Learning topic
Internal Forces in Bending
Learn how to construct beam shear-force Q and bending-moment M diagrams using support reactions, the method of sections, sign conventions, and diagram checks.
In transverse beam bending, the main internal force resultants are the shear force $Q$ and the bending moment $M$. They are determined by the method of sections, and their variation along the beam is represented graphically by diagrams.
Shear force and bending moment
The shear force $Q$ represents the resultant internal tangential action in a cross-section. The bending moment $M$ represents the resultant effect of internal normal forces that causes curvature of the beam axis.
To determine $Q$ and $M$, make an imaginary cut at the required section and write equilibrium equations for one of the two resulting beam parts.
- Determine the beam support reactions from equilibrium equations for the entire structure.
- Divide the beam into segments. Segment boundaries occur at supports, concentrated forces or moments, and the start or end of distributed loads.
- Make an imaginary cut within the required segment at coordinate $x$.
- Consider equilibrium of the left or right cut portion. Replace the removed part by the internal resultants $Q(x)$ and $M(x)$.
- Use $\sum F_y=0$ to determine the shear force $Q(x)$, and a moment equilibrium equation to determine the bending moment $M(x)$.
- Calculate characteristic values at segment boundaries and at points where $Q(x)=0$. Use these values to construct the $Q$ and $M$ diagrams.
- Check the shapes and slopes of the diagrams using the differential relations between $q$, $Q$, and $M$.
Sign conventions
Use one consistent sign convention throughout the calculation. A positive bending moment is commonly associated with sagging, in which the lower fibers are in tension and the upper fibers in compression. The sign of $Q$ follows the selected convention for the cut portion. A negative calculated value means that the actual internal resultant acts opposite to the assumed positive direction.
$Q$ and $M$ diagrams
A diagram shows how an internal force resultant varies along the beam axis. Before constructing diagrams, divide the beam into segments at supports, concentrated forces and moments, and the start or end points of distributed loads.
For a beam under a distributed load $q(x)$, shear force and bending moment are related by:
$$\frac{dQ}{dx}=-q(x),\qquad \frac{dM}{dx}=Q(x).$$
- $q(x)$ — distributed load intensity;
- $Q(x)$ — shear force;
- $M(x)$ — bending moment;
- $x$ — longitudinal beam coordinate.
Therefore, if $q=0$ over a segment, $Q$ is constant and $M$ varies linearly. If $q$ is constant, $Q$ varies linearly and $M$ varies quadratically. At a section where $Q$ changes sign, $M$ has a local extremum.
Characteristic diagram features
- a concentrated transverse force causes a jump in the $Q$ diagram;
- a concentrated applied moment causes a jump in the $M$ diagram;
- where distributed load is absent, $Q$ is constant and $M$ varies linearly;
- under a uniform distributed load, $Q$ varies linearly and $M$ is parabolic;
- where $Q$ crosses zero, $M$ has a local maximum or minimum.
Short example
Consider a cantilever beam of length $L$ carrying a concentrated force $F$ at its free end. No distributed load acts between the free end and the fixed support, so the shear force is constant in magnitude: $|Q|=F$. The bending moment varies linearly from zero at the free end to the maximum magnitude $|M|_{\max}=FL$ at the fixed end. The signs depend on the adopted convention.
Calculation check
After constructing the diagrams, verify overall equilibrium, jumps at concentrated forces and moments, the expected curve shape on every segment, and consistency between the slope of the $M$ diagram and the sign of $Q$. These checks reveal many common calculation errors.
Learning outcome
After studying this topic, you should be able to determine support reactions, divide a beam into calculation segments, derive $Q(x)$ and $M(x)$, locate characteristic and extreme values, and construct shear-force and bending-moment diagrams.
About this topic
Accurate determination of internal forces is the primary step in beam bending calculations. This section covers the method of sections for evaluating shear forces Q and bending moments M across beam segments. Detailed explanations include sign conventions, span segmentation rules, critical section identification, and differential consistency checks.