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3 Flat stress

Lecture



The most common in problems of resistance of materials is a plane stress state: during torsion, bending, bending with torsion, etc. Let us dwell on it in more detail.

9.3.1. Derivation of formulas for stresses on incline platforms

Select the box from the body (Fig. 9.8). Under the action of forces applied to its faces, the parallelepiped is in equilibrium. The lengths of the edges of the parallelepiped are assumed to be infinitesimal and equal 3 Flat stress .

3 Flat stress

Figure 9.8

Consider inclined platforms perpendicular to the unloaded parallelepiped faces. We cut the elementary parallelepiped, shown in Fig. 9.8, with an oblique section perpendicular to the plane 3 Flat stress , selecting an elementary triangular prism from it (Fig. 9.9a).

3 Flat stress

Figure 9.9

The slope of the site with the desired stress will be determined by the angle 3 Flat stress which forms the external normal to this site with the axis 3 Flat stress . From figure 9.9 it follows that

3 Flat stress3 Flat stress (9.6)

The system of forces shown in Fig.9.9 is a plane arbitrary system. The equilibrium of such a system of forces is described by three independent equations. Compose these equations.

3 Flat stress . (9.7)

From:

3 Flat stress . (9.8)

Expression (9.8) is the law of paired tangential stresses: tangential stresses acting on any two mutually perpendicular sites are equal in magnitude and opposite in sign .

In the case of a plane stress state, only two variants of the action of shear stresses are possible (Fig. 9.10).

3 Flat stress

Figure 9.10

To determine the stresses on an inclined platform, we design the forces acting on the prism (Fig. 9.9) on the axis 3 Flat stress and 3 Flat stress . We get:

3 Flat stress ; (9.9)

3 Flat stress . (9.10)

Substituting in (9.9) - (9.10) instead 3 Flat stress and 3 Flat stress from expression (9.6), we reduce all terms to 3 Flat stress . Further, given that according to (9.8) 3 Flat stress , but 3 Flat stress and 3 Flat stress we find:

3 Flat stress ; (9.11)

3 Flat stress . (9.12)

We represent formula (9.9) in a slightly different form, using the equalities known from trigonometry:

3 Flat stress . (9.13)

Substituting (9.13) into (9.11), we obtain:

3 Flat stress . (9.14)

Find out the relationship between normal voltages 3 Flat stress and 3 Flat stress acting on two mutually perpendicular sites (Fig. 9.11).

3 Flat stress

Fig. 9.11

Voltage 3 Flat stress is determined by the formula (9.14). Voltage 3 Flat stress we get if we substitute in this formula 3 Flat stress :

3 Flat stress

or

3 Flat stress . (9.15)

Adding (9.14) and (9.15), we conclude:

3 Flat stress . (9.16)

Expression (9.16) is called the invariance condition for the sum of normal stresses acting on two mutually perpendicular sites: at this point, the algebraic sum of normal stresses acting on any two mutually perpendicular sites is a constant value . This condition is used to verify the correctness of the solution of problems in the study of the stress state at a point.

2. Calculation of the values ​​of the main stresses and determination of the position of the main sites

We study the expression for normal stresses (9.14) at the extremum. To do this, take the partial derivative of the voltage 3 Flat stress by 3 Flat stress and equate to zero:

3 Flat stress , (9.17)

Where 3 Flat stress The angle that is normal to the area under consideration with a positive axis direction 3 Flat stress and at which normal voltage 3 Flat stress reaches the highest value for a given point.

Expression (9.17) represents the value of the shear stress in the main area 3 Flat stress . Thus, the shear stress in the area under consideration ( 3 Flat stress ) is zero. From this we conclude: the platform, the normal to which is the angle 3 Flat stress with positive axis direction 3 Flat stress is the main site.

Equating the expression in parentheses of formula (9.17) to zero, we find the tangent of a double angle that defines the slope of the main sites:

3 Flat stress . (9.18)

Expression (9.18) gives two mutually perpendicular directions with tilt angles 3 Flat stress and 3 Flat stress over which the main stresses act (Fig. 9.12).

To determine the values ​​of the principal stresses, we substitute the formula (9.14) 3 Flat stress . Enduring 3 Flat stress beyond the bracket, we get:

3 Flat stress . (but)

3 Flat stress

Figure 9.12

From trigonometry it is known:

3 Flat stress . (b)

Sign 3 Flat stress set because cosines of angles 3 Flat stress and 3 Flat stress have opposite signs. Substituting (9.18) in (b) and (a), we obtain:

3 Flat stress3 Flat stress .

In this formula, the plus sign corresponds to the maximum principal voltage 3 Flat stress , and the “minus” ним the minimum 3 Flat stress . Thus, we finally have:

3 Flat stress . (9.19)

From the above conclusion it follows that at any initial voltage 3 Flat stress at this point there is a parallelepiped, on the faces of which only normal stresses act.

We return to the formula (9.18). It gives two main directions, but does not indicate which of them acts 3 Flat stress , and in which 3 Flat stress . To solve this problem, it would be necessary to study the sign of the second derivative 3 Flat stress at 3 Flat stress and 3 Flat stress . However, this problem can be solved using expressions similar to those used to determine the direction of the main axes of inertia in the section “Geometric characteristics of plane figures”:

3 Flat stress . (9.20)

Here: 3 Flat stress The angle that should be set aside from the positive axis direction 3 Flat stress to the normal to the site in which the maximum voltage 3 Flat stress ; 3 Flat stress The angle that should be set aside from the positive axis direction 3 Flat stress to the normal to the site in which the minimum voltage 3 Flat stress . The positive angle should be set counterclockwise, the negative should be counterclockwise.

To control the accuracy of determining the position of the main sites, you can use another method given in [2]. Based on the fact that with the rotation of the site in the direction of the tangent stress vector, the normal stress at the site is algebraically increased, the following rule is formulated in [2]: 3 Flat stress always passes through two quarters of the coordinate axes, in which the arrows of tangential stresses 3 Flat stress and 3 Flat stress converge.

3. Extreme tangential stresses

We take as the initial site, in which the main stresses act (Fig. 9.13).

3 Flat stress

Figure 9.13

Counting the angle 3 Flat stress from direction 3 Flat stress write the expressions for 3 Flat stress and 3 Flat stress using formulas (9.12), (9.14), setting in them 3 Flat stress , 3 Flat stress , but 3 Flat stress :

3 Flat stress ; (9.21)

3 Flat stress . (9.22)

From formula (9.22) it follows that for 3 Flat stress double angle sine 3 Flat stress , tangential stresses have extreme values:

3 Flat stress . (9.23)

Extreme tangential stresses at a point are equal to the half-difference of the main stresses and act on sites inclined to the main sites at an angle of 45 0 (Fig. 9.13, a).

Substituting (9.19) into (9.23), we obtain the expression 3 Flat stress through the source voltage 3 Flat stress and 3 Flat stress :

3 Flat stress . (9.24)

In the particular case when two main stresses act on the boundaries of the prism 3 Flat stress (Fig. 9.13b), the extreme tangential stresses (9.23) are numerically equal to the principal stresses:

3 Flat stress ,

and normal stresses on sites with extreme tangential stresses in this case are equal to zero. Such a case of a stress state is called a pure shear , and the areas on which one shear stress acts are called pure shear sites .

9.3.4. Examples of investigation of plane stress at a point

Example 9.2 Normal stresses on the sites 3 Flat stress MPa 3 Flat stress MPa, shear stress 3 Flat stress MPa Identify normal 3 Flat stress , 3 Flat stress and tangents 3 Flat stress , 3 Flat stress stresses in areas normal to which is inclined with respect to the axis 3 Flat stress at angles respectively 3 Flat stress and 3 Flat stress , if a 3 Flat stress = 3 Flat stress , 3 Flat stress = 3 Flat stress (Fig. 9.14).

3 Flat stress

Figure 9.14

Decision:

To determine the normal voltage in the site 3 Flat stress we use the expression (9.14):

3 Flat stress

3 Flat stress MPa

Normal voltage on site 3 Flat stress we find using expression (9.15):

3 Flat stress

3 Flat stress MPa

To verify, we use the invariance condition (9.16):

3 Flat stress ; 3 Flat stress .

Tangential stress 3 Flat stress we define from the expression (9.12):

3 Flat stress MPa

Shear stresses acting on the site 3 Flat stress :

3 Flat stress MPa

In accordance with the law of paired tangential stresses (9.8):

3 Flat stress .

Therefore, the problem is solved correctly. Direction of normal and tangential stresses acting on the sites 3 Flat stress and 3 Flat stress We will show in Fig. 9.15.

3 Flat stress

Figure 9.15

Example 9.3 Determine the magnitude of the main stresses 3 Flat stress and 3 Flat stress and directions of the main stresses (Fig. 9.16, a). Draw the main pads and the main stresses in the figure.

3 Flat stress

Figure 9.16

Decision:

1. Determine the maximum normal stresses from the expression (9.19):

3 Flat stress =

3 Flat stress MPa

3 Flat stress

3 Flat stress MPa

To verify, we use the invariance condition (9.16):

3 Flat stress ; 3 Flat stress .

We find the direction of the main stresses using expressions (9.20):

3 Flat stress ; 3 Flat stress ;

3 Flat stress ; 3 Flat stress .

To verify the correctness of the solution, we add the absolute values ​​of the angles 3 Flat stress and 3 Flat stress . Since the main axes are mutually perpendicular, the sum should be an angle of 90 0 :

3 Flat stress3 Flat stress .

The decision is correct. We postpone the found angles in the figure (Fig. 9.16, b) and put down the values ​​of the main stresses.

Example 9.4 Determine the normal, tangential, and principal stresses at point A, shown in the cross-section of the bent beam, if the bending moment in the section is 3 Flat stress kNm, shear force - 3 Flat stress kN Find the position of the main sites, depict them in the figure, show the directions of the main stresses.

3 Flat stress

Figure 9.17.

Decision:

1. Calculate the moment of inertia of the cross section relative to the neutral section line 3 Flat stress shown in Fig. 9.17a and determine the magnitude of the normal stresses and shear stresses at point A of the section:

3 Flat stress cm 3 ; 3 Flat stress MPa;

3 Flat stress MPa

2. Cut out an elementary area around point A and apply normal and tangential stresses acting at point A to its faces (Fig. 9.17b).

3. Determine the main stresses at point A:

3 Flat stress MPa;

3 Flat stress MPa

To verify, we use the invariance condition (9.16):

3 Flat stress ; 3 Flat stress .

We find the direction of the main stresses using expressions (9.20):

3 Flat stress ; 3 Flat stress ;

3 Flat stress ; 3 Flat stress .

To verify the correctness of the solution, we add the absolute values ​​of the angles 3 Flat stress and 3 Flat stress . Since the main axes are mutually perpendicular, the sum should be an angle of 90 0 :

3 Flat stress3 Flat stress .

The decision is correct. We postpone the found angles on the figure (Fig. 9.18) and put down the values ​​of the main stresses.

3 Flat stress

Fig. 9.18

created: 2019-11-22
updated: 2021-03-13
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Fundamentals of the theory of stress-strain state

Terms: Fundamentals of the theory of stress-strain state