Calculating stress and strain in complex loading scenario

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In summary, calculating stress and strain in complex loading scenarios involves analyzing how materials respond to various forces and moments. This includes understanding the types of stresses (tensile, compressive, shear) and strains (elastic, plastic) that occur under different loading conditions. Engineers utilize principles from mechanics, material science, and mathematics to apply the theory of elasticity, plasticity, and failure criteria. Techniques such as finite element analysis (FEA) may be employed to model and predict material behavior, ensuring the design meets safety and performance standards. Accurate calculations are essential for preventing structural failures and optimizing material use.
  • #1
Bert2000
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Homework Statement
Stress and strain in complex loading scenario
Relevant Equations
Stress = F / A
Strain = change in length / original length
A bar is 100mm long and has a 20mm by 10mm cross section. It is subject the following complex loading a tensile load of 10,000N along its length

a compressive load of 100,000N on its 100mm by 20mm faces a tensile load of 100,000N on its 100mm by 10mm faces
Calculate the stress and strain on each axis
The Young's modulus, E = 200 GPa

The Poisson's ratio, v = 0.28
Stress = F / cross sectional area

Im ok with that.

Strain = change in length / original length which we don’t know.
 
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  • #2
Welcome to PF.

What does your textbook say about how to handle multiple stresses on the same member in calculations like this?
 
  • #3
Welcome, Bert!
The material seems to be steel, which is isotropic.
You have the original unloaded dimensions on each axis.
For the each delta L, you will need to use both, the given Young's modulus and Poisson's ratio.
Consider that, for each axis, the final value of delta L is affected by the longitudinal and perpendicular loads.

Please, see:
https://en.m.wikipedia.org/wiki/Young's_modulus

https://en.m.wikipedia.org/wiki/Poisson's_ratio
 
  • #4
Bert2000 said:
Homework Statement: Stress and strain in complex loading scenario
Relevant Equations: Stress = F / A
Strain = change in length / original length

A bar is 100mm long and has a 20mm by 10mm cross section. It is subject the following complex loading a tensile load of 10,000N along its length

a compressive load of 100,000N on its 100mm by 20mm faces a tensile load of 100,000N on its 100mm by 10mm faces
Calculate the stress and strain on each axis
The Young's modulus, E = 200 GPa

The Poisson's ratio, v = 0.28
Stress = F / cross sectional area

Im ok with that.

Strain = change in length / original length which we don’t know.
You forgot about the stress-strain relationship. For uni-axial loading, $$\sigma=E\epsilon$$ where ##\sigma## is the axial stress and ##\epsilon## is the axial strain. You also neglected to list the relationship between the axial strain, the transverse strain, and Poisson's ratio. What would you get if you only subjected the bar to the 10000 N axial load alone?
 

FAQ: Calculating stress and strain in complex loading scenario

What is stress and strain in the context of complex loading scenarios?

Stress is the internal force per unit area within materials that arises from externally applied forces, temperature changes, or other factors. Strain is the deformation or displacement of material that results from this stress. In complex loading scenarios, these forces and deformations can occur in multiple directions and modes, making the calculations more intricate.

How do you calculate stress in a complex loading scenario?

To calculate stress in a complex loading scenario, you need to determine the internal forces acting on the material. This often involves using tensor calculus to account for multi-axial stresses. The stress tensor, which is a 3x3 matrix, represents the state of stress at a point within the material. Each component of the tensor corresponds to the normal and shear stresses on different planes.

What methods are used to determine strain in complex loading conditions?

Strain in complex loading conditions is typically determined using strain gauges, digital image correlation, or finite element analysis (FEA). Strain gauges measure the deformation directly on the material's surface, while digital image correlation uses optical methods to track changes in the material's shape. FEA simulates the material's response to loading conditions using computational models to predict strain distribution.

How do you account for material properties in stress and strain calculations?

Material properties such as Young's modulus, Poisson's ratio, and yield strength are critical for accurate stress and strain calculations. These properties are incorporated into the constitutive equations that relate stress and strain. For example, Hooke's Law for linear elastic materials uses Young's modulus and Poisson's ratio to relate stress and strain in three dimensions.

What are the common challenges in calculating stress and strain for complex loading scenarios?

Common challenges include accurately modeling the material behavior, especially if it is non-linear or anisotropic, dealing with boundary conditions, and ensuring numerical stability in computational methods like FEA. Additionally, complex geometries and loading conditions can make it difficult to obtain precise measurements and require advanced techniques and tools to resolve.

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