2 Definition of Stress Stress Transformation PDF

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Stress transformation lecture 02 notes...

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Definition of Stress at a point Stress is defined as the force intensity of the internal force acting on specific plane acting on a specific plane area passing through the point. The stress is a vector and has magnitude and direction.

Consider a point P on the plane. A force below.

is acting on the area

as shown in the figure

Generally the force may not be perpendicular to the surface. Resolve this force into two components, one perpendicular to the plane ( ) and along the plane ( ). Then the stress at this point perpendicular to the plane can be defined as

This stress is called Normal stress and denoted by Greek alphabet The stress along the plane due to

.

is defined as

This stress is called shear stress and denoted by Greek alphabet

.

The stresses defined on this plane are not unique, since we can cut the body by passing plane through the point P in infinite directions.

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In order to define a stress at a point uniquely, the stress at a point is defined by considering an infinitesimally small cubic element whose planes are perpendicular to the reference coordinate axes. Normal and shear stresses are defined on all the planes of cubic element. If stresses are required to be determined with respect to any other coordinate system rotated with respect to original reference coordinate axes, these stresses can be obtained by transformation of the stresses. By considering a cubical element whose planes are perpendicular to reference coordinate axes are defined below. The normal and shear stresses at a point are defined by considering an infinitesimal cubic differential element whose faces are perpendicular to the reference coordinate directions as shown in the figure. On a face perpendicular to x-axis, the stress normal to the surface is called normal stress and denoted by Greek alphabet σxx. The first suffix refers to the axis that is perpendicular to the plane. The second suffix refers to the direction of the stress.

Similarly, on a face perpendicular to x-axis two shear stresses that act along the plane of the surface in y-direction and z-direction are defined and denoted by τxy and τxz.. The stresses that act on all other faces of cubic differential elements are defined similarly

Considering Static Equilibrium of the element, we have * Normal and shear stress components acting on opposite sides of an element must be equal in magnitude and opposite in direction * Shear stress components satisfy moment equilibrium

τ xy = τ yx ; τ xz = τ zx ; τ yz = τ zy 2

Note: In the text book only one suffix is used for normal stresses. is normal stress on the cubical element that act on a plane perpendicular to x-axis and so on. From now on we also use single suffix for normal stresses.

Plane Stress or Biaxial Stresses State of stress at a point in a material is completely defined by the stresses acting on all the planes of a cubical volume element whose edges are parallel to coordinate directions. These stresses are referred to as three dimensional stresses. This state of stress however is not often encountered in engineering practice. Most of machine elements involve that one of the surfaces of differential volume element is completely free of stresses or only normal stress exist with zero shear stresses as shown below.

Figure (a) Figure (b) In the figure (a), no stresses are acting on a plane perpendicular to Z-axis. The stresses are shown only on planes perpendicular to x and y axes. This state of stress is called plane stress state. We can represent this stress state on a square lying on x-y plane as shown in Figure (b).

Stress Transformation The state of stress at appoint in the body is same whether the stresses are defined on a cubic element whose planes are perpendicular to x, y and z axes or on a cubical element whose planes are perpendicular to axes rotated with respect to x, y and z axes. Only the magnitudes and directions are different. Knowing the magnitudes of the stresses in x, y and z axes, we can obtain the magnitudes of stresses in the rotated coordinate axes system.

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In order to determine the stresses with respect to rotated coordinate system ( ) consider plane stress system shown in the figure. Section the square element by an inclined plane such that normal to that plane makes angle θ with x-axis and isolate the angular section. The areas of the inclined plane and horizontal and vertical planes are shown in Figure (b). On the horizontal plane Normal stress ( ) in y-direction and shear stress ( ) along the horizontal plane in x-direction are acting. On a vertical plane normal stress ( ) in x-direction and shear stress ( ) in y-direction is acting. On the inclined plane Normal stress act perpendicular to the plane in the and a shear stress act in direction. Corresponding forces that act on all the planes are shown Figure (c)

(b) (a)

(c)

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Now considering the free body diagram (c) and using the force equilibrium equations in direction we obtain the normal stress and as

1 1 σ x + σ y ) + (σ x − σ y ) cos 2θ + τxy sin 2θ ( 2 2 1 τ x′y ′ = − (σ x − σ y )sin 2θ + τ xy cos 2θ 2

σx′ =

The Other normal stress

σ y′ =

can be obtained by substituting θ as θ +90o.

1 1 σ x + σ y ) − (σ x − σ y ) cos 2θ − τ xy sin 2θ ( 2 2

The counter clockwise angular rotation is taken as positive.

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and

PRINCIPAL STRESSES and MAXIMUM IN-PLANE SHEAR STRESS In engineering practice, it is important to determine maximum Principal stress and maximum Shear stress for the design of any component. Principal stress is defined as normal stress acting on a plane over which no shear stress acts. Such a plane is called Principal plane. If the normal to that plane is at an angle θP such that on that plane no shear stress is present, we can obtain the magnitudes of Principal stresses by considering stress transformation equation that we have derived.

Using stress transformation equation, the normal stress can be written as.

along

1 1 σ x + σ y ) + (σ x − σ y ) cos 2θ p + τ xy sin 2θ p ( 2 2 1 τ x′y ′ = − (σ x − σ y )sin 2θ p + τ xy cos 2θ p 2

σ x′ =

axis and the shear stress

(A)

will become principal stress if = 0. By using this condition, we can the orientation of axis for which will become principal stress. The orientation of the principal stress axis with respect to x-axis is given by

tan 2θ p =

τ xy (σ x − σ y ) / 2

Graphically, we can represent the above as below.

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We can found

Substituting the above equations into equation (A), we obtain

Mohr’s Circle For Stresses The stress transformation equation is again shown below.

1 1 σ x + σ y ) + (σ x − σ y ) cos 2θ + τ xy sin 2θ ( 2 2 1 = − (σ x − σ y ) sin 2θ + τ xy cos 2θ 2

σ x′ = τ x′y′

(B)

The same results can be expressed graphically using Mohr’s Circle. Rearranging the equations (B), we obtain

1   1 σ x′ − 2 (σ x + σ y ) = 2 (σ x − σ y ) cos 2θ + τ xy sin 2θ   1 τ x′y ′ = − (σ x −σ y ) sin 2θ +τ xy cos 2θ 2

Squaring the above two equations, then adding, give 2

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1   1  2 2 σ x′ − 2 (σ x + σ y ) + τ x ′y ′ =  2 (σ x − σ y ) + τxy 7

This is the equation of a circle whose center is at 2 1   2   (σ x − σ y ) + τ xy   2 

1/2

1  with radius given by  2 ( σ x + σ y ) ,0   

Every point on the circle defines the stress state acting on planes at any angle θ from the original x or y axis. For the correct construction of Mohr’s circle, certain rules are followed and a consistent handling of positive and negative stress is essential, only if proper orientation of planes is desired. No such concern is required if only the magnitudes of the principal stresses are sought. Although various conventions are in use, we follow the convention given in Hibbler’ Book. 1. Normal stresses are plotted to scale along the abscissa (horizontal axis) with tensile stresses considered positive and compressive stresses negative. 2. Shear stresses are plotted along the ordinate (vertical axis) with positive direction downward to the same scale as used for normal stresses. A shear stress that would tend to cause counter-clock wise rotation of the stress element in the physical plane is considered positive while negative shear stress tends to cause clockwise rotation.

3. Angle between lines of direction on the Mohr plot is twice the indicated angle on the physical plane. The angle ‘2θ' on the Mohr circle is measured in the same direction as the angle θ for the orientation of the plane in physical plane.

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points D and B (where τ = 0) are the principal stresses. The radius of the Mohr’s circle gives the maximum in-plane shear stress Three -Dimensional Mohr’s Plot • •

As mentioned previously, Mohr’s circle can be drawn to determine principal stresses only if one of the three principal stresses is known. Since the known principal stress is also a normal stress, it can be plotted on s axis and circles can be drawn between all the principal stresses as shown.

Then the maximum absolute shear stress is equal to radius of largest Mohr’s circle. Absolute maximum shear stress is given as

τ max =

1 (σ1 − σ3 ) 2

Or

τ max = 9

1 (σ pmax − σ pmin ) 2...