A bending moment causing concavity upward will be taken as _____ and called as ______ bending moment.
positive, sagging
Bending Moment Conventions
In structural analysis, understanding the sign convention for bending moments is fundamental to correctly analyze how a beam deforms under load. A bending moment is an internal force that causes a beam to curve or bend. The direction of this curvature, or concavity, helps engineers classify the bending moment as either positive or negative.
Understanding Concavity and Bending Moment
The term "concavity" describes the shape a beam takes when it bends.
- Concavity Upward: This occurs when the beam bends in a 'U' shape, like a smile. In this state, the top surface of the beam is typically under compression, while the bottom surface is under tension. This type of bending is associated with a sagging bending moment.
- Concavity Downward: This happens when the beam bends like an inverted 'U' or a frown. Here, the top surface is in tension, and the bottom surface is in compression. This type of bending is associated with a hogging bending moment.
Sagging Bending Moment Explained
A sagging bending moment is the type of bending moment that results in the beam bending with its concavity directed upward. Consider a simple beam supported at both ends with a load applied in the middle; it will bend downwards at the center, forming an upward concave shape.
- Convention: By standard engineering convention, a sagging bending moment is designated as positive.
- Stress Distribution: When a beam experiences a sagging bending moment, its top fibers are subjected to compressive stress, and its bottom fibers are subjected to tensile stress.
Hogging Bending Moment Explained
Conversely, a hogging bending moment causes the beam to bend with its concavity directed downward. An example would be a cantilever beam (fixed at one end, free at the other) with a load applied at its free end, which will cause the beam to curve upward relative to the support, creating a downward concave shape.
- Convention: A hogging bending moment is conventionally taken as negative.
- Stress Distribution: In a beam under a hogging bending moment, the top fibers are in tension, and the bottom fibers are in compression.
Connecting Concavity to Bending Moment Type
The relationship between the concavity of a beam's deformation and the sign/type of its bending moment is summarized below:
| Concavity | Bending Moment Sign | Bending Moment Type | Fiber Stress |
|---|---|---|---|
| Upward | Positive (+) | Sagging | Top: Compression, Bottom: Tension |
| Downward | Negative (-) | Hogging | Top: Tension, Bottom: Compression |
Based on this established convention, if a bending moment causes concavity upward, it means the beam is undergoing a "sagging" deformation. According to the widely accepted sign convention in structural engineering, this type of bending moment is considered positive.
Therefore, for the statement: "A bending moment causing concavity upward will be taken as _____ and called as ______ bending moment."
- Concavity upward indicates a positive bending moment.
- This specific type of bending moment is known as a sagging bending moment.
Thus, the correct terms to fill the blanks are "positive" and "sagging".
Important Questions from Bending Moment
Slope and deflection of a cantilever beam carrying a moment M at the free end is given by:
Which of the following beams is likely to have the point of contraflexure?
The point of contraflexure is the point at which ___________ changes its sign.
The maximum bending moment of the center of laminated spring of span L due to load W is given by-
If a simply supported beam is loaded with point load W at the centre then what is the ratio of bending moment at the support to the bending moment at the centre?
