A welded moment splice in a continuous beam

Look at the joint over the middle support B. The section steps down — a deep 610UB125 arriving from the left, a shallower 410UB53.7 leaving to the right — and across the interface the drawing shows exactly one clean, heavy line. That line is the site’s language for a welded moment splice: a full-strength butt weld joining flange to flange and web to web. No plate, no bolts, no cope — the two sections are fused into one piece of steel.

Compare it with the fin plate of the Gerber example, where unconnected flanges meant no moment path and statics alone could finish the job. Here the flanges are connected, so bending moment crosses the joint at full strength. The structural model is one continuous beam over three supports — statically indeterminate, one redundancy — and we will need more than statics to solve it.

One redundancy calls for one compatibility equation, and for continuous beams the classic is Clapeyron’s three-moment theorem. In its general form the stiffness of each span appears explicitly — every term carries its own I. For our two spans (ends simply supported, so , with a UDL w on both):

Now put in what this beam actually is: = 6 m and the same w on both spans, but — that is the whole point of the splice. The equation becomes

Look at both sides: the stiffness factor appears on the left and on the right — so it cancels exactly, whatever the two values are:

That is the lesson of this page in one line: for equal spans under equal load, the I’s cancel and the stiffness step does not move the moments at all. We never used 988.3 or 188.4 — the answer is the same as if the beam were prismatic.

Cut the beam at B and take span A–B as a free body: a simply supported 6 m span under its UDL plus the known hogging moment at the B end. Moments about B:

Span B–C carries the same load and the same end moment, so by the same free body kN. The middle support takes the rest:

Global check: 2 × 33.75 + 112.5 = 180 kN = 15 × 12 ✓. Every reaction is symmetric — a first hint that the moments do not know one span is five times stiffer than the other.

Walk the shear from the left: it starts at +33.75, the UDL bleeds it away, and over B it jumps by the full 112.5 kN reaction, then plays the mirror image out to C. It crosses zero where the sagging moment peaks:

The moment diagram hogs to −67.5 kN·m over B and rises to twin sagging peaks of 37.97 kN·m — at 2.25 m in the deep span and, mirrored, at 9.75 m in the shallow one. Perfectly symmetric moments on a beam that is anything but symmetric.

So does the splice change nothing? Look at the deflected shape. Curvature is : both spans carry identical moments, but the 410UB53.7 has an I about five times smaller than the 610UB125 — so at every matching station it curves about five times as hard. The deep span barely moves; the shallow span sags visibly under the very same bending moments.

And note what does not happen at B: no kink. The welded splice keeps the slope continuous through the support — the exact opposite of the Gerber page, where the fin-plate hinge announced itself with a slope break. The moments are symmetric; the deflections are not. That asymmetry is the stiffness step, finally showing itself in the one place it can.