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Along a beam, derivatives of internal shear and bending moment are related to distributed load and shear force.
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Formula
q in N/m; V in N; M in N·m; x in m
| Symbol | Meaning | Unit |
|---|---|---|
| Distributed transverse load under the stated sign convention | N/m | |
| Internal shear force at a section | N | |
| Internal bending moment at a section | N·m | |
| Coordinate along the beam | m |
\frac{dV}{dx}=-q(x),\qquad \frac{dM}{dx}=V(x)The differential relations follow from force and moment equilibrium of an infinitesimal beam element. Sign conventions differ; this entry uses V′=−q and M′=V. A constant distributed load therefore gives a linear shear diagram and quadratic moment diagram. Point loads and couples create jumps in internal-force diagrams. Boundary conditions are needed to integrate the relations.
The differential relations follow from force and moment equilibrium of an infinitesimal beam element. Sign conventions differ; this entry uses V′=−q and M′=V. A constant distributed load therefore gives a linear shear diagram and quadratic moment diagram. Point loads and couples create jumps in internal-force diagrams. Boundary conditions are needed to integrate the relations.
Along a beam, derivatives of internal shear and bending moment are related to distributed load and shear force.
Base equation for the target quantity
Preserve signs, boundary conditions and units when rearranging.
Work through the numerical example for “Distributed load, shear-force and bending-moment relations” and check its unit and magnitude.
A uniform q=2 kN/m over 3 m changes V by −6 kN.
If V(0)=0, then V(3 m)=−6 kN; integrate V to obtain the moment change.
Substituting a related quantity that shares the same symbol.
Use variable definitions and check: q in N/m; V in N; M in N·m; x in m.
Formula cluster
Along a beam, derivatives of internal shear and bending moment are related to distributed load and shear force.
A uniform q=2 kN/m over 3 m changes V by −6 kN. If V(0)=0, then V(3 m)=−6 kN; integrate V to obtain the moment change.
Construct and check beam shear-force and bending-moment diagrams.
Here is how to work through a typical Distributed load, shear-force and bending-moment relations (V′ = −q; M′ = V) task step by step:
Task
Work through the numerical example for “Distributed load, shear-force and bending-moment relations” and check its unit and magnitude.
Solution path
A uniform q=2 kN/m over 3 m changes V by −6 kN.
If V(0)=0, then V(3 m)=−6 kN; integrate V to obtain the moment change.
The sign follows the chosen positive load and shear directions. This page uses dV/dx=−q; other texts may define q or V oppositely. Write the convention by the diagram and keep it through integration, jumps and reactions instead of mixing signs from different sources. These differential relations come from equilibrium of a small beam element, but sign conventions vary. This entry uses V′=−q and M′=V. Point forces and couples create jumps, so distinguish continuous loading from concentrated actions when sketching diagrams. Finally check units, assumptions, measurement conditions and reference quantities against the problem before using the result.
Integrating dM/dx=V shows that bending-moment change over a segment is the integral of shear force. Constant shear produces a linear moment change; signed area under the shear diagram gives that change. A jump in V affects M piecewise. These differential relations come from equilibrium of a small beam element, but sign conventions vary. This entry uses V′=−q and M′=V. Point forces and couples create jumps, so distinguish continuous loading from concentrated actions when sketching diagrams. Finally check units, assumptions, measurement conditions and reference quantities against the problem before using the result.
A concentrated transverse force causes a jump in shear at its location by the force magnitude under the sign convention. With a point force alone, bending moment stays continuous but its slope changes. An applied couple instead creates a jump in the moment diagram. These differential relations come from equilibrium of a small beam element, but sign conventions vary. This entry uses V′=−q and M′=V. Point forces and couples create jumps, so distinguish continuous loading from concentrated actions when sketching diagrams. Finally check units, assumptions, measurement conditions and reference quantities against the problem before using the result.
Write the quantity and unit from the variable list next to every value: q in N/m; V in N; M in N·m; x in m. Substitute only after converting to a consistent unit system, and state sign or reference direction explicitly. The example A uniform q=2 kN/m over 3 m changes V by −6 kN. If V(0)=0, then V(3 m)=−6 kN; integrate V to obtain the moment change. is a calculation check: result, unit and order of magnitude must fit the described situation. Round after intermediate calculations so small contributions are not lost prematurely. These differential relations come from equilibrium of a small beam element, but sign conventions vary. This entry uses V′=−q and M′=V. Point forces and couples create jumps, so distinguish continuous loading from concentrated actions when sketching diagrams. Finally check units, assumptions, measurement conditions and reference quantities against the problem before using the result.
The background states this starting condition: The differential relations follow from force and moment equilibrium of an infinitesimal beam element. Sign conventions differ; this. That limits when the result is a valid model prediction. Compare the problem’s boundary conditions with this assumption before substituting values; real components, material properties or loads can introduce additional effects. The cited reference supports the relation used here but does not remove the need to check whether the specific case fits the model. These differential relations come from equilibrium of a small beam element, but sign conventions vary. This entry uses V′=−q and M′=V. Point forces and couples create jumps, so distinguish continuous loading from concentrated actions when sketching diagrams. Finally check units, assumptions, measurement conditions and reference quantities against the problem before using the result.
Quanta exam set
Curated exam set for "Distributed load, shear-force and bending-moment relations":
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Which formula describes Distributed load, shear-force and bending-moment relations?
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How do you rearrange Distributed load, shear-force and bending-moment relations for Base equation for the target quantity?
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Which common mistake happens with Distributed load, shear-force and bending-moment relations?
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