CELE Strength of Materials — Shear and Moment DiagramsSummary

When a beam is cut at any section and one side is isolated, the cut face must supply two internal effects to maintain equilibrium: (1) Shear force V – the transverse (vertical) internal force balancing transverse loads on that side, and (2) Bending moment M – the internal moment balancing the moment of those loads about the section. These are found by applying equilibrium equations to the free body of one isolated side: V = ΣF_transverse and M = ΣM_about_section.

Shear is positive when the resultant force to the left of a section acts upward (equivalently, the internal shear pair acts clockwise on an element). Bending moment is positive when it causes the beam to be concave up (sagging)—compression on top, tension at the bottom—like a smile. Negative moment is concave down (hogging)—tension on top, compression at bottom—like a frown. Memory aid: positive moment = 'smile', negative = 'frown'.

Before drawing any diagram, solve for all support reactions using the three equilibrium equations: ΣF_x = 0, ΣF_y = 0, ΣM = 0. Reactions depend on support type: roller (1 vertical reaction), pin (vertical + horizontal), fixed (vertical + horizontal + moment). Statically determinate beams (simply supported, cantilever, overhanging) have exactly 3 unknowns; indeterminate beams require additional methods (slope-deflection, flexibility, etc., covered in Structural Theory).

From equilibrium of a differential beam element: dV/dx = −w(x) and dM/dx = V(x), where w(x) is the load intensity. In practical (integral) form: ΔV = −(area under load diagram), and ΔM = area under shear diagram. These allow sketching without equations: (1) slope of shear diagram = −load intensity, (2) slope of moment diagram = shear, (3) where V = 0, moment reaches a local extreme (critical for locating M_max).

Integration raises the degree of a curve by one: a point load creates constant shear (degree 0 polynomial) and linear moment (degree 1). A uniformly distributed load creates linear shear (degree 1) and parabolic moment (degree 2). A uniformly varying (triangular) load creates parabolic shear (degree 2) and cubic moment (degree 3). Moment is always at least one degree higher than shear (since M is the integral of V).

A downward concentrated force causes a sudden vertical drop (downward jump) in the shear diagram equal to the load magnitude. A concentrated couple (applied moment) causes a sudden vertical jump in the moment diagram equal to the couple magnitude but does not affect shear (no transverse force). These jumps are the signature of concentrated effects; distributed effects produce smooth curves.

Since dM/dx = V, the moment is stationary (slope = 0) wherever V = 0. This is where internal moment reaches a local maximum or minimum. For simply supported beams with symmetric loading, M_max is at midspan. For unsymmetric loading or cantilevers, you must solve for the point where V crosses zero. This is the pivotal step in many board problems.

Simply supported, cantilever, and overhanging beams have 3 unknowns and 3 equilibrium equations → statically determinate (solvable immediately). Fixed, continuous, and propped cantilever beams have more than 3 unknowns → statically indeterminate (require compatibility equations, covered in Structural Theory). The PRC exam assumes familiarity with determinate cases at this level; indeterminate beams appear in Structural Theory & Analysis topics.

Systematic procedure: (1) find all reactions, (2) divide the beam at every point where loading changes (supports, point loads, start/end of distributed loads, applied couples), (3) for each segment, cut at distance x and isolate one free body, (4) apply ΣF_y = 0 for V(x) and ΣM = 0 for M(x), (5) plot V(x) and M(x) over each segment. This always works but is slower than the area method for simple beams.

The type of support (roller, pin, fixed) determines how many and what kinds of reactions develop. A roller provides only a vertical reaction perpendicular to the rolling surface; a pin provides vertical and horizontal; a fixed support provides vertical, horizontal, and moment. These reactions are the boundary conditions that determine the magnitude and shape of the SFD and BMD.

Load is the rate of change of shear (dV/dx = −w). Integrating load over a region gives the total change in shear. This relationship allows area-based calculations and is the core of the fast-sketch method.

Shear is the rate of change of moment (dM/dx = V). Integrating shear over a region gives the total change in moment. Critically, where shear equals zero, moment is at a local extremum. This is the primary tool for locating maximum moment without calculus.

The three diagrams are connected by successive integration: load is integrated to get shear, shear is integrated to get moment. Each diagram encodes information about the lower-level one: the shape (concavity) of the shear diagram reflects the load, and the shape of the moment diagram reflects the shear.

Simply supported (2 reactions) beams typically have positive (sagging) moment in the span and zero moment at supports. Cantilevers (1 reaction + 1 moment at fixed end) have negative (hogging) moment throughout. Overhanging beams have both sagging and hogging regions; the largest moment is often at an interior support (hogging), not in the span.

Determinate beams (simply supported, cantilever, overhanging) have exactly 3 unknowns and are solvable by equilibrium alone. Indeterminate beams (fixed, continuous, propped cantilever) have more than 3 unknowns and require additional compatibility equations (part of Structural Theory & Analysis, not core Strength of Materials).

ACI 318 requires engineers to design flexural reinforcement based on the maximum bending moment M_u (factored). The SFD and BMD are drawn first (often at service loads or factored loads depending on the design stage); the maximum moment value and its location are then used to determine the required area of tension steel A_s = M_u / (φ f_y d). Shear diagram is used to place shear reinforcement (stirrups). Forgetting to locate the actual peak moment (not just scanning the diagram) results in undersized beams or unnecessary steel.

AISC 360 requires selection of the lightest steel section (I-beam, wide-flange, channel) that resists the maximum moment without exceeding allowable stress: σ = M / S ≤ σ_allow, where S is the elastic section modulus. The SFD and BMD are used to find M_max; the location of maximum shear determines where lateral bracing must be provided. Incorrect moment values lead to either over-designed (uneconomical) or under-designed (unsafe) members.

Methods such as integration of the moment curvature (double integration method), conjugate beam method, and virtual work all depend on knowing M(x) along the entire span. The SFD and BMD are the starting point; from M(x) you compute curvature, which is integrated to find slope and deflection. Without correct SFD and BMD, deflection calculations are meaningless.

When analysing the internal moment in footing beams or retaining wall cantilevers, the same SFD and BMD methods apply. A cantilever footing under soil pressure (distributed load) has maximum hogging moment at the face of the column. Retaining walls under lateral earth pressure (triangular load distribution) require finding the maximum moment, which is not at the base but at the point where shear (lateral force resultant) equals zero.

The Philippine National Structural Code (NSCP 2015) requires SFD and BMD as part of structural design drawings. The code mandates specific design loads, load combinations, and safety factors (e.g., load factors of 1.2 for dead load, 1.6 for live load in ultimate limit state). The diagrams must be drawn at the factored load level for ultimate design and at service load level for serviceability checks (deflection, cracking). Correct interpretation of NSCP load combinations directly affects the shape and magnitude of the diagrams.

For bridges and long-span structures, concentrated loads (vehicles, point masses) and distributed loads (self-weight, pavement, distributed live load) both contribute to the SFD and BMD. The diagrams may be highly irregular, with multiple peaks and valleys. Overhanging sections (cantilever arms beyond supports) create hogging moments that can exceed mid-span sagging moments. The fast area-based method becomes essential for hand calculations when diagrams are complex.