beam deflection M/EI explained
So far the question was whether the beam breaks, which is strength. Not breaking is not the end. If a diving board rocks too much, or a floor beam sags so you can see it, it is useless even if it does not break. So you have to ask how far the beam actually bends, which is stiffness. That bent shape is the deflection curve, and the way to it is simple. As seen in B3, the curvature, how hard the beam bends, is proportional to the moment at that place and inversely proportional to the bending stiffness. Curvature = MEI. Curvature is the deflection differentiated twice, so integrating MEI twice brings the deflection out. Integration produces constants, and the supports' boundary conditions fix them. When several loads act together, solve them apart and add. The supports fix the last constants, and that is why the same curvature can still give two shapes.
Look first at the deflection itself. Press the tip of a cantilever fixed at one end, and the beam bends downward in a smooth curve. That is the deflection curve y(x), how far each place has dropped from where it was. Press the tip harder and it sags deeper. Inside the elastic range the deflection is also proportional to the load, which is Hooke's law followed all the way here. For a tip load on a cantilever the tip deflection is δ = PL³(3EI). What to notice is L³. Double the length and the deflection is eight times. A long beam therefore sags much more, and in a deflection problem the length is the most frightening variable.
Peel one layer off what deflection is. Take one piece of a bent beam and it is curved like a small arc. How hard it is curved is the curvature, the reciprocal of the radius R. The central relation is here. κ = MEI. The larger the bending moment M at that place, the more it curves, and the larger the bending stiffness EI, the less it curves. Move M and the piece curls rounder as the moment grows. EI grows as the section gets deeper and the material gets stiffer, and even at the same moment a large EI hardly curves. If the section modulus S was the lead in strength, EI is the lead in stiffness. Curvature is the seed of deflection, and these small bends, fixed place by place, pile up into the whole deflection curve. The seed grows into the curve.
Now pile the small curvatures into a deflection. In mathematics the curvature is the deflection y differentiated twice. EI·y″ = M. So go backwards. Integrate MEI once and the slope of the beam y′ comes out, and integrate once more and the deflection y comes out. For a tip load on a cantilever the moment is linear in position, so MEI is linear too, the slope after one integration is a parabola, and the deflection after another is a curve of degree 3. Move P and the three curves grow together, and the degree climbs by one with each integration. These two integrations are the engine of deflection analysis. With only the moment diagram, two integrations give the whole shape the beam bends into.
Integration has one trap. Each integration sticks on an unknown constant, and two integrations mean two constants. Leave them unset and the deflection curve floats up and down or tilts as a whole. What pins the constants is the boundary conditions, the facts the supports give. For a cantilever the deflection is 0 at the fixed end and the slope is 0 too, because it is gripped by the wall. Those two conditions fix the two constants. For a simple beam the deflection is 0 at both ends instead. Change the supports with the button and, under the same load, the deflected shape changes completely. Even with the same curvature, a different boundary condition gives a different result. So in a beam problem what the supports are is half the problem. The supports are half the shape.
Last, a gift that makes practice easy. Everything is linear, so deflection adds. If several loads sit on the beam, find the deflection each load would make alone and then just add them. That is superposition. Switch the buttons among load A alone, B alone, and both, and the deflection with both is exactly the deflection of A alone plus the deflection of B alone. A complicated load is therefore nothing to fear. The deflection formulas for the basic cases are gathered in a table, and a real problem is solved by taking that table and adding. Deflection analysis turns, in the end, on two integrations and an addition. Taking the table and adding is the whole of practice. There is no need to integrate from scratch every time. Knowing the basic cases is enough.