Appendix A - Structural Analysis of Beams

The structural analysis of beam can be confusing especially for those without a background in structural or mechanical design. However, simple structures can be analyzed with a little backgound. This appendix explains the steps for a analyzing bending in beams.

As one might remember from elementary mechanics the first step in analyzing a mechanical system starts with a free body diagram and determination of moments and forces. Figure A1 shows the free body diagram for a simple beam loaded at one end.

After developing the free body diagram there are four more steps to find the displacement of the beam as a function of the applied force.

• Find the shear along the beam as a function of distance x. Shear is the some of the forces to the left of position. Mathematically it is For the simply loaded beam in Figure A1 the shear is just Fa over the intire length of the beam. Mathematically one can think of the F(0) as an impulse of Fa.

  
  

• Find the bending moment along the beam. The bending moment is the sum of the moments from supports plus the integral of the shear to the left of position. Mathematically it is,
• Find the slope of the beam. The slope is found using a small angle approximation that sin(theta)= theta and tan(theta)= theta. Then, the slope can be found by, where epsilon is the strain (or relative change in length) and c is the distance from the centroid axis to the top surface of the beam. Using the following relationships, where sigma is the bending stress, E is young's modulus of elasticity and I is the centroid moment of inertia. Finally, the slope equation is, This equation means that the slope of the beam can be found by integrating the moment over the EI product over the distance downthe beam.

  
  

• Now, the displacement can be found by integrating the slope along the length of the beam.

Figure A2. shows the solution to the beam in Figure A1.