Beams and Frames

1. Beams and Frames

2. beam theory can be used to solve simple beams • complex beams with many cross section changes are solvable but lengthy • many 2-d and 3-d frame structures are better modeled by beam theory

3. One Dimensional Problems The geometry of the problem is three dimensional, but the variation of variable is one dimensional Variable can be scalar field like temperature, or vector field like displacement. For dynamic loading, the variable can be time dependent

4. Element Formulation • assume the displacement w is a cubic polynomial in a1, a2, a3, a4 are the undetermined coefficients • L = Length • I = Moment of Inertia of the cross sectional area • E = Modulus of Elsaticity • v = v(x) deflection of the neutral axis • = dv/dx slope of the elastic curve (rotation of the section F = F(x) = shear force M= M(x) = Bending moment about Z-axis

5. `

6. Applying these boundary conditions, we get • Substituting coefficients ai back into the original equation for v(x) and rearranging terms gives

7. The interpolation function or shape function is given by

8. strain for a beam in bending is defined by the curvature, so • Hence

10. the stiffness matrix [k] is defined To compute equivalent nodal force vector for the loading shown

11. Equivalent nodal force due to Uniformly distributed load w