The load that chooses its own length

Almost every load in this collection arrives over a length that somebody chose. A bearing is detailed at 150 mm because that is what the drawing says; a stiffener is 200 mm wide; a bay is 7.2 m. The length is an input, and the calculation divides by it.

A wheel is different. So is a launching bridge on its rollers, a column landing on an unstiffened beam, and a girder being lifted on slings. The load arrives over a length so short that the flange under it bends, and the bending spreads the load along the web — over a length that nobody decided and that has to be computed.

Which free body produced the number #

Take the flange as a beam sitting on the web, loaded over the bearing length and supported continuously along the web line beneath it.

Push hard enough and the flange forms plastic hinges: one under each edge of the bearing and one further out where the bending has run out. The load between the outer hinges is delivered to the web, and the distance between them is the effective loaded length $\ell_y$.

Doing that yield-line calculation gives

$$\ell_y = s_s + 2t_f\left(1 + \sqrt{m_1 + m_2}\right), \qquad m_1 = \frac{f_{yf}\,b_f}{f_{yw}\,t_w}, \qquad m_2 = 0.02\left(\frac{h_w}{t_f}\right)^{\!2}$$

where $m_1$ compares the flange’s plastic strength with the web’s, and $m_2$ carries the web’s slenderness. For the girder above, $m_1 = 37.5$ and $m_2 = 72.0$, so the square root is 10.46, and

$$\ell_y = 100 + 2 \times 20 \times 11.46 = 559\;\text{mm}$$

Note where the bearing length appears: as an additive constant, and only there. Everything else is a property of the section. A bearing of zero length still spreads the load over 459 mm, because the flange is still there.

Three failures, one curve #

The web now has a length to be loaded over, and three separate things can go wrong with it.

It can yield. The load simply crushes the web over its effective length: $F_y = f_{yw}\,t_w\,\ell_y = 1{,}586$ kN. This is web crushing, and it is the answer for a stocky web — the same crushing limit a bolt in bearing meets against the plate it presses into.

It can buckle elastically. The web is a plate loaded on one edge, and its critical load is $F_{cr} = 0.9\,k_F\,E\,t_w^3/h_w = 510$ kN, with $k_F$ from the panel geometry. This is the answer for a very slender web.

And in between it does something that is neither, which is the case every real girder is in.

$\chi = 0.5/\lambda \le 1$, so $F_{Rd} = \chi F_y = 450$ kN. The web could yield at 1,586 kN and buckle at 510, and it does neither: it fails at 450, below both, which is exactly the pattern the plate curve and the column curve share and that the imperfect column explains and for the same reason — imperfections and residual stress make the transition region worse than either bound.

The web thickness, squared #

The exponent is worth deriving rather than reading off, because the route is short and the answer is surprising.

$F_y \propto t_w \ell_y$ and $F_{cr} \propto t_w^3$, so $\lambda = \sqrt{F_y/F_{cr}} \propto \sqrt{\ell_y}/t_w$ and $\chi = 0.5/\lambda \propto t_w/\sqrt{\ell_y}$. Therefore

$$F_{Rd} = \chi F_y \;\propto\; \frac{t_w}{\sqrt{\ell_y}} \cdot t_w \ell_y = t_w^2 \sqrt{\ell_y}$$

and $\ell_y$ depends only weakly on $t_w$. The measured exponent over 5 to 25 mm of web is 1.95.

That is a strong dependence and it is not what a designer’s intuition supplies. A shear check on the same web is linear in the thickness; a bending check does not contain it at all to first order. Here, taking the web from 8 to 12 mm — a half as much again — multiplies the resistance by 2.20, and $1.5^{1.95} = 2.20$ to three figures.

The flange thickness, hardly at all #

This is the result that inverts the expectation, and the mechanism is in the two yield-line terms.

$\ell_y$ contains $2t_f\sqrt{m_1 + m_2}$. The $m_1$ term is independent of $t_f$, so its contribution grows linearly with the flange. The $m_2$ term is $0.02(h_w/t_f)^2$, so $\sqrt{m_2} \propto 1/t_f$ and its contribution to $2t_f\sqrt{m_2}$ is $0.283\,h_w$ — a constant, with no flange thickness in it at all.

So a thicker flange grows one half of the spread and leaves the other alone, and since $m_2$ dominates for a slender web, most of the effective length does not know how thick the flange is. Add the square root in $F_{Rd} \propto t_w^2\sqrt{\ell_y}$ and the flange’s influence is halved again.

The practical version: improving a patch-load resistance is a web problem. Doubling the flange from 20 to 40 mm buys 18%. Adding half again to the web buys 120%. And the cheapest answer of all is not a thickness at all.

The answer that is not a thickness #

A bearing stiffener under the load changes the problem completely rather than improving it. The load is delivered into a member designed to carry it as a short column, the web’s own patch resistance stops being the check, and the calculation becomes a stub-column one.

That is why the check matters most where a stiffener cannot be put: under a moving wheel, which is not at a stiffener; under a launching bridge’s rollers, which travel; at a support during erection before the stiffeners are welded on.

Where the web has already gone #

There is a case where the web has buckled and the girder is still working, and it is worth putting beside this one because the two look similar and are not.

The asymmetry is worth naming. Shear buckling of a web is a serviceability event with a strength reserve behind it. Patch-load failure of a web is not: the web folds locally under the load, the flange follows it down, and there is no second mechanism. Which is why the reduction curve is used as a strength limit rather than as a warning.

Four places it decides something #

A crane girder is the case the check was written for. A wheel load of a hundred kilonewtons or more travels the whole length of the girder, arrives on a rail bolted to the top flange, and cannot be met with a stiffener because it is never in the same place twice. The girder’s web thickness is very often set by this check and not by shear or bending — which is a member sized by a load case that occupies a few hundred millimetres of it.

A launched bridge is worse, because the girder passes over the rollers nose first with its full self-weight behind it, and every section of the web is subjected to the reaction in turn. Launching noses and roller spacings are designed around this, and a launched girder frequently carries a thicker web over its whole length for the sake of a condition that exists for a few weeks.

A column landing on a beam puts a concentrated load into an unstiffened flange, and the same yield-line spread applies. Here the answer is usually a stiffener and the interest is in what happens if one is left out — which happens, because the column above is drawn on one drawing and the beam below on another.

And a beam sitting on a bearing is the same problem from below. The reaction is not applied to the member; the member is resting on something, and the length over which that something answers is decided by the same flange bending. It is the one case where a designer has usually written down a length — the bearing plate’s — and it is not the length the web sees.

None of the four has a stiffener available at the point where the load actually is, and that is what they have in common. Where a stiffener can be put, this check is replaced by a different and easier one; where it cannot, the web’s own resistance is the whole answer, and it is a function of a length nobody drew.

And a beam sitting on a bearing is the same problem from below. The reaction is delivered into the bottom flange over the bearing’s length, spreads through the flange, and loads the web — so a support is a patch load too, and it is the one place a designer is most likely to have assumed the length was the bearing plate’s.

The one number worth carrying #

If a single figure survives from this page it should be the ratio 5.6, and it should be carried as an order of magnitude rather than a value.

A concentrated load on a girder flange is delivered to the web over a length several times the bearing that produced it, and the multiple is largest for exactly the girders that most need help — a deep slender web has a large $m_2$, which lengthens the spread. The plate that looks least able to take a wheel is the one whose flange spreads the wheel furthest.

That is a genuinely unusual arrangement, and it is worth stating as a general shape: a mechanism that relieves the member in proportion to how badly it needs relieving is rare, and this is one. The spread is not a design allowance being claimed; it is a plastic mechanism that happens whether anybody counts it or not, and counting it is the difference between a resistance of 284 kN and one of 450.

Where the model stops #

The expressions are semi-empirical. The yield-line derivation gives the form of $\ell_y$; the constants — the 0.02 in $m_2$, the 0.9 in $F_{cr}$, the 0.5 in $\chi$ — are fitted to tests. They are not derivable and they differ between codes by more than the difference between a 20 and a 40 mm flange.

Bending is assumed absent. A girder carrying a patch load is usually also carrying a moment, and the two interact — the web’s compression from bending reduces what is left for the patch load. The interaction is checked separately and is not in any figure here.

The load is assumed perpendicular to the flange. A wheel on a crane girder also applies a horizontal surge, and a launching bridge’s rollers apply friction. Both put a horizontal force into the top flange at exactly the point where the vertical one is being spread.

And the flange is assumed to stay flat. A wide thin flange under a concentrated load dishes, so the load is delivered to the web over less width than the yield-line model assumes and the flange itself may fail before the web does.

What the pictures cannot show #

The spread figure draws the yield lines as two straight dashes from the bearing edges to the web. The real mechanism is a set of hinge lines across a plate in two dimensions, and the drawing is a section through it.

The reduction curve places every girder in the two sweeps as a dot, which makes the curve look densely evidenced. It is one expression evaluated many times, not a set of measurements, and the tests behind the expression scatter by more than the width of the line.

And nothing here draws the failure. A web that crumples under a wheel does so over a length of a few hundred millimetres and a depth of a few tens, and the deformed shape is a local fold in a plate — which no elevation of a 1,200 mm girder can show at a scale where the girder is visible.

The ladder from here #

Later rungs on this anchor: the interaction between patch loading and bending, which is where the check actually bites on a crane girder. Bearing stiffeners as stub columns, including the effective web width that acts with them. Launching, where the patch load travels the whole length of the girder and every section is checked at the worst position. Web crippling in cold-formed sections, where the same problem has an entirely empirical treatment because no yield-line model fits a rounded corner. Wheel loads on rails, where the rail spreads the load further still and the rail-to-flange interface becomes the model. And the historical case: the plate girder’s web was made thin long before anybody could compute what a wheel did to it, and the resulting failures are why crane gantries carry stiffeners at every wheel position on drawings from the 1900s.