Unit - 2
Compound Stresses and Strains
1. Two Dimensional System:
A two-dimensional Cartesian coordinate system is shaped by two jointly perpendicular axes. The axes intersect on the factor O, that's referred to as the origin. The x-coordinate of the factor is referred to as the abscissa of the factor, and the y-coordinate is referred to as its ordinate
2. Member subjected to normal stresses in two mutually perpendicular direction accompanied by a simple shear stress:
Consider a rectangular element ABCD of uniform cross-sectional area and unit thickness as shown in fig.
It is subjected to two normal stresses 
and 
and a shear stress 
.
Consider a plane BE inclined at an angle 
to face BC as shown in fig.
Horizontal force on face BC = Stress on BE 
Area = 
(BC 
1)
Vertical force on CE = 
(CE 
1)
Vertical force on BC due to,
Shear Stress 
= 
BC 
1
Horizontal force on CE due to shear stress =
CE 
1
From fig.
Resultant normal forces,
= 
(BC
)
+ 
(CE 
1)
+ 
(BC 
+ 
(CE 
1)
…………(i)
= 
= 
+ 
+ 
+ 
= 
+ 
+ 
sin2
……………(ii)
Resultant tangential force on plane BE, along x-axis.
= 
= 
- 
- 
+ 
= 
sin2
- 
Resultant stress, 
= 
Angle of obliquity 
= 
3. Member subjected to direct stresses on two mutually perpendicular plane accompanied with shear stress.
Select the co-ordinate system representing direct stress on x-axis and shear stress on y-axis with origin at O.
Locate point P and Q, OP = 
and OQ = 
Locate point R (
and S (
)
Joint RS intersecting the x-axis at point C
CP = CQ = 
Radius of circle, R = 
The circle intersects abscissa at point A and B shear stress is zero.
1. Stress at a point:
If you deal with a factor as an infinitely small dice, it's far apparent that a dice has six faces, or 3 pairs of planes as faces. It is simplest vital to recollect 3 faces, as the alternative 3 parallel faces are same in natures.
As proven within side the diagram the pressure on every face may be proven in phrases of three separate pressure vectors, and the stresses skilled may be expressed in nine pressure or stress vectors.
2. Stress on a plane:
When stress is carried out to a plane, it could be expressed in phrases of 1 regular pressure (the pressure performing perpendicular to the plane), and shear stresses (the pressure performing parallel to the plane) as formerly mentioned, simplification of the hassle is required. The plane version works properly whilst best thinking about a unique plane. This may be taken one step in addition to expect that the pressure at an unmarried factor desires to be calculated, because the pressure in the course of an extent can vary. To calculate the pressure at a factor we take the simplification that a factor is in reality an infinitely small cube, and that is dealt with within side the subsequent section.
Key takeaway:
We have studied the effect of simple stresses which were either normal or tangential acting on particular plane only.
Most of cases on actual member, more complex condition of stresses are developed.
We study the analytical and graphical method to find the stresses acting on an inclined plane of the member subjected combined stresses.
The plane on which the normal stress or the shear stress reaches their maximum intensity has particularly significant on the materials.
1. Concept Principal Plane:
The plane on which only normal stresses acts and no shear stress is called principal plane.
The plane AB, BC, CD and AD as shown in fig carries only normal stress and no shear stress. These planes are called as Principal planes.
2. Principal stress:
3. Major Principal stress:
4. Minor Principal Stress:
5. Major Principal Plane:
6. Minor Principal Plane:
Key takeaway:
A graphical method of finding of principle stresses normal longitudinal and resultant stresses with the help of circle is called as Mohr’s circle method
In this method direct stress (
) is represent on x axis and shear stress (
) on y axis
Tensile direct stress is taken on positive x- axis and compressive direct stress on negative x- axis
Shear stress produced clockwise moment in the elements is consider positive and anticlockwise moment is negative
Centre of circle is found by 
Where 
are dierect stresses
Radius of circle 
In Mohr’s circle the angle between the planes will be represented by double the angle
For example, when circle draw from center c with radius R it cuts x- axis at point A and B where shear stress is zero
The angle between AOB is equal to 180 degree
But angle between two principal plans is 90 degree
At point A and B since, there is no shear stress OA and OB represents principal stresses
Radius of circle indicates the maximum shear stress which is located at 2
w.r.to x- axis i.e., at 2 x 45 degree = 90 degree
Key takeaway:
Once we circulate into dimensions, matters are a chunk extra complicated. There is a limitless range of in another way orientated cloth strains in a surface. Lines with unique orientations will display unique values of s and γ
The maximum intuitive manner to symbolize stress in 2D is to assume a circle earlier than deformation, and to study its form after deformation. In general (for homogeneous stress) the circle turns into an ellipse - the stress ellipse.
The stress ellipse is the made of a finite stress carried out to a circle of unit radius. It is an ellipse whose radius is proportional to the stretch s in any direction.
A deformed round item has the equal form (all even though not, strictly, the equal size) because the stress ellipse.
Application:
The application of the stress ellipsoid is to be seemed handiest as a convenient way wherein a given structural circumstance is made understandable.
The stress ellipsoid is a strong or a 3-dimensional conception, and its application must be made with that reality in mind Intersecting shear planes in go segment imply simpler upward relief than lateral, a circumstance that can exist at intensity in addition to on the surface
Key takeaway:
A two -dimensional pressure (stress) system is one wherein the stresses at any factor in a frame act within side the same aircraft. Consider a skinny square block of material, abcd, faces of which can be parallel to the xy-plane
Stress within side the Lithosphere – isostatic equilibrium (or Archimedes principle). In the mantle below the lithosphere, stress ought to be equal. Thus, the burden of the overlying rocks ought to be equal. The thick low density crust floats better at the as lithosphere. The continental crust is be neat anxiety due to the fact the pressure at any given intensity in the crust is better than within side the adjoining mantle. The low-density crust needs to unfold out over the mantle.
Thrust Sheets - Shear pressure consequences from the frictional resistance to movement at the fault. Shear pressure is associated with the burden of overburden and the coefficient of friction.
Stress in Two Dimensions - State of pressure in 2D may be defined through ordinary stresses and 1 shear pressure (the 2 shear stresses ought to be equal or the frame might rotate).
Normal and shear pressure on any arbitrary aircraft may be decided from the 2 ordinary stresses (x- and y-axes) and one shear pressure and the attitude among the ordinary to the plane and the x-axis.
Principal axes of pressure are ordinary to planes of 0 shear pressure. The important axes are orthogonal.
Maximum shear pressure is 45° from the important pressure direction. Maximum shear pressure is one 1/2 of the difference of the important stresses.
Principal Strain:
Maximum and minimal normal strain feasible for a particular factor on a structural element
Shear strain is zero on the orientation where major strain occurs.
Principle strain is also known as st. vanant theory
The principal strain causes that the material subjected to conjugate stresses will falls when the maximum principal strain in the reaches the value of strain at yield point in a simple tensile test at materials
Principal axis of Strain
Principal stress axes in a strained material, 3 jointly perpendicular axes (special X, Y, and Z) which can be parallel to the instructions of greatest, intermediate, and least elongation, and which describe the kingdom of stress at any specific point.
The maximum intuitive manner to represent pressure strain in 2D is to assume a circle before deformation and to appearance at its form after deformation. In general (for homogeneous pressure) the circle turns into an ellipse - the pressure ellipse. The pressure or strain ellipse is the fabricated from a finite pressure implemented to a circle of unit radius.
A graphical method of finding of principle stresses normal longitudinal and resultant stresses with the help of circle is called as Mohr’s circle method
In this method direct stress (
) is represent on x axis and shear stress (
) on y axis
Tensile direct stress is taken on positive x- axis and compressive direct stress on negative x- axis
Shear stress produced clockwise moment in the elements is consider positive and anticlockwise moment is negative
Centre of circle is found by 
Where 
is direct stress
Radius of circle 
In Mohr’s circle the angle between the planes will be represented by double the angle
For example, when circle draw from center c with radius R it cuts x- axis at point A and B where shear stress is zero
The angle between AOB is equal to 180 degree
But angle between two principal plans is 90 degree
At point A and B since, there is no shear stress OA and OB represents principal stresses
Radius of circle indicates the maximum shear stress which is located at 2
w.r.to x- axis i.e., at 2 x 45 degree = 90 degree
2. Ellipse of strain:
Once we circulate into dimensions, matters are a chunk extra complicated. There is a limitless range of in another way orientated cloth strains in a surface.
Lines with unique orientations will display unique values of s and γ.
The maximum intuitive manner to symbolize stress in 2D is to assume a circle earlier than deformation, and to study its form after deformation.
In general (for homogeneous stress) the circle turns into an ellipse - the stress ellipse.
The stress ellipse is the made of a finite stress carried out to a circle of unit radius.
It is an ellipse whose radius is proportional to the stretch s in any direction.
A deformed round item has the equal form (all even though not, strictly, the equal size) because the stress ellipse.
Key takeaway:
References:
