Chapter 1 - Theories of Stress and Strain_2

8/6/2019 Chapter 1 - Theories of Stress and Strain_2

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EME3046

MECHANICS OF MATERIALS

CHAPTER 1

Theories of Stress and Strain

by Low Kean Ong

2009/2010 Session, Trimester 3Faculty Of Engineering And Technology


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Theories of Stress and Strain

1.4 Transformation of stress, Principal stresses andother properties.

Transformation of stress.

Principal stresses.

Octahedral stress.

Plane stress.

Mohrs circle in two dimensions. Mohrs circle in three dimensions.

CHAPTER 1

2EME3046 MECHANICS OF MATERIALS


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Transformation of stress

When referring to stress components, it is often necessary or

convenient to change the coordinate system.

Example:

- to determine stresses in fiber-reinforced composite.

- to analyze the integrity of a weld on a thin-walled pressure

vessel.



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Let (x, y, z) and (X, Y, Z) denote two rectangular coordinate

systems with a common origin.

The angles xX, xY, xZ, are measured from the (x, y, z) axes to

the (X, Y, Z) axes.



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The cosines of the angles between the coordinates are listed as

follows:

For example, l1 = cos xX, l2 = cos xY,



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The direction cosines must satisfy the following relations:



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The stress components XX, XY, XZ,

are defined with reference to the (X, Y, Z)

axes in the same manner as xx, xy, xz,

for (x, y, z).

Recall, the normal stress PN on the plane P

is the projection of the vector P in the

direction of that is:



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The shear component XY is the component of the stress vector

in the Y direction on a plane perpendicular to the X axis. Thus, it

is formulated by the scalar product of vector X with a unit vector

parallel to the Y axis,



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Similarly,

(cont)



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Shear stress in the (X, Y, Z) axes:

(cont)



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Shear stress in the (X, Y, Z) axes:



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Note:

Equations in slide 8 and 12 determine the stress components

relative to axes (X, Y, Z) in terms of the stress components

relative to axes (x, y, z).

That is, they determine how the stress components transform

under a rotation of rectangular axes.



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Principal stresses

In engineering practice, it is important to determine:

- The orientation of the planes that cause the normal stress to be

a maximum and minimum.

- The orientation of the planes that cause the shear stress to be a

maximum.

For any general state of stress at any point in a body, there exist

three mutually perpendicular plane on which the shearing

stresses vanish.

These planes are known as the principal planes and the

corresponding stresses are called the principal stresses.



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Principal stresses

The stress vector on principal planes is given by

where N is the unit normal to a principal plane.

(cont)



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Principal stresses

Equating the components on both sides of the equations:

Or in matrix form:

(cont)



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Principal stresses

Since the equation are linear homogeneous equations in (l, m, n)

and the trivial solution l = m = n = 0 is impossible because,

Thus, from the theory of linear algebraic equations, it is

consistent only and only if the determinant of the coefficients of

(l, m, n) vanishes identically. So, for nontrivial solutions:

(cont)



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Principal stresses

The invariants of stress can be calculated by:



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Principal stresses

The three roots of the equation,

are the principal stresses.

The stress invariants can be expressed in term of principal

stresses:



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Principal stresses

Example:



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Principal stresses

Solution:



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Principal stresses



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Principal stresses

Solve the equation for principal stresses.

since

The principal stresses are



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Principal stresses

The principal directions is found by solving the equation

simultaneously:

and



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Principal stresses

Principal direction for = 1 = 4



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Principal stresses

Solve the four equations simultaneously:



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Principal stresses

Thus, the principal direction corresponding to 1 is:



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Principal stresses

The principal direction corresponding to = 2 = 1 is:

The principal direction corresponding to = 3 = -2 is:



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Principal stresses (Recap - EME1066)



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Orientation and expression for principal stresses.


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Orientation and expression for maximum shearing stresses.


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The orientation between the principal stresses and maximum

shearing plane.


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Octahedral stresses

A plane that is equally inclined to

all three principal axes (X, Y, Z) is

called an octahedral plane.

There are eight such planes

exist.

The normal and shear stress

components associated withthese planes are called the:

- octahedral normal stress, oct.

- octahedral shear stress, oct.



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Octahedral stresses

Let the unit normal vector to the octahedral plane in (X, Y, Z)

axes be:

Then,

Thus, on octahedral planes,

or



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Octahedral stresses

Recall:

In (x, y, z) axes, the stress vector at

point P on an arbitrary plane is:

where



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Octahedral stresses

Thus, in principal axes (X, Y, Z), the stress vector at a point P on

an arbitrary plane is:

where



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Octahedral stresses

The octahedral normal stress is:



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Octahedral stresses

The octahedral shear stress is:



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Octahedral stresses

Since ( ) are invariants under the rotation of axes, oct and

oct may be expressed with respect to (x, y, z) axes.

Normal stress

Shear stress



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Plane stress

Certain approximations may be applied to

simplify three-dimensional stress array.

Consider a thin plane with thickness, h.

Since the plate is not loaded on surfaces,

If the plate is thin (h is small), it can be

assumed throughout the plate thickness that,

This state of stress is known as plane stress.



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Plane stress

Stress components for plane stress:

The remaining stress are assumed to be independent of z:



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Plane stress

With this approximation, it is called the stress tensor for plane

stress:



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Plane stress

Consider a transformation from the (x, y, z) axes to the (X, Y, Z)

axes for the condition that the z axis and the Z axis remain

coincident.



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Plane stress

For such state of plane stress in the (x, y) plane, the direction

cosines between the corresponding axes are:



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Plane stress

Therefore, the normal and shear stress components reduce to:



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Plane stress

Manipulating using trigonometric double angle formulas, the

equations can be expressed as:



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Mohrs circle in two dimensions

The stress equations in Slide

53 can be represent

graphically using Mohrs

circle.

Stress components XX and

XY act on the incline face BE

that is located at positive

(counterclockwise) angle

from face BC on which stress

components xx and xy act.



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Normal stress:



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Shear stress:



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Eq. (1) + Eq. (2):

This is an equation for a circle with

center: (average stress)

radius:



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Mohrs circle:


Principal stress occurwhen = 1 and 1 + /2,

measuredcounterclockwise.


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Magnitudes and direction of the principal stresses:



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Mohrs circle in two dimensions(Recap - EME1066)


Note: Different notation used for shear stress, xy = xy


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Mohrs circle in two dimensions(Recap - EME1066)



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Mohrs circle in three dimensions

The maximum normal stress is equal to the maximum of the

three principal stresses 1, 2 and 3.

Arrange the principal stresses in descending order:

Construct the Mohrs circle using any two of the principal

stresses.



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Stress in principal axes.

Consider a plane P whose unit

normal vector with respect to the

principal axes is:

The stress vector on plane P is:



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Normal stress.



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Shear stress.



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Arranging the principal stresses such that



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The inequalities may be rewritten in the form:



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Or

where



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Example 1



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Example 1



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Example 1



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Example 1



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Example 1



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Example 1


E l


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Example 1


E l 2


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Example 2


E l 2


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Example 2


E l 2


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Example 2


E l 2


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Example 2


E l 2


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Example 2


E l 2


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Example 2


E l 2


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Example 2



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To be continued (1.5) .


REFERENCES


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REFERENCES

F. P. Beer, E. R. Johnston Jr. and J. T. DeWolf, "Mechanics ofMaterials", 6th ed., McGraw-Hill, 2006.

A. P. Boresi and R. J. Schmidt, Advanced Mechanics of

Materials, 6th ed., John Wiley & Sons, Inc., 2003.

R. C. Hibbeler, "Mechanics of Materials", 6th ed., Prentice Hall,2005.

H. T. Toh, Mechanics of Materials lecture notes, MMU, 2008.


Supplement question 1


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Supplement question 1


Solution 1


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Solution 1



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Solution 2


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Solution 2

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Chapter 1 - Theories of Stress and Strain_2