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.
Introduction to External Loads
Unlock Audio Lesson
Today, we begin by discussing how external loads affect the internal stress state of a body. When a load is applied, like in an external traction t₀, these forces must be balanced within the material.
What type of loads are considered external in this context?
Great question! External loads include any force applied to the surface of a body, such as weight, pressure, or distributed loads over an area. These loads influence the stress state throughout the material.
How do we determine the stress at the boundary due to these loads?
We use traction forces to understand the stress state. The traction can be thought of as the force per unit area acting on the boundary, directly informing us of internal stress distribution.
Can we consider this as a balance of forces?
Exactly! It’s about balancing internal and external forces. We'll see later how this connects to the stress tensor.
To recap, remember that external loads lead to internal stresses that we quantify via traction forces. This balance is essential for understanding structural integrity.
Applying Newton’s Second Law
Unlock Audio Lesson
Now, let’s consider how we can apply Newton’s second law to analyze a small element, or 'pillbox', within our body under load.
What does a 'pillbox' mean in this context?
A 'pillbox' is a tiny, box-like volume we examine to see how forces acting on it translate to internal stresses. By applying Newton's second law, we can express the sum of forces acting on this element.
What are the forces we need to consider?
We account for internal tractions from surrounding elements, the external loads like t₀, and any body forces acting on the volume, such as gravity.
Are these forces all balanced?
Yes! By setting up the equilibrium equation equal to zero, we unravel the relationship that links these applied loads to stress. This is key to our analysis.
In summary, applying Newton's law allows us to relate external forces directly to stress distributions in our body, which forms the foundation for our next steps.
Limit and Traction Forces
Unlock Audio Lesson
Next, let’s focus on how taking limits of our equations helps us derive relationships between externally applied traction and internal stress.
Do we use limits to determine stress as the box shrinks?
Exactly! By shrinking the dimensions of our 'pillbox' while keeping the area constant, we can eliminate terms that become negligible, simplifying our equations.
What do we find at the end of this process?
We arrive at the relationship t₃ = σₑ, which tells us that the internal stress at the boundary must equal the external traction applied at that point.
Does this mean external loads directly inform our stress state?
Yes, it indicates that the body's internal response is governed by these external conditions, allowing us to solve further stress equilibrium equations effectively.
In summary, the key takeaway is that as we shrink our analysis volume, the rules of stress equilibrium directly relate our external loads to internal stresses.
Stress Tensor Symmetry
Unlock Audio Lesson
Finally, let's discuss the symmetry of the stress tensor and why it holds in our analysis.
What does symmetry mean in this context?
Symmetry in the stress tensor means that the shear stress components are equal across the different planes, which is a crucial property.
How does this symmetry relate to external loading?
Good point! As we derive our stress relationships, the consistency of internal forces reflects their equal nature across different faces, leading to symmetrical equations overall.
Does this suggest anything about how we handle stresses in calculations?
Absolutely! Knowing that the stress tensor is symmetric simplifies many of our calculations, as we only need to solve a set of independent equations when finding stress states.
To summarize, remember that the symmetry of the stress tensor is a result of the conditions we established and is vital for efficiently solving stress equilibrium equations.
Introduction & Overview
Read a summary of the section's main ideas. Choose from Basic, Medium, or Detailed.
Quick Overview
Standard
The section delves into the concept of stress within a body subjected to external loads, demonstrating the connection between traction forces and stress matrices using equilibrium conditions. It also highlights how traction applied by external agents informs the stress state at the boundary of the body.
Detailed
Detailed Summary
In this section, we address the relationship between externally applied distributed loads and the stress tensor in solid mechanics. When a body experiences a load distributed over its surface, understanding how this load translates to internal stress is crucial for analyzing structural performance.
- Setting up the Problem: We begin by considering a body with an external load, expressed in terms of traction t₀ on its surface. Through equilibrium equations, we seek to establish how this external load is represented within the body’s internal stress state.
- Newton's Second Law Application: The equilibrium of a small element, or 'pillbox', is examined. The forces acting on this element include internal traction from surrounding material, external loads, and body forces. By applying Newton's second law, we derive a critical relationship pertaining to stress under these conditions.
- Finding Stress Relationships: After taking limits as the dimensions of our element approach zero, we uncover the fundamental connection: the traction on the body at the surface is directly related to the stress tensor components. This leads to the equation t₃ = σₑ, illustrating that the internal stress state at the boundary reflects the applied load, thus acting as boundary conditions for stress equilibrium.
- Symmetry in Stress Matrix: Lastly, the stress tensor must remain symmetric due to equilibrium and consistency in its components. We elaborate on how this property arises, linking it back to external application conditions and boundary constraints.
Audio Book
Dive deep into the subject with an immersive audiobook experience.
Introduction to Externally Applied Load
Unlock Audio Book
Signup and Enroll to the course for listening the Audio Book
Suppose we have an arbitrary body which is clamped at some part of the boundary and a load is applied on some part of the boundary by an external agent. As the load is usually distributed over an area of the boundary of the body, it has the unit of traction which we write as t₀ as shown in Figure 3.
Detailed Explanation
In this section, we start by considering a body that has been fixed or clamped at certain parts, meaning it cannot move. An external load, represented as 't₀', is applied on some parts of its surface. This load does not act at a single point but is spread out over an area, creating what we call 'distributed load'. The significance of understanding this load is that it directly influences the stress that develops in the body.
Examples & Analogies
Imagine a heavy book resting on a table. The weight of the book exerts a downward force on the table surface, which is similar to how an external load acts on the body. Just as the table supports the book's weight, the body we are discussing supports the distributed load applied to it.
Finding Stress at Applied Load Points
Unlock Audio Book
Signup and Enroll to the course for listening the Audio Book
Our aim is to relate the state of stress (at the point where external load is acting) to the applied load itself. If we solve the stress equilibrium equations, we can find stress everywhere in the body. But at points on the boundary where the external load acts, we can find the state of stress partially without solving the stress equilibrium equations.
Detailed Explanation
The goal here is to establish a connection between the external load applied to the body and the internal stresses that develop within it. Normally, to find stress throughout the entire body, one would solve the stress equilibrium equations. However, at the surfaces where external loads are applied, we can directly relate the applied load to the stress present without having to go through the entire equilibrium analysis.
Examples & Analogies
Consider a sponge being pressed down by your hand. Even as you press, you can feel how much stronger the sponge feels at the point of contact. This sensation can be likened to stress; the pressure your hand applies is the external load that directly influences the internal stress of the sponge.
Coordinate System Setup
Unlock Audio Book
Signup and Enroll to the course for listening the Audio Book
Let us set up our coordinate system such that we have e₃ axis along the surface normal as shown in Figure 3. We have drawn the part of the body near the surface. On the top part of this surface, external load t₀ acts. We are looking at an infinitesimal part of the body, so we can safely assume that t₀ is constant on the top surface.
Detailed Explanation
To analyze the stresses correctly, we position our coordinate system with the e₃ axis aligned with the normal to the surface where the load is applied. By doing this, we simplify our analysis. Since the load t₀ is applied to an infinitely small area of the surface, we assume that this load remains consistent across that area when making calculations.
Examples & Analogies
Think of it like stacking books on a table. If you place a single book evenly on the table, the weight from that book is distributed consistently across the surface it touches. Setting up a coordinate system in this way helps us analyze how the weight (or stress) travels through the table (or body) in relation to where the force is applied.
Newton’s Second Law Application
Unlock Audio Book
Signup and Enroll to the course for listening the Audio Book
The total force on this small part of the body will be the vector sum of the traction forces applied from the remaining part of the body, externally applied load, and the body force.
Detailed Explanation
Next, we apply Newton’s second law to this small section or 'pillbox' we are analyzing. According to Newton's second law, the total force acting on an object is equal to its mass times its acceleration. In our case, we sum up all the forces acting on our infinitesimal section, from external loads to internal traction forces.
Examples & Analogies
Imagine holding a small box with weights inside. The total force you feel is a combination of your grip strength and the weight of the contents. Similarly, the total force on our infinitesimal part involves contributions from both outside forces and the material forces acting within it.
Limit Process to Find Stress Relationship
Unlock Audio Book
Signup and Enroll to the course for listening the Audio Book
We now divide both sides of the above equation by Δ∆ and then take the limit as Δ₃ approaches zero.
Detailed Explanation
To derive a meaningful relationship between the applied load and the stress, we continue by simplifying our forces and then conducting a limit process. By dividing by the area elements and shrinking our analysis down to a single point, we enable ourselves to extract information about the stress directly from the external load applied.
Examples & Analogies
Imagine you're squishing a balloon. When you press down on one part, the air inside redistributes. By focusing on smaller areas as you press harder (eventually narrowing down to a single point), you can see how that pressure influences the entire balloon's structure at that point. This limit process helps us connect the narrow area of focus to the broader effects of the external load.
Traction and Stress Relation
Unlock Audio Book
Signup and Enroll to the course for listening the Audio Book
As mentioned earlier, this equation is also used as boundary condition for solving the stress equilibrium equation.
Detailed Explanation
The relationship formed through our analysis shows that the internal forces (traction) that develop in response to an external load are equivalent to the stress at the surface. This is crucial as it serves as a boundary condition when solving complex stress equilibrium equations, providing a concrete foundation to ensure our stress analysis is accurate.
Examples & Analogies
Think of a sports team. The performance of each player impacts the outcome of the game. Here, the external load (like an opponent) directly influences how the internal dynamics (team strategy) evolve. The relationship we found ensures that our stress calculations are grounded in real external conditions, akin to assessing how each player needs to adjust based on the team strategy during a game.
Definitions & Key Concepts
Learn essential terms and foundational ideas that form the basis of the topic.
Key Concepts
-
External Load: The forces acting on the surface of a body that cause internal stresses.
-
Stress Tensor: A matrix representation of stress, with components that describe how forces are distributed internally.
-
Equilibrium Condition: The requirement that the sum of forces on a body is zero, allowing for static analysis.
-
Traction Forces: The forces applied per unit area at the surface of the body, representing external loads.
-
Symmetry of Stress Matrix: A property of the stress tensor indicating that shear forces are balanced across different planes.
Examples & Real-Life Applications
See how the concepts apply in real-world scenarios to understand their practical implications.
Examples
-
If a beam is loaded uniformly along its length, the traction applied at any given point reflects the internal bending stress experienced at that location.
-
In a column under axial loading, the traction at the top surface directly corresponds to the axial stress at that surface, illustrating the relationship between external force and internal stress.
Memory Aids
Use mnemonics, acronyms, or visual cues to help remember key information more easily.
🎵 Rhymes Time
-
Loads on a surface lead stress to resurface, traction's the force per area we traverse.
📖 Fascinating Stories
-
Imagine a cube in space, feeling all kinds of forces acting with grace. Each grain feels the touch of external loads, redistributing stress in tiny roads.
🧠 Other Memory Gems
-
Remember T.E.S. for the concepts: Traction, Equilibrium, and Stress.
🎯 Super Acronyms
T.R.A.C. - Traction Represents Applied Constraints.
Flash Cards
Review key concepts with flashcards.
Glossary of Terms
Review the Definitions for terms.
-
Term: Traction
Definition:
The force per unit area exerted on a surface.
-
Term: Stress Tensor
Definition:
A mathematical representation of stress across different orientations in a material.
-
Term: Equilibrium
Definition:
A state where the sum of forces acting on an object is zero.
-
Term: Pillbox
Definition:
An infinitesimal volume element used to analyze stresses in a solid body.
-
Term: Symmetric Stress Matrix
Definition:
A stress tensor where the shear components are equal across axes, indicating balanced internal forces.