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.

Figure 1. Tapered two-noded element.

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

$\mathbf {K} ={\cfrac {E(A_{01}+A_{02}}{2l_{0}}}{\begin{bmatrix}1&-1\\-1&1\end{bmatrix}}$

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.
   1. Find the fundamental natural frequency of the bar.
   2. Find the first three modal frequencies for a load of 40,000 lbf.
2. 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.
   1. Find the fundamental natural frequency of the bar.
   2. Find the first five modal frequencies for a load of 10,000 lbf.

Nonlinear finite elements