7.7:
Bending Moment Diagram
Consider an overhanging beam AC under a distributed load, supported at two joints. Construct its bending moment diagram.
Draw the free-body diagram of the beam, and by using the equilibrium equations, the reaction forces at each support can be calculated.
Apply the method of section to the beam.
Next, consider an arbitrary distance x within region AB. The resultant force acts at the midpoint of the section. By using the equilibrium equation on the free-body diagram and substituting the values, the equation of moment is obtained.
Similarly, consider an arbitrary point within the region BC. By drawing the free-body diagram and using the equilibrium equation, the moment equation for section BC is determined.
Substituting the position values in the moment equations, the bending moment values are calculated.
The bending moment diagram shows the graphical representation of the bending moment values along the beam's length.
By recalling the moment equation and applying the differentiation method, the local maxima for each section are obtained.
The bending moment diagram is obtained by joining these points with parabolic curves.
7.7:
Bending Moment Diagram
A bending moment diagram is a graphical representation of the bending moments experienced by a beam under load along the beam length. It is an essential tool for engineers and designers to analyze structures and ensure they can withstand applied forces. The steps to create the bending moment diagram for a beam are listed below.
Determine reactive forces and couple moments: Calculate all the reactive forces and couple moments acting on the beam. In certain cases, when the beam is inclined at an angle, these forces are resolved into their components acting perpendicular and parallel to the beam's axis.
Section the beam and draw the free-body diagram: Consider an arbitrary distance from the beam's left end. This arbitrary distance can be extended in different regions along the beam's length. At each assumed distance, section the beam and create a free-body diagram of one of the segments. Ensure that the bending moment (M) is shown acting in their positive sense, according to the established sign convention.
Calculate bending moment: Apply the moment equilibrium equations and obtain the bending moment (M) about the sectioned end of the segment.
Plot the moment diagram: Create the moment diagram (M versus x) from the obtained bending moment along the beam length. If computed values of the functions describing M are positive, plot the values above the x-axis. Conversely, if the values are negative, plot them below the x-axis.
Following this procedure, you can construct accurate shear and bending moment diagrams for any beam. These diagrams are crucial in designing structural elements, such as shelving arms, that resist the forces and moments induced by applied loads.
Suggested Reading
- Hibbeler, R.C. (2016). Engineering Mechanics ‒ Statics and Dynamics. Hoboken, New Jersey: Pearson Prentice Hall. pp 362.