Exsercises

1. Write 10 questions to each part of the text.

2. Write out of the text the sentences with the verbs in the Passive voice.

3. Translate any part of the text (1500 signs) in writing.

4. Retell part II.

5. Speak on Moulding¹⁸ methods

Unit II. Tension and compression

I. Tension and compression

I. Longitudinal19 Strain. Stress. Hooke’s Law

Take a prismatic rod of constant cross-sectional area A m². Mark two thin lines mm apart on its surface using a sharp needle. Now apply two equal and opposite forces, each of P kN, at the ends of the rod so that these forces will act precisely along the axis of the rod. The rod, being in equilibrium under the action of the tensile forces, will elongate in the longitudinal direction and its transverse dimensions will somewhat reduce.

We shall assume that all plane sections normal to the axis of the rod remain plane and normal to its axis after deformation. This hypothesis is known as the hypothesis of plane sections. It is supported by experimental evidence for sections sufficiently far removed from the point of application of the force P, by accepting this hypothesis it is assumed that all longitudinal elements of the rod are stretched in the same manner.

By measuring carefully the distance between the two lines marked on the surface, we find it increased and equal to mm. The elongation of the rod in the portion is.

This increment of the length of the rod is called the total or absolute elongation in tension; in the case of compression it is called the total or absolute contraction. In the latter case the quantity has a negative sign.

The absolute elongation depends obviously on the original length of the rod. Therefore, a more convenient measure of deformation is the elongation per unit of original length of the rod. The ratio

,

is termed the longitudinal strain or the unit elongation. The unit elongation has no dimension; it is a pure number and is often expressed as a percentage of the original length

.

To determine the stress at a transverse section, i. e., at a section perpendicular to the axis of the rod, we apply the general method accepted in strength of materials.

Imagine the rod cut into two parts by a transverse section and the right-hand part removed. To hold the remaining left-hand part in equilibrium, we apply, in the plane of the section, internal elastic forces normal to the plane of the section. These forces replace the action exerted by the removed right-hand part on the left-hand part of the rod. The resultant elastic force will act along the axis of the rod and will be equal to P kN. Accepting the hypothesis of plane sections, we thereby assume that in tension the elastic forces are uniformly distributed over the whole section, therefore, the stress at any point in the cross section is given by the formula

kN/m².

This stress will be normal since it acts, like the force P, perpendicular to the plane of the cross section. If the force P is measured in kilometers-force and the area A in square meters (centimeters), then the stress will have the dimension kN/m² (kN/cm²).

In the case of compression the stress is calculated by the same formula, since only the direction of the forces is reversed here.

The magnitude of the stress in tension or compression is independent of the choice of section along the length of the rod. At any cross section the distribution of elastic forces is assumed to be uniform, and only at sections near the point of application of the external force uniform stress distribution is not to be expected. The determination of stresses at such locations is a difficult problem which is beyond the scope of a course in strength of materials.

The loads and the deformations produced in a rod are closely related. This relationship between load and deformation was first formulated by R. Hooke in 1678. According to Hooke’s law deformation is proportional to load. This is one of the fundamental laws in strength of materials. For a rod in tension or compression, Hooke’s law expresses direct proportionality between stress and strain .

This proportionality is violated when the stress exceeds a certain limit called the proportional limit. The proportional limit for materials is established by experiment.

The factor E appearing in formula is known as the modulus of elasticity of the first kind or Young’s modulus, after the name of the physicist who introduced it into the science. From formula it is seen that the dimension of the modulus of elasticity E is the same as that of stress since is a dimensionless²⁰ quantity, i. e., E is expressed in kN/m² (kN/cm²). For one and the same stress, the strain will be smaller for a material for which E is larger. Consequently the modulus of elasticity characterizes the stiffness²¹ of the material, i. e., its ability to resist deformation, a fact which follows from, formula

.

The magnitude of the modulus of elasticity of materials is established experimentally. In table 1 are given average values of E for some materials at room temperature.

Table 1

Module of Elasticity

Material	Steel	Cast iron	Bronze	Titanium	Aluminum	Copper	Wood
E, MPa

For materials which do not obey Hooke’s law, such as stone, cement, leather, cast iron, etc., a power relation is used: . The exponent m, which is sometimes close to unity, is chosen experimentally. Formula, which expresses Hooke’s law, may be written in an alternate form substituting the appropriate expressions for and

and ;

we then obtain .

From this formula it follows that the elongation (contraction) of the rod is directly proportional to the tensile (compressive) force and the length of the rod, and inversely proportional to the cross-sectional²² area and the modulus of elasticity of the material. Sometimes the module in compression and tension are not equal (cast iron).

The product in the denominator of formula, i. e., is termed the stiffness in tension (compression). The greater the stiffness of the rod, the smaller is the deformation for one and the same length of the rod. The stiffness characterizes simultaneously the physical properties of the material and the geometric dimensions of the section. Formula for stress and Hooke’s law or are fundamental formulas in design for tension and compression.