Beams - Bending Moments and Shear Forces - Topic 9 - Lecture Notes

Oct 27, 2026

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Equilibrium of beams under externally applied transverse loads and couples is developed through the internal action results at a cut section: the bending moment M and the shear force V. The notes establish sign conventions for M and V on left and right faces, relate internal forces to normal and shear stresses, and use equilibrium relations to connect loading p(x) and concentrated loads/couples with diagram-ready expressions. Applications include analysis of simply supported beams by computing reactions and deriving V(x) and M(x) over a span, supporting accurate bending-moment and shear-force diagrams.

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Equilibrium in beams: bending moments and shear forces

Objectives

Background: internal moment and shear at a cut

Lecture topics

Applications: beams under transverse loads

Sign conventions for bending moments and shear forces

Sign conventions for external loadings on beams

Example 9.2: simply-supported beam reactions and internal forces

Equilibrium relations for bending moments and shear forces

9. Equilibrium in beams: bending moments and shear forces Objectives: To understand the distribution of internal bending moments and internal shear forces at cross sections along the length of a beam with externally-applied transverse loads and couples. Background: • If a cut is made through the cross section of a beam, a bending moment M and shear force V must be applied at the cut in order to maintain equilibrium of the beam: at cut • V and M are force/couple resultants of the normal and shear stresses acting on the face of a cross section. Lecture topics: a) Sign conventions for bending moments and shear forces. b) Equilibrium relations for bending moments and shear forces. c) Bending-moment and shear-force diagrams. Applications Beams are structural members that are designed to support transverse loads, that is, loads that are perpendicular to the longitudinal axis of the beam. A beam resists the applied loads by a combination internal transverse shear force and bending moment. Beams: Bending moments and shear forces Topic 9: 2 Mechanics of Materials Lecture Notes a) Sign conventions for bending moments and shear forces Sign conventions to be used in this course for internal bending moments and shear forces (see following figure): • A positive bending moment M on the left face (negative x-face) of a section is CW. A positive bending moment M on the right face (positive x-face) of a section is CCW. Such a positive bending moment creates a concave curvature in the deflection of the beam. • A positive shear force V on the left face (negative x-face) of a section is in positive y-direction. A positive shear force V on the right face (positive x-face) of a section is in negative y-direction. y M + M + V + V + positive bending positive shear moment force When making a cut through a cross section of the beam, the positive sign conventions for the bending moment and shear force are as shown below: y x positive bending moment and shear force at cut in beam Beams: Bending moments and shear forces Topic 9: 3 Mechanics of Materials Sign conventions for external loadings on beams: • Positive EXTERNAL distributed loads p(x) and EXTERNAL concentrated loads P0 act in the “+”y-direction: • Positive EXTERNAL couples are in the “+” z-direction (CCW by the right hand rule): P0 ( positive UPWARD) Beams: Bending moments and shear forces Topic 9: 4 Mechanics of Materials Example 9.2 For the simply-supported beam loaded as shown: a) Determine the reactions on the beam at A and D. b) Determine the internal shear force and bending moment, V (x) and M (x) , for an arbitrary location x between A and B. Beams: Bending moments and shear forces Topic 9: 5 Mechanics of Materials b) Equilibrium relations for bending moments and shear forces | applied loading | | FBD | | | | | key rela1onship(s) | | --- | --- | --- | --- | --- | --- | --- | --- | | p(x)

x | | V(x)
p(x) Δx V(x + Δx)

M(x) M(x + Δx)Δx | | | | |
| | | P 0 | V (x − ) | | P 0

V (x+) | | | V (x+ ) = V (x − ) + P0 | | x | | | | | | | | | M0
x

| | M0 | | | | | M (x+) = M (x − ) − M0
| | | |
M(x − )

| |
| | M(x+) | | The derivations ofthe above key relationships are to be added below: Beams: Bending moments and shear forces Topic 9: 6 Mechanics of Materials

FAQs

Why must bending moment M and shear force V be applied at a beam cut to maintain equilibrium?

A cut through a beam’s cross section requires internal actions that balance the stress resultants on that section. The internal bending moment M and shear force V are the force/couple resultants needed to keep the beam in equilibrium.

What are the sign conventions for a positive bending moment and positive shear force on a section face?

A positive bending moment M on the left face (negative x-face) is clockwise (CW), while on the right face (positive x-face) it is counterclockwise (CCW). A positive shear force V on the left face acts in the positive y-direction, and on the right face it acts in the negative y-direction.

How do equilibrium relations connect loading to changes in shear force and bending moment?

The notes present key relationships linking p(x), concentrated loads P0, and couple effects to the step changes between x− and x+. Specifically, shear changes by the applied concentrated load term, and bending moment changes according to the moment effect at the cut.

Oct 27, 2026

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