Reading shear-force and bending-moment diagrams
SFD and BMD make the load path visible. Read their sign convention and boundary conditions before interpreting peaks. A diagram’s smooth appearance is not proof of equilibrium or of acceptable stress.
Worked numbers are illustrative. Use your own operating conditions and documented material/process data.
Required inputs
- Span, support positions and point/distributed loads with directions.
- Applied moments and their sign convention.
- The actual cross-section for later stress calculation.
Governing equations
ΣFy = 0; ΣM = 0
dV/dx = −w; dM/dx = V under the stated convention
ΔM = ∫V dx; σ = M/Z
Variables and units
- x: m; w: kN/m; V: kN; M: kN·m.
- Positive upward shear on the left cut and positive sagging moment are used in this example.
Assumptions
- Static loading and a simply supported straight beam for the worked example.
- Positive w denotes downward loading; support/point-force jumps are handled separately.
Calculation workflow
- Solve reactions from force and moment balance.
- Move along the beam, applying point-force shear jumps and distributed-load slopes.
- Integrate signed shear area to obtain moment changes, and check boundary moments and internal extrema.
Worked example
- L = 4 m; central point load P = 25 kN; full-span UDL w = 5 kN/m. Total downward force is 45 kN, giving RA = RB = 22.5 kN.
- Before the point load, V = 22.5 − 5x and M = 22.5x − 2.5x². At x = 2 m, M = 35 kN·m.
- The point load changes V from +12.5 to −12.5 kN; M stays continuous. At the right support M returns to zero.
Limitations
- Diagram sign conventions vary; compare physically equivalent magnitudes and directions.
- BMD/SFD do not establish buckling, connection design, load combinations or code compliance.
Common mistakes
- Assuming moment must jump at a point force.
- Ignoring an applied moment because it has no net vertical force.
- Using the drawing scale to read a value instead of the calculated ordinate.
Practical interpretation
Under UDL, shear is linear and moment is quadratic. A point force changes shear discontinuously but not moment; an applied concentrated couple changes moment. An extremum may occur where shear crosses zero or at a load discontinuity, so checking sampled nodes alone can miss the exact location.
Related formulas
The governing equations, units and calculation steps above provide the formula basis for this guide. The related workspace also shows its calculation trace.
Elementary beam moment and deflection cases →Related workspace
Open Beam Analysis →Sources and reference basis
Equations and worked examples are independently explained here. References reviewed 2026-10-17; no copyrighted standard tables are reproduced.