Shear Lag: Why Wide Flanges Don't Pull Their Weight

The mechanism: bending is carried by shear that diffuses inward

Elementary beam theory (Euler–Bernoulli) says the flange of a beam in bending carries a uniform axial stress σ = My/I, constant across the flange width. That relies on the assumption that plane sections remain plane. For a narrow flange it's nearly true. For a wide one it fails, because of how the flange actually gets loaded.

The flange is not pushed axially by some external agent. Its axial (bending) force is fed in from the web through the flange-to-web weld or fillet line, purely as in-plane shear flow q = VQ/I along that junction. To reach the outer corners of a wide flange, that force must travel sideways across the plate, and it can only do so by shearing the plate. Shear stress produces shear strain (τ = Gγ), so the flange deforms in its own plane — the far material "lags" behind the material next to the web. The corners therefore see less axial strain, hence less stress, than plane-sections predicts.

• Near the web: fully engaged, σ higher than nominal (the concentration).
• At the flange tip: only partially recruited, σ lower than nominal.
• The width over which recruitment happens scales with a shear-diffusion length, roughly the span or the distance between load/reaction points, not the flange width.

Because it is a shear-diffusion phenomenon, shear lag is worst where the axial force is changing fastest — at concentrated loads, at supports, and near the ends of a member — and it relaxes toward the uniform plane-sections state far from those disturbances (a direct expression of Saint-Venant's principle).

The governing equation and the effective-width fix

The honest way to describe the flange is a warping-plate problem: the in-plane displacement u(x,z) obeys a governing equation of the form G·∂²u/∂z² + E·∂²σ-related terms = 0, and analytical solutions (Reissner, Kuhn, Karman) give a stress that decays across the half-width b as approximately σ(z) ≈ σ_web·[1 − (z/b)²·k] — a parabolic-ish dip toward the tips. Solving the full plate equations for every member is impractical, so engineers collapse it into one number.

The design device is the effective width, b_eff. You replace the true flange width b with a narrower strip that, carrying the peak stress uniformly, delivers the same total flange force:

• b_eff = (1/σ_max) · ∫ σ(z) dz — the effective width is the area under the true stress curve divided by the peak stress.
• You then design with σ_max = M·c / I_eff, where I_eff uses b_eff, so the real peak is captured while keeping the tidy σ = Mc/I bookkeeping.

The reduction factor ψ = b_eff/b is a function of the aspect ratio b/L_e (flange width to an effective span between moment zeros) and the loading type. Reissner's classic result for a wide flange under sinusoidal moment gives ψ ≈ 1/(1 + n·(b/L)²) with n on the order of 4–6, so a flange with b/L ≈ 0.3 already loses ~25–35% of its width. As b/L → 0, ψ → 1 and shear lag vanishes — narrow beams obey plane sections, exactly as taught.

The controlling variables and the design trade-offs

Three geometric/loading knobs set the severity:

• Aspect ratio b/L_e — the dominant driver. Double the flange width at fixed span and the shear-lag penalty roughly quadruples (the (b/L)² term). This is why very wide top flanges on box girders are penalized hard.
• Load gradient — concentrated loads and support reactions create sharp moment gradients and the worst lag; uniformly distributed load is milder. Under a point load b_eff can drop to 0.3–0.5 b locally.
• Position along the member — worst at supports/loads, best at mid-span of a uniformly loaded simple beam. Codes give different b_eff for sagging (mid-span) and hogging (over-support) regions precisely for this reason.

The trade-off is unavoidable: wide flanges are what you want for stiffness and section modulus (I ∝ A·d² for flange material far from the neutral axis), but the wider you go the less of that width actually works. Beyond a point, adding flange plate buys diminishing return — you are paying for steel that stress gauges show is idling at half-utilization. Material efficiency, not just strength, drives the effective-width limits. The same logic caps the useful width of a composite concrete deck acting with a steel beam.

Quantitative sizing: what the codes actually say

Design codes package shear lag into simple width limits so you never solve the plate equation:

• AISC 360 (steel tension members), §D3 — the shear-lag factor U reduces the net area: effective net area A_e = U·A_n. For a bolted W-shape connected only through its flanges, U = 1 − x̄/L, where x̄ is the connection eccentricity (centroid-to-faying-surface) and L is the connection length. A typical single-angle in tension might get U ≈ 0.6–0.8; get the connection too short and U can fall below 0.6, slashing capacity.
• Eurocode EN 1993-1-5 §3 (steel plated members) — effective width b_eff = β·b₀, with β a function of κ = α₀·b₀/L_e. For sagging spans β can stay near 1.0 for slender girders but drops to 0.2–0.4 over interior supports of wide box girders.
• Composite beams, EN 1994 / AASHTO / AISC — the concrete deck effective width is commonly limited, on each side of the beam, to the least of L/8, one-half the beam spacing, or the distance to the slab edge — so the total effective width is at most L/4. On a 12 m simple span the code cap is b_eff ≈ 2·(12/8) = 3.0 m total, but if only a 2.4 m deck strip is available the geometry governs and all 2.4 m is effective.

Order-of-magnitude: for a plate girder with a 800 mm flange on a 20 m span, b/L ≈ 0.04 and shear lag is minor (ψ ≈ 0.95). For a stiff, wide box girder deck with b ≈ 4 m over a 15 m support region, ψ can fall to 0.4–0.6, and the flange stress next to the web can be 1.5× the naive My/I value — the number a fatigue check must use.

Where it bites: real hardware and structures

Shear lag is not an academic footnote; it sizes real steel:

• Steel box-girder bridges — the wide top and bottom flanges are the textbook case. The Merrison rules (post the 1970s box-girder collapses at Milford Haven, West Gate, Koblenz) formalized effective-width and flange-buckling checks after shear lag and plate buckling were both implicated in under-strength flanges.
• Composite steel–concrete bridge and building decks — the slab is a very wide flange for the steel beam; effective-width rules decide how much slab you may count toward positive-moment capacity and stiffness.
• Bolted/welded tension connections — angles, tees, channels and W-shapes connected through only part of their cross-section: the unconnected legs lag, captured by the U-factor. This is the most common place a practicing engineer meets shear lag by name.
• Aircraft and aerospace — stiffened skin panels, wing box covers and stringer-stiffened shells: Paul Kuhn's NACA work in the 1930s–40s on stringer sheet-metal structures is the historical root of the term "shear lag," where skin between stringers carries less than the beam-theory share.
• Tall-building framed tubes — the "tube" concept treats the perimeter columns as a giant hollow box beam resisting wind. Shear lag makes corner columns pick up far more axial load than mid-face columns, so the face doesn't act as a perfect flange — a first-order effect in framed-tube design.

Failure modes, limits, and best practice

Shear lag rarely causes a clean single-event failure; it sets the stage for others by hiding a stress peak:

• Fatigue cracking — the flange-to-web junction sees the concentrated stress AND is a welded detail (a low fatigue category). Using the plane-sections stress there underpredicts the true range Δσ by 30–60%, exactly where cracks start. Always apply b_eff to fatigue-critical welded flanges.
• Tension-member rupture — an over-optimistic net-section (U = 1 assumed) can rupture at the connection below the design load. AISC's U-factor exists specifically to prevent this; short connections are the trap.
• Serviceability / deflection — shear lag reduces effective I, so real deflections exceed plane-sections predictions by up to ~15–20% for wide-flanged members; a stiffness check that ignores it can fail an L/360 limit in service.

Best practice: (1) use the code effective-width or U-factor — don't hand-wave; (2) lengthen tension connections (more bolt rows) to raise U toward 1.0; (3) keep flange width sensible relative to span; (4) for unusual wide-flange geometry or box decks, run a shell/plate finite-element model and read the true σ distribution rather than trusting a beam element, which by formulation cannot see shear lag; (5) place the fatigue check at the web line, not the flange centroid, using the peak stress.

Elementary beam theory vs. real (shear-lag) flange behavior in a wide-flanged bending member

Aspect	Euler–Bernoulli / plane sections	Real flange with shear lag
Flange normal stress σ_x	Uniform across width, σ = My/I	Peaks at web junction, decays toward tips
Assumption violated	None (by definition)	Plane sections do NOT remain plane in-plane
Width used in σ = Mc/I	Full geometric width b	Reduced effective width b_eff = ψ·b
Worst case	—	Wide flange, short span, point/concentrated load
Design consequence	Optimistic (unconservative)	Higher peak σ, lower stiffness, fatigue hotspot
Typical error if ignored	0%	20–60% underestimate of peak flange stress