Nonlinear finite elements/Homework 9/Solutions

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Problem 1: Total Lagrangian[edit]

Given:

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

Figure 1. Tapered two-node element.

The reference (initial) cross-sectional area is

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

Solution[edit]

Part 1[edit]

Using the total Lagrangian formulation, develop expressions for the internal nodal forces.

The displacement field is given by the linear Lagrange interpolation expressed in terms of the material coordinate.

where . The strain measure is evaluated in terms of the nodal displacement,

which defines the matrix to be

The internal nodal forces are then given by the usual relations.

Integrating the above integral with to obtain

Part 2[edit]

What are the internal nodal forces if the reference area and the nominal stress are constant over the element?

Part 3[edit]

Assume that the body force is constant. Develop expressions for the external nodal forces for that case.

The external body forces arising from the body force, , are obtained by the usual procedure.

Part 4[edit]

What are the external nodal forces if the reference area and the nominal stress are constant over the element?

Part 5[edit]

Develop an expression for the consistent mass matrix for the element.

The element mass matrix is

Part 6[edit]

Obtain the lumped (diagonal) mass matrix using the row-sum technique.

Lumped mass matrix is given by

Part 7[edit]

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.

The above equation can be rewrite as

which only has a solution if

Solving the above determinant for , we have


Problem 2: Updated Lagrangian[edit]

Given:

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.,

Solution[edit]

Part 1[edit]

Using the updated Lagrangian formulation, develop expressions for the internal nodal forces.

The velocity field is

In term of element coordinates, the velocity field is

The displacement is the time integrals of the velocity, and since is independent of time

Therefore, since

where is the current length of the element. For this element, we can express in terms of the Eulerian coordinates by

So can be obtained directly, instead of through the inverse of .

The matrix is obtained by the chain rule

Using (146) in Handout 13, we have

Integrating the above equation to obtain

Part 2[edit]

Assume that the body force is constant. Develop expressions for the external nodal forces for that case.

The external forces are given by


Problem 3: Modal Analysis[edit]

Given: Consider the axially loaded bar in problem VM 59 of the ANSYS Verification manual. Assume that the bar is made of Tungsten carbide.

The input file for ANSYS is as shown in VM 59 except the following material properties are used: Msi and lbsin.

Solution[edit]

Find the fundamental natural frequency of the bar.

Find the first three modal frequencies for a load of 40,000 lbf.


Given: 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.

The input file for ANSYS is as shown in VM 55 except the following material properties are used: Msi and lbsin, and the modes are expanded to 5.

Solution[edit]

Find the fundamental natural frequency of the bar.

Find the first five modal frequencies for a load of 10,000 lbf.