Pull an Axial Member and It Stretches by δ=PL/AE
Earlier you saw the stress σ and strain ε at a single point. But in practice what you really want to know is how much the whole bar stretches: how far a cable sags, how much a column settles. Call that stretch δ. Remarkably, δ is set by just four things: how hard you pull (P), how long the bar is (L), how thick it is (A), and what material it is (E). Tie them into one line and you get δ = PLAE, one of the most used formulas in mechanics of materials. And turn this equation around, and it reveals that the bar was really a spring of stiffness k = AEL all along.
Start with the link between pull and stretch. Drag P to pull the bar and its end advances by δ. Pull twice as hard and δ doubles. The little graph beside it shows P and δ joined by a straight line, direct proportion. This is really Hooke's law σ = E·ε scaled up to the whole bar: since stress was proportional to strain, force is proportional to elongation. Inside the elastic range it is always this clean straight line. So measure the stretch at half the pull, and you immediately know the stretch at double the pull.
Now see exactly what sets the stretch: δ = PLAE. Take the four letters one by one and each makes sense. Bigger P stretches more, longer L stretches more, thicker A stretches less, stiffer E stretches less. Here drag only the length L. Hold P, A, E fixed and just lengthen the bar, and δ grows in proportion. Why? If a 1 m bar stretches 1 mm, that means it stretches 1 mm for every 1 m. So 2 m gives 2 mm, 3 m gives 3 mm. For the same strain, more length piles up more absolute stretch. The strain-and-length link from A2 is sitting right here.
Now the denominator. In δ = PLAE, A and E sit below, so the larger they are the more they cut the stretch. Drag the area A to make it thicker: the same force spreads over a wider section, the stress drops, and the stretch shrinks. Switch the material with the buttons too. Steel barely stretches, aluminum a bit more, wood far more. It helps to think of the product A·E as the stiffness of that bar. Thicker and stiffer means more stiffness, so it budges less under the same force. The reason a bridge cable uses thick steel wire instead of thin thread lives right here in this denominator.
Real members are not uniform in thickness. Picture a bar whose section changes in steps. The same force P runs equally through every segment, but since each has a different thickness, each stretches differently. The total stretch is simply the sum of the segment stretches: δ = Σ PLAE, computed segment by segment and added up. Drag P. You will see the thinnest segment glow reddest and stretch the most, because for the same force the deformation crowds into the narrow section. So in design, one weakest segment governs the whole behavior, the way a chain breaks at its weakest link.
Finally, turn δ = PLAE around. Written as P = (AEL)·δ, it is exactly the shape of the spring law F = kδ. So an axial member is a spring of stiffness k = AEL. Drag P and the spring extends, with δ = Pk following along. A large k is a stiff spring that barely stretches; a small k is soft and stretches easily. What makes this view powerful is that you can replace a complex structure with a set of springs and solve it. Connect them in series and the stiffnesses combine one way; place them in parallel and they combine another. The moment you see a bar as a spring, structural analysis turns into a familiar spring problem.