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Lecture 1Dr Muhammad Adil Khattak

Plane Stress

Stress Transformation in Plane Stress

Principal Stresses & Maximum Shear Stress

Stress Transformation

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Stress Analysis - Review1. Determine critical point

2. Solve for internal forces at that point (orreduce to cantilever)

3. Solve for stresses at that point

4. Add like stresses, i.e.:

Wtotal = W1 +W2 +W3 + ...

Xtotal = X1 +X2 +X3 + 5. Summarize stresses at that point on a stress

element.

6. May be necessary to use Stress

Transformation or Mohrs circle to get max

Chapter 8

Chapter 9

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Drill Bit Isolator (

Thrust Load = 8,000 to

10,000 lb

Bending Load= 125 lbs)=K*H

Torsion Load= 300 lb-ft

Drill Rod

Chuck

Isolator

Drill Bit

HOT

SPOT!

Recall from Chapter 8, already did steps 1 5:

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Now What????

Solve for stress at a

point using standard

Equations.

Summarize these

stresses on an initial

stress element or

alignedstress element.

Must find MAXIMUM

stresses at that point,may be different then

the applied stresses can

occur at some other

orientation plane or

angle.

Compare

max stresses

to material

allowablesto

determine:

Is it safe,

will it fail???

step 6

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Introduction

We have learned Axially

In Torsion

In bending

These stresses act on cross sections of

the members. Larger stresses can occur on inclined

sections.

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Concrete Beam subject to Bending and Shear

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Ductile material failed in Axial loading

Brittle Material failed in Axial Loading

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Ductile Material failed in Torsion

Brittle Material failed in Torsion

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State of stresses varies in the body

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6Stress Components

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Plane Stress Plane Stress The state of stress

when we analyzed bars in tension

and compression, shafts in torsion,and beams in bending.

Consider a 3 dimensional stresselement

Material is in plane stress in the xy

plane Only the x and y faces of the element

are subjected to stresses

All stresses act parallel to the x and yaxis

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Plane Stress

Normal stress

subscript identifies the face on which thestress acts

Sign Convention

Tension positive

compression negative

xW

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Plane Stress

Shear Stress - Two subscripts

First denotes the face on which the stress acts

Second gives the direction on that face

xyX

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Plane Stress

A2-dimensionalview can depict therelevant stressinformation, fig. 9.1c

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A. Given Plane Stress State: B. What are new stresses at

element rotation ofU??:

Note, positivestress

directionsshown.

Note, positive angle (ccw)

shown.

Sign convention

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Stresses onInclined Planes

First we know Wx, Wy, andXxy,

Consider a new stresselement Located at the same point in

the material as the original

element, but is rotatedabout the z axis

x and y axis rotatedthrough an angle U

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Stresses onInclined Planes

The normal and shear stresses acting

on they new element are:

Using the same subscript designations

and sign conventions described.

Remembering equilibrium, we knowthat:

'''' ,,

yxyxXWW

''''xyyx

XX !

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Stresses onInclined Planes

The stresses in the xyplane can be expressed interms of the stresses onthe xy element by usingequilibrium.

Consider a wedge shapedelement

Inclined face same as the xface of inclined element.

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Stresses onInclined Planes

Construct a FBD showing all theforces acting on the faces The sectioned face is (A.

Then the normal and shearforces can be represented onthe FBD.

Summing forces in the x and ydirections and rememberingtrig identities, we get:

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UX

U

WWWWW

2sin2cos22' xyyxyx

x

!

UXUWW

X 2cos2sin2

''xy

yx

yx

!

UXUWWWWW 2sin2cos22

' xy

yxyx

y

!

(9.1)

(9.2)

(9.3)

1.990 equationinforSubstitute UUU !

0'

!Y

F

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Stresses onInclined Planes

These are called the transformation equationsfor plane stress.

They transfer the stress component form one setof axes to another.

The state of stress remains the same.

Based only on equilibrium, do not depend onmaterial properties or geometry

There are Strain Transformation equations thatare based solely on the geometry of deformation.

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Mohrs Circle for Plane Stress

The transformation equations for plane

stress can be represented in a graphicalformat known as Mohrs circle.

This representation is useful invisualizing the relationships betweennormal and shear stresses acting onvarious inclined planes at a point in astressed body.

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Mohr s CircleRules

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Construction of Mohrs circle 1.Determine the point on the body in which the principal stresses are to be determined. 2. Treating the load cases independently and calculated the stresses for the point

chosen. 3. Choose a set of x-y reference axes and draw a square element centered on the axes. 4.Identify the stresses x, y, and xy = yx and list them with the proper sign. 5.Draw a set of - coordinate axes with being positive to the right and being

positive in the upward direction. Choose an appropriate scale for the each axis. 6.Using the rules on the previous page, plot the stresses on the x face of the element in this coordinate system (point V).Repeat the process for the y face

(point H). 7.Draw a line between the two point V and H. The point where this line crosses the axis establishes the center of the circle.

8.Draw the complete circle. 9. The line from the center of the circle to point V identifies the x axis or reference axis for angle measurements (i.e. = 0). Note: The angle between the reference axis and the axis is equal to 2p.

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