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…
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.
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Q1: What is the first step in drawing shear and bending moment diagrams?
Start by drawing a free-body diagram of the entire beam, including all concentrated loads, distributed loads, and reaction forces at the supports. Apply equilibrium equations (sum of forces and moments equal zero) to determine the reaction forces. This foundation is essential before analyzing internal forces in different sections.
Q2: How does shear force behave in different sections of a loaded beam?
Shear force remains constant between concentrated loads and reaction forces, creating horizontal segments on the shear diagram. In sections with a distributed load, shear varies linearly with a constant slope. Understanding this relationship between the distributed load and shear helps predict diagram shape without calculating every point.
Q3: Why is the area under the shear curve important for finding bending moments?
The area under the shear curve between two points equals the change in bending moment between those same points. This relationship allows you to calculate bending moments at various locations by integrating shear values. Starting from a known bending moment (typically zero at a free end), you can determine moments throughout the beam.
Q4: How do you divide a beam into sections for analysis?
Divide the beam based on load distribution: create separate sections between concentrated loads, between concentrated loads and reaction forces, and within distributed load regions. Draw a free-body diagram for each section and apply equilibrium equations to calculate shear forces. This systematic approach ensures accurate internal force determination across the entire beam.
Q5: What shapes appear in a bending moment diagram?
Connect known bending moment points with straight lines in regions where shear is constant, and parabolic curves in regions where shear varies linearly. The diagram shape reflects the underlying shear distribution: constant shear produces linear moment segments, while linear shear produces parabolic segments, creating a visual representation of internal moment distribution.
Q6: Why is sectioning the beam necessary for solving shear and bending moment problems?
Sectioning isolates portions of the beam to reveal internal forces and moments at specific locations. By drawing free-body diagrams of each section and applying equilibrium equations, you can calculate shear and bending moment values that vary along the beam's length. This method transforms a complex problem into manageable steps for internal loadings structural members problem solving.
Q7: How do concentrated loads affect the shear diagram?
Concentrated loads create vertical jumps in the shear diagram at their points of application. The magnitude of each jump equals the load value, and the direction indicates whether the load acts upward or downward. Between concentrated loads, shear remains constant, producing horizontal segments that simplify diagram construction and interpretation.