Short Answer:
Bending stress is the internal stress developed in a material when it is subjected to a bending moment. It occurs due to the bending of a beam or structural member under transverse loading. The bending stress is responsible for causing compression on one side of the neutral axis and tension on the other side.
In simple words, when a beam bends under load, the upper layers are compressed while the lower layers are stretched. The stress developed due to this bending action is known as bending stress. It is an important factor in designing beams and shafts to prevent failure or permanent deformation.
Detailed Explanation :
Bending Stress
When a beam or any structural member is subjected to a transverse load (a load acting perpendicular to its axis), it tends to bend. During bending, some fibers (or layers) of the material are stretched, while others are compressed. This phenomenon produces internal stresses within the material, known as bending stresses.
These stresses resist the external bending moment and help the beam return to its original shape after the removal of the load, as long as the stress remains within the elastic limit of the material. The bending stress acts normal (perpendicular) to the cross-sectional area of the beam and varies linearly from the top to the bottom surface.
Concept of Bending in Beams
When a beam bends under an external load:
- The top fibers (layers above the neutral axis) undergo compression, shortening in length.
- The bottom fibers (layers below the neutral axis) undergo tension, elongating in length.
- Between these two regions lies a layer that does not change its length — this layer is known as the neutral axis.
The stress at the neutral axis is zero, and the bending stress increases linearly as the distance from the neutral axis increases. The maximum stress occurs at the outermost fibers (topmost and bottommost layers).
This distribution of stress follows a linear variation because, under elastic bending, strain is directly proportional to distance from the neutral axis.
Assumptions in Theory of Pure Bending
To calculate bending stress accurately, certain assumptions are made, which form the theory of simple bending or pure bending. These are:
- The material of the beam is homogeneous and isotropic (same properties in all directions).
- The beam is initially straight and has a uniform cross-section.
- The material obeys Hooke’s law (stress is proportional to strain within the elastic limit).
- Plane sections before bending remain plane even after bending.
- The radius of curvature is large compared to the beam depth.
- The beam is subjected to pure bending moment without shear force.
These assumptions ensure a simple and linear relationship between stress, strain, and bending moment.
Derivation of Bending Stress Formula
Consider a beam subjected to a bending moment (M) that causes curvature.
Let:
- = distance of a fiber from the neutral axis,
- = bending stress at that fiber,
- = modulus of elasticity of the material,
- = radius of curvature of the neutral layer,
- = moment of inertia of the beam section about the neutral axis.
According to the bending theory,
Also, from the bending moment relation,
By equating the two equations:
This is the bending equation, also called the flexural formula, and it represents the relationship between bending stress, bending moment, and beam geometry.
Where,
- = bending stress (N/mm²),
- = bending moment (N·mm),
- = distance from the neutral axis (mm),
- = moment of inertia (mm⁴).
Meaning of the Bending Equation
- Linear Variation of Stress:
The bending stress increases linearly with distance from the neutral axis. - Maximum Bending Stress:
The maximum bending stress occurs at the outermost fiber, where .
- Nature of Stress:
- Above the neutral axis → Compressive stress
- Below the neutral axis → Tensile stress
Thus, the beam experiences both tension and compression due to bending.
Moment of Resistance
The total resisting moment developed by the internal stresses across the beam section is called the moment of resistance (M).
It balances the external bending moment applied on the beam, ensuring equilibrium.
This concept is used to determine the strength of beams and to design cross-sections that can resist the applied bending moments safely.
Factors Affecting Bending Stress
- Magnitude of Bending Moment (M):
Higher bending moments cause higher bending stresses. - Cross-Sectional Shape:
Beams with a larger moment of inertia (I), such as I-beams, resist bending better. - Material Properties:
Materials with higher modulus of elasticity (E) can withstand higher bending stresses without excessive deflection. - Beam Depth and Neutral Axis Distance (y):
Bending stress increases with the distance from the neutral axis. - Type of Loading and Support Condition:
The bending stress distribution changes depending on whether the beam is simply supported, cantilever, or fixed.
Applications of Bending Stress Analysis
- Structural Design:
Used in designing beams, bridges, and building frameworks to ensure safe load carrying capacity. - Machine Components:
Helps in designing shafts, levers, axles, and connecting rods which are subjected to bending. - Material Selection:
Determines suitable materials that can resist bending stresses without failure. - Safety Analysis:
Helps locate the maximum stress regions to prevent cracking or permanent deformation. - Optimization:
Used in optimizing cross-sectional shapes like I, T, and rectangular beams for strength and weight efficiency.
Conclusion
In conclusion, bending stress is the internal resistance developed within a beam or structural member when it is subjected to a bending moment. It varies linearly from zero at the neutral axis to a maximum at the outer surfaces. The bending stress is calculated using the flexural formula . It plays a crucial role in beam and structural design, ensuring that materials and sections are strong enough to resist bending safely without failure. Understanding bending stress helps engineers design efficient, durable, and safe mechanical and civil structures.