4. ROLLING LOAD AND INFLUENCE LINE, 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 🏗️
Semester: 4th Semester 📚

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4. ROLLING LOAD AND INFLUENCE LINE 🚗

Rolling load refers to a moving load that passes over a structure like a beam. This load can be a single concentrated load or multiple loads that move across the beam. Influence lines are diagrams that help us understand how the loads affect different parts of the beam (like reactions, shear forces, and bending moments) as they move along the structure.

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4.1 Introduction to Influence Line Diagram for Simply Supported Beams 📊

An influence line diagram shows how the value of a quantity (like reaction, shear force, or bending moment) at a particular point in a structure changes as a moving load passes over it.

For a Simply Supported Beam:

A simply supported beam has two supports — one at each end. When a load moves across it, the reaction, shear force, and bending moment at various points on the beam change. The influence line shows how these quantities change as the load moves.

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4.1.1 Reaction ⚖️

The reaction at a support is the force that the support exerts to keep the beam in equilibrium.

• Influence line for reaction: The reaction will change as the moving load passes across the beam. The influence line will be used to find how much reaction force is at each support at any given moment when the load is moving.

Example: If we have a simply supported beam with a concentrated load moving from left to right, the reaction at the left support will increase when the load is near it, and it will decrease when the load moves to the right.

Diagram: Here’s a basic influence line for reaction at the left support:

     |    /        
     |   /        
     |  /        
-----|/--------  (Support A, Left) 
     |
     |  

• The line rises towards the left side, indicating the change in reaction force at support A.

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4.1.2 Shear Force 🔨

Shear force refers to the internal force that resists the sliding or shearing of one part of the beam relative to the other.

• Influence line for shear force: The shear force at any point of the beam changes depending on where the load is placed. The influence line helps to determine how much shear force is at any point when a moving load is applied.

Example: When a concentrated load is at the middle of the beam, the shear force is maximum at the support closest to the load.

Diagram: For a simply supported beam with a moving load, the influence line for shear looks like this:

       /      
      /    
-----/-------- (Support A, Left) ------
     | 
     |

• The shear force varies as the moving load crosses the beam. At each point, the shear force will either be positive or negative.

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4.1.3 Bending Moment 🌀

The bending moment is the internal moment that resists the bending of the beam. It is responsible for the curvature of the beam.

• Influence line for bending moment: The influence line for bending moment shows how the moment changes at a particular point as the moving load moves across the beam.

Example: The bending moment is highest when the load is at the center of the beam. When the load is at the ends, the bending moment will be zero.

Diagram: For a simply supported beam, the bending moment influence line looks like this:

     /\
    /  \
   /    \
--/------\-- (Beam with Moving Load)  

• The diagram shows the variation of bending moment, with the maximum value in the middle of the beam.

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4.2 Drawing of Maximum Bending Moment Diagram (B.M.D.) and Shear Force Diagram (S.F.D.) for Simply Supported Beam for Rolling Loads 📏

We will now look at different cases of rolling loads and how to draw the maximum Bending Moment Diagram (BMD) and Shear Force Diagram (SFD).

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4.2.1 Single Concentrated Load 💪

When a single concentrated load moves across a simply supported beam, we calculate the Shear Force (SFD) and Bending Moment (BMD) at various positions as the load moves.

• Shear Force: The shear force will be zero at the center of the beam and maximum at the supports.
• Bending Moment: The bending moment will be highest at the midpoint of the beam when the load is at the center.

Diagram: Here’s an example of a simply supported beam with a moving concentrated load $P$:

     |        |  
     |        |  
-----P-------- (Beam with Load)
     |        |

• As the load moves across the beam, the SFD and BMD change. The maximum BMD occurs when the load is at the center of the beam.

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4.2.2 Two Point Loads ⚖️⚖️

When there are two point loads moving across a simply supported beam, we calculate the effects of both loads at various points.

• Shear Force: The shear force will change more frequently as the two loads move across the beam.
• Bending Moment: The bending moment will change based on the position of the two loads.

Diagram: Here’s an example of a beam with two point loads $P_1$ and $P_2$:

     |      |      |
-----P1----P2----- (Beam with Two Loads)
     |      |      |

• The shear force and bending moment diagrams will be more complex and will depend on the relative positions of the loads.

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4.2.3 Series of Point Loads 🚚🚚🚚

A series of point loads refers to multiple loads placed at different points along the length of the beam.

• Shear Force: The shear force will vary as each load passes over a point on the beam.
• Bending Moment: The bending moment will change, and the maximum bending moment occurs when a load is near the middle.

Diagram: Here’s an example of a beam with a series of point loads $P_1, P_2, \dots, P_n$:

  |    |    |    |    |
--P1---P2---P3---Pn---- (Beam with Series of Loads)
  |    |    |    |    |

• The shear force and bending moment diagrams will show variations depending on the sequence and position of the loads.

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Revision Points 📚

• Rolling Load: A moving load that affects the reaction, shear force, and bending moment as it passes over a structure.
• Influence Line Diagram: A graph that shows how the reaction, shear force, and bending moment vary as a moving load moves along the beam.
• Reaction: The force exerted by the support to maintain equilibrium.
• Shear Force: The internal force that resists shearing of the beam.
• Bending Moment: The internal force that resists bending of the beam.
• Drawing SFD and BMD: For different types of loads, the shape of the shear force and bending moment diagrams will change. Single concentrated loads, two point loads, and series of point loads all create different patterns.

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Example Questions ❓

1. Question: For a simply supported beam with a moving concentrated load, how does the reaction change as the load moves from left to right?

   • Answer: The reaction at the left support will increase as the load moves towards it and decrease as the load moves away from it. The reaction at the right support behaves in the opposite way.

2. Question: Where does the maximum bending moment occur for a simply supported beam with a single concentrated load?

   • Answer: The maximum bending moment occurs at the midpoint of the beam when the load is at the center.

3. Question: How would the shear force diagram look for a beam with two point loads?

• Answer: The shear force diagram would have changes in slope as the two point loads move across the beam, and the shear force would be higher near the supports.

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