Stress, Strain and Deflection

In this tutorial, you will understand the basic concepts of stress and strain and gain a deeper understanding of when and how they affect the materials they act on. You will see the different types of stresses and laws which govern their relationship with strain.

Contents:

  1. What is Stress?
  2. What is Strain?
  3. The Hooke’s Law
  4. The Young’s Modulus
  5. Shear Stresses and Shear Strain
  6. The Shear Modulus
  7. Stresses due to Bending Moment
  8. Stresses due to Torsional Moment

What is Stress?

Stress is a force per unit area within materials that results from externally applied forces, unequal heating, or persistent deformation and allows for precise description and prediction of elastic, plastic, and fluid behavior in physical sciences and engineering.

  • A body that is exposed to an external load is considered. Internal forces are induced within the body because of the imposed stress. This internal force is referred to as the resistive force, and its direction is opposite that of the imposed load.
  • The generated resistive force prevents the body from deforming. We get a quantity called stress when we take this resisting force on a unit basis. The resistive force created in the body is proportional to the deformation caused within a particular range.
  • Mathematically stresses are given by the following equation
    σ=\(\frac{Resistive\, Force}{Cross Section \,Area}\)
  • Since the unit of the load is N and the unit of cross-section area is m2 or cm2 or mm2, the SI unit of stress is N/m2 or N/cm2 or N/mm2. Generally, in the numerical problems, we use N/ m2 or N/mm2.
  • There are various types of stresses depending on the application of the force. Two of the most common ones seen in axial applications are tensile and compressive forces.
  • Tensile stress occurs when a person’s body is exposed to two equal and opposite pulls. Tensile tension causes the area of the body to expand in length while decreasing in cross-section.
  • When a body is exposed to two equal and opposing pushes, the resulting tension is known as compressive stress. Compressive stress causes the body’s cross-section area to grow while its length decreases.

What is Strain?

We’ve just looked at stress within structural parts so far. When you put stress on anything, it deforms. Consider a rubber band: when you tug on it, it lengthens and stretches. Strain is the ratio between the deformation and the original length, while deformation is the measure of how much an item is stretched.

  • Consider strain in terms of % elongation: how much larger (or smaller) the item is after it has been loaded.
  • Strain is a unitless measurement of how much an item expands or contracts as a result of a load put to it.
  • The Greek symbol epsilon denotes normal strain, which happens when an item elongates in response to a normal tension (i.e. perpendicular to a surface).
  • A positive number indicates tensile strain, whereas a negative value indicates compressive strain.
  • Tensile strain is the strain created in a body as a result of tensile force. The tensile force always causes the body’s length to increase while its cross-section area decreases. Tensile strain is the ratio of the increase in length to the initial length in this situation.
  • Compressive strain is the strain that develops as a result of a compressive force. The body’s dimension shrinks as a result of compressive force. Compressive strain is defined as the ratio of the reduction in body length to the original length.

The Hooke’s Law

Stress and strain are related. Stress and strain are related by a constitutive law, and we can determine their relationship experimentally by measuring how much stress is required to stretch a material.

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  • This measurement can be done using a tensile test. In the simplest case, the more you pull on an object, the more it deforms, and for small values of strain, this relationship is linear. The following figure shows the Stress-Strain graph for a tensile test
    Stress Strain Graph
  • This linear, elastic relationship between stress and strain is known as Hooke’s Law. If you plot stress versus strain, for small strains this graph will be linear, and the slope of the line will be a property of the material known as Young’s Elastic Modulus.
  • The following assumptions are made in the analysis of stress and strain
    • The material is homogeneous.
    • The load is gradually applied.
    • The line of action of force P passes through the geometric axis of the cross-section.
    • The cross-section is uniform.
    • There is no stress concentration.

The Young’s Modulus

Young’s modulus is a measurement of a material’s capacity to resist length changes when subjected to longitudinal tension or compression. Young’s modulus, often known as the modulus of elasticity, is equal to the longitudinal stress divided by the strain.

  • The Young’s Modulus of a substance is an unchangeable basic characteristic of all materials. Temperature and pressure, however, have a role.
  • The Young’s Modulus (or Elastic Modulus) is a measure of a material’s stiffness. To put it another way, it refers to how readily it may be bent or stretched.
  • When the material reaches a particular level of stress, it starts to deform. It’s up to the point where the material’s structure stretches rather than deforms. However, if the material is stressed above this point, the molecules or atoms within begin to distort, irreversibly altering the substance.
  • An Excellent comparison is a rubber band: when you stretch a rubber band, you are extending it rather than deforming it. The rubber band, on the other hand, will begin to degrade or distort if you pull it too hard. When this happens, it usually doesn’t take long for it to fracture.

Shear Stresses and Shear Strain

When the external force acting on a component tends to slide the adjacent planes concerning each other, the resulting stresses on these planes are called direct shear stresses. Shear stresses distort the original right angles. The shear strain () is defined as the change in the right angle of a shear element. The following figure shows an element loaded in pure shear and the resulting strain.

Shear Stress and Shear Strain

Within the elastic limit, the stress-strain relationship is given by the following equation
τ=Gγ
where,
γ = shear strain (radians)
τ = shear stress
G = the constant of proportionality known as shear modulus or modulus of rigidity

The Shear Modulus

Shear modulus, also known as modulus of rigidity, is a measure of a material’s elastic shear stiffness that is defined as the ratio of shear stress to shear strain in materials science. The derived SI unit of shear modulus is the pascal (Pa).

  • The shear modulus is concerned with a solid’s deformation when a force parallel to one of its surfaces is applied while the other face is subjected to an opposing force (such as friction). In the case of a rectangular prism, the item will deform into a parallelepiped.
  • Metals’ shear modulus is typically found to decrease as temperature rises. The shear modulus appears to rise with the applied pressure at high pressures.

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Stresses due to Bending Moment

When a beam is subjected to a combination of tensile stress on one side of the neutral axis and compressive stress on the other. Such a stress distribution can be visualized by bending a thick leather belt. Cracks will appear on the outer surface, while folds will appear on the inside. Therefore, the outside fibers are in tension, while the inside fibers are in compression.
The bending stress is given by
σb = \(\frac{M_by}{I}\)
where,
σb = bending stress at a distance of y from the neutral axis (N/mm2 or MPa)
Mb = applied to bending moment (N-mm)
I = moment of inertia of the cross-section about the neutral axis (mm4)

The bending stress is maximum in a fiber, which is farthest from the neutral axis. The distribution of stresses is linear and the stress is proportional to the distance from the neutral axis.
The equation is based on the following assumptions:

  • The beam is straight with a uniform cross-section.
  • The forces acting on the beam lie in a plane perpendicular to the axis of the beam.
  • The material is homogeneous, isotropic, and obeys Hooke’s law.
  • Plane cross-sections remain plane after bending.

Stresses due to Torsional Moment

A transmission shaft, subjected to an external torque, is shown in the following figure. The internal stresses, which are induced to resist the action of twist, are called torsional shear stresses.

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Torsional Stresses

The Torsional Shear Stress is given by
τ=\(\frac{M_tr}{J}\)
Where,
τ = torsional shear stress at the fiber (N/mm2 or MPa)
Mt = applied torque (N-mm)
r = radial distance of the fiber from the axis of rotation (mm)
J = polar moment of inertia of the cross-section about the axis of rotation (mm4)
The stress is maximum at the outer fiber and zeroes at the axis of rotation. The angle of twist is given by
θ = \(\frac{M_t*L}{JG}\)
Where,
θ = angle of twist (radians)
L = length of the shaft (mm)
These equations are based on the following assumptions

  • The shaft is straight with a circular cross-section.
  • A plane transverse section remains plane after twisting.
  • The material is homogeneous, isotropic, and obeys Hooke’s law.

Key Points to Remember

Here is the list of key points we need to remember about “Stress, Strain, and Deflection”.

  • Stress is a force per unit area within materials that results from externally applied forces
  • Tensile tension causes the area of the body to expand in length while Compressive stress causes its length to decrease.
  • Strain is a unitless measurement of how much an item expands or contracts as a result of a load put to it.
  • The linear, elastic relationship between stress and strain is known as Hooke’s Law.
  • Young’s modulus is a measurement of a material’s capacity to resist length changes when subjected to longitudinal tension or compression.
  • Shear stresses distort the original right angles. The shear strain is defined as the change in the right angle of a shear element.

If you find any mistake above, kindly email to [email protected]

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Manish Bhojasia - Founder & CTO at Sanfoundry
Manish Bhojasia, a technology veteran with 20+ years @ Cisco & Wipro, is Founder and CTO at Sanfoundry. He lives in Bangalore, and focuses on development of Linux Kernel, SAN Technologies, Advanced C, Data Structures & Alogrithms. Stay connected with him at LinkedIn.

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