Answer to Question #163775 in Physics for hamza

Question

 steel tube with wall thickness w which is small in comparison with the diameter of tube . the tube is under tension, caused by force T, parallel to axis of tube. to reduce stress in material of tube, it is proposed to thicken the wall. the tube diameter and tension being constant which wall thickness gives half of stress? 

Answer 1

$$\sigma=\frac{F}{A}$$

The steel tube is hollow in the centre with its wall having thickness w. When we calculate stress, we need to consider the area of the steel material, not the hollow space.

Suppose the length of the tube is L and the diameter is d. Imagine that we have cut the steel tube (along the length) so that it is now in the form of a cuboid (a rectangular sheet with thickness w).

$$A=2\pi (0.5d)w=\pi dw$$

Since π and d are constant, the area of the annulus is proportion to w.Because the wall is thin in comparison with the diameter of the tube, the area of the annulus (representing the wall thickness) is proportional to w.

The tension (force) is kept constant. So, the stress is inversely proportional to the area. To obtain half the stress, the area should be doubled.

But we have seen above that the area is directly proportional to w.

Thus, if the area is doubled, w also needs to be twice greater.