Suppose that w and are approximated as φx (3.7) where(W are the nodal values of and = 1, 2) are the associated j,Φj ) (w, φx) ψj(α)(x) (α interpolation functions. Substitution of (7) for w and and = and = into φx, δw ψi(1) δφx ψi(2) (3.5) yields the finite element model (3.8) where Kij11 = `Kij12 = Kij22 = (3.9) 3.1.2 Reduced Integration element (RIE): For a linear interpolation of w and and exact evaluation of the integrals of (3.9),(3.8) takes the ϕx form (3.10) Where (3.11a) (3.11b) In the thin beam the first and third equations of (3.10) imply the limit, i.e Ω0, following relation among (W1,W2,Φ1,Φ2) : (3.12) Which is equivalent to the Kirchoff +dw/dx =0 (or shear strain constraint φx γxz=0). The second and fourth equations of (10), in view of (12), yield the constraint (3.13) This is equivalent to which is an incorrect condition to satisfy as it forces the curvature and dφx/dx =0, hence bending energy to zero. Thus, (3.10) in an effort to satisfy the constraints (3.12) and (3.13), will yield the trivial solution the element locks). W1=W2=Φ1=Φ2=0 (i.e., The Kirchoff condition (3.12) suggests that w and be interpolated such that dw/dx is a φx polynomial of the same order as If w is approximated using a linear polynomial ( a minimum φx. requirement), then should be a constant. Since the minimum continuity requirement on is φx φx also linear, it follows that w be approximated using a quadratic polynomial. This is a consistent interpolation. Unless the weak form of Timoshenko beam theory is modified, we have no
Suppose that w and φx are approximated as (3.7) where(W j,Φj ) are the nodal values of (w, φx) and ψj(α)(x) (α = 1, 2) are the associated interpolation functions. Substitution of (7) for w and φx, and δw = ψi(1) and δφx = ψi(2) into (3.5) yields the finite element model (3.8) where Kij11 = `Kij12 = Kij22 = (3.9) 3.1.2 Reduced Integration element (RIE): For a linear interpolation of w and ϕx and exact evaluation of the integrals of (3.9),(3.8) takes the form (3.10) Where (3.11a) (3.11b) In the thin beam limit, i.e Ω0, the first and third equations of (3.10) imply the following relation among (W1,W2,Φ1,Φ2) : (3.12) Which is equivalent to the Kirchoff constraint φx +dw/dx =0 (or shear strain γxz=0). The second and fourth equations of (10), in view of (12), yield the constraint (3.13) This is equivalent to dφx/dx =0, which is an incorrect condition to satisfy as it forces the curvature and hence bending energy to zero. Thus, (3.10) in an effort to satisfy the constraints (3.12) and (3.13), will yield the trivial solution W1=W2=Φ1=Φ2=0 (i.e., the element locks). The Kirchoff condition (3.12) suggests that w and φx be interpolated such that dw/dx is a polynomial of the same order as φx. If w is approximated using a linear polynomial ( a minimum requirement), then φx should be a constant. Since the minimum continuity requirement on φx is also linear, it follows that w be approximated using a quadratic polynomial. This is a consistent interpolation. Unless the weak form of Timoshenko beam theory is modified, we have no
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