Stress at a PointStress at a Point

Lecture 3Lecture 3

Engineering 473Engineering 473Machine DesignMachine Design

PurposePurpose

The stress state at critical locations in a machine component is required to evaluate whether the component will satisfy strength design requirements.

The purpose of this class is to review the concepts and equations used to evaluate the state of stress at a point.

2D Cartesian Stress 2D Cartesian Stress ComponentsComponents

xxσxxσ

yyσ

yyσ

xyτ

xyτ

yxτ

yxτ

xyτ

X

Y

i�j�

Face Direction

NotationNotation

σ Normal Stress

τ Shear Stress

Moment equilibrium requires thatyxxy ττ =

Tensor Sign ConventionTensor Sign Convention

xxσxxσ

yyσ

xyτ

xyτ

yxτ

yxτ

Stresses acting in a positive coordinate direction on a positive face are positive.

Shear stresses acting in the negative coordinate direction on a negative face are positive.

Xi�j�

Y

Posit

ive

Face

PositiveFace

NegativeFace

Neg

ativ

eFa

ce

This sign convention must be used to satisfy the differential equilibrium equations and tensor transformation relationships.

2D Mohr�s Circle 2D Mohr�s Circle Sign ConventionSign Convention

xxσxxσ

yyσ

xyτ

xyτ

yxτ

yxτ

Y

Xi�j�

The sign convention used with the 2D Mohr�s circle equations is slightly different.

A positive shear stress is one that tends to create clockwise (CW) rotation.

2D Mohr�s Circle2D Mohr�s Circle(Transformation of Axis)(Transformation of Axis)

xxσ

yyσ

xyτ

yxτ

στ φ

φ

x

yAll equations for a 2-D Mohr�s Circle are derived from this figure.

dxdy ds

ΣF in the x- and y-directions yields the transformation-of-axis equations

( ) ( )

( ) ( )2φcosτ2φsin2σσ

τ

2φsinτ2φcos2σσ

2σσ

σ

xyyyxx

xyyyxxyyxx

+−

−=

+−

++

=

2D Mohr�s Circle2D Mohr�s Circle(Principal Stress Equations)(Principal Stress Equations)

2xy

2yyxx

21

2xy

2yyxxyyxx

21

τ2σσ

τ,τ

τ2σσ

2σσ

σ,σ

+���

����

� −±=

+���

����

� −±

+=

The transformation-of-axis equations can be used to find planes for which the normal and shear stress are the largest.

We will use these equations extensively during this class.

2D Mohr�s Circle2D Mohr�s Circle(Graphical Representation)(Graphical Representation)

Shigley, Fig. 3.3

Note that the shear stress acting on the plane associated with a principal stress is always zero.

2xy

2yyxx

21

2xy

2yyxxyyxx

21

τ2σσ

τ,τ

τ2σσ

2σσ

σ,σ

+���

����

� −±=

+���

����

� −±

+=

Comments on Shear Stress Comments on Shear Stress Sign ConventionSign Convention

xxσxxσ

yyσ

xyτ

xyτ

yxτ

yxτ

xxσxxσ

yyσ

xyτ

xyτ

yxτ

yxτ

TensorTensor

2D Mohr�s 2D Mohr�s CircleCircle

2xy

2yyxx

21

2xy

2yyxxyyxx

21

τ2σσ

τ,τ

τ2σσ

2σσ

σ,σ

+���

����

� −±=

+���

����

� −±

+=

The sign convention is important when the transformation-of-axis equations are used.

The same answer is obtained when computing the principal stress components.

3D Stress Components3D Stress Components

x

y

z

i�j�

k�

xxσ

zzσ

yyσ

xyτ

xzτ

yxτ

zxτzyτ

Note that the tensor sign convention is used.

There are nine components of stress.

Moment equilibrium can be used to reduce the number of stress components to six.

zyyz

zxxz

yxxy

ττττ

ττ

==

=

Cauchy Cauchy Stress TensorStress Tensor

���

�

�

���

�

�

=≈

zzzyzx

yzyyyx

xzxyxx

στττστττσ

σ

is known as the Cauchy stress tensor. Its Cartesian components are shown written in matrix form.

Tensors are quantities that are invariant to a coordinate transformation.

A vector is an example of a first order tensor. It can be written with respect to many different coordinate systems.

jnijmimn β σ βσ =

Tensor Transformation Tensor Transformation EquationEquation

Tensor Transformation Tensor Transformation EquationEquation ≈

σ

Cauchy Formula

ΣF in the x,y,and z directions yields the Cauchy Stress Formula.

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=��

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�

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���

�

z

y

x

zzzyzx

yzyyyx

xzxyxx

TTT

nml

στττστττσ

This equation is similar to the Mohr�s circle transformation-of-axis equation

x

z

y

A

B

C

xxσxyτ

xzτ

zzσ

zyτzxτ

yyσyxτ yzτ

k�nj�mi�ln� ++=

P

T�

n�

x

z

y

A

B

CP

T�

n�nσuτ

vτ

3D Principal Stresses3D Principal Stresses

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�

=��

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�

��

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�

���

���

�

z

y

x

zzzyzx

yzyyyx

xzxyxx

TTT

nml

στττστττσ The shear stress on planes

normal to the principal stress directions are zero.

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=��

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nml

σnml

στττστττσ

zzzyzx

yzyyyx

xzxyxx

( )( )

( ) ��

��

�

��

��

�

=��

��

�

��

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�

���

���

�

−−

−

000

nml

σστττσστττσσ

zzzyzx

yzyyyx

xzxyxx

We need to find the plane in which the stress is in the direction of the outward unit normal.

This is a homogeneous linear equation.

3D Principal Stresses3D Principal Stresses((Eigenvalue Eigenvalue Problem)Problem)

( )( )

( ) ��

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=��

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−−

−

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σστττσστττσσ

zzzyzx

yzyyyx

xzxyxxA homogeneous linear equation has a solution only if the determinant of the coefficient matrix is equal to zero.

( )( )

( )0

σστττσστττσσ

zzzyzx

yzyyyx

xzxyxx

=−

−−

This is an eigenvalue problem.

3D Principal Stresses3D Principal Stresses(Characteristic Equation)(Characteristic Equation)

( )( )

( )0

σστττσστττσσ

zzzyzx

yzyyyx

xzxyxx

=−

−− The determinant can be

expanded to yield the equation

0IσIσIσ 322

13 =−+−

2xyzz

2zxyy

2yzxxzxyzxyzzyyxx3

2zx

2yz

2xyxxzzzzyyyyxx2

zzyyxx1

τστστσττ2τσσσI

τττσσσσσσI

σσσI

−−−+=

−−−++=

++=

I1, I2, and I3 are known as the first, second, and third invariants of the Cauchy stress tensor.

3D Principal Stresses3D Principal Stresses

0IσIσIσ 322

13 =−+−

There are three roots to the characteristic equation, σ1, σ2, and σ3.

Each root is one of the principal stresses.

The direction cosines can be found by substituting the principal stresses into the homogeneous equation and solving.

The direction cosines define the principal directions or planes.

Characteristic EquationCharacteristic Equation

3D Mohr�s Circles3D Mohr�s Circles

σ

τ

σ1σ2σ3

τ1,2

τ1,3

τ2,3

Note that the principal stresses have been ordered such that .

Maximum shear stressesMaximum shear stresses

2σστ

2σστ

2σστ

311,3

322,3

211,2

−=

−=

−=

321 σσσ ≥≥

Octahedral StressesOctahedral Stresses

( ) ( )

( )( ) ( ) ( )[ ]( ) ( ) ( )

( )21

2xz

2yz

2xy

2xxzz

2zzyy

2yyxx

212

132

322

21

212

1,322,3

2oct

zzyyxx3211oct

τττ6

σσσσσσ31

σσσσσσ31

τττ32τ

σσσ31σσσ

31I

31σ

1,2

��

�

�

��

�

�

+++

−+−+−=

−+−+−=

++=

++=++==

Note that there eight corner planes in a cube. Hence the name octahedral stress.

AssignmentAssignment

Derive the Cauchy stress formula. Hint: Ax=A l, Ay=A m, Az=A n

Verify the that the terms in the 3D characteristic equation used to compute the principal stresses are correct.

Draw a Mohr�s circle diagram properly labeled, find the principal normal and maximum shear stresses, and determine the angle from from the x axis to σ1.σxx=12 ksi, σyy=6 ksi, τxy=4 ksi cw.

Use the Mohr�s circle formulas to compute the principal stresses and compare to those found using the Mohr�s circle graph.

Write the stress components given above as a Cauchy stress matrix. Use MATLAB to compute the principal stresses. Compare the answers to those found using Mohr�s circle. Note that tensor notation is required.

Read chapter 4 � Covers Mohr�s Circle in detail.