Question

Which of the following equations is the correct expression for bending moment (M) in an element, given the modulus of elasticity (E), moment of inertia (I), element length (le), shape function (ξ) and displacement (q) in a uniformly distributed load on a simply supported beam?

(a) M=\(\frac{EI}{(le)^2}\)[6ξq1+(3ξ-1)leq2-6ξq3+(3ξ+1)leq4]

(b) M=\(\frac{EI}{(le)^3}\)[6ξq1-(3ξ-1)leq2-6ξq3-(3ξ+1)leq4]

(c) M=\(\frac{EI}{(le)^2}\)[6ξq1+(3ξ-1)leq2-6ξq3-(3ξ+1)leq4]

(d) M=\(\frac{EI}{(le)^3}\)[6ξq1+(3ξ-1)leq2-6ξq3+(3ξ+1)leq4]

The question was posed to me in my homework.

My question comes from Shear Force & Bending Moment in chapter Beams and Frames of Finite Element Method

Answer 1

Right answer is (a) M=\(\frac{EI}{(le)^2}\)[6ξq1+(3ξ-1)leq2-6ξq3+(3ξ+1)leq4]

Explanation: The correct expression is given by

 M=\(\frac{EI}{(le)^2}\)[6ξq1+(3ξ-1)leq2-6ξq3+(3ξ+1)leq4]

Here E is the ratio of normal stress to normal strain. Here q1, q2, q3, q4 are the four displacements at the two supported nodes of the simply supported beam. The value of shape function (ξ) varies between -1 to +1.