The stress acting at a point is represented by the stresses acting on the faces ϑ the element enclosing the point, the stresses on the faces ϑ the element vary as the regular position ϑ the element changes. From the figure ϑ variation ϑ stress components, applying the conditions ϑ equilibrium to axis chosen as in figure (d) we obtain: A σ = (σx A cosθ)cosθ + (σy A sinθ)sinθ - (τxy A cosθ)sinθ - (τyx A sinθ)cosθ ......(a) A τ = (σx A cosθ)sinθ - (σy A sinθ)cosθ + (τxy A cosθ)cosθ - (τyx A sinθ)sinθ .....(b) since the common term A can be canceled and since τyx is numerically equal to , we use the relations cos2 θ = , sin2 θ = also sinθ cosθ = sin2θ sub. the above into equations a&b σ = cos2θ - τxy sin2θ .......(5) and τ = sin2θ + τxy cos2θ ......(6) the planes defining max. or min. normal stresses:
The stress acting at a point is represented by the stresses acting on the faces ϑ the element enclosing the point, the stresses on the faces ϑ the element vary as the regular position ϑ the element changes. From the figure ϑ variation ϑ stress components, applying the conditions ϑ equilibrium to axis chosen as in figure (d) we obtain: A σ = (σx A cosθ)cosθ + (σy A sinθ)sinθ - (τxy A cosθ)sinθ - (τyx A sinθ)cosθ ......(a) A τ = (σx A cosθ)sinθ - (σy A sinθ)cosθ + (τxy A cosθ)cosθ - (τyx A sinθ)sinθ .....(b) since the common term A can be canceled and since τyx is numerically equal to , we use the relations cos2 θ = , sin2 θ = also sinθ cosθ = sin2θ sub. the above into equations a&b σ = cos2θ - τxy sin2θ .......(5) and τ = sin2θ + τxy cos2θ ......(6) the planes defining max. or min. normal stresses:
5_2018_04_24!08_36_07_PM.docx