2. Slope and Deflection, Theory of Structures, CE 4003 (Same as CC 4003)

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Subject: Theory of Structures (CE 4003 Same as CC 4003)

Branch: Civil Engineering 🏗️
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2. Slope and Deflection

Slope and deflection are critical concepts in structural analysis as they help in understanding how a beam behaves under loading. Let's break it down step by step:

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2.1 Concept of Slope and Deflection 🌐

• Slope: The slope of a beam refers to the angle of rotation of a point on the beam due to the applied load. It is the rate of change of the angle with respect to the length of the beam.
  • Mathematically, the slope at a point is the derivative of the angle of deflection.
  • The slope helps in understanding how much the beam is bending in response to the load.
• Deflection: Deflection is the displacement of a point on the beam under the applied load. It refers to the amount of vertical displacement the beam experiences as a result of bending.
  • Deflection is important because excessive deflection can lead to structural failure or serviceability issues.
  • It is typically measured at the midpoint or any other critical location along the beam.

Example: When you push a stick at the center, it bends. The angle by which the stick bends is the slope. The distance it moves down from its original position is the deflection.

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2.2 Stiffness of Beams 🏋️‍♂️

• Stiffness of a beam is a measure of its resistance to bending. A stiffer beam will experience less deflection when subjected to a load.

• The stiffness depends on:

  1. Moment of Inertia (I): It’s a geometric property that measures how the beam's cross-section resists bending. Larger I means greater stiffness.
  2. Modulus of Elasticity (E): The material property that defines the material's ability to resist deformation. Higher E means the material is stiffer.

• Formula for Stiffness: $\text{Stiffness} = \frac{EI}{L^3}$ where:

  • $E$ = Modulus of Elasticity
  • $I$ = Moment of Inertia
  • $L$ = Length of the beam

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2.3 Relation among Bending Moment, Slope, Deflection, and Radius of Curvature 🔄

The relationship between these quantities helps us understand beam behavior under load:

• Bending Moment (M): The internal moment that resists the bending of a beam.
• Slope (θ): The angle of rotation at a point on the beam due to bending.
• Deflection (δ): The displacement at a point due to the bending.
• Radius of Curvature (ρ): The radius of the curve formed by the deflected beam.

The relationship is governed by the following equation (without derivation):

$$\frac{d^2y}{dx^2} = \frac{M(x)}{EI}$$

Where:

• $y$ = deflection at a point
• $x$ = position along the beam
• $M(x)$ = bending moment at position $x$
• $E$ = modulus of elasticity
• $I$ = moment of inertia of the beam's cross-section

This means that the rate of change of deflection (i.e., the slope) is directly related to the bending moment.

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2.4 Double Integration Method to Find Slope and Deflection 🧮

The Double Integration Method is a common approach to find the slope and deflection of beams under different types of loads.

2.4.1 Concentrated Load (Point Load) ⚡

When a concentrated load $P$ is applied at a certain point on the beam, the deflection and slope can be found using the double integration method.

• For a cantilever beam with a load $P$ at the free end:
  • Slope at the free end: $$\theta = \frac{P L^2}{3EI}$$
  • Deflection at the free end: $$\delta = \frac{P L^3}{3EI}$$
• For a simply supported beam with a load $P$ at the center:
  • Slope at the supports: $$\theta = \frac{P L}{16EI}$$
  • Deflection at the center: $$\delta = \frac{P L^3}{48EI}$$

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2.4.2 Uniformly Distributed Load (UDL) over Entire Span 📏

When a uniformly distributed load (UDL) $w$ is applied over the entire span of a beam, the following formulas are used:

• For a cantilever beam with a UDL:

  • Slope at the free end: $$\theta = \frac{w L^2}{6EI}$$
  • Deflection at the free end: $$\delta = \frac{w L^4}{8EI}$$

• For a simply supported beam with a UDL:

  • Slope at the supports: $$\theta = \frac{w L^2}{8EI}$$
  • Deflection at the center: $$\delta = \frac{5w L^4}{384EI}$$

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Diagrams

1. Cantilever Beam with Point Load:

   |---------------------|<--- L
   |      P (Point Load) |
   |---------------------|       
   
   Free end deflection and slope occur at the point of load.
   

2. Simply Supported Beam with Uniform Load:

   |----------- L --------|
   |                       |
   |--- w (Uniform Load) ---|
   |-----------------------|
   

   Maximum deflection happens at the center of the beam.

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Example Questions and Answers 📚

1. Q: What is the deflection at the free end of a cantilever beam subjected to a concentrated load $P$?

   • A: The deflection is given by: $$\delta = \frac{P L^3}{3EI}$$ where $L$ is the length, $P$ is the applied load, $E$ is the modulus of elasticity, and $I$ is the moment of inertia.

2. Q: How do you calculate the slope for a simply supported beam with a UDL?

   • A: The slope at the supports is given by: $$\theta = \frac{w L^2}{8EI}$$ where $w$ is the UDL, $L$ is the span, $E$ is the modulus of elasticity, and $I$ is the moment of inertia.

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Revision Points 📌

1. Slope and deflection are key measures of how a beam bends under load.
2. Stiffness is a measure of resistance to bending, and it depends on the beam's material and cross-sectional properties.
3. The relationship between bending moment, slope, deflection, and radius of curvature is important for understanding beam behavior.
4. The Double Integration Method is a powerful tool to calculate slope and deflection for both cantilever and simply supported beams under different loads.
5. Concentrated load and uniformly distributed load (UDL) have different effects on the deflection and slope of beams.
6. Always check the units and consistency of the parameters like $E$, $I$, and $L$ when calculating slope and deflection.

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