Deflection Is the Curvature M/EI Integrated Twice
So far you have asked whether a beam breaks, that is, its strength. But not breaking is not everything. A diving board that bounces too much, or a floor beam that visibly sags, is unusable even if it never breaks. So you also have to ask how much the beam actually bends, its stiffness. That bent shape is the deflection curve, and the route to it is surprisingly simple. As you saw in B3, how sharply a beam bends, its curvature, is proportional to the moment there and inversely proportional to the bending stiffness: curvature = MEI. Since curvature is the deflection differentiated twice, going the other way, integrating MEI twice gives the deflection. Each integration brings a constant, and those are fixed by the boundary conditions the supports impose. And when several loads act together, you simply solve each separately and add.
First, look at the deflection itself. Press the tip of a cantilever fixed at one end. The beam bends downward into a smooth curve. This is the deflection curve y(x), showing how far each position has dropped from where it started. Press harder and it sags deeper. Within the elastic range the deflection too is proportional to the load, Hooke's law following all the way here. For a cantilever with a tip load, the tip deflection is δ = PL³(3EI). The thing to notice is the L³. Double the length and the deflection grows eightfold. So a longer beam sags far more, and in deflection problems length is the most fearsome variable.
Peel back one layer of what deflection really is. Take a single slice of the bent beam and it is curved like a small circular arc. How sharply it curves is the curvature κ, the reciprocal of the radius R. The key relation is here: κ = MEI. The larger the bending moment M at that spot, the more it curves; the larger the bending stiffness EI, the less. Drag M. As the moment grows, the slice rolls into a tighter arc. EI grows with a deeper section and a stiffer material, and for the same moment a large EI barely curves. Where the section modulus S was the star of strength, EI is the star of stiffness. And curvature is the very seed of deflection: these small bends, set at each location, accumulate into the whole deflection curve.
Now stack those small curvatures up into a deflection. In math, curvature is the deflection y differentiated twice: EI·y″ = M. So go the other way. Integrate MEI once and you get the beam's slope y′; integrate once more and you get the deflection y. For a cantilever with a tip load, the moment is linear in position, so MEI is linear, the once-integrated slope is a parabola, and the twice-integrated deflection is a cubic. Drag P and the three curves grow together. You can watch the degree rise by one with each integration. This double integration is the engine of deflection analysis. Given only the moment diagram, two integrations hand you the whole shape into which the beam bends.
Integration has one trap. Each integration brings an unknown constant, so two integrations bring two. Without fixing them, the deflection curve floats up and down or tilts as a whole. What pins these constants down is the boundary conditions, the facts the supports give you. For a cantilever, at the fixed end both the deflection and the slope are zero, since it is clamped into the wall, and those two conditions fix the two constants exactly. For a simply supported beam, instead the deflection is zero at both ends. Toggle the supports. For the very same load, the deflected shape changes completely. Even with the same curvature distribution, different boundary conditions give different results. So in a beam problem, what the supports are is half the answer.
Finally, a gift that makes practice very easy. Because everything is linear, deflections add. If several loads act on a beam, find the deflection each load makes on its own, then simply add them. This is called superposition. Toggle between load A alone, B alone, and both. The deflection with both is exactly the deflection from A alone plus the deflection from B alone. So even complicated loading is nothing to fear. The back of a textbook tabulates deflection formulas for basic cases, the cantilever tip load, the uniform load, the central point load, and a real problem is solved by pulling those out of the table and adding. Deflection analysis runs, in the end, on two integrations and addition.