Module 10 · Lesson 10.1
The girder, and the web it can afford
Depth is what carries moment, and depth is cheap in a welded girder. The price is a web too thin to behave like a plate in shear.
Why this matters
Rolled sections stop. The largest available is around a metre deep, and beyond that span or load a designer either uses several of them or welds plates into the shape that is actually wanted. The welded girder is startlingly efficient — the numbers below are better than most people expect. But the efficiency comes from making the web as thin as possible, and a thin web does something a rolled web never does: it buckles in shear, at a stress far below yield, and then carries on carrying load by a completely different mechanism.
By the end of this lesson you should be able to
- Quantify what a plate girder buys over a rolled section
- Say why the web can be so thin, and what that costs
- Explain shear buckling and tension-field action physically
- Read the effect of stiffener spacing and end-post detailing
What you should already know
- Plate buckling and the (t/b)² dependence (Module 5)
- Shear resistance and the shear-buckling threshold (Module 6)
- Cross-section classification, and what a Class 3 or 4 web means (Module 5)
- The relative-slenderness pattern λ̄ = √(resistance/critical) (Modules 7 and 8)
Why depth is worth so much
Second moment of area goes with the square of the distance from the neutral axis. Move a flange twice as far out and it contributes four times as much. A welded girder lets you choose that distance directly instead of accepting whatever the rolling mill produces.
The comparison in the worked example is the argument in one line: a 1550 mm deep girder weighing 275 kg/m has 44% more second moment of area than the largest rolled UB, which weighs 481 kg/m. More stiffness, on 57% of the steel.
And where does the material go? For that girder, the flanges supply 80% of the second moment of area while using 57% of the material. The web is not carrying much moment at all. Its job is to hold the two flanges apart and to carry shear — and since it is barely contributing to bending, there is no reason to make it thick.
So the designer makes it thin. On that girder the web is 1500 mm deep and 10 mm thick.
hw/tw = 150. That is not a plate you can push on.
The web buckles — and then does not fail
This is the part worth getting right, because it is genuinely surprising.
A web panel in shear is, at 45°, in tension along one diagonal and compression along the other. When the panel buckles, it is the compression diagonal that goes: it wrinkles out of plane and sheds its load.
The tension diagonal does not. Tension does not buckle.
So what forms is a band of tension running diagonally across the panel, anchored at its ends. The girder reorganises itself into a truss:
- the tension field is a diagonal tie
- the transverse stiffeners are the vertical posts
- the flanges are the chords
The girder keeps carrying load, often a great deal more than the load at which the web first buckled. This is post-buckling reserve, and there is nothing like it in a column — a column that buckles is finished.
But the tension has to pull against something. At an internal stiffener the tension fields on either side balance each other. At the end of the girder there is nothing on the far side, so the end detail has to anchor the whole diagonal pull. A rigid end post — in effect a small vertical beam made of two stiffeners with the web between them — can do it. A single end stiffener cannot, and the end panel then gets no tension-field benefit at all.
This is why the shear resistance of a girder depends on a detail at its very end that has nothing to do with the panel being checked.
Try it
Web thickness, stiffener spacing, end post
A 1500 mm deep web with 400 × 25 flanges, all S355. Three levers, and the mass alongside each so the comparison is fair. The reference case is a 10 mm web with stiffeners at 3 m and a rigid end post.
End post
- Web thickness tw
- 10 mm
- Web slenderness hw/tw
- 150
- Panel aspect a/hw
- 2.00
- Shear buckling factor kτ
- 6.34
- Critical shear stress τcr
- 53 N/mm²
- Web slenderness λ̄w
- 1.958
- Reduction χw
- 0.515
- Shear resistance Vb,Rd
- 1585 kN
- Plastic resistance Vpl,Rd
- 3074 kN
- Available fraction
- 52 %
- Girder mass
- 275 kg/m
- Gain over the reference
- 1.00 ×
- Extra mass over the reference
- 1.00 ×
- Worth of a rigid end post here
- 22 %
λ̄w = 1.96 gives χw = 0.515, so only 52% of the plastic shear resistance is available. The web buckled at τcr = 53 N/mm² against a shear yield of 205 — but it did not fail there, because a tension field formed. With a rigid end post that field can be anchored, and the resistance reflects it.
Things worth trying
- Start at the reference: 10 mm web, 3 m stiffeners, rigid end post. 1585 kN against a plastic resistance of 3074 — only 52% is available.
- Take the stiffener spacing to 1.5 m. The resistance rises to 1821 kN — 15% for twice as many stiffeners, and the girder's mass has not changed at all.
- Put the spacing back to 3 m and take the web to 20 mm instead. 5214 kN, a factor of 3.3, for 1.4 times the mass. Web thickness is usually the better lever on a girder, and this is why.
- Watch the available fraction as you thicken the web: 52% at 10 mm, 85% at 20 mm. The gain compounds, because the area and χw both improve.
- Switch to a non-rigid end post at 10 mm. The resistance drops to 1303 kN — the tension field has nothing to pull against in the end panel.
- Now switch the end post at 25 mm. Almost nothing happens, because at that thickness the web is stocky enough that there is no tension field to anchor. A rigid end post is worth a great deal on a slender web and nothing on a stocky one.
- Take the stiffener spacing past 4.5 m and watch kτ flatten out towards 5.34. Beyond about a/hw = 3 there is nothing left for stiffener spacing to give.
Worked example
A 1550 mm welded girder in shear
Given
- Web 1500 × 10 mm, flanges 400 × 25 mm, all S355 (no thickness reduction at 25 mm)
- Transverse stiffeners at 3.0 m centres
- Rigid end posts
- Compare with the largest rolled UB, 1016 × 305 × 487
Find
What the girder buys, and how much of its shear resistance survives.
Assumptions
- Section properties computed from plate geometry — for a welded girder the weld fillets really are negligible, unlike the root radii of a rolled section
- χw, kτ and η are calibrated and unverified here
- The web is Class 4 in bending; the bending resistance below uses the elastic modulus and takes no account of the effective web
Predict first
The girder's web is thickened from 10 mm to 20 mm. The web area doubles. What happens to the shear buckling resistance?
Practice
A web is 1500 mm deep and 10 mm thick. What is hw/tw?
Practice
The web of that girder has τcr = 53.5 N/mm² and fyw = 355 N/mm². What is λ̄w? Take τy = fyw/√3.
Practice
With χw = 0.515, what is Vb,Rd for a 1500 × 10 mm web in S355? Take γM1 = 1.0 and give the answer in kN.
Practice
The same girder has a non-rigid end post, giving Vb,Rd = 1303 kN instead of 1585 kN. By what percentage does the rigid end post increase the resistance?
Check yourself
What actually happens when a girder's web buckles in shear?
Summary
- A welded girder puts the flanges where you want them — 44% more I than the largest UB at 57% of its weight
- Its flanges carry 80% of the second moment on 57% of the material
- So the web is made thin: hw/tw = 150 on the worked example
- τcr goes with (t/h)², so this web buckles at 53.5 N/mm² against a shear yield of 205
- λ̄w = 1.96, χw = 0.515 — only 52% of the plastic shear resistance is available
- But the web does not FAIL: a tension field forms and the girder acts as a truss
- The tension field must be anchored — a rigid end post was worth 22%
- Thickening the web from 10 to 20 mm more than TRIPLED the resistance, because χw improved too
This is educational material. It uses simplified examples to teach principles, and must not be relied on for real design or safety-critical decisions. Module overview and checkpoint