Stretching and contraction, Determination of normal...

Stretching and Compression

Determination of normal force

Central stretching (compression) is one of the simplest types of loading. The method of cross-sections in the cross-section of the beam reveals only one internal force factor-the normal force. Its vector is perpendicular to the cross section and is directed along the longitudinal axis of the beam. The stretch-squeeze beam is commonly referred to as a rod .

According to the cross-section method, the magnitude and direction of the longitudinal force are determined from the equilibrium equation compiled for the cut-off part of the bar:

(2.9)

Thus, the longitudinal (normal) force σ of an arbitrary cross-section of the bar is numerically equal to the algebraic projectile suspension on the longitudinal axis of all external (active and reactive) forces applied to the cut off part.

In the general case

(2.10)

where is the intensity of the load distributed along the axis of the beam from 0 to .

The longitudinal force is considered positive if it causes tension, i. e. is directed away from the cross section. In the cross section of the beam, it is the resultant of the internal normal forces arising in this section.

The graph of the function is called normal force diagram . It follows from (2.10) that

(2.11)

i.e. the intensity of the distributed load in each section is equal in magnitude and sign to the tangent of the slope angle of the tangent to the diagram in the corresponding section of the diagram.

Normal stresses and strains

When stretching (compressing) the beam in the cross sections, only normal stresses arise. In order for the problem to be determined by known N A to have a unique solution, it is necessary to establish the distribution law σ ( x ) over the cross section. For this, the hypothesis of flat sections (Bernoulli's conjecture) is used: the bar sections, flat and normal to its axis before deformation, remain flat and normal to its axis and under deformation. The cross sections only move along the axis, remaining parallel to each other.

Suppose the beam consists of an infinitely large number of longitudinal fibers. From the Bernoulli hypothesis it follows that all fibers are deformed identically. Since, according to Hooke's law, equal stresses correspond to equal deformations, then under tension (compression) of the beam, the normal stresses are uniformly distributed over the cross section, that is, ;

As you know, . Since , then . Hence

(2.12)

Positions are the directions , corresponding to the stretching.

In the sections of the beam adjacent to the place of application of external forces and to the anchorages, the stress distribution depends on

Fig. 2.7

from the way the load is applied and can be uneven. Therefore, the hypothesis of flat sections in these places is incorrect.

Consider a homogeneous stress state of the bar when the stresses do not change along the length (Figure 2.7).

Changing the linear dimensions is called absolute elongation ; the ratio - relative elongation or linear deformation .

In the case of an inhomogeneous stress state, the linear deformation is determined by the expression , where is the increment of the segment .

Between linear strains and their stresses there is a relationship due to the elastic properties of the material. This relationship is determined by Hooke's law:

(2.13)

where E - is the modulus of elasticity of the material.

Consider the expression . According to the formula (2.13), we obtain ; because

Hence the change in the length of the entire beam

(2.14)

The product AT is called the rigidity of the bar when stretched (compressed).

If the laws of variation N and A are different for individual sections of the bar, then

(2.15)

where - the number of sites.

In the particular case when N and A are constant along the length of the bar, we obtain Hooke's formula in the form

(2.16)

So, moving the i - th section with the coordinate x relative to the fixed section

(2.17)

Similarly, you can write

(2.18)

where - moving the initial section relative to the seal.

Let the cross-section of the beam (see figure 2.7) be in the form of a rectangle with sides a and b, then the perimeter of the beam decreases. The value characterizes the relative change in the perimeter of the cross section and is called transverse deformation . If the section is round, then . The ratio of the transverse strain to the linear value is constant for a given material and is called the Poisson's ratio :

(2.19)

For steel and most metal materials . In the general case, .

thematic pictures

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