Length of Rod given Strain Energy Stored Calculator

Formula Used:

\[ L = \frac{U \times 2 \times A \times E}{P^2} \]

Strain Energy in Rod or Shaft (U):

Joule

Cross Sectional Area of Rod (A):

m²

Modulus of Elasticity of Rod or Shaft (E):

Pascal

Axial Force on Beam (P):

Newton

Length of Rod or Shaft (L):

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1. What is the Length of Rod given Strain Energy Stored Formula?

The Length of Rod given Strain Energy Stored formula calculates the length of a rod or shaft based on the strain energy stored, cross-sectional area, modulus of elasticity, and axial force applied. This formula is derived from Castigliano's theorem in structural mechanics.

2. How Does the Calculator Work?

The calculator uses the formula:

\[ L = \frac{U \times 2 \times A \times E}{P^2} \]

Where:

\( L \) — Length of Rod or Shaft (m)
\( U \) — Strain Energy in Rod or Shaft (Joule)
\( A \) — Cross Sectional Area of Rod (m²)
\( E \) — Modulus of Elasticity of Rod or Shaft (Pascal)
\( P \) — Axial Force on Beam (Newton)

Explanation: The formula calculates the length of a rod or shaft by considering the strain energy stored due to axial loading, material properties, and cross-sectional dimensions.

3. Importance of Length Calculation

Details: Accurate length calculation is crucial for structural analysis, design optimization, and ensuring the structural integrity of rods and shafts under axial loading conditions.

4. Using the Calculator

Tips: Enter strain energy in Joules, cross-sectional area in m², modulus of elasticity in Pascal, and axial force in Newton. All values must be positive and valid.

5. Frequently Asked Questions (FAQ)

Q1: What is strain energy?
A: Strain energy is the energy stored in a material when it is deformed elastically under load. It represents the work done on the material to cause deformation.

Q2: What is modulus of elasticity?
A: Modulus of elasticity (Young's modulus) is a measure of a material's stiffness, defined as the ratio of stress to strain in the elastic deformation region.

Q3: When is this formula applicable?
A: This formula applies to rods and shafts undergoing axial loading with linear elastic material behavior and uniform cross-section.

Q4: What are the limitations of this formula?
A: The formula assumes homogeneous material, constant cross-section, and linear elastic behavior. It may not be accurate for materials with non-linear elasticity or varying cross-sections.

Q5: How is this related to Castigliano's theorem?
A: This formula is derived from Castigliano's theorem, which states that the partial derivative of the strain energy with respect to a force gives the displacement in the direction of that force.