# Nonlinear finite elements/Homework 9

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## Problem 1: Total Lagrangian[edit]

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

Assume that the nominal (engineering) stress is also linear in the element, i.e.,

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

- with
- where is the Young's modulus and is the initial length of the element.

## Problem 2: Updated Lagrangian[edit]

Consider the tapered two-node element shown in Figure 1.

The current cross-sectional area is

Assume that the Cauchy stress is also linear in the element, i.e.,

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

## Problem 3: Modal Analysis[edit]

- Consider the axially loaded bar in problem VM 59 of the ANSYS Verification manual. Assume that the bar is made of Tungsten carbide.
- Find the fundamental natural frequency of the bar.
- Find the first three modal frequencies for a load of 40,000 lbf.

- 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.
- Find the fundamental natural frequency of the bar.
- Find the first five modal frequencies for a load of 10,000 lbf.