Shear Force and Bending Moment Diagrams — Formulas & Examples
Shear force diagrams (SFD) and bending moment diagrams (BMD) are fundamental tools in structural analysis. This reference covers the key formulas and diagram shapes for common beam loading conditions.
Basic Sign Convention
| Quantity | Positive | Negative |
|---|---|---|
| Shear force | Upward on left face | Downward on left face |
| Bending moment | Sagging (concave up) | Hogging (concave down) |
| Load direction | Upward | Downward |
Key Relationships
The fundamental differential equations governing beams:
dV/dx = -w(x) (shear rate = distributed load intensity)
dM/dx = V(x) (moment rate = shear)
d²M/dx² = -w(x) (curvature proportional to load)
Consequence: Under UDL, SFD is linear and BMD is parabolic. Under point loads only, SFD is step-wise and BMD is piecewise linear.
Simply Supported Beam — Point Load at Midspan
P
↓
|------•------|
A B
RA = RB = P/2
SFD: +P/2 -------- -P/2 (step at load point)
BMD: 0 /-------\ 0 (peak = PL/4 at midspan)
Maximum moment: M_max = PL/4 (at load)
Simply Supported Beam — UDL
w (kN/m)
↓↓↓↓↓↓↓↓↓
|---------|
A B
RA = RB = wL/2
SFD: +wL/2 \ / -wL/2 (linear, zero at midspan)
BMD: 0 (----) 0 (parabola, peak = wL²/8 at midspan)
Maximum moment: M_max = wL²/8 (at midspan) Zero shear (location of max moment): x = L/2
Cantilever — Point Load at Free End
P
↓
[========•
Fixed Free
Reaction at fixed end: R = P (upward), M_fix = PL (hogging)
SFD: P --------- P (constant)
BMD: 0 (slope up) PL (linear, max at fixed end)
Maximum moment: M_max = PL (at fixed support, hogging)
Cantilever — UDL
w (kN/m)
↓↓↓↓↓↓↓↓↓
[==========
Fixed Free
Reaction: R = wL, M_fix = wL²/2
SFD: wL --(linear)--> 0 (varies from wL at wall to 0 at tip)
BMD: wL²/2 --(parabola)--> 0
Maximum moment: M_max = wL²/2 (at fixed support, hogging)
Propped Cantilever — UDL
w
↓↓↓↓↓↓↓↓↓
[==========]
Fixed Pin
RA (pin) = 3wL/8
RB (fixed) = 5wL/8
M_B (fixed) = wL²/8 (hogging)
Point of zero shear: x = 3L/8 from pin
Max +ve moment: M = 9wL²/128 at x = 3L/8 from pin
Fixed-Fixed Beam — UDL
w
↓↓↓↓↓↓↓↓↓
[===========]
Fixed Fixed
End moments: M_end = wL²/12 (hogging)
Midspan moment: M_mid = wL²/24 (sagging)
Reactions: R = wL/2
SFD: Linear, ±wL/2 at ends, 0 at center
BMD: wL²/12 (hogging) → wL²/24 (sagging) → wL²/12 (hogging)
Point of Contraflexure
The point where moment changes sign (M = 0) is the point of contraflexure. It marks where the beam transitions from hogging to sagging (or vice versa).
For fixed-fixed beam with UDL:
- x_contraflexure = L/4 from each support
- Between these points: beam sags (positive moment)
- Outside these points: beam hogs (negative moment)
Maximum Moment Summary Table
| Beam Type | Loading | M_max | Location |
|---|---|---|---|
| Simply supported | Point load P at midspan | PL/4 | Midspan |
| Simply supported | UDL w | wL²/8 | Midspan |
| Cantilever | Point load P at tip | PL | Fixed end |
| Cantilever | UDL w | wL²/2 | Fixed end |
| Fixed-fixed | Point load P at midspan | PL/8 | Midspan & ends |
| Fixed-fixed | UDL w | wL²/12 at ends, wL²/24 at midspan | Ends |
| Propped cantilever | UDL w | wL²/8 at fixed end | Fixed end |
Fixed-end conditions significantly reduce midspan moments but introduce hogging moments at supports that must be designed for.
Moment Diagrams and Reinforcement
For reinforced concrete:
- Positive moment (sagging): tension at BOTTOM → bottom steel required
- Negative moment (hogging): tension at TOP → top steel required
The moment diagram is the reinforcement demand diagram — more moment = more steel needed.
Numerical Example
Problem: Simply supported beam, L=6m, w=15kN/m + P=30kN at 2m from left support.
Reactions:
- ΣMB = 0: RA×6 - 15×6×3 - 30×4 = 0 → RA = 65kN
- RB = 15×6 + 30 - 65 = 55kN
Shear at key points:
- x=0⁺: V = 65kN
- x=2⁻: V = 65 - 15×2 = 35kN (just left of P)
- x=2⁺: V = 35 - 30 = 5kN (just right of P)
- x=6⁻: V = 5 - 15×4 = -55kN ✓ (= -RB)
Zero shear: at x = 2 + 5/15 = 2.33m from left (max moment location)
Max moment: M = 65×2.33 - 15×2.33²/2 - 30×0.33 = 151.45 - 40.83 - 9.90 = 100.8 kN·m
→ Calculate beam reactions, SFD and BMD online →
Frequently Asked Questions
Where is the maximum bending moment in a simply supported beam with UDL? The maximum moment occurs at midspan (x = L/2) and equals wL²/8. This is where shear force equals zero — the zero-shear point always marks the maximum moment location. For UDL loading, the shear diagram is linear with +wL/2 at the left support decreasing to −wL/2 at the right support, crossing zero at exactly midspan.
How do I find the location of maximum moment for an off-center point load? Maximum moment occurs at the point load location. For a load P at distance a from the left support, the moment there equals R_A × a = (Pb/L) × a = Pab/L. The shear is constant between supports (with a step at the load), so the moment diagram is piecewise linear with a peak directly under the load.
What is a point of contraflexure and where does it occur? A point of contraflexure is where the bending moment equals zero and changes sign (from sagging to hogging or vice versa). For a fixed-fixed beam with UDL, contraflexure points occur at x = L/4 from each support — within these inner two points the beam sags, outside them it hogs. These points are structurally significant because a pinned connection at the contraflexure point introduces no additional moment.
Why does fixing a beam's ends reduce the midspan moment? End fixity introduces negative (hogging) moments at the supports that partially cancel the positive (sagging) moment at midspan. For a fixed-fixed beam with UDL, end moments = wL²/12 hogging, reducing midspan moment to wL²/24 — one-third of the simply supported wL²/8. However, the support connections must be designed for the full fixed-end moment; if they yield, the structure redistributes toward simply supported behavior.
What is the difference between sagging and hogging moments in reinforced concrete? Sagging (positive) moments put the bottom fiber in tension — reinforcement must be placed at the bottom of the beam. Hogging (negative) moments put the top fiber in tension — reinforcement must be placed at the top. Continuous beams have both: positive midspan moments requiring bottom steel and negative support moments requiring top steel. The moment diagram is the demand map for reinforcement placement.
Run This Calculation
→ Beam Calculator — compute reactions, shear force diagrams, and bending moment diagrams for any loading pattern.
→ Beam Deflection Calculator — calculate deflections for simply supported, cantilever, and fixed beam configurations.
→ Continuous Beam Calculator — reactions, moments, and deflections for multi-span beams with fixed or pinned ends.
Related pages
- Beam Calculator
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- Beam Capacity Calculator
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- W-Shape Beam Sizes — Dimensions, Sx, Ix, Zx Properties Table
- How Far Can a Steel Beam Span? — W-Shape Span Guide
- Steel Beam Load Tables — W-Shape Allowable Uniform Load
- How to verify calculator results
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