20.1.1 - Flexure
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Understanding Flexure and Bending Moments
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Today, we're going to discuss how beams behave under bending, which is referred to as flexure. Can anyone tell me what happens to a beam when it is subjected to bending moments?
The beam bends, right? But why does it do that?
Exactly! When a beam is subjected to bending moments, it deforms. The internal forces try to maintain equilibrium, causing different strains across its cross-section. We often refer to the neutral axis.
What’s the neutral axis?
The neutral axis is the line in the beam where there's no longitudinal stress during bending. Strains vary based on your distance from this axis.
So, if strains vary, does that mean the stress does too?
Yes! Stress varies linearly as well. What's important is learning how we calculate these values. Remember the equation for bending moment: M = y * dA, where dA is the differential area.
Can you give us a quick example of how we’d use that equation?
Sure! If we know the stress-strain relation of the material, we can compute the moment by integrating over the beam’s cross-section.
To wrap up, today we learned about flexure, bending moments, and the importance of the neutral axis. Any questions before we move on?
Stress Distribution in Beams
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Now that we understand bending moments, let's discuss how stress distributes across the beam when it's bending. How do you think stress looks in a bent beam?
I guess it would be higher at the top and lower at the bottom?
Good guess! Stress distribution varies. In a bent section, the top fibers experience compressive stress while the bottom fibers experience tensile stress.
What other factors affect this distribution?
The shape of the cross-section and the load type play crucial roles. For example, W sections tend to carry shear force effectively through the web, while the flanges resist bending.
So, when calculating, do we consider this distribution?
Absolutely! We typically use the stress-strain diagram rotated by 90 degrees to factor in this distribution when performing calculations.
To conclude, remember that understanding stress distribution is key to analyzing beam behavior effectively. Can anyone summarize what we discussed?
Shear Stress Distribution
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Next, we will cover shear stresses in beams. Does anyone know how shear stress is distributed in a typical beam?
I think there's a maximum shear stress at the web, right?
Exactly, well done! In fact, the average shear stress can be calculated using the formula τ = VQ/(Ib), where V is shear force, Q is the first moment of area, and I is the moment of inertia.
How do we find Q?
Q is found by integrating the area above or below the point of interest multiplied by the distance to the centroid of that area from the neutral axis.
What’s the importance of knowing this shear distribution?
Knowing it helps in designing beams to resist shear forces and aids in preventing failures due to shear. Remember, different cross-sections like rectangular and W sections behave differently under shear loads.
As we finish, keep in mind how crucial it is to understand shear stress distribution in beam design.
Introduction & Overview
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Quick Overview
Standard
Flexure involves the bending of originally straight beams subjected to certain external moments. The section describes how strains and stresses vary within the beam's cross-section, and introduces key calculations necessary for determining the flexural strength of steel beams.
Detailed
Detailed Summary
The section on Flexure focuses on the bending behavior of steel beams. When a beam is subjected to bending moments, it deforms, leading to different strain levels across its cross-section. The neutral axis is critical in this analysis, as it marks the location where there is no longitudinal stress during bending. Strains vary linearly with respect to the distance from this neutral axis, which is the foundation for calculating the bending moment (M).
The bending moment is defined mathematically where the moment M is equal to the product of the distance to the neutral axis and the differential area (dA), allowing the evaluation of the moment if the material's stress-strain relations are known. Furthermore, the formulas governing shear stress distribution are introduced, emphasizing the importance of the web in W sections where most shear force is carried. Additionally, the section indicates the requirement of utilizing certain equations for estimating nominal strengths and outlines potential failure modes related to flexural design.
Audio Book
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Concept of Pure Bending
Chapter 1 of 5
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Chapter Content
Fig.20.1 shows portion of an originally straight beam which has been bent to the radius \( \rho \) by end couples \( M \), thus the segment is subjected to pure bending.
Detailed Explanation
Pure bending occurs when a straight beam is subjected to forces (couples) at its ends, causing it to bend into an arc shape. In this situation, the internal forces distribute uniformly across the cross-section, leading to predictable stress and strain patterns.
Examples & Analogies
Think of a bent paper clip. When you apply a force at both ends, it bends smoothly. Similarly, a beam subjected to bending forces behaves in a controlled manner, allowing engineers to predict how it will respond to loads.
Assumptions of Beam Behavior
Chapter 2 of 5
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Chapter Content
It is assumed that plane cross-sections normal to the length of the unbent beam remain plane after the beam is bent.
Detailed Explanation
This assumption implies that as the beam bends, the shapes of the cross-sections do not distort; they merely move and rotate. This is crucial in structural analysis because it simplifies the calculations related to stress and strain in the beam.
Examples & Analogies
Imagine a stack of pancakes. When one is slightly tilted, the others remain flat and level. Similarly, in a bent beam, each cross-section is still flat like a pancake, even though the beam itself has changed shape.
Strain Distribution in Bending
Chapter 3 of 5
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Chapter Content
Therefore, considering the cross-sections AB and CD a unit distance apart, the similar sectors Oab and bcd give \( y = \frac{\rho (b-c)}{d} \). Where \( y \) is measured from the axis of rotation (neutral axis). Thus strains are proportional to the distance from the neutral axis.
Detailed Explanation
In a beam under bending, the strain experienced at any point varies depending on its distance from the neutral axis, which is the line along the beam where no elongation occurs during bending. The further away from this axis, the greater the strain (and consequently the stress). This relationship is crucial for determining how much load a beam can safely handle.
Examples & Analogies
Think of a rubber band stretched from both ends. The part of the band in the middle (neutral axis) stays at its original length, while the sections on either side get stretched more. Just like the strain in the rubber band increases the further you move from the center.
Stress Distribution Over Cross-Section
Chapter 4 of 5
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Chapter Content
The corresponding variation in stress over the cross-section is given by the stress-strain diagram of the material rotated 90° from the conventional orientation, provided the strain axis \( \epsilon \) is scaled with the distance \( y \).
Detailed Explanation
As strain varies, so does stress. By referencing the stress-strain relationship, engineers can determine how much stress each section of the beam can handle. When the beam bends, the stress distribution across the cross-section becomes non-uniform, with the highest tensile stress on one side and the highest compressive stress on the other.
Examples & Analogies
Consider a tightly held rope. While it’s pulled from both ends, the middle remains unchanged, but the ends experience different levels of stress based on how much they are being jerked or twisted. Similarly, a bent beam exhibits varying stress levels across its section.
Calculate Bending Moment
Chapter 5 of 5
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Chapter Content
The bending moment M is given by \( M = \int y \sigma dA \), where \( dA \) is an differential area a distance \( y \) from the neutral axis. Thus the moment M can be determined if the stress-strain relation is known.
Detailed Explanation
To compute the bending moment, engineers integrate the product of the distance from the neutral axis \( y \) and the stress at that distance \( \sigma \) over the entire cross-sectional area. This equation is fundamental in understanding how external loads applied to the beam translate into internal moments that cause bending.
Examples & Analogies
Think of balancing a seesaw with a child on one end. The moment created is a product of how far the child is from the pivot point (y) and their weight (stress). The farther they are, the greater the moment, just like how a bending load creates moments in the beam.
Key Concepts
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Bending Moment: The measure of internal moment that leads to the bending of a beam.
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Stress Distribution: Varies across the beam's cross-section, crucial for calculating load effects.
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Neutral Axis: A line where there is zero longitudinal stress while bending occurs.
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Shear Force: The force acting parallel to the beam's cross-section that can cause shear stress.
Examples & Applications
Example of a simple beam subjected to a central load illustrating bending moments and strains.
A W-section beam exhibiting shear force calculations and stress distribution across its web and flanges.
Memory Aids
Interactive tools to help you remember key concepts
Rhymes
When bending's in play, stress goes away, at the neutral axis, it’s zero today.
Stories
Picture a beam like a tree bending in the wind. At its center, it feels no pressure, flexing smoothly without a crack.
Memory Tools
Bend Smart! (Bending, Strain, Moment) - Remember the order of analysis: bending leads to strain which results in moment.
Acronyms
BMS - Bending, Moment, Stress. Remember these three key concepts in flexure.
Flash Cards
Glossary
- Flexure
The deformation of a beam caused by bending moments.
- Neutral Axis
The axis in a beam where there is no longitudinal stress during bending.
- Bending Moment
A measure of the internal moment that causes a beam to bend.
- First Moment of Area (Q)
The product of the area above or below a point and the distance to its centroid from the neutral axis.
- Shear Stress (τ)
The stress acting parallel to the cross-section of the beam.
Reference links
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