4.1 Shear Forces and Bending Moments

4.1 Shear Forces and Bending Moments

Introduction to Internal Forces

• When external loads are applied to a structural member (like a beam), they induce internal forces within the member to maintain equilibrium.

• These internal forces are typically resolved into components: Axial Force, Shear Force, and Bending Moment.

• Understanding the variation of these forces along the length of the member is fundamental to structural analysis and design, as it determines stress distribution and required member strength.

• This section establishes the definitions, sign conventions, and methods for determining these critical internal actions.

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1. Internal Force Components

• To analyze a beam, an imaginary cut (section) is made. The internal forces that must act at this cut to maintain equilibrium of either segment are categorized as follows.

1.1 Axial Force (N)

• Core Definition: The internal force component acting perpendicular to the cross-section, parallel to the longitudinal axis of the member. It tends to elongate or shorten the member.

  $\boldsymbol{N}$

• Sign Convention:

  • Positive (+): Tension. Force pulling away from the cut section, elongating the member.

  • Negative (-): Compression. Force pushing into the cut section, shortening the member.

• Physical Effect: Produces uniform normal stress across the cross-section.

  $\boldsymbol{\sigma_{axial} = \frac{N}{A}}$

  where, $\boldsymbol{\sigma_{axial}}$ is the axial stress and $\boldsymbol{A}$ is the cross-sectional area.

• Typical Members: Truss members, columns, tension rods.

1.2 Shear Force (V)

• Core Definition: The internal force component acting parallel to (in the plane of) the cross-section. It tends to cause one part of the beam to slide vertically relative to the adjacent part.

  $\boldsymbol{V}$

• Sign Convention (Common for Beams):

  • Positive (+): Causes a clockwise rotation of the beam segment on which it acts.

  • Negative (-): Causes a counter-clockwise rotation of the beam segment on which it acts. (Visual Aid: For the left segment of a cut, downward shear is positive; for the right segment, upward shear is positive).

• Physical Effect: Produces shear stress distributed non-uniformly across the cross-section.

• Primary Cause: Transverse loads (point loads, distributed loads) applied perpendicular to the member's axis.

1.3 Bending Moment (M)

• Core Definition: The internal moment (couple) acting about the neutral axis of the cross-section. It tends to bend the member by causing curvature.

  $\boldsymbol{M}$

• Sign Convention (Sagging/Hogging):

  • Positive (+): Sagging Moment. Tension in the bottom fibers, compression in the top fibers. Beam curvature is concave upwards.

  • Negative (-): Hogging Moment. Tension in the top fibers, compression in the bottom fibers. Beam curvature is concave downwards.

• Physical Effect: Produces linearly varying normal (bending) stress across the depth of the section.

  $\boldsymbol{\sigma_{bending} = \frac{M y}{I}}$

  where, $\boldsymbol{\sigma_{bending}}$ is the bending stress, $\boldsymbol{y}$ is the distance from the neutral axis, and $\boldsymbol{I}$ is the second moment of area (moment of inertia).

• Primary Cause: Any load that does not pass through the shear center, creating a rotational effect.

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2. Relationship Between Load, Shear, and Moment

• Differential relationships exist between the applied transverse load intensity, the shear force, and the bending moment. These are derived from equilibrium considerations on a differential beam element.

2.1 Differential Relationships

Consider a beam element of length $\boldsymbol{dx}$ under a distributed load $\boldsymbol{w(x)}$ (positive upwards).

• Load - Shear Relationship: The rate of change of shear force equals the negative of the load intensity.

  $\boldsymbol{\frac{dV}{dx} = -w(x)}$

• Shear - Moment Relationship: The rate of change of bending moment equals the shear force.

  $\boldsymbol{\frac{dM}{dx} = V(x)}$

• Load - Moment Relationship (Second Derivative): Combining the above gives the relationship between bending moment and load intensity.

  $\boldsymbol{\frac{d^2M}{dx^2} = -w(x)}$

2.2 Integral (Graphical) Relationships

• The change in shear force between two points is equal to the negative of the area under the load diagram between those points.

  $\boldsymbol{V_B - V_A = -\int_{A}^{B} w(x) \, dx}$

• The change in bending moment between two points is equal to the area under the shear force diagram between those points.

  $\boldsymbol{M_B - M_A = \int_{A}^{B} V(x) \, dx}$

• Key Points from Relationships:

  • At a point load location, the shear force diagram has a jump discontinuity equal to the magnitude of the point load.

  • At a concentrated moment location, the bending moment diagram has a jump discontinuity equal to the magnitude of the moment.

  • The shear force is zero at points of local maximum or minimum bending moment.

  • The slope of the shear diagram at a point equals the negative load intensity at that point.

  • The slope of the moment diagram at a point equals the shear force at that point.

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3. Load Superposition

• Principle of Superposition: For linear elastic structures subjected to small deformations, the total internal force (or deflection) at any point due to a set of loads acting simultaneously is equal to the algebraic sum of the internal forces caused by each load acting separately.

• Condition for Validity: The material must obey Hooke's Law (linear stress-strain), and deformations must be small enough that the geometry of the structure does not change significantly under load.

• Application: This principle is extremely powerful for analyzing complex loading conditions.

  • Solve for shear ($\boldsymbol{V_i}$) and moment ($\boldsymbol{M_i}$) diagrams for each individual load case.

  • The final diagrams are obtained by algebraic addition: $\boldsymbol{V_{total} = \sum V_i}$ and $\boldsymbol{M_{total} = \sum M_i}$.

• Limitation: Does not apply to structures where load path or stiffness changes with deformation (e.g., cables, some buckling problems).

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4. Diagram Interpretation and Key Features

• Drawing and interpreting shear force and bending moment diagrams is a core skill. The diagrams tell a story about how the beam resists the applied loads.

4.1 Common Loading Patterns and Their Diagrams

Load Type on Segment	Shear Force Diagram (V)	Bending Moment Diagram (M)
No Load ($\boldsymbol{w=0}$)	Constant value (horizontal line).	Linear (inclined straight line). Slope = V.
Uniform Load ($\boldsymbol{w=c}$)	Linear (inclined straight line). Slope = -w.	Parabolic (second-degree curve). Concavity faces opposite load direction (down for +w).
Point Load	Vertical jump at load point. Magnitude of jump = load value.	A sharp change in slope (kink) at load point.
Concentrated Moment	No change.	Vertical jump at moment location. Magnitude of jump = moment value.

4.2 Procedure for Constructing Diagrams

1. Calculate Reactions: Use equilibrium equations ($\boldsymbol{\sum F_y = 0}$, $\boldsymbol{\sum M = 0}$) on the entire beam to find support reactions.

2. Section the Beam: Make a cut at a variable distance $\boldsymbol{x}$ from one end.

3. Apply Equilibrium: Use equilibrium on the free-body diagram of one segment to write equations for $\boldsymbol{V(x)}$ and $\boldsymbol{M(x)}$.

4. Plot the Functions: Plot the equations over the relevant domains. Alternatively, use the integral/graphical method based on area calculations.

5. Identify Critical Points: Locate points where $\boldsymbol{V=0}$, as these correspond to local maxima/minima in the moment diagram.

4.3 Information from Diagrams

• Shear Force Diagram:

  • Magnitude of maximum shear, critical for designing against shear failure.

  • Points of zero shear indicate locations of maximum bending moment.

• Bending Moment Diagram:

  • Magnitude and location of maximum positive and negative moments. This is the primary input for flexural design (sizing beams).

  • The shape indicates regions of tension and compression.

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Summary

• Axial Force (N) acts along the member's axis, causing uniform tension or compression.

• Shear Force (V) acts transversely across the section, resisting sliding failure, and is the integral (negative area) of the load diagram.

• Bending Moment (M) acts to bend the member, creating flexural stresses, and is the integral (area) of the shear force diagram.

• The differential relationships ($\boldsymbol{dV/dx = -w}$, $\boldsymbol{dM/dx = V}$) provide a fundamental link between load, shear, and moment.

• Superposition allows the analysis of complex loads by breaking them into simpler, independent cases.

• Correctly interpreting Shear Force and Bending Moment Diagrams is essential for identifying critical sections, maximum values, and ultimately for the safe and efficient design of beams and other flexural members.