Introduction about 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.
FAQs on Bending Equation Derivation
1. What is Bending Theory?
Bending theory, also termed as flexure theory, involves the concept of axial deformation of a homogenous beam resulting from the application of a perpendicular load on a longitudinal axis. This theory has a lot of application in applied mechanics. For any given substance the flexural strength is described as the stress that is received from the yield slightly before the flexure test. It denotes the greatest stress experienced within the material at the point of its yield. 𝜎 is used for the representation of flexural strength.
2. What do E and σ stand for in the Bending Equation?
In the bending equation derivation, E denotes Young’s Modulus of elasticity and σ signifies stress of the fibre at a distance ‘y’ from the neutral/centroidal axis.
3. What is the Bending Equation?
The axial deformation of the beam due to external load that is applied perpendicularly to a longitudinal axis is called the Bending Theory. The bending equation stands as σ/y = E/R = M/T.
4.What are different types of Stress?
Stress is the quantity that represents the magnitude of forces that cause deformation in a body. Some types of stress are-
Tensile Stress – Tensile Stress is the stress that acts when forces pull an object and force its elongation. For example- stretching rubber bands.
Compressive Stress – Compressive Stress is the stress that acts when the forces cause the object to compress.
Bulk Stress – Bulk Stress is seen when an object is squeezed from all sides. For example - a submarine in the deep ocean.
Shear Stress – Shear Stress is that type of stress where the deforming stress operates tangentially to the object’s surface.
5.What is Plastic Deformation in Crystalline and Amorphous objects?
The permanent distortion happening when a material is subjected to tensile, compressive, bending, or shear stresses that exceed its yield strength is called Plastic Deformation. This causes the object to elongate, buckle, bend, compress, or twist. The transition from elastic to plastic state is determined by the yield strength of the material. The plastic deformation system varies in the case of crystalline and amorphous materials. In the case of crystalline materials, deformation occurs through a process called the slip, which involves the movement of dislocations. In the case of amorphous materials, deformation occurs by the sliding of atoms and ions with no directionality.
6.Explain the different elastic constants?
Elastic constants are constant values that determine the deformation produced by the stress system operating on the materials. The different types of Elastic Constants are-
Young’s modulus/ Modulus of Elasticity (E) - Hooke’s law states that when a body is exposed to tensile stress or compressive stress, the stress involved is directly proportional to the strain w the elastic limits of that body. The ratio of the applied tensile stress to the tensile strain experienced is constant and is known as Young’s modulus.
Bulk modulus (K)- Bulk Modulus comes up when a body is exposed to mutually perpendicular direct stresses which are, within its elastic limits, alike and equal, the ratio of the change in pressure to the corresponding volumetric strain is always constant. This ratio is represented by the letter 'K' with Newton per meter square as its unit.
Shear modulus/ modulus of rigidity (G) - Shear Modulus is observed when a body is exposed to shear stress and the shape of the body gets changed. The ratio of shear stress to the corresponding shear strain experienced on the object is called rigidity modulus or modulus of rigidity. It is denoted by the letter 'G' with the unit being Pascal (Pa).
Poisson’s Ratio (µ)- Poisson’s Ratio comes up when a body is subjected to tensile stress within the object's elastic limits and there's a change in the dimensions of the body in the direction of the load and also in the opposite direction.It has no unit.
