transverse shear in beams
In C1 the shear V of the beam was found. But that V is only the resultant the whole section carries, and how the shear stress spreads inside the section has to be looked at separately. A result that fights intuition is waiting. Bending stress was largest at the surface, but shear stress is the opposite, 0 at the surface and largest at the neutral axis in the middle. The distribution is not a straight line. It is a parabola. Why? Inside the beam the layers try to slide on each other horizontally. What stops that slide is the horizontal shear, and the formula that gives its size is τ = VQIt. A new guest called Q appears here, and that guest sets the shape of the distribution. The middle being the busiest is the heart of this lesson.
First, why horizontal shear appears. Stack several thin boards, span them like a beam, and press. Each board bends on its own and at the ends they step apart like a stair. The upper layers slide forward and the lower layers slide back. That is evidence that bending makes a horizontal slide between layers. Now suppose the boards are glued tight. They can no longer step apart, and the force that wanted to slide has to be carried by the glue, which is the inside of the section. That is the horizontal shear stress. Switch between the loose stack and the glued beam, and inside the glued beam a horizontal shear is flowing that stops the slide. Glue turns the slide into a force, and that force flows inside the section.
The formula that gives the size of that horizontal shear stress is τ = VQIt. A fine fact hides here. At a point, the urge to slide horizontally and the urge to slide up and down on the vertical section are always equal, and that is called complementary shear. So this one formula also gives the shear stress on the transverse section as it stands. V is the shear force of that section found in C1, I is the second moment of area, and t is the width of the section at that height. Move V and the whole distribution grows in proportion to V. When the shear force grows, the shear stress grows in the same ratio, and only the size changes while the shape stays. The piece that sets the shape is the remaining Q.
Q is the 1st moment of area. The words are heavy and the meaning is simple. At the height where you want the stress, cut the section across and take only the area above that cut line. Multiply that area by the distance from its centroid to the neutral axis, and that product is Q. Q = A′·ȳ′. Move the cut line and cut at the very top, and the area above is nearly 0, so Q is 0 too. Come down and cut at the neutral axis, and the area above is half the section and the lever arm is ample, so Q is largest. So Q is largest at the neutral axis and 0 at the surface. 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-stress distribution is set by Q as it stands.
Now draw the distribution. Since Q is largest at the neutral axis and 0 at the surface, the shear stress of a rectangular section is a parabola. It is largest at the neutral axis in the middle and falls gently to 0 toward the top and bottom surfaces. Move the probe up and down and the parabola is read. Set it beside bending and it sticks. Bending stress was linear in the depth, largest at the surface, and 0 at the neutral axis. Shear stress is the opposite, a parabola, largest at the neutral axis, and 0 at the surface. So even on the same beam, the dangerous place differs. Bending worries about the outer fibers, and shear worries about the neighborhood of the neutral axis. A long beam is governed by bending, and a short thick 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 jumps where the section is narrow. An I-beam has a thin web in the middle and wide flanges above and below. Move V and look at the distribution. On the wide flange t is large and the stress is small, and the moment you enter the thin web t drops and the stress leaps. So nearly all the shear is carried by the web. Bending stress, the other way, is carried by the flanges far from the neutral axis. The roles are split. The flanges take the bending and the web takes the shear. When an I-beam is designed, the web is checked for shear and the flanges for bending, separately. Two lessons sit together in one section. Web and flange split the work.