Example 2.7

Given: P = 50 kN, k = 200 kN/m, L = 3 m,

E = 210 GPa, I = 2×10-4 m4.

Find: Deflections, rotations and reaction forces.

Solution:

The beam has a roller (or hinge) support at node 2 and a spring support at node 3. We use two beam elements and one spring element to solve this problem.

The spring stiffness matrix is given by,

Adding this stiffness matrix to the global FE equation (see

Example 2.5), we have

‘Deleting’ the first three and seventh equations (rows and columns), we have the following reduced equation,

Solving this equation, we obtain the deflection and rotations at node 2 and node 3,

Lecture Notes: Introduction to Finite Element Method Chapter 2. Bar and Beam Elements

The influence of the spring k is easily seen from this result. Plugging in the given numbers, we can calculate

θ2 −0.002492 rad

v3 = −0.01744 mθ3 −0.007475 rad

From the global FE equation, we obtain the nodal reaction forces as,

Checking the results: Draw free body diagram of the beam

FE Analysis of Frame Structures

Members in a frame are considered to be rigidly connected. Both forces and moments can be transmitted through their joints. We need the general beam element (combinations of bar and simple beam elements) to model frames.

Find: Displacements and rotations of the two joints 1 and 2.

Solution:

For this example, we first convert the distributed load to its equivalent nodal loads.

Lecture Notes: Introduction to Finite Element Method Chapter 2. Bar and Beam Elements

In local coordinate system, the stiffness matrix for a general 2-D beam element is

In general, the transformation matrix T is,

We have

l = 0, m = 1

for both elements 2 and 3. Thus,

Using the transformation relation,

k = TT k'T

we obtain the stiffness matrices in the global coordinate system for elements 2 and 3,

Assembling the global FE equation and noticing the following boundary conditions,

u3 = v3 = θ3 = u4 = v4 =θ4 = 0

we obtain the condensed FE equation,

Lecture Notes: Introduction to Finite Element Method Chapter 2. Bar and Beam Elements

To calculate the reaction forces and moments at the two ends, we employ the element FE equations for element 2 and element 3. We obtain,

Lecture Notes: Introduction to Finite Element Method Chapter 2. Bar and Beam Elements

Check the results:

Draw the free-body diagram of the frame. Equilibrium is maintained with the calculated forces and moments.