Nonlinear finite elements/Homework 9

Problem 1: Total Lagrangian
Consider the tapered two-node element shown in Figure 1. The displacement field in the element is linear.

The reference (initial) cross-sectional area is

A_0 = (1-\xi)~ A_{01} + \xi~ A_{02} ~. $$ Assume that the nominal (engineering) stress is also linear in the element, i.e.,

P = (1-\xi)~ P_1 + \xi~ P_2 ~. $$


 * 1) Using the total Lagrangian formulation, develop expressions for the internal nodal forces.
 * 2) What are the internal nodal forces if the reference area and the nominal stress are constant over the element?
 * 3) Assume that the body force is constant. Develop expressions for the external nodal forces for that case.
 * 4) What are the external nodal forces if the reference area and the nominal stress are constant over the element?
 * 5) Develop an expression for the consistent mass matrix for the element.
 * 6) Obtain the lumped (diagonal) mass matrix using the row-sum technqiue.
 * 7) Obtain the lumped (diagonal) mass matrix using the row-sum technqiue.
 * 8) Find the natural frequencies of a single element with consistent mass by solving the eigenvalue problem

\mathbf{K}~\mathbf{u} = \omega^2~\mathbf{M}~\mathbf{u} $$
 * with

\mathbf{K} = \cfrac{E(A_{01} + A_{02}}{2 l_0} \begin{bmatrix} 1 & -1 \\ -1 & 1 \end{bmatrix} $$
 * where $$E$$ is the Young's modulus and $$l_0$$ is the initial length of the element.

Problem 2: Updated Lagrangian
Consider the tapered two-node element shown in Figure 1.

The current cross-sectional area is

A = (1-\xi)~ A_1 + \xi~ A_2 ~. $$ Assume that the Cauchy stress is also linear in the element, i.e.,

\sigma = (1-\xi)~ \sigma_1 + \xi~ \sigma_2 ~. $$


 * 1) Using the updated Lagrangian formulation, develop expressions for the internal nodal forces.
 * 2) Assume that the body force is constant.Develop expressions for the external nodal forces for that case.

Problem 3: Modal Analysis

 * 1) Consider the axially loaded bar in problem VM 59 of the ANSYS Verification manual. Assume that the bar is made of Tungsten carbide.
 * 2) Find the fundamental natural frequency of the bar.
 * 3) Find the first three modal frequencies for a load of 40,000 lbf.
 * 4) Consider the stretched circular membrane in problem VM 55 of the ANSYS Verification manual. Assume that the membrane is made of OFHC (Oxygen-free High Conductivity) copper.
 * 5) Find the fundamental natural frequency of the bar.
 * 6) Find the first five modal frequencies for a load of 10,000 lbf.