axial deformation PL/AE
Earlier we looked at the stress and the strain at a single point. What practice actually wants is how much the whole bar lengthens. How many mm a cable sags, how far a column settles. Call that lengthening δ. Surprisingly, δ is fixed by exactly four things. How hard you pull (P), how long the bar is (L), how thick it is (A), and what material it is (E). Tied into one line, δ = PLAE. It is one of the formulas mechanics of materials uses most. Turn the formula over and the bar turns out to have been a spring of stiffness k = AEL. Those four things fix the lengthening of the whole bar.
Start with the relation of pull and lengthening. Move P and pull the bar, and the end advances by that much. Pull twice as hard and the lengthening doubles too. The small graph beside it shows the pull and the lengthening joined by a straight line, a direct proportion. This is Hooke's law σ = E·ε, grown up to the whole bar. Stress was proportional to strain, so force is proportional to lengthening. Inside the elastic range the line is always this clean. Measure the lengthening at half the pull and you already know the lengthening at twice the pull. P and the lengthening walking side by side is the whole picture.
Now see exactly what fixes the lengthening. δ = PLAE. Take the four letters one by one and each one makes sense. A larger P lengthens more, a longer L lengthens more, a thicker A lengthens less, and a stiffer E lengthens less. Here only the length L is moved. Leave P, A, E as they are and only make the bar longer, and the lengthening grows in proportion. Why? If a 1m bar grows 1mm, that means it grows 1mm in every 1m. So 2m means 2mm, and 3m means 3mm. Even at the same strain, a longer bar piles up a larger absolute lengthening. The relation of strain and length seen in A2 sits here as it stands.
Now the denominator. In δ = PLAE, A and E sit underneath, so the larger they are the less the lengthening. Move the area A thicker and the same force spreads over a wider section, the stress falls, and the lengthening falls too. Change E with the material button. Steel hardly lengthens, aluminum lengthens a bit more, and wood lengthens much more. Think of the product A·E as the stiffness of that bar. The thicker and stiffer it is, the larger the stiffness, and the less it moves under the same force. The reason a bridge cable uses thick steel wire instead of a thin thread sits in this denominator.
A real member is not the same thickness all the way. Think of a bar whose section changes like a stair. The same force P runs through every segment alike, but each segment has a different thickness, so each lengthens differently. The total lengthening is simply the sum of the lengthenings of the segments. δ = Σ PLAE, computed segment by segment and added. Move P and the thinnest segment is the one that lengthens the most. The same force, and still the deformation gathers where the section is narrow. When you design a member, the single weakest segment governs the whole behavior. A chain breaks at its weakest link.
Last, turn δ = PLAE over. Written as P = (AEL)·δ, it has exactly the shape of the spring law F = kδ. An axial member is therefore a spring of stiffness k = AEL. Move P and the spring lengthens, following δ = Pk. When k is large it is a stiff spring and hardly lengthens, and when k is small it is soft and lengthens easily. This view is powerful because a complicated structure can be turned into several springs and solved that way. Joined in series the stiffnesses combine one way, and set in parallel they combine another. The moment you see a bar as a spring, structural analysis becomes a familiar spring problem. A bar seen as a spring is the useful picture.