CELE Strength of Materials — Stresses in BeamsSummary

For a linearly elastic beam bent by moment M, plane sections remain plane, so strain—and stress—varies linearly through depth. At distance y from the neutral axis (NA), the bending stress is σ = My/I, where I is the second moment of inertia about the centroidal axis. The maximum bending stress occurs at the extreme fiber (y = c): σ_max = Mc/I = M/S. Sagging (positive) moment places the bottom fiber in tension and the top in compression. This is the foundational equation in strength of materials and beam design.

The section modulus S = I/c combines moment of inertia and distance to extreme fiber into a single design parameter. It simplifies the flexure formula to σ_max = M/S. For a rectangular section (width b, depth h): S = bh²/6. For a solid circle (diameter d): S = πd³/32. In design, the required section modulus is simply S_req = M_max/σ_allow, after which standard shapes or custom sections are selected to match or exceed S_req. Section modulus grows with the square of depth for rectangles—why I-beams and deep sections are far more bending-efficient than flat plates.

The neutral axis (NA) is the surface in a beam where bending stress is zero. For symmetric sections under bending alone, it coincides with the centroidal axis. For unsymmetric sections (e.g., T-beams, L-channels), the NA must be located by calculating the centroid: ȳ = Σ(A_i × y_i) / Σ A_i. Once the NA is found, distances c_top and c_bottom to the extreme fibers may differ, leading to unequal maximum stresses in top and bottom fibers—a crucial point in reinforced concrete T-beam design where the larger tension controls reinforcement.

A transverse shear force V produces shear stress τ given by τ = VQ/(Ib), where Q is the first moment of area of the section above (or below) the level of interest, taken about the NA; I is the moment of inertia of the entire section; and b is the width at that level. Shear stress is zero at top and bottom fibers (where Q = 0) and maximum at the neutral axis. For a rectangular section: τ_max = 3V/(2A). For a solid circle: τ_max = 4V/(3A). For a wide-flange I-beam, approximately τ_max ≈ V/A_web because the web carries nearly all shear. The distribution is parabolic, not linear—opposite to bending stress.

Q is the area above (or below) the level of interest multiplied by the distance from that area's centroid to the neutral axis: Q = A_above × d_centroid. For a rectangular section at distance y from the NA, Q = b × [(c² − y²)/2]. Q is maximum at the neutral axis and zero at the extreme fibers. Correct calculation of Q is essential; many students mistakenly use the entire section's Q or measure distances from the wrong reference, leading to wrong shear stresses.

When a beam is fabricated from separate pieces joined by fasteners (nails, bolts, welds) or adhesive, the connectors must resist the horizontal shear flow q = VQ/I at each joint. The shear flow has units of force per unit length (e.g., N/mm or kN/m). If each connector has capacity F (in newtons), the required spacing s (in mm) is s = F/q. This formula is used to specify nail/bolt spacing in built-up wood beams, weld size in plate girders, and glue-line strength in laminated members—all common practical problems.

When a member carries both axial force P and bending moment M (as in a short column with eccentric load, M = P×e), the total stress at a fiber is the superposition of axial and bending components: σ = P/A ± Mc/I. The extreme fibers see σ_max,min = P/A ± Mc/I. If the load is applied within the kern (core) of the section, all fibers remain in compression (or tension, depending on load sign). The kern is defined by the region where no bending moment would cause stress reversal; for a rectangle, the kern extends h/6 each way from the centroid ('middle-third rule'). Load outside the kern produces tension—often undesirable in masonry, concrete footings, and retaining walls.

A common source of confusion: bending stress σ = My/I is linear through depth (zero at NA, maximum at extreme fibers), while shear stress τ = VQ/(Ib) is parabolic (maximum at NA, zero at extreme fibers). For a rectangular section: bending stress varies as a triangle, shear stress as a parabola. At the extreme fibers, bending stress is maximum but shear is zero. At the neutral axis, bending stress is zero but shear is maximum. This inverse relationship is crucial: the bottom fiber of a simple beam carries the largest bending tension but no shear; the neutral axis carries the largest shear but no bending stress.

A systematic approach ensures nothing is missed: (1) Construct SFD and BMD to find M_max and V_max at critical sections. (2) For flexure: calculate S_req = M_max / σ_allow, then select a section (from tables or custom) meeting or exceeding S_req. (3) For shear: verify that τ_max = VQ/(Ib) ≤ τ_allow (using 3V/2A for rectangular sections as a quick check). (4) If deflection limits apply, check maximum deflection per Chapter 5. (5) In short, heavily loaded beams, shear often governs; in long beams, flexure or deflection usually dominates. Always design for both; one may be hidden until checked.

Real beams often have unsymmetric cross-sections (T-beams in composite floors, channel sections, L-angles). The neutral axis is not at mid-depth. The procedure: (1) Find the centroid (and thus the NA location) by ȳ = Σ(A_i y_i) / Σ A_i. (2) Calculate moment of inertia about the NA using the parallel-axis theorem: I = Σ[I_local + A_i d_i²]. (3) Calculate c_top and c_bottom as distances from NA to extreme fibers—these are different. (4) Compute σ_top = Mc_top/I and σ_bottom = Mc_bottom/I separately; they are not equal. In reinforced concrete, the larger tension stress (usually at the bottom in a simply supported T-beam) governs the reinforcement design.

Rather than derive τ = VQ/(Ib) for each section every time, useful closed-form maxima exist: For a rectangle (width b, depth h, area A = bh): τ_max = 3V/(2A) = 3V/(2bh). For a solid circle (diameter d, area A = πd²/4): τ_max = 4V/(3A) = 16V/(3πd²). For a wide-flange I-beam, the web carries nearly all shear, so τ_max ≈ V/A_web (an approximation useful for quick checks). These formulas are time-savers during design and board exams; students should memorize them.

Section modulus S is derived from moment of inertia: S = I/c. Both measure a section's bending efficiency. However, S is the practical design parameter because σ_max = M/S eliminates the need to separately know I and c. I is fundamental for shear-stress calculations τ = VQ/(Ib) and deflection (Chapter 5).

The neutral axis coincides with the centroidal axis. Locating the centroid (ȳ) determines the NA location. Once NA is known, distances c to extreme fibers are defined, which then determine both I (about the NA) and S = I/c. For unsymmetric sections, this chain of dependencies is critical: centroid → NA → c values → I → S and stress calculations all follow in sequence.

Shear force V from the SFD drives shear stress via τ = VQ/(Ib). Q depends on the geometry above/below the level of interest. The higher the Q (further the sheared area from NA), the higher the τ at that point. This explains why τ_max occurs at the NA (where Q is maximum): τ_max = V × Q_max / (I × b).

The shear stress distribution τ(y) = VQ(y)/(Ib) defines the internal shear stress at each level y. At a joint (interface between pieces in a built-up beam), the total shear force per unit length is the shear flow q = VQ/I. The connector at that joint must resist q; spacing s = F/q ensures adequacy. Understanding τ distribution is the key to sizing connectors.

Superposition allows σ_total = σ_axial + σ_bending = P/A ± Mc/I. This extends the flexure formula to members experiencing both forces simultaneously (e.g., short columns, eccentrically loaded footings). The kern is the region where load can act without changing sign of stress; it emerges naturally from the combined formula.

In a given beam, either flexural stress or shear stress may be the limiting factor. Short beams with heavy loads often fail in shear first; long, slender beams fail in bending first. The design process checks both and accepts the more restrictive limit. Understanding when each governs helps engineers make rapid judgments about proportions (e.g., deeper sections for bending, wider webs for shear).

A complete causal chain: choice of width, depth, shape, etc., → I, c calculations → S = I/c → stress calculations σ = M/S, τ = VQ/(Ib) → comparison with allowables → accept or redesign. Every design iteration cycles through this chain. Understanding the relationships allows engineers to judge whether increasing width or depth is more effective for the problem at hand.

Reinforced concrete beams use the flexure formula to determine the tension and compression stresses in concrete and steel. The neutral axis location (determined by the area of reinforcement) affects the moment arm and thus the moment capacity. Once bending stress is found, reinforcement is proportioned to carry tension in the bottom fiber of a simply supported beam. Shear stress τ = VQ/(Ib) at the critical section determines whether shear reinforcement (stirrups) is required per ACI 318 Section 11.1. Many problems involve unsymmetric sections (T-beams, flanged sections) where c_top and c_bottom differ significantly.

Steel beams are designed using the flexure formula to ensure that σ_max = M/S does not exceed the flexural strength F_b (often based on lateral-torsional buckling, per AISC 360 Chapter F). Section modulus S is a key parameter in steel design; tables of standard shapes (W, S, C, L sections) list S for rapid selection. Shear capacity is checked using τ = V/A_web (for I-sections, the web carries most shear). Built-up plate-girder beams must have connection details (welds or bolts) sized using shear flow q = VQ/I to ensure the flange-to-web connection can carry the horizontal shear.

Timber beams are proportioned using the flexure formula σ_max = M/S ≤ F_b (allowable bending stress, adjusted for load duration, moisture, etc.). For rectangular timber (the most common form), S = bh²/6 is applied directly. Shear is checked using τ_max = 3V/(2A) ≤ F_v (allowable shear). In built-up timber beams (e.g., laminated beams or beams made from multiple planks), nailing patterns are designed using shear flow q = VQ/I; nail spacing is set so that s = F_nail / q ensures nails carry the required shear.

Composite beams (steel section acting together with a concrete slab) have a transformed neutral axis determined by the steel area and concrete area weighted by the modular ratio n = E_steel / E_concrete ≈ 8–10. The neutral axis location differs from either material alone, requiring careful calculation of I about the transformed axis. The flexure formula then yields bending stresses in both steel and concrete. Shear transfer between steel and concrete (shear studs) is designed using shear flow: connector spacing is set by s = F_connector / q, where q = VQ/I_transformed.

Short, thick members (footings, retaining wall stems) often carry eccentric loads. The combined stress formula σ = P/A ± Mc/I applies. For a footing, the pressure (stress) should not exceed soil bearing capacity and, in concrete, should not cause excessive tension (which concrete is weak in). The kern criterion ensures that eccentric loads remain within the middle third (kern) to avoid tension. For a rectangular footing of width B, the kern extends B/6 each side; if the load is further than B/6 from centerline, tension appears and uplift may occur.

Engineers rapidly select beams by calculating S_req = M_max / σ_allow, then consulting standard shape tables (e.g., AISC shapes, timber sections) to find the lightest section with S ≥ S_req. This shortcut avoids calculating I and c separately. Once a candidate section is chosen, shear is verified using τ_max = V/A_web (steel) or τ_max = 3V/(2A) (timber). If either criterion is not met, the next larger section is selected and checked again.

When a beam is fabricated by riveting, bolting, or welding separate plates, or by laminating timber planks, the connectors must resist the horizontal shear flow q = VQ/I at each joint. For a steel plate girder, the welds joining flange to web are sized to carry q (typically a fillet weld size is specified per AISC 360 J2.4). For timber laminations, adhesive thickness or nail spacing is set by s = F / q. The critical regions are near the supports where V is large and Q (of the connected part) is significant.

Shear failure is sudden and brittle; unlike bending, which may give warning through deflection or cracking, shear can cause catastrophic collapse. Equations τ_max = 3V/(2A) for rectangles and τ_max = 4V/(3A) for circles allow rapid checks. In reinforced concrete, stirrups (vertical shear reinforcement) are required when τ exceeds τ_c (concrete contribution); the stirrup spacing is set by ACI 318 equations based on τ and the stirrup strength. Engineers must never neglect shear design, especially in short, heavily loaded beams and corbels.

Many real beams have unsymmetric sections (T-beams in composite construction, channels, L-sections). The procedure is systematic: (1) locate the centroid and thus the NA, (2) calculate I about the NA, (3) measure c_top and c_bottom, (4) find σ_top and σ_bottom separately. In a reinforced concrete T-beam in positive bending, c_bottom > c_top, so σ_bottom > σ_top; the tension in the bottom governs reinforcement. This is why many T-beams have bottom reinforcement only (the top is in compression and concrete handles it). Engineers must not assume σ_top = σ_bottom in unsymmetric sections; doing so leads to underdesign.