Fixed-Fixed Beam (UDL)

stressmass-cost

Take a beam and build both ends rigidly into the walls — no rotation, no deflection at either support — then load it with a uniform load $w$. A simply-supported beam rests on pins that are free to rotate; a propped cantilever clamps one end and props the other. Clamp both ends and you get the stiffest of the lot, and the most indeterminate. Each wall now supplies a vertical reaction and a fixing moment, so there are four unknown reactions — $R_A, M_A$ at the left wall, $R_B, M_B$ at the right — but statics still offers only two equations. The beam is statically indeterminate to the second degree, and equilibrium alone is two equations short.

This is the workhorse end condition of real structures: a beam welded to columns at both ends, a machine bed bolted down at each end, a bridge deck continuous over rigid piers, the fixed-fixed guide rails inside a press. Fixing the ends is what a designer does to make a member stiff.

The missing equations are compatibility — two of them

Being indeterminate to the second degree means two extra equations are needed, and both come from geometry. Release the right end to a plain cantilever fixed at $A$ (the primary structure). Its free end $B$ would, under the load, both drop and rotate — but the real wall allows neither. So we impose two conditions at $B$ at once:

• Zero deflection: the UDL drops $B$ by $wL^4/8EI$, the redundant force $R_B$ lifts it by $R_B L^3/3EI$, and the redundant couple $M_B$ contributes $M_B L^2/2EI$ — together they must cancel.
• Zero slope: a built-in end cannot rotate either, so the tip rotations $wL^3/6EI$, $R_B L^2/2EI$, and $M_B L/EI$ must also cancel.

Four equations now — two equilibrium, two compatibility — for four unknowns. By symmetry the answer is clean:

$R_A = R_B = \tfrac{1}{2} w L, \qquad M_A = M_B = \tfrac{1}{12} w L^2$

The build does not hand you these to trust. It certifies that the four relations form a system that is linear in the four unknowns, solves that $4\times4$ system exactly at build time, and checks the solution back through every relation. This is the solveLinear capability at the top of its range — the propped cantilever is the reference $3\times3$ case, and this clamped beam is the four-unknown one. What the machine proves and what still rests on a book is on the verification page.

The ends govern — and the whole beam is stiffer for it

Fixing the ends does two good things at once, both visible in the widget. First, the peak bending moment moves to the walls and shrinks: the hogging wall moment $wL^2/12$ is only two-thirds of a simply-supported beam’s $wL^2/8$ under the same load, and it is exactly twice the sagging moment at midspan, $wL^2/24$. Because the walls carry the larger moment, first yield happens there — which is why the safety factor is taken at the wall’s outer fiber. Second, the midspan deflection collapses to

$\delta_{max} = \frac{w L^4}{384\,E I}$

— one-fifth of the simply-supported beam’s $5wL^4/384EI$. Clamping the ends is the single most effective thing you can do to a beam short of changing its section.

The reactions do not care what the beam is made of

Look at the compatibility steps in the derivation below. Every deflection and slope term carries the same flexural rigidity $EI$, and because the beam is prismatic, $EI$ divides out completely — the reaction equations have no $E$ and no $I$ in them. That cancellation is machine-checked, and it is the lesson: the reactions are material-blind. Switch the material in the widget from A36 steel to Ti-6Al-4V and watch $R_A$, $R_B$, $M_A$, and $M_B$ not move at all, even as:

• Young’s modulus $E$ changes the midspan deflection $\delta_{max} = wL^4/384EI$ — titanium is about half as stiff ($E \approx 110$ vs $200$ GPa), so it deflects nearly twice as far.
• Yield strength $\sigma_y$ changes the safety factor $\mathrm{SF} = \sigma_y/\sigma_{max}$.
• Density $\rho$ changes the mass.

A redundant restraint redistributes load by geometry, not by material. (Make the beam non-prismatic — a tapered section, or two materials spliced together — and $EI$ no longer cancels; the reactions would then follow the stiffness distribution. That is the composite-bar story, a later THING.)

What governs, and the sign convention

Throughout: $x$ runs from wall $A$ to wall $B$, downward load and deflection are positive, $R_A$ and $R_B$ are the upward reactions, and $M_A$, $M_B$ are the magnitudes of the hogging fixing moments at the two walls. The bending moment is largest at the walls: $|M|_{max} = wL^2/12$ (hogging) beats the sagging midspan value $wL^2/24$ by a factor of two, so the walls’ outer fibers are where first yield happens and what the safety factor guards. The midspan deflection $wL^4/384EI$ is the true maximum, exactly at the centre by symmetry.

Try it

Governing relations

Derivation

Steps marked modeling step are where physics enters by citation — every other line is machine-proven to follow from them, and the cited models are independently re-derived in the test pipeline where possible. See what is and isn't machine-verified.

How it fails

The widget’s safety factor guards one thing — first yield at the wall’s outer fiber, where the hogging moment $M_A = wL^2/12$ peaks. A real fixed-fixed beam has more exits, and the most interesting ones are specific to it being statically indeterminate:

• Support settlement or rotation — the assumption that can quietly move every number. The whole solution rests on two premises per wall: zero deflection and zero rotation. Let one support settle by $\Delta$ or one built-in end rotate slightly (a column that flexes, a connection that is not truly rigid) and the compatibility equations pick up those movements. The reactions redistribute, the wall moments can swing sharply, and the “material-blind” result stops being material-blind — now $EI$ does appear, multiplied by the movement. In a determinate beam a support that moves just goes along for the ride; in a doubly-clamped one, support movement is a load case in its own right, and a stiff structure is a temperamental one.

• Yielding at the walls, then a collapse mechanism. Because the peak moment is at the built-in ends, plastic hinges form there first. Unlike a simply-supported beam, the fixed-fixed beam does not collapse when the ends yield — the wall hinges cap their moment and the span picks up more, until a third hinge forms at midspan and the beam becomes a mechanism. That reserve (three hinges to collapse, not one) is a genuine advantage of the clamped beam, and it is entirely beyond the elastic formulas here. The $\sigma_{max} > \sigma_y$ warning marks where they stop telling the truth.

• Thermal stress with no room to expand. This is the clamped beam’s signature failure. Pin-ended beams expand freely; a beam built into rigid walls at both ends cannot. Heat it and it cannot grow, so it develops a compressive axial stress $\sigma = E\,\alpha\,\Delta T$ that this bending-only page never computes — enough to buckle rails in summer heat (the classic “sun kink”) or crush a fixed-fixed member that a simply-supported one would have shrugged off. Fixing both ends trades flexibility for a thermal liability.

• Lateral-torsional buckling and thin webs. A tall, narrow section ($h \gg b$) can twist and flop sideways before it yields — a stiffness failure, not a strength one — and the high shear near the walls can buckle a thin web.

• Fatigue at the fixed ends. A fixed-fixed beam that vibrates cycles its largest stress at the walls, usually at a welded or bolted connection full of stress raisers. Fatigue strength sits far below static yield, and nothing on this page addresses cyclic life.

• Short, deep beams and brittle materials. For $L < 10h$ (the validity warning) shear deflection, neglected by Euler–Bernoulli theory, is no longer negligible. And a brittle material such as gray cast iron has no yield plateau — rate it against ultimate strength with a generous factor instead of the yield-based safety factor shown here.

Sources

• Gere, J. M., & Goodno, B. J., Mechanics of Materials, 9th ed., Cengage, 2018 — ch. 10 (Statically Indeterminate Beams): the force (flexibility) method, fixed-fixed (clamped) beam reactions by superposition; §5.5 flexure formula; App. E section properties.
• Hibbeler, R. C., Mechanics of Materials, 10th ed., Pearson, 2017 — ch. 12 (Deflection of Beams and Shafts): statically indeterminate beams by the method of superposition; equilibrium of the fixed-fixed beam.