Maximum Moment & Shear — Identify maximum moment and shear locations

The ability to locate maximum internal shear and maximum bending moment within a structural member is one of the most consequential skills an engineer can develop. The entire history of structural mechanics has, in a sense, been driven by the need to understand where beams, bridges, and frames are most susceptible to failure. Ancient builders relied on empirical rules and generous proportions, but as the demand for longer spans, lighter sections, and more economical use of material grew throughout the Industrial Revolution, engineers required rigorous analytical tools to predict exactly where internal forces would reach their critical values.

The central question this lesson addresses is deceptively simple: given a loaded beam with known support reactions, at which cross-section does the internal shear force reach its maximum, and at which cross-section does the bending moment reach its maximum? Answering this question is the essential prerequisite for sizing structural members, selecting cross-sections, and verifying that a design can safely carry its intended loads.

Before locating maximum values, we need to establish the foundational relationships that govern how shear and moment vary along a beam. These principles arise directly from the equilibrium of an infinitesimal beam element and provide the mathematical backbone for constructing and interpreting shear–moment diagrams. A firm grasp of these relationships transforms the task of finding maximum internal forces from a tedious section-by-section analysis into a systematic, almost mechanical procedure.

The following diagram illustrates a simply supported beam carrying a single concentrated load P at a general position along its span. Below the beam schematic, the corresponding shear (V) and bending moment (M) diagrams are drawn to scale, with the locations of maximum shear and maximum moment clearly annotated. This canonical case forms the template from which more complex loading patterns are analyzed by superposition.

Observe the direct correspondence between the two diagrams. In the shear diagram, the horizontal segments indicate that no distributed load acts between the supports and the concentrated force. The shear is positive (upward resultant on the left face) from A to the load, then negative from the load to B. The moment diagram is piecewise linear (since V is constant in each segment), increasing with a positive slope of +Pb/L from A to the load, then decreasing with a slope of −Pa/L from the load to B. The peak of the triangle coincides precisely with the point where the shear crosses zero — this is not a coincidence but a direct consequence of the relationship dM/dx = V(x). For design purposes, the maximum bending stress will occur at the cross-section under the load, and the maximum shear stress will occur at either support reaction.

The differential relationships between load, shear, and moment form a trio of equations that are derived from the equilibrium of an infinitesimal beam element of length dx subjected to a distributed load w(x). These relationships may also be expressed in integral form, which is especially convenient for constructing diagrams graphically. Together they provide a complete analytical framework for locating critical sections.

A critical subtlety arises for beams with distributed loads: the location where V(x) = 0 may fall between supports or load application points, requiring you to solve the algebraic equation V(x) = 0 for x. For example, consider a uniformly distributed load w₀ over the full span L of a simply supported beam. The shear varies linearly as V(x) = w₀L/2 − w₀x, which equals zero at x = L/2. Substituting back, M_max = w₀L²/8, occurring at midspan — a result that should be committed to memory.

Practicing engineers often work from a catalog of known loading cases, each with a well-established location for maximum shear and maximum moment. Knowing these standard results accelerates the design process and provides a reliable sanity check for computational analyses. The following table summarizes the most common configurations encountered in statics and introductory structural analysis courses.

Notice that for the cantilever the maximum shear and maximum moment both occur at the fixed support. This is fundamentally different from the simply supported beam case, where V_max and M_max generally occur at different locations. The parabolic shape of the moment diagram in the cantilever case arises because the shear varies linearly — integrating a linear function yields a quadratic. This is a direct application of the integral relationship M(x₂) − M(x₁) = ∫V dx.

Consider a simply supported beam of span L = 6 m. A uniformly distributed load of w₀ = 10 kN/m acts over the left half (0 ≤ x ≤ 3 m), and a concentrated load P = 20 kN is applied at x = 4.5 m. Determine the locations and magnitudes of the maximum internal shear force and maximum bending moment.

The rule that maximum bending moment occurs where the shear crosses zero is extraordinarily powerful for statically determinate beams with well-defined loading, but it is important to understand its scope and the situations where it must be applied with additional care. The table below contrasts the strengths and limitations of this approach.

The principles developed for determinate beams extend naturally into more advanced structural analysis topics. In statically indeterminate beams (e.g., propped cantilevers, continuous beams over multiple supports), the fundamental relationship dM/dx = V(x) still holds, and maximum moment still occurs where V = 0 or at boundaries with moment fixity. The difference is that computing the reactions and constructing the shear diagram requires solving compatibility equations or using methods like moment distribution, slope-deflection, or matrix stiffness analysis.

Another important extension involves influence lines, which graphically depict how the moment or shear at a fixed section varies as a unit load traverses the span. The maximum ordinate of the influence line for moment at a particular section indicates how sensitive that section is to load placement. By combining influence lines with actual load patterns (such as a truck crossing a bridge), engineers construct moment envelopes and shear envelopes that capture the worst-case internal forces at every section — a critical step in bridge and infrastructure design. The conceptual foundation for all of this, however, remains exactly what you have learned in this lesson: the interplay between V and M as expressed by dM/dx = V.