Mohr’s circles

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. 

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... 

Total and effective Mohr s circles. For a specimen exhibiting positive pore water pressure, (a) Element A at failure (b) Total and effective Mohr s circles for element A at failure. 

It is sometimes useful to visualize the changes in a two-dimensional set of strain components by using a construction called Mohr s circle. Consider the rotation of the axes pictured in Fig. 2.20. If a = 0, then a, =022=cos0, a,2=— 21 = sin 0, 1 and the remaining direction cosines are zero. If x, and... 

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. 

Resolve og into components o-j, 0 2, and Ti2 using Mohr s circle. 

Application of Mohr s circle to these strains then yields the thermal expansion at 30° to the fibres. 

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. 

It can be seen that the Equations (4.27) to (4.29) are directly analogous to Mohr s circle analysis in the determination of the basic... 

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. 

Fig. 1.21 Construction of aMohr s circle, a and b from Fig. 1.17a, 1.17b c is the Mohr s circle construction for the two-dimensional case...

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]

Summarizing this section on the Mohr s circle, the following feamres have been described ... 

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. 

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