Differential relations between beam load, shear and bending moment

The slope of the shear diagram is negative distributed load under the chosen convention. Constant line load gives linear shear; with no line load, V is piecewise constant. A point force creates a jump in V. Use one consistent sign convention; point loads and couples create jumps. Application: Derive section-force diagrams from load distributions and check their consistency. The numerical example For constant q=2 kN/m, V has slope −2 kN/m along x; the slope of M at each point equals the local V anchors the idea in a concrete measurement case.

Moment slope equals shear: dM/dx=V. Where V is zero, M may have a local extremum, a useful check. A concentrated couple causes a jump in the moment diagram. Use one consistent sign convention; point loads and couples create jumps. The corresponding units are q in N/m; V in N; M in N·m; x in m. For constant q=2 kN/m, V has slope −2 kN/m along x; the slope of M at each point equals the local V. This checks that the reference quantities are compatible.

Sign conventions define positive load direction and cut-face directions differently. A source may therefore write dV/dx=q instead of −q. The physics is unchanged if load, V and M use one consistent convention. Use one consistent sign convention; point loads and couples create jumps. Scope: The relations follow from force and moment equilibrium of an infinitesimal beam element. Signs of q, V and M depend on the section convention; concentrated forces cause jumps in V, and applied couples cause jumps in M. They apply within classical beam theory and underlie load, shear-force and bending-moment diagrams. A practical application is Derive section-force diagrams from load distributions and check their consistency.

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 For constant q=2 kN/m, V has slope −2 kN/m along x; the slope of M at each point equals the local V. 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. Use one consistent sign convention; point loads and couples create jumps.

The background states this starting condition: The relations follow from force and moment equilibrium of an infinitesimal beam element. Signs of q, V and. 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. Use one consistent sign convention; point loads and couples create jumps.