Deflection of Beams

# Deflection of Beams | Strength of Materials (SOM) - Mechanical Engineering PDF Download

The deformation of a beam is usually expressed in terms of its deflection from its original unloaded position. The deflection is measured from the original neutral surface of the beam to the neutral surface of the deformed beam. The configuration assumed by the deformed neutral surface is known as the elastic curve of the beam.
Slope of a Beam: Slope of a beam is the angle between deflected beam to the actual beam at the same point.
Deflection of Beam: Deflection is defined as the vertical displacement of a point on a loaded beam. There are many methods to find out the slope and deflection at a section in a loaded beam.

• The maximum deflection occurs where the slope is zero. The position of the maximum deflection is found out by equating the slope equation zero. Then the value of x is substituted in the deflection equation to calculate the maximum deflection

Differential Equation of the Deflection Curve of Beam

### Methods of Determining Beam Deflections

Numerous methods are available for the determination of beam deflections. These methods include:

1. Double Integration Method
• This is most suitable when concentrated or udl over entire length is acting on the beam. A double integration method is a powerful tool in solving deflection and slope of a beam at any point because we will be able to get the equation of the elastic curve.
• A double integration method is a powerful tool in solving deflection and slope of a beam at any point because we will be able to get the equation of the elastic curve.
• In calculus, the radius of curvature of a curve y = f(x) is given by:

• In the derivation of flexure formula, the radius of curvature of a beam is
ρ = EI / M

• Deflection of beams is so small, such that the slope of the elastic curve dy/dx is very small, and squaring this expression the value becomes practically negligible, hence:

• If EI is constant, the equation may be written as:
EIy′′ = M
where x and y are the coordinates shown in the figure of the elastic curve of the beam under load.

• y is the deflection of the beam at any distance x.

• E is the modulus of elasticity of the beam,

• I represent the moment of inertia about the neutral axis, and

• M represents the bending moment at a distance x from the end of the beam.

The product EI is called the flexural rigidity of the beam.

Integrating one time:

The first integration y'(dy/dx) yields the Slope of the Elastic Curve.
Second Integration:

EI y = - ∫∫M

The second integration y gives the Deflection of the Beam at any distance x.

• The resulting solution must contain two constants of integration since EI y" = M is of second order.
• These two constants must be evaluated from known conditions concerning the slope deflection at certain points of the beam.
• For instance, in the case of a simply supported beam with rigid supports, at x = 0 and x = L, the deflection y = 0, and in locating the point of maximum deflection, we simply set the slope of the elastic curve y' to zero
2. Area Moment Method (Mohr's Method)
• Another method of determining the slopes and deflections in beams is the area-moment method, which involves the area of the moment diagram.The moment-area method is a
• The moment-area method is a semi graphical procedure that utilizes the properties of the area under the bending moment diagram. It is the quickest way to compute the deflection at a specific location if the bending moment diagram has a simple shape.

(a) Theorems of Area-Moment Method

• Theorem 1: The angle between the tangent of the deflection curve of two points A and B is equal to the negative area of M/EI diagram between the points.
• Theorem 2: The deviation of B from tangent at A is equal to the negative of the statical moment (or the first moment) with respect to B, of the M/EI diagram area between A and B.
3. Method of Superposition

The method of superposition, in which the applied loading is represented as a series of simple loads for which deflection formulas are available. Then the desired deflection is computed by adding the contributions of the component loads(principle of superposition).

• Mostly direct formula is used in questions, hence it is advised to look for the beam deflection formula which are directly asked from this topic rather than going for long derivations.

(i) Concentrated load at the free end of cantilever beam (origin at A)

• Maximum Moment: M = −PL
• Slope at end: θ = PL/ 2EI
• Maximum deflection: δ = PL/ 3EI
• Deflection Equation (y is positive downward): EIy = (Px2)(3L − x) / 6

(ii) Concentrated load at any point on the span of cantilever beam

• Maximum Moment: M = -wa
• Slope at end: θ = wa/ 2EI
• Maximum deflection: δ = wa3(3L − a) / 6EI
• Deflection Equation (y is positive downward):
(i) EIy = Px2(3a − x) / 6 for 0 < x <a
(ii) EIy = Pa2(3x − a) / 6 for a < x <L

(iii) Uniformly distributed load over the entire length of cantilever beam

• Maximum Moment: M = −wL/ 2
• Slope at end: θ = wL/ 6EI
• Maximum deflection: δ = wL/ 8EI
• Deflection Equation (y is positive downward): EIy = wx2(6L2− 4Lx + x2) / 120L

(iv) Triangular load, full at the fixed end and zero at the free end

• Maximum Moment: M = −wL/ 6
• Slope at end: θ = wL/ 24EI
• Maximum deflection: δ = wL/ 30EI
• Deflection Equation (y is positive downward): EIy = wx2(10L3− 10L2x + 5Lx− x3) / 120L

(v) Moment load at the free end of cantilever beam

• Maximum Moment: M = −M
• Slope at end: θ = ML / EI
• Maximum deflection: δ = ML/ 2EI
• Deflection Equation (y is positive downward): EIy = Mx/ 2

(vi) Concentrated load at the midspan of simple beam

• Maximum Moment: M = PL / 4
• Slope at end: θ= θB = WL2/ 16EI
• Maximum deflection: δ = PL3/ 48EI
• Deflection Equation (y is positive downward): EIy = Px{(3 / 4)L− x2)} / 12 for 0 < x < L / 2

(vii) Uniformly distributed load over the entire span of simple beam

• Maximum Moment: M = wL/ 8
• Slope at end: θL= θ= wL/ 24EI
• Maximum deflection: δ = 5wL/ 384EI
• Deflection Equation (y is positive downward): EIy = wx(L− 2Lx+ x3) / 24

(viii) Triangle load with zero at one support and full at the other support of simple beam

• Maximum Moment: M = woL/ 9√3
• Slope at end:
(i) θL= 7wL/ 360EI
(ii) θ= 8wL/ 360EI
• Maximum deflection: δ = 2.5wL/ 384EI at x = 0.519L
• Deflection Equation (y is positive downward): EIy = wx(7L− 10L2x + 3x) / 360L

(ix) Triangular load with zero at each support and full at the midspan of simple beam

• Maximum Moment: M = wL/ 12
• Slope at end: θL= θ= 5wL/ 192EI
• Maximum deflection: δ = wL/ 120EI
• Deflection Equation (y is positive downward): EIy = wox(25L− 40L2x+ 16x4) /960L    for 0 < x < L / 2

### Conjugate Beam method (Method of elastic weights)

Rule.1: The slope at any point of a real beam, relative to the original axis of the beam, is equal to the shear force at the corresponding point of the conjugate beam.

Rule.2: The deflection at any point of a real beam, relative to the original axis of the beam is equal to the bending moment at the corresponding point of the conjugate beam.

Support conditions for the real and conjugate beam:

Deflections by Castigliano's Theorem

Beam Deflection Formula
• Cantilever Beams
• Simply supported Beams
The document Deflection of Beams | Strength of Materials (SOM) - Mechanical Engineering is a part of the Mechanical Engineering Course Strength of Materials (SOM).
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## FAQs on Deflection of Beams - Strength of Materials (SOM) - Mechanical Engineering

 1. What is deflection in beams?
Ans. Deflection in beams refers to the bending or deformation of a beam under the action of external loads. It is the measure of how much the beam deviates from its original position or shape. Deflection is usually expressed in terms of the vertical displacement at a specific point along the beam.
 2. How is beam deflection calculated?
Ans. Beam deflection can be calculated using various methods, including analytical equations, numerical methods, or software programs. For simple cases, engineers often use the Euler-Bernoulli beam theory, which provides analytical equations to calculate deflection based on the beam's material properties, dimensions, and applied loads. More complex cases may require finite element analysis or other advanced computational techniques.
 3. What factors affect beam deflection?
Ans. Several factors influence beam deflection, including the magnitude and distribution of the applied loads, the beam's length, cross-sectional shape, and material properties. In general, a stiffer and stronger beam will experience less deflection compared to a weaker and more flexible beam. Additionally, the position and type of supports or restraints along the beam's length also impact its deflection.
 4. How does beam deflection affect structural integrity?
Ans. Beam deflection can have significant implications for the structural integrity of a system. Excessive deflection can cause the beam to exceed its maximum allowable limits, leading to structural failure or reduced performance. It can also induce additional stresses and strains in the beam, which can result in fatigue, buckling, or other failure modes over time. Therefore, engineers must consider deflection limits and ensure that the beam's design can withstand the expected deflection under anticipated loads.
 5. What are some practical applications of beam deflection analysis?
Ans. Beam deflection analysis is crucial in various engineering fields and applications. It is commonly used in the design and analysis of structures such as bridges, buildings, and aerospace components. By understanding the deflection behavior, engineers can ensure that the structures meet safety and performance requirements. Beam deflection analysis is also essential in the design of mechanical systems, such as conveyor belts, cranes, and robotic arms, to ensure they function properly and avoid excessive deflection that may affect their operation.

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