Define Shear Force: Formula, Examples & Calculation
Learn to define shear force and bending moment, understand key formulas, and discover how to calculate shear force with a simple beam example.
STRUCTURAL ENGINEERING
10/9/20267 min read
Shear Force: Definition, Shear Force and Bending Moment, and How to Calculate Shear Force
Introduction
In civil engineering and structural analysis, understanding shear force and bending moment is essential for designing safe and stable structures. Beams in buildings, bridges, and other structures carry different types of loads, including dead loads, live loads, and point loads. These loads create internal forces that help us understand how a structural member behaves under loading.
If you are a civil engineering student, site engineer, or preparing for a technical interview, learning how to calculate shear force is an important skill. In this guide, we will explain the definition of shear force, the relationship between shear force and bending moment, important formulas, and a practical calculation example.
What Is Shear Force?
Shear force is the internal force that acts parallel to a cross-section of a structural member and resists the tendency of one part to slide relative to the adjacent part.
In a beam, shear force develops due to applied loads and support reactions. It is commonly measured in newtons (N) or kilonewtons (kN).
Table of Contents
Define Shear Force
What Causes Shear Force in a Beam?
Define Shear Force and Bending Moment
Difference Between Shear Force and Bending Moment
How to Calculate Shear Force
Shear Force Calculation Example
Shear Force Diagram (SFD)
Types of Loads and Their Effects on Shear Force
Applications of Shear Force in Civil Engineering
Frequently Asked Questions
Conclusion
1. Define Shear Force
Shear force is the internal transverse force that develops within a structural member when external loads and support reactions act on it. It resists the relative sliding of one part of the member against another.
For example, consider a simply supported beam carrying a point load at its centre. The supports provide upward reactions, while the applied load acts downward. These forces create internal shear forces within the beam.
Shear force is determined by considering the equilibrium of the forces acting on one side of a section cut through the beam.
Shear Force Formula
The general relationship is:
Shear force = Algebraic sum of vertical forces on one side of a section
[
V=\sum F_y
]
Where:
(V) = internal shear force
(F_y) = vertical forces, including loads and support reactions
(\sum F_y) = algebraic sum of the vertical forces
The sign of shear force depends on the sign convention used in the analysis.
Why Is Shear Force Important?
Shear force analysis helps engineers determine the internal forces acting on beams and other structural members. It is used to:
Design beams and structural elements safely.
Identify sections with high shear forces.
Calculate the required shear reinforcement in reinforced concrete beams.
Draw shear force diagrams.
Support structural analysis and bending moment calculations.
2. What Causes Shear Force in a Beam?
Shear force develops when external loads and support reactions produce a net transverse force across a section of a beam.
Common causes include:
Point loads: Concentrated loads acting at specific locations.
Uniformly distributed loads (UDL): Loads spread evenly over a length.
Uniformly varying loads (UVL): Loads whose intensity changes along the beam.
Support reactions: Forces developed at supports to maintain equilibrium.
The internal shear force at any section must balance the net vertical force acting on one side of that section.
3. Define Shear Force and Bending Moment
Shear force and bending moment are two important internal actions that develop in structural members under loading. They are closely related but describe different effects.
What Is a Bending Moment?
A bending moment is the internal moment that resists the tendency of a structural member to bend under applied loads.
It is calculated by taking the algebraic sum of the moments of external forces about a section.
[
M=\sum M_{\text{section}}
]
Where:
(M) = internal bending moment
(\sum M_{\text{section}}) = algebraic sum of external moments about the selected section
Bending moment is usually measured in N·m or kN·m.
Relationship Between Shear Force and Bending Moment
Shear force and bending moment are related through the change in bending moment along a beam.
[
\boxed{V=\frac{dM}{dx}}
]
Where:
(V) = shear force
(M) = bending moment
(x) = distance along the beam
This relationship applies using a consistent sign convention. In practical terms, the shear force at a section represents the rate at which the bending moment changes along the beam.
For a uniformly distributed load (w), the relationship is:
[
\frac{dV}{dx}=-w
]
under the common convention where downward distributed load is positive.
5. How to Calculate Shear Force
To calculate shear force in a beam, follow these steps.
Step 1: Identify the Beam and Its Supports
Determine whether the beam is simply supported, cantilevered, fixed, or continuous. The support conditions affect the reactions and internal forces.
Step 2: Identify All Applied Loads
List the point loads, distributed loads, and applied moments, along with their magnitudes and positions.
Step 3: Calculate the Support Reactions
Use the equations of static equilibrium:
[
\sum F_y=0
]
[
\sum M=0
]
For a simply supported beam, the sum of upward and downward vertical forces must be zero, and the sum of moments about any point must also be zero.
Step 4: Select a Section
Choose the location where you want to determine the shear force. Imagine cutting the beam at that point.
Step 5: Apply Vertical Force Equilibrium
Consider all vertical forces on either the left or right side of the cut.
[
V=\sum F_y
]
Use a consistent sign convention to determine the algebraic shear force.
Step 6: Repeat Along the Beam
Calculate shear force at important positions, especially immediately before and after point loads and at locations where distributed loads begin or end.
Step 7: Draw the Shear Force Diagram
Plot the calculated shear force values along the beam length. The resulting graph is called the Shear Force Diagram (SFD).
6. Shear Force Calculation Example
Consider a simply supported beam with a span of 6 m and a 12 kN point load at its centre.
Given:
Beam span, (L=6,m)
Point load, (P=12,kN)
Load location = 3 m from either support
Left support = A
Right support = B
Step 1: Calculate Support Reactions
Because the load is at the centre and the beam is symmetrical, both support reactions are equal.
Taking moments about support A:
[
R_B(6)-12(3)=0
]
[
6R_B=36
]
[
\boxed{R_B=6,kN}
]
Using vertical equilibrium:
[
R_A+R_B-12=0
]
[
R_A+6-12=0
]
[
\boxed{R_A=6,kN}
]
Step 2: Calculate Shear Force Before the Point Load
Consider a section between support A and the point load.
The only vertical force on the left side is the upward reaction of 6 kN.
[
V=R_A
]
[
\boxed{V=+6,kN}
]
Therefore, the shear force remains +6 kN from A up to the point load.
Step 3: Calculate Shear Force After the Point Load
Immediately after the 12 kN downward point load:
[
V=R_A-P
]
[
V=6-12
]
[
\boxed{V=-6,kN}
]
The shear force remains −6 kN between the point load and support B.
Step 4: Check at Support B
At support B, the upward reaction is 6 kN.
[
V=-6+6=0
]
The shear force returns to zero at the end of the beam, satisfying equilibrium.
Shear Force Results
Beam positionShear forceJust to the right of A+6 kNJust to the left of the point load+6 kNJust to the right of the point load−6 kNJust to the left of B−6 kNJust to the right of B0 kN
This example shows that a concentrated downward load causes an instantaneous drop in the shear force equal to the magnitude of that load.
7. Shear Force Diagram (SFD)
A Shear Force Diagram is a graphical representation of how the internal shear force varies along a beam.
For the 6 m beam in our example:
From A to the centre, the shear force is +6 kN.
At the centre point load, it drops by 12 kN.
From the centre to B, the shear force is −6 kN.
At B, the support reaction brings the shear force back to zero.
The SFD has a horizontal line between concentrated loads when no distributed load acts on the beam.
A useful related rule is that the change in bending moment between two sections equals the signed area under the shear force diagram between those sections, provided the same sign convention is used.
8. Types of Loads and Their Effects on Shear Force
Different loads produce different shapes in a shear force diagram.
Point Load
A point load causes a sudden vertical jump in the shear force diagram. The size of the jump equals the load magnitude, with the sign determined by the direction of the load and the chosen convention.
Uniformly Distributed Load (UDL)
A UDL causes shear force to vary linearly along the loaded length.
For a constant load intensity (w):
[
V=V_0-wx
]
Where (V_0) is the shear force at the beginning of the loaded region, using the convention that downward loading reduces shear force.
Uniformly Varying Load (UVL)
A linearly varying distributed load generally produces a curved, parabolic shear force diagram because the accumulated load changes quadratically with distance.
Applied Moment
An applied concentrated moment changes the bending moment diagram by a jump but does not directly cause a jump in the shear force diagram.
9. Applications of Shear Force in Civil Engineering
Shear force calculations are commonly used in:
Building construction: Analysing beams that support floors, roofs, and walls.
Bridge engineering: Evaluating the internal forces caused by vehicle and pedestrian loads.
Reinforced concrete design: Checking shear resistance and designing stirrups or other shear reinforcement.
Steel structures: Checking the shear capacity of beams, connections, and structural members.
Structural analysis: Preparing shear force and bending moment diagrams before completing member design.
Engineers must use the applicable design code, load combinations, and material properties when moving from basic shear force calculations to final structural design.
10. Frequently Asked Questions
Q1. Define shear force in simple words.
Shear force is the internal force that acts parallel to a beam's cross-section and resists the tendency of adjacent parts to slide relative to each other.
Q2. How to calculate shear force?
First calculate the support reactions, then select a section and find the algebraic sum of all vertical forces on one side of that section.
[
V=\sum F_y
]
Q3. What is the difference between shear force and bending moment?
Shear force is an internal transverse force, while bending moment is an internal turning effect that resists bending. Shear force is measured in kN, whereas bending moment is measured in kN·m.
Q4. What is the unit of shear force?
The SI unit is the newton (N). In structural engineering, shear force is commonly expressed in kilonewtons (kN).
Q5. Can shear force be negative?
Yes. A negative shear force indicates that the internal force acts in the direction opposite to the assumed positive direction under the selected sign convention.
Q6. Where is shear force zero in a beam?
It depends on the loading and support conditions. In some beams, zero shear occurs at the location where the bending moment reaches a local maximum or minimum. For example, the shear force changes sign at the centre of a simply supported beam carrying a central point load, although the shear force is not uniquely zero across that concentrated load.
11. Conclusion
Shear force is a fundamental concept in civil engineering that helps engineers understand the internal forces developed in beams and other structural members. To define shear force, remember that it is the internal transverse force resisting relative sliding across a section.
Understanding the relationship between shear force and bending moment makes structural analysis easier. By calculating support reactions, selecting sections, applying equilibrium equations, and drawing a Shear Force Diagram, students and engineers can determine how loads affect a beam.
Practising different examples involving point loads, uniformly distributed loads, and varying loads will help you develop a strong foundation in structural analysis and beam design.