A beam can be perfectly safe and completely unacceptable. Plaster cracks, doors bind, floors bounce — and none of that is a strength problem. Deflection is checked separately, against limits written as fractions of the span (L/360 for a plastered ceiling is the classic), and on long spans it governs the design far more often than stress does.

Simply supported, one load at midspan: $\delta = \dfrac{P L^{3}}{48 E I}$ — read aloud delta equals P L cubed over forty-eight E I. $\delta$, the Greek letter delta, is the maximum deflection at midspan, in mm. $P$ is the point load in N, $L$ the span, $E$ is Young's modulus — the material's stiffness, 200 GPa for structural steel — and $I$ the second moment of area. $E$ and $I$ always travel together as the product $EI$, the flexural rigidity: one number for what the material brings and what the shape brings.

Simply supported under a uniform load: $\delta = \dfrac{5 w L^{4}}{384 E I}$ — delta equals five w L to the fourth over three-eighty-four E I, with $w$ the load per unit length. The 5 is not decoration and the 384 is not a typo; both fall out of integrating the curvature, and dropping either wrecks the answer by a fixed, embarrassing ratio.

The powers are the headline. Deflection runs as $L^{3}$ for a point load and $L^{4}$ for a spread one, so doubling a span multiplies the sag by eight or by sixteen. Nothing else in beam design punishes ambition that hard. And note where $I$ sits — underneath — which is why the cure for a bouncy floor is always a deeper joist and never a stronger grade of timber.

One last habit: deflection answers arrive in MILLIMETRES. If your arithmetic hands you metres, or micrometres, a unit slipped somewhere upstream — usually $E$, entered as 200 000 MPa into a slot that already wanted 200 GPa.