Beam Analysis: Shear and Moment Diagrams
Key Takeaways
- Shear force V and bending moment M vary along a beam and are found by cutting the beam and applying equilibrium.
- The shear diagram shows V(x); the moment diagram shows M(x). Key relationships: dV/dx = -w(x) and dM/dx = V(x).
- At a concentrated load, the shear diagram has a vertical jump equal to the load.
- At a concentrated moment, the moment diagram has a vertical jump equal to the applied moment.
- Maximum bending moment occurs where V = 0 (shear crosses zero) or at a support.
- For simply supported beams with a single central load P: Mmax = PL/4 at midspan.
Last updated: March 2026
Beam Analysis: Shear and Moment Diagrams
Sign Convention
| Quantity | Positive | Negative |
|---|---|---|
| Shear (V) | Causes clockwise rotation | Causes counterclockwise rotation |
| Moment (M) | Causes concave-up bending (smile) | Causes concave-down bending (frown) |
Relationships Between Load, Shear, and Moment
These imply:
- Shear at a point = negative integral of the distributed load
- Moment at a point = integral of the shear
- Slope of shear diagram = negative of the distributed load intensity
- Slope of moment diagram = value of the shear
Key Rules for Drawing Diagrams
| Loading | Shear Diagram | Moment Diagram |
|---|---|---|
| No load | Constant (horizontal) | Linear (straight line) |
| Uniform distributed load (w) | Linear (sloping) | Parabolic |
| Triangular distributed load | Parabolic | Cubic |
| Concentrated force P | Vertical jump of P | Change in slope (kink) |
| Concentrated moment M₀ | No change | Vertical jump of M₀ |
Critical Points
- V = 0: Maximum or minimum bending moment
- Change in sign of V: Local extremum in moment
- Supports: Check reactions at supports
Common Beam Configurations
Simply Supported Beam — Central Point Load P
| Quantity | Formula |
|---|---|
| Reactions | RA = RB = P/2 |
| Max Shear | V = P/2 (at supports) |
| Max Moment | Mmax = PL/4 (at midspan) |
| Max Deflection | δmax = PL³/(48EI) (at midspan) |
Simply Supported Beam — Uniform Distributed Load w
| Quantity | Formula |
|---|---|
| Reactions | RA = RB = wL/2 |
| Max Shear | V = wL/2 (at supports) |
| Max Moment | Mmax = wL²/8 (at midspan) |
| Max Deflection | δmax = 5wL⁴/(384EI) (at midspan) |
Cantilever Beam — End Point Load P
| Quantity | Formula |
|---|---|
| Reactions | R = P (upward), M = PL (CCW at wall) |
| Max Shear | V = P (constant along beam) |
| Max Moment | Mmax = PL (at fixed support) |
| Max Deflection | δmax = PL³/(3EI) (at free end) |
Cantilever Beam — Uniform Distributed Load w
| Quantity | Formula |
|---|---|
| Reactions | R = wL (upward), M = wL²/2 (CCW at wall) |
| Max Shear | V = wL (at fixed support) |
| Max Moment | Mmax = wL²/2 (at fixed support) |
| Max Deflection | δmax = wL⁴/(8EI) (at free end) |
Test Your Knowledge
A simply supported beam of length 8 m carries a uniform distributed load of 5 kN/m. What is the maximum bending moment?
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B
C
D
Test Your Knowledge
Where does the maximum bending moment typically occur?
A
B
C
D