CH.4. STRESS - UPC Universitat Politècnica de...
Transcript of CH.4. STRESS - UPC Universitat Politècnica de...
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CH.4. STRESSContinuum Mechanics Course (MMC) - ETSECCPB - UPC
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Overview
Forces Acting on a Continuum Body
Cauchy’s Postulates
Stress Tensor
Stress Tensor Components Scientific Notation Engineering Notation Sign Criterion
Properties of the Cauchy Stress Tensor Cauchy’s Equation of Motion Principal Stresses and Principal Stress Directions Mean Stress and Mean Pressure Spherical and Deviatoric Parts of a Stress Tensor Stress Invariants
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Overview (cont’d)
Stress Tensor in Different Coordinate Systems Cylindrical Coordinate System Spherical Coordinate System
Mohr’s Circle
Mohr’s Circle for a 3D State of Stress Determination of the Mohr’s Circle
Mohr’s Circle for a 2D State of Stress 2D State of Stress Stresses in Oblique Plane Direct Problem Inverse Problem Mohr´s Circle for a 2D State of Stress
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Overview (cont’d)
Mohr’s Circle a 2D State of Stress (cont’d)
Construction of Mohr’s Circle Mohr´s Circle Properties The Pole or the Origin of Planes Sign Convention in Soil Mechanics
Particular Cases of Mohr’s Circle
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Ch.4. Stress
4.1. Forces on a Continuum Body
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Forces acting on a continuum body: Body forces. Act on the elements of volume or mass inside the body. “Action-at-a-distance” force. E.g.: gravity, electrostatic forces, magnetic forces
Surface forces. Contact forces acting on the body at its boundary surface. E.g.: contact forces between bodies, applied point or distributed
loads on the surface of a body
Forces Acting on a Continuum Body
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,V Vt dV f b x
,S Vt dS
f xt
body force per unit mass
(specific body forces)
surface force per unit surface (traction vector)
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Ch.4. Stress
4.2. Cauchy’s Postulates
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1. Cauchy’s 1st postulate.The traction vector remains unchanged for all surfaces passing through the pointand having the same normal vector at .
2. Cauchy’s fundamental lemma (Cauchy reciprocal theorem)
The traction vectors acting at point on opposite sides of the same surface are equal in magnitude and opposite in direction.
Cauchy’s Postulates
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t
nPP
,Pt t n
REMARK The traction vector (generalized to internal points) is not influenced by the curvature of the internal surfaces.
, ,P P t n t n
P
REMARK Cauchy’s fundamental lemma is equivalent to Newton's 3rd law (action and reaction).
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Ch.4. Stress
4.3. Stress Tensor
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The areas of the faces of the tetrahedron are:
The “mean” stress vectors acting on these faces are
The surface normal vectors of the planes perpendicular to the axes are
Following Cauchy’s fundamental lemma:
Stress Tensor
1 1
2 2
3 3
S n SS n SS n S
1 2 3
1 * 2 * 3 ** * * * *1 2 3
* *
ˆ ˆ ˆ( ), ( , ), ( , ), ( , )
1,2,3 ; mean value theoremi
S S S S
S i SS i S
t t x t t x e t t x e t t x e
x x
REMARK
The asterisk indicates an mean value over the area.
T1 2 3n , n ,nnwith
1 1 2 2 3 3ˆ ˆ ˆ; ; n e n e n e
not
ii iˆ ˆ, , i 1, 2,3t x e t x e t x
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Let be a continuous function on the closed interval , and differentiable on the open interval , where .
Then, there exists some in such that:
I.e.: gets its“mean value” at the interiorof
Mean Value Theorem
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: a, bf R
a, b a,b a,b
a b*x
* 1 df x f x
: a, bf R
*f x a,b
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From equilibrium of forces, i.e. Newton’s 2nd law of motion:
Considering the mean value theorem,
Introducing and ,
Stress Tensor
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1 2 3
1 2 3 V S S S S V
dV dS dS dS dS dV b t t t t a
1 2 3* * * * * *1 2 3( ) V S S S S ( ) V b t t t t a
resultant body forces
resultant surface forces
1, 2,3i iS n S i 13
V Sh
1 2 3* * * * * *1 2 3
1 1( ) S S S S ( )3 3
h S n n n hS b t t t t a
i i ii i V V V V
m dV dS dV dVdm
R f a b t a a
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If the tetrahedron shrinks to point O,
The limit of the expression for the equilibrium of forces becomes,
Stress Tensor
, iiO n t n t 0 1 2 3* * * * * *
1 2 31 1( ) ( )3 3
h n n n h b t t t t a
* *
h 0 h 0
1 1lim ( ) lim ( )3 3
h h
b a 0
, t nO
1 t 2 t 3 t
(h 0)
i i
i i* * *S S i
* * *S S
h 0
h 0
ˆlim i 1,2,3
lim , ,
O
O
O,
O
x x t x t e
x x t x n t n
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P
Considering the traction vector’s Cartesian components :
In the matrix form:
( )
,
,
,
ii
ij j i i ij
ij
P n
P n n
P P
t n t
t n t
t n n
( ) ˆ ˆ( )
, 1,2,3i i
j j ij j
iij j
P t Pi j
P t P
t e e
ˆ ˆ e eij i jCauchy’s Stress Tensor
{1,2,3}
Tj i ij ji i
T
t n n
j
t n
t1
t2
t3
11 21 31
12 22 32
13 23 33
n1
n2
n3
1t 2t 3t
11t
12t
13t
Stress Tensor
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REMARK 2 The Cauchy stress tensor is constructed from the traction vectors on three coordinate planes passing through point P.
Yet, this tensor contains information on the traction vectors acting on any plane (identified by its normal n) which passes through point P.
Stress Tensor
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REMARK 1 The expression is consistent with Cauchy’s postulates: ,P P t n n
,P t n n ,P t n n , ,P P t n t n
11 12 13
21 22 23
31 32 33
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Ch.4. Stress
4.4.Stress Tensor Components
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Cauchy’s stress tensor in scientific notation
Each component is characterized by its sub-indices: Index i designates the coordinate plane on which the component acts. Index j identifies the coordinate direction in which the component acts.
Scientific Notation
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11 12 13
21 22 23
31 32 33
ij
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Cauchy’s stress tensor in engineering notation
Where: is the normal stress acting on plane a. is the tangential (shear) stress acting on the plane perpendicular to
the a-axis in the direction of the b-axis.
Engineering Notation
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x xy xz
yx y yz
zx zy z
a ab
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The stress vector acting on point P of an arbitrary plane may be resolved into:
a vector normal to the plane
an in-plane (shear) component which acts on the plane.
The sense of with respect to defines the normal stress character:
The sign criterion for the stress components is:
Tension and compression
( )n n
( ; )n n
n
n n<0 compressive stress (compression)>0 tensile stress (tension)
ij a ortensile stresscompressive stress
positive (+)negative (−)
ab
positive (+)negative (−)
positive direction of the b-axisnegative direction of the b-axis
n
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Ch.4. Stress
4.5.Properties of the Cauchy Stress Tensor
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Consider an arbitrary material volume, Cauchy’s equation of motion is:
In engineering notation:
Cauchy’s Equation of Motion
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1,2,3ijj j
i
V
b a jx
b a x
yxx zxx x
xy y zyy y
yzxz zz z
b ax y z
b ax y z
b ax y z
*
, t V
, t V
b x x
t x x
REMARK Cauchy’s equation of motion is derived from the principle of balance of linear momentum.
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For a body in equilibrium , Cauchy’s equation of motion becomes
The traction vector is now known at the boundary
The stress tensor symmetry is derived from the principle of balance of angular momentum:
Equilibrium Equations
0 1,2,3ijj
i
V
b jx
b 0 x
a 0
*
*
, , ,1,2,3i ij j
t t t Vn t j
n x x t x x equilibrium equation at the boundary
internal equilibrium equation
, 1,2,3T
ij ji i j
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Taking into account the symmetry of theCauchy Stress Tensor, Cauchy’s equation of motion
Boundary conditions
Cauchy’s Equation of Motion
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*
, t V
, t V
b x x
t x x
1,2,3ij jij j j
i i
V
b b a jx x
b b a x
*
*
( , ), , 1,2,3i ij ji i j
t Vn n t t V i j
n n t x xx x
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Regardless of the state of stress, it is always possible to choose a special set of axes (principal axes of stress or principal stress directions) so that the shear stress components vanish when the stress components are referred to this system.
The three planes perpendicular to the principal axes are the principal planes.
The normal stress components in the principal planes are the principal stresses.
Principal Stresses and Principal Stress Directions
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1
2
3
0 00 00 0
1 2 3
12
3
1x2x
3x
1x2x
3x
1x
2x
3x
1113
1222
2321
333231
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12
3
1x2x
3x
1x2x
3x
1x
2x
3x
1113
1222
2321
333231
The Cauchy stress tensor is a symmetric 2nd order tensor so it will diagonalize in an orthonormal basis and its eigenvalues are real numbers. For the eigenvalue and its corresponding eigenvector :
Principal Stresses and Principal Stress Directions
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1 2 3
v v v v 0 1
det 0 not
1 1
1 1
2 2
3 3
3 21 2 3( ) ( ) ( ) 0I I I σ σ σ characteristic
equation
INVARIANTS
REMARK The invariants associated with a tensor are values which do not change with the coordinate system being used.
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Given the Cauchy stress tensor and its principal stresses, the following is defined: Mean stress
Mean pressure
A spherical or hydrostatic state of stress:
Mean Stress and Mean Pressure
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1 2 31 1 13 3 3m iiTr
1 2 313mp
1 2 3 0 0
0 00 0
1
REMARK In a hydrostatic state of stress, the stress tensor is isotropic and, thus, its components are the same in any Cartesian coordinate system.As a consequence, any direction is a principal direction and the stress state (traction vector) is the same in any plane.
11 12 13
21 22 23
31 32 33
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The Cauchy stress tensor can be split into:
The spherical stress tensor: Also named mean hydrostatic stress tensor or volumetric stress tensor or
mean normal stress tensor. Is an isotropic tensor and defines a hydrostatic state of stress. Tends to change the volume of the stressed body
The stress deviator tensor: Is an indicator of how far from a hydrostatic state of stress the state is. Tends to distort the volume of the stressed body
Spherical and Deviatoric Parts of a Stress Tensor
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sph
1 1:3 3sph m iiTr 1 1 1
dev m 1
REMARK The principal directions of a stress tensor and its deviator stress component coincide.
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Principal stresses are invariants of the stress state: invariant w.r.t. rotation of the coordinate axes to which the stresses are
referred.
The principal stresses are combined to form the stress invariants I :
These invariants are combined, in turn, to obtain the invariants J :
Stress Invariants
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1 1 2 3iiI Tr
22 1 1 2 1 3 2 3
1 :2
I I
3 detI
1 1 iiJ I
22 1 2
1 1 12 :2 2 2ij jiJ I I
33 1 1 2 3
1 1 13 33 3 3 ij jk kiJ I I I I Tr
REMARK The J invariants can be expressed in the unified form:
1 1, 2,3iiJ Tr i
i
REMARK The I invariants are obtained from the characteristic equation of the eigenvalue problem.
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The stress invariants of the stress deviator tensor:
These correspond exactly with the invariants J of the same stress deviator tensor:
Stress Invariants of the Stress Deviator Tensor
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1 1 0J I
22 1
12
J I 2 212 :2
I I
33 1
13
J I 1 23I I 3 31 133 3 ij jk kiI I Tr
1 0I Tr 2
2 11 :2
I I 12 12 13 13 23 23
2 2 23 11 22 33 12 23 13 12 33 23 11 13 22
1det 23 ij jk kiI
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Ch.4. Stress
4.6. Stress Tensor in Different Coordinate Systems
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The cylindrical coordinate system is defined by:
The components of the stress tensor are then:
Stress Tensor in a Cylindrical Coordinate System
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´ ´ ´ ´ ´
´ ´ ´ ´
´ ´ ´ ´ ´
x x y x z r r rz
x y y y z r z
x z y z z rz z z
cos ( , , ) sin
x rr z y r
z z
x
dV r d dr dz
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The cylindrical coordinate system is defined by:
The components of the stress tensor are then:
Stress Tensor in a Spherical Coordinate System
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2 sen dV r dr d d
sen cos
, , sen sen cos
x rr y r
z r
x
´ ´ ´ ´ ´
´ ´ ´ ´
´ ´ ´ ´ ´
x x y x z r r r
x y y y z r
x z y z z r
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Ch.4. Stress
4.7. Mohr´s Circle
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Introduced by Otto Mohr in 1882. Mohr´s Circle is a two-dimensional graphical representation of
the state of stress at a point that: will differ in form for a state of stress in 2D or 3D. illustrates principal stresses and maximum shear stresses as well as stress
transformations. is a useful tool to rapidly grasp
the relation between stresses for a given state of stress.
Mohr’s Circle
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Ch.4. Stress
4.8. Mohr´s Circle for a 3D State of Stress
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Consider the system of Cartesian axes linked to the principal directions of the stress tensor at an arbitrary point P of a continuous medium: The components of the stress tensor are
The components of the traction vector are
where is the unit normal to the base associated to the principal directions
Determination of Mohr’s Circle
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with1
2
3
0 00 00 0
1 2 3
1 1 1 1
2 2 2 2
3 3 3 3
0 0 0 0 0 0
n nn nn n
t n
n
1
2
31x
2x
3x1n
2n
3n
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The normal component of stress is
The modulus of the traction vector is
The unit vector must satisfy
Locus of all possible points?
Determination of Mohr’s Circle
12 2 2
1 1 2 2 3 3 2 1 1 2 2 3 3
3
, ,T
nn n n n n n n
n
n
t
t n
1n 2 2 21 2 3 1n n n
n n
n
2 2 2 2 2 2 21 1 2 2 3 3 2 2 2 2 2 2 2 2
1 1 2 2 3 32 2 2 : : n
n n nn n n
t t t
t τ
,
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Mohr's 3D problem half space
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The previous system of equations can be written as a matrix equation which can be solved for any couple
A feasible solution for requires that for theexpression to hold true.
Every couple of numbers which leads to a solution , will be considered a feasible point of the half-space. The feasible point is representative of the traction vector on a
plane of normal which passes through point P.
The locus of all feasible points is called the feasible region.
Determination of Mohr’s Circle
2 2 2 2 2 21 2 3 1
21 2 3 2
231 1 1
1
nnn
x bA
1 2 3, ,T
n n n n
212223
0 10 10 1
nnn
x
,
2 2 21 2 3 1n n n
2 2 21 2 3, ,
Tn n n x
,
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The system
can be re-written as
Determination of Mohr’s Circle
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1 2 2 3 1 3A with
2 2 21 3 1 3 1
1 3
2 2 22 3 2 3 2
2 3
2 2 21 2 1 2 3
1 2
( ) 0
( ) 0
( ) 0
AI n
AII n
AIII n
2 2 2 2 2 21 2 3 1
21 2 3 2
231 1 1
1
nnn
x bA
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Consider now equation :
It can be written as:
which is the equation of a semicircle of center and radius :
Determination of Mohr’s Circle
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2 2 2a R
3C 3R
3 1 2
2 23 1 2 2 3 1 3 3
1 , 02
14
C
R n
2 2 2
1 2 1 2 31 2
0A n
1 2 2 3 1 3A with( )III
1 2
2 21 2 2 3 1 3 3
12
1 4
a
R n
with
23
23
0
1
n
n
REMARK A set of concentric semi-circles is obtained with the different values of
with center and radius :3C 3 3R n3n
3 1 2
3 1 2 3
1212
min
max
R
R
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41
Following a similar procedure with and , a total of three semi-annuli with the following centers and radii are obtained:
Determination of Mohr’s Circle
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( )II( )I
1
1 2 3
a
1C [ ,0]2
2 3
1 1
12
a
min1
max1
R
R
2
2 1 3
a
1C [ ,0]2
1 3
2 2
12
a
max2
min2
R
R
3
3 1 2
a
1C [ ,0]2
1 2
3 3
12
a
min3
max3
R
R
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Superposing the three annuli,
Determination of Mohr’s Circle
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Superposing the three annuli,
The final feasible region must be the intersection of these three semi-annuli.
Determination of Mohr’s Circle
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Superposing the three annuli,
The final feasible region must be the intersection of these semi-annuli
Every point of the feasible region in the Mohr’s space, corresponds to the stress (traction vector) state on a certain plane at the considered point
Determination of Mohr’s Circle
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Ch.4. Stress
4.9. Mohr´s Circle for a 2D State of Stress
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2D State of Stress
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3D general state of stress 2D state of stress
x xy xz
yx y yz
zx zy z
00
0 0
x xy
yx y
z
x xy
yx y
REMARK In 2D state of stress problems, the principal stress in the disregarded direction is known (or assumed) a priori.
3D problem
2D (plane) problem
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Given a plane whose unit normal forms an angle with the axis, Traction vector
Normal stress
Shear stress
Tangential stress is now endowed with sign Pay attention to the “positive” senses given in the figure
Stresses in a oblique plane
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sincos
m
cos sincoscos sinsin
x xy x xy
xy y xy y
n
t n
cos 2 sin 22 2
x y x yxy
t n
sin 2 cos 22
x yxy
t m
x
n
cossin
n
( 0 0)or
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Direct Problem: Find the principal stresses and principal stress directions given in a certain set of axes.
Inverse Problem: Find the stress state on any plane, given the principal stresses and principal stress directions.
Direct and Inverse Problems
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equivalent stresses
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In the and axes, then,
Using known trigonometric relations,
Direct Problem
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sin 2 cos 2 02
x yxy
222
2 22
1sin 211
tg 2 2
1 2cos 21 tg 2
2
xy
x yxy
x y
x yxy
x́ ´y 0
This equation has two solutions:
1.
2.
These define the principal stress directions.(The third direction is perpendicular to the plane of analysis.)
1 ( " ")sign
2 1 ( " ")2
sign
tan 2
2
xy
x y
x y
2 x y
2cos 2 xy sin 2
x y
2sin 2 xy cos 2
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The angles and are then introduced into the equation
to obtain the principal stresses (orthogonal to the plane of analysis):
Direct Problem
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1 2
22
1
22
2
2 2
2 2
x y x yxy
x y x yxy
cos 2 sin 22 2
x y x yxy
x y
2 x y
2cos 2 xy sin 2
x y
2sin 2 xy cos 2
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Given the directions and principal stresses and , to find the stresses in a plane characterized by the angle : Take the equations
Replace , , and to obtain:
Inverse Problem
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1 2
x 1 y 2 xy 0
1 2 1 2
1 2
cos 22 2
sin 22
x y
2 x y
2cos 2 xy sin 2
x y
2sin 2 xy cos 2
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Considering a reference system and characterizing the inclination of a plane by , From the inverse problem equations:
Squaring both equations and adding them:
Mohr’s Circle for a 2D State of Stress
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2 221 2 1 2
2 2
´ ´x y
1 2 1 2
1 2
cos 22 2
sin 22
REMARK This expression is valid for any value of .
1 2 , 02
C
1 2
2R
Eq. of a circle with center and radius .
Mohr’s CircleRC
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The locus of the points representative of the state of stress on any of the planes passing through a given point P is a circle.
(Mohr’s Circle) The inverse is also true: Given a point in Mohr’s Circle, there is a plane passing through
P whose normal and tangential stresses are and , respectively.
Mohr’s Circle for a 2D State of Stress
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1 2 , 02
C
1 2
2R
,
1 2
1 2
2cos 2
2
aR
1 2
sin 2
2R
Mohr's 2D problemspace
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Interactive applets and animations: by M. Bergdorf:
from MIT OpenCourseware:
from Virginia Tech:
From Pennsilvania State University:
Construction of Mohr’s Circle
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http://www.zfm.ethz.ch/meca/applets/mohr/Mohrcircle.htm
http://ocw.mit.edu/ans7870/3/3.11/tools/mohrscircleapplet.html
http://web.njit.edu/~ala/keith/JAVA/Mohr.html
http://www.esm.psu.edu/courses/emch13d/design/animation/animation.htm
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A. To obtain the point in Mohr’s Circle representative of the state of stress on a plane which forms an angle with the principal stress direction :
1. Begin at the point on the circle (representative of the plane where acts).
2. Rotate twice the angle in the sense . 3. This point represents the shear and normal stresses at the desired plane
(representative of the stress state at the plane where acts).
Mohr’s Circle’s Properties
1
1
1.
3.
2.
( 1,0)
1
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B. The representative points of the state of stress on two orthogonal planes are aligned with the centre of Mohr’s Circle:
This is a consequence of property A as .
Mohr’s Circle’s Properties
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2 1 2
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C. If the state of stress on two orthogonal planes is known, Mohr’s Circle can be easily drawn:
1. Following property B, the two points representative of these planes will be aligned with the centre of Mohr’s Circle.
2. Joining the points, the intersection with the axis will give the centre of Mohr’s Circle.
3. Mohr’s Circle can be drawn.
Mohr’s Circle’s Properties
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3.
2.1.
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D. Given the components of the stress tensor in a particular orthonormal base, Mohr’s Circle can be easily drawn:
This is a particular case of property C in which the points representative of the state of stress on the Cartesian planes is known.
1. Following property B, the two points representative of these planes will be aligned with the centre of Mohr’s Circle.
2. Joining the points, the intersection with the axis will give the centre of Mohr’s Circle.
3. Mohr’s Circle can be drawn.
Mohr’s Circle’s Properties
x xy
xy y
3.
2.1.
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The radius and the diametric points of the circle can be obtained:
Mohr’s Circle’s Properties
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x xy
xy y
2x y 2
xy2R
1 a R x y
2
x y
2
2
xy2
2 a R x y
2
x y
2
2
xy2
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Note that the application of property A for the point representative of the vertical plane implies rotating in the sense contrary to angle.
Mohr’s Circle’s Properties
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x xy
xy y
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The point called pole or origin of planes in Mohr’s circle has the following characteristics: Any straight line drawn from the pole will intersect the Mohr circle at a
point that represents the state of stress on a plane parallel in space to that line.
The Pole or the Origin of Planes
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1.
2.
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The point called pole or origin of planes in Mohr’s circle has the following characteristics: If a straight line, parallel to a given plane, is drawn from the pole, the
intersection point represents the state of stress on this particular plane.
The Pole or the Origin of Planes
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2.
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The sign criterion used in soil mechanics, is the inverse of the one used in continuum mechanics: In soil mechanics,
But the sign criterion for angles is the same: positive angles are measured counterclockwise
Sign Convention in Soil Mechanics
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continuum mechanics
soil mechanics
tensile stresscompressive stresspositive (+)
negative (−)
negative (-)positive (+) counterclockwise rotation
clockwise rotation
*
*
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For the same stress state, the principal stresses will be inverted.
The expressions for the normal and shear stresses are
The Mohr’s circle construction and properties are the same in both cases
Sign Convention in Soil Mechanics
continuum mechanics soil mechanics
* * * ** *2 1 2 1
* * * ** *1 2 1 2 1 2 1 2
* * * *1 2 * * *2 1 1 2
*
*
cos 2
sin 2
cos 22 2
cos 2 cos 22 2 2 2
sin 2 sin 2 sin 22 2 2
*
like incontinuum mechanics
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Ch.4. Stress
4.10. Particular Cases of Mohr’s Circle
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Hydrostatic state of stress
Mohr’s circles of a stress tensor and its deviator
Pure shear state of stress
Particular Cases of Mohr’s Circles
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sph 1 m 1
2 m 2
3 m 3
sph m 1
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Ch.4. Stress
Summary
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Forces acting on a continuum body:
Body forces:
Surface forces:
Cauchy’s Postulates:1.2.
The traction vector:
Summary
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,V Vt dV f b x
,S Vt dS
f xt
,Pt t n
, ,P P t n t n
ˆ( ) , 1,2,3iij jP i j t e
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Cauchy stress tensor:
The stress vector acting on point P of an arbitrary plane may be resolved into:
a component normal to the plane .
a shear component which acts on the plane .
Summary
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11 12 13
21 22 23
31 32 33
x xy xz
yx y yz
zx zy z
Scientific notationEngineering notation
n
n
n
n
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12
3
1x2x
3x
1x2x
3x
1x
2x
3x
1113
1222
2321
333231
Cauchy’s equation of motion:
Boundary conditions
The principal stress directions correspond to the set of axes that make the shear stress components vanish when the stress components are referred to this system. The normal stress components in the principal planes are the principal stresses.
Summary
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V b b a x
*( , )t V n n t x x
1
2
3
0 00 00 0
1 2 3
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Mean stress:
Mean pressure:
A spherical or hydrostatic state of stress:
The Cauchy stress tensor can be split into:
Stress invariants:
Summary
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1 2 31 1 13 3 3m iiTr
mp
1 2 3 1
sph :sph m 1dev m 1
spherical stress tensor:stress deviator tensor:
1 1 2 3iiI Tr
22 1 1 2 1 3 2 3
1 :2
I I
3 detI
1 1 iiJ I
22 1 2
1 1 12 :2 2 2ij jiJ I I
33 1 1 2 3
1 1 13 33 3 3 ij jk kiJ I I I I Tr
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The components of the stress tensor in the cylindrical coordinate system:
The components of the stress tensor in the spherical coordinate system:
Summary
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´ ´ ´ ´ ´
´ ´ ´ ´
´ ´ ´ ´ ´
x x y x z r r rz
x y y y z r z
x z y z z rz z z
dV r d dr dz
´ ´ ´ ´ ´
´ ´ ´ ´
´ ´ ´ ´ ´
x x y x z r r r
x y y y z r
x z y z z r
2 sen dV r dr d d
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Mohr’s Circle is a two-dimensional graphical representation of the state of stress at a point.
For a 3D state of stress:
For a 2D state of stress:
Summary
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1 2 , 02
C
1 2
2R
2 2 2C R
Equation for Mohr’s circle (2D) sin 2
R
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Mohr’s circle is a two-dimensional graphical representation of the state of stress at a point. For a 3D state of stress:
For a 2D state of stress:
Summary
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1 2 , 02
C
1 2
2R
2 2 2C R
Equation for Mohr’s circle (2D)
C1 12 2 3
a1
,0
2 3
1 1
12
a
min1
max1
R
R
C2 12 1 3
a2
,0
1 3
2 2
12
a
max2
min2
R
R
C3 12 1 2
a3
,0
1 2
3 3
12
a
min3
max3
R
R
sin 2R
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2D Mohr’s circle can be used to solve: Direct problem
Inverse problem
Summary
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22
1
22
2
2 2
2 2
x y x yxy
x y x yxy
1 2 1 2
1 2
cos 22 2
sin 22
2
22
1sin 211
tg 2 2
xy
x yxy
principal stresses principal stress directions
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Any straight line drawn from the pole or origin of planes will intersect the Mohr circle at a point that represents the state of stress on a plane parallel in space to that line. The inverse is also true.
Particular cases:
Summary
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Hydrostatic state of stress
A tensor and its deviator
Pure shear state of stress
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Sign criterion
The sign criterion used in soil mechanics, is the inverse of the one used in continuum mechanics, except the criterion for angles, which remains the same (positive angles are measured counterclockwise).
Summary
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t n<0 compressive stress>0 tensile stress
ij aortensile stresscompressive stress
positive (+)negative (−)
ab
positive (+)negative (−)
positive direction of the b-axisnegative direction of the b-axis
continuum mechanics
soil mechanics
continuum mechanics
soil mechanics