mechanics of materials lab - university of washington

Jiangyu Li, University of Washington

Lecture 5Stress

Mechanical Behavior of Materials Sec. 6.1-6.5

Jiangyu LiUniversity of Washington

Mechanics of Materials Lab

Force Vectors

• A force, F , is a vector (also called a "1st-order tensor")

• The description of any vector (or any tensor) depends onthe coordinate system used to describe the vector

100 lbf

+x

+y

100 lbf

F = 100 lbf (y-direction)

F = 86.6 lbf (x"-direction) + 50 lbf (y"-direction)

F = 70.7 lbf (x'-direction) + 70.7 lbf (y'-direction)

100 lbf

+x'+y'

45 degs 60 deg

+x"

+y"

Prof. M. E. TuttleUniversity of Washington

2

Normal and Shear Forces

• A "normal" force acts perpindicular to a surface

• A "shear" force acts tangent to a surface

P = Normal ForceV = Shear Force

3


Forces Inclined to a Plane

• Since forces are vectors, a force inclined to a plane can alwaysbe described as a combination of normal and shear forces

P = Normal Force

V = Shear Force

Inclined Force


4

Moments• A moment (also called a "torque" or a "couple") is a force

which tends to cause rotation of a rigid body

• A moment is also vectoral quantity (i.e., a 1st-order tensor)...

M

M

5


Static Equilibrium• A rigid solid body is in "static equilibrium" if it is either

- at rest, or - moves with a constant velocity

• Static equilibrium exists if: Σ F = 0 and Σ M = 0

(BALL ACCELERATES)

50 lbf

50 lbf

(40 lbf)

(30 lbf)

(40 lbf)

(30 lbf)

50 lbf

50 lbf

(40 lbf)

(30 lbf)

(40 lbf)

(30 lbf)

60 lbf

(NO ACCELERATION)

+x

+y


6

Free Body Diagramsand Internal Forces• An imaginary "cut" is made at plane of interest• Apply Σ F = 0 and Σ M = 0 to either half to determine

internal forces

"cut"

F

F

F

R (= F)

F

R (= F)

(or)


7

Free Body Diagramsand Internal Forces• The imaginary cut can be made along an arbitrary plane

• Internal force R can be decomposed to determine the normaland shear forces acting on the arbitrary plane

"cut"

F

F F

R (= F) PV

F 8


Stress:Fundamental Definitions

• Two "types" of stress:normal stress = σ = P/Ashear stress = τ = V/A

where P and V must be uniformly distributed over A

P = Normal Force

V = Shear Force

σ = P/A

τ = V/A

A = Cross-Sectional Area


9

Distribution of Internal Forces• Forces are distributed over the internal plane...they may

or may not be uniformly distributed

"cut"

F

F F

"cut"

M

M M

σ = F/A σ = ?


10

Infinitesimal Elements• A free-body diagram of an "infinitesimal element" is used to

define "stress at a point"

• Forces can be considered "uniform" over the infinitesimallysmall elemental surfaces

+x

+y+z

dx

dydz

11



Stress Element

Labeling Stress Components• Two subscripts are used to identify a stress component, e.g.,

"σxx" or "τxy" (note: for convenience, we sometimes write σx = σxx, or σxy = τxy)

• 1st subscript: identifies element face

• 2nd subscript: identifies "direction" of stress

+x

+y

σxxσxx

τxy

12


Admissable ShearStress States

+x

+y

+x

+y

+x

+y

τxy τxy τxyτxy τxy

ΣF = 0ΣM = 0

(inadmissable)

ΣF = 0ΣM = 0

(inadmissable)ΣF = 0ΣM = 0

(admissable)

τyx

τyx

τyx= τxyIf:

13


Stress Sign Conventions

• The "algebraic sign" of a cube face is positive if the outwardunit normal of the face "points" in a positive coordinatedirection

• A stress component is positive if:

• stress component acts on a positive face and "points" ina positive coordinate direction, or

• stress component acts on a negative face and "points"in a negative coordinate direction.

14

Stress Sign Conventions

15

σxx

σyy

σyy

σxx

τxy

+x

+y

All Stresses Positive

σxx

σyy

σyy

σxx

τxy

+x

+y

σxx and Positive

Negative

σyy

τxy

3-Dimensional Stress States• In the most general case, six independent components of

stress exist "at a point"

F1

F2

F3

F4

M1

M2

+x

+y+z σxx

σyy

σzzτxy

τzyτxz

16


Plane Stress• If all non-zero stress components exist in a single plane, the

(3-D) state of stress is called "plane stress"

+y

+z+x

σyy

σxx

τxy

σxx

σyy

σxx

σyy

σyy

σxx

τxy

+x

+y

17


Uniaxial Stress

• If only one normal stress exists, the (3-D) state of stress iscalled a "uniaxial stress"

+y

+z+x

σyy

σyy

σyy

σyy

+x

+y


16

Free Body Diagram Defines the Coordinate System

"cut"

F

F F

F

F

σyy

+x

+y

+x

+y


17

Free Body Diagram Definesthe Coordinate System

"cut"

F

F F

P V x'

y'

σy'y' τx'y'

F

x'

y'


18

Free Body Diagram Definesthe Coordinate System

"cut"(a plane)

F

x"

y"

F

z"

Py"y"

Vy"x"Vy"z"

y"

x"

F

z"

σy"y"

τy"x"τy"z"

F

21


Stress TransformationsWithin a Plane

• Given stress components in the x-y coordinate system (σxx,σyy, τxy), what are the corresponding stress components inthe x'-y' coordinate system?

+x

+y

σxx

σyy

τxy

? +x'

+y' σx'x'

σy'y' τx'y'

θ


20

Stress Transformations

• Stress components in the x'-y' coordinate system may berelated to stresses in the x-y coordinate system using a freebody diagram and enforcing Σ F = 0

+x

+y

σxx

σyy

τxy

+x'+y'

σx'x'

τx'y' θ

"cut"

σxx

σyy

τxy

ΣFx' = 0

23


Stress TransformationEquations

• By enforcing ΣFx' = 0, ΣFy' = 0, it can be shown:

σ x' x' =σ xx + σ yy

2+

σ xx − σ yy

2cos2θ + τxysin2θ

σ y' y' =σ xx + σ yy

2−

σ xx − σ yy

2cos2θ − τxysin2θ

τx'y' = −σ xx − σ yy

2sin 2θ + τ xy cos2θ


22


Extreme Values

Normal Stress

yx

xy22tan

σστ

θ−

=p

2xy

2yxyx1,2 )

2(

2τ

σσσσσ +

−±

+=

Shear Stress

2xy

2yx2,1 )

2( τ

σστ +

−±=

xy

yx2

2tanτσσ

θs−

−=


Mohr’s Circle

Shear stress tending to rotate the element clockwise are plotted above the axis

2xy

2yx )2

( τσσ

+−

=R


Three-Dimensional Stress

"Stress":Summary of Key Points

• Normal and shear stresses are both defined as a (force/area)

• Six components of stress must be known to specify the stateof stress at a point (stress is a "2nd-order tensor")

• Since stress is a tensoral quantity, numerical values ofindividual stress components depend on the coordinatesystem used to describe the state of stress

• Stress is defined strictly on the basis of static equilibrium;definition is independent of:

material properties strain 27



Morh’s Circle


Mohr’s Circle

©2001 Brooks/Cole, a division of Thomson Learning, Inc. Thomson Learning™ is a trademark used herein under license.


Assignment

• Mechanical behavior of materialsHW 6.1, 6.5

mechanics of materials lab - university of washington

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