What Are Shear Force and Bending Moment?
Shear force and bending moment describe the internal forces that a loaded beam must resist to stay in equilibrium. Structural and mechanical engineers calculate them at every point along a beam to size it safely and avoid failure.
Shear force (V) is the internal force acting perpendicular to a beam's axis, while bending moment (M) is the internal moment that bends the beam; both are found from static equilibrium and vary along the beam's length depending on the loading.
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Step-by-step worked examples
A simply supported beam spans 8 m with a 10,000 N point load at midspan. Find the maximum bending moment.
M_max = P·L / 4 M_max = 10,000 × 8 / 4 M_max = 80,000 / 4 = 20,000 N·m = 20 kN·m
For the same beam, find the maximum shear force.
By symmetry, each support carries half the load: V_max = P / 2 V_max = 10,000 / 2 = 5,000 N = 5 kN
A cantilever beam of length 3 m has a 2,000 N point load at its free end. Find the maximum bending moment (at the fixed support).
For a cantilever with an end load: M_max = P × L M_max = 2,000 × 3 = 6,000 N·m = 6 kN·m
Flashcards
Quick quiz
Q1.For a simply supported beam with a central point load P over span L, what is M_max?
Q2.Shear force acts...
Q3.What is the relationship between shear force V and bending moment M?
Q4.For a cantilever beam with a point load P at the free end (length L), where is the bending moment maximum?
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Common mistakes
Assuming bending moment is maximum at the supports of a simply supported beam. — Correct: For a simply supported beam under a central load, M is zero at the supports and maximum at midspan.
Confusing shear force units (N) with bending moment units (N·m). — Correct: Shear force is a force (N); bending moment is a force times distance (N·m).
Forgetting that V and M relate by V = dM/dx. — Correct: Where the shear diagram crosses zero, the bending moment diagram has a local maximum or minimum.
Using the simply supported beam formula (PL/4) for a cantilever. — Correct: A cantilever with an end load has M_max = PL, not PL/4 — the loading and support conditions change the formula.
FAQ
What is shear force and bending moment?
Shear force is the internal transverse force in a beam, and bending moment is the internal moment causing it to bend — both found from equilibrium.
What is the bending moment formula?
For a simply supported beam with a central point load, M_max = PL/4; formulas change with loading and support type.
What are examples of shear force and bending moment problems?
Simply supported beams with point or distributed loads, cantilevers, and overhanging beams are classic examples.
How do you calculate maximum bending moment?
Draw the shear force diagram from equilibrium, then integrate (or use standard formulas like PL/4) to get the bending moment diagram and read off its peak.