Basic equations of the theory of limiting equilibrium. Graphical determination of stresses using a pore circle Graphical representation of a plane stress state Stress circle

Direct problem in a plane stress state. Circle of stress (Mohr's circle)

The analytical solution of the direct problem is given by formulas (3.2) - (3.5).

Let's analyze the stress state using a simple graphical construction. To do this, we introduce into consideration the geometric plane and refer it to the rectangular coordinate axes and. The calculation procedure will be described using the example of the stress state shown in Fig. 3.5, a.

Having chosen a certain scale for stresses, we put on the abscissa axis (Figure 3.5, b) the segments

Draw a circle with the center at a point as on the diameter. The constructed circle is called circle of stress or mora's circle.

The coordinates of the points of the circle correspond to the normal and shear stresses at different sites. So, to determine the stress on the site, drawn at an angle (Fig. 3.5, a). from the center of the circle (Figure 3.5, b) draw the beam at an angle until it intersects with the circle at a point (set the positive angles counterclockwise). The abscissa of a point (segment) is equal to the normal stress, and its ordinate (segment) is equal to the shear stress.

We find the voltage on the site perpendicular to the considered one by drawing the beam at an angle and obtaining a point at the intersection with the circle. Obviously, the ordinate of the point corresponds to the shear stress, and the abscissa of the point corresponds to the normal stress.

Drawing from a point a line parallel (in our case, a horizontal) to the intersection with a circle, we find a pole - a point. The line connecting the pole to any point on the circle is parallel to the direction of normal stress on the site to which this point corresponds. So, for example, the line is parallel to the main voltage. Obviously, the line is parallel to the direction of the main voltage.

Inverse problem in a plane stress state.

In practical calculations, normal and shear stresses are usually determined at some two mutually perpendicular areas. Let, for example, the stresses are known, (Fig. 3.6, a). Based on these data, it is required to determine the values \u200b\u200bof the main stresses and the position of the main sites.

Let's first solve this problem graphically. Let us assume that\u003e, a\u003e.

In the geometric plane in the coordinate system, draw a point with coordinates, and a point with coordinates, (Fig. 3.6, b). Having connected the points and, we find the center of the circle - a point - and draw a circle with a radius. The abscissas of the points of its intersection with the axis - segments and - will give respectively the values \u200b\u200bof the principal stresses and.

To determine the position of the main sites, we will find the pole and use its property. Let's draw a line from a point parallel to the line of voltage action, i.e., horizontal. The point of intersection of this line with the circle is the pole. Connecting the pole to the points and, we obtain the directions of the principal stresses. The main areas are perpendicular to the found directions of the main stresses.

Figure: 3.6

We use the constructed circle to obtain analytical expressions for the principal stresses and:

Formula (3.10) defines the only value of the angle by which the normal must be rotated in order to obtain the direction of the algebraically greater principal stress. A negative value corresponds to a clockwise rotation.

If one of the main stresses turns out to be negative and the other positive, then they should be denoted by and. If both main stresses turn out to be negative, then they should be designated and.

Inverse problem.

Direct task

Constructing Mohr's circles

A graphical method for studying the stress state at a point.

It can be shown that the equations represent the equation of the circle in parametric form. Therefore, for the graphical method of studying the stress state, stress circles are used, called Mohr circles.

In the theory of the stress state, two main tasks can be distinguished:

Direct task:at the point the position of the main sites and the corresponding principal stresses are known; it is required to determine the normal and shear stresses along the sites inclined to the main ones at an angle a.

Inverse problem: at the point, the normal and shear stresses are known, acting on two mutually perpendicular areas passing through this point, it is required to determine the main stresses and the position of the main areas.

Let's consider the solution of these tasks by a graphical method.

The analytical solution of the direct problem is determined by formulas (4.6) - (4.9).

For a graphical solution, Mohr's circle is built on a plane in s-t coordinates

(fig. 4.9) in the following sequence.

Figure: 4.9

A rectangular coordinate system is selected so that the abscissa axis is parallel to the largest of the principal stresses s 1, along this axis in the selected scale segments OA and OB are plotted, numerically equal to the stresses s 1 and s 2, and on their differences (on the segment AB) as on diameter, draw a circle centered at point C.

From the extreme left point (B) of the circle, draw a ray parallel to the outer normal to the area under consideration, i.e. at an angle a to the s axis. The point of intersection of this ray with the circle (D a) has as its coordinates the segments D a K a and OK a, numerically equal to the tangent t a and normal s a stresses acting on the site under consideration.

SC α \u003d SC β \u003d CD α cos2α \u003d cos2α

The point D b, which lies at the opposite end of the diameter from the point D a, characterizes the stresses s β and t b, acting on an inclined area perpendicular to the first.

The performed transformations are carried out taking into account that 1 + cos2α \u003d 2cos 2 α., 1-cos2α \u003d 2sin 2 α.

The expressions obtained for s a, s b, τ α and τ β completely coincide with analytical formulas (4.6) - (4.9).

In conclusion, it should be noted that each point of the Mohr circle has its own coordinates of stresses acting on the corresponding site, therefore, knowing the principal stresses for a plane stress state, using the Mohr circle, it is possible to determine the stresses acting on various sites passing through this point. The maximum shear stress corresponds to point D c and is equal to the radius of the circle.

Quite often it is necessary to solve the inverse problem, i.e., by the stresses on arbitrary sites s a, t a, s b, t b, determine the magnitude and direction of the principal stresses. Easier this task is solved graphically, that is, using the Mohr circle (Fig. 4.10). Let's consider the order of its construction.

We choose a rectangular coordinate system s, t so that the abscissa axis is parallel to the larger of the normal stresses (let s a\u003e

parallel to the larger of the normal stresses (let s a\u003e s b). On the s axis, in the selected scale, plot the segments OK a, OK b, which are numerically equal to s a and s b. From points К a and К b draw perpendiculars К a D a, К b D b, which are numerically equal to t a and τ β, respectively (К a D a \u003d t a, К b D b \u003d τ β \u003d - t a). On the segment D a D b, as on the diameter, construct a circle centered at point C. The rightmost point of intersection of the circle with the s axis is denoted by the letter A, the leftmost by the letter B. The tangential stresses at these points are zero, therefore, OA \u003d s 1, ОВ \u003d s 2 - principal stresses (according to the direct problem).

From Fig. 6.10 we determine the radius of the circle R and the size of the OS segment (4.12)

Taking into account expressions (4.12), (4.13), we obtain the following formulas for the principal stresses

ОА \u003d σ I \u003d ОС + R \u003d + (4.14)

ОВ \u003d σ II \u003d ОВ - R \u003d - (4.15)

To determine the direction of the main stress s 1, draw the ray through the leftmost point of the circle B and point D a ¢, which is symmetric to point D a about the s axis. The direction of the ray ВD a ¢ coincides with the direction s 1, the direction s 2 is perpendicular to it. The angle a 0 is determined from the triangle VK a D a ¢ (Fig. 6.10):

The angle a 0 is considered positive if it is plotted counterclockwise from the s axis.

In an elementary parallelepiped, on the edges of which all three principal stresses act, consider an arbitrary area a, the normal to which makes angles α 1 α 2 α 3 with the coordinate axes 1,2,3 (Fig. 4.11). On this site, the total stress p α will act, making an angle α with the normal n. Let us define its projections onto the normal to the area - σ α and onto the area itself - τ α.

Figure 4.11

Normal stress, using the principle of superposition, can be represented by the expression \u003d,

where is the stress on the site under consideration, caused by the action, and, respectively, from the stresses and. To calculate these quantities, we use the formula for the linear stress state: \u003d, \u003d, \u003d.

Taking these values \u200b\u200binto account, the normal stresses on an arbitrary site are determined by the equality

To derive the formula for shear stresses τ α, one should consider its vector value. Since, then.

Omitting the conclusions that follow from the equilibrium equations of the considered three-sided pyramid (Fig. 3.11), we write the formula in the final form for the total stress vector on the site n α:

Given this expression

As an example, consider the stresses at a site equally inclined to all main sites. Such a site is called octahedral, and the stresses acting on this site are called octahedral.

Since for such a site, and considering that always

That. Therefore (4.20)

Just as in the case of a plane stress state, in a volumetric stress state, the sum of normal stresses over three mutually perpendicular areas passing through the point under consideration is a constant value.

Consider a graphical method for analyzing the stress state at a point with a volumetric stress state.

First of all, we determine the stresses on the sites parallel to one of the main stresses (Figure 4.12)

		s 2
		Mohr's circle corresponding to this case is shown in Fig. 4.13 circle "a". The stresses in the family of sites parallel to s 2 are determined along the circle "b", and in the family of sites parallel to s 3 - using the circle "c". In the theory of elasticity, it is proved that the areas in general position correspond to points lying in the shaded area (Fig. 4.13). From the presented figure it follows that the smallest and largest normal stresses are equal to the smallest and largest principal stresses,. The largest shear stresses are equal to the radius of the largest circle and act on an area equally inclined to the areas of the maximum and minimum of the principal stresses ().

Consider a graphical method for analyzing the stress state at a point with a volumetric stress state.

First of all, we determine the stresses on the sites parallel to one of the main stresses (Figure 4.12)

On areas parallel to s 1 (Fig. 4.12, a), the stresses depend only on s 2 and s 3 and do not depend on s 1, since.
, then according to (4.18)

Mohr's circle corresponding to this case is shown in Fig. 4.13 circle "a".

The stresses in the family of sites parallel to s 2 are determined along the circle "b", and in the family of sites parallel to s 3 - using the circle "c".

In the theory of elasticity, it is proved that the areas in general position correspond to the points lying in the shaded area (Fig. 4.13).

From the presented figure it follows that the smallest and largest normal stresses are equal to the smallest and largest principal stresses
,
.

The largest shear stresses are equal to the radius of the largest circle

and act on a site equally inclined to the sites of the maximum and minimum of the main stresses (
).

Strains under volumetric stress state.

Generalized Hooke's Law

Considering the issues of strength in bulk and plane stressed states, it is necessary, in accordance with the main hypotheses, to assume that the material is isotropic, follows Hooke's law, and the deformations are small.

Studying the central tension, compression, it was found that the relative longitudinal and transverse deformations are determined by the expressions

,
(4.12)

These equalities express Hooke's law for simple stretching or compression, i.e. at a linear stress state (Fig. 4.14).

Let us consider the relationship between stresses and strains in the case of a bulk stress state.

Applying the principle of superposition, we represent the volumetric stress state as a sum of three linear stress states (Fig. 4.15). In this case, the deformation in the direction of the first principal stress s 1 can be written
,Where , , - relative elongations in

direction s 1, respectively caused by the action of only

stresses s 1, s 2, s 3.

Because the is the longitudinal deformation for stress s, and , - transverse deformations, then from formulas (4.12) it follows:

,
,
. (4.13)

Adding these values, we get.

Expressions for the other two main extensions are obtained similarly. As a result

(4.14)

.

These formulas are called the generalized Hooke's law for an isotropic body, that is, they determine the relationship between linear deformations and principal stresses in the general case of a bulk stress state. From these formulas it is easy to obtain Hooke's law for a plane stress state. For instance,
:

Expressions (4.14) are valid not only for principal deformations, but also for relative deformations in any three mutually perpendicular directions.

In deriving the analytical expression for the generalized Hooke's law in this case, we will

proceed from the condition that angular deformations do not depend on normal stresses, and linear deformations do not depend on shear stresses. In this case, the relative elongation in the direction of the axis xwill be due to stress σ x and is equal to ... Stresses
elongations will correspond in this direction
and
By analogy, we obtain the same expressions for and .

In this way,

(4.15)

.

Angular deformations are determined by the corresponding shear stresses

(4.16)

The set of deformations arising from differentth directions and in different planes passing through a given point is called deformed state at a point.

Along with linear and angular deformation in the resistance of materials, it is sometimes necessary to consider volumetric deformation, i.e., the relative change in volume at a point. Linear dimensions of the edges of an elementary parallelepiped
as a result, the deformations change and become equal. The absolute increase in volume is determined by the difference

-
.

Expanding the brackets and neglecting the products of linear deformations as quantities of the second order of smallness, we obtain

.

The relative change in volume is indicated by the letter e and is determined from the ratio

e

.

Replacing the deformations with their expressions according to Hooke's law, we obtain

e
(4.17)

This relation, along with formulas (4.14) - (4.16), refers to the generalized Hooke's law.

4.8 P potential strain energy

in general stress state

The potential energy accumulated in an elementary volume is determined by the sum of the work of forces distributed over the surface of this volume (Figure 4.16). Normal strength
on the face of the perpendicular axis x
equal to

where - relative linear deformation along the axis xcaused by all the acting forces.

The rest of the normal forces acting along the faces perpendicular to the axes will perform similar work. atand x:
,
.

Tangential force dxdz on the site perpendicular to the axis ywill do the work on the move
equal to
... Similar expressions of works are given by the

energy and will be equal to

Using the expressions of Hooke's law for deformations (4.15), (4.16), we finally obtain (4.18)

For main stresses. (4.19)

Mohr's circle is a pie chart that gives a visual representation of the stresses in various sections passing through a given point. Named after Otto Christian Mohr. It is a two-dimensional graphical interpretation of the stress tensor.

Karl Kuhlman was the first person to create a graphical representation of stresses for longitudinal and transverse stresses in a bending horizontal beam. Mohr's contribution is to use this approach for planar and volumetric stress states and to define a strength criterion based on a circular stress diagram.

Encyclopedic YouTube

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  Internal forces arise between particles of a solid deformable body in response to applied external forces: surface and volume. This reaction is consistent with Newton's second law, applied to particles of material objects. The magnitude of the intensity of these internal forces is called mechanical stress. Because the body is considered to be solid, these internal forces are distributed continuously throughout the entire volume of the object under consideration.

  cos 2 \u2061 θ \u003d 1 + cos \u2061 2 θ 2, sin 2 \u2061 θ \u003d 1 - cos \u2061 2 θ 2, sin \u2061 2 θ \u003d 2 sin \u2061 θ cos \u2061 θ (\\ displaystyle \\ cos ^ (2) \\ theta \u003d (\\ frac (1+ \\ cos 2 \\ theta) (2)), \\ qquad \\ sin ^ (2) \\ theta \u003d (\\ frac (1- \\ cos 2 \\ theta) (2)) \\ qquad (\\ text ( ,)) \\ qquad \\ sin 2 \\ theta \u003d 2 \\ sin \\ theta \\ cos \\ theta)

  Then you can get

  σ n \u003d 1 2 (σ x + σ y) + 1 2 (σ x - σ y) cos \u2061 2 θ + τ xy sin \u2061 2 θ (\\ displaystyle \\ sigma _ (\\ mathrm (n)) \u003d (\\ frac (1) (2)) (\\ sigma _ (x) + \\ sigma _ (y)) + (\\ frac (1) (2)) (\\ sigma _ (x) - \\ sigma _ (y)) \\ cos 2 \\ theta + \\ tau _ (xy) \\ sin 2 \\ theta)

  Shear stress also acts on a site with an area d A (\\ displaystyle dA)... From the equality of the projections of the forces on the axis τ n (\\ displaystyle \\ tau _ (\\ mathrm (n))) (axis y ′ (\\ displaystyle y ")) we get:

  ∑ F y ′ \u003d τ nd A + σ xd A cos \u2061 θ sin \u2061 θ - σ yd A sin \u2061 θ cos \u2061 θ - τ xyd A cos 2 \u2061 θ + τ xyd A sin 2 \u2061 θ \u003d 0 τ n \u003d - (σ x - σ y) sin \u2061 θ cos \u2061 θ + τ xy (cos 2 \u2061 θ - sin 2 \u2061 θ) (\\ displaystyle \\ (\\ begin (aligned) \\ sum F_ (y ") & \u003d \\ tau _ ( \\ mathrm (n)) dA + \\ sigma _ (x) dA \\ cos \\ theta \\ sin \\ theta - \\ sigma _ (y) dA \\ sin \\ theta \\ cos \\ theta - \\ tau _ (xy) dA \\ cos ^ ( 2) \\ theta + \\ tau _ (xy) dA \\ sin ^ (2) \\ theta \u003d 0 \\\\\\ tau _ (\\ mathrm (n)) & \u003d - (\\ sigma _ (x) - \\ sigma _ (y )) \\ sin \\ theta \\ cos \\ theta + \\ tau _ (xy) \\ left (\\ cos ^ (2) \\ theta - \\ sin ^ (2) \\ theta \\ right) \\\\\\ end (aligned)))

  It is known that

  cos 2 \u2061 θ - sin 2 \u2061 θ \u003d cos \u2061 2 θ, sin \u2061 2 θ \u003d 2 sin \u2061 θ cos \u2061 θ (\\ displaystyle \\ cos ^ (2) \\ theta - \\ sin ^ (2) \\ theta \u003d \\ cos 2 \\ theta \\ qquad (\\ text (,)) \\ qquad \\ sin 2 \\ theta \u003d 2 \\ sin \\ theta \\ cos \\ theta)

  Then you can get

  τ n \u003d - 1 2 (σ x - σ y) sin \u2061 2 θ + τ xy cos \u2061 2 θ (\\ displaystyle \\ tau _ (\\ mathrm (n)) \u003d - (\\ frac (1) (2)) ( \\ sigma _ (x) - \\ sigma _ (y)) \\ sin 2 \\ theta + \\ tau _ (xy) \\ cos 2 \\ theta)

  CIRCLES OF MORA

  Mohr's circles are pie charts that give a visual representation of the stresses in different sections passing through a given point. The coordinates of the points of the circle correspond to the normal and shear stresses at different sites. We postpone from the  axis from the center C a ray at an angle of 2 (\u003e 0, then against the hour p.), We find point D,

  whose coordinates are:  ,  . You can graphically solve both direct and inverse problems.

  ABOUT bulk stress

  Stresses in any site with known principal stresses  1,  2,  3:

  where  1,  2,  3 are the angles between the normal to the site under consideration and the directions of the main stresses.

  Highest shear stress:
  .

  It acts on an area parallel to the main voltage  2 and inclined at an angle of 45 ° to the main stresses  1 and  3.

  TO
  mora's circle for volumetric stress.

  The points that are the tops of the circles correspond to diagonal areas inclined at 45 ° to the main stresses:
  

  ,
  (sometimes called principal shear stresses).

  Plane stress state is a special case of volumetric one and can also be represented by three Mohr circles, while one of the principal stresses should be equal to 0. For tangential stresses, as in the plane stress state, pairing law: the components of shear stresses along mutually perpendicular areas, perpendicular to the line of intersection of these areas, are equal in magnitude and opposite in direction.

  H stresses along the octahedral site.

  Octahedral platform (ABC) is a platform equally inclined to all main directions.

  ;

  The octahedral normal stress is the average of the three principal stresses.

  or
  , The octahedral shear stress is proportional to the geometric sum of the principal shear stresses. Stress intensity:

   x +  y +  z \u003d  1 +  2 +  3 - the sum of normal stresses acting on any three mutually perpendicular areas is a constant value equal to the sum of the principal stresses (first invariant).

  Strains under volumetric stress state.

  Generalized Hooke's Law (Hooke's law for volumetric stress):

  
  1,  2,  3 - relative elongations in the main directions ( main lengthening). If any of the stresses  i are compressive, then they must be substituted into the formulas with a minus sign.

  Relative volumetric deformation:

  
  

  The volume change does not depend on the ratio between the principal stresses, but depends on the sum of the principal stresses. Those. an elementary cube will receive the same volume change if the same average stresses are applied to its faces:
  then
  , where K \u003d
  - bulk modulus... When a body is deformed, the material of which has Poisson's ratio  \u003d 0.5 (for example, rubber), the volume of the body does not change.

  Potential strain energy

  With simple tension (compression), the potential energy U \u003d
  .

  Specific potential energy - the amount of potential energy accumulated per unit volume: u \u003d
  ;
  ... In the general case of a volumetric stress state, when three main stresses act:

  or

  The total strain energy accumulated per unit volume can be considered as consisting of two parts: 1) the energy uo accumulated due to the change in volume (i.e. the same change in all dimensions of the cube without changing the cubic shape) and 2) the energy u f with a change in the shape of the cube (i.e., the energy spent on turning the cube into a parallelepiped). u \u003d u о + u f.

  ;

  - stress tensor (matrix of the third order).

  On passing to principal stresses, the stress tensor takes the form:

  

  J 1 \u003d  x +  y +  z;

  J 2 \u003d  x  y +  y  z +  y  z -  2 xy -  2 zx -  2 yz;

  J 3 \u003d  x  y  z -  x  2 yz -  y  2 zx -  z  2 xy + 2 xy  zx  yz.

  ... When the coordinate system is rotated, the tensor coefficients change, the tensor itself remains constant. Three invariants of the stress state:

  Similar dependences arise when considering the deformed state at a point. Comparison of the dependences of the stressed and deformed plane states (analogy):

  
  

  
  

    - relative deformation,   - shear angle.

  The same analogy holds for the bulk state. Therefore, we have the invariants of the deformed state:

  J 1 \u003d  x +  y +  z;

  J 2 \u003d  x  y +  y  z +  z  x -  2 xy -  2 yz -  2 zx;

  - strain tensor.

   x,  y,  z,  xy,  yz,  zx are the components of the deformed state.

  For the axes coinciding with the directions of the principal deformations  1,  2,  3, the strain tensor takes the form:
  .