 


15.72 


2.62 


1.83 


1.31 


2.00 


1.42 


60,000 


30.00 


36,000 


18.00 


3.00 


1.96 


(( 


2* 


1.60 


2.16 


1.62 


67,400 


3,1.70 


40,440 


20.22 


3.37 


2.07 


(( 


2K^ 


1.68 


2.27 


1.61 


754)00 


87.50 


45,000 


22.60 


3.76 


2.19 


U 


2V| 


1.87 


2.40 


1.70 


82,400 


41.20 


40,440 


24.72 


4.12 


2.29 


u 


2% 


2.06 


2.51 


1.78 


90,000 


45.00 


54,000 


27.00 


4.50 


2.40 


it 


Sj" 


2.26 


2.63 


1.86 


97,400 


48.70 


58.440 


29.22 


4.87 


2.49 


*t 




2.43 


2.73 


1.93 


105,000 


52.60 


63,000 


31.50 


6.25 


2.59 


(C 


3V^ 


2.62 


2.84 


2.01 


113,400 


66.20 


67,440 


33.72 


6.62 


2.67 


" 


^1 


2.81 


2.93 


2.08 


120,000 


6aoo 


72,000 


36.00 


6.00 


2.77 


«t 


4 


3.00 


ao3 


2.i6 


127,400 


63.70 


76,440 


38.22 


6.37 


2.85 


** 


4^ 


3.18 


ai2 


2.21 


135 000 


67.60 


81,000 


40.50 


6.75 


2.93 


u 


4V» 


zm 


3.22 


24i8 


142,400 


71.20 


85,440 


42.72 


7.12 


3.01 


tt 


^74 


3.66 


3U» 


2.34 


150,000 


7600 


oa/w 


4&60 


7.50 


8.10 


(( 


5 


S.75 


3^39 


2.40 



y Google 



18i JOINTS OE CX)HirECTIOHS IN IRON CONSTRUCTION. 

JOINTS OR CONNECTIONS IN IRON CONSTRUCTION. 

Proportions of Bolts, Nuts, Rivets, Ac. 

Reji'rence. 
A = Sectional area of bolt, rivet, or pin, 
Ai= Sectional area of all rivets in a joint. 
Ai= Sectional area of one plate. 
D = Diameter of bolt, rivet, or pin. 
S= Ultimate resistance to shearing of material. 
T= Ultimate resistance to tearing of material. 
Ti= Tensional strain on joint, &c. 
a = Number of times that a bolt, <&c., will have to be sheared* 

(See 2 on Fig. 258.) 
d = Distance between centres of rivets. 

k = Factor of safety. 

I = Overlap, approximately If cf to If c2. 
m = Number of rivets in a joint. 
n = Number of lines of rivets in a joint at right angles to strain. 

t =■ Thickness of a plate. 



Fig. 257. 



Rivets. 
For tension in direction of rivet: 



2> = J_ZL^_ 
^Z T 0.785 



K7864 
For shearing at riglit angles : 



If at one place D= / ^ — 

^ N ^ 0.7854 



If at two places i)= / ?L^ 

^ '^ S 1.5708 

Approximately : I =: 3t D = St 

Digitized by LnOOQlC 



joints or cjokfectioirs in iron construction. 

Pin, &c., in Tie Bars. 
Fig, 258. 



186 




PlaUb Joints. 

No. I. — Plate Joint OveHapped, four lines of Bivets, 
Fig. 259. 




"O. 2. — Plate Joint Overlapped, single line of Bivet 
Fig. 260. (Same as No. 1.) 



o 
o 
o 
o 



y Google 



2MB 



99IVT8 OR OOVNEOTIOKS IS lEOlST COVSTS.UCTIOV. 



No. 3» — Plate Joint Overlappedt kvo lines of Bivett. 
Fig, 261. (Same as No. 1.) 




No. 4. — Fish Joints^ two lines of Bivets, 
Fig. 262. 



o 
o 



kZJ 



One fish plate. (Same as No. 1.) 

Two fish plates. 

Thickness of each fish plate ^ } 1 



^ ^ /Sf 1.5708 
(Otherwise same as No. 1.) 



y Google 



DIHEBSIOBS Qg BOLia ABD VUT8, 



167 



DIMENSIONS OF BOLTS AND NUTS. 

(Wbitworth's proportions.) 

.%». 263, 264, 265, 266, 267, 268, 269, 270, »nd 271. 



Dia-of 
Bolt. 



Inch. 
3 



2i 
2i 
2 

li 
If 
If 
li 
1| 

u 
1* 

1 

i 
f 

* 

i 

A 

i 

A 

i 



Dimension of Nuts and Heads. 



Dia. of No. Threads 
Core. per inch^ 




Inch. 
4J 
4* 
3i 
3i 
3 

2J 

2| 

^ 

21 

2 

li 

If 

IJ 

lA 

lA 
1 

f 

i 

f 

A 

A 

I 



Inch. 
6.18 
4.76 
4.33 
3.89 
3.46 
3.17 
3.03 
2.88 
2.69 
2.30 
2.16 
1.87 
1.73 
1.51 
1.38 
1.15 
1.01 
0.86 
0.86 
0.64 
0.50 
0.43 



i — >l 

Inch. 

5 

4J 
H 
3| 

3| 
3 



2J 
2i 
2 

1* 
1« 
IJ 

lA 
1 

i 
i 
i 

A 
A 






k— — 
Inch. 
7.07 
6.37 
5.83 
5.30 
4.76 
4.24 
3.89 
3.71 
3.53 
a. 18 
2.82 
2.64 
2.29 
2.12 
1.86 
1.67 
1.41 
1.23 
1.06 
1.06 
0.79 
0.79 



Inch, 
3 



Inch. 
2.67 
2.35 
2.13 
1.91 
1.69 
1.58 
1.47 
1.36 
1.25 
1.14 
1.08 
0.92 
0.81 
0.70 
0.59 
0.48 
0.42 
0.37 
A 0.31 
i 0.26 
A 0.20 
J 0.15 



2i 
2t 
2 

IJ 

li 
It 
ij 
It 
U 
U 
1 

i 
I 

i 

A 

i 



3.5 
S,& 

4.0 

4.0 

4.5 

4.5 

5.0 

5.0 

BO 

6.0 

7.0 

7.0 

8.0 

9.0 

10.0 

11.0 

11.0 

12.0 

14.0 

la.o 

18.0 



1.50 
1.75 
2.00 
2.12 
2.25 
2.37 
2.50 
2.75 
3.00 
3.25 
3.50 
4.00 
5.00 
6.00 
6.00 
7.00 
7.00 
8.00 
8.00 
9.00 
9.00 



20.0 10.00 



y Google 



188 



STRAINS IN HORIZONTAL AND SLOPING BEAMS. 




Approximate proportions of bolts, nuts, and 
heads in inches: 

d = 1.4 i> + 0.25 = Inscribed circle. 

h = D =i Height of nut. 

Ai= 0.7 D = Height of head. 



I!i!i|i|!|i|i 

!:i:i; 



^ 



1*1 • 



COMPOUND STRAINS IN HORIZONTAL AND SLOPING 
BEAMS. 

(Load equally distributed or between supports.) 

Area of Cross-section necessary to resist a Orosi-hreahing and 
Compressive Strain in Beams acting as a Boom in Trusses^ dtc.y 
or Beams acting as Rafters^ dc. 

Inference, 
ffi ^ Bending moment (See Page 100.) 
C=s Compressive strain. (See Roof and Simple Trusses.) 
g = A factor depending on form of cross-section. 
2= Moment of inertia of cross-section. 
s ^ Distance from neutral axis to most compressed fibres. 
A = Sectional area of beam, &c. 
h = Depth of beam, Ac. 
p = Resistance to compression with safety per square inch of 

section. 
TF= Total load. 
I = Length of beam, &c< 



2 = - 



h*A 



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STRAINS IV HOBIZOKTAL AND SLOPING BEAMS. 189 

For horizontal beams, &c. : 

For sloping beams, <&c., v = angle between horizontal and beam: 
. W r 1 f 1 , . \ , ^ COS. v-i 

W r 1 / I , . \ . I COS. v-i 

Bafteb OF A Roof Truss. 
Mg. 273. 



Example. 

Reference. 

Tr= 2.5 tons. C= 2.8 tons. Z = 10 feet. v = 26*» 30^ 

p = 5 tons per square inch. 

We will assume a Phoenix Co*s six-inch beam of the following 

dimensions: ^ = 6 inches; ii = 4 inches; 7=22.5 

22 5 

Q = Zir = 0.312 

^ 0.5X6«X4 

showins that the six-inch beam has a greater sectional area than 
required. 

If the load is concentrated at the apex of roof, the compressive 
strain C^ 2.8 tons, and the area necessary to resist this strain 

2 8 
would be (taking p al five tons per square inch) — ^ =■- 0.56 sq. 

o 

inches, provided this is able to resist buckling. 

By comparing this with the above result, it will be seen how 
much greater the sectional area will have to be to resist a cross - 
breaking strain, caused by the load being distributed. These 
remarks also apply to simple trusses. ,ig,,,, ^y ^uuy it: 



1^ strains nr hobizoktal and sloping beams. 

Simple T«u8s, {Beam continuous oveb Stbut.) 
Fig. 274. 



Example. 

Reference. 

Tr = 20 ions. I^m2ld #set t^ss Id^ ji «b^ tsot per sq. inch. 



We will assume « Pte&iz Oo*s imAn4sA hnm of the fol- 
low iDg dimensioiM: 

\ SB 12 hidiefi. J« t^M 

A «e 12.5 ifi«h««. « 3s 6 incbis. 

2Y5.92 _^_3^ 



^ ~" 0.5 X 12? X 12.5 

m=0.0703x IX 120'=s84^ (See Beaction of Supports.) 
(7s 23.32 tottSu 

1 / 84.36 N . „« o^ 46.26 ^ „^ . , 

Consequently the sectional area of the twelve-inch beam ii 
amply sufficient. 

/NoTK.— The formtilas for horiaostsl beams are also itoplicable to raften 
roof trusses, m and C being given. For the bending moments f«») 
the yarious distances are the horizontal projections of those on the rafter 
from abutment to ridge. 

The toregoing formulas also appfy to beams under a ^ess-Drealang «ad 
tensional strain. If the truss {Fig. 274) is Inverted, the horisontal member 
will be in tension. Hence, insert the resistance of the material to tension 
instead of compression, and put temloDail for eom^^ressive vtrmiii ; other* 
wise, the formulas remain the same.] 



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WEIGHT OP MOVING LOADS. 



IW 



WEIGHT OF MOVING LOADS. 

Variable and Accidental Loads. 

{Wel^t of construction not jncluded.) 



ChaTftct*»r of 
BtraetitFe. 


How loaded. 


Weight in Iba. per sctimre 
foot of sarfMB. 


Street bridges for 
horse cars and 


Crow'd wii& per- 
sons. 


MiBkaum.^ 

Maximum 


40 lbs. 
120 " 


heavy traffid. 


Average.. ^ 


80 " 






Street bridges for 
jjeneral traflBc, 
oot passengers, 


PerBons, animals, 
and wagons. 


Public travel.... 

Private travel..." 

Heavy business 

wagons 


80 lbs. 
40 " 

•80 ** 




Ligiit business^ 
wagons. ,. 


40 " 










Crowded public 
places. 

Dwelling 


Minimum... ^ 

Maximum 

Average 


401ha, 
120 •• 
80 »• 






40 " 




Churches, coart- 
Tooms, theatres, 
«knd ball-rooms. 

Storage o«f grain... 




80 " 


Floore, Ac - 




100 ** 


General merclian-' 
dise 




200 " 




Wiupehottses 




250 " 




Factories^ 




f200 
\ to ** 




Hay-lofte 




(400 
^0 " 











y Google 



192 



8TATI0 AND MOVING LOADS ON BRIDGES. 



STATIC AND MOVING LOADS ON BRIDGES OF 
WROUGHT IRON. 

The following table ^ives an approximate weight per lineal 
foot in poands of the static load or weight of construction complete 
for Single-Line BaUway Bridaes, supported at the ends, from ten 
to four hundred feet span ; also the weight of the moving load 
per lineal foot of span, oased on the assumption that the heaviest 
locomotives exert a pressure of three thousand poands per lineai 
foot between their extreme bearings. 

The table is applicable in computing the strains in all trusses 
with parallel booms mentioned in this work. 

Weight of OonstrticUon and Moving Load of Wrought- Iron Single- 
Line BaUway Bridges for the heaviest traffic. 
(From 20 to 400 feet span.) 



Wei 


ght of constraction complete. 


Weight of moving load equal t^ 3,000 


including cross-ties and rails. 


lbs 


. per lineal foot ot load. 


a 


Weight in 
lbs. per 


d 


Weight in 
lbs. per 


a 


Weight in 
lbs. per 


a 


Weight 
in lbs. 


1 


lineal foot 
of span. 


1 


lineal foot 
of span. 


1 

OQ 


lineal foot 
of span. 


1 

OQ 


per lin. 
foot of 
span. 


10 


427 


210 


1.891 


10 


6,300 


210 


2,535 


20 


500 


220 


1.964 


20 


5.370 


220 


2,495 


30 


573 


230 


2,037 


30 


4.250 


230 


2,455 


40 


646 


240 


2,110 


40 


3.780 


240 


2,375 


50 


719 


250 


2.183 


50 


3,550 


250 


2,335 


60 


792 


260 


2,256 


60 


3,400 


260 


2.290 


70 


865 


270 


2,329 


70 


3,300 


270 


2.245 


80 


938 


280 


2,402 


80 


3.250 


280 


2.200 


90 


1,011 


290 


2.475 


90 


3,180 


290 


2.160 


100 


1,084 


300 


2,548 


100 


3,120 


300 


2.120 


110 


1,157 


310 


2,621 


110 


3,050 


310 


2,080 


120 


1,230 


320 


2.694 


120 


3,000 


320 


2,045 


130 


1.303 


330 


2,767 


130 


2,930 


330 


2,010 


140 


1,380 


340 


2,840 


140 


2.880 


340 


1,975 


160 


1,453 


350 


2.913 


150 


2,820 


350 


1,940 


160 


1,526 


360 


2,986 


160 


2,760 


360 


1,910 


170 


1,599 


370 


3.059 


170 


2,700 


370 


1.880 


180 


1,672 


380 


3.132 


180 


2,655 


380 


1,850 


190 


1.745 


390 


3,205 


190 


2,615 


390 


1.820 


200 


1,818 


400 


3,278 


200 


2.575 


400 


1.800 



d by Google 



BTATIO AND MOVING LOADS OF BEIDQES. 



193 



The following gives the actual weight of some well-known 
Bridges (single line) in America, Germany, and England: 



Name of Bridge. 


System. 


a 

d 

1 




6S 

ftl 


is 




Lbs. 


Lbs. 


Lbs. 


"Brenz," near 
Konigsbronn... 

"Colomak" 

" Iser," near Mu- 
nich 


1 Open Web, "i 
\ parallel booms, f 

(-"PauliVpar-) 
< abolic arcned V 
I booms. J 

<( 
Latticei.^.,. 


63.0 

111.0 

164.7 

178.0 

179.0 

345.0 

455.0 

264.0 

36.0 

36.0 

124.0 


760 

1.090 

1.770 

1,954 

1.324 

2,170 

4,418 

3,225 

566 

580 

770 


3,131 
3.067 
3,656 
3,312 
2,783 
1.970 
2,240 


7,530 
9,516 
8,532 
8,532 
10,390 
11,660 
9,954 


"Donau," near 
Ingolstadt 

"Elb," near Mei- 
ssen ... 


"Rhine," near 
Mainz 


"Royal Albert," 
near Saltash... 

"Boyne" 


"Leven" 


(t 
Truss 




"Kent" 




''Harper's Ferry" 









13 



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y Google 



MISCELLANEOUS. 



(196) 

Digitized by CjOOQIC 



yGopgle 



I^J 



aEOMETRY. 



LONQIMETBY AND PLANIMETRY. 
(Twines and Areas.) 

Beference. 
A =s Area. 

r = Periphery of circle = 3.14159 when diameter = 1. 
r = Radius of circle, 
c = Length of cord of segment. 
p = Circumference of circle for given diameter. 
I = Length of circle arc, &c. 
h ^ Height of segment. 

V = Angles, expressed in decimals, as 15® 30'':= 15.5. 
For other designations, see Figni es. 

[Note.— Always use the same unit for dimensions.] 



j/- "^= 





Values 


cfn. 




7r:i\l4159 
2^ = 6.28319 






^ =1.04720 
o 


— = 0.31831 






-^= 0.78540 


^ = 0.15915 






-^= 0.52360 
o 


-V = 0.10132 

TT 






7r«= 9.86960 
rs = 31.00628 


— =0.63662 

TT 






x/^= 1.77245 
^F= 1.46459 


TT , ^ 






/ 1 



! 1.57080 



(197) 



0.56419 



y Google 



198 



LONGIMETBT. 



Fiq, 275. 


p^=Ttd 




360° ^^ 360° 180° 
1;= — 180° 
180° I 


Fig. 277. 


V = 2(180° — vi) 


Fig. 278. 


c2+ W 62 
'"" 8A ~" 2A 




«=r-j-'-(i)' 



J%. 279. 



y Google 



LOSrOIMBT&T. 



199 



Fig. 280. 








Fig. 281. 
Ellipse. 





a — b 
a+b' 



^x/a^—b^ 



^y/ai^c' 



^x^b'2+c' 



26 



^i^. 284. 



c2=a2+62+2W 
(^^a^--b^ 



d = 



26 



d by Google 



200 



PLA-iriMBTET. 



J^ig 



ae.) 



Fig. 286. (Circle ring.) 



Fig. 287. (Sector.) 



7:d^ 

A = 7rr2== --- = 0.7654<ia 

4 

r = jA = 0.564V J~ 

J A A 



= 7r(ri+rj,)(ri — rj) 



11 V 

^ "2" ^'"^ T ^^'''^360^ '^^^ 

= 0.008727 vr». v = -:^ 360<= 
TTra 






jPt^r. 288. (Segment.) 



-4 = (v — sin. v] 



2 
sm. 



= (-1805~"''-'')-2- 
= (0.017453 1> — sin. ») — 



i^i^.289. (Circle ring sector) 



2 



= -360^(^^'-^»') 
= 0.008727 v(ri« — rja) 



d by Google 



PLAHIMBTBT. 



201 



Zt^. 290. (Ellipse.) 


' A^nah 


Fig. 291. (Square.) 


A^a^ 

a^x/A 


Fig. 292. (Rectangle.) 




!■■[ 


A=^ah 


Fig. 293. (Parallelogram.) 


A=:ab sin. v 


Fig. 294. (Triangle.) 


c^ sin.vsin. Vi 






2 sin. Vj 
When the three sides are given: 
Let a -^^ h + c := B 



d by Google 



202 



OBNTEE OF OEAVITT OF PLANES. 



CENTER OF GRAVITY OF PLANES. 
Jteference. 

X = Distance from a fixed base to center of gravity. 
r = Radius. 
c = Chord. 
h^p, ^ := Dimeusions. 
A = Area. 
V = Angle. 



Fig. 295. (Quadrangle.) 



a and b parallel. * 

A h / b — a \ 

^ — "2 6"U + aJ 



Fig, 296. (Triangle.) 



Fig. 297. (Half circle, or 
elliptic plane.) 




=5 radius = r 

2 

X as 0.4244r 



Fig. 298. (Concentric ring.) 



I 



4 sin.Jt; r^ — V 



d by Google 



CENTER OF GBAVITY OP PLANES. 203 



Fig, 299, (Circle, or elliptic 
arc.) 





1 




rc 2 sin. iv 

X = = — r 

p V 


Fi 


9^.300. (Halfcircumfe 
Qce* of circle or ellipse 


r- 

•) 


2 
x = — r = 0.6366r 


F 


xf. 301. (Circle sectoi 


r.) 


4 sin. hv 


m 


g. 302. (Circle segraen 


t.) 


A = Area. 
"=124 














2h 
"~ 5 



d by Google 



204 



CENTER OF GRAVITY OP PLANES. 




Fig, 305. 



Of any eection, composed of any 
number of simple figures : 

Additional Reference. 

A, il/, ^//= Sectional areaof simple 
figures. 
X = Distance from center of 
^avity of whole sec- 
tion to axis mn, 
a;, a!/, a;//= Distance from center of 
gravity of a simple 
Dgure to a fixed axis 
mn. 



X = 



Ax + A,x, + A^,x, + Ac. 



A A^-\- A^f + Ac. 



y Google 



TRIGONOMETEICAL JOBMULAS. 205 

TRIGONOMETEICAL FORMULAS. 

Beference, 

a^h,c^ Length of sides. 
AtB,0=^ Angles opposite to o, 6, c respectively. 

Right Angle Triangle. 



c 



sin. 
b m n ^ sin. C __ 1 

6 = a COS. (7 COS. (7 1 

ft = ccot.a Cotang.(7=-^;^=^^j^ 

6 = a sm. J5 o 

6 = c tang. 5 1 

c = 6 tang. C Secant C = ■ g 
e=:a sin. C 

Sin.C7=— Co8ec,C=-T-- 



a 

Co8.C7=5i 
a 



sin. (7 



Oblique Angle Triangle. 

Fig. 307. 



e sin. A 
sin. C 
c sin. A 



■n C •— - ( •! 

c sm. B V 2 V 



o = %/62 + c* — 26c COS. A ^ ^ (h (^ + ^)(^'~^h 



'-4-( 



®^°* ^ r,. „ c sin. 5 c sin. A 

6 + 



(a+c)(a-.c)N S^'^-^^ b - a" 



„. ^ a sin. C 
Sin.il =s- 



c 

Digitized 



by Google 



^06 



TMQOirOMlTBICAIi rUNOTIOHS. 
Natural Sink 



Natubai, Cosimb. 











Mii^ates. 










Deg. 























5 


10 


15 


20 


25 


30 


4 





.00000 


.00145 


.00291 


.00436 


.00582 


.00727 


.00873 


1 


.01745 


.01891 


.02036 


.02181 


.02327 


.02472 


.02618 




2 


.03490 


.03635 


.03781 


.03926 


.04071 


,04217 


.04362 




3 


.05234 


.06379 


.05624 


i06609 


.66814 


.05960 


.06105 




4 


.06976 


.07121 


.07266 


^411 


.07556 


i)77©l 


.07846 




6 


.08716 


.08860 


.(»9005 


.09160 


j002^ 


.09440 


.09585 




6 


.10453 


.10597 


.10742 


.10887 


.11031 


.11176 


.11320 




7 


.12187 


.12331 


.12476 


.12620 


02764 


.12908 


.13053 




8 


.13917 


.14061 


.14205 


.14349 


.14493 


.14637 


.14781 




9 


.15643 


.15787 


.16931 


.16074 


.16218 


.16361 


.16505 




10 


.17366 


.17508 


.17651 


.17794 


.17937 


.18081 


.18224 




11 


.19081 


.19224 


.19366 


.19509 


.19652 


.19794 


.19937 




12 


.20791 


.20933 


.21076 


.21218 


.21360 


.21602 


.21644 




13 


.22495 


.22637 


.22778 


.22920 


.23062 


.23203 


.23:^46 




14 


.24192 


.24333 


.24474 


.24615 


.24756 


.24897 


.25038 




16 


.25882 


i8(5022 


.26163 


.26303 


.26443 


.28584 


.26724 




16 


.27564 


.27704 


.27843 


.27983 


.28123 


.28262 


.28402 




17 


.29237 


.29376 


.29615 


.29664 


.29793 


.29932 


.30071 




18 


.30902 


.31040 


.31178 


.31316 


.31454 


.31693 


.31730 




19 


.32557 


.32694 


.32882 


.32969 


.33106 


.33244 


.33381 




20 


.34202 


.34339 


.34475 


.34612 


.34748 


.34884 


.35021 




21 


.35837 


.35973 


.36108 


.86244 


.36379 


.36615 


.36650 




22 


.37461 


.37595 


.87730 


.37865 


.37999 


.38134 


.38268 




23 


.39073 


.39207 


.39341 


.39474 


.39608 


.39741 


.39875 




24 


.40674 


.40806 


.40939 


.41072 


.412 4 


.41337 


.41469 


4 


25 


.42262 


.42394 


.42526 


.42657 


.42788 


.^2920 


.43051 




26 


.43837 


.43968 


.44098 


.44229 


.44359 


.44494 


.44620 




27 


.45899 


.45529 


.45658 


.46787 


.45917 


.46046 


.46175 




28 


.46947 


.47076 


.47204 


.47332 


.474:60 


.47588 


.47716 




29 


.48481 


.48608 


.48735 


.48862 


.48989 


.49116 


.49242 




30 


.50000 


.60126 


.60252 


.60377 


,50603 


.60628 


.60754 




31 


.51604 


.61623 


.61768 


.51877 


.62002 


.62123 


.62260 




32 


.52992 


.63115 


.63238 


.6a361 


.63484 


.53607 


.63730 




83 


.54464 


.54586 


.64708 


.64829 


.64951 


.55072 


.65194 




34 


.65919 


.66040 


.56160 


.56280 


.56401 


.56521 


.66641 




36 


.57358 


.57477 


.67596 


.67715 


.67883 


.SlVf2 


.68070 




36 


.68779 


.58869 


.59014 


Ji9131 


.6flBn8 


.5^66 


.59482 




37 


.60182 


.60298 


.60414 


.60039 


.60646 


.60761 


.60876 




38 


.61666 


.61681 


.61795 


.61900 


.62024 


.62138 


.62251 




39 


.62932 


.63045 


.63168 


.6^71 


.63383 


.63496 


.63608 




40 


.64279 


.64390 


.64501 


.64612 


.64728 


.64884 


.64945 




41 


.65606 


.65716 


.65826 


.65935 


.66044 


.66163 


.66262 




42 


.66913 


.67221 


.67129 


.67237 


.87344 


.67462 


.•67559 




43 


.68200 


.68306 


.68412 


.68518 


.68624 


.68780 


.68835 


i 


44 


.69466 


.69670 


.69676 


.69779 


.60883 


.69987 


.70091 




60 


55 


50 


45 


40 


35 


30 




Deg. 


























Minutes. 






" 





y Google 



TBiaOHOMETKIOAL FUNCnOWS. 
Natural Sine. 



207 



MinuteB. 


Deg. 














35 


40 


46 


50 


55 


60 




.01018 


.01164 


.01309 


J01454 


.01600 


.01745 


89 


.02763 


.08908 


.03054 


J03199 


.03345 


.03490 


88 


.04607 


.04653 


.04798 


J04943 


.06088 


.05234 


87 


.06250 


.06396 


.06540 


X)6685 


.06831 


.06976 


86 


.07991 


.08136 


.08281 


X)8426 


.68571 


.08716 


85 


.09729 


.09874 


.10019 


.10164 


.10308 


.10453 


84 


.11466 


.11609 


.11764 


.11898 


.12043 


.12187 


83 


.13197 


.13341 


.13485 


.13629 


.13802 


.13917 


82 


.14925 


.15069 


45212 


.15356 


.15500 


.15643 


81 


.16648 


.16792 


.16985 


.17078 


.17222 


.17366 


80 


.18367 


.ia509 


.18652 


.18796 


.18938 


.19081 


79 


.20079 


.20222 


.20364 


.20507 


.20649 


.20791 


78 


.21786 


.21928 


.22070 


.22212 


.22363 


.22495 


77 


.23486 


.23627 


.23769 


.23910 


.21051 


.24192 


76 


.25179 


.25320 


.25460 


^5601 


.26741 


.25882 


76 


.2S864 


.27004 


.27144 


.27284 


i!7421 


.27564 


74 


..28541 


.28680 


.28820 


.28959 


.29098 


.29237 


73 


.30209 


.30348 


.30486 


*30625 


.30763 


.30902 , 


72 


.31868 


.32006 


.32144 


^2282 


.32419 


.32.557 


71 


.33518 


.33655 


.33792 


*33929 


.34065 


.34202 


70 


.35157 


.35293 


.85429 


.35565 


.36701 


.35837 


69 


.36786 


.36921 


.37056 


.37191 


.37326 


.37461 


68 


.38403 


.38637 


.38671 


.38805 


.38939 


.39073 


67 


.40008 


.40141 


.40276 


.40408 


.40541 


.40674 


66 


.41602 


.41734 


.41866 


.41998 


.42130 


.42212 


65 


.43182 


.43313 




.43676 


A3706 


.43837 


04 


.44760 


.44880 


.45010 


.45140 


.45269 


.46399 


63 


.46304 


.46433 


.46661 


.46690 


.46819 


.46947 


02 


.47844 


.47971 


.48099 


.48226 


.48354 


.48481 


01 


^309 


.49496 


.49622 


*49748 


.49874 


.60000 


60 


.60879 


.61004 


.61129 


.61264 


.61379 


.61504 


69 


JJ2374 


U52498 


.62621 


.62746 


.62869 


.62992 


68 


.63863 


.68976 


.64097 


.64220 


.64342 


.64464 


57 


.66315 


.66436 


Ji5667 


.66678 


.65799 


.65919 


60 


.66760 


.66880 


.57000 


.67119 


.67238 


,57.358 


55 


.68189 


.68307 


.58425 


.68643 


.68661 


.58779 


54 


.60699 


.59716 


.698:^ 


.69949 




,60182 


63 


.60991 


.61107 


.61222 


.61337 


.61451 


.61566 


62 


.62365 


.62479 


.62595 


.62706 


.62819 


.62932 


61 


.63r2a 


.63832 


.63944 


.64066 


.64167 


.64279 


50 


.66055 


.66166 


.65276 


.65386 


.65496 


.65606 


49 


.66371 


.66480 


.66688 


.66697 


.66805 


.66913 


48 


.67666 


.67773 


.67880 


.67987 


.68093 


.68200 


47 


.68941 


.69046 


.69151 


.6925tf 


.69361 


.69466 


46 


.70195 


.70298 


.70401 


.70505 


.70608 


.70711 


45 


23 


2v') 


15 


10 


5 





Deg. 


















Minn 


tPS. 









- Natural (Jomnk. 



y Google 



208 



T&iaONOMETBICA.L FUNOTIONS. 
Natubal Sixk. 





Minutes. 


Deg. 





















6 


10 


16 


20 


26 


30 


46 


.70711 


.70813 


.70916 


.71019 


.71121 


.71223 


.71326 


46 


.71934 


.72035 


.72136 


.72236 


.72337 


.72437 


.72537 


47 


.73136 


.73214 


.73333 


.73432 


.73581 


.73629 


.7.3728 


48 


.74314 


.74412 


.74509 


.74606 


.747<)3 


.74799 


.74896 


49 


.76471 


.75666 


.75661 


.75756 


.75851 


.75946 


.76041 


60 


.76604 


.76698 


.76791 


.76884 


.76977 


.77070 


.77162 


61 


.77715 


.77806 


.77897 


.77988 


.78079 


.78170 


.78261 


62 


.78801 


.78891 


,78980 


.79069 


.79158 


.79247 


.79335 


63 


.79864 


.79961 


.80038 


.80125 


.80212 


.80299 


.80388 


64 


.80902 


.80987 


.81072 


.81157 


.81212 


.81327 


.81412 


66 


.81916 


.81999 


.82082 


.82165 


.82218 


.82330 


.82413 


66 


.82904 


.82985 


.8.3066 


.83147 


.83228 


.83308 


.83389 


67 


.83867 


.83946 


.84025 


.84104 


.84182 


.84261 


.84339 


68 


.84805 


.84882 


.84959 


.85035 


.85112 


.85188 


.85264 


60 


.86717 


.85792 


.85866 


.85941 


.86015 


.86089 


.86163 


60 


.86003 


.86676 


.86748 


.86820 


.86892 


.86964 


.87036 


61 


.87462 


.87632 


.87603 


.87673 


.87743 


.87812 


.87882 


62 


.88295 


.88363 


.88431 


.88499 


.88566 


.88634 


.88701 


63 


.89101 


.89167 


.89232 


.89298 


.89363 


.89428 


.89493 


64 


.85879 


.89943 


.90007 


.90070 


.90133 


.90196 


.90259 


65 


.90631 


.90692 


.90763 


.90814 


.90876 


.90936 


.90996 


66 


.91365 


.91414 


.91472 


.91531 


.91590 


.91648 


.91706 


67 


.92Jo0 


.92107 


.92164 


.92220 


.92276 


.92332 


.92388 


68 


.92718 


.92773 


.92827 


.92381 


.92935 


.92<)88 


.93042 


69 


.93358 


.93410 


.93462 


.93514 


.93566 


.93616 


.93667 


70 


.93969 


.94019 


.94068 


.94118 


.94167 


.94215 


.94264 


71 


.94552 


.94599 


.94646 


.94693 


.94740 


.94786 


.94832 


72 


.95106 


.95150 


.95191 


.95240 


.9.5284 


.95328 


.95372 


73 


.95630 


.95673 


.95715 


.95767 


.95799 


.95841 


.95882 


74 


.96126 


.96166 


.96206 


.96246 


.96285 


.96324 


.96363 


76 


.96593 


.96630 


.96667 


.96705 


.96742 


.96778 


.96816 


76 


.97030 


.97065 


.97100 


.97134 


.97169 


.97203 


.97237 


77 


.97437 


.97470 


.97502 


.97534 


.97566 


.97598 


.97630 


78 


.97815 


.97845 


.97875 


.97905 


.97934 


.97963 


.97992 


79 


.98163 


.98190 


.98218 


.98246 


.98272 


.98299 


.98323 


80 


.98481 


.98506 


.98531 


.98506 


.98580 


.98604 


.98629 


81 


.98769 


.98791 


.98814 


.98836 


.98858 


.98880 


.98902 


82 


.99027 


.99047 


.99067 


.99087 


.99106 


.99125 


.99144 


83 


.992S5 


.99272 


.99290 


.99307 


.99324 


.99341 


.99357 


84 


.99452 


.99467 


.99482 


.99497 


.99511 


.99526 


.99540 


86 


.99619 


.99632 


.99644 


.99657 


.99668 


.99680 


.99692 


86 


.99756 


.99766 


.99776 


.99786 


.99796 


.99804 


.99813 


87 


.99863 


.99870 


.99878 


.99886 


.99892 


.99898 


.99905 


88 


.99939 


.99944 


.99949 


.99963 


.99958 


.99962 


.99966 


89 


.99985 


.99987 


.99989 


.99991 


.99993 


.99995 


.99996 




60 


65 


60 


45 


40 


35 


30 


Deg. 


















Minutes. 





Natubal Cosinb. 



y Google 



TSI<K)NOMETRIGAL FUNCTIONS. 
NATU&A.L Sink. 



209 



Minutes. 
















Deg. 


*35 


40 


45 


60 


65 


60 




.71427 


.71529 


.71630 


.717.32 


.71833 


.71934 


44 


.72637 


.72737 


.72837 


.72937 


.73036 


.73185 


43 


.73826 


.73924 


.74022 


.74123 


.74217 


.74314 


42 


.74992 


.75088 


.75184 


.75280 


,75375 


.75471 


41 


.76135 


.76229 


.763-23 


.76417 


.76511 


.76604 


40 


.77255 


.77347 


.77439 


.77531 


.77023 


.77715 


39 


.78351 


.78442 


.78532 


.78622 


.78711 


.78801 


38 


.79424 


.79512 


.79300 


.79088 


.70776 


.79864 


37 


.80472 


.80558 


.80644 


.80730 


.80816 


.80902 


36 


.81496 


.81580 


.81664 


.81748 


.81832 


.81915 


35 


.82495 


.82577 


.82659 


.82741 


.82822 


.82904 


34 


.83469 


.83549 


.836-29 


.83708 


.83788 


.83867 


aa 


.84417 


.84495 


.84573 


.84650 


.84728 


•84805 


32 


.85340 


.85416 


.85491 


.85507 


.86642 


.85717 


31 


.86237 


.80317 


.86384 


.86457 


.86530 


.86603 


30 


.87107 


.87178 


.87250 


.87321 


.87391 


.87462 


29 


.87959 


.88020 


.88089 


.88158 


.88226 


.88295 


28 


.88768 


.88835 


•88902 


.88968 


.89035 


.89101 


27 


.89558 


.89023 


.89687 


.89762 


,89816 


.89879 


26 


.90321 


.90383 


.90446 


.90507 


.90669 


.90631 


25 


491056 


.91116 


.91176 


.91236 


.91295 


.91355 


24 


.91764 


.91822 


.91879 


.91936 


.91994 


.92050 


2a 


.92444 


.92499 


.92554 


.92609 


.92064 


.92718 


22 


.93095 


.93148 


.93201 


.93253 


.93306 


.93358 


21 


.93718 


.93769 


.93819 


.93869 


.93919 


.93969 


20 


.94313 


.94361 


.94409 


.94457 


.94504 


.94552 


19 


.94878 


.94924 


.94970 


.95016 


.95061 


.95106 


IS 


.95415 


.95459 


.95502 


.95545 


.96688 


.95630 


17 


.95923 


.95964 


.96005 


.96C46 


.96086 


.96126 


la 


.96402 


.96440 


.96479 


.96517 


.96555 


.96593 


15 


.90851 


.90887 


.96923 


.96959 


.96994 


.97030 


14 


.97271 


.97304 


.97338 


.97371 


.97404 


.97437 


la 


.97661 


.97692 


.97723 


.97754 


.97784 


.97815 


12 


.98021 


.98050 


.98079 


.98107 


.98135 


.98103 


11 


.98352 


.98378 


.98404 


.98430 


.98455 


.98481 


la 


.98052 


.98676 


.98700 


.98723 


.98746 


.98709 


9 


.98923 


.98944 


.98905 


.98986 


.99006 


.99027 


a 


.99163 


.99182 


.99200 


.99219 


.99237 


.99255 


7 


.99374 


.93390 


.99406 


.99421 


.99437 


.99452 


6 


.99553 


.99507 


.99580 


.99594 


.99607 


.99619 


5 


.99703 


.99714 


.99725 


.99736 


.99746 


.99756 


4 


.99822 


.99831 


.99839 


.99847 


.99855 


.99863 


3. 


.99911 


.99917 


.99923 


.99929 


.99934 


.99939 


2 


.99969 


.99973 


.99976 


.99979 


.99982 


' .99985 


1 


.99997 


.99998 




1.00000 


1.00000 


1.00000 


0. 


25* 


20 


15 


10 


5 





I>09<^ 


Minutes. 





14 



Natural Cobinx. 



y Google 



210 



TBiQOirOMETRIOAL FUVOTIONS. 
Natusal Tanobkt. 





Minates. 


Deg. 














, 







5 


10 


16 


20 


26 


30 





0.0000 


0.0014 


0.0029 


0.0044 


0.0058 


0.0073 


0.0087 


1 


0.0175 


0U)189 


0.0204 


0.0218 


0.02:« 


0.0247 


0.0262 


2 


0.0349 


0.0364 


0.0378 


0.0393 


0.0407 


0.0422 


0.0437 


3 


0.0624 


0.0539 


0.0553 


0.0568 


0.a'j82 


0.0597 


0.0612 


4 


0.0099 


0714 


0.0728 


0.0743 


0.0768 


0.0772 


0.0787 


6 


0.0875 


0.0889 


0.09<»4 


0.0919 


0.0933 


0.0948 


a0963 


6 


0.1051 


0.1066 


0.1080 


0.1095 


0.1110 


0.1126 


0.1139 


7 


0.1228 


0.1243 


0.1267 


0.1272 


0.1287 


0.1302 


0.1316 


8 


0.1406 


0.1420 


0.1435 


0.1450 


0.1466 


0.1480 


0.1495 


9 


0.1684 


0.1599 


0.1614 


0.1629 


0.1644 


0.1658 


0.1673 


10 


0.1703 


0.1778 


0.1793 


0.1808 


0.1823 


0.1838 


0.1853 


11 


0.1944 


0.1959 


0.1974 


0.1989 


0.2004 


0.2019 


0.2034 


12 


0.2126 


0.2141 


0.2156 


0.2171 


0.2180 


0.2202 


0.2217 


13 


0.2309 


0.2324 


0.2339 


0.2:^55 


0.2370 


0.2385 


0.2401 


14 


0.2493 


0.2509 


0.2524 


0.2540 


0.2555 


0.2571 


0.2686 


16 


0.2079 


0.2695 


0.2711 


0.2726 


0.2742 


0.2758 


0.2773 


16 


0.2867 


0.2883 


0.2899 


0.<J916 


0.2930 


0.2946 


0.2962 


17 


0.3057 


0.3073 


0.3089 


0.3105 


0.3121 


0.3137 


0.3153 


18 


0.3249 


0.3266 


0.3281 


0.3297 


0.3314 


0.3.320 


0.3346 


19 


0.3443 


0aJ400 


0.3476 


0M02 


0.3508 


0.3525 


0.3541 


20 


0.3040 


0.3056 


0.3073 


0.3089 


0.3706 


0.3722 


0.8739 


21 


0.3839 


03855 


0.:^872 


0.3889 


0.3!K)5 


0.3922 


0.3939 


22 


0.4040 


0.4057 


0.4074 


0.4091 


0.4108 


0.4125 


0.4142 


23 


0.4245 


0.4202 


0.4279 


0.4206 


0.4314 


0.4331 


0.4348 


24 


0.4452 


0.4470 


0.4487 


0.45»>5 


0.4522 


0.4540 


0.4557 


25 


0.40G3 


0.4G81 


0.4698 


0.4716 


0.4734 


0.4752 


0.4770 


26 


0.4877 


0.4895 


0.4913 


0.4931 


0.4950 


0.4968 


0.4986 


27 


0.5096 


0.6114 


0.5132 


0.5150 


0.5169 


0.5187 


0.6206 


26 


0.5317 


0.5336 


0.6354 


0.6373 


05392 


0.5411 


0.5430 


29 


0.6543 


0.6502 


0.5581 


0.6600 


5619 


0.5638 


0.5658 


30 


0.5774 


0.5793 


0.5812 


0.6832 


0.5851 


0.5871 


0.5891 


31 


0.6008 


0.6028 


0.0048 


0.0068 


0.6088 


0.6108 


0.G128 


32 


0.6249 


0.6269 


0.6289 


0.6309 


0.6:i30 


0.6350 


0.6371 


33 


0.045)4 


0.6516 


0.6635 


0.0556 


0.6577 


^ 0.6598 


0.6619 - 


34 


0.6746 


0.0706 


0.6787 


0.6809 


0.6830 


0.0851 


0.6873 


35 


0.7002 


0.7024 


0.7046 


0.7007 


0.7089 


0.7111 


0.7133 ' 


36 


0.7265 


0.7288 


0.7310 


0.7332 


0.7356 


0.7377 


0.7400 


37 


0.75.36 


0.7558 


0.7581 


0.7004 


0.7627 


0.7650 


0.7673 ' 


88 


0.7813 


0.7836 


0.7800 


0.7883 


0.7907 


0.7931 


0.7954 


89 


0.8098 


0.8122 


0.8146 


0.8170 


0.8195 


0.8219 


0.8243 


40 


0.8391 


0.8416 


0.8441 


0.8466 


0.8491 


0.8516 


0.8541 - 


41 


0.8693 


0.8718 


0.8744 


0.8770 


0.8795 


0.8821 


0.8847 - 


42 


0.9004 


0.9030 


0.9057 


0.9083 


0.9110 


0.9137 


0.9163 - 


43 


0.9326 


0.9362 


0.9380 


0.9407 


0.9434 


0.9462 


0.9490 ^ 


44 


0.9667 


0.9685 


0.9713 


0.9742 


0.9770 


0.9798 


0.9827 L 




00 


65 


60 


46 


40 


36 


30 


Beg. 


















Minutes. 



Natueal Cotakoekt. 



y Google 



TBIOOIRWirrBIOAL TmtaswvB, 



211 



Minutes. 


Defr 














35 


40 


45 


60 


55 


60 




0.0102 


0.0116 


0.0181 


0.0146 


0.0160 


0.^75 


89 


0.0276 


0.0291 


0.0306 


0.0320 


0.0336 


a0340 


88 


0.0451 


0.0466 


0.0480 


0.0496 


0.0509 


0.0524 


87 


0.0626 . 


0.0641 


0.0656 


0.0670 


0.0686 


0.0699 


86 


0.0802 


0.0816 


a0831 


0.084B 


0.0660 


0.6875 


86 


0.0978 


0.0992 


0.1007 


0.1022 


0.1036 


0.1051 


84 


0.1154 


0.1169 


0.1184 


0.1198 


0.1218 


0.1228 


88 


0.1331 


0.1346 


0.1361 • 


0.1376 


0.1891 


0.1406 


82 


0.1509 


0.1524 


al539 


0.1564 


0.1669 


0.1684 


81 


0.1688 


0.1703 


0.1718 


0.1733 


0.1748 


0.1763 


80 


0.1868 


0.1883 


ai899 


0.1914 


0.1929 


0J944 


79 


0.2050 


0.2866 


0.2080 


0.2096 


0.2110 


0.2126 


78 


0.2232 


2247 


0.2263 


0.227B 


0.2293 


0.2309 


77 


0.2416 


0.2432 


a2447 


0.2462 


0.2478 


0.2498 


76 


2002 


0.2617 


0.2633 


0.2648 


0.2664 


0.2679 


75 


0.2789 


0.2805 


a2820 


0.2836 


0.28^2 


0.2867 


74 


0.2978 


0.2994 


0.3010 


0.8026 


0.8041 


0.305T 


73 


0.3169 


0.3185 


O.3201 


0.8217 


0.3283 


0.3249 


72 


0.3362 


0.8378 * 


0.3394 


0.3411 


0.3427 


0.3443 


71 


0.3558 


0.3674 


0.3690 


0.8607 


0.3623 


0.3640 


70 


0.3765 


0.3772 


0.3789 


0.8805 


0.3822 


0.8839 


69 


0.3956 


0.3973 


0.3990 


0.4006 


0.4023 


0.4040 


68 


0.4159 


0.4176 


0.4193 


0.4210 


0.4228 


0.4246 


6T 


0.^65 


0.4383 


a4400 


0.4417 


0.4485 


0.44S2 


66 


0.4675 


0.4692 


0.4610 


0.4628 


0.4646 


0.4668 


66 


0.4788 


0.4805 


a4823 


0.4841 


0.4889 


0.4877 


64 


0.5004 


0.5022 


0.6040 


0.5059 


0.5077 


0.5095 


63 


0.5224 


0.5243 


0.5261 


0.6280 


0.5298 


0.6317 


62 


0.5448 


0.5467 


0.5486 


0.5505 


0.5524 


06548 


61 


0.5677 


0.6696 


0.5715 


0.5735 


0.5754 


0.6774 


CO 


J . 0.5910 


0.5930 


0.6949 


0.5969 


0.5989 


0.6008 


69 


■" a6148 


0.0168 


aci88 


0.6206 


0.0228 


0;6249 


68 


5? 0.6391 


0.6412 


0.6432 


0.6468 


0.6473 


0.6404 


67 


T- 0.6640 


0.6661 


0.6682 


0.6703 


0.6724 


0.6746 


66 


iu 0.6894 


0.6916 


0.6937 


0.6950 


0.6980 


0.7002 


56 


>^ 0.7155 


0.7177 


0.7199 


0.7221 


0.7S43 


0.7265 


54 


~'>t 0.7422 


0.7445 


0.7467 


0.7490 


0.7618 


0.7636 


58 


••' 0.7696 


0.7720 


0.7743 


0.7766 


0.7789 


0.7818 


52 


'Jf 0.7976 


0.8002 


0.8026 


0.8050 


0.8074 


0.8098 


51 


37 0.8268 


0.8292 


0.8317 


0.8341 


0.8366 


0.8391 


50 


i>d 0.8566 


0.8591 


0.8617 


0.86^ 


0.8667 


0-8693 


49 


i» 1 0.8873 


0.8899 


0.8925 


0.8961 


0.8978 


0.9004 


48 


V 0.9190 


0.9217 


0.9244 


0.92n 


0.9298 


0.9326 


47 


t. 0.9617 


0.9546 


0.9573 


0.9601 


09629 


0.9667 


46 


iru 0.9866 


0.9884 


0.9913 


0.9942 


0.9071 


1.0000 


46 


26 


20 


15 


10 


5 





Deg. 


Minotet. 





NAtVBAI. GovANOxm. 



y Google 



212 




TWOOKOMBTEICAL FUH0TI0H8. 














Natural Tangent. 
















Minutes. 








Deg. 























5 


10 


16 


20 


25 


30 




45 


1.0000 


1.0029 


1.0058 


1.0088 


1.0117 


1.0146 


1.0176 




46 


1.0356 


1.0386 


L0416 


1.0446 


1.0477 


1.0507 


1.0538 




47 


1.0724 


1.0755 


1.0786 


10818 


1.0850 


1.0881 


1.0913 




48 


11106 


1.1139 


1.1171 


1.1204 


1.1237 


1.1270 


1.1303 




49 


1.1504 


1.1537 


1.1571 


1.1606 


1.1640 


1.1674 


1.1708 




60 


1.1917 


1.1953 


1.1988 


1.2024 


1.2059 


1.2095 


1.2131 




61 


1.2349 


1.2386 


1.2423 


1.2460 
1.291§ 


1.2497 


1.2534 


1.2572 




62 


1.2799 


1.2838 


1.2876 


1.2954 


1.2993 


1.8032 




68 


1.3270 


1.3311 


1.3351 


1.3802 


1.3432 


1.3472 


1.3514 




54 


13764 


1.3806 


1.3848 


1.3891 


1.39:^ 


lJt976 


1.4019 




55 


1.4281 


1.4326 


1.4370 


1.4415 


1.4460 


1.4505 


1.4550 




66 


1.4826 


1.4872 


1.4919 


1.4966 


1.5013 


1.5061 


1.5108 




67 


1.5399 


1.5448 


1.5497 


1.5547 


1.5597 


1.5647 


1.5697 




58 


1.6003 


1.6055 


1.6107 


1.6160 


1.6212 


1.6265 


1.6318 




59 


1.6643 


1.6698 


1.6763 


1.6808 


1.6864 


1.6920 


1.6976 




60 


1.7320 


1.7879 


1.7437 


1.7496 


1.7556 


1.7G15 


1.7675 




61 


1.8040 


1.8102 


1.8165 


1.8228 


1.8291 


1.8354 


1.8418 




62 


1.8807 


1.8873 


1.8940 


1.9007 


1.9074 


1.9142 


1.9210 




63 


1.9620 


1.9697 


1.9768 


1.9840 


1.9912 


1.9984 


2.0057 




64 


2.0503 


2.0579 


2.0655 


2.0732 


2.0809 


2.0887 


2.0966 




65 


2.1446 


2.1527 


2.1609 


2.1692 


2.1775 


2.1859 


2.1943 




60 


2.2460 


2.2549 


2.2637 


2.2727 


2.2817 


2.2907 


2.2998 




67 


2.3558 


2.3654 


2.3750 


2.3847 


2.3045 


2.4043 


2.4142 




68 


2.4751 


2.4856 


2.4960 


2.5066 


2.5171 


2.6279 


2.5386 




69 


2.6051 


2.6166 


2.6279 


2.6394 


2.6511 


2.6628 


2.6746 




70 


2.7475 


2.7600 


2.7725 


2.7852 


2.7980 


2.8109 


2.8239 




71 


2.9042 


2.9180 


2.9319 


2.9456 


2.9600 


2.0743 


2.9886 




72 


3.0777 


3.0930 


3.1084 


8.1240 


3.1397 


3.1656 


8.1716 




73 


3.2708 


a2879 


a3052 


3^3226 


3.3402 


8.8580 


8.3759 




74 


3.4874 


3.5067 


3.5261 


3.5457 


8.5656 


3.5856 


a6059 




76 


3.7320 


3.7639 


8.7760 


3.7083 


8.8208 


8.8436 


3.8667 




76 


4.0108 


4.0358 


4.0611 


4.0867 


4.1126 


4.1388 


4.1653 




n 


4.3315 


4.3604 


4.3897 


4.4194 


4.4494 


4.4799 


4.5107 




78 


4.7046 


4.7385 


4.7729 


4.8077 


4.8430 


4.8788 


4.9152 




79 


5.1445 


5.1848 


5.2257 


5.2671 


5.3093 


5.3521 


6.3955 




80 


5.6713 


5.7199 


5.7694 


5.8197 


6.8708 


5.9228 


5.9758 




81 


6.3137 


6.3737 


6.4348 


6.4971 


6.5605 


6.6252 


6.6912 




82 


7.1154 


7.1912 


7.2687 


7.3479 


7.4287 


7.5U3 


7.5957 




83 


8.1443 


8.2434 


8.3450 


8.4490 


8.6555 


8.6648 


8.7769 




84 


9.5144 


9.6493 


9.7882 


9.9310 


10.0780 


10.2290 


10.3850 




$6 


11.4300 


11.6250 


11.8260 


12.0350 


12.2610 


12.4740 


12.7060 




86 


14.3010 


14.6060 


14.9240 


16.2570 


15.6050 


15.9690 


10.3500 




87 


19.0810 


19.6270 


20.2060 


20.8190 


21.4700 


22.1640 


22.9040 




88 


28.6360 


29.8820 


81.2420 


32.7300 


34.3680 


36.1780 


38.1880 




89 


57.2900 


62.4990 


68.7500 


76.3900 


85.9480 


98.2180 


114.5900 






60 


55 


60 


45 


40 


86 


30 




Deg. 


























Minutes. 
















Natubai 


. COTANO« 


KT. 
















Digitized b\ 


GooqI 


e 





TBIGONOMETBIOAL FUUCTIONS. 
Natubal Tangent. 



213 







Minutes. 






Deg. 














85 


40 


46 


60 


66 


60 




1.0206 


1.0236 


1.0265 


1.0296 


1.0326 


1.0855 


44 


1.0568 


1.0590 


1.0630 


1.0661 


1.0692 


1.0724 


43 


1.0945 


1.0977 


1.1009 


1.1041 


1.1074 


1.1106 


42 


1-1386 


1.1369 


1.1403 


1.1436 


1.1470 


1.1504 


41 


1.1743 


1.1778 


1.1812 


1.1847 


1.1882 


1.1917 


40 


1.2167 


1.2203 


1.2239 


1.^2276 


1.2312 


1.2349 


39 


1.2609 


1.2647 


1.2686 


1.2723 


1.2761 


1.2799 


38 


1.3071 


1.3111 


1.3161 


1.3190 


1.3230 


1.3270 


37 


1.3555 


1.3597 


1.3638 


1.3680 


1.3722 


1-3764 


36 


1.4063 


1.4106 


1.4150 


1.4193 


1.4237 


1.4281 


36 


1.4595 


1.4641 


1.4687 


1.4733 


1.4779 


1.4826 


34 


1.5156 


1.6204 


1.5252 


1.6301 


1.5350 


1.5399 


83 


1.6747 


1.5798 


1.5849 


1.5900 


1.5952 


1.6003 


32 


1.6372 


1.6426 


1.6479 


1.0534 


1.6588 


1.6643 


31 


1.7033 


1.7090 


1.7147 


1.7205 


1.7263 


1.7320 


30 


1.7735 


1.7795 


1.7856 


1.7917 


1.7979 


1.8040 


29 


1.8482 


1.8546 


1.8611 


1.8676 


1.8741 


1.8807 


28 


1.9278 


1.9347 


1.9416 


1.9486 


1.9556 


1.9626 


27 


2.0130 


2.0204 


2.0278 


2.0353 


2.0428 


2.0503 


26 


2.1044 


2.1123 


2.1203 


2.1283 


2.1364 


2.1445 


25 


212028 


2.2113 


2.2199 


2.2286 


2.2373 


2.2460 


24 


2.3090 


2.3183 


2.3276 


2.3.369 


2.3464 


2.3558 


23 


2.4242 


2.4342 


2.4443 


2.4545 


2.4648 


2.4751 


22 


2.5495 


2.5605 


2.5716 


2.5826 


2.6938 


2.6051 


21 


2.6865 


2.6985 


2.7106 


2.7228 


2.7351 


2.7475 


20 


2.8370 


2.8502 


2.8636 


2.8770 


2.8905 


2.9042 


19 


3.0032 


3.0178 


3.0326 


3.0475 


3.0625 


3.0777 


18 


3.1877 


3.2041 


3.2205 


3.2371 


3.2539 


3.2708 


17 


3.3941 


3.4124 


3.4308 


3.4495 


3.4684 


3.4874 


16 


3.6264 


3.6471 


3.6680 


3.6891 


3.7105 


3.7320 


15 


3.8900 


3.9186 


3.9375 


3.9616 


3.9861 


4.0108 


14 


4.1921 


4.2193 


4.2468 


4.2747 


4.3029 


4.3315 


13 


4.5420 


4.6736 


4,6057 


4.6382 


4.6712 


4.7046 


12 


4.9620 


4.9894 


5.0273 


6.0658 


5.1049 


5.1445 


11 


6.4397 


6.4845 


5.5301 


5.5764 


6.6234 


5.6713 


10 


6.0296 


6.0844 


6.1402 


6.1970 


6.2549 


6.3137 


9 


6.7584 


6.8269 


6.8969 


6.9682 


7.0410 


7.1154 


8 


7.6821 


7.7703 


7.8606 


7.9530 


8.0476 


8.1443 


7 


8.8918 


9.0098 


9.1309 


9.2553 


9.3831 


9.6144 


6 


10.5460 


10.7120 


10.8830 


11.0590 


11.2420 


11.4300 


5 


12.9470 


13.1970 


13.4570 


13.7270 


14.0080 


14.3010 


4 


16.7500 


17.1690 


17.6110 


18.0750 


18.5640 


10.0810 


3 


23.6940 


24.5420 


26.4520 


26.4320 


27.4900 


286360 


2 


40.4360 


42.9640 


45.8290 


49.1040 


52.8820 


67.2900 


1 


187.5100 


171.8800 


229.1800 


343.7700 


687.6500 







25 


20 


15 


10 


6 





Deg. 


Minutes. 





Nattbal Cotanobnt. 



y Google 



214 



TBIOOVOKXTBIOiJ. FUIKZUOITB. 
Natdbal Ssciavt. 











Minutes. 








I>^- 





















6 


10 


W 


20 


26 


89 


• 


1.0000 


1.0000 


l.onoo 


1.0000 


1.0000 


1.0000 


1.0000 


1 


1.0001 


1.0002 


1.0002 


1.00C2 


1.0003 


1.0003 


1.00O3 


2 


1.0006 


1.0007 


1.0007 


1.0008 


1.0008 


1.0009 


1.00(19 


3 


1.0014 


1.0014 


1.0016 


1.0016 


1.0017 


1.0018 


IXXttO 


4 


1.0021 


1.0025 


1.0023 


1.0027 


1.0029 


1.0030 


1.0081 


6 


1.0038 


1.0039 


1.0041 


1.0042 


1.0043 


1.0046 


1,0046 


6 


1.0055 


1.0067 


1.0068 


1.0060 


1.0061 


1.0063 


14)005 


7 


1.0076 


1.0077 


1.0079 


1.0080 


1.0082 


1.0084 


1.0086 


8 


1.0098 


1.0100 


1.0102 


1.0104 


1.0107 


1.0109 


1.0111 


9 


1.0125 


1.0127 


1.0129 


1.0132 


1.0134 


1.0136 


1.0139 


10 


1.0164 


1.0167 


10169 


1.0162 


1.0166 


1.0167 


1.0170 


11 


1.0187 


1.0190 


1.01S3 


1.0196 


1.0199 


1.0202 


1.0205 


12 


10223 


1.0223 


1.0229 


1.0233 


1.02:16 


1.0239 


1.0243 


13 


1.0263 


1.0266 


1.0270 


1.0274 


1.0277 


1.0280 


1.0284 


14 


1.0306 


1.0310 


1.0314 


1.0317 


1.0321 


1.0325 


1.0329 


16 


1.0363 


1.0367 


1.0361 


1.0366 


1.0369 


1.0373 


i.r — 


16 


1.0403 


1.0407 


1.0412 


1.0416 


1.0420 


1.0426 


l.( 


17 


1.0467 


1.0461 


1.0466 


1.0471 


1.0476 


1.0480 


l.( 


18 


1.0515 


1.0620 


1.0626 


1.0630 


1.0535 


1.0640 


l.( 


19 


1.0677 


1.0581 


1.0587 


1.0698 


1.0598 


1.0C03 


l.( 


ao 


1.0642 


1.0647 


1.0663 


1.0669 


1.0664 


1.0670 


l.( 


21 


1.0711 


1.0717 


1.0723 


1.0729 


1.0736 


1.0742 


l.( 


22 


1.0785 


1.0792 


1.0798 


1.0804 


1.0811 


1.0817 


l.( 


23 


1.0864 


1.0870 


1.0877 


1.0884 


1.0891 


1.0897 


l.( 


24 


1.0946 


1.0968 


1.0961 


1.0968 


1.0976 


1.0982 


l.( 


26 


1.1034 


1.1041 


1.1049 


1.1066 


1.1064 


1.1072 


1. 


20 


1.1126 


1.1134 


1.1142 


1.1160 


1.1168 


1.1166 


1. 


27 


..^ 


1.1231 


1.1240 


■1 tntn 


1.1257 


1.1'265 


1. 


28 




1.1334 


''"43 




1.1361 


1.1370 


1. 


29 




1.1443 


62 




1.1471 


1,1480 


1. 


30 




1.1667 


66 




1.1586 


1.1696 


1. 


31 




1.1676 


87 




1.1707 


1.1718 


1. 


82 




1.1802 


30 




1.1835 


1.1846 


1. 


83 




1.1935 


46 




1.1969 


1.1980 




84 




1.2074 


68 




1.2110 


1.2122 


1.2134 


85 




1.2220 


33 




1.2268 


1.2270 


1.2283 


86 




^ tvtitA 


87 




1.2413 


1.2427 


1.2440 


87 






49 




1.2577 


1.2591 


1,2605 


88 






19 




1.2748 


1.2763 


1.2778 


39 






98 




1.2929 


1.2944 


1.2960 


40 






86 


-.uiuj 


1.3118 


1.3134 


1.3161 


41 






84 


1.3301 


1.3318 


1.3335 


1.3362 


42 






92 


1.3509 


1.3507 


1.3640 


1.3663 


43 






10 


1.3729 


1.3748 


1.3767 


1.3786 


44 






41 


1.3960 


1.3980 


1.4000 


1.4020 




60 


65 


60 


46 


40 


86 


80 


Peg. 
























Minates. 









Natvbai. ConoAMf.. 



y Google 



TiaOOFOMSTBIOAL FUHOTIONS. 
Natubai. Skgamt. 



2ia 



Minutes. 
















De* 


35 


40 


45 


50 


66 


60 




IcaK) 


1.0001 


1.0001 


1.0001 


1.0001 


1.0001 


89 


1.0004 


1.0004 


1.0005 


1.0005 


1.0005 


1.0006 


88 


1.0010 


1.0011 


1.0011 


1.0012 


1.0013 


1.0014 


87 


1.0019 


1.0020 


1.0021 


1X)022 


1.0023 


1.0024 


86 


1.0032 


1.0033 


1,0034 


1.0036 


1.0037 


1.0038 


85 


1.0048 


1.0049 


1.0050 


1.0052 


1.0053 


1.0055 


84 


1.00G6 


1.0068 


1.0070 


1.0071 


1.0073 


1.0076 


83 


1.0088 


1.0090 


1.0092 


1.0094 


1.0096 


1.0098 


82 


1.0113 


1.0116 


1.0118 


1.0120 


1.0122 


1.0125 


81 


1.0141 


1.0145 


1.0146 


1.0149 


1.0152 


1.0154 


80 


1.0173 


10176 


1.0179 


1,0181 


1.0184 


1.0187 


79 


10208 


1.0211 


1.0214 


1,0217 


1.0220 


1.0223 


78 


1.0246 


1.0248 


1.0253 


1,0260 


1.0260 


1.0263 


77 


1.0288 


1.0291 


1.0295 


1.0298 


1.0302 


1.0306 


76 


1.0333 


1.0337 


1.0341 


1.(345 


1.0349 


1.0353 


75 


1.0382 


10386 


1.0390 


1.0394 


1J0399 


1.0403 


74 


1.0434 


1.0438 


1.0443 


1,0448 


1.0452 


1.0457 


73 


1.0490 


1.0495 


1.0500 


1.0505 


1.0510 


1.0515 


72 


1.0660 


1.0656 


1.0560 


1.0565 


1.0571 


1.0577 


71 


1.0644 


1.0619 


1.0625 


1.0630 


1.06.36 


1.0642 


70 


1.0682 


1.0688 


1.0694 


1,0699 


1.0705 


1.0711 


69 


1.0764 


1.0760 


1.0T66 


1.0773 


1.0779 


1.0785 


68 


1.08S0 


1.0837 


1.0844 


1X)860 


1.0857 


1.0864 


67 


1.0911 


1.0918 


1.0925 


1.0932 


1.0939 


1.0946 


66 


1.0997 


1.1004 


1.1011 


1.1019 


1.1026 


1.1034 


66 


1.1087 


1.1096 


1.1102 


1.1110 


1.1118 


1.1126 


64 


1.1182 


1.1190 


1.1198 


1.1207 


1.1216 


1.1223 


63 


1.1282 


1.1291 


1.1299 


1.1308 


2.1317 


1.1326 


62 


1.1388 


1.1397 


1.1406 


1.1415 


1.1424 


1.1433 


61 


1.1499 


1.1508 


1.1518 


1.1628 


1.1537 


1.1547 


60 


1.1616 


1.1626 


1.1636 


1.1646 


1.1666 


1.1666 


59 . 


1.1739 


1.1749 


1.1760 


1.1770 


1.1781 


1.1792 


58 


1.1868 


1.1879 


1.1819 


1.1901 


1.1912 


1.1923 


57 


1.2004 


1.2015 


1.2027 


1.2039 


1.2050 


1.2062 


56 


1.2146 


1.2158 


1.2171 


1.2183 


1.2196 


1.2208 


55 


1.2296 


1.2309 


1.2322 


1.2335 


1.2318 


1.2361 


54 


1.2453 


1,2467 


1.2480 


1.2494 


L2508 


1.2521 


53 


1.2619 


1.26a3 


1.2647 


1.2661 


1.2676 


1.2690 


52 


1.2793 


1.2807 


1.2822 


1.283T 


1.2852 


1.2867 


51 


1.2976 


1.2991 


1.3006 


1,3022 


1.3038 


1.3054 


50 


1.3167 


1.3184 


1.3200 


1..3217 


1.3233 


1.3250 


49 


1.3;i69 


1.3386 


1.3404 


1.3421 


1.3439 


1.345G 


48 


1.3581 


1.3600 


1.3618 


1.3636 


1.3055 


l.,%73 


47 


1.3805 


1.3824 


1.3843 


1.3863 


1.3882 


1.3902 


46 


1.4040 


1.4056 


1.4081 


1.4101 


1.4122 


1.4142 


45 


26 


20 


15 


10 


5 





D€>g. 


Minutes. 





Natural Coseoaxt. 



y Google 



216 



TBIOONOHETBICAL FIT50TI0H3. 
Natural Skcamt. 





Minutes. 


Deg. 





















5 


10 


15 


20 


25 


30 


45 


1.4142 


1.4163 


1.4183 


1.4204 


1.4225 


1.4246 


1.4267 


46 


1.4395 


1.4417 


1.4439 


1.4401 


1.4483 


1.4505 


1.4527 


47 


1.4663 


1.4686 


1.4709 


1.4732 


1.4755 


1.4778 


1.4802 


48 


14945 


1.4969 


1.4993 


1.5018 


1.5042 


1.5007 


1.5092 


49 


1.5242 


1.5-268 


1.5-294 


15319 


1.5346 


15371 


1.5398 


50 


1.5557 


1.5584 


1.5011 


1.5639 


1.5666 


1.5694 


1.6721 


51 


1.5890 


1.5919 


1.5947 


1.5076 


1.6005 


1.6034 


1.6064 


52 


1.6243 


1.6273 


1.6303 


1.6334 


1.6366 


1.6396 


1.6427 


53 


1.6616 


1.6648 


1.0681 


1.6713 


1.6746 


1.6779 


1.6812 


54 


1.7013 


1.7047 


1.7081 


1.7116 


1.7151 


1.7185 


1.7220 


55 


1.7434 


1.7471 


1.7507 


1.7544 


1.7681 


1.7018 


1.7665 


56 


1.7883 


1.7921 


1.7960 


1.7999 


1.8039 


1.8078 


1.8118 


57 


1.8361 


1.8402 


1.8443 


1.8485 


1.8527 


1.8569 


1.8611 


58 


1.8871 


1.8915 


1.8959 


1.9004 


1.9048 


1.9093 


1.9139 


59 


1.9416 


1.9463 


1.9510 


1.9558 


1.9606 


1.9654 


1.9703 


60 


2.0000 


2.0050 


2.0102 


2.0152 


2.0204 


2.0256 


2.0308 


61 


2.0627 


2.0681 


2.0735 


2.0790 


2.0846 


2.0001 


2.0967 


62 


2.1300 


2.1359 


2.1418 


2.1477 


2.1536 


2.1590 


2.1657 


63 


2.2027 


2.2090 


2.2153 


Z2217 


2.2-282 


2.2346 * 


2.2411 


64 


2.2812 


2.2880 


2.2949 


2.3018 


2.3087 


2.3158 


2.32-'>^ 


65 


2.3602 


2.3730 


2.3811 


2.3886 


2.3961 


2.4037 


2.4U4 


66 


2.4580 


2.4C66 


2.4748 


2.48-29 


2.4912 


2.4995 


2.5078 


67 


2.5593 


2.5081 


2.5770 


2.5859 


2.5949 


2.C040 


2.6131 


68 


2.CG95 


2.0701 


2.0888 


2.6986 


2.7085 


2.7185 


2.7285 


69 


2.7904 


2.8010 


2.8117 


2.8225 


2.8334 


2.8444 


2.8554 


70 


2.9238 


2.9355 


2.9474 


2.9593 


2.9713 


2.9835 


2.9957 


71 


3.0715 


3.0846 


3.0977 


3.1110 


3.1244 


3.1379 


3.1515 


72 


a2301 


3.2506 


3.2653 


3.2801 


3.2951 


3.3102 


3.3256 


73 


3.4203 


3.4366 


3.4532 


3.4697 


3.4867 


3.5037 


3.5209 


74 


3.6276 


3.6464 


3.0051 


3.6840 


3.7031 


3.7224 


3.7420 


• 76 


3.8037 


3.8848 


3.9001 


3.9277 


3.9495 


3il7l6 


3.9939 


76 


4.1.336 


4.1578 


4.1824 


4.2072 


4.2324 


4.2579 


4.S836 


77 


4.4454 


4.4730 


4.5021 


4.5331 


4.5604 


4.5901 


4.6202 


78 


4.8097 


4.8429 


4.8765 


4.9106 


4.9452 


4.9802 


5.0158 


70 


5.2408 


5.2903 


5.3205 


5..3012 


5.4026 


64447 


5.4874 


80 


5.7588 


5.8007 


5.8554 


6.9049 


6.9554 


6.9963 


6.0688 


81 


C.3924 


6.4517 


6.5121 


6.5736 


6.6363 


6.7003 


6.7655 


82 


7.1853 


7.2604 


7.:«72 


7.4156 


7.4957 


7.5776 


7.6613 


83 


8.2055 


8.3C39 


8.4046 


8.5079 


8.6138 


8.7223 


8.8337 


84 


9.5668 


9.7010 


9.8391 


9.9812 


10.1270 


10.2780 


10.4390 


85 


11.4740 


11.6080 


11.8680 


12.0760 


12.2910 


12.6140 


12.7450 


86 


14.3350 


14.0400 


14.9580 


15.2900 


15.6370 


16.0000 


16.3800 


87 


19.1070 


19.6530 


20.2300 


20.8430 


21.4940 


22.1860 


22.9250 


88 


28.6540 


29.8990 


31.2570 


32.7450 


34.3820 


36.1910 


38.2010 


89 


57.2990 


62.6070 


68.7570 


76.3960 


85.9460 


98.2230 


114.5900 




60 


65 


60 


46 


40 


35 


30 


Deg. 


















Minutes. 



Natural Coseoaxt. 



y Google 



TBIGONOMETRICAL FUNCTIOHS. 
Natural Seoamt. 



217 



Minutes. 
















Deg. 


35 


40 


46 


60 


66 


60 




1.4288 


i.4;ao 


1.4331 


1.4352 


1.4374 


1.4395 


44 


1.4550 


1.4572 


1.4595 


1.4617 


1.4640 


1.4663 


43 


1.4825 


1.4849 


1.4878 


1.4897 


1.4921 


1.4946 


42 


1.511G 


1.5141 


1.5106 


1.5192 


1.5217 


1.6242 


41 


1.5424 


1.6450 


1.6477 


1.5503 


1.6630 


1.5557 


40 


1.6749 


1.5777 


1.5805 


1.5833 


1.5862 


1.5890 


39 


1.C093 


1.6123 


1.6153 


1.0182 


1.6212 


1.6243 


38 


1.6458 


1.6489 


1.6621 


1.6552 


1.6584 


1.6616 


37 


1.6845 


1.6878 


1.6912 


1.6945 


1.6979 


1.7013 


36 


1.7256 


1.7291 


1.7327 


1.7362 


1.7398 


1-7434 


36 


1.7693 


1.7730 


1.7768 


1.7806 


1.7844 


1.7883 


34 


1.8168 


1.8198 


1.8238 


1.8279 


1.8320 


1.8361 


33 


1.8654 


1.8697 


1.8740 


1.8783 


1.8827 


1.8871 


32 


1.9184 


1.9230 


1.9276 


1.9322 


1.9309 


1.9416 


31 


lJbl62 


1.9801 


1.9850 


L9900 


1.9950 


2.0000 


30 


2.0360 


2.0413 


2.0406 


2.0619 


2.0573 


2.0G27 


29 


2.1014 


2.1070 


2.1127 


2.1185 


2.1242 


2.1300 


28 


2.1717 


2.1778 


2.1840 


2.1902 


2.1964 


"2.2027 


27 


2L24T7 


2.2543 


2.2610 


2.2676 


2.2744 


2.2812 


26 


2.3299 


2.3371 


2.3443 


2.3516 


2.3588 


2.3002 


25 


2.4191 


2.4269 


2.4347 


2.4426 


2.4506 


2.4586 


24 


2.5163 


2.5247 


2.5333 


2.5419 


2.6606 


2.6593 


23 


2.6223 


2.6316 


2.6410 


- 2.6504 


2.6599 


2.6095 


22 


2.7386 


2.7488 


2.7591 


2.7694 


2.7799 


2.7904 


21 


2.8666 


28778 


2.8892 


2.9006 


2.9122 


2.9338 


20 


3.0081 


30206 


3.0331 


3.0458 


3.0586 


3.0715 


19 


3.1653 


3.1792 


3.1932 


3.2074 


3.2216 


3.2301 


18 


3.3400 


3.:i565 


3.3722 


3.3881 


3.4041 


3.4203 


17 


3.5383 


3.5559 


3.5736 


3.5916 


3.6096 


3.6279 


16 


3.7617 


3.7816 


3.8018 


3.8222 


3.8428 


3.8637 


15 


4.0165 


4.0394 


4.0625 


4.0859 


4.1096 


4.1336 


14 


4.3098 


4.3362 


4.3630 


4.3901 


4.4176 


4.4454 


13 


4.6507 


4.6817 


4.7130 


4.7448 


4.7770 


4.8097 


12 


6.0520 


6.0886 


5.1258 


6,1636 


6.2019 


6.2408 


11 


6.5308 


5.5749 


6.6197 


5.6653 


5.7117 


6.7588 


10 


6.1120 


6.1061 


6,2211 


6.2772 


6.3343 


6.3924 


9 


6.8320 


6.8U93 


6.9690 


7.0396 


7.1117 


7.1853 


8 


7.7469 


78344 


7.9240 


7,9971 


8.1094 


8.2055 


7 


8.9479 


9.0051 


9.1855 


9.3092 


9.4362 


9.5668 


6 


10.5930 


10.7580 


10.9290 


11.1040 


11.2080 


11.4740 


5 


12.9860 


13.2350 


13.4940 


13.7630 


14.0430 


14.3350 


4 


16.7790 


17.1980 


17.0390 


18.1030 


18.6910 


19.1070 


3 


23.7160 


24,5020 


26.4710 


20.1600 


27.5080 


28.6540 


2 


39.9780 


42.0760 


45.8400 


49.1140 


52.8910 


57.2990 


1 


137.6100 


171.8900 


229.1800 


343.7700 


687.5600 


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CIRCUMFERENCE, AREA, AKfi CUBIC CONTENTS OF CIRCLES. 223 



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221 



8FE0IFI0 aBAVITIES OF MATERIALS. 



SPECIFIC GRAVITIES OF MATERIALS. 



Gases at 32^ Fahr., and under the pressure of one 
atmosphere of 2116.4 lbs. on the square foot: 

Air 

Carbonic acid , 

Hydrogen 

Oxygen , 

Nitrogen 

Steam (ideal) 

iEther vapor fideal) 

Bisulphuret-ol-carbon vapour (ideal) 

Olefiant gas .* ^... 



Weight of 
a cubic foot 

in lbs. 
avoirdupoia 



0.080728 

0.12344 

0.005592 

0.080256 

0.078696 

0.05022 

0.2093 

2137 

0.0795 



Liquids at 32® Fahr. ^except water, 
which is taken at 39°.4 Fahr.): 

Water, pure, at 39°.4 

" sea, ordinary 

Alcohol, pure 

•• proof spirit 

.^ther 

Mercury 

Naphtha 

Oil, linseed 

" olive 

•' whale 

•' of turpentine 

Petroleum 

Solid Mineral Substances, non- 
metallic : 

Basalt.. 

Brick 

Brickwork 

Chalk 

Clay 

Coal, anthracite 

•* bituminous 

Coke 

Felspar 

Flint 



Weight of a 

cubic foot in lbs. 

avoirdupois. 



62.425 
64.05 
49.38 
57.18 
44.70 

848.75 
52.94 
58.68 
57.12 
57.62 
54.31 
54.81 



187.3 

125 to 135 

112 

117 to 174 

120 

100 

77.4 to 89.9 

62.43 to 103.6 

162.3 

164.2 



Specific 

gravity, pare 

water — » 1. 



1.000 
1.026 
0.791 
0.916 
0.716 
13.596 
0.848 
0.940 
0.915 
0.923 
0.870 
0.878 



3.00 

2 to 2.167 

1.8 

1.87 to 2.78 

1.92 

1.602 

1.24 to 1.44 

l.OOtol.66 

2.6 

2.63 



d by Google 



SPEOinO GEAVITIES OP MATERIALS. 



225 



Solid Mineral Substances — con- 
tinued : 

Glass, crown, average 

" flint 



" green 

" plate 

Granite 

Gypsum , 

Limestone, (including marble).. 

" magnesian 

Marl.. ..;. 

Masonry 

Mortar 

Mud 

Quartz 

Sand (damp) 

" (dry) -J 

Sandstone, average 

•' various kinds 



Slate., 
Trap.. 



Metals, solid: 
Brass, cast 

" wire 

Bronze 

Copper, cast 

*• sheet 

" hammered 

Gold 

Iron, cast, various 

" average 

Iron, wrought, various.. 
'* average.. 

Lead 

Platinum 

Silver 

Steel 

Tin 

Zinc 



Timber:* 
Ash 

Bamboo . 
Beech. 



15 



Weight of a 

cubic root in lbs. 

avoirdupois. 



156 
187 
169 
169 

164 to 172 
143.6 

169 to 175 
178 

100 to 119 

116 to 144 
109 
102 
165 
118 
88.6 
144 

130 to 157 
162 

175 to 181 
170 



487 to 524.4 

533 

524 

537 

549 

556 

1186 to 1224 

434 to 456 

444 

474 to 487 

480 

712 

1311 to 1373 

655 

487 to 493 

456 to 468 

424 to 449 



47 
43 



Specific 

gravity, pure 

water <— 1. 



2.5 
3.0 
2 7 
2.7 
2.63 to 2.76 
2.3 

2.7 to 2.8 

2.86 

1.6tol.9 

1.85 to 2.3 

1.75 

1.63 

2.65 

1.9 

1.42 

2.3 

2.08 to 2.52 

2 6 

2.8 to 2.9 

2.72 



7.8 to 8.4 

8.54 

8.4 

8.6 

8.8 

8 9 

19tol9!6 

6.95 to 7.3 

7.11 

7.6 to 7.8 

7.69 

11.4 

21 to 22 

. 10.5 

7.8 to 7.9 

7.3 to 7.5 

6.8 to 7.2 



0.753 

0.4 

0.6a 



d by Google 



226 



8PECIPIC GBAVITIE8 OP MATEBIALS. 



Timber :*— continued. 

Birch 

Blne-gnm 

Box ^ 

Bullet-tree 

.Cabacalli 

Cedar of Lebanon 

Chestnnt 

Cowrie 

Ebony, West Indian 

Elm 

Fir, red pine 

" sprnce 

•' American yellow pine.. 

" larch 

Greenhart..^ 

Hawthorn 

Hazel 

Holly 

Hornbeam 

Laburnnm 

Lancewood 

Larch. (See "fir".) 

Lignum- vitaB....i 

Locust 

Mahogany, Honduras 

" Spanish 

Maple 

Mora 

Oak, European 

•* American red 

Poon 

Saul.. 



Sycamore 

I'eak, Indian.., 
" African . 

Tonka 

Water-gum 

Willow 

Yew 



Weight of a 

cubic loot in lbs. 

avoirdupois. 



41.4 
52.5 

60 
65.3 
56.2 
30.4 
33.4 
36.2 
H.5 

34 
30 to 44 

30 to 44 

29 

31 to 35 
62.5 

57 
54 
47 
47 
57 

42 to 63 

41 to 83 
44 
35 
53 
49 
67 

43 to 62 

54 
36 
60 
37 

41 to 56 
61 

62 to 66 

62.5 

25 

50 






SpC«fflc 

grsrity, pure 

water — 1. 



0.711 

0.843 

0.96 

1.046 

0.9 

0.48^ 

0.135 

#.579 

1.193 

0.544 

0.48 to 0.7 

0.48 to 0.7 

0.46 

0.5 to 0.56 

1.001 

0.91 

0.86 

0.76 

0.76 

0.92 

0.675 to 1.01 

0.65 to 1.33 
0.71 
0.56 
0.85 
0.79 
0.92 

0.69 to 0.99 
0.87 
0.58 
0.96 
0.59 

0.66 to 0.88 
0.98 

0.99tol.06 

1.001 

0.4 

0.8 



*The timber in erery caM ii supposed to be dry. 



y Google 



WKOfiT OF A(8UPBB9I0£&L IHCffi, BTO. 



227 



WEIGHT OF A SUPERFICIAL INCH OF WROUGHT AOT 
CAST IRON. 

(From one-iixt«6B4h to one-lnoh tbicknesa.) 



B 


WBOU0HT IbOH. 


Cast Ibov. 




Cubic foot = 480 lbs. 


Cubic foot =: 450 lbs. 


B% 






1*^ 


Weight In lbs. 


Weight is lbs. 

/ / ... 








A 


0.017356 


0.0163 


i 


0.0347 


0.0326 


A 


0.0520 


0.0489 


i 


0.0694 


0.0652 


A 


0.0867 


0.0816 


* 


0.1041 


O.0978 


A 


0.1214 


0.1141 


4 


0.1388 


0.1304 


ft 


0.1562 


0.1467 


1 


0.1735 


0.1630 


tt 


0.1909 


0.179S 


t 


0.2082 


0.1956 


« 


0.2256 


0.2119 


t 


0.2429 


0.2282 


« 


0.2603 


0.2445 


1 


0.2777 


0.2608 



y Google 



223 



WSiaST PEB S^UABB FOOT TS POTTHDB AYOIBDUFOIS. 



WEIGHT PER SQUARE FOOT IN POUNDS AVOIRDUPOIS. 





Wrought 
Iron. 


Cast Iron. 


Copper, 
sheet 


Lead. 


Zinc 


SB 


480 lbs. per 
cubic foot 


450 lbs. per 
cubic foot. 


549 lbs. per 
cubic foot 


712 lbs. per 
cubic foot 


436 lbs. per 
cubic foot 


A 


2.60 


2.34 


2.86 


3.71 


2.27 


i 


5.00 


4.69 


5.72 


7.42 


4.54 


A 


7.50 


7.03 


8.58 


11.12 


6.81 


t 


10.00 


9.37 


11.44 


14.83 


9.08 


A 


12.50 


11.72 


14.30 


18.54 


11.35 


* 


15.00 


14.06 


17.16 


22.25 


13.62 


A 


17.50 


16.41* 


20.02 


25.96 


15.89 


i 


20.00 


18.75 


22.88 


29.66 


18.16 


A 


22.50 


21.09 


25.74 


33.37 


20.43 


t 


25.00 


23.44 


28.60 


37.10 


22.70 


H 


27.50 


25.78 


31.46 


40.79 


24.97 


f 


30.00 


28.12 


34.32 


44.50 


27.24 


H 


32.50 


30.47 


37.18 


48.20 


29.51 


* 


35.00 


32.81 


40.04 


51.91 


31.78 


« 


37.50 


35.16 


42.90 


55.62 


34 05 


1 


40.00 


37.50 


45.75 


59.33 


36.33 



y Google 



\?BIGHT OP A LIHEAL POOT, ETC. 



229 



WEIGHT OF A LINEAL FOOT OF FLAT AND SQUARE 
BAR IRON IN POUNDS AVOIRDUPOIS. 

(480 pounds per cubic foot) 



<D 2 
art 




0.104 
0.208 
0.208 
0.416 
0.832 
0.312 
0.624 
0.937 
1.249 
1.562 
1.874 
0.416 
0.833 
1.249 
1.667 
2.089 
2.500 
2.916 
3.333 
0.521 
1.041 
1.562 
2. 089 
2.603 
3.124 
3.646 
4.166 
4.687 
6.728 
0.624 
1.250 
1.875 
2.500 
3.125 
3.750 
4.375 



n 



If 



2t 






5.000 


21 


^ 


1.875 


5.625 






2.813 


6.250 


»« 


. . 


3.750 


6.874 


•« 




4.687 


7.500 


it 




5.624 


0.739 


i< 




6.562 


1.459 


II 




7.500 


2.187 


i< 


Ii 


8.437 


2.916 


t( 




9.374 


3.646 


i< 


1 


10.310 


4.375 


ii 


IJ 


11.250 


5.103 


it 


1- 


12.190 


5.833 


«t 


1^ 


13.120 


6.562 


»• 


1 


14.060 


7.291 


It 


o 


15.000 


8.020 


ft 


2i 


15.940 


8.750 


»• 


2} 


17.810 


9.478 


2J 


1.041 


10.930 


it 




2.089 


833 


It 




3.125 


1.667 


ft 


, . 


4.166 


2.500 


•* 




5.208 


3.333 


«l 


; . 


6.250 


4.166 


It 




7.291 


5.000 


It 




8.333 


5.833 


II 


1 


9.398 


6.666 


" 


10.410 


7.500 


It 


1 


11.460 


8.333 


ft 


1; 


12.500 


9.156 


It 




13.540 


10.000 


" 


1 


14.580 


10.830 


It 


If 


15.620 


11.660 


It 


2 


16.660 


12.500 


ft 


2J 


17.710 


13.330 


tt 


2 
2J 


18.750 


0.937 


tt 


20.820 


Dig 


tized by V_ 


:.UU>^ 


l^C 



WKBKT OF A USXA& IVWi; ] 



3} 



19 

1 

2 

3 

4 

5 

6 

8 

9 

10 

11 

12 

13 

14 

16 

17 

18 

19 

20 

21 

22 

24 

25 

2 

6 

7 

10 
12 
16 
17 
20 
22 
25 
27 
30 
2 
5 
8, 
10. 
13. 
16. 
18. 



.800 
146 

.292 
437 
.583 
.729 
.874 
.020 
.154 
.310 
.460 
.600 
.750 
.900 
.030 
.190 
.330 
.480 
.620 
.770 
.910 
.060 
.200 
.600 
.000 
.500 
.000 
.500 
.000 
.500 
.000 
.500 
.000 
.500 
.000 
.708 
416 
124 
830 
600 
250 
950 



H 



3J 



H 



2i 
2| 
3 

H 

i 
i 
} 

1 

n 

!| 

2 
2J 



1 
U 

II 

2 

^ 

2| 

3 

31 

It 



1 
u 



21.660 
24.370 
27.080 
29.790 
32.600 
24.200 
2.916 
5.833 
8.750 
11.660 
14.580 
17 500 
20.430 
23.330 
26.250 
29.160 
32.080 
35.000 
37.910 
40.830 
3.125 
6.250 
9.375 
12.500 
15.620 
18.750 
21.870 
25.000 
28.120 
31.250 
34.370 
37.500 
40.620 
43.750 
46.860 
3.330 
6.660 
10.000 
13.830 
16.660 
20.000 
23.330 



4 ' 



H 



4i 



2 


26.660 


^ 


30.000 


2} 


33.330 


^ 


36.660 


3 


40.000 


31 


43.330 


^ 


46.660 


3f 


50.000 


4 


63.330 


\ 


3.541 


\ 


7.082 


10.620 


1 


14:160 


u 


16.800 


H 


21:330 


If 


24.780 


2 


28.330 


2J 


31.870 




35.410 


2f 


38.950 


3 


42.500 


3i 


46.030 


3i 


49.570 


3} 


53.120 


4 


56.660 


4} 


60.200 


; • 


3.750 


1 * 


7.500 


11.250 


1 


15.000 


\\ 


18.750 


u 


22.500 


1} 


26.250 


2 


30.000 


2| 
2} 


33.750 


37.500 


2f 


41.250 


3 


46.000 


3} 


48.750 


3| 

3} 


52.600 


66.250 


4 


60.000 



d by Google 



WEIGHT OP h LUTEAL FOOT, ETa 





a 






i^ 


1% 




d 

is 


Is 


%a 




11 


£.*«! 


JS 






« 


H 




« 


4J 


4V 


63.750 


51 




^ 


67.500 




4i 


\ 


3.953 






T 


7.910 




II 


a 


11.860 




l< 


1 


15.830 




II 


u 


19.760 




II 


^ 


23.750 




II 


If 


27.700 




fl 


2 


31.670 




II 


2J 


35.620 




(1 


2i 


39.580 




II 


2J 


43.540 




II 


3 


47.500 




II 


H 


51.460 




(( 


3 


55.410 




II 


3 


59.370 




tl 


4 


63.330 




II 


4t 


67.290 




II 


4 


71.250 




II 


4 


75.200 


5t 


& 


; . 


4.166 




II 


, . 


8.330 




«« 




12.500 




II 


1 


16.660 




II 
II 
11 


1 
1 


20.830 
25.000 
29.160 




l< 


2 


33.330 




11 


21 


37.500 




1« 


2i 


41.660 




II 


21 


45.830 




tl 


3 


50.000 




II 


3t 


54.160 




II 


3} 


68.330 




II 


3i 


62.500 




II 


4 


66.660 




II. 


^ 


70.830 




II 


^ 


75.000 




«l 


4} 


79.160 




II 


5 


83.330 




6t 


I 


4.376 





i 
■i 
1 

u 
IJ 

2 

2t 

2} 

21 

3 

3i 

H 

3f 

4 

4t 

P 

i 
i 

H 
I* 

3 
I 

4 

t* 

4; 
5 

5J 
6} 



8.753 


51 


X 


4.788 


13.130 






j 


9.587 


17.500 






14.370 


21.870 






1 


19.160 


26.250 






IJ 


23.950 


30.620 








28.750 


35.000 






33.540 


39.370 






2 


88.330 


43.750 






2J. 


43.120 


48.110 








47.910 


52.500 






52.700 


56.680 






3 


57.500 


61.250 






3} 


62.300 


65.620 






3i 


67.080 


70.000 






3| 


71.860 


74.370 






4 


76.650 


78.750 






^ 


81.450 


83.110 






4} 


86.240 


87.500 






4| 


91.030 


91.860 






5 


95.820 


4.587 






^ 


100.600 


9.164 






5J 


105.400 


13.750 






5f 


119.700 


18.330 






1 '■ 


10:000 


22.900 






1 


20.000 


27.500 






H 


30.000 


32.080 






2 


40.000 


36.660 






2J 


50.000 


41.250 






3 


60.000 


45.830 






3J 


70.000 


50.310 






4 


80.000 


55.000 






4} 


90.000 


59.570 






6 


100.000 


64.160 






5} 


110.000 


68.740 






6 


120.000 


73.330 


e 


i 


i 


10.830 


77.910 






1 


21.660 


82.500 






n 


32.500 


87.080 






2 


43.330 


91.560 






2i 


54.160 


96.240 






3 


65.000 


100.600 






Si 


76.830 



Digitized by VjUU^ It: 



232 



WEIGHT OF A LIVEAL FOOT, ETC. 





a 


d 




d 


d 




c 


S 






£ 






£ 






^ 


is 


u 


4« 


a 

is 


u 


^ 


a 

5| 


s<» 


a 


So 


5 « 




Is 


1!| 
2 


X3 




1-2 


1 


« 


H 


^ 


n 


H 


^ 


a" 


^ 


f^ 


6} 


4 


86.66 


8 


4 


106.60 


9 


255.00 


(( 


4i 


97.50 




* 


4J 


120.00 




9 


270.00 


i< 


5 


108.30 




♦ 


5 


133.30 


9J 


i 


15.83 


«« 


5} 


119.10 




1 


5} 


146.60 




1 


31.66 


(i 


6 


130.00 




1 


6 


160.00 




H 


47.50 


(1 


6i 


140.80 




1 


6} 


173.30 




2 


63.33 


7 


1 


11.66 




i 


7 


186.60 




2} 


79.16 


II 


1 


23.33 




* 


^ 


200.00 




3 


95.00 


«i 


ij 


35.00 




• 


8 


213.30 




3J 


110.80 


II 


2 


46.66 


fi 


li 


i 


14.16 




4 


126.60 


i« 


2} 


58.33 






1 


28.33 




4} 


142.50 


II 


3 


70.00 




' 


H 


42.48 




5 


158.30 


i< 


3i 


81 66 




' 


2 


56.66 




5i 


174.10 


II 


4 


93.33 




• 


2J 


70.83 




6 


190.00 


K 


4J 


105.00 




' 


3 


85.00 




6J 


205.80 


l< 


5 


116.60 




i 


3J 


99.16 


«« 


7 


221.60 


II 


5} 


128.30 




» 


4 


113.30 




7i 


237.60 


II 


6 


140.00 




t 


4i 


127.50 




8 


253.30 


II 


6J 


151.60 




tl 


5 


141.60 




8J 


269.10 


l< 


7 


163.30 




» 


6i 


155.80 


II 


9 


285.00 


^ 


} 


12.50 




II 


6 


170.00 




9} 


300.80 


i( 


1 


25.00 




" 


6} 


184.10 


10 


1 


16.66 


11 


U 


37.50 




CI 


7 


198.30 




1 


33.33 


II 


2 


50.00 




II 


7J 


212.50 




u 


50.00 


II 


2} 


62.50 




•« 


8 


226.60 




2 


66.66 


II 


3 


75.00 


11 


8} 


240.70 




2J 


83.33 


II 


3i 


87.50 


9 


i 


15.00 




3 


100.00 


II 


4 


100.00 


M 


1 


30.00 




3i 


116.60 


II 


4i 


112.50 




li 


45.00 




4 


133.30 


II 


5 


125.00 




2 


60.00 




4i 


150.00 


II 


5i 


137.50 




2i 


75.00 




5 


166.60 


II 


6 


150.00 




3 


90.00 




5} 


183.30 


K 


6J 


162.50 




3} 


105.00 




6 


200.00 


•1 


7 


175.00 




4 


120.00 




6} 


216.60 


II 


7J 


187.50 




4J 


135.00 




7 


233.30 


8 


2^ 


13.33 




5 


150.00 




7J 


250.00 


II 


1 


26.66 




5} 


165.00 




8 


266.60 


11 


ij 


40.00 




6 


180.00 




8J 


283.30 


II 


2 


53.33 




6} 


195.00 




9 


800.00 


«i 


2J 


66.66 




7 


210.00 




9J 


316.60 


II 


3 


80.00 




7} 


225.00 




10 


333.30 


II 


3J 


93.33 




i 


8 


240.00 


lOJ 


^ 


17.50 



y Google 



WEIdHT OP A LINEAL FOOT, ETC. 



233 








ID 




o 


X 




d 


« 




•M 


£ 






£ 






^ 


d 


s« 


fl 


d 


«S 


a 


d 


S" 


d 


is 


s^- 




5| 


§.s 


•4. 


5| 


£| 




HaJB 

«8 « 


|.s 


1 


■§1 


1.2 


§ 


«l 


2.2 


1, 


®.S 






o a 






« d 




•Sr 




!S 


o 


U"^ 


!a 


? 


t,-'^ 


3 


P 


«*" 


Eh 


^ 


« 


H 


^ 


PQ 


H 


^ 


lOJ 


1 


35.00 


11 


IJ 


55.00 


lU 


li 


57.50 


i» 


IJ 


52.50 




2 


73.33 


" 


2 


76.66 


u 


2 


70.00 




2J 


91.56 


•I 


^ 


95.83 


i( 


2J 


87.50 




3 


110.00 


*• 


3 


115.00 


(i 


3 


105.00 




3i 


128.30 


(C, 


3i 


134.10 


tl 


3i 


122.50 




4 


146.60 


(1 


4 


153.30 


<l 


4 


140.00 




4} 


165.00 


i( 


4i 


172.50 


ft 


4J 


157.50 




5 


183.30 


li 


5 


191.60 


t( 


5 


175.00 




5} 


201.60 


(1 


5i 


210.80 


•u 


5i 


192.50 




6 


220.00 


(1 


6 


230.00 


<l 


6 


210.00 




^ 


238.30 


li 


•6i 


249.10 


l( 


6J 


227.50 




1 


256.60 


ti 


7 


268.30 


II 


7 


245.00 




n 


275.00 


It 


7i 


287.50 


li 


7i 


262.50 




8 


293.30 


*• 


8 


306.60 


11 > 


8 


280.00 




8J 


311.60 


i< 


8i 


325.80 


II 


8i 


297.50 




9 


330.00 


II 


9 


345.00 


•» 


9 


315. m) 




9} 


348.30 


li 


9i 


364.10 


It 


9i 


332.50 




10 


366.60 


II 


10 


383.30 


II 


10 


350.00 




10} 


385.00 


ti 


10} 


402.50 


II 


10} 


367.50 




11 


403.30 


It 


11 


421.60 


11 


J 


18.33 


Hi 


} 


19.16 


It 


Hi 


440.70 


II 


1 


36.66 


(1 


1 


38.33 


12 


12 


480.00 



y Google 



284 



WEIQHT or A UREAL F001^, BTO. 



WEIGHT OF A LINEAL FOOT OF ROLLED ROUND 
IRON IN POUNDS AVOIRDUPOIS. 

(480 pounds fmv cubic foot) 



52 



^ 



1-^ 



1-2 
3 



^ 









t 
t 

t 
t 
t 



U 
H 
If 
i| 
i| 
if 
i| 

2 
21 



0.010 
0.041 
0.091 
0.163 
0.255 
0.3*68 
0.501 
0.655 
0.828 
1.022 
1.237 
1.473 
1.728 
2.004 
2.301 
2.618 
3.310 
4.094 
4.950 
5.885 
6.911 
8.018 
9.205 
10.470 
11.820 
13.250 



14.77 

16.36 
18.04 
19.80 
21.64 
23.56 
25.56 
27.64 
29.82 
32.07 
34.39 
36.81 
39.30 
41.88 
44.57 
47.28 
50.10 
53.02 
56.03 
59.05 
62.17 
65.49 
68.71 
72.13 
75.65 
79.17 



82.79 
86.52 
90.34 
94.26 
98.18 
102.20 
106.40 
110.60 
114.9^ 
119.30 
123.-70 
128.30 
132.90 
137.60 
142.30 
147.30 
152.20 
157 20 
162.40 
167.50 
172.80 
178.20 
183.60 
189.10 
194.80 
200.40 



9} 
10 
lOJ 

m 
m 

loi 
io| 
lol 
io| 

II 

lU 
lU 

111 

111 
111 

11* 
llj 

12 



206.2 
21*2.2 
21«.0 
22^.9 
230.1 
236.2 
242.5 
24«.9 
255.2 
261.7 
288.4 
275.0 
281.8 
288.6 
2!i5.6 
302.5 
309.5 
316.8 
323.9 
331.3 
338.7 
346.2 
353.7 
301.5 
369.1 
376.9 



y Google 



BOLTS^ AUTB, AXIk HBABS. 

BOLTS. NUTS, ANB HEADS. 
(Whitworih^fl PropOTtiona.) 









Weight in lbs 


. of Heads and Nuts. 






t4 


Hexagonal. 


Square. 


Hexagonal. 


Square. 


13 


























ll 


Head, 


Nut. 


Head. 


Nut. 


Two 


Head 

A Nut. 


Two 
Heads. 


Head 

A Nut. 


i 


0.008 


0.005 


0.022 


0.019 


0.017 


0.013 


0.044 


0.041 


i^ 


0.014 


0.007 


0.027 


0.021 


0.029 


0.022 


0.056 


0.048 


0.029 


O.OlV 


0.061 


0.049 


0.057 


0.046 


0.122 


0.110 


t 


0.069 


0.040 


0.069 


0.050 


0.119 


0.101 


0.138 


0.119 


0.068 


0.041 


0.104 


0.076 


0.136 


0.109 


0.208 


0.181 




O.104 


0.065 


0.157 


0.118 


0.208 


0.169 


0.316 


0.276 


1 


0.151 


0.097 


0.246 


0.193 


0.302 


0.248 


0.493 


0.440 


4 


0.254 


0.161 


0.362 


0.269 


0.508 


0.415 


0.724 


0.631 


1 


0.367 


0.219 


0.551 


0.408 


0.734 


0.586 


1.102 


0.969 




0.546 


0.326 


0.683 


0.463 


1.092 


0.872 


1.366 


1.146 






0.724 


411 


1.109 


0.797 


1.448 


1.136 


2.217 


1.906 






1.060 


0.630 


i.^m 


0.971 


2.120 


1.690 


2.800 


2.371 






1.330 


0.759 


1.949 


1.379 


2.660 


2.088 


3.898 


3.328 






1.840 


1.098 


2.625 


1.883 


3.680 


2.938 


5.260 


4.508 




. 


2.460 


1.517 


3.136 


2.192 


4.920 


3.977 


6.270 


6.327 






2.920 


1.742 


3.704 


2.532 


6.840 


4.662 


7.409 


6.236 




. 


3.440 


1.991 


4.725 


3.276 


6.880 


6.431 


9.450 


8.001 


2 


4.370 


2.611 


6.384 


4.625 


8.740 


6.981 


12.77 


11.00 


H 


6.150 


3.645 


.8.858 


6.353 


12.30 


9.795 


17.71 


16.21 


4 


8.480 


5.045 


11.91 


8.476 


16.96 


13.62 


23.82 


20.39 


4 


11.3^ 


6.747 


16:. 59 


9.019 


22.64 


18.06 


31.18 


24.61 


3 


14.72 


8.783 


21.00 16.06 


29.44 


23.60 


42.00 


36.06 



y Google 



236 



WEIGHT. nrpouiTDa or eouwd jboh, etc. 



WEIGHT IN POUNDS OF ROUND IRON FOR 



i:« 



5« 



Length in inches. 



Vs 



% 



Ji 



0.002 

0.003 

0.004 

0.005 

0.007 

0.009 

0.011 

0.015 

0.021 

0.0270. 

0.035 

0.043 

0.053 

0.062 

0.072 

0.084 

0.097 

0.11] 

0.140 

0.174 

0.209 

0.250 



'.003 

,005 

,0070, 

.0100. 

.014 

,017 

0220. 

031 

042 

055 

070 

.087 

104 

124 

143 

168 

194 

221 



0.2800. 
0, 



347 
418 
5000. 



0.005 
0.008 

Oil 

.016 
0.021 
0.026 

032 
0.046 
0.063 
083 
0.105 
0.131 
0.157 
0.186 
0.2150. 
0.253 
0.291 
0.3320. 

.420 
0.521 
0.627 

750 



0070, 



Oil 

015 

021 

,027 

035 

,043 

062 

,084 

1100. 

140 

173 

2090, 

249 

287 

337 

389 

,442 

660 

695 

8361 

,000 



».008 
0.013 
0.019 
0.026 
0.034 
0.043 
0.054 
0.077 
0.105 

.138 
0. 
0.; 

•.261 
0.311 
0.358 
0.421 
0.486 
0.553 
0.700 
0.869 

.045 
1.250 



1850. 
2170 



0.0100. 

0.0160. 

0.023 

0.031 

0.041 

0.0520, 

0.0650. 

0.093 

0.126 

0.1650. 

1.2100. 

.2620. 
0.314 
0.373 
0.4300. 
0.506 
0.583 
0.663 
0.8400. 
1.042 
1.254 
1.500 



0120.014 
.021 



.0190. 



L027 
•.0360. 



.048 

.061 

.076 

.108 

.148 

,193 

,2450. 

,304 
0.3660. 
0.435 

5020. 
0.5900. 
0.6800. 
0.7740. 

980 

216 

463 

760 



0.031 

,0420. 
0.055 
0.069 
0.087 
0.124 
0.1700. 
0.221 

2800. 
0.347 

.418 
0.497 

.584 

.677 

•.778 

.884 
1.1202 
1.390 
1.673 
1.9903. 



027 
.043 
062 
,084 



0.041 
0.064 
0.093 
0.126 



0.1100.166 
0. 



139 
174 
249 
,338 
,442 
,560 
,695 
836 
995 
168 
354 
,655 
770 



0.208 
0.261 
0.373 
0.508 
0.663 
0.840 
1.043 
1.265 
1.493 
1.752 
2.032 
2.333 
2.654 



2403.360 



781 
346 
981 



4.172 
5.019 
5.972 



Example. — Recruited, the weight of a bolt 1 J inches diameter, 
4 inches between inside of head and nnt. 

Weight of bolt = 1.39 
Weight of square head ■= 1.40 
Weight of hexagonal nut = 1.06 taken as a hexagonal head 

Ans. 3.85 lbs. 



y Google 



WEIGHT nr POUHDS OP EOUKD lEOK, BTO. 



m 



BOLTS, ETC., BETWEEN HEAD AND NUT. 



ii 


Length in inches. 


It 

g a 


4 


6 


6 . 


7 


8 


9 


10 


11 


12 


J 


0.055 


0.069 


0.082 


0.096 


0.110 


0.124 


0.137 


0.161 


0.165 


A 


0.086 


0.107 


0.128 


0.150 


0.171 


0.192 


0.214 


0.235 


0.25'? 


f 


0.124 


0.155 


0.186 


0.217 


0.248 


0.279 


0.311 


0.342 


0.37a 


"A 


0.167 


0.209 


0.251 


0.293 


0.335 


0.377 


0.419 


0.461 


1 ■ 


0.221 


0.276 


0.331 


0.386 


0.442 


0.497 


0.552 


0.607 


0.663[ 


1 r 


0277 


0.347 


0.416 


0.486 


0.555 


0.624 


0.694 


0.763 


0.833 




0.347 


0434 


0.521 


0.608 


0.695 


0.782 


0.869 


0.956 


1.043 




0.497 


0.622 


0.746 


0.871 


0.995 


1.119 


1.244 


1.368 


1.493 




0.677 


0.846 


1.016 


1.X85 


1.354 


1.524 


1.693 


1.862 


2.032 


1 - 


0.884 


1.105 


1.326 


1.548 


1.769 


1.990 


2.211 


2.432 


2.654 


IJ 


1.120 


1.400 


1.680 


1.960 


2.240 


2.520 


2.800 


3.080 


3.360 


li 


1.390 


1.738 


2.085 


2.433 


2.781 


3.128 


3.476 


3.823 


4.172 


If 


1.673 


2.091 


2.510 


2.928 


3.346 


3.765 


4.182 


4.601 


6.019 


li 


1.990 


2.488 


2.985 


3.483 


3.981 


4.478 


4.976 


4.973 


5.972 


If 


2.336 


2.920 


3.504 


4.088 


4.673 


5.257 


5.841 


6.425 


7.010 


1 


2.709 


3.386 


4.064 


4.741 


5.418 


6.096 


6.773 


7.450 


8.128 


l| 


3.111 


3.888 


4.666 


5.334 


6.221 


6.999 


7.777 


8.547 


9.333 


2 


3.538 


4.423 


6.307 


6.192 


7.077 


7.961 


8.846 


9.730 


10.610 


2} 


4.480 


5.600 


6.720 


7.840 


8.960 


10.080 


11.200 


12.320 


13.440 




5.562 


6.953 


8.343 


9.734 


11.120 


12.610 


13.910 


15.290 


16.690 


6.692 


8.365 


10.040 


11.710 


13.380 


15.060 


16.730 


18.400 


20.070 


8 


7.962 


9.953 


11.940 


13.930 


15.920 


17.910 


19.910 


21.890 


23.890 



y Google 



WIBiaHT OF KATBBIALS USED IH BVILDnrO. 

WEIGHT OF MATERIALS USED IN BUILDING. 

tPer nqnarefoot from one inch thickness to a onbic foot.) 

St(me$t Earths^ &c. 



Stones^ Sarthst <&e. 



y Google 



DIVISIONS OF A FOOT, ETC. 



OQ 

o 
p 

H 

W 

Hi 

I— I 



Oh 
M 

O 
O 



PC4 

O 

GQ 

O 
I— 1 
OQ 

I— I 
P 



W 



C0t^t^00QO'-<C<lC0C0»0«5CDt^000> 



£5'^'^i5^'=C)OOa>QOi-iMcr5'^iOCD 
OO^t^Oii-lcOlOt^OC^ "^S 00 o ca S 

CO CO tH T»< »0 in CD <fflt»-^ 00 0$ 05 o o ^ 
OOOOQOQOOOOOOOOOOOQOQO-QOOOOiaaaa 



r-l «-< C^ CO ^ »j 

lOC _ . >^ ^^ 

iClOcScOt^J^-OOOOOSOaOOi-ir-iCMC^ 
Jr»t*t*t^t*£*t«»t>-£^r^0O0O0O0O0OOO 



>oooc5njQOc3oo?JiSi>a>»-5 

iOCDf-'COr-«CD<Mt'-(Ml>fMOO 



COt^t^OpOCi-KMC^lCOTjilCCDt-OOa) 

coi>i^-ooooo5a>oo^^<M<SiMco^ 



i^sssas 

coooc6oo^®Tj<a>ioouboiO»^coi-i 
coooai0500»-if-«(Ncoco'«t<'^miocD 

lOlOuDUDCOCOCCCOCOCDCOQDCOCDCO^ 



i-H^^O^CO-^AOCOCOI^-OOOOT-tOiCO 
<M'^<;0000<NT*<COQOOC?5Si>CDr-H 
lOO»OOCO»-<^'-^CO<NI>.(NJC«-CNIOO 



COJC^JC^OOOOi-KNCOCOIdOCDI^OOOi 
CD00O<N»Ct^05»-iC0»0^^05r-4C0l0t* 
CDi-tl>C<H>.<Mt^CO00C00OCOOSTHO>'^ 
T-i(M(MC0eO^THlO»O':OC01>J>000005 
^T* ^t* ^T* ^T* ^T* ^^ "^^ ^T* 'm' "^^ "^^ ^T^ ^iH "^^ ^s^ ^^ 



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cpooocq»5i>OirHeo»Ot*05i-HeoiOi> 

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gitizedby^^UUyH:: 



240 



TABLE FOB OOMFABING ICEASUBES AND WEIGHTS. 



TABLE FOR COMPARING MEASURES AND WEIGHTS 
OF DIFFERENT COUNTRIES. 

Weights. 



Umitco 
States and 
Eholand. 


Pewbia. 


AVBTBIA. 


Baden AifD 
Switzerland. 


Fbanob. 


Pound. 


Poand, Z. F. 


Pound. 


Pound. 


Kilogra'e. 


1 
1.1023 
1.2346 
1.2346 
2.2046 


0.9072 

1 
1.1200 
1.1200 
2.0000 


0.8100 
0.8928 
1 
0.9999 

1.7857 


Same as 
Prussia. 


0.4536 
0.5000 
0.5600 
0.5600 

1 



Measures of Length. 



Foot. 


Foot 


Foot. 


Foot. 


Meter. • 


— 12 inches. 
1 
1.0297 
1.0371 
0.9843 
3.2809 


— 12 inches. 
0.9711 

1 
1.0072 
0.9559 
3.1862 


— 12 inches. 
0.9642 
0.9929 

1 
0.9490 
3.1635 


— 10 inches. 
1..0160 
1.0462 
1.0537 

1 
3.3333 


- 100 Centi. 
0.3048 
0.3138 
0.3161 
0.3000 
1 



Measures of Surface— Square Measure. 



Square foot. 


Square foot. 


Square foot 


Square foot 


Sq. Meter. 


1 

1.0603 

1.0756 

0.9688 

10.7643 


0.9451 

1 

1.0144 

0.9137 

10.1519 


0.9297 
0.9858 

1 

0.9007 

10.0074 


1.0322 

1.0945 

1.1103 

1 

11.1111 


0.0929 
0.0985 
0.0999 
0.0900 

1 



y Google 



HbLB foe dOMFAKIl^a Jf BAStBBS AND WEiaHTS. 241 

Cubio Measure. 



UiimD 

fiV^TBS AND 
BSOLAMD. 


Fbubbea. 


AmxniA. 


Baden AUD 

SWITZESLAND. 


Pranot. 


Cabio foot. 


Cubic foot. 


Cubio foot. 


Cubic foot. 


Cubic metet 


1 

1.0918 

1.1156 

0.9535 

35.3166 


0.9159 

1 

1.0217 

0.8733 

32.3459 


0.8964 
0.9787 

0.8548 
31.6578 


1.0487 

1.1450 

1.1699 

1 

37.0370 


0.0283 
0.0309 
0.0316 
0.0270 

1 





Weight 


per Unit of Lengtii. 




Lbs. per 
lineal foot. 


Lbs. per 
lineal foot 


Lb&per 
lineal foot 


LbB.por 
lineal foot 


Kil. per 
lineal meter 


1 

1.0705 
1.1904 
1.1199 
0.6720 


0.9342 

1 
1.1120 
1.0462 
0.6277 


0.8400 
0.8993 

1 
1.9408 
0.5645 


0.8929 
0.9559 
1.0629 

1 
0.6000 


1.4882 
1.6931 
1.7716 
1.6667 

1 



Weiglit per Unit of Surface. 



Lbs. per 
square inch. 


Lbs. per 
square Inch. 


Lbs. per 
square inch. 


Lbs. per 
square inch. 


Kil. per 
square cent 


1 

1.0396 

1.1478 

0.7902 

14.2223 


0.9619 

1 

1.1041 

0.7601 

13.6811 


0.8712 
0.9057 

1 

0.6884 

12.3910 


1.2656 

1.3157 

1.4526 

1 

18.0000 


0.0703 
0.0731 
0.0807 
0.0566 

1 



16 



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SA2 BtEaiBTAJfOK X0 OBOSB-BBEAEIH0* 

RESISTANCE TO CROSS-BREAKINa. 

To Out the Strongest and Stiff est BectangulaT Beam from a Lag^ 

Fig, 308. (Strongest.) 



The diameter aa = d, divided into three eqnal parts, with per« 
pendiculars i d from a erected thereon, intersecting the circle at 
0, will give section for greatest capacity. 



Fig.SOd. (Stiffeet.) 



The diameter aa = cf, divided into four eqnal parti, with per- 
pendiculars J d from a erected thereon, intersecting the circle, at 
0, will give section with least deflection, but less capacity than 
Mg.BoS. 



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I N I> E X . 



VAQM. 

Atfea, circumference, and cubic contents af circles 218 

AJLis, neutral 4 

Bars, tie rods, <fec 181 

resistance of, to tearing,... 2 

Beams, capacity and strengin of 29 

of rolled 39 

of cast-iron 6Y 

IT of rolled l-sbaped ;... 39 

and strength of parabolic arched 153 

cast-iron * 63 

iron ties, struts, and 3 

sloping rafters and..*.. 102 

strains in trussed 122 

horizontal and sloping 188 

strength of wooden 88 

Bolts and nuto, dimensions of. 187 

nuts, and heads.... 235 . 

Boom derricks, strains in 114 

Booms, strains in trusses with parallel 126 

Bow-string ^rders , 147 

Bridges, static and moving loads, of wrought iron 192 

Oamber ....f....... 2 

Capacity , , 2 

and strength of beams 29 

TFof roUSi l-shaped beams 39 

of rolled beams..... 41 

of cast-iron beams 57 

and strength of parabolic arched ^aims;........ 153 

Cast-iron beams 3, 53 

Center of gravity of planes 202 

Circumference, area, and cubic contents of circles 218 

Columns, pillars, and struts, strength of 110 

Oompoflition and resolution of forces. ......< Ill 

GcNUpound strains in horizontal and sloptBg beamsw..;.*. 188 

Compression 1 

OcHnpressive strain and pressure on supportSv...*.. 102 

OMiraction and expansion •••«mm.«*m«*^»^»».*.«...^ m,.vv.v..... 4 
(243) 

Digitized by ^^jUU^ It: 



2H XS9BS. 



ConBtants for strain in trasses 117 

roof trusses 174 

Connections in iron construction, joints or. 184 

Cross-breaking 2 

and shearing, resistance to 29 

Crushing, resistance to , 103 

direct 1 

Deflection 2 

Derricks, strains in boom 114 

Dimensions of bolts 187 

Divisions of a foot, expressed in equivalent decimals 239 

Expansion and contraction • 4 

External forces 1 

Factors of safety , 29 

Forces external 1 

internal 1 

composition and resolution of. , Ill 

parallelogram of * Ill 

Frame, strains in polygonal 154 

Functions, trigonometrical 207 

Geometry.. 197 

Girders, strains in parabolic and bow-string 147 

Gravities of materials, specific 224 

Heads, nuts, and bolts ^ 235 

Horizontal and sloping beams, compound strains in 188 

Howe truss 129 

Inertia and resistance o various sections, moments of. 5 

Internal forces 1 

Iron beams, capacity of cast 57 

cast , 53 

bridges, static and moving loads, of wrought. 192 

construction, joints or connections in ..-. 184 

ties, struts, or beams 3 

Joints or connections in iron construction 184 

Lattice truss 139 

with vertical members 131 

Longimetry and planimetry 197 

Materials, Ac., streiigth of 26 

Miscellaneous .^.^,».,... , , 195 



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Modalns of mptiire 4 

Moment of inertia and resistance of yarions sections 5 

Moving loads, weight of...... 191 

Natural sine, cosine, &o 306 

Neutral axis 4 

^ Nuts, heads, and bolts 235 

dimensions of... 187 

Parallelogram of forces..... Ill 

Parallel booms, strains in trusses with 126 

Parabolic arched beams, capacity and strength of. 153 

curved trusses, strains in 147 

Planimetry, longimetry, Ac 197 

Pillars, columns, and struts, strength of 110 

Pins, Ac., in tie bars 185 

Polygonal frame, strains in 154 

Pressure on supports 100 

compressive strain and 102 

of snow on roofs 178 

of wind on roofs 180 

Rafters, Ac., sloping beams 102 

Reactions of supports 100 

Resistance to direct crushing 1 

of bars, Ac., to tearing 2 

to cross-breaking ana shearing 29 

crushing 103 

Resolution of forces, composition, Ac Ill 

Rolled beams, capacity or. 41 

l-shaped beams, capacity of. 39 

Rods and bars, tie 181 

Roof trusses 3 

strains in 156 

constants for strains in 174 

Roofe, pressure of wind on 178 

of snow on , 180 

Rupture, modulus of. 4 

Shearing 2 

, and cross-breaking, resistance to 29 

Sloping beams, rafters, Ac 102 

and horizontal beams, compound strains in 188 

Specific gravities of materials 224 

Static and moving loads of wrought-iron bridges 192 

Strength of materials 26 

wooden beams.. 98 

columns, pillars, and struts 110 



Digi 



itizedby^jUUyH:: 



rAoi; 

Strength of beams, capacity, Sus r... ...«.«.«•....... 2B 

Strains in frames « ^.^#....4... «.«.«.*.. 112 

boom derricks • « 114 

trusses 115 

trussed beams 122 

trusses with parallel booms 126 

parabolic curved trusses, or bow-string girders.... 147 

polygonal frame 154 

roof trusses 156 

constants for , 174 

trusses, constants for 117 

Strongest and stiffest rectangular beam from a log, to cut the.. 242 

Struts and beams, iron ties , 3 

Supports, reaction of. 100 

compressive strain and pressure On 102 

Table for comparing measures and weights 240 

Tearing, resistance of bars, Ac., to 2 

Tension 1 

Tie rods and bars 181 

Trigonometrical functions 207 

formulas 205 

Truss, Howe 12^ 

Warren 132 

Whipple 144 

lattice 139 

with vertical members 131 

Tfusses parallel booms, strains in 126 

parabolic curved, or bow-string 147 

constants for strains in roof 174 

constants for strains in 117 

strains in 115 

roof. 156 

Trussed beams, strains in 122 

Warren truss 132 

Weight of moving loads 191 

static and moving loads of wroughi-iron bridges... 192 

a lineal foot of flat or square bar iron 229 

rolled round iron ,. 234 

materials used in building » 238 

superficial inch of wrought and cast iron..........^. 227 

rolled round iron for bolts........... 236 

heads and nuts ...w^**..*.^ 235 

per square foot of metals.....^. >. 228 

Whipple truss 144 

Wooden beams, strength of. >.«...»...•..••.•• . 98 

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