Equilibrium of Deformable Body

Review – Static EquilibriumIf a body is in static equilibrium under the action appliedIf a body is in static equilibrium under the action applied

external forces, the Newton’s Second Law provides us six scalar equations of equilibrium

0; 0; 0x yF F Fz 0; 0; 0x y zM M M

•The sum of all the forces in x, y, and z coordinate directions are zero

•The sum of moments of all the forces about axes x,y, and z are zero

Considering Static Equilibrium of the element, we have

* Normal and shear stress components acting on opposite p g ppsides of an element must be equal in magnitude and opposite in direction

* Shear stress components satisfy moment equilibrium

; ;xy yx xz zx yz zy ; ;xy yx xz zx yz zy

PRINCIPAL STRESSES

State of stress at a point in a material iscompletely defined by the stresses acting onthe planes of a cubical volume element whose

Often, it is important to determine the state of stress on a plane

the planes of a cubical volume element whoseedges are parallel to coordinate directions.

, p pat some angle to the coordinate axes. Mostly, the state of stress on a plane on which no shear stresses act is important for design purposespurposes.

A plane, where no shear stresses act is called a principal plane and the normal stress that acts on such a plane is called principal stress.

Mohr’s Circle - Its Use and Limitations

Consider a stress state at a point is such that on one of theConsider a stress state at a point is such that on one of thecoordinate plane, there are no shear stresses and only normalstress (may be zero or not) is present. That normal stress is onef th i i l t h i fiof the principal stress as shown in figure.

You can see that on an plane perpendicular to z-axis only sz ( may be tensile, compressive or zero) is present. Then remaining two principal stresses can be determined using biaxial stress field and p c p s esses c be de e ed us g b s ess e d dMohr Circle.

Biaxial stress field in xy-plane can be shown as above.

Using stress transformation equation, the normal stress along axis and the shear stress (from last terms Mech Of solids) can

x xx y axis and the shear stress (from last terms Mech. Of solids) can

be written as.

1 12 i 2

x y

cos2 sin 22 2

1sin 2 cos2

2

x x y x y xy

x y x y xy

(A)

will become principal stress if = 0. Using this condition, the remaining principal stresses can be determined as.

x x y

2 21 1

2 2p x y x y xy

and the orientation of the principal planes with respect to x-axis is given by

22tan 2 xy

x y

The same results can be expressed graphically using Mohr’s Circle. Rearranging the equations (A), we obtaing g q ( ),

1 1cos2 sin 2

2 2

1sin 2 cos2

x x y x y xy

x y x y xy

Squaring the above equations, then adding, gives

2x y x y xy

2 2

2 21 1 2 21 1

2 2x x y x y x y xy

This is the equation of a circle whose center is at d h di i i b

1,0

2 x y 1/ 2

and whose radius is given by

1/ 2221

2 x y xy

Every point on the circle defines the stress state acting on planes at any angle q from the original x or y axis.

For the correct construction of Mohr’s circle, certain rules are followed and a consistenthandling of positive and negative stress is essential, only if proper orientation of planes isdesired. No such concern is required if only the magnitudes of the principal stresses areq y g p psought.

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 tend to cause clockwisethe physical plane is considered positive while negative shear stress tend to cause clockwise rotation.

3.Angle between lines of direction on the Mohr plot are twice the indicated angle on the physical plane.g p y p

The angle ‘2' on the Mohr circle is measured in the same direction as the angle for the orientation of the plane in physical plane.

According definition, the values of corresponding to thepoints D and B (where = 0) are the principal stresses. Theradius of the Mohr’s circle gives the maximum in-plane shearstress

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

max min

1

2 pmax p

Static Failure Theories

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At Point A

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