Shear, Moment, and Bending Moment Equations in Beam Analysis

Shear and Moment Diagrams

To draw shear and moment diagrams in beam analysis, we need to follow several steps. First, we need to solve each loading condition separately, determining the magnitude and direction of shear and moment at each loading position. Then, we add these values algebraically as we go along the beam. The shear at the support is equal to the reaction, while the moment is zero. At a concentrated load, both shear and moment experience abrupt shifts equal to the load magnitude. In the case of a uniformly distributed load, shear changes linearly, and the moment changes quadratically with the load.

By using this information, we can create the shear and moment diagrams, which provide us with a visual representation of the internal forces acting on the beam.

Bending Moment Equation of Segment CD

The bending moment equation for segment CD of the beam can be determined by analyzing the moment diagram. In this specific case, the bending moment in segment CD is constant and equals -18 kN.m. Therefore, the bending moment equation for this segment is M(x) = -18 kN.m.

Selecting the Most Economical W Section

To select the most economical W section based on the allowable bending stress of 205 MPa, we need to calculate the required section modulus. This can be done using the formula: Z = I/c, where I represents the moment of inertia of the section, and c is the distance from the neutral axis to the extreme fiber. By comparing the required section modulus (determined by the bending moment and allowable stress) with the section modulus values of various W sections, we can identify the most cost-effective option.

In this case, the W310x97 section would be the optimal choice as it has a section modulus of 3,030 cm3, meeting or exceeding the required value.