Shear Force & Bending Moment

# Shear Force & Bending Moment | Solid Mechanics - Mechanical Engineering PDF Download

## Shear Force and Bending Moment Diagrams

Beam is one of the most important structural components. Beams are usually long, straight, prismatic members and always subjected forces perpendicular to the axis of the beam

• A Shear Force Diagram (SFD) indicates how a force applied perpendicular to the axis (i.e., parallel to cross-section) of a beam is transmitted along the length of that beam.
• A Bending Moment Diagram (BMD) will show how the applied loads to a beam create a moment variation along the length of the beam.

### Types of Supports

1. Roller Support: Resists vertical forces only
2. Hinge support or pin connection: Resists horizontal and vertical forces

Hinge and roller supports are called as simple supports
3. Fixed support or built-in end

The distance between two supports is known as “span”.

### Types of beams

Beams are classified based on the type of supports:

1. Simply supported beam: A beam with two simple supports
2. Cantilever beam: Beam fixed at one end and free at other
3. Overhanging beam
4. Continuous beam: More than two supports

Shear Force
Shear force has a tendency to slide the surface, it acts parallel to surface.
∑Fvert = 0
V - qdx - (V+dV) = 0

Only for distributed load not for point load.

Bending Moment
Any moment produced by forces acting on the beam must be balance by an equal opposite moment produced by internal forces acting in beam at the section. This moment is called bending moment.
∑M = 0
-M - qdx(dx / z) - (V + dV)dx + M + dm = 0
dM/dx = V ⇒ Ma - MA = ∫V dx
Only for distributed and concentrated load not for couple.
The necessary internal forces to keep the segment of the beam in equilibrium are
∑Fx = 0 ⇒ P
∑Fy = 0 ⇒ V
∑Fz = 0 ⇒ M

### Differential equations of equilibrium

Sign Conventions:
[∑Fx = 0 → +]
[∑Fy = 0 ↑ +]
So the differential equations would be:

From equation dV/dx = -P, we can write

From equation dy/dx = -M, we can write
MD - MC = -∫Vdx

Statically Determinate Beam
A beam is said to be statically determinate if all its reaction components can be calculated by applying three conditions of static equilibrium i.e.,

∑ V = 0, ∑ H = 0 and ∑ M = 0

Statically Indeterminate Beam
When the number of unknown reaction components exceeds the static conditions of equilibrium, the beam is said to be statically indeterminate.

The document Shear Force & Bending Moment | Solid Mechanics - Mechanical Engineering is a part of the Mechanical Engineering Course Solid Mechanics.
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## Solid Mechanics

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## FAQs on Shear Force & Bending Moment - Solid Mechanics - Mechanical Engineering

 1. What are shear force and bending moment diagrams?
Ans. Shear force and bending moment diagrams are graphical representations of the internal forces and moments experienced by a beam under various loading conditions. The shear force diagram shows the variation of the shear force along the length of the beam, while the bending moment diagram shows the variation of the bending moment along the same length.
 2. How are shear force and bending moment diagrams useful in structural analysis?
Ans. Shear force and bending moment diagrams are essential tools in structural analysis as they provide valuable information about the internal forces and moments acting on a beam. They help engineers determine the maximum shear force and bending moment at any given section of the beam, which is crucial in designing and dimensioning the beam.
 3. What factors affect the shape of shear force and bending moment diagrams?
Ans. The shape of shear force and bending moment diagrams is influenced by various factors, including the type and magnitude of the applied loads, the beam's support conditions, and the beam's geometry. Different loadings, such as point loads, distributed loads, or moments, will result in different shapes of the diagrams.
 4. How do you construct shear force and bending moment diagrams for a beam?
Ans. To construct shear force and bending moment diagrams, one needs to follow a systematic approach. First, the reactions at the supports are determined using equilibrium equations. Then, the shear force diagram is built by considering the loadings and support conditions. Finally, the bending moment diagram is obtained by integrating the shear force diagram.
 5. What are the key features to look for in shear force and bending moment diagrams?
Ans. When analyzing shear force and bending moment diagrams, engineers should look for key features such as the locations of maximum and minimum shear forces and bending moments. These points indicate critical sections where the beam is most susceptible to failure or deformation. Additionally, engineers should pay attention to sudden changes in the diagrams, which may indicate the presence of concentrated loads or moments.

## Solid Mechanics

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