Civilax4U: Complex stress and strains

Plane stress: A Plane stress system is one in which , the normal stress Z - direction σZ and shear stresses ΤxZ and Τyz must be zero σx,σy, Τxy may have non zero values.

Ex : thin members subjected to loading.

Plane strain : In a plane strain problem, the normal strain in the Z-direction Shear strains y and y must be zero. The normal stains e , , y may have non zero values.

Ex : A retaining wall can be considered as a plane strain problem.

It should not be inferred from the similarities in the definitions of plane stress and plane strain, that both occurred simultaneously.
If an element in plane stress is subjected to equal and opposite normal stresses (i.e. when c =), the normal strain in the z- direction is equal to zero.in this exceptional case plane stress and plane strain occur simultaneously.
If a material has Poisson’s ratio (p. = O) zero, It also will be in a state of both plane stress & plane strain.

Complementary shear stress:

For equilibrium t = t without t the element ill not be in equilibrium.

State of simple Shear of Pure Shear:

Element subjected to shear as shown above. As a result of pure shear, one diagonal (AC) will be subjected to compression and other (BD) tension of same magnitude as that of pure shear stress (t).

Note : Linear strain of diagonal is equal to half the shear Strain i.e. Τ / 2G.

Stresses Induced by Direct Stresses on a Plane inclined at O with the Verticals shown in figure:

Principal Stresses and Principal Planes:

Planes across which only normal stress and no shear stress is acting.
The normal stress across the principal plane is principal stress.
At any point three such planes and three corresponding principal stresses will be there. Only major and minor principal stresses are generally considered and third is neglected for 2 D case.
General case : Let ‘σx’ and ‘σy’ be two normal stresses (both tensile) and Τxy be the shear stress.

Then principal stresses are σ1 = maximum principal stress

σ2 = minor principal stress

Position of principal planes (with respect to vertical plane) are

where α = angle of major principal plane

α1= angle of minor principal plane

Maximum value of shear stress

(for in plane, 2D, stresses only)

For a system wherein three dimensional (plane stress) behavior is more pronounced, such as thin cylinders, shells, boilers.

Whichever is higher.

The planes carrying maximum shear stress will be inclined at 45° to those of principle stresses.

i.e. (α + 45°)&(α +135°)

On the plane of maximum shear stress normal stress will also be acting that is,σ = σ1 + σ2 /2

Mohr’s Circle: Named after Christian Otto Mohr Can be used for stresses, strains, and inertia. The transformation equations for plane stress be represented in graphical form by a plot known as Mohr’s Circle.

This graphical representation is extremely’ useful because it enables you to visualize the relationships between the normal and shear stresses acting on various inclined planes at a point in a stressed body.

Using Mohr’s Circle principal stresses maximum shear stresses and stresses on inclined planes can be

calculated.

Ends of Mohr circle represent principal stresses.
Distance from y-axis to centre of Mohr circle is,σavg = σ1 + σ2 /2
Radius of Mohr circle is maximum shear stress

Stress trajectories: Stress trajectories are the curves that give the directions of principle stresses along a beam.

There are two systems of stress trajectories

Principle tensile stress trajectories
Principle compressive stress trajectories

The above two systems are orthogonal to each other
These trajectories cause the neutral surface at 45 degrees
At the top and bottom surfaces of the beam, where the shear stress is zero, the trajectories becomes either horizontal or vertical.

Note:

The above fig shows stress trajectories for cantilever and simply supported rectangular beams
Thick lines show principle tensile stress trajectories
Dotted lines show principle compressive stress trajectories
The above stress trajectories are for combined flexure and transverse shear only

STRESS CONTOUR: This is a imaginary curve that connects points of equal magnitude of principles stress. Typical stress contour (tensile principle stress for a cantilever beam is shown below)

Analysis of Strain : If σ1,σ2 , and σ3 are three principal stresses and all are alike, the principal strains are

Computation Of Principal Stresses from Principal Strains:

For general case

Strains in an Inclined Direction:

Let εx, = Strain in x — direction,

εy= Strain in y — direction

φxy,= Shear strain relative to ‘x & y planes.

εϴ = Direct strain on inclined plane,

φϴ = Shearing strain along any direction inclined at an angle O with X — direction.

Now

Principal Strains in 2D system : (If Shear Strain φxy = 0)

Note: Greatest shear strain is equal to the difference of major and minor principal strains.

φmax = εmax - εmin

Strain gauges: An electrical-resistance strain gauge is a device for measuring normal strains (E) on the surface of a stressed object. The gauges are small (less than 1/2 inch) made of wires that are bonded to the surface of the object. Each gauge that is stretched or shortened when the object is

strained at the point, changes its electrical resistance. This change in resistance is converted into a measurement of strain. From three measurements it is possible to calculate the strains in any direction. A group of three gauges arranged in a particular pattern is called a strain rosette. Because the rosette is mounted in the surface of the body, where the material is in plane stress, we can use the transformation equations for plane strain to calculate the strains in various directions.