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AERSP 301Finite Element Method

Beams

Jose Palacios

July 2008

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Today

No class Friday

Final Rod FEM example

Beam Bending Elements

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Sample Problem 3:

FEM of a more complex system

(loaded axially)

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Beams Under Bending Load

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Euler-Bernoulli Beam Theory assumes:

Plane sections perpendicular to the mid-plane remain plane and

perpendicular to the beam axis after deformation (i.e. no shear)

Long slender beams (Timoshenko theory for short beams)

Consider the displacement

of point P (to P)

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Thus, we can write the axial and vertical displacements of generic

point P as:

Use these displacements to get strains:

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That leaves us with,

And the stress:

Now consider the Resultant axial force on a cross section:

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And the Resultant Bending Moment on a cross-section:

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From the above expressions, it is seen that Extension & Bending are

decoupled:

Recall displacement of generic point, P:

But for pure bending problem uoterm vanishes, so:

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Recall, Strain Energy:

This comes from:

For the beam bending problem:

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External Work,W

, for the beam bending problem:

We can use the Finite Element Method to analyze the beam under

bending loads.

First, discretize the beam into a number of elements.

Strain Energy and Work can be written for a single element:

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Before we can obtain the element stiffness matrix, , and elementload vector, , we need to decide:

What should the nodal D.O.F.s be?

What should the assumed displacement with the element be?

To answer the first question, we have to keep in mind that thedisplacement must be continuous from element to element (i.e. atelement boundaries).

For axial (bar) problem, only displacement was u. By having u1 and u2as elemental NODAL DOFs and assigning u2

k-1and u1kto the same

global DOF (during assembly) we ensured that the displacement wascontinuous at element boundaries.

For beam bending, both u and w have to be continuous at boundaries.

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Choose w and as NODAL DOFs

Then, assign local DOFs of adjacent elements to same global DOFs

(during assembly) such that:

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Second Question: What should assumed displacement be with

element?

Recall for bar problem, displacement had to be written in terms of u1&

u2. A linear function was chosen.

For beam bending problem, assumed displacement w (within the

element) has to be written in terms of:

You can choose a cubic function:

How do we calculate the a coefficients in terms of nodal displacements(shape functions)? As with the rod elements we must rewrite the

assumed displacement function in terms of nodal displacements and

shape functions.

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For a beam bending element, assumed displacement can be written

as:

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Element Strain Energy:

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Now we need to determine the element load vector

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The element strain energy was used to derive the element stiffness

matrix.

Similarly, the external work (over an element) is used to derive the

element load vector:

where,

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Sample Problem 1

Calculate the tip displacement and rotation of the beam due to tip

load P, using a single element.

Calculate the reaction force, R, and the reaction Moment, M, at the

clamped boundary.

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Sample Problem 2

Analyze the structure below using 2 elements. Calculate the

deflection at the joint of the two beams.

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Sample Problem 3

Look at the previous problem, except now the two beams are

connected through a hinge.

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Sample Problem 4

How does the linear spring shown below affect the problem?

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Sample Problem 5

Show the affect of the linear and rotational spring to the finite

element method.

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Coupled Axial-Bending Problems

We can also combine our rod and beam elements.

Look at elements individually (1 is a rod, 2 is a beam, etc)

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Coupled Axial-Bending Problems

Example 6

11. fem - hw 6c

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