Chapter 6 - COMPLEX STRESS.ppt

7/27/2019 Chapter 6 - COMPLEX STRESS.ppt

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CHAPTER 6

COMPLEX

STRESS


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INTRODUCTION

The most general state of stress at apoint may be represented by 6

components,

,,:(Note

stressesshearing,,

stressesnormal,,

xzxzyyzyxxy

zxyzxy

zyx

Same state of stress is represented by adifferent set of components if axes arerotated.

9.1 Plane Stress Transformati

In practice, approximations andsimplificationsare done to reduce the stress components

to a single plane.


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3

Plane Stress

Plane Stress state of stress in which two facesof the cubic element are free of stress. For theillustrated example, the state of stress isdefined by

.0,, andxy

zyzxzyx

State of plane stress occurs in a thin platesubjected to forces acting in the midplane of theplate.

State of plane stress also occurs on the freesurface of a structural element or machinecomponent, i.e., at any point of the surface notsubjected to an external force.



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Plane Stress Transformation

The material is then said to be subjected toplane stress. For general state of plane stress at a pt, werepresent it via normal-stress components, x,

yand shear-stress

component xy.

Transforming stress components from one orientationto the other is similar in concept to how we transform

force components from one system of axes to theother.

Note that for stress-component transformation, weneed to account for- the magnitude and direction of each stress

component, and- the orientation of the area upon which each

component acts. 9.1 Plane Stress Transformati


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Procedure of Analysis

(Method of Equilibrium)

If state of stress at a pt is known for a given orientation of an element ofmaterial, then state of stress for another orientation can be determined.



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Transformation of Plane Stress

consider the stress element has rotated through anangle as shown.

As linear equations, the transformedstresses are given by

145


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Sign Convention

Positive normal stresses, xand y, actsoutward from all faces

Positive shear stress xyacts upward on theright-hand face of the element.

The orientation of the inclined plane isdetermined using the angle .

Establish a positive xand yaxes usingthe right-hand rule.

Angle is positive if it movescounterclockwise from the +xaxis to the

+xaxis.

General Equation


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Transformation of Plane Stress

2cos2sin2

2sin2cos22

2sin2cos22

xy

yx

yx

xy

yxyx

y

xy

yxyx

x

The equations may be rewritten to yield

General Equation

1

2

3


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9

Procedure of Analysis

General Equation

To apply equations -1 and -2, just substitute the knowndata for x, y, xy, andaccording to established signconvention.

Ifxandxy are calculated as positive quantities, thenthese stresses act in the positive direction of the xandyaxes.

Tip: For your convenience, equations -1 to -3 can beprogrammed on your pocket calculator.


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Example 6.1

The state of plane stress at a point is represented by

the element shown in figure. Determine the state ofstress at the point on another element oriented 30clockwise from the position shown.

25 MPa

80 MPa

50 MPa

147

AD

CB


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119.2 General Equation


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Principal Stresses

Principal stresses,1/2 - the maximum and minimum normal

stresses. Principal planes - the planes on which the principalstresses act.

the principal stresses will occur on mutually perpendicular

planes. the shear stresses are zero on the principal planes.

There are two way to determine the principal stresses

(a) Calculation method

(b) Mohrs Circle method

148


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Principal Stresses The previous equations are

combined to yield parametricequations for a circle,

22

222

22

where

xyyxyx

ave

yxavex

R

R

Principal stressesoccur on theprincipalplanes of stresswith zero shearingstresses.

9.3 Principle Stresses and Max. In-Plane Shear


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MaximumShearing Stress

Maximum shearing stressoccurs for avex

Principle Stresses and Max. In-Plane ShearStress


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22

22tan

22

21

212

2

m a x

yx

ave

xy

yx

s

xy

yxR

Mohr's Circle for Plane Stress

a graphical representation of the stress transformationequations

all stresses on Mohr's circle are in-plane stresses

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Steps in constructing Mohr's circle:1. Draw axes. ( positive to the right and

xy positive down)

2. Plot first point on circle. (x , xy )3. Plot second point on circle. (y , - xy)4. Draw a line between the two points.

This line is the diameter of the circle,

and it passes through the center ofthe circle.5. Draw a circle through the two points.

152


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Stress Transformations

Principal Stresses and Maximum Shear Stresses

153


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Example 6.2

The state of plane stress at apoint on a body is shown onthe element in the figure given.Represent this stress in termsof the :(a) principal stresses.(b) maximum in-plane shear

stress and associatedaverage normal stress



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Example 6.3


For the state of plane stressshown, determine (a) theprincipal planes, (b) the

principal stresses, (c) themaximum shearing stress andthe corresponding normalstress.

SOLUTION:

Find the element orientation for theprincipal stresses from

yx

xyp

22tan

Calculate the maximum shearing

stress with2

2

max2

xyyx

2

yx

Determine the principal stressesfrom

22

minmax,22

xyyxyx


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SOLUTION:

Find the element orientation for theprincipal stresses from

1.233,1.532

333.11050

40222tan

p

yx

xyp

6.116,6.26p

Determine the principal stressesfrom

22

22

minmax,

403020

22

xy

yxyx

MPa30

MPa70

min

max

MPa10

MPa40MPa50

x

xyx



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MPa10

MPa40MPa50

x

xyx

2

1050

2

yxave

The corresponding normal

stress is

MPa20

Calculate the maximum shearingstress with

22

22

max

4030

2

xyyx

MPa50max

45 ps

6.71,4.18s



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Exercise 1

The state of plane stress at a point is represented bythe element shown in figure. Represent this state ofstress on an element oriented 30 counterclockwise fromthe position shown.

6 MPa

8 MPa

12 MPa

157

EXAMPLE FOR L SHAPED


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EXAMPLE FOR L SHAPED

QUESTION:

A single horizontal force Pof 600N magnitude is applied to end D of lever

ABD. Determine (a) the normal and shearing stresses on an element at pointHhaving sides parallel to the x and y axes, (b) the principal planes and

principal stresses at the point H.


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SOLUTION:

1. Determine an equivalent force-couple system at the center ofthe transverse section passingthrough H.

2. Evaluate the normal and shearingstresses at H.

3.Determine the principal planes and

calculate the principal stresses.


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x

P 600N

T 600N 0.45m 270 N.m

M 600N 0.25m 150 N.m

Evaluate the normal and shearing stresses at H.

y 414

xy 412

150N.m 0.015mMy

I 0.015m

56.6 MPa

270 N.m 0.015mTc

J 0.015m

50.9 MPa

a)


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x y xy0 56.6 MPa 50.9 MPa

Note that, the shearing force P Does not cause any

shearing stress at point Hb) Determine the principal planes and calculate the principal

stresses. xy

p

x y

p

p

2 2 50.9tan 2 1.8

0 56.6

2 61.0 ,119

30.5 , 59.5


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2

x y x y 2

max, min xy

2

2

max

min

2 2

0 56.6 0 56.6 50.92 2

86.5 MPa

29.9 MPa


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Example 2 (L shape) Refer to word file

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Chapter 6 - COMPLEX STRESS.ppt