In the wide top and bottom flanges of a box girder, the bending stress concentrates next to the webs and "lags" toward the middle. Adjust the flange width, span length and plate thickness to see the shear-lag factor and effective flange width update in real time, and design without missing the true peak stress at the webs.
Parameters
Flange width (centreline) b
mm
Full width of the top flange spanning between the two webs
Span length L
mm
Distance between the girder supports
Flange thickness t
mm
Thickness of the flange plate
Flange stress at the web σ_max
MPa
Peak longitudinal stress, highest next to the webs
Results
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Flange-width/span ratio b/L
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Shear-lag factor ψ
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Effective flange width (mm)
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Mean flange stress (MPa)
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Flange axial force (kN)
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Shear-lag verdict
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Box-girder section and flange stress distribution
The actual stress distribution across the top flange (high at the webs, sagging at mid-flange) is overlaid with the uniform "effective width" rectangle that carries the same total force. Effective and ineffective widths are marked with arrows.
ψ is the simplified shear-lag factor, b/L the flange-width-to-span ratio, and b_eff the effective flange width. The wider the flange relative to the span (the larger b/L), the smaller the effective width b_eff.
The mean flange stress σ_mean is the peak stress σ_max reduced by the shear-lag factor; the flange axial force N_flange is the total force the peak stress carries over the effective width and plate thickness.
What is shear lag?
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What is "shear lag" in a box girder? When you bend a bridge girder, isn't the stress simply set by the top and bottom of the section, like the textbook says?
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Good question. Elementary beam theory assumes the cross-section stays plane when it bends. From that, the bending stress varies linearly with depth and is uniform across the width of a flange — that's the textbook picture. For a narrow flange it's good enough. But for a really wide plate, like the top flange of a box girder, that assumption breaks down. That breakdown is where shear lag starts.
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Why does a wide flange stop being uniform?
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The bending moment enters the flange along its edges — the junctions with the webs — as a shear flow. For that shear to travel inward and stress the middle of the flange, it needs distance, because the plate is not infinitely stiff in shear. So the material near the webs picks up its stress immediately, but toward the centre the stress "lags" lower. Drag the "flange width b" slider on the left wider, and the distribution in the chart below shifts to a shape that is high at the webs and sags in the middle.
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I see — the stress piles up next to the webs. Why is that a problem?
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The problem is that if you design assuming uniform stress, you under-estimate the real peak stress next to the webs. Cracks and yielding always start where the stress is highest — the web-to-flange junction. Treat that lightly and you miss the most dangerous spot of the bridge. So in design we handle it with the idea of an "effective flange width".
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The effective flange width — how does that idea work?
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Dealing with that curved stress profile directly is awkward, right? So we replace the real flange with a narrower equivalent flange in which the peak stress acts uniformly. We pick its width so that this narrow flange carries the same total axial force as the real one. Then we can keep using the simple uniform-stress formula and still capture the peak stress at the webs. The wider the flange relative to the span, the stronger the shear lag and the smaller the effective width — which is why wide, short-span box-girder bridges are where shear lag drives the design.
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So the "shear-lag factor ψ" in this tool is the number used to get that effective width.
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Exactly. ψ = 1 means no shear lag; the smaller ψ, the stronger the lag. This tool uses a simplified parametric model, ψ = 1/(1 + 5(b/L)²). The effective width is b_eff = ψ·b and the mean stress is σ_mean = ψ·σ_max. In practice you would evaluate the effective width precisely with code provisions or finite-element analysis, but a simple model like this is great for building a feel for how much the lag matters.
Frequently Asked Questions
Shear lag is the phenomenon that, in a wide flange such as the top or bottom plate of a box girder, the longitudinal bending stress is not uniform across the flange width. The bending moment is fed into the flange as a shear flow from the webs along its edges, and that stress takes distance — time — to diffuse toward the middle. As a result the stress is highest next to the webs and lags to a lower value toward the centre of the flange. This tool computes the non-uniformity with a simplified parametric model.
Instead of dealing with the awkward curved stress profile caused by shear lag, the effective flange width replaces the real flange with a narrower equivalent flange that carries the peak (web) stress uniformly and delivers the same total axial force. This lets engineers keep using the simple uniform-stress formulas while still capturing the true peak stress at the web junctions. The wider the flange relative to the span, the smaller the effective width.
Elementary beam theory assumes a cross-section stays plane when it bends. From that assumption the bending stress varies linearly with depth and is uniform across the width of any flange. For a narrow flange that is fine. But for the wide top and bottom plates of a box girder the shear flow cannot diffuse across the width, so the plane-section assumption breaks down. Designing with uniform stress under-estimates the true peak stress at the web junctions, where cracks and yielding would start.
Shear lag is most pronounced in bridges whose flange is wide relative to the span — that is, wide, short-span box-girder bridges. The larger the flange-width-to-span ratio b/L, the smaller the shear-lag factor and the smaller the effective flange width. For wide, short-span steel box-girder bridges and girders with large deck overhangs, shear lag is a defining design consideration and is built into modern bridge design codes as effective-width provisions.
Real-World Applications
Steel box-girder and plate-girder bridge design: In highway and railway steel box-girder bridges, the wide top and bottom flanges carry the bending. Where the flange is wide relative to the span, designers reduce the section properties with an effective flange width and check the stress peak at the webs. Bridge design codes (such as AASHTO, the Japanese Specifications for Highway Bridges and the Eurocodes) state explicit effective-width rules, so shear lag is quantitatively built into design.
PC box girders and concrete decks: In prestressed-concrete box girders and in wide decks cantilevering off the girder, shear lag also lowers the effectiveness of the deck centre. For composite girders with large overhangs, the effective width of the concrete deck is narrowed to evaluate the stiffness and stress of the composite section. Ignore this and the deflection and the midspan stress are under-estimated.
Support sections of continuous girders: Near the interior supports of a continuous box girder the moment is negative, so a wide bottom flange is in compression. Shear lag raises the compressive stress next to the webs, which makes it important alongside the buckling check. Stress gradients are steep over the supports, and shear-lag effects tend to be more pronounced there than at midspan.
Pre-study and verification of finite-element analysis: Before running a detailed shell-element FEM, a simple model like this tool gives the order of magnitude of the effective width. Checking that the FEM stress distribution is high at the webs and low at mid-flange, and that the effective width is not orders of magnitude off the estimate, catches mesh or boundary-condition mistakes early. The simple calculation works well as a sanity check on the detailed analysis.
Common Misconceptions and Pitfalls
The biggest misconception is "shear lag is just a uniform reduction of the bending stress". The essence of shear lag is a non-uniformity of stress, not a simple discount. The width-averaged flange stress does drop, but the stress next to the webs can actually be higher than the elementary beam-theory value. The effective flange width is a "convenient way to take an average"; what must be checked in design is still the peak stress at the webs. Look only at the mean stress and you miss the most dangerous junction.
Next, "the effective width is a single fixed value, the same everywhere". The real effective flange width varies with the position along the span (near a support versus at midspan), the type of loading (uniform versus concentrated) and the type of member (simple versus continuous girder). This tool is a simplified model driven only by the flange-width-to-span ratio, but in a real bridge the effective width is smaller over supports and larger at midspan — it depends on position. Code effective-width rules also distinguish these positions and loadings in tables.
Finally, over-confidence that "the coefficient of a simple formula can be applied to a real bridge as-is". The ψ = 1/(1 + 5(b/L)²) used here is a simplified parametric model for grasping the trend of shear lag, and the coefficient 5 is a representative value chosen for illustration. The real effective width depends on the section shape, the plate-thickness distribution, the stiffeners and the support conditions, and it is found precisely with series solutions of elasticity theory or shell FEM. The simple model is useful for first-cut judgement — does it matter, which design is unfavourable — but the final section check should be done with the provisions of the applicable design code and with FEM.
How to Use
Enter flange width (b) in mm—typical range 800–2400 mm for steel box girders
Set span length (L) in meters; common values 20–60 m for bridge applications
Input web thickness (t) in mm, typically 10–25 mm for welded steel boxes
Specify applied shear force (s) in kN at the cross-section of interest
Click Calculate to obtain shear-lag factor ψ, effective flange width, and stress distribution
Review the b/L ratio and verdict to assess whether full flange participation is achievable
Worked Example
Steel box girder: flange width b=1200 mm, span L=40 m, web thickness t=16 mm, applied shear force s=450 kN. The simulator computes b/L=0.030, yielding shear-lag factor ψ=0.82 (indicating 18% stress lag near flange midpoint). Effective flange width becomes 984 mm. Mean flange stress calculates to 156 MPa under shear action, with total flange axial force 187 kN concentrated within 984 mm effective zone instead of full 1200 mm. Verdict: moderate shear lag; consider stiffeners or wider webs for 0.40 m+ span reduction.
Practical Notes
Shear-lag factor ψ below 0.90 signals non-uniform stress distribution; design flange reinforcement or add longitudinal stiffeners in the lag zone (outer 10–15% of flange)
For concrete box girders (fc'=35 MPa), shear lag is typically 5–10% higher than steel due to lower transverse shear stiffness; increase web depth or reduce flange width accordingly
Fatigue-critical applications require ψ ≥ 0.95 to minimize local stress peaks; verify by finite-element analysis if ψ drops below 0.85