frame me15resch11006 me15mtech11033

ASSIGNMENT -1 ME 5130 - FINITE ELEMENT ANALYSIS

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FINITE ELEMENT PROGRAM FOR FRAME

Description of the problem:

Consider a portal frame structures shown below.

a) Write a finite element program to compute the displacements and rotations at the point of application of load.

b) Verify your results by solving the same problem using symmetry/antisymmetry boundary conditions at the point of application of the load/moment.

c) Does the solution improve if more than one element is used to discretize each member of the frame.

Figure 1: portal Frame

Solution:

Assumptions used for the finite element program –

Material and section properties:

Following material and sectional properties are used in the program:

Elastic Modulous, E=1

Cross-sectional area, A=1

Length of each element, L=1

Loading and boundary conditions:

Frame is fixed at two location, as shown in the figure 1. And an unit load is applied at the center of the frame.

Unit system: No specific unit system is used for this problem, as all the inputs to the program are considered as unity.


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Strategy followed for the finite element program:

The program is divided into three sections :

1) Pre-‐processing: At this stage the frame is discretized into elements, typically the problem is solved using 1 element per frame member. Then nodes and elements were defined, node definition includes providing the coordinates of each node. There are 5 nodes and 4 elements to start with. Next stiffness, force and displacement matrices are initialized with zero values, the stiffness matrix is of the size 3*number of nodes X 3*number of nodes. Also based on the boundary conditions the active number of degrees of freedom are calculated.

Next The values of A,E, and I are defined, local stiffness matric is defined. The local stiffness matrix for the frame is combination of the stiffness matrix of beam and bar.

Since the members of frame are at angles to each other, coordinate transformation is required to calculate the global stiffness, displacement and force matrices.

2) Processing : Global stiffness matrix is assembled from local stiffness matrices, boundary conditions are applied, loads are applied and displacement matrix is calculated.

3) Post-‐processing: The results from the finite element program i.e. displacement and rotations at each node is presented.

4) Verification of the results: At first the calculation is performed considerint the complete frame, the displacement and rotation are as given below:

Figure 2: Displacement and Rotations of the portal frame – Application of force


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Figure 3: Displacement and Rotations of the portal frame with symmetric boundary conditions.

From figure 2 & 3 it can be observed that the deformations and rotations of the frame remain unchanged if symmetric boundary conditions are used. Next each frame member was divided into 2 elements to study the effect of increasing number of elements on the results, figure 4 shows the results from the FEM program with 2 elements per member of frame. And it can be observed from the results that the deformations and rotations are unchanged with increase in the number of elements.

Figure 5 shows the results when force is replaced with a moment.

Figure 4: Displacement and Rotations of the portal frame with 2 elements per frame member


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Figure 5: Displacement and Rotations of the portal frame – Application of moment, M3=+1 unit

Analysis of run time for the program:

Case 1 : Full frame, one element per member = 0.2783 secs

Case 2 : Half symmetric frame, one element per member = 0.2676 secs

Case 3: Full frame, two element per member = 0.3485 sec

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frame me15resch11006 me15mtech11033

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