Mohr Stress circle

These equations show the relationship between the normal and shear t3q>es of stress on the principal types of stress and the angle a that represents a force balance for the volume element. The equations can be represented the Mohr stress circle (Figure 12.35) with a radius of (o-j — o-2)/2 and its center at (ci + 0 2)/ on the abscissa. The shear stress T y is plotted on the ordinate and the normal forms of stress r and Tyy are plotted on the abscissa. Tbe points of the circle intersection with the abscissa give the principal t3q>es of stress ai and a2- The normal stress and the associated shear stress, t, are fixed by the radius arm which is rotated through an angle 2a from the abscissa. The normal stress r, now appears as the projection of the radius arm onto the abscissa, and the shear stress appears as the projection onto the ordinate. It can be seen from the Mohr circle that the shear stress has its maximum values for a = 45° and 135°. 

FIGURE 12J3S The Mohr stress circle (a) is a representatioii of a two-dimensional state of stress in a powder compact (b). The Coulomb 3neld a-iterion is also plotted as a straight line in the Mohr stress circle. 

In view of the experimental errors normally affecting shear cell measurements and the amount of personal judgement required to draw Mohr stress circles tangential to a curved yield locus, there is always some uncertainty in the flow function derived from the Jenike-type shear yield loci method. A direct measurement therefore offers considerable advantage and, besides possibly giving better accuracy, may prove to be more rapid and reproducible. 

Fig. 11 shows a cr, t-diagram. The curve represents the maximum shear stress x the sample can support under a certain normal stress o it is called the yield locus. Parameter of a yield locus is the bulk density Ai,. With higher preconsolidation loads the bulk density Ai, increases and the yield loci move upwards. Each yield locus terminates at point E in direction of increasing normal stresses a. Point E characterizes the steady state flow which is the flow with no change in stresses and bulk density. Two Mohr stress circles are shown. The major principal stresses of the two Mohr stress circles are charcteristic of a yield locus, Oi is (he major principal stress at steady state flow, called major consolidation stress, and cTc is the... 

Mohr s circle A graphical representation of the stresses acting on the various planes at a given point. 

If the normal and shear stresses on all planes through a point within a soil mass are plotted on coordinate axes, a circle is formed, known as Mohr s circle. This circle is used graphically to show the relationship between principal stresses and the normal and shear stresses at a point. It is used to define the failure plane from data gotten through lab tests. 

Fig. 13 shows the value of the principal normal stress Sigma 1 under which the sample has been consolidated its value is obtained by drawing Mohr s circle through the end point of the yield locus (point at no volume change), tangential to the locus. 

The lines OA and OB are tangent to all the stress circles for any value of Py provided the material is noncohesive. They form the Mohr rupture envelope. With cohesive solids or solid masses the tangents forming the envelope do not pass through the origin but intercept the vertical axis at points above and below the horizontal axis. ... 

For an NC clay or a sandy clay in the grormd, a relation between S, tive friction angle (( ) ) can be derived as follows Consider a soil element at A in Figure 8.37. The major and minor effective principal stresses at A can be given by and K, a vo/ respectively. K is the coefficient of earth pressure at rest. Let the soil element be subjected to an unconsolidated rmdrained (UU) triaxial test. The total and effective stress Mohr s circles for this test, at failure, are shown in Figure 8.38. A review of this figure shows that at failure the total major principal stress is Oj = -I- AOj the total minor principal stress... 

For a uniaxial compression test, determine the plane on which the maximum shear stress will occur for a material that obeys the Mohr-Coulomb yield criterion. Suggest an approach for measuring the material parameters (d>, Tq) in the Mohr-Coulomb yield criterion. (Hint construct the Mohr s circle (see Fig. 2.23) for various values of the normal stress.)... 

The hardness test can be considered to be similar to uniaxial compression (o-, = cr) but with the constraint that o-2=(Ty 0. If the applied stress required to yield the material is three times the uniaxial yield stress, determine 0-2 and CT-j using the von Mises and Tresca yield criteria. Illustrate your answer using the Mohr s circle construction (see Fig. 2.23). 

Once elastic constants are known relative to the axes 1,2, and 3, Mohr s circle can be used to predict deformations resulting from stress applied in any direction. Suppose a tensile stress Og is applied to the 1-2 plane, along a line at angle 6 to axis 1 (Figure 6.13). The procedure is as follows. 

When a tensile stress is applied in an arbitrary direction in the 1-2 plane (Figure 6.13), the failure stress can be predicted with the aid of a simple assumption. Failures under stresses o-j, 0-2, and T]2 are assumed to occur independently of each other. Consider the case shown, of tensile loading at angle 6 to axis 1. Hie stress o-g can be resolved into components 0-1, o-j, and Tjj by application of Mohr s circle. 

Equation (1.47) of the circle can be redrawn by replacing coordinates a, b, and r with Cave, and R, as dehned in Eq. (1.47a). In this circle, the points along the abscissa (at zero shear x y ) are the principal stresses, as indicated in Eq. (1.46), in which is the principal stress designated as Pythagorean theorem. Such constructions serve as the basis for Mohr s circle, yielding the particular stress at each point. Bear in mind that the normal and shear stress components in the z direction are zero or negligible. 

Figure 1.19 demonstrates that the relation found in Eq. (1.33) Ci + O = (Tx + o y = I (an invariant) was applied for the sake of simplicity. However, to draw Mohr s circle, the accepted procedure is as follows. Consider Fig. 1.17a, redrawn in Fig. 1.20a. The plot is in Cartesian coordinates the abscissa is for normal stresses and the ordinate for shear stresses. Two points on Fig. 1.20a are the coordinates on the diameter of the circle as indicated A —x y) and B...

Figure 1.21 shows the construction of a Mohr s circle with a counter-clockwise rotation of an element. It intersects the axis at two points, C and D. The stresses at these two end points of the horizontal diameter are Ui and an, the principal stresses. In Fig. 1.21, the equation is basically that of Eq. (1.39), defined like Eq. (1.46) for 2R. 

Mohr s circle may be used in the transformation of stresses from one coordinate system to another. Figure 1.21 may also be used for this purpose. Consider Fig. 1.21a or 1.17a representing the normal and shear stresses, ffy and x y acting on the respective planes in the body characterized by the coordinate system, X and y. The stresses acting in the new coordinate system, x and y, after rotation to an angle 0, from x towards x, are indicated in Fig. 1.21b. The previous Mohr s circle shows the stress state of Fig. 1.21a at points A and B with coordinates Txy and ffy, Tyx, respectively. Now, a line may be drawn between these two points, and then rotated to angle 20, which is twice the angle 0 between x and x and in the opposite direction of 0. A Une drawn after the rotation between the two new points, E and F, provides the new stresses, [Pg.34]

A short exercise can illustrate how to use a Mohr s circle to get the principal stresses. A priori the angles are not needed for this. Rgure 1.22 indicates the method when no rotation of the coordinate system has occurred. The magnitudes of ffx [Pg.34]

In this case, a Mohr s circle is constructed as follows. A horizontal axis is drawn for the normal stresses, like in the figure, while the vertical axis, represents shear stresses. Two points, A and B, are indicated by the coordinates ([Pg.35]

A Mohr s circle represents all stress states, namely normal and shear that can exist on the surface of an elementary cube as it is being rotated. 

As in the two-dimensional case, the direct stresses are on the horizontal axis and the shear stresses are on the vertical axis. For the construction of the Mohr s circle, three circles are required. The stresses on any plane at any rotation, when plotted in the three-dimensional Mohr s circle diagram, are represented by a point located either on one of the three circles or within the area between the largest and the two smaller circles. The maximum shear stress is given by the radius of the largest circle. When constructing the Mohr s circle, the angle of rotation is double that of the real stress system. Shear stresses are positive if they cause clockwise rotation,... 

The stress state at the point upon which the Mohr s circle is based was discussed earlier in Sects. 1.2-1.2.5. 

The residual stresses in the joint are determined by FEA. The stresses ozz and age are the maximum principal stresses taken from the free surface in the vicinity of the ceramic-metal joint (i.e., ai and 03 respectively). These stresses can be converted into a maximum shearing stress for comparison with the measured joint strength by a Mohr s circle analysis. 

The mathematical stress analysis of the flow of unaerated powders in a hopper requires the use of principal stresses. We therefore need to use the Mohr s stress circle in order to determine principal stresses from the results of the shear tests. 

The Mohr s circle represents the possible combinations of normal and shear stresses acting on any plane in a body (or powder) under stress. Figure 10.11 shows how the Mohr s circle relates to the stress system. Further information on the background to the use of Mohr s circles may be found in most texts dealing with the strength of materials and the analysis of stress and strain in solids. 

---