# Shift and Torsion, Shift - Applied Mechanics

## Shift

Shift - This is the type of simple deformation of the bar, in which only forces in the plane of the section act in its cross-sections from the internal force factors. These forces are called transverse shifting) . They cause shear stresses or shear stresses.

In the process of stretching a bar of mild steel in the region of plastic deformation, shear deformations are observed, caused by the sliding of some parts of the material along the others. In a pure form, it is difficult to make a shift by external influences, since it is often accompanied by bending and other deformations.

Fig. 2.18

The shear phenomenon can be observed when cutting the strip with scissors (Figure 2.18, a). It follows from the figure that the shift of one part relative to the other occurs when the shoulder h little. With a large shoulder h , the shift is accompanied by a bend. As the forces F increase, the shear deformation terminates by cutting the strip. We fix the strip by a plane along the line 1-4 and consider the shifted element in the form shown in Fig. 2.18, b.

The action of the discarded right-hand side on the left is represented by shearing forces, the resultant of which is reduced to the transverse force Q, equal in magnitude to the external force F.

Tangential stresses arise in the section. Summarizing them over the entire area A, we obtain the transverse force

(2.44)

If the law of distribution of tangential stresses is known, then from the expression (2.44) we can find the magnitude of tangential stresses at any point of the section.

The distribution of tangential stresses across the section is nonuniform, but for small thicknesses δ it can be considered as uniform, and then

(2.45)

The shear stresses are calculated from (2.45). Generally speaking, normal bending stresses arise, which we neglect because of their smallness.

When the force F is applied, the plane (see Figure 2.18, b) moves vertically relative to the seal by < img src="images/image788.jpg"> (Figure 2.18, c), called the absolute shift. It is assumed that the plane remains flat, and the longitudinal fibers are straight, rotating relative to the initial position by an angle , called the relative shift:

(2.46)

Neglecting small quantities, we can assume that the shift does not change the volume, but only the shape changes: the rectangle 1234 turns into a parallelogram. The angle of the shift determines the change in shape - the distortion of the angles of the original parallelepiped.

Similarly to Hooke's law, when stretching within elasticity, the shear stress at shear is directly proportional to the relative shift :

(2.47)

There is a relationship between the shear elastic modulus G and the tensile modulus of elasticity E:

(2.48)

where μ is Poisson's ratio.

For steel , i.e. the shear resistance is almost two times weaker than the tensile strength.

Substituting relations (2.45) and (2.46) into the formula (2.47), we obtain

(2.49)

Equation (2.49) is externally similar to the Hooke's formula for stretching, but it is approximate, since in reality τ is variable in cross section height, which is felt for significant thicknesses

The work of the transverse force A or the work of internal elastic forces under shear is calculated similarly to the stretching:

(2.50)

Formula (2.50) expresses the potential energy of deformation under the shift . Introducing the relation (2.49) into (2.50), we obtain

(2.51)

Specific potential energy of deformation under shear

(2.52)

A pure shift is a case of a plane stress state in which an elementary parallelepiped with lateral faces under the action of only tangential stresses can be distinguished in the vicinity of a given point (Fig. 2.19).

The shear strength condition has the form

(2.53)

where is the allowable tangential shear stress,

The expression (2.53) can be represented in the form

Calculation of the shear strength is subject to bolts, rivets, eyes, welds and other types of joints working on shear (shear).

Fig. 2.19

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