In our examination so far, loads have been applied in a direction directly in line with the structural member’s longitudinal axis. The compressive force has pushed downwards directly in line with the column’s axis; the tensile force has pulled down directly in line with the cable from which it was sus- pended. The stress we investigate now—shear force—is applied transversely:
at a 90-degree angle to the structural member it is stressing.
The structural member we have in mind now is a slender beam, rect- angular in its cross section, laid horizontally over a gap, each end firmly attached. Assume that you are standing in the gap, staring at the side of the beam. Let us name the beam’s three dimensions: the length is the longest dimension, extending horizontally in front of you. Depth is the up-down distance. And width is the distance in the directions to and from you (figure 3.5). To make it resemble a simple beam bridge, we make the beam’s depth greater than its width. This reflects the discovery that wood bridge builders made long ago, that if the wooden planks were laid flat (the width of each Figure 3.4. From original size (0), the cable stretches proportionately to applied stress, until yield stress (A). Beyond that threshold, the cable deforms permanently (B), and eventually snaps (C).
Stress
Yield Stress
Strain
plank greater than its depth), they would flap up and down with the passing load. But if enough planks were available and were laid next to each other while resting on their narrow edges (width now much narrower than depth), the structure would become much stiffer—it could resist far greater loads.
A stiff beam exhibits a kind of stress behavior that most of us would not think of. Called shear, it occurs when forces push in opposite directions:
not at directly opposed points on the same cross-sectional plane (that would just be compression) but at opposite points on adjacent planes (figure 3.6).
For example, consider the effects of a pair of pliers on a sheet of metal as compared to a pair of shears. As you press together with the pliers on the sheet, forces converge from opposite directions on the same area in the metal, causing ordinary compressive forces. By contrast, if you press with the shears on the sheet, the area the downward force presses is separated by a very small space from the area the upward force presses. The sheet of metal will be cut—or sheared—apart.
But even a beam on which no scissors act, one that bears only its embodied dead load, undergoes shear strain. This is important, so we should try to picture it. Imagine that the beam is an assemblage of cards glued together and placed horizontally across two bricks, directly in front of us, left to right. As we follow the cards left to right, we see some cards that are resting directly on the left brick, until we reach the first card suspended
Width
Length Depth
Figure 3.5. Terms for a beam’s three dimensions, from an observer’s point of view.
over the gap. As compared to the previous card over the brick, the first card over the gap will tend to slip downward. An analogous process occurs in a beam laid across columns above a river: the beam’s fibers adjacent to the column undergo shear strain (figure 3.6).
The shear strain occurs even if we have in mind just the dead load.
If we put a live load (a paper weight) on the card bridge just past the left brick, the shear strain is even greater.
As the applied load increases, we can observe an equivalent to Young’s modulus, but now for shear. As the load increases, the shear strain at first reacts proportionately, slipping in direct proportion to the weight placed on top of it. Beyond a threshold however, the slippage becomes excessive, endangering the bridge.
Our pack of cards gives a sense of the stress undergone by a beam extending from one riverbank across a column in the middle of the river to the other bank. The beam experiences high shear strains at fibers extending just past the left bank, just past each edge of the column, and just before the right bank.
A B
Figure 3.6. (A) Shear forces applied to a component. (B) Shear strain experienced by a beam at its juncture with a column.