Deflection of Beams

# Deflection of Beams | Solid Mechanics - Mechanical Engineering PDF Download

## Deflections of Beam

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.

1. Slope of a Beam: Slope of a beam is the angle between deflected beam to the actual beam at the same point.
2. 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

### 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.
• 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
ρ = 1 / d2y / dx= 1 / y"
Thus, EI / M = 1 / y"
y" = M / EI = (1 / EI)M
• 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.
(EI)(d2y / dx2) = -M
Integrating one time
EI(dy / dx) = -∫M

The first integration y'(dy / dx) yields the Slope of the Elastic Curve
Second Integration
EIy = -∫∫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 boundary 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

• 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 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.

(i) Theorems of Area-Moment Method

• Theorem 1
The change in slope between the tangents drawn to the elastic curve at any two points A and B is equal to the product of 1 / EI multiplied by the area of the moment diagram between these two points
θAB = (1 / EI)(Area between A and B)
• Theorem 2
The deviation of any point B relative to the tangent drawn to the elastic curve at any other point A, in a direction perpendicular to the original position of the beam, is equal to the product of 1 / EI multiplied by the moment of an area about B of that part of the moment diagram between points A and B.
tB/A = (1 / EI)(Area between A and B)⋅X¯B
and
tA/B = (1 / EI)(Area between A and B)⋅X¯A

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.

1. Concentrated load at the free end of cantilever beam

• Maximum Moment, M = −PL
• Slope at end, θ = PL/ 2EI
• Maximum deflection, δ = PL3 / 3EI
• Deflection Equation (y is positive downward), EIy = (Px2)(3L − x) / 6

2. Concentrated load at any point on the span of the cantilever beam

• Maximum Moment. M =  -Pa
• Slope at end, θ = Pa/ 2EI
• Maximum deflection, δ = Pa3(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

3. Uniformly distributed load over the entire length of the cantilever beam

• Maximum Moment, M = −woL/ 2
• Slope at end, θ = woL/ 6EI
• Maximum deflection, δ = woL/ 8EI
• Deflection Equation (y is positive downward), EIy = wox2(6L− 4Lx + x2) / 120L

4. Triangular load, full at the fixed end and zero at the free end

• Maximum Moment, M = −woL2/ 6M
• Slope at end, θ = woL/ 24EI
• Maximum deflection, δ = woL/ 30EI
• Deflection Equation (y is positive downward), EIy = wox2(10L3− 10L2x + 5Lx− x3) / 120L

5. Moment load at the free end of the 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

6. Concentrated load at the midspan of simple beam

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

7. Uniformly distributed load over the entire span of simple beam

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

8. 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 = 7woL/ 360EI
(ii) θR = 8woL/ 360EI
• Maximum deflection, δ = 2.5woL/ 384EI at x = 0.519L
• Deflection Equation (y is positive downward), EIy = wox(7L− 10L2x + 3x) / 360L

9. Triangular load with zero at each support and full at the midspan of simple beam

• Maximum Moment, M = woL/ 12
• Slope at end, θL = θR = 5woL/ 192EI
• Maximum deflection, δ = w0L/ 120EI
• Deflection Equation (y is positive downward), EIy = wox(25L− 40L2x+ 16x4) / 960L for 0 < x < L / 2

### Beam Deflection Formula

• Cantilever Beams:
• Simply supported Beams:
The document Deflection of Beams | Solid Mechanics - Mechanical Engineering is a part of the Mechanical Engineering Course Solid Mechanics.
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## FAQs on Deflection of Beams - Solid Mechanics - Mechanical Engineering

 1. What is beam deflection and why is it important?
Beam deflection refers to the bending or displacement of a beam under an applied load. It is important because it helps engineers and designers determine the structural integrity and stability of a beam. By calculating the deflection, they can ensure that the beam can safely support the expected loads without exceeding its maximum deflection limits.
 2. How is beam deflection calculated?
Beam deflection can be calculated using various methods, including the use of mathematical equations, formulas, and computer simulations. One commonly used equation is the Euler-Bernoulli beam theory, which considers the beam's length, material properties, applied loads, and support conditions to calculate the deflection at any given point along the beam.
 3. What factors affect the deflection of a beam?
Several factors can influence the deflection of a beam. The most significant ones include the beam's length, cross-sectional shape, material properties (such as modulus of elasticity), applied loads, and support conditions. For example, a longer beam will generally have a larger deflection compared to a shorter one under the same load.
 4. How can beam deflection be minimized?
There are several ways to minimize beam deflection. One approach is to use materials with higher modulus of elasticity, such as steel, which are stiffer and less prone to bending. Increasing the beam's cross-sectional area or changing its shape can also help reduce deflection. Additionally, proper support conditions, such as fixing one end of the beam or adding additional supports, can limit the beam's deflection.
 5. What are the limitations of beam deflection calculations?
Beam deflection calculations are based on simplified assumptions and mathematical models, such as the Euler-Bernoulli beam theory. These models may not accurately represent real-world conditions, especially for complex beam geometries or materials with nonlinear behavior. Additionally, other factors like temperature changes, creep, and vibration may affect the deflection of a beam but are not always considered in basic calculations. Therefore, it is crucial to validate the results of deflection calculations through experimental testing or using more advanced analysis methods.

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