CHAPTER V.

STRESSES IN FRAMES.

186. Rigid Bodies. By a rigid body is meant one whose parts maintain the same relative positions no matter what forces may act on the body.

No such bodies have however been actually experimented with; some become deformed when small forces act on them, others require large forces before any change of shape is perceptible by ordinary means. The more delicate our means of measuring change of shape the fewer exceptions there appear to be to the law that every body alters its shape on the application of force.

187. Solid bodies are distinguished from liquids and gases by the fact that they return either partially or wholly to their original shape on the withdrawal of the force. (This assumes that the force has not been sufficiently great to fracture the body.)

If the return to the original shape is only partial, the Elastic limit of the material is said to have been passed.

Only solid bodies will be considered here and the forces applied will always be supposed such that the Elastic limit has not been reached.

188. Strain. The alteration in the form of a body due to the application of force is called the Strain.

189. Stress. The change in the relative position of the different parts of a body sets up various pushing and pulling forces between the molecules and the whole set of these intermolecular forces constitute the stress in the body. This stress may consist of the forces called into play by lengthening, shortening, bending or twisting the body in some way or other.

190. Consider the simple case of a vertical rod or wire supporting a weight W.

The weight elongates the wire and the elongation per unit length is the measure of the strain. If land l' denote the initial and final lengths of the rod or wire, then

B

Now the wire supports the weight because the various particles of which it is composed resist being pulled apart. If we consider any imaginary right section of the wire as BB, all the particles on the under side are pulling downwards on those of the upper side, and all those on the upper side are pulling upwards on those of the lower side.

B

Since there is no motion after the elongation has taken place, these two sets of forces are in equilibrium. The sum of the forces in one set measures the total stress. The total stress divided by the area of the section gives the stress per unit area, which for any given material must not exceed a certain value called the working stress of the material.

Fig. 123.

Neglecting the weight of the wire in comparison with W, we may say (if A denote the cross-sectional area) that measures the stress and is the same for all cross-sections.

W

A

This case is an example of uniform tensile stress, the forces at any section BB acting away from one another

and being uniformly distributed. If the forces had been towards one another, then the stress would have been compressive.

191. Hooke's Law and Young's Modulus. It can be shewn by experiment that for elongated or shortened bars the stress is proportional to the strain. This is known as Hooke's Law.

where E is a constant known as Young's Modulus.

E depends on the material of the rod.

192. If a body be in equilibrium under the action of any system of forces, then, whether the body has been deformed by the forces or not, it may be treated as a rigid body for that particular set of forces: for, by supposition, the various parts of the body retain their relative position unaltered and it can make no difference if the various parts be supposed rigidly connected.

In the cases of equilibrium about to be considered, it will be supposed that the various elongations and compressions due to the forces applied to the body have taken place.

193. We proceed now to shew how the stresses in framed structures such as those used in girders for bridges and for roof supports may be graphically determined. At first the forces due to the weights of the bars will be neglected and in all cases the hinges or joints between the various bars will be supposed perfectly smooth.

194. The following theorem proved in § 177 is of fundamental importance in this subject.

If a body is in equilibrium under the action of two forces only whose points of application are A and B, then the forces must be equal in magnitude, opposite in sense, and must act along the line AB.

Consider any one bar of a framework which is pinjointed at the ends, the joints being smooth and of circular

section.

Whatever the number of bars jointed on to one end of this bar may be, it is clear the force due to each must pass through the centre of the pin and therefore the resultant force of the pin on the bar must also pass through the centre. A similar argument applies to the pin-centre at the other end. Hence the force transmitted by the bar must lie in the line joining the centre points of the pins at the ends.

We may therefore in construction treat the bars of a framework as lines, these lines being the directions of the forces.

195. Simple Cantilever. Let ABC be a framework supported by a wall and carrying a load W at A.

Then we know that the forces at A due to the stresses in the bars must act along the bars AB, AC and must be in equilibrium with W. The vector-polygon for the forces at A must therefore be a closed figure, viz. a triangle.

Using the notation of § 180, i.e. numbering the spaces between the lines in which the forces act, we draw the vector-polygon as follows:

Set off a line vertically downward (12) representing W to scale and through the end points draw lines parallel to the bars AB (13) and AC (23).

This may be done in two ways, as indicated, but only one, viz. the one on the left, gives a consistent notation.

We thus see that the lines must be drawn by method. For the stress in the bar 13, draw the vector 13, starting from 1, parallel to the bar, and for the stress in the bar 23 draw the vector 23 starting from 2. The point (3) of intersection of these vectors is thus uniquely determined and the lengths of the vectors give to scale the stresses in the bars.

Since there is equilibrium, the vector-polygon for these forces at A must be closed, the sense of 12 is known, viz. from 1 to 2, hence the sense of 23 is from 2 to 3, and that of 31 from 3 to 1. These senses are shewn by the arrowheads in the figure.

The bar 13 therefore pushes at A, whilst the bar 23 pulls, hence the former pushes at B, and the latter pulls at C. The bar 13 has been therefore compressed and tends by its elasticity to lengthen, while 23 has been elongated and tends to shorten. 13 is thus in compression and 23 in tension, the former is called a strut, the latter a tie-bar.

The vector-polygon with its arrowheads as drawn is quite correct as regards the forces acting at A, but if we require the force acting at B, the arrowhead of 23 points the wrong way. The lines 23 and 31 of the link polygon ought therefore to have two arrowheads pointing in opposite ways, if we wish to represent the forces at all the joints. This would only confuse, so avoid arrowheads altogether, indicate the bars which are in compression by fine lines drawn parallel to the bars.

196. Since the stresses in the bars of the framework can be found from the Vector-Polygon, the latter is called the Stress Diagram.