Internal Force Diagrams

Internal force diagrams are graphical representations of the shear force V(x) and bending moment M(x) along the length of a beam. They are used to locate the critical sections where the maximum shear and maximum bending moment occur, which govern the design of the cross-section. Two sign conventions are commonly used. In the standard convention, positive shear force acts downward on the left face and upward on the right face of a cut, and positive bending moment causes sagging (compression at the top, tension at the bottom). The differential relationships between load, shear, and moment are the foundation of the diagrams: dV/dx = -w(x) and dM/dx = V(x), where w(x) is the distributed load intensity. From these, the change in shear between two points equals the negative of the area under the load diagram, and the change in moment equals the area under the shear diagram. A point load causes a step change in the shear diagram equal to the magnitude of the load and a change in slope (kink) in the moment diagram. A concentrated moment causes a step change in the moment diagram but does not change the shear. A uniformly distributed load (UDL) produces a linear shear diagram and a parabolic moment diagram. A linearly varying load produces a parabolic shear diagram and a cubic moment diagram. At a point where the shear force is zero, the bending moment has a local maximum or minimum, which is often the critical design section. At a free end or a simple support, the bending moment is zero. At a fixed support, both the shear and moment are generally nonzero. The area method is a convenient way to construct the diagrams: starting from a known value of V or M, add the appropriate area from the load or shear diagram to find the value at the next point. The moment diagram can also be drawn by remembering that the slope of the moment diagram at any point equals the shear force at that point. For design, the maximum absolute values of shear and moment are used to select the beam section and check shear and flexural capacity.