Nonlinear finite elements/Homework 6/Solutions/Problem 1/Part 1

Problem 1: Part 1
What is the continuum-based approach for the finite element analysis of beams? Draw figures to elucidate.

In the continuum-based approach, instead of starting off with simplified governing equations, a beam (or shell) element is developed directly from a continuum element by imposing structural assumptions.

The kinematic assumptions (for example, normals to midsurface remain normal) are imposed on the discretized equations derived from the weak form. A plane stress assumption in the direction normal to the beam is also enforced.

In the continuum-based approach for beams (in two dimensions), we start with a parent element which is a six-noded square (see Figure 1).

The parent element maps to the actual beam element in the reference and current configurations. The reference line of the beam element maps to the $$\eta = 0$$ line in the parent element. Master nodes are placed at the intersection of the fibers connecting the slave nodes with the reference line.

The momentum equation is solved on the master nodes and the resulting velocities are projected to the slave nodes. A set of quadrature points is set up along the fiber and the stresses are computed at these points. The plane stress assumption is imposed and the internal forces are computed at the slave nodes. The internal forces are interpolated to the master nodes and the process is repeated for each load step.