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Mohr Circle for stress. In 2D space (e.g., on the s 1 s 2 , s 1 s 3 , or s 2 s 3 plane), the normal stress ( s n ) and the shear stress ( s s ), could be given by equations (1) and (2) in the next slides

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mohr circle for stress
Mohr Circle for stress
  • In 2D space (e.g., on the s1s2 , s1s3, or s2s3 plane), the normal stress (sn) and the shear stress (ss), could be given by equations (1) and (2) in the next slides
  • Note: The equations are given here in the s1s2 plane, where s1 is greater than s2.
  • If we were dealing with the s2s3 plane, then the two principal stresses would be s2 and s3
normal stress
Normal Stress

The normal stress, sn:

sn= (s1+s2)/2 + (s1-s2)/2 cos2q (1)

  • In parametric form the equation becomes:

sn = c + r cosω

Where

  • c = (s1+s2)/2 is the center, which lies on the normal stress axis (x axis)
  • r = (s1-s2)/2 is the radius
  • ω= 2q
sign conventions
Sign Conventions

sn is compressive when it is “+”, i.e., when sn>0

sn is tensile when it is “-”, i.e., when sn< 0

sn= (s1+s2)/2+(s1-s2)/2 cos2q

NOTE:

qis the angle froms1to the normal to the plane!

sn = s1 at q = 0o(a maximum)

sn = s2 at q = 90o(a minimum )

  • There is no shear stress on the three principal planes (perpendicular to the principal stresses)
shear stress
Shear Stress

The shear stress

ss = (s1-s2)/2 sin2q (2)

  • In parametric form the equation becomes:

ss = r sinωwhere ω= 2q

ss > 0 ‘+’ shear stress represents left-lateral shear

ss < 0 ‘-’ shear stress represents right-lateral shear

ss = 0 (a min)at q = 0o or 90o or 180o

ss = (s1-s2)/2 (a max) at q = + 45o

  • The maximum ss is 1/2 the differential stress (diameter), i.e., it is the length of the radius!
construction of the mohr circle in 2d
Construction of the Mohr Circle in 2D
  • Plot the normal stress, sn, vs. shear stress, ss, on a graph paper using arbitrary scale (e.g., mm scale!)
  • Calculate:
    • Center c = (s1+s2)/2
    • Radius r = (s1-s2)/2
  • Note: Diameter is the differential stress(s1-s2)
  • The circle intersects the sn (x-axis) at the two principal stresses (s1 ands2)
construction of the mohr circle
Construction of the Mohr Circle
  • Multiply the physical angle q by 2
  • The angle 2q is from the cs1 line to any point on the circle
  • +2q (CCW) angles are read above the x-axis
  • -2q (CW) angles below the x-axis, from the s1 axis
  • The sn andss of a point on the circle represent the normal and shear stresses on the plane with the given 2q angle
  • NOTE: The axes of the Mohr circle have no geographic significance!
maximum minimum normal stresses
Maximum & Minimum Normal Stresses

The normal stress:

sn= (s1+s2)/2 + (s1-s2)/2 cos2q

NOTE: q (in physical space) is the angle from s1 to the normal to the plane

When q = 0othencos2q = 1 and sn=(s1+s2)/2 + (s1-s2)/2

which reduces to a maximum value:

sn= (s1+s2 +s1-s2)/2  sn= 2s1/2  sn= s1

When q = 90othencos2q = -1 and sn= (s1+s2)/2 - (s1-s2)/2

which reduces to a minimum

sn= (s1+s2 -s1+s2)/2  sn= 2s2/2  sn= s2

special states of stress uniaxial stress
Special States of Stress - Uniaxial Stress
  • Uniaxial Stress (compression or tension)
    • One principal stress (s1 or s3) is non-zero, and the other two are equal to zero
  • Uniaxial compression

Compressive stress in one direction: s1 > s2=s3 = 0

| a 0 0|

| 0 0 0|

| 0 0 0|

  • The Mohr circle is tangent to the ordinate at the origin (i.e., s2=s3= 0) on the + (compressive) side
uniaxial tension
Uniaxial Tension

Tension in one direction:

0 = s1 = s2 > s3

|0 0 0|

|0 0 0|

|0 0-a|

  • The Mohr circle is tangent to the ordinate at the origin on the - (i.e., tensile) side
special states of stress axial stress
Special States of Stress - Axial Stress
  • Axial (confined) compression: s1 > s2 = s3 > 0

|a 0 0|

|0 b 0|

|0 0 b|

  • Axial extension (extension): s1 = s2 > s3 > 0

|a 0 0|

|0 a 0|

|0 0 b|

  • The Mohr circle for both of these cases are to the right of the origin (non-tangent)
special states of stress biaxial stress
Special States of Stress - Biaxial Stress
  • Biaxial Stress:
    • Two of the principal stresses are non-zero and the other is zero
  • Pure Shear:

s1 = -s3 and is non-zero (equal in magnitude but opposite in sign)

s2 = 0 (i.e., it is a biaxial state)

    • The normal stress on planes of maximum shear is zero (pure shear!)

|a 0 0 |

|0 0 0 |

|0 0 -a|

  • The Mohr circle is symmetric w.r.t. the ordinate (center is at the origin)
special states of stress triaxial stress
Special States of Stress - Triaxial Stress
  • Triaxial Stress:
    • s1, s2,ands3have non-zero values
    • s1 > s2 >s3and can be tensile or compressive
  • Is the most general state in nature

|a 0 0 |

|0 b 0 |

|0 0 c |

  • The Mohr circle has three distinct circles
two dimensional cases general stress
Two-dimensional cases: General Stress
  • General Compression
    • Both principal stresses are compressive
      • is common in earth)
  • General Tension
    • Both principal stresses are tensile
    • Possible at shallow depths in earth
isotropic stress
Isotropic Stress
  • The 3D, isotropic stresses are equal in magnitude in all directions (as radii of a sphere)
  • Magnitude = the mean of the principal stresses

sm= (s1+s2+s3)/3 = (s11+s22+s33 )/3

P = s1= s2= s3when principal stresses are equal

  • i.e., it is an invariant (does not depend on a specific coordinate system). No need to know the principal stress; we can use any!
  • Leads to dilation (+ev& -ev); but no shape change
  • ev=(v´-vo)/vo= v/vo[no dimension]

v´and vo are final and original volumes

stress in liquids
Stress in Liquids
  • Fluids (liquids/gases) are stressed equally in all directions (e.g. magma); e.g.:
  • Hydrostatic, Lithostatic, Atmospheric pressure
  • All of these are pressure due to the column of water, rock, or air, respectively:

P = rgz

  • z is thickness
  • ris density
  • g is the acceleration due to gravity
hydrostatic pressure hydrostatic tension
Hydrostatic Pressure- Hydrostatic Tension
  • Hydrostatic Pressure: s1 = s2 = s3 = P

|P 0 0|

|0 P 0|

|0 0 P|

  • All principal stresses are compressive and equal (P)
  • No shear stress exists on any plane
  • All orthogonal coordinate systems are principal coordinates
  • Mohr circle reduces to a point on the sn axis
  • Hydrostatic Tension
    • The stress across all planes is tensile and equal
    • There are no shearing stresses
    • Is an unlikely case of stress in the earth
deviatoric stress
Deviatoric Stress
  • A total stress sTcan be divided into its components:
  • isotropic (Pressure) or mean stress (sm)
    • Pressure is the mean of the principal stresses (may be neglected in most problems). Only causes volume change.
  • deviatoric (sd) that deviates from the mean
    • Deviator’s components are calculated by subtracting the mean stress (pressure) from each of the normal stresses of the general stress tensor (not the shear stresses!). Causes shape change and that it the part which we are most interested in.

sT=sm+sd or sd=sT-sm

confining pressure
Confining Pressure
  • In experimental rock deformation, pressure is called confining pressure, and is taken to be equal to the 2 and3 (uniaxial loading)
  • This is the pressure that is hydraulically applied around the rock specimen
  • In the Earth, at any point z, the confining pressure is isotropic (lithostatic) pressure:

P = rgz

decomposition of matrix
Decomposition of Matrix
  • Decomposition of the total stress matrix into the mean and deviatoric matrices
  • The deviatoric part of total stress leads to change in shape
example deviatoric mean stress
Example - Deviatoric & Mean stress

Given: s1 = 8 Mpa, s2 = 5 Mpa, and s3 = 2 Mpa

Find the mean and the diviatoric stresses

The mean stress (sm):

sm = (8 + 5 + 2) / 3 = 5 MPa

The deviatoric stresses (sn):

s1= 8-5 = 3 Mpa (compressive)

s2 = 5-5 = 0 Mpa

s3 = 2-5 = -3 Mpa (tensile)

differential stress
Differential Stress
  • The difference between the maximum and the minimum principal stresses (s1-s2)
  • Is always positive
  • Its value is:
    • twice the radius of the largest Mohr circle
    • It is twice the maximum shear stresses

Note: ss = (s1-s2)/2 sin2q

ss = (s1-s2)/2 at q = + 45o(a maximum)

  • The maximum ss is 1/2 the differential stress
  • Is an invariant of the stress tensor
effective stress
Effective Stress
  • Its components are calculated by subtracting the internal pore fluid pressure (Pf) from each of the normal stresses of the external stress tensor
  • This means that the pore fluid pressures opposes the external stress, decreasing the effective confining pressure
  • The pore fluid pressure shifts the Mohr circle toward lower normal stresses. This changes the applied stress into an effective stress
effective stress29
Effective Stress
  • (applied stress - pore fluid pressure)= effective stress

|s11s12s13 | | Pf 0 0 | |s11- Pf s12s13 |

|s21s22s23 | - | 0 Pf 0 |=|s21 s22 – Pf s23 |

|s31 s32s33 | | 0 0 Pf | |s31 s32 s33- Pf |

  • Mechanical behavior of a brittle material depends on the effective stress, not on the applied stress