Bending Equation Derivation

Bending equation is a subsection within the purview of bending theory. This theory, in turn, primarily suggests that a beam is subject to deformation when a force acts upon a point that passes through the longitudinal axis of the beam. Therefore, bending theory refers to a study of axial deformation caused due to such stresses and consequently also known as flexure theory. 

What is the Bending Stress Equation? 

Bending stress equation, or simply bending equation implies a mathematical equation that aims to find the amount of stress on the beam. However, the bending moment equation stipulates a set of assumptions that one has to take into account to arrive at the exact data of flexure stresses. 

The comprehensive assumptions of bending equation are thus as follows – 

• The beam has to be straight. Besides, it has to possess a constant cross-section without aberrations. 

• The construction of the beam has to be with a homogenous material. It must also possess a symmetrical longitudinal plane. 

• The bending moment equation derivation states that the point of the applied load has to lie on its longitudinal plane of symmetry. 

• One of the most essential assumptions in the bending equation is that failure should be a result of buckling and not bending. 

• ‘E’ or the elastic limit remains constant for both tension and compression. 

• The plane cross-section continues to be a plane throughout the bending process. 

What are the Factors in Bending Equation Derivation? 

The factors or bending equation terms as implemented in the derivation of bending equation are as follows – 

• M = Bending moment. 

• I = Moment of inertia exerted on the bending axis. 

• σ = Stress of the fibre at a distance ‘y’ from neutral/centroidal axis. 

• E = Young’s Modulus of beam material. 

• R = Curvature radius of this bent beam.

However, if the distance to the remotest element c replaces y, then 

\[\frac{M}{I}\]=\[\frac{\sigma max}{c}\]

\[therefore \sigma max\]=\[\frac{MC}{I}\]=\[\frac{M}{Z}\]

Where \[Z=\frac{I}{c}\]. This Z is the section modulus of this beam

How is Bending Stress Formula Derivation Done? 

Bending stress formula derivation fundamentally computes the figure of bending stresses that develops on a loaded beam. 

Therefore, the bending equation of stress includes the following steps – 

Strain in fibre AB=\[\frac{change in lenght}{orginal length}\]

\[\frac{A'B'-AB}{AB}\] but AB = CD and CD=C’D’

Therefore,strain=\[\frac{A'B'-C'D'}{C'D'}\]

With the presence of CD and C’D’ on neutral axis, the stress on neutral axis comes to be zero. Thus, this neutral axis is devoid of any strain from the applied force.

\[\frac{(R+y)\theta -R\theta }{R\theta }\]=\[\frac{R\theta +y\theta -R\theta }{R\theta }\]=\[\frac{y}{R}\]

Yet, \[\frac{Stress}{Strain}\]=E(E=Young’s Modulus of elasticity)

Thus, equation of the two strains based on the two relations is \[\frac{\sigma}{y}\]=\[\frac{y}{R}\]

Or \[\frac{\sigma}{y}\]=\[\frac{E}{R}\]................(i)

On the other hand, let us assume any arbitrary cross-section of the beam. Strain on the fibre is at a distance of ‘y’ from the N.A. Thus, the following expression is -

\[\sigma=\frac{E}{R}y\]

Hoever, if the shaped strip has an area of ‘dA’, the following equation denotes force on strip -

F=\[\sigma\delta A=\frac{E}{R}y\delta A\]

Consequently, moment of the bending equation on the neutral axis will amount to -

\[Fy=\frac{E}{R}y^2\delta A\]

Therefore, the total moment for the entire cross-section equals to -

M=\[\sum \frac{E}{R}y^2\delta A\]=\[\frac{E}{R}\sum y^2\delta A\]

Here, Σy²δA is the beam material’s property and suggest the second moment of area of cross-section.

The symbol I further denotes  it.

As a result,

M=\[\frac{E}{R}l\]..................(ii)

Thus, when we combine equation (i) and (ii), we arrive at the following bending equation -

\[\frac{\sigma }{y}\]=\[\frac{M}{T}\]=\[\frac{E}{R}\]

The above equation thus refers to bending equation derivation. It is, however, pure bending because the bending results despite the lack of a force.

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Different regions of the Stress-Strain Graph 

The different regions in the stress-strain graph are:

• Proportional Limit- The proportional limit is the region of the Stress-Strain Graph that follows Hooke’s Law,  which means that, in this region, the stress-strain ratio shows a constant proportionality. This constant value is called Young’s modulus. 

• Elastic Limit- Elastic Limit is that point in the Stress-Strain graph, up to which the material returns to its initial position when a load is acting on it, is completely removed. Further Elastic limit, plastic deformation starts to appear in it.

• Yield Point-The yield point is the point on the Stress-Strain graph at which the material starts to bend plastically. The passing of the yield point denotes that permanent plastic deformation has occurred.

•  Ultimate Stress Point- Ultimate Stress Point is the point on the Stress-Strain graph that describes the maximum stress that the given material can endure before the ultimate failure. 

• Fracture or Breaking Point- Breaking Point is the point in the Stress-Strain Graph at which the collapse of the material takes place which means that it is broken.