Shear Force and Bending Moment
i. Beam: A beam is a structural element that is designed to resist bending and support loads along its length. Beams are commonly used in construction to span between two supports and distribute loads evenly. They can be made of various materials, such as wood, steel, or concrete, and can be solid or hollow in cross-section.
ii. Load: A load is any force, weight, or pressure that is applied to a structure or material. Loads can be classified as either point loads, which are concentrated loads at a single point, or distributed loads, which are loads that are spread out over a certain length or area. Loads can also be classified as live loads, which are loads that are produced by moving people or vehicles, or dead loads, which are loads that are permanent, such as the weight of the structure itself.
iii. Support: A support is a component that provides a stable foundation for a structure and resists the loads applied to it. Supports can be classified as either fixed supports, which do not allow for any movement, or pinned supports, which allow for rotational movement. Supports can also be classified as roller supports, which allow for translational movement, or simple supports, which allow for both rotational and translational movement. The type of support used in a structure depends on the loads and movements that it will be subjected to.
The sign convention of shear force and bending moment is a set of rules used to determine the direction and magnitude of these internal forces in a beam. These conventions are important for understanding the behaviour of a beam under load and for analyzing and designing structures.
Shear force sign convention: Shear force is the internal force that acts perpendicular to the longitudinal axis of a beam and resists the tendency of the beam to shear or deform transversely. The direction of the shear force is determined by the following sign convention:
- If the shear force is acting upwards, it is considered positive.
- If the shear force is acting downwards, it is considered negative.
Bending moment sign convention: Bending moment is the internal force that acts along the longitudinal axis of a beam and resists the tendency of the beam to bend. The direction of the bending moment is determined by the following sign convention:
- If the bending moment is creating compression on the top fibres and tension on the bottom fibres of the beam, it is considered positive.
- If the bending moment is creating tension on the top fibres and compression on the bottom fibres of the beam, it is considered negative.
It is important to note that these sign conventions are arbitrary and have been established for the purpose of analysis and design. By following these conventions consistently, engineers can effectively communicate and analyse the behaviour of beams under load.
The relationship between shear force, bending moment, and load in a beam can be understood through the concept of equilibrium. When a beam is subjected to external loads, it experiences internal forces that resist the deformation caused by the loads. These internal forces include shear force and bending moment.
Shear force and bending moment can be related to the external loads acting on the beam through the principle of equilibrium. The principle of equilibrium states that the sum of all the forces acting on an object must equal zero, and the sum of all the moments acting on an object must also equal zero.
In the case of a beam, the shear force and bending moment can be related to the loads acting on the beam as follows:
- Shear force: The shear force at any cross-section of a beam is equal to the algebraic sum of all the external loads acting to the right and left of that cross-section.
- Bending moment: The bending moment at any cross-section of a beam is equal to the algebraic sum of all the external loads multiplied by their distances from the cross-section.
In other words, the shear force and bending moment can be determined by summing the contributions of all the external loads acting on the beam and considering the distances of the loads from the cross-section of interest. This information can be used to analyze the behaviour of the beam under load and to design beams to withstand the loads they will experience.
The procedure for drawing Shear Force and Bending Moment diagrams involves the following steps:
- Determine the reactions at the supports: The first step is to determine the reactions at the supports using the principle of moments or equilibrium.
- Determine the distributed loads: The next step is to determine the distributed loads acting on the beam.
- Plot the Shear Force Diagram: The Shear Force Diagram (SFD) is plotted by starting from one end of the beam and working towards the other end. The SFD is plotted against the distance along the beam.
- Plot the Bending Moment Diagram: The Bending Moment Diagram (BMD) is plotted by determining the bending moments at various points along the beam. The BMD is plotted against the distance along the beam.
- Label critical points: Critical points on the SFD and BMD such as the points of maximum shear force and maximum bending moment can be identified and labelled.
It is important to note that the SFD and BMD are not independent of each other, but are related through the relation between shear force, bending moment, and load. By analysing the SFD and BMD, information about the behaviour of the beam under loading can be determined, including the maximum shear force, maximum bending moment, and the points at which these occur.
The shear force and bending moment in a cantilever beam can be described and calculated using the principles of mechanics and statics. A cantilever beam is a type of beam that is supported at only one end and is free at the other end. This type of beam is often used in construction, such as in bridges and building structures.
To calculate the shear force and bending moment in a cantilever beam, we need to consider the loads that are applied to the beam and the reactions that are generated at the support. The shear force is defined as the algebraic sum of the forces acting on a cross-section of the beam. The bending moment is defined as the torque or twisting force that acts on the cross-section of the beam.
To calculate the shear force and bending moment, we need to consider the distribution of forces along the length of the beam. This is done by dividing the beam into small segments and calculating the forces acting on each segment. The resulting values are then plotted on a shear force and bending moment diagram, which is a graphical representation of the shear force and bending moment values at each point along the length of the beam.
The procedure for drawing a shear force and bending moment diagram can be described as follows:
- Determine the loads that are applied to the beam, including their magnitude and point of application.
- Determine the reactions at the support, which will equal the sum of the applied loads.
- Divide the beam into small segments and calculate the forces acting on each segment.
Plot the shear force and bending moment values at each point along the length of the beam, using the following formulas:
Shear force: V = ∑ Fy, where Fy is the net force in the y-direction
- Bending moment: M = ∑ Vx, where x is the distance from the cross-section to the line of action of the force
- Label the points of maximum shear force and bending moment, and use these values to determine the design of the beam and the strength of the materials that are used.
By following these steps, we can accurately calculate the shear force and bending moment in a cantilever beam and determine the design and strength of the beam, which is critical for ensuring the safety and stability of the structure.
A good example of a cantilever beam is a balcony. A balcony is supported on one end only, the rest of the beam extends over open space; there is nothing supporting it on the other side. Other examples would be the end of a continuous beam of a high-rise building floor or the cantilevered girders of a bridge segment.
Let’s consider a cantilever beam of length L, fixed at one end and free at the other end. A concentrated load of magnitude P is applied at a distance from the fixed end. The beam has a cross-section of width b and height h.
To find the shear force and bending moment at any point x along the length of the beam, we need to first calculate the reactions at the fixed end.
Since the beam is fixed at one end, the reaction at that end is a moment reaction (M) in the opposite direction of the applied load. Therefore:
M = -Pa
The reaction at the free end is a vertical reaction (V) equal to the magnitude of the applied load:
V = P
A simply supported beam is a type of beam that has supports at both ends and is free to rotate at the supports. It is also known as a simply supported span or a simply supported structure. When a load is applied on a simply supported beam, the beam experiences both shear force and bending moment.
Shear Force: It is the force that acts transversely to the longitudinal axis of the beam. It is responsible for causing the beam to shear or fracture. In a simply supported beam, the shear force is maximum at the supports and decreases to zero at the midpoint of the beam.
Bending Moment: It is the rotational force that acts on the beam and causes it to bend. In a simply supported beam, the bending moment is maximum at the midpoint and decreases to zero at the supports.
The shear force and bending moment diagrams can be determined using the equations of equilibrium. The equations of equilibrium state that the total force acting on the beam must be equal to zero and the total moment about any point must also be equal to zero.
The procedure for drawing shear force and bending moment diagrams for a simply supported beam is as follows:
- Determine the loading on the beam.
- Determine the support reactions.
- Plot the shear force and bending moment diagrams.
- Draw the straight line representing the shear force, and label the maximum and minimum values.
- Draw the curved line representing the bending moment, and label the maximum and minimum values.
It is important to note that the maximum shear force and bending moment values determine the critical points of the beam and the strength of the material must be sufficient to withstand these forces.
Describe Shear force and Bending Moment For an Overhanging Beam and calculation of Point of Contraflexure
Shear Force and Bending Moment in an Overhanging Beam:
Shear force refers to the internal force within a structural element such as a beam that acts perpendicular to the longitudinal axis of the element. In other words, it is a force that tends to cause a structural element to break apart or shear along its length. In a beam, the shear force is a result of the applied loads and the beam’s own weight that are not balanced by the beam’s reactions.
Bending moment refers to the internal moment within a structural element that causes it to bend. It is the torque or rotational force experienced by a beam due to the loads and forces acting upon it. The bending moment is a measure of the resistance of the beam to bending and is proportional to the product of the load and the distance from the beam’s neutral axis.
An overhanging beam is a type of beam that extends beyond its supports on one or both ends. This creates a cantilever structure that is unsupported at one or both ends. The overhanging beam is subject to bending moments and shear forces due to the loads and its own weight.
Calculation of Point of Contraflexure:
The point of contraflexure is the location along a beam where the direction of the bending moment changes from compression to tension. It is the point along the beam where the beam switches from bending in one direction to bending in the other direction. To calculate the point of contraflexure, the bending moments at various points along the beam must be calculated, and the point where the bending moment changes sign is identified as the point of contraflexure.