3. Shear Force and Bending Moment, CE 3003 notes in English, Mechanics of Materials notes in English

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3. Shear Force and Bending Moment

3.1 Types of Supports, Beams, and Loads

Types of Supports:

1. Pinned Support (or Hinge Support): It can resist both vertical and horizontal forces, but it cannot resist any moment. It allows rotation and provides a reaction in both vertical and horizontal directions.

   • Reaction forces: $R_y$ (Vertical), $R_x$ (Horizontal)

2. Roller Support: It can resist only a vertical force and allows free horizontal movement. It prevents vertical displacement but not horizontal displacement.

   • Reaction force: $R_y$ (Vertical)

3. Fixed Support: It resists both vertical and horizontal forces, as well as moments. A fixed support prevents both translation and rotation of the beam at the point of support.

   • Reaction forces: $R_x$, $R_y$ (Vertical and Horizontal), and a Moment $M$

Types of Beams:

1. Cantilever Beam: A beam that is fixed at one end and free at the other. The beam can bend under the applied load.
2. Simply Supported Beam: A beam supported at both ends (usually with a pinned and roller support).
3. Overhanging Beam: A beam that extends beyond its supports on one or both sides.
4. Continuous Beam: A beam that is supported at more than two points.

Types of Loads:

1. Point Load (Concentrated Load): A load applied at a specific point along the length of the beam.
2. Uniformly Distributed Load (UDL): A load that is distributed uniformly over a certain length of the beam.
3. Uniformly Varying Load (UVL): A load that varies linearly along the length of the beam.
4. Couple (or Moment Load): A pair of equal and opposite forces acting at a distance from each other, creating a moment without producing any resultant force.

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3.2 Concept and Definition of Shear Force and Bending Moment

Shear Force (V):

Shear Force at a section of a beam is the internal force that acts perpendicular to the beam’s axis, resisting the applied external loads. It is defined as the algebraic sum of the external forces (both upward and downward) to the left or right of the section.

• Shear Force Equation:

$$V = \text{Sum of forces to the left (or right)} \text{ of the section}$$

Bending Moment (M):

Bending Moment at a section of a beam is the internal moment that resists the bending caused by external loads. It is the algebraic sum of the moments about the section of all external forces to the left or right of the section.

• Bending Moment Equation:

$$M = \text{Sum of moments to the left (or right)} \text{ of the section}$$

3.3 Relation Between Load, Shear Force, and Bending Moment (with Derivation)

The relationship between load (w), shear force (V), and bending moment (M) in a beam can be derived using basic principles of equilibrium.

1. Shear Force and Load: The rate of change of the shear force is equal to the load applied on the beam:

   $$\frac{dV}{dx} = w(x)$$

   where:

   • $w(x)$ is the load per unit length at a given point.

2. Bending Moment and Shear Force: The rate of change of the bending moment is equal to the shear force at that section:

   $$\frac{dM}{dx} = V(x)$$

   where:

   • $V(x)$ is the shear force at a point.

   Integrating the equations gives the relationship:

   $$M(x) = \int V(x) dx$$ $$V(x) = \int w(x) dx$$

These relationships allow for the construction of shear force and bending moment diagrams for various loading conditions.

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3.4 Shear Force and Bending Moment Diagrams

3.4.1 Cantilever Beam (Subjected to Point Load)

Consider a cantilever beam subjected to a point load $P$ at the free end.

• Shear Force Diagram (SFD):

  • The shear force is constant along the beam length and equal to the applied load $P$ at the free end.
  • The diagram is a horizontal line at $P$ from the fixed support to the free end.

• Bending Moment Diagram (BMD):

  • The bending moment increases linearly from the fixed support to the free end.
  • The maximum bending moment occurs at the fixed support and is given by: $$M_{\text{max}} = P \times L$$ where $L$ is the length of the beam.

3.4.2 Simply Supported Beam (Subjected to Point Load)

Consider a simply supported beam with a point load $P$ applied at the center.

• Shear Force Diagram (SFD):

  • The shear force is constant on each side of the point load.
  • The diagram shows two constant values: $R_A = P/2$ at the left support and $R_B = P/2$ at the right support.

• Bending Moment Diagram (BMD):

  • The bending moment increases linearly from the left support to the point of load application and then decreases linearly to the right support.
  • The maximum bending moment occurs at the center of the beam: $$M_{\text{max}} = \frac{P L}{4}$$

3.4.3 Beam with Uniformly Distributed Load (UDL)

Consider a simply supported beam with a uniformly distributed load $w$ per unit length.

• Shear Force Diagram (SFD):

  • The shear force varies linearly along the length of the beam, starting at $R_A$ at the left support and ending at $R_B$ at the right support.

• Bending Moment Diagram (BMD):

  • The bending moment is parabolic, with the maximum moment at the center of the beam.

  For a UDL of $w$ over the entire length $L$, the maximum bending moment is:

  $$M_{\text{max}} = \frac{w L^2}{8}$$

3.4.4 Combination of Point Load and UDL

When a beam is subjected to both a point load and a uniformly distributed load, the shear force and bending moment diagrams are a combination of the diagrams for each individual load.

• Shear Force Diagram (SFD):

  • The shear force is calculated by adding the contributions from both the point load and the UDL.

• Bending Moment Diagram (BMD):

  • The bending moment is calculated by summing the moments from both the point load and the UDL. The diagram will reflect the contributions from both loads.

3.4.5 Point of Contra Flexure

The point of contra flexure is the point on a beam where the bending moment changes sign, i.e., where the bending moment becomes zero. It occurs in certain cases, such as when a beam has multiple loads and moments, and is typically found between points of support or loading.

To find the point of contra flexure, set the bending moment equation equal to zero and solve for the location along the beam.

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Summary of Shear Force and Bending Moment Relationships:

• Shear Force (V): Represents the internal resistance to shearing forces, typically changes abruptly with applied point loads.
• Bending Moment (M): Represents the internal resistance to bending, changes gradually with applied loads and shows a maximum value at certain points.

Diagrams (for Cantilever and Simply Supported Beams):

• Shear Force Diagram (SFD): Horizontal or linear plots showing internal shear forces along the beam.
• Bending Moment Diagram (BMD): Parabolic or linear curves showing the internal bending moments.