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Beam Deflection Formula

Formula

Beam Deflection Formula

Reviewed by:

Gaurang Shah

Beam Deflection

The degree to which a structural structure can be displaced by a significant amount of load is known as deflection. It is also known as the angle or the distance. The slope of the deflected form of the body under the load is directly proportional to the distance of deflection of a member under a load. It can be calculated by integrating the function that describes the member's slope under that load.

The Beam deflection equation is a long piece of a body capable of supporting a weight while resisting bending. Light deflection is the deflection of a beam in a certain direction when a force is applied to it.

Types of Beam

The most common way structural engineers identify beams is by their support configuration. There are several different kinds of support configurations, but these are the four most common:

Simply Supported
Cantilever
Continuous
Fixed-Fixed

Simply Supported Beam

This is commonly thought to be the most basic kind of beam. This is a very typical kind of beam that is determinate since there are only three uncertain reactions and three equilibrium equations. One from the roller support and two from the pinned support.

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Cantilever Beam

Fixed Supports are used to support Cantilever Beams from one end.

This is the only form of support that can be used in this situation because it provides the necessary moment resistance to keep the beam steady. The moment restraint required by the beam will not be provided by a pinned or roller support.

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Continuous Beam

Multi-spanned continuous beams have several supports running the length of the beam.

A single beam backed by a series of columns over its length is an example of a continuous beam.

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Fixed Beam

Fixed Beams have fixed supports on both ends, offering moment resistance.

Since the two fixed supports prevent rotation, this form of the beam can be used when the designer wants to control the deflection at the mid-span.

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Beam Bending Moment Diagram and Share Force Diagram

A beam's shear force diagram (SFD) and bending moment diagram (BMD) represent the difference in shear force and bending moment over its length.

These diagrams come in handy when modelling beams for a variety of applications.

A shear force F, which is defined as the algebraic sum of all vertical forces acting on a segment from the left or right. A shear force diagram, abbreviated as S.F.D., is a graphical representation of the variation of shear force along the length of a beam.

A bending moment M, which is defined as the algebraic sum of all vertical force moments to the left or right of a segment. A bending moment diagram, abbreviated as B.M.D., is a graphical representation of the variance of the bending moment along the length of the beam.

The beam bending equations moment of a beam is required for beam design as well as slope and deflection formula beam calculations.

The Beam Deflection Formula

The displacement of a beam or node from its original position is related to deflection in structural engineering.

It occurs as a result of the pressures and loads imposed on the body. Externally attached loads or the weight of the body structure itself may cause deflection, which is also known as displacement. It can occur in beams, wooden beams, frames, and just about any other type of body structure.

Where,

w = uniform load (force/length units)

V = shear

I = moment of inertia

E = modulus of elasticity

d = deflection

M = moment

Beam Deflection Formula and Formula Table

Name of Beam deflection	Diagram	Formulas
Pinned-Pinned Beam With Uniform Load	(Image will be uploaded soon)	\[V = w \times (\frac{L}{2} -x) \] \[M = \frac{wx}{2} (L-x)\] \[\delta = \frac{wx}{24EI} (L^{3} -2Lx^{2}+x^{3}) \]
Fixed-Fixed Beam With Uniform Load	(Image will be uploaded soon)	\[V = w \times (\frac{L}{2} -x) \] \[M = \frac{w}{12} (6Lx-L^{2}-6x^{2}) \] \[\delta = \frac{wx^2}{24EI}(L-x)^2\]
Pinned-Fixed Beam With Uniform Load	(Image will be uploaded soon)	\[V = w \times (\frac{3L}{8} -x) \] \[M = wx \times (\frac{3L}{8} - \frac{x}{2}) \] \[\delta=\frac{wx}{48EI}(L^{3}-3Lx^)\]
Free-Fixed Beam With Uniform Load	(Image will be uploaded soon)	\[V = -wx\] \[M=-\frac{wx^2}{2}\] \[\delta = \frac{w}{24EI} (x^{4}-4L^{3}+3L^{4})\]
Pinned-Pinned Beam With Point Load	(Image will be uploaded soon)	\[V = \frac{Pb}{L} - P(x-a)^0\] \[M = \frac{Pbx}{L} - P(x-a)^1\] \[\delta=\frac{P}{6EI}[\frac{bx^3}{L}-\frac{abx}{L(2L-a)(x-a)^3} ]\]
Fixed-Fixed Beam With Point Load	(Image will be uploaded soon)	\[V=\frac{Pb^2}{L^3}(L+2a)-P(x-a)^0\] \[M=-\frac{Pab^2}{L^2}+\frac{Pb^2x}{L^3(L+2a)-P(x-a)^1}\] \[\delta=\frac{P}{6EI}[\frac{b^2x^3}{L^3(L+2a)}-\frac{3ab^2x^2}{L^2-(x-a)^3}]\]
Pinned-Fixed Beam With Point Load	(Image will be uploaded soon)	\[V=\frac{Pb^2}{2L^3}(2L+a)-P(x-a)^0\] \[M=\frac{Pb^2x}{2L^3}(2L+a)-P(x-a)^1\] \[\delta=\frac{P}{6EI}[\frac{b^3x^3}{2L^3(2L+a)}-\frac{3ab^2x}{2L-(x-a)^3}]\]
Free-Fixed Beam With Point Load	(Image will be uploaded soon)	\[V = -P(x-a)^0\] \[M = -P(x-a)^1\] \[\delta = \frac{P}{6EI} [3b^2x-2L^3-a^3-(x-a)^3]\]

Solved Examples

Ex.1. A pinned-pinned beam of length 80 cm is put under uniform load of 70 gms having x as 5 cm. Determine the shear.

Solution:

Given parameters are

Length L = 80 cm

Uniform load w = 70 gms

Displacement x = 5 cm

The shear is given by \[V = w \times (\frac{L}{2}-x) \]

= 0.07 kg \[\times (\frac{0.6}{2} -0.05) \]

=0.07 \[\times\] 0.25

=0.0175kg/m

Ex.2. A fixed beam of length 30 cm is put under a uniform load of 600 gm having x as 40 cm. Determine the moment.

Solution:

Given parameters are

Length L = 30 cm,

Uniform load w = 600 gms,

Displacement x = 40 cm

The moment is

\[M = -\frac{wx^2}{2} \]

\[M = -\frac{0.6\times0.4^2}{2}\]

\[M = 0.048 kg/m^2\]

Hence the moment is \[0.048 kg/m^2\]

FAQs on Beam Deflection Formula

1. Who discovered the slope deflection equation?

The slope deflection method, invented by George A. M in 1914, is a structural analysis method for beams and frames.

2. What is the deflection rate?

The rate at which automated and self-help resources serve service demand that would otherwise be handled by supported retail workers is known as deflection. In the IT industry, the average rate of case deflection is 23%.

3. What is the formula for beam deflection?

In general, deflection is determined by dividing M(x) by EI (Young's Modulus x Moment of Inertia), which is the double integral of the Bending Moment Equation.