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The vertical shear at C in the figure shown in previous section (also shown to the right) is taken as
VC = (ΣFv )L = R1 − wx
where R1 = R2 = wL/2
The moment at C is
If we differentiate M with respect to x:
thus,
| dM/dx = V |
Thus, the rate of change of the bending moment with respect to x is equal to the shearing force, or the slope of the moment diagram at the given point is the shear at that point.
Differentiate V with respect to x gives
dv/dx = 0-w
thus,
| dV/dx= Load |
Thus, the rate of change of the shearing force with respect to x is equal to the load or the slope of the shear diagram at a given point equals the load at that point.
Properties of Shear and Moment Diagrams
The following are some important properties of shear and moment diagrams:
- The area of the shear diagram to the left or to the right of the section is equal to the moment at that section.
- The slope of the moment diagram at a given point is the shear at that point.
- The slope of the shear diagram at a given point equals the load at that point.
- The maximum moment occurs at the point of zero shears. This is in reference to property number 2, that when the shear (also the slope of the moment diagram) is zero, the tangent drawn to the moment diagram is horizontal.
- When the shear diagram is increasing, the moment diagram is concave upward.
- When the shear diagram is decreasing, the moment diagram is concave downward.
Sign Convention
The customary sign conventions for shearing force and bending moment are represented by the figures below. A force that tends to bend the beam downward is said to produce a positive bending moment. A force that tends to shear the left portion of the beam upward with respect to the right portion is said to produce a positive shearing force.
An easier way of determining the sign of the bending moment at any section is that upward forces always cause positive bending moments regardless of whether they act to the left or to the right of the exploratory section.
Instruction:
Without writing shear and moment equations, draw the shear and moment diagrams for the beams specified in the following problems.
Give numerical values at all change of loading positions and at all points of zero shear. (Note to instructor: Problems 403 to 420 may also be assigned for solution by semi-graphical method describes in this article.)
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FAQs on Relationships between Distributed Load and Shear and Moment - Additional Study Material for Mechanical Engineering
| 1. What is a distributed load and how does it affect shear and moment in a structure? |  |
Ans. A distributed load refers to a load that is spread over a length or area, such as a uniform load acting on a beam. This load affects the shear and moment in a structure by creating internal forces and bending moments along the length of the structure. The magnitude and distribution of the load determine how the shear and moment vary along the structure.
| 2. How can I calculate the shear force at a specific point due to a distributed load? | |
Ans. To calculate the shear force at a specific point due to a distributed load, you need to determine the area of the load distribution that is located before the point of interest. Multiply this area by the intensity of the load, and the result will give you the shear force at that point. If the load distribution is not uniform, you may need to integrate the load intensity over the relevant section of the structure.
| 3. Is there a relationship between the slope of a shear diagram and the magnitude of a distributed load? | |
Ans. Yes, there is a relationship between the slope of a shear diagram and the magnitude of a distributed load. The slope of the shear diagram at any point is directly proportional to the intensity of the distributed load at that point. Therefore, a higher magnitude of the distributed load will result in a steeper slope of the shear diagram, indicating a higher rate of change in shear force.
| 4. How does a distributed load affect the bending moment in a beam? | |
Ans. A distributed load affects the bending moment in a beam by creating internal moments that cause the beam to bend. The magnitude and distribution of the load determine the shape of the bending moment diagram. A higher intensity or concentrated load at a specific location will result in a higher magnitude of bending moment at that point, while a uniform distributed load will create a gradually increasing or decreasing bending moment along the length of the beam.
| 5. Can a distributed load cause a negative moment in a structure? | |
Ans. Yes, a distributed load can cause a negative moment in a structure. A negative moment refers to a bending moment that induces compressive stresses on the top side of a beam or structure. Depending on the distribution and magnitude of the load, certain sections of the structure may experience negative moments. These negative moments can be critical in the design and analysis of the structure, as they affect its overall stability and strength.
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