Shear in a Beam Section Peaks at the Neutral Axis
In C1 you found the beam's shear V. But that V is only the resultant the whole section carries; how the shear stress spreads inside the section is a separate question, and a counterintuitive answer is waiting. The bending stress was largest at the surface. The shear stress is the opposite: zero at the surface and maximum at the neutral axis in the middle. And its distribution is not a straight line but a parabola. Why does it come out this way? The heart of it is that the layers inside a beam try to slide horizontally past one another. What resists that slip is a horizontal shear, and the formula that gives its size is τ = VQIt. A new guest, Q, appears here, and it is what shapes the distribution.
First, why a horizontal shear arises at all. Stack several thin planks loosely, lay them across like a beam, and press down: what happens? Each plank bends on its own and they stagger at the ends, sliding into a staircase. The top layer slides forward, the bottom backward. That is proof that bending inherently creates a horizontal slip between the layers. Now imagine gluing those planks firmly together. They can no longer stagger. Instead, the glue, that is, the interior of the section, has to take up that sliding tendency. That is exactly the horizontal shear stress. Toggle between the stacked planks and the bonded beam. Inside the bonded beam runs a horizontal shear that prevents the slip.
The formula that gives the size of that horizontal shear stress is τ = VQIt. A neat fact hides here: at any point, the tendency to slide horizontally and the tendency to slide up and down on a vertical face are always equal. This is called complementary shear. So this one formula gives the shear stress on the cross-section too. V is the section's shear force you found in C1, I is the second moment of area, and t is the width of the section at that height. Drag V. The whole distribution scales in proportion to V: double the shear force, double the shear stress. The shape stays the same while only the size changes. What sets that shape is the remaining piece, Q.
Q is the first moment of area. The name sounds hard, but the meaning is simple. At the height where you want the stress, cut the section horizontally and take only the area above that cut line. Multiply that area by the distance from its centroid to the neutral axis, and you get Q. Q = A′·ȳ′. Drag the cut line. Cut near the top and the area above is almost nothing, so Q is nearly zero. Come down to the neutral axis and the area above is half the section, with a healthy arm too, so Q reaches its maximum. So Q is largest at the neutral axis and zero at the surfaces. In τ = VQIt, V and I are fixed for the whole section, and t is constant for a rectangle, so the shape of the shear distribution is set entirely by Q.
Now draw the distribution. Since Q is largest at the neutral axis and zero at the surfaces, the shear stress in a rectangular section is a parabola: greatest at the central neutral axis, falling smoothly to zero toward the top and bottom surfaces. Drag the probe up and down to read off that parabola. Comparing with bending makes it stick. The bending stress σ was linear in depth, maximum at the surface, zero at the neutral axis. The shear stress τ is the opposite: parabolic, maximum at the neutral axis, zero at the surfaces. So even for the same beam, where it is at risk differs. Bending threatens the outer fibers, shear the region near the neutral axis. Usually a long beam is governed by bending, a short stubby beam by shear.
All of this explains why an I-beam looks the way it does. In τ = VQIt, t is in the denominator, so the shear stress shoots up where the section is narrow. An I-beam has a thin web in the middle and wide flanges at top and bottom. Drag V to see the distribution. Across the wide flange t is large so τ is small, then the instant you enter the thin web t drops sharply and τ jumps up. So the shear is carried almost entirely by the web. Bending stress, by contrast, is handled by the flanges far from the neutral axis. The division of labor is perfect: the flanges take the bending, the web takes the shear. That is why in I-beam design the web is checked for shear and the flanges for bending. A single section holds two lessons at once.