Shape functions for beam elements in finite element analysis

This video introduces the displacement function for a beam element in terms of its shape functions and corresponding nodal displacements and rotations. 0:00 - Introduction, use of Eqn. of the Elastic Curve to show that the displacement function must be cubic 2:10 - Boundary conditions to solve for unknown coefficients in beam displacement function 3:13 - Rearrange the displacement function in terms of shape functions and their corresponding displacements 3:53 - Shape function, N1, for the displacement at the first node, v1 4:57 - Shape function, N2, for the rotation at the first node, theta1 5:33 - Shape function, N3, for the displacement at the second node, v2 6:08 - Shape function, N4, for the rotation at the second node, theta2 6:45 - Reflection questions Answers to reflection questions 1.) The difference is that the displacement function is written in terms of shape functions with their corresponding degrees of freedom. The shape functions represent the influence of their associated degree of freedom on the displacement profile throughout the element. 2.) The slope of N2 should equal 1.0 because N2 represent the rotation at the first node of the beam element. The displacement should equal zero at each end. The slope should also equal zero at the second node for N2. 3.) For a beam element, the shape function can be represented by a cubic profile. 4.) For a beam element, the displacement function also varies according to a cubic profile. The displacement function and shape function both are the same order polynomial.