# Strain Energy Formula Derivation | Importance Of Strain Energy (2023)

## Strain Energy Formula

Strain energy is defined as the energy stored in a body due to deformation. The strain energy per unit volume is known as strain energy density and the area under the stress-strain curve towards the point of deformation. When the applied force is released, the whole system returns to its original shape. It is usually denoted by U.

The strain energy formula is given as,

U = Fδ / 2

Where,

δ = compression,

F = force applied.

When stress σ is proportional to strain ϵ, the strain energy formula is given by,

Where,

(Video) Basic Concept of Strain energy with formula derivation | Strain Energy | Strength of Materials

σ = stress

V = volume of body

Regarding Young’s modulus E, the strain energy formula is given as,

U = σ2 / 2E × V.

Where,

σ = stress,

E = Young’s modulus,

(Video) Strain energy and strain energy density introduced

V = volume of body.

### Solved Examples

Example 1

When a force of 1000 N is applied on a body, it gets compressed by 1.2 mm. Determine the strain energy.

Solution:

Given:

Force F = 1000 N,

Compression δ = 1.2 mm

Strain energy formula is given by,

U = Fδ / 2

= 1000 ×1.2×10−3 / 2

(Video) Introduction to Strain Energy - Strain Energy - Strength of Materials

Therefore, U = 0.6 J.

Example 2

A rod of area 90 mm2 has a length of 3 m. Determine the strain energy if the stress of 300 MPa is applied when stretched. Young’s modulus is given as 200 GPa.

Solution:

Given:

Area A = 90 mm2

Length l = 3m

Stress σ = 300 MPa

Young’s modulus E = 200 GPa

Volume V is given by the formula

(Video) Strain Energy & Strain Energy Per Volume

V = area*length

= (90 × 10−6) × 3

V = 270x10−6 m3

The strain energy formula is given as,

U = σ2 / 2E× V

= (300×106)2 / 2 x 200×109 x 270 x 10-6

Therefore, U = 83.3 x 106 J

Therefore, the strain energy of the rod is83.3 x 106 J

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## FAQs

### What is the derivation of formula for strain energy? ›

δ = compression, F = force applied. Regarding Young's modulus E, the strain energy formula is given as, U = σ2 / 2E × V.

What is strain energy answer? ›

Strain energy is a particular form of potential energy which is stored within materials which have been subjected to strain, i.e. to some change in dimension.

What is the formula of strain method? ›

Hint: The strain is the ratio of change in the parameter (length, angle, volume) by the original content (length, angle, volume).

What is strain energy examples? ›

Strain Energy is the energy stored in any substance when it undergoes deformation. This energy can be reversed back into kinetic energy. A perfect example is the stretching of a rubber band. The elongation causes a deformation which results in storing of energy in the rubber band.

What is the derived unit of strain? ›

The unit of strain is it is dimensionless. It is the ratio of the change of length to the original length, hence it is unitless.

What is the derivation and unit of energy? ›

The joule, J, is the derived unit of energy or work. One joule is the work done by a force of one newton exerted over a distance of one meter, i.e. J = N m = kg m2/s2. The watt, W, is the derived unit of power or energy per unit time. One watt is one joule per second, i.e., W = J/s = N m/s = kg m2/s3.

What is strain energy method? ›

What is Strain Energy Method? Strain energy is calculated by the work done by the structure's member to deflect the member under the action of external loads. Hence, this strain energy can be used to calculate the deflection at any point in the member caused due to external loads.

What is strain energy also called? ›

The strain energy is defined as the energy stored in any object which is loaded within its elastic limits. In other words, the strain energy is the energy stored in anybody due to its deformation. The strain energy is also known as Resilience. The unit of strain energy is N-m or Joules.

How do you calculate total strain? ›

The equation for change in length is traditionally rearranged and written in the following form: FA =Y ΔLL0. Δ L L 0 . The ratio of force to area,FA, is defined as stress (measured inN/m2 ), and the ratio of the change in length to length,ΔLL0, Δ L L 0 , is defined as strain (a unitless quantity).

What is true strain formula? ›

where true stress = σ; true strain = ε, n is the n-value (work hardening exponent or strain hardening exponent), and the K-value is the true stress at a true strain value of 1.0 (called the Strength Coefficient).

### Why do we calculate strain? ›

Strain measurement also plays a vital role in Low-Cycle Fatigue testing that is used to determine the durability of materials subject to alternating strains during service (e.g. engine parts). Devices designed to measure strain are referred to as extensometers.

How do you calculate true strain example? ›

true strain, not engineering stress or strain. True stress = (engineering stress) * exp(true strain) = (engineering stress) * (1 + engineering strain) where exp(true strain) is 2.71 raised to the power of (true strain).

What is strain energy measured? ›

Strain energy is measured in joules and is given by the equation; SE = 1/2kd2, where SE is strain energy, k is a spring constant representing a material's ability to store energy on deformation, and d is the distance over which the material has been deformed.

What is the formula of strain energy due to self weight? ›

(viii) Formula to calculate the strain energy, if the applied tension load isgiven: U = P²L / ( 2AE ) Where, P = Applied tensile load. L = Length of the member A = Area of the member E = Young's modulus.

Is strain energy always positive? ›

The strain energy is always positive, due to the square on the force P, regardless of whether the bar is being compressed or elongated.

How do you derive the dimension of strain? ›

Therefore, the strain is dimensionally represented as [M0 L0 T0] = Dimensionless Quantity.

What is the formula for strain and stress? ›

stress = (elastic modulus) × strain. stress = (elastic modulus) × strain. As we can see from dimensional analysis of this relation, the elastic modulus has the same physical unit as stress because strain is dimensionless.

What is the unit of strain answer? ›

Explanation: The strain is defined as the ratio of change in dimension to the original dimension. If 'δl' changes in the length and 'l' is the original length, then strain = δl/l. Hence the unit of Strain is mm/mm.

What is the formula of derived unit? ›

A derived unit is a SI unit of measurement comprised of a combination of the seven base units. Like SI unit of force is the derived unit, newton or N where N=s21×m×kg.

How is energy derived? ›

Where does the energy that makes life possible come from? Humans obtain energy from three classes of fuel molecules: carbohydrates, lipids, and proteins. The potential chemical energy of these molecules is transformed into other forms, such as thermal, kinetic, and other chemical forms.

### What is the formula of energy method? ›

Energy U = Fx/2 = F2/2k where F is the applied force, x is the distance moved in the direction of the force at its point of application and k is the elastic stiffness of the part, again in the direction of the force at its point of application.

How do you find the strain energy from a graph? ›

We can derive the strain energy density (ρe) in a material by calculating the area under its stress - strain graph. The definition of the density of energy is analogous to the definition of the density of mass. It is the energy stored per unit volume (how many joules are stored in 1m3 of the material).

What is the partial derivative of strain energy? ›

The partial derivative of the strain energy, considered as a function of the applied forces (and moments) acting on a linear elastic structure, with respect to one of these forces (or moments), is equal to the displacement (or rotation angle) in the direction of the force (or moment) of its point of application.

How is strain defined? ›

Strain is the deformation of a material from stress. It is simply a ratio of the change in length to the original length. Deformations that are applied perpendicular to the cross section are normal strains, while deformations applied parallel to the cross section are shear strains.

What are the three types of strain? ›

The four types of strain are longitudinal strain, lateral strain, volumetric strain and shear strain.

What is strain energy due to bending? ›

Strain energy due to bending is: U = ∫ M 2 2 E I d x. 2.

How do you calculate change in strain? ›

The equation for change in length is traditionally rearranged and written in the following form: FA =Y ΔLL0. Δ L L 0 . The ratio of force to area, FA, is defined as stress (measured in N/m2), and the ratio of the change in length to length,ΔLL0, Δ L L 0 , is defined as strain (a unitless quantity).

How do you calculate strain distribution? ›

The basic equation establishes relations among the variables, namely deformation, elastic force, geometrical stiffness and modulus of elasticity of material. The deformation is proportional to the mean value of the distributed elastic force and inverse to the geometrical stiffness and modulus of elasticity, D = F/ER.

How do you calculate strain energy release rate? ›

The units of G are J/m2. The quantity Gc is the fracture energy, and is considered to be material properties that are independent of the applied loads and the geometry of the body. where E is Young's modulus, and E = E′ for the plane stress, and E ′ = E / ( 1 − v 2 ) the plane strain.

How do you calculate strain percentage? ›

Percent Elongation - The strain at fracture in tension, expressed as a percentage = ((final gage length – initial gage length)/ initial gage length) x 100. Percent elongation is a measure of ductility.

### Is strain energy the same as kinetic energy? ›

Kinetic Energy is energy due to motion, Potential Energy is energy due to position while Strain Energy is stored elastic energy (elastic potential energy).

What does strain value mean? ›

R-value (also known as “Plastic Strain Ratio”) is a measurement of the drawability of a sheet metal. Simply put, it measures the resistance of a material to thinning or thickening when put tension or compression.

What is the formula for strain energy for simply supported beam? ›

For a simply supported rectangular beam loaded, with single central load, The strain energy resulting from the bending moments is [l2 /h2]/3 times that due to traverse shear loading. For a typical beam of l/h ratio = 10 the bending shear energy is 33 times the traverse force shear energy.

How do you calculate natural strain? ›

Engineering strain is the amount that a material deforms per unit length in a tensile test. Also known as nominal strain. True strain equals the natural log of the quotient of current length over the original length.
...
True stress: σt =F/A.
σ =F/A0Engineering Stress
ε =δ/L0Engineering Strain
εt = ln (L/L0)True Strain
1 more row

What is the formula of elastic potential energy in terms of stress and strain? ›

Elastic Potential Energy Stored in a wire under stress

ΔU=2LAYl2 (This is the energy stored in the wire). All the work done on the wire is stored as its potential energy, thus there is no heat produced during elongation.

Which of the following statement is correct for strain energy? ›

Which of the following statement is correct? The energy stored in a body, when strained within elastic limit is known as strain energy. The maximum strain energy which can be stored in a body is termed as proof resilience.

Can strain energy be negative? ›

In a condition of static equilibrium, U = W, and the minimum value of Π is −U (or −W), which is the negative of the strain energy (and the work done) on the solid.

How do you know if a strain is positive or negative? ›

In some applications, the change (decrease) in volume or in length for compression is taken to be negative, whereas the change (increase) for dilation or tension is designated as positive. Compressive strains, by this convention, are negative, and tensile strains are positive.

How is the strain dimension derived? ›

Therefore, the strain is dimensionally represented as [M0 L0 T0] = Dimensionless Quantity.

How do you derive work energy theorem? ›

The work W done by the net force on a particle equals the change in the particle's kinetic energy KE: W=ΔKE=12mv2f−12mv2i. The work-energy theorem can be derived from Newton's second law. Work transfers energy from one place to another or one form to another.

### How is energy dimensional formula derived? ›

Or, E = [M] × [L1 T-1]2 = M1 L2 T-2. Therefore, energy is dimensionally represented as M1 L2 T-2.

What is strain definition formula? ›

Ans: Strain is defined as a change in the shape or size of a body caused by a deforming force. It is given by the formula. ε = Change in dimension/Original dimension = Δx/x.

What are the derived dimensions? ›

The dimensions of all other quantities may be found to be combinations of quantities expressible in terms of the basic or primary dimensions. These are known as derived or secondary dimensions. For example, area may be represented as a length times a length or L2.

What is the dimensional formula of strain energy? ›

Strain energy is the energy stored in a body as a result of deformation. It is represented by the symbol U. It's unit of measurement is J. The dimensional formula of strain energy is given by [M1L2T-2].

What is derivation in work-energy and power? ›

Work Energy Theorem Derivation

s = displacement of the object. This is the derivation of the Work-Energy Theorem. Thus, we can say that the work done on an object is equal to the change in the kinetic energy of the object.

How do you solve work-energy theorem problems? ›

Work done on an object transfers energy to the object. The translational kinetic energy of an object of mass m moving at speed v is KE=12mv2 KE = 1 2 m v 2 . The work-energy theorem states that the net work Wnet on a system changes its kinetic energy, Wnet=12mv2−12mv20 W net = 1 2 m v 2 − 1 2 m v 0 2 .

What are the formulas for work and energy? ›

Overview of Work, Energy and Power
What is Work, Energy and Power?
Work
FormulaThe energy stored in an object due to its position and height is known as potential energy and is given by the formula: P.E. = mgh
UnitThe SI unit of energy is Joules (J).
Power
8 more rows

How do you derive energy balance? ›

The complete energy equation looks like this:
1. Energy balance = energy input – energy output.
2. Weight gain = energy input > energy output.
3. Weight loss = energy input < energy output.
Oct 24, 2017

What is the derivation of energy density? ›

= [M1 L2 T-2] × [M0 L3 T0]-1 = [M1 L-1 T-2]. Therefore, the energy density is dimensionally represented as [M1 L-1 T-2].

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