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Mohr Circle of Stress -- GEOTECHNICAL ENGINEERING- Prof. G.P. Raymond 97

Figure 1. Vectordirection of

positive stresses.

Figure 2.Pure shear.

Figure 3.

MOHR CIRCLE OF STRESS

SYMBOLSNotation Dimensional Analysis

x = distance from origin in x direction Lx = infinitesimal length in x direction L

y = distance from origin in y direction L

y = infinitesimal length in y direction L= the normal stress on a plane M L-1 T-2

= the shearing stress on a plane M L-1 T-2

, , and are angles Angle

Subscriptx = relates to the plane perpendicular to the x directiony = relates to the plane perpendicular to the y directionxy = relates to the plane perpendicular to the x direction actingin the y directionyx = relates to the plane perpendicular to the y direction actingin the x direction

1. STRESS

Stress is an engineering concept defined as the force per unit area asthe area reduces to zero (i.e. in the limit of area --> 0). Stress cannot bemeasured directly. Either the overall force and area are measured and

uniformity is assumed, or deformations are used to infer stress by meansof an established or assumed relationship between stress and strain(strain being defined as the change in length per unit length).

Direct or normal stresses (given the symbol ) are defined as thosestresses that have a direction perpendicular to the plane on which theyact. Shear stresses (given the symbol ) are defined as those stresseswhich have a direction parallel to the plane on which they act.

2. SIGN CONVENTION

In geotechnical engineering, since soils androck are generally considered to be unable to taketension it is common practice to definecompressive stresses as positive, the opposite ofstructural mechanics. To be consistent shearingstresses and angular directions are also given anopposite sign convention. Thus in an analyticalsolution if the direction of the stress is oppositeto the axis direction the stress is positive as

illustrated in Figure 1. Similarly anticlockwiseangular directions are positive.

3. COMPLIMENTARY SHEARINGSTRESS

When considering two planes at right anglesthe shear stresses on the both planes must be"complimentary". This is illustrated below.Considering rotational equilibrium of the elementin Figure 2, then:

(1)

(2)

4. STRESS COMPONENTS ON ANYPLANE

Consider the case of a point (i.e. smallelement) subject to plane stress (stresses in twodimensions only). Assume there are three knowncomponents of stress on planes at right to eachother x, y and xy as shown inFigure 3. For equilibrium normal to AC

(3)

substituting equation (2) in the above

(4)

(5)

For equilibrium parallel to AC

(6)

from which

(7)

Thus n and n for any plane may be analytically calculated. Note thatthese relationships are quite general and valid for any material. Furtheranalysis can express these values in terms of principle stress as:

(8)

(9)

5. MOHR'S CIRCLE OF STRESS

It is generally more convenient to communicated stresses at a pointvia graphical representation. Such a method is known as "Mohr'sCircle". The method enables the stresses on all planes to be quickly andsimply visualized.

Writing equations (5) and (7) as

(10)

(11)

then squaring and adding the two equations gives

(12)

which is the equation for a circle of

(13)

(14)

Noting the above derivation, Mohr devised a geometricalrepresentation of the state of stress at a point (as shown in Figure 3) byplotting the normal stress as abscissa and shear stress as ordinate.

Examination of equations (10) and (11) in terms of the Mohr Circleindicates that

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Mohr Circle of Stress -- GEOTECHNICAL ENGINEERING- Prof. G.P. Raymond 98

Figure 4 Mohr circle.

Figure 5

Figure 6 Pole construction

(a) for two planes physically degrees apart the stresses on the MohrCircle will subtend an angle of 2 degrees at the centre of thecircle and

(b) the complimentary shear stresses, which were equal in theanalytical equation (2) given earlier, have different signs.

The latter fact requires a new definition of the sign convention for shearstresses. For construction of the Mohr Circle, shear stresses which causeanticlockwise rotation are defined herein as positive. Thus

(15)

(16)

All other stress and angular measurements retain the sign convention ofan analytical solution.

By constructing the angle 2 atthe centre of the circle from thepoint on the circumference of thecircle representing the planestresses x, xy in the same angulardirection as going from that planeto the plane on which the stresses

, act (i.e. anticlockwise) givesthe representation on thecircumference of the stresses and. The is illustrated in Figure 4.

Note is measured from the planeon which the stresses ( x, xy) areacting to the plane on which thestresses ( , ) are acting or visaverse. The doubled angle at the centre must go in the same circulardirection. Alternatively point ( , ) may be located by what is knownas the Pole Construction as given in the next section. Note the majorprincipal stress is, by definition, the maximum positive stress, and is thefurthest point to the right of the diagram and on the circle.

6. RESULTANT STRESSES ON A PLANE

The resultant stress on a plane may be simply obtained from a Mohr'scircle by measurement of the distance from the origin (0,0) to thegraphical representation of the stresses on that plane ( , ). The anglebetween this resultant and the -axis will give the angle between theresultant and the normal and if measured from the resultant has aclockwise (positive) rotation on the Mohr Circle but anticlockwiserotation on the stress elemental diagram when the shearing stress is

positive. Note this difference in angular rotation.

7. POLE CONSTRUCTION

The pole construction,method of determining stressesfrom a Mohr Circle is based onthe geometric fact that the anglebetween three points on thediameter of a circle is half theangle subtended by the two outerpoints and the centre of thecircle. This may be simplydemonstrated by use of Figure 5.The triangles AOB and AOC areboth isosceles triangles thus thebase angles of triangle AOB areequal (say ) as are the baseangles of AOC (say ). Since the angles of a triangle add up to 180 the

apex angles BOA + AOC must be (360 - 2 - 2 ) and thus the angleBOC outside the two triangles must equal 2( + ). This is twice theangle BAC.

This permits the construction shown on Figure 6. The anglebetween ( y, yx), (POLE) and ( , ) is from the above proof half theangle between ( y, yx), (CENTRE) and ( , ) shown as 2 . The poleconstruction consists of drawing a line parallel to the plane on whichgiven known stresses ( y, yx) act through the point on the Mohr Circlerepresenting these stresses. Where this line cuts the Mohr Circle is

known as the "Pole". Thus reversing the procedure any line through thePole parallel to a plane for which required stresses are desired (i.e. , )will cut the Mohr Circle at the point representing those stresses.

Note that if Figure 3 is rotated in relation to Figure 6 the location ofthe pole will change but the final location of ( , ) in relationship to ( y,

yx) will not.

8. SUMMARY OF MOHR CIRCLE SIGN CONVENTION(1) Compressive normal stresses: positive(2) Anticlockwise shearing stresses: positive(3) Anticlockwise angles between stress planes: positive(4) Clockwise angles on the Mohr Circle from the resultant to the -

axis indicates positive shearing stress on plane and anticlockwiseangles between resultant and normal on elemental diagram:positive.