Max Obliquity Of Soil
3.3 The Möhr circle diagram
Figure 3.3a shows a major principal plane, acted upon by a major principal stress, <71, and a minor principal plane, acted upon by a minor principal stress, 03.
By considering the equilibrium of an element within the stressed mass (Fig. 3.3b) it can be shown that on any plane, inclined at angle 9 to the direction of the major principal plane, there is a shear stress, r, and a normal stress, crn. The magnitudes of these stresses are:
These formulae lend themselves to graphical representation, and it can be shown that the locus of stress conditions for all planes through a point is a circle (generally called a Mohr circle). In order to draw a Mohr circle diagram a specific convention must be followed, all normal stresses (including principal stresses) being plotted along the axis OX while shear stresses are plotted along the axis OY. For most cases the axis OX is horizontal and OY is vertical, but the diagram is sometimes rotated to give correct orientation. The convention also assumes that the direction of the major principal stress is parallel to axis OY, i.e. the direction of the major principal plane is parallel to axis OX.
To draw the diagram, first lay down the axes OX and OY, then set off OA and OB along the OX axis to represent the magnitudes of the minor and major principal stresses respectively, and finally construct the circle with diameter AB. This circle is the locus of stress conditions for all planes passing through the point A, i.e. a plane passing through A and inclined to the major principal plane at angle 8 cuts the circle at D. The coordinates of the point D are the normal and shear stresses on the plane (Fig. 3.4).
Fig. 3.3 Stress induced by two principal stresses, o\ and 03, on a plane inclined at 6 to <73.
Fig. 3.3 Stress induced by two principal stresses, o\ and 03, on a plane inclined at 6 to <73.
Proof
Normal stress = an = OE = OA + AE — 03 + AD cos 9
— <73 + AB cos2 9 = <73 + (o\  03) cos2 0 Shear stress = r = DE = DC sin (180°  29) = DC sin 29
In Fig. 3.4, OE and DE represent the normal and shear stress components of the complex stress acting on plane AD. From the triangle of forces ODE it can be seen that this complex stress is represented in the diagram by the line OD, whilst the angle DOB represents the angle of obliquity, a, of the resultant stress on plane AD.
Limit conditions
It has been stated that the maximum shearing resistance is developed when the angle of obliquity equals its limiting value, <fi. For this condition the line OD becomes a tangent to the stress circle, inclined at angle <j> to axis OX (Fig. 3.5).
An interesting point that arises from Fig. 3.5 is that the failure plane is not the plane subjected to the maximum value of shear stress. The criterion of
failure is maximum obliquity, not maximum shear stress. Hence, although the plane AE in Fig. 3.5 is subjected to a greater shear stress than the plane AD, it is also subjected to a larger normal stress and therefore the angle of obliquity is less than on AD which is the plane of failure.
Strength envelopes
If <j> is assumed constant for a certain material, then the shear strength of the material can be represented by a pair of lines passing through the origin, O, at angles +</> and —<p to the axis OX (Fig. 3.6). These lines comprise the Mohr strength envelope for the material.
In Fig. 3.6 a state of stress represented by circle A is quite stable as the circle lies completely within the strength envelope. Circle B is tangential to the strength envelope and represents the condition of incipient failure, since a slight increase in stress values will push the circle over the strength envelope and failure will occur. Circle C cannot exist as it is beyond the strength envelope.
Relationship between <p and 6
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