Fixed Beam with Uniform Load

Reference

Clamp both ends of a uniformly loaded span and the moments migrate to the supports: the interior relaxes and the deflection collapses to a fifth.

The equation

Mend = wL2/12 hogging at each end
Mmid = wL2/24 at midspan
R = wL/2 at each end
δmax = wL4/384EI at midspan
  • w is the load per unit length, in kN per meter
  • L is the clear span, in meters
  • E and I are the modulus and second moment of area
  • δ is the vertical deflection
w = 10 kN/mL = 6 mBending moment, sagging plotted upwL2 /24 = 15 kNmwL2 /12 = 30 kNm hogging at each endcalculatgeek.com
Clamping the ends sends two thirds of the moment to the supports and cuts the deflection to a fifth: the ends do the work, provided the structure can actually deliver that fixity.

Worked example

A 6 m span carrying 10 kN/m with EI = 16000 kNm2, ends fully fixed. End moments are 10 × 62/12 = 30 kNm hogging, midspan is 10 × 62/24 = 15 kNm, and the deflection is 10 × 64/(384 × 16000) = 2.1 mm, exactly one fifth of the 10.5 mm the same beam deflects simply supported. An independent numerical solve of the clamped beam returns the same three numbers.

When to use it

Both ends must genuinely refuse to rotate: stiff columns above and below, or symmetry inside a continuous run. Real connections are partially rigid, so the true moments sit somewhere between this case and the pinned one, and support settlement redistributes them further. Use the two cases as brackets, and the continuous analyzer when the honest answer lives in the middle.

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