Professional Courses
Industry-relevant training in Business, Technology, and Design to help professionals and graduates upskill for real-world careers.
Categories
Interactive Games
Fun, engaging games to boost memory, math fluency, typing speed, and English skills—perfect for learners of all ages.
Typing
Memory
Math
English Adventures
Knowledge
Enroll to start learning
You’ve not yet enrolled in this course. Please enroll for free to listen to audio lessons, classroom podcasts and take practice test.
Interactive Audio Lesson
Listen to a student-teacher conversation explaining the topic in a relatable way.
Understanding Internal Work
Unlock Audio Lesson
Today, we are going to discuss internal work. Can anyone tell me what they think internal work in structural analysis refers to?
Is it about the energy stored in materials when they deform?
Exactly! Internal work refers to the strain energy stored in a structural system due to deformation. The first law of thermodynamics states that the work done by external forces is equal to the internal strain energy in an adiabatic system. This can be summarized with the formula W = U.
Can you explain how we calculate this strain energy?
Sure! The strain energy density for an infinitesimal element is given by dU = 1/2 * σ * ε, where σ is stress and ε is strain. Would you like to see how we apply this concept to axial and torsional members?
Calculating Strain Energy in Axial Members
Unlock Audio Lesson
Now, let's focus on axial members. We can calculate total strain energy using the relationship U = ∫(σ²/(2E) dVol). Can someone remind me what E represents?
It's the modulus of elasticity, right?
Correct! Modulus of elasticity helps us understand how much a material will deform under stress. If we apply this calculation to an axial member, we can understand how much internal energy it can store.
And what if we have a variable stress along the length?
Good question! We would integrate along the length of the member to account for the varying stress, just like we did in the previous formula.
Calculating Strain Energy in Torsional Members
Unlock Audio Lesson
Let's now consider torsional members. The strain energy for torsional members is calculated with U = ∫(τ² * Vol/(2G)), where τ is the shear stress. Who can tell me what G stands for?
That's the shear modulus!
Right! As we apply torque to a member, the internal strain energy is stored based on the shear stress and volume. This approach allows us to ensure stability in structural applications.
So, the principles for calculating strain energy are similar but adjusted for the type of stress?
Exactly! The foundational principles remain consistent, but the specifics such as stress type and material properties will guide our calculations.
Significance of Internal Work in Structural Analysis
Unlock Audio Lesson
Now that we've covered the calculations, who can summarize why understanding internal work is crucial for structural engineers?
It's important for predicting how materials will behave under loads and ensuring they don't fail.
Absolutely! By accurately calculating strain energy, engineers can design safer and more efficient structures. Remember, knowledge of these concepts forms the backbone of structural integrity.
Does this apply to all structures?
Yes! Whether it’s bridges or buildings, understanding internal work helps us make informed decisions in design and safety.
Introduction & Overview
Read a summary of the section's main ideas. Choose from Basic, Medium, or Detailed.
Quick Overview
Standard
In the context of structural analysis, internal work refers to the strain energy contained within a structure when subjected to external loads. This section explains how strain energy is calculated for different types of structural members, emphasizing the relationships between stress, strain, and the resulting internal energy.
Detailed
Internal Work
In structural analysis, internal work is concerned with the calculation of strain energy within members of a structure when subjected to loads. The following key concepts are central to understanding internal work:
- Strain Energy Definition: The internal work done on an infinitesimal element of a structure under load is expressed as the strain energy density. It is calculated using the relationship:
$$dU = \frac{\sigma \epsilon}{2}$$
where \(\sigma\) is the stress and \(\epsilon\) is the strain. The total strain energy \(U\) then becomes:
$$U = \int \frac{\sigma \epsilon}{2} \text{dVol}$$
- Application in Axial Members: For axial members subjected to a uniform stress, the total strain energy can be represented as:
$$U = \int \frac{\sigma^2}{2E} dVol$$
where \(E\) is the modulus of elasticity.
- Calculation for Torsional Members: The strain energy for torsional members is similarly calculated, taking into account torsional stresses and the geometric properties of the section:
$$U = \int \frac{\tau^2 \text{Vol}}{2G}$$
where \(\tau\) is the shear stress, and \(G\) is the shear modulus.
Understanding internal work is crucial for accurately analyzing structures and ensuring that they can withstand applied loads without failure.
Youtube Videos
Audio Book
Dive deep into the subject with an immersive audiobook experience.
Strain Energy of an Element
Unlock Audio Book
Signup and Enroll to the course for listening the Audio Book
Considering an infinitesimal element from an arbitrary structure subjected to uniaxial state of stress, the strain energy can be determined with reference to Fig. 9.2. The net force acting on the element while deformation is taking place is P = (σ) dydz. The element will undergo a displacement u = ε dx. Thus, for a linear elastic system, the strain energy density is dU = 1/2 (σ) ε.
Detailed Explanation
In this chunk, we start by analyzing a very small portion of a structure, known as an infinitesimal element. This element is under a one-dimensional stress state. The net force acting on the element can be calculated using the stress (c3) and the dimensions of the element (dydz). When this element is deformed, it experiences a displacement defined by the strain (b5) and the original length (dx). The strain energy density, which is the energy stored in this element per unit volume, is calculated using the relation where the total change in energy is a half-multiple of stress and strain. This lays the foundation for calculating the total strain energy stored in a larger structure by integrating over its volume.
Examples & Analogies
Imagine pulling on a rubber band. The more you stretch it, the more energy is stored in it. Each tiny slice of the rubber band experiences some stress and strain as you pull it. The energy density relates to how much energy each little part of the band can store as you stretch it, similar to how we calculate strain energy in structures.
Total Strain Energy Calculation
Unlock Audio Book
Signup and Enroll to the course for listening the Audio Book
And the total strain energy will thus be
U = ∫ (1/2) σ ε dVol.
Detailed Explanation
The total strain energy (U) in any structural member can be calculated by integrating the strain energy density equation over the entire volume of the material. The factor (1/2) reflects the relationship between stress and strain for elastic materials. This integral accounts for how stress and strain may vary throughout the volume of the member, providing a comprehensive picture of the energy stored in the entire structure.
Examples & Analogies
Think of filling a balloon with air. Just as you need to know how much air (energy) fills the whole volume of the balloon for it to maintain its shape, we need to integrate the energy density across the entire structural member to figure out how much strain energy (or potential energy) is stored within it.
Specific Cases of Strain Energy for Structural Members
Unlock Audio Book
Signup and Enroll to the course for listening the Audio Book
When this relation is applied to various structural members it would yield:
- Axial Members:
U = ∫ (σ^2 / (2E)) dVol
- Torsional Members:
U = ∫ (1/2) (τ) (γ) dVol.
Detailed Explanation
This chunk describes how the general formula for total strain energy can be applied to specific types of structural members, like axial members and torsional members. For axial members, the energy is calculated using stress and Young's modulus (E), accounting for how axial stress varies along the length of the member. For torsional members, shear stress (τ) and shear strain (γ) are used, showing how elements twisting or rotating store energy differently. Each formula captures the essence of how different loads and types of deformation influence the energy within structural elements.
Examples & Analogies
Consider a metal rod being pulled from both ends (axial member) versus a cylindrical can being twisted (torsional member). Each scenario transforms energy differently within the material. Just as you can calculate how much energy is stored in a stretched rubber band versus a twisted piece of paper, we can determine the specific strain energy in each situation based on the types of forces applied to structural members.
Definitions & Key Concepts
Learn essential terms and foundational ideas that form the basis of the topic.
Key Concepts
-
Strain Energy: The energy stored in materials due to stress and strain during deformation.
-
Modulus of Elasticity (E): The constant that describes material's elastic behavior under stress.
-
Shear Modulus (G): The property that describes material's response to shear stress.
Examples & Real-Life Applications
See how the concepts apply in real-world scenarios to understand their practical implications.
Examples
-
When a steel rod is stretched by a force, the internal work done translates into stored strain energy, which can be calculated using U = ∫(σ²/(2E) dVol).
-
In a twisted rod, the strain energy can be calculated with U = ∫(τ² * Vol/(2G)), illustrating how torsion impacts structural integrity.
Memory Aids
Use mnemonics, acronyms, or visual cues to help remember key information more easily.
🎵 Rhymes Time
-
Energy from strain, in loads we gain; Merging stress and strain, to avoid structural pain.
📖 Fascinating Stories
-
Imagine a bridge made of flexible material. As cars pass over it, the bridge stretches, storing energy. This strain energy is crucial for the bridge's design, ensuring it doesn't snap under pressure.
🧠 Other Memory Gems
-
For Axial [A] - U = ∫σ²/[2E], and for Torsional [T] - U = ∫τ²/[2G]. Remember 'A and T' for axial and torsional calculations.
🎯 Super Acronyms
STRENGTH
- Stress
- Torque
- Resilience
- Energy
- New geometry
- Total strain
- Hybrid dynamics.
Flash Cards
Review key concepts with flashcards.
Glossary of Terms
Review the Definitions for terms.
-
Term: Strain Energy
Definition:
The energy stored in a material due to deformation from an applied load.
-
Term: Stress
Definition:
Force per unit area within materials, arising from externally applied forces, uneven heating, or permanent deformation.
-
Term: Strain
Definition:
The deformation experienced by a material in response to applied stress.
-
Term: Modulus of Elasticity (E)
Definition:
A measure of a material's ability to resist deformation under load; a constant specific to each material.
-
Term: Shear Modulus (G)
Definition:
A measure of a material's response to shear stress.