Stress-Strain -Mechanical Engineering

 Stress and Strain

Stress

The term stress

 (σ) is used to express the loading in terms of force applied to a certain cross-sectional area of an object. From the perspective of loading, stress

 is the applied force or system of forces that tends to deform a body. From the perspective of what is happening within a material, stress

 is the internal distribution of forces within a body that balance and react to the loads applied to it. The stress

 distribution may or may not be uniform, depending on the nature of the loading condition. For example, a bar loaded in pure tension will essentially have a uniform tensile

 stress

 distribution. However, a bar loaded in bending will have a stress

 distribution that changes with distance perpendicular to the normal axis.

Simplifying assumptions are often used to represent stress

 as a vector

 quantity for many engineering calculations and for material property determination. The word "vector" typically refers to a quantity that has a "magnitude" and a "direction". For example, the stress

 in an axially loaded bar is simply equal to the applied force divided by the bar's cross-sectional area.

Some common measurements of stress

 are:
Psi

 = lbs/in2 (pounds per square inch)
ksi or kpsi = kilopounds/in2 (one thousand or 103 pounds per square inch)
Pa = N/m 2 (Pascals or Newtons per square meter)
kPa = Kilopascals (one thousand or 103 Newtons per square meter)
GPa = Gigapascals (one million or 106 Newtons per square meter)
*Any metric prefix can be added in front of psi

 or Pa to indicate the multiplication factor

It must be noted that the stresses in most 2-D or 3-D solids are actually more complex and need be defined more methodically. The internal force acting on a small area of a plane can be resolved into three components: one normal to the plane and two parallel to the plane. The normal force component divided by the area gives the normal stress

 (σ), and parallel force components divided by the area give the shear

 stress

 (τ). These stresses are average stresses as the area is finite, but when the area is allowed to approach zero, the stresses become stresses at a point. Since stresses are defined in relation to the plane that passes through the point under consideration, and the number of such planes is infinite, there appear an infinite set of stresses at a point. Fortunately, it can be proven that the stresses on any plane can be computed from the stresses on three orthogonal planes passing through the point. As each plane has three stresses, the stress

 tensor has nine stress

 components, which completely describe the state of stress

 at a point.

Strain

Strain

 is the response of a system to an applied stress

. When a material is loaded with a force, it produces a stress

, which then causes a material to deform. Engineering strain

 is defined as the amount of deformation in the direction of the applied force divided by the initial length of the material. This results in a unitless number, although it is often left in the unsimplified form, such as inches per inch or meters per meter. For example, the strain

 in a bar that is being stretched in tension is the amount of elongation

 or change in length divided by its original length. As in the case

 of stress

, the strain

 distribution may or may not be uniform in a complex structural element, depending on the nature of the loading condition.

If the stress

 is small, the material may only strain

 a small amount and the material will return to its original size after the stress

 is released. This is called elastic deformation

, because like elastic

 it returns to its unstressed state. Elastic deformation

 only occurs in a material when stresses are lower than a critical stress

 called the yield strength

. If a material is loaded beyond it elastic limit

, the material will remain in a deformed condition after the load is removed. This is called plastic deformation

.

Engineering and True Stress and Strain

The discussion above focused on engineering stress

 and strain

, which use the fixed, undeformed cross-sectional area in the calculations. True stress

 and strain

 measures account for changes in cross-sectional area by using the instantaneous values for the area. The engineering stress-strain curve does not give a true indication

 of the deformation characteristics of a metal

 because it is based entirely on the original dimensions of the specimen, and these dimensions change continuously during the testing used to generate the data.

Engineering stress

 and strain

 data is commonly used because it is easier to generate the data and the tensile

 properties are adequate for engineering calculations. When considering the stress-strain curves in the next section, however, it should be understood that metals and other materials continues to strain-harden until they fracture

 and the stress

 required to produce further deformation also increase.

Stress Concentration

When an axial

 load is applied to a piece of material with a uniform cross-section, the norm al stress

 will be uniformly distributed over the cross-section. However, if a hole is drilled in the material, the stress

 distribution will no longer be uniform. Since the material that has been removed from the hole is no longer available to carry any load, the load must be redistributed over the remaining material. It is not redistributed evenly over the entire remaining cross-sectional area but instead will be redistributed in an uneven pattern that is highest at the edges of the hole as shown in the image. This phenomenon is known as stress concentration

.

Mechanical Engineering- Loading conditions

Loading Conditions

The application of a force to an object is known as loading. Materials can be subjected to many different loading scenarios and a material’s performance is dependent on the loading conditions. There are five fundamental loading conditions; tension, compression

, bending, shear

, and torsion

. Tension is the type of loading in which the two sections of material on either side of a plane tend to be pulled apart or elongated. Compression

 is the reverse of tensile

 loading and involves pressing the material together.  Loading by bending involves applying a load in a manner that causes a material to curve and results in compressing the material on one side and stretching it on the other.  Shear

 involves applying a load parallel to a plane which caused the material on one side of the plane to want to slide across the material on the other side of the plane. Torsion

 is the application of a force that causes twisting in a material.

If a material is subjected to a constant force, it is called static loading. If the loading of the material is not constant but instead fluctuates, it is called dynamic

 or cyclic loading. The way a material is loaded greatly affects its mechanical properties

 and largely determines how, or if, a component will fail; and whether it will show warning signs before failure actually occurs.

Mohr's circle

 

Before start with Mohr's circle we will go through some basics , and importance to study Mohr's circle.

STRESS IN 1-D SYSTEM

When stress acts in one direction  its called 1-D stress system,

ex-

 we know that, Stress=resisting forcecross−section area

 *Stress will be maximum when   cross-section  area would be minimum (for constant resisting force)

*cross-section area will be minimum in vertical plane only , so we can conclude that material will fail  at vertical plane when the applied load exceed the ultimate tensile/shear/compressive strength of material.

STRESS IN 2-D SYSTEM 

When stress acts in more than one direction its called 2-D stress system.

ex-

*In 1-D stress system its easy to say from which plane material will fail, but in the case of 2-D stress system it's difficult to say from which plane material will fail, so we have to find out the plane in which normal and shear stress will be maximum.

COMPLEX STRESS

When stress acts in more than one direction and its difficult to say from which plane material will fail.

PRINCIPAL STRESS

Principal stresses are defined as the maximum and minimum normal stresses that acts on a body.

PRINCIPAL PLANE

Plane at which principal stresses acts.There are 2 principal plane

1-MAJOR PRINCIPAL PLANE

Plane at which normal stress is maximum.

2-MINOR PRINCIPAL PLANE

Plane at which normal stress is minimum.

SHEAR PLANE

Plane at which maximum shear stress acts.

*There are 2 methods to determine principal stress, principal plane, max shear stress and shear plane.

1-ANALYTICAL METHOD

2-GRAPHICAL METHOD 

Mohr's is based on the second method.

MOHR'S CIRCLE

*It is a graphical way to determine principal stress,principal plane,shear stress and shear plane.

*We can also calculate shear stress and normal stress for a particular plane.

*We can also calculate the angle at which principal plane will occur with respect to the vertical or horizontal plane.

NOTES

*Any point on mohr's circle will have 2 co-ordinates,normal stress and shear stress.

*Radius  mo mohr's circle represent plane in the material also radius represents the max shear stress value.

*All the angle taken in the material will be double in mohr's circle.

*Principal plane are separated by an angle of 90 degree.

*Shear stress is zero at principal plane.

*Normal stress is not zero at shear plane.

*There are 2 shear plane and shear plane are separated by an angle of 90 degree.

*Angle between principal plane and shear plane is 45 degree.

We discuss all these notes in analytical method in another blog.

WE WILL DRAW MOHR'S CIRCLE FOR THE GIVEN PROBLEM:

*Plot a graph taking normal stress in x-axis and shear stress in y-axis.

*Mark the value of normal stress and shear stress in X  and Y axis respectively.

sign convention 

*tensile stress is taken as positive where as compressive stress is taken as negative.

*Shear stress which  tends generate a clock wise moment taken as positive, and shear stress which tends to generate  anti clock wise moment is taken as negative.

*Join the 2 points  and named the intersection point as p.

*Draw the circle taking the join line as diameter and point p as center.

*In vertical place sigma x and tao x,y are  acting similarly  in horizontal plane sigma y and tao x,y are acting. As per the notes each radius of mohr's circle represent a plane.

* Find in which plane  in mohr's circle  sigma x and tao x,y are  acting  and name them as  vertical plane(v.p) and  same way name the horizontal plane(h.p).And name all points.

*XA- represent  magnitude of major principal stress.

*XB- represent  magnitude of minor principal stress.

*PA- represent major principal plane.

*PB-represent major principal plane.

*XY-represent magnitude of maximum shear stress.

*PY-represent major shear plane.

*PT-represent vertical plane.

*PG-represent horizontal plane.

*XF-represent magnitude of normal stress at angle 2 ፀ from  v.p.

*XE-represent magnitude of shear stress at angle 2 ፀ from  v.p.

In this way we can draw mohr's circle for various case and can determine various parameters.