Shear and Moment in Beams

Beams are structural members designed primarily to resist transverse (perpendicular) loads. These external loads, along with support reactions, create internal Shear Forces (VV) and Bending Moments (MM) that vary along the longitudinal axis. To safely design a beam, engineers must plot these variations to locate the points of maximum internal stress.

Types of Beams and Supports

Beam Types & Support Conditions

  • Simply Supported: Pinned at one end, Roller at the other. Statically Determinate.
  • Cantilever: Fixed at one end, Free at the other. Statically Determinate.
  • Overhanging: Supported at two points with one or both ends extending beyond the supports. Statically Determinate.
  • Continuous: Supported at more than two points. Statically Indeterminate.
  • Propped Cantilever: Fixed at one end, Roller at the other. Statically Indeterminate.

Method of Sections (Equation Method)

The most fundamental way to determine internal shear and moment at any point in a beam is the Method of Sections. This involves cutting the beam at a specific point and applying equilibrium equations to the isolated segment.

Sign Convention and Procedure

Standard Sign Convention:
  • Positive Shear (VV): Tends to rotate the cut segment clockwise. (e.g., pointing DOWN on the right face of a left-hand segment).
  • Positive Moment (MM): Tends to bend the segment concave upward, creating a "smiling" shape. This puts the top fibers in compression and bottom fibers in tension.
Procedure:

Procedure

  1. Calculate Reactions: Draw a Free Body Diagram (FBD) of the entire beam and use static equilibrium (∑F=0,∑M=0\sum F = 0, \sum M = 0) to find support reactions.
  2. Section the Beam: Make an imaginary cut at a distance xx from the origin. Make a new cut whenever the loading condition changes (e.g., after a point load or at the start of a uniform load).
  3. Draw FBD of Segment: Draw the isolated segment (usually the left side for convenience). Place the unknown internal shear V(x)V(x) and moment M(x)M(x) at the cut face acting in their positive directions.
  4. Apply Equilibrium:
    • ∑Fy=0\sum F_y = 0 yields the equation for V(x)V(x).
    • ∑Mcut=0\sum M_{cut} = 0 yields the equation for M(x)M(x).

Load, Shear, and Moment Relationships (Area Method)

Writing equations for every segment is tedious. Because load, shear, and moment are mathematically related by differential calculus, we can construct the diagrams graphically using the Area Method.

Differential Relationships

The relationship between Load (ww), Shear (VV), and Moment (MM) is defined by differential calculus:
1. Slope of Shear Diagram = Load Intensity
dVdx=w(x)\frac{dV}{dx} = w(x)
(Note: Sign convention matters. Downward load usually corresponds to a negative slope).
2. Slope of Moment Diagram = Shear Force
dMdx=V(x)\frac{dM}{dx} = V(x)
This means the shear value at any point is the slope of the moment curve at that point.

Area Method Rules

  • Change in Shear: V2−V1=∫x1x2w dxV_2 - V_1 = \int_{x_1}^{x_2} w \, dx (Area under load diagram).
  • Change in Moment: M2−M1=∫x1x2V dxM_2 - M_1 = \int_{x_1}^{x_2} V \, dx (Area under shear diagram).
  • Degree of Curve: If load is degree nn, Shear is degree n+1n+1, Moment is degree n+2n+2.
    • Point Load (deg 0, singular) →\rightarrow Constant Shear (deg 0) →\rightarrow Linear Moment (deg 1).
    • Uniform Load (deg 0) →\rightarrow Linear Shear (deg 1) →\rightarrow Parabolic Moment (deg 2).

Maximum Bending Moment

Because dMdx=V(x)\frac{dM}{dx} = V(x), the bending moment reaches a local maximum or minimum at points where the shear force is zero (V=0V = 0) or where it passes through zero. Locating the points of zero shear is critical for beam design.

Point of Contraflexure (Inflection Point)

The Point of Contraflexure (or Inflection Point) is the location on a beam where the bending moment is exactly zero (M=0M = 0).
At this point, the beam's curvature changes from concave upward (positive moment, sagging) to concave downward (negative moment, hogging).
Significance in Design:
  • At this point, flexural stress is zero.
  • In reinforced concrete, this point dictates where top longitudinal reinforcement (for negative moment) can be terminated or shifted to the bottom (for positive moment).
  • It is a common location to place splices or hinges in continuous structures, as they will only need to transmit shear forces, not bending moments.

Moving Loads and Absolute Maximum Bending Moment

Moving Loads

While static loads are fixed in position, moving loads (like trucks on a bridge or crane hoists on a runway beam) change position over time. To design structures carrying moving loads, engineers must find the absolute maximum internal forces (shear and moment) that occur as the load system traverses the entire span.
For a system of moving point loads, the maximum shear usually occurs at the supports when the heaviest load is positioned directly over or adjacent to the support.

Absolute Maximum Bending Moment

For a single point load, the maximum moment clearly occurs when the load is at the midspan of a simply supported beam. However, for a train of wheel loads, the Absolute Maximum Bending Moment occurs under a specific load when the entire load system is positioned such that:
Rule: The centerline of the beam span perfectly bisects the distance between the Resultant of all loads and the specific Wheel Load under consideration.
Procedure:

Procedure

  1. Calculate the magnitude and position (centroid) of the Resultant force (RR) of the entire moving load system.
  2. Assume the absolute maximum moment will occur under one of the heaviest loads near the Resultant (call this load PiP_i).
  3. Position the load system on the beam so that the centerline of the beam is exactly halfway between PiP_i and the Resultant RR.
  4. Calculate the beam reactions for this specific position.
  5. Calculate the bending moment under the load PiP_i.
  6. Repeat steps 2-5 for other heavy loads near the resultant to find the absolute largest value.

Macaulay's Method (Singularity Functions)

Singularity functions (also called Macaulay functions) provide a single mathematical expression for the load, shear, and bending moment over the entire length of a beam, regardless of discontinuous loads.

Singularity Brackets \langle x - a \rangle^n

A singularity function is defined by pointed brackets. It effectively "turns on" only when x>ax > a.
Definition:
  • If x<ax < a: ⟨x−a⟩n=0\langle x - a \rangle^n = 0
  • If x≥ax \ge a: ⟨x−a⟩n=(x−a)n\langle x - a \rangle^n = (x - a)^n
Integration Rules:
For n≥0n \ge 0:
∫⟨x−a⟩ndx=⟨x−a⟩n+1n+1\int \langle x - a \rangle^n dx = \frac{\langle x - a \rangle^{n+1}}{n+1}
For concentrated moments (n=−2n = -2) or forces (n=−1n = -1), integration increments the power by 1 but without dividing by the new exponent:
∫⟨x−a⟩−2dx=⟨x−a⟩−1\int \langle x - a \rangle^{-2} dx = \langle x - a \rangle^{-1}
∫⟨x−a⟩−1dx=⟨x−a⟩0\int \langle x - a \rangle^{-1} dx = \langle x - a \rangle^0

Interactive Tool: Beam Analysis

Experiment with different beam supports and loads to see how the Shear and Moment diagrams change in real-time.

Shear & Moment Diagram Generator

Simply supported beam with a single concentrated point load.

10 kN
RAR_A = 5.0
RBR_B = 5.0
10 m

Shear Force Diagram (VV)

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Bending Moment Diagram (MM)Max: 25.0 kN·m

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Key Takeaways
  • Shear (VV) is the algebraic sum of the vertical forces to one side of the section.
  • Moment (MM) is the algebraic sum of the moments of all forces to one side of the section.
  • Sign Convention: Shear is positive when it causes clockwise rotation; Moment is positive when it causes sagging.
  • Differential Relationships: Load is the derivative of Shear; Shear is the derivative of Moment.
  • Zero Shear: The point where the shear diagram crosses zero corresponds to a local maximum or minimum bending moment.
  • Point of Contraflexure: The point where the bending moment is zero (M=0M=0) and curvature reverses, which is critical for reinforcement detailing.
  • Moving Loads: Require calculating the envelope of maximum effects. The Absolute Maximum Bending Moment occurs when the beam centerline bisects the distance between the Resultant and the critical load.
  • Singularity Functions allow a single equation to define shear or moment across an entire beam, handling discontinuous loads systematically.
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