Module 5: Shear and Moment in Beams

Locating the maximum internal forces for structural design.
Beams are the fundamental horizontal structural members in almost every building. They primarily resist loads applied laterally (perpendicularly) to their longitudinal axis. This module explores how to systematically analyze these beams under various loading conditions, an absolutely essential skill for sizing floor joists, steel girders, and concrete lintels in architectural design.

Key Concepts

Checklist

Beam

A structural element that primarily resists loads applied laterally to its longitudinal axis. Its primary mode of deflection is by bending.

Types of Beams and Loads

Classification based on supports and loading
Beams are classified in structural analysis based on their support conditions:
  • Simply Supported Beam: Supported by a pin on one end (restricting vertical and horizontal translation) and a roller on the other (restricting only vertical translation). It is statically determinate.
  • Cantilever Beam: Fixed rigidly at one end (restricting all translation and rotation) and completely free at the other. It is statically determinate.
  • Overhanging Beam: Supported at two points (like a simply supported beam) but extending continuously beyond one or both of those supports.
  • Continuous Beam: Supported at more than two points. This is statically indeterminate and requires advanced analysis methods (like Three-Moment Equation or Moment Distribution) covered in higher-level courses.
Loads applied to beams can be categorized as:
  • Concentrated Loads (Point Loads): A force applied at a single, specific point on the beam (e.g., a column resting on a girder, represented as 10kN10 kN).
  • Uniformly Distributed Loads (UDL): A force spread evenly over a length of the beam (e.g., the dead weight of a concrete floor slab, represented as 5kN/m5 kN/m).
  • Uniformly Varying Loads (UVL): A distributed load whose intensity varies linearly along the beam (like the hydrostatic pressure of water against a retaining wall, or a triangular snow load).
  • Concentrated Moments (Couples): A pure turning force applied directly at a point on the beam, creating rotation without a net lateral force.
Key Takeaways
  • Beams are classified by supports (simply supported, cantilever, overhanging, continuous).
  • Loads can be concentrated (point), uniformly distributed (UDL), uniformly varying (UVL), or concentrated moments.

Internal Shear and Bending Moment

Forces acting within the beam's cross-section
When a beam is loaded externally, it must not move (static equilibrium). To maintain this equilibrium, internal forces and moments develop continuously throughout every cross-section of its length to resist the external loads.
If you conceptually cut a loaded beam at any point xx, the segment must remain in equilibrium. This requires an internal vertical force (VV) and an internal rotational moment (MM) acting on the cut face.

Shear Force (VV)

The algebraic sum of all the vertical forces acting to one side (either left or right) of a specific section of a beam. It represents the tendency for one part of the beam to slide vertically past the adjacent part.

Bending Moment (MM)

The algebraic sum of the moments of all the forces acting to one side (either left or right) of a specific section of a beam about an axis passing through the centroid of that section. It represents the internal resistance to the beam's tendency to bend or flex.
Key Takeaways
  • Internal shear (VV) prevents vertical sliding; internal moment (MM) prevents bending.
  • These internal forces vary continuously along the length of the beam.

Relationship Between Load, Shear, and Moment

The calculus of beam analysis
The mathematical relationships between distributed load intensity w(x)w(x), internal shear force V(x)V(x), and internal bending moment M(x)M(x) are the foundational principles of beam analysis and the basis for constructing accurate diagrams.

Load and Shear Relationship

The rate of change of the shear force with respect to distance (xx) is equal to the negative of the downward distributed load intensity.
dVdx=−w(x) \frac{dV}{dx} = -w(x)
Visually, this means the change in shear (ΔV\Delta V) between two points is equal to the negative of the area under the load curve between those same two points. If there is no distributed load between two points, the shear is constant (a horizontal line on the diagram).

Shear and Moment Relationship

The rate of change of the bending moment with respect to distance (xx) is equal to the shear force at that point.
dMdx=V(x) \frac{dM}{dx} = V(x)
This means the change in bending moment (ΔM\Delta M) between two points is equal to the area under the shear curve between those two points.
Crucially, from calculus, a function reaches a local maximum or minimum where its derivative is zero. Therefore, when the shear force crosses zero (V=0V=0), the bending moment reaches a critical maximum or minimum value. Finding where V=0V=0 is the most important step in beam design.
Key Takeaways
  • The change in shear (ΔV\Delta V) is the area under the load diagram.
  • The change in moment (ΔM\Delta M) is the area under the shear diagram.
  • Maximum (or minimum) bending moment always occurs exactly where the shear force is zero (V=0V=0).

Moving Loads

Analyzing vehicles and cranes
The static analysis we perform on fixed point loads and UDLs is highly useful for stationary architectural components like walls and dead load weights. However, beams such as highway bridge girders, monorail tracks, and warehouse gantry cranes are subjected to moving loads—a series of concentrated wheel loads that travel along the span.
As a multi-axle vehicle moves across a simply supported beam, the shear forces and bending moments at every single cross-section constantly change. The engineering challenge is not simply to find the maximum moment for one load position, but to find the absolute maximum shear and absolute maximum moment that will ever occur anywhere on the beam during the entire transit of the vehicle.

Absolute Maximum Shear

For a simply supported beam carrying a moving vehicle (like a truck with two or three axles), the absolute maximum shear force will always occur directly at the reaction support (either the left or right support) when one of the heaviest outer wheels is positioned exactly over or infinitesimally close to that support.

Absolute Maximum Moment

Finding the absolute maximum bending moment is more complex. The maximum moment will always occur directly under one of the concentrated wheel loads (usually the heaviest one nearest the center of gravity of the entire vehicle group).
The exact position of the vehicle that produces this absolute maximum moment is governed by a specific geometric rule:
The absolute maximum bending moment occurs under a specific load when the centerline of the beam span exactly bisects the distance between that specific load and the resultant center of gravity of the entire moving load group.
Key Takeaways
  • Moving loads create constantly changing internal forces as a vehicle transits a beam.
  • Absolute maximum shear always occurs at a support when a heavy wheel is over it.
  • Absolute maximum moment occurs under a specific wheel when the beam centerline bisects the distance between that wheel and the resultant load group's center of gravity.

Graphical Method for Drawing S&M Diagrams

The Area Method
While writing out continuous equations for V(x)V(x) and M(x)M(x) works, the graphical "Area Method" based on the calculus relationships above is far faster and less prone to error for engineers.
Key Takeaways
  • Use the Area Method to quickly draw S&M diagrams without writing complex calculus equations.
  • Follow the degree rule: constant load →\rightarrow linear shear →\rightarrow parabolic moment.

Singularity Functions (Macaulay's Method)

A single mathematical expression for shear and moment

Macaulay's Method

While drawing shear and moment diagrams section-by-section is intuitive, it requires writing separate equations for each segment of the beam between concentrated loads or reactions. Singularity Functions (or Macaulay's Brackets) allow us to write a single, continuous mathematical expression for the shear force (VV) and bending moment (MM) across the entire length of the beam.
A Macaulay bracket ⟨x−a⟩n\langle x - a \rangle^n is defined as zero if x<ax < a, and (x−a)n(x - a)^n if x≥ax \ge a. This acts as an "on-switch" for loads, mathematically introducing them only when moving past their point of application (x=ax = a). This method is particularly powerful for finding beam deflections via integration.
Key Takeaways
  • Singularity functions use Macaulay brackets ⟨x−a⟩n\langle x - a \rangle^n to write a single equation for shear or moment across an entire beam.
  • The bracket evaluates to zero if the expression inside is negative, and acts as a normal parenthesis if positive.
PreviousModule 4: Torsion - Theory & Concepts
NextShear and Moment in Beams