7.10:
Shear and Bending Moment Diagram: Problem Solving
Consider a beam supporting two concentrated loads and a distributed load. Draw the shear and bending moment diagram for the beam.
First, draw a free-body diagram of the beam and, using the equilibrium equation, the reaction forces are obtained.
Next, divide the beam into different sections and draw the free-body diagram of each section. Applying the equation of equilibrium for the sections, the shear for individual sections can be determined.
Shear remains constant between concentrated loads and reaction forces while it varies linearly with a constant slope in the distributed load section.
The area under the shear curve between two points equals the change in bending moment between the same two points.
Considering the bending moment zero at the beam's end and recalling the relation between the change in bending moment and area under the shear curve, the bending moment at each point is calculated.
The bending moment diagram is drawn by connecting the known points with straight lines for regions with constant shear and a parabolic line for regions with linear shear.
7.10:
Shear and Bending Moment Diagram: Problem Solving
When analyzing a beam supporting concentrated loads and a distributed load, drawing the shear and bending moment diagrams is essential. These diagrams help understand the internal forces and moments acting on the beam, which is crucial for designing safe and efficient structures. Follow these steps to create the shear and bending moment diagrams:
Draw a Free-Body Diagram: Start by drawing a free-body diagram of the entire beam, including the concentrated loads, distributed load, and reaction forces at the supports. By applying the equilibrium equations (sum of forces and moments equal to zero), the reaction forces at the supports can be determined.
Divide the Beam into Sections: Divide the beam into different sections based on the load distribution. For example, separate sections can be created for areas between concentrated loads, concentrated loads and reaction forces, and within the distributed load. Draw a free-body diagram for each section.
Determine Shear Forces: Apply the equation of equilibrium to each section's free-body diagram to calculate the shear force for individual sections. Note that the shear force remains constant between concentrated loads and reaction forces while it varies linearly with a constant slope in the distributed load section.
Calculate Bending Moments: To find the bending moments at various points along the beam, recall that the area under the shear curve between two points equals the change in bending moment between those points.
Draw Shear and Bending Moment Diagrams: Finally, create the shear force and bending moment diagrams by plotting the calculated values.
Suggested Reading
- Beer, F.P.; Johnston, E.R.; Mazurek, D.F; Cromwell, P.J. and Self, B.P.(2019). Vector Mechanics for Engineers ‒ Statics and Dynamics. New York: McGraw-Hill. Pp 394-395.