Bending moment vs shear force comes down to this: shear force is the net internal transverse (vertical) force acting across a beam section, measured in newtons (N), while bending moment is the net internal turning effect about that section, measured in newton-metres (N·m). Shear force tends to slide one part of the beam vertically past the other; bending moment tends to rotate and curve the beam. They are different internal reactions at the same cut, and they are mathematically linked: the bending moment is the integral of the shear force along the beam.
What is shear force?
Imagine cutting a loaded beam at any section and looking at the portion to the left of the cut. The shear force (V) is the algebraic sum of all vertical forces on that portion. Using the standard sign convention, shear force is positive when the net force on the left portion is upward (and the right portion correspondingly acts downward). It is measured in newtons (N) or kilonewtons (kN).
- Caused by transverse loads acting perpendicular to the beam axis.
- Plotted as the Shear Force Diagram (SFD) along the span.
- A point load produces a sudden vertical jump in the SFD.
What is bending moment?
At the same cut, the bending moment (M) is the algebraic sum of the moments of all forces on the left portion, taken about the section. Using the standard convention, bending moment is positive for sagging — when the beam bends concave-up, like a smile, with the top fibres in compression and the bottom fibres in tension. A hogging (concave-down) curvature is negative. It is measured in newton-metres (N·m) or kilonewton-metres (kN·m).
- Caused by the lever-arm effect of loads about the section.
- Plotted as the Bending Moment Diagram (BMD) along the span.
- Governs bending stress and the choice of beam cross-section.
Bending moment vs shear force: comparison table
| Aspect | Shear Force (V) | Bending Moment (M) |
|---|---|---|
| Definition | Net internal transverse force at a section | Net internal turning moment at a section |
| Physical effect | Slides one side vertically past the other | Bends and curves the beam |
| SI unit | Newton (N) or kN | Newton-metre (N·m) or kN·m |
| Sign convention | Positive when left portion net force is upward | Positive for sagging (concave-up) |
| Diagram | Shear Force Diagram (SFD) | Bending Moment Diagram (BMD) |
| Governing relation | dV/dx = −w (load intensity) | dM/dx = V (shear force) |
| Design role | Shear stress, web/connection design | Bending stress, section sizing |
How are bending moment and shear force related?
The two quantities are tied together by differential relationships derived from equilibrium of a small beam element:
- dV/dx = −w — the rate of change of shear force equals the negative of the distributed load intensity (w).
- dM/dx = V — the rate of change of bending moment equals the shear force.
Three consequences follow directly. The bending moment is maximum (or minimum) where the shear force is zero, because dM/dx = V = 0 at that point. Where there is no distributed load, the SFD is constant and the BMD varies linearly. Where the load is uniform, the SFD varies linearly and the BMD is parabolic.
Worked example: simply supported beam with a central point load
Take a simply supported beam of span L carrying a single central point load W at mid-span. By symmetry, each support reaction is:
- RA = RB = W/2
Shear force. From the left support to the centre, the only force on the left portion is the upward reaction W/2, so V = +W/2 (constant). At the centre the load W acts down, and from the centre to the right support V = +W/2 − W = −W/2 (constant). The SFD is a step that jumps by W at mid-span, crossing zero exactly under the load.
Bending moment. Because dM/dx = V, the moment rises linearly under the constant positive shear and falls linearly under the constant negative shear. It is zero at both simple supports and peaks where V changes sign — at the centre. The maximum sagging bending moment is:
- Mmax = WL/4, occurring at mid-span.
The BMD is therefore a triangle, rising from zero at each support to WL/4 at the centre — consistent with the SFD crossing zero at that same section.
How is this shown and measured in a teaching lab?
In a Strength of Materials laboratory, students verify these relationships experimentally rather than only on paper. Typical apparatus includes:
- A shear force apparatus, where the beam is cut at a section and a spring-balance or load-cell mechanism reads the internal shear; students apply known weights and compare the measured shear with W/2.
- A bending moment apparatus, which measures the internal moment at a section via a balanced moment arm, letting students confirm M = WL/4 at mid-span and plot the BMD.
- Hanger weights, a graduated span, and dial indicators to record reactions, deflections, and the load positions used to build SFD and BMD plots.
By moving the load along the span and re-reading the instruments, students directly observe that the bending moment peaks where the shear force passes through zero — turning the dM/dx = V relationship into a hands-on result.
Scientico India manufactures ISO 9001:2015 and CE certified shear force and bending moment teaching apparatus for engineering colleges and universities. Explore the full range under our Strength of Materials Lab Equipment category.
Frequently Asked Questions
What is the main difference between bending moment and shear force?
Shear force is the net internal transverse (vertical) force at a beam section, measured in newtons (N), and it tends to slide one part of the beam vertically past the other. Bending moment is the net internal turning effect at that section, measured in newton-metres (N·m), and it bends or curves the beam. They are distinct internal reactions at the same cut but are linked by dM/dx = V.
What is the relationship between shear force and bending moment?
They are connected by two differential relationships from equilibrium: dV/dx = −w (the rate of change of shear equals the negative distributed load intensity) and dM/dx = V (the rate of change of bending moment equals the shear force). A key result is that the bending moment is maximum or minimum where the shear force is zero.
What are the sign conventions for shear force and bending moment?
Under the standard convention, shear force is positive when the net force on the left portion of the section acts upward (right portion downward). Bending moment is positive for sagging — a concave-up curvature like a smile, with bottom fibres in tension and top fibres in compression.
What is the maximum bending moment for a simply supported beam with a central point load?
For a simply supported beam of span L with a central point load W, each reaction is W/2. The shear force is +W/2 from the left support to the centre, then −W/2 to the right support. The maximum sagging bending moment occurs at mid-span and equals WL/4.
How are shear force and bending moment measured in a lab?
Strength of Materials labs use a dedicated shear force apparatus and bending moment apparatus. The beam is cut at a section, and a spring-balance or load-cell mechanism reads the internal shear or balanced moment as known hanger weights are applied. Students compare readings against W/2 and WL/4 and plot the SFD and BMD.
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