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Past Year Questions: Deflection of Beams | Solid Mechanics - Mechanical Engineering PDF Download

Q1: For the 6m long horizontal cantilever beam PQR shown in the figure. Q is the mid-point. Segment PQ of the beam has flexural rigidity EI = 2 × 105 kN,m2 whereas the segment QR has infinite flexural rigidity. Segment QR is subjected to uniformly distributed, vertically downward load of 5 kN/m. [2024, Set-II]
Past Year Questions: Deflection of Beams | Solid Mechanics - Mechanical Engineering

The magnitude of the vertical displacement (in mm) at point Q is _____ (rounded off to 3 decimal places)
Ans: 1.176 to 1.186
Past Year Questions: Deflection of Beams | Solid Mechanics - Mechanical EngineeringVertical deflection atPast Year Questions: Deflection of Beams | Solid Mechanics - Mechanical Engineering
Past Year Questions: Deflection of Beams | Solid Mechanics - Mechanical Engineering
ΔQ = 6.75 × 10−4 + 5.0625 × 10−4m 
ΔQ = 1.181 mm

Q2: The beam shown in the figure is subjected to a uniformly distributed downward load of intensity q q between supports A and B.  [2024, Set-I]
Past Year Questions: Deflection of Beams | Solid Mechanics - Mechanical EngineeringConsidering the upward reactions as positive, the support reactions are
(a) RA = ql/2 ; RB = ql ; RC = ql/2
(b) RA = ql/2 ; RB = 5ql/2 ; RC = 0
(c) RA = -ql ; RB = 5ql/2 ; RC = ql/2
(d) RA = ql/2 ; RB = 5ql/2 ; RC = -ql
Ans: (d)
Past Year Questions: Deflection of Beams | Solid Mechanics - Mechanical EngineeringBMP = 0
Past Year Questions: Deflection of Beams | Solid Mechanics - Mechanical Engineering
Past Year Questions: Deflection of Beams | Solid Mechanics - Mechanical Engineering
Past Year Questions: Deflection of Beams | Solid Mechanics - Mechanical Engineering
From 1, 2 and 3
RA = ql/2 ; RB = 5ql/2 ; RC = -ql

Q3: For the frame shown in the figure (not to scale), all members (AB, BC, CD, GB, and CH) have the same length, L and flexural rigidity, El. The joints at B and C are rigid joints, and the supports A and D are fixed supports. Beams GB and CH carry uniformly distributed loads of w per unit length. The magnitude of the moment reaction at A is wL2/k. What is the value of k k (in integer) ? ____ [2023, Set-II]
Past Year Questions: Deflection of Beams | Solid Mechanics - Mechanical Engineering(a) 4
(b) 6
(c) 8
(d) 10
Ans: (b)
Past Year Questions: Deflection of Beams | Solid Mechanics - Mechanical EngineeringDue to symmetry, slope at mid of BC is zero and can be taken as slider at point O
Distribution factors:
Past Year Questions: Deflection of Beams | Solid Mechanics - Mechanical Engineering
∴ Past Year Questions: Deflection of Beams | Solid Mechanics - Mechanical Engineering
Past Year Questions: Deflection of Beams | Solid Mechanics - Mechanical Engineering
∴ K = 6

Q4: When a simply-supported elastic beam of span L and flexural rigidity EI(E is the modulus of elasticity and I I is the moment of inertia of the section) is loaded with a uniformly distributed load w per unit length, the deflection at the mid-span isPast Year Questions: Deflection of Beams | Solid Mechanics - Mechanical Engineering
If the load on one half of the span is now removed, the mid-span deflection ____  [2023, Set-II]
(a) reduces to Δ0/2 
(b) reduces to a value less than Δ0/2 
(c) reduces to a value greater than Δ0/2
(d) remains unchanged at Δ0
Ans: (a)
Past Year Questions: Deflection of Beams | Solid Mechanics - Mechanical EngineeringPast Year Questions: Deflection of Beams | Solid Mechanics - Mechanical EngineeringAs Case-I and II are mirror image of each other,. 
Hence, ΔC,1 = ΔC,II  
Also, ΔC = ΔC,1 + ΔC,1  
⇒ ΔC,I = ΔC,II = ΔC/2 

Q5: A single story building model is shown in the figure. The rigid bar of mass 'm' is supported by three massless elastic columns whose ends are fixed against rotation. For each of the columns, the applied lateral force (P) and corresponding moment (M) are also shown in the figure. The lateral deflection (δ) of the bar is given by δ= PL3/12EI , where L is the effective length of the column, E is the Young's modulus of elasticity and I is the area moment of inertia of the column cross-section with respect to its neutral axis. [2021, Set-II]
Past Year Questions: Deflection of Beams | Solid Mechanics - Mechanical EngineeringFor the lateral deflection profile of the columns as shown in the figure, the natural frequency of the system for horizontal oscillation is
(a)Past Year Questions: Deflection of Beams | Solid Mechanics - Mechanical Engineering

(b)Past Year Questions: Deflection of Beams | Solid Mechanics - Mechanical Engineering
(c)Past Year Questions: Deflection of Beams | Solid Mechanics - Mechanical Engineering

(d)Past Year Questions: Deflection of Beams | Solid Mechanics - Mechanical Engineering
Ans: (a)
Past Year Questions: Deflection of Beams | Solid Mechanics - Mechanical Engineering

As the deflection will be same in all the 3 columns, so it represents a parallel connection.
Past Year Questions: Deflection of Beams | Solid Mechanics - Mechanical Engineeringkeq = 3k = 36EI/L3 

Past Year Questions: Deflection of Beams | Solid Mechanics - Mechanical Engineering

Natural frequency Past Year Questions: Deflection of Beams | Solid Mechanics - Mechanical Engineering
Past Year Questions: Deflection of Beams | Solid Mechanics - Mechanical Engineering

Q6: The equation of deformation is derived to be y = x2 − xL for a beam shown in the figure. [2021, Set-I]
Past Year Questions: Deflection of Beams | Solid Mechanics - Mechanical Engineering
The curvature of the beam at the mid-span (in units, in integer) will be ______________  [2020, Set-I]
(a) 1
(b) 2
(c) 3
(d) 4
Ans: (b)
Past Year Questions: Deflection of Beams | Solid Mechanics - Mechanical EngineeringGiven, y = x2 − xL
1/R = d2y/dx2
Curvature at mid section is dy/dx = 2x - 1
d2y/dx2 = 2

Q7: A cantilever beam PQ of uniform flexural rigidity (EI) is subjected to a concentrated moment M at R as shown in the figure.
Past Year Questions: Deflection of Beams | Solid Mechanics - Mechanical EngineeringThe deflection at the free end Q is
(a) M L2/6EI
(b) M L2/4EI
(c) 3M L2/8EI
(d) 3M L2/4EI
Ans: (c)
Past Year Questions: Deflection of Beams | Solid Mechanics - Mechanical EngineeringPast Year Questions: Deflection of Beams | Solid Mechanics - Mechanical Engineering

Q8: The figure shows a simply supported beam PQ of uniform flexural rigidity El carrying two moments M and 2M
Past Year Questions: Deflection of Beams | Solid Mechanics - Mechanical EngineeringThe slope at P will be [2018 : 2 Marks, Set-I]
(a) 0
(b) ML/(9EI)
(c) ML/(6EI)
(d) ML/(3EI)
Ans: (c)
Method-l

Past Year Questions: Deflection of Beams | Solid Mechanics - Mechanical Engineering

Past Year Questions: Deflection of Beams | Solid Mechanics - Mechanical Engineering

Method-ll
Moment area method:

Past Year Questions: Deflection of Beams | Solid Mechanics - Mechanical Engineering

Past Year Questions: Deflection of Beams | Solid Mechanics - Mechanical Engineering

δQ/P = Defleciton of point Q wrt to tangent at point P

Past Year Questions: Deflection of Beams | Solid Mechanics - Mechanical Engineering

Past Year Questions: Deflection of Beams | Solid Mechanics - Mechanical Engineering

Q9: Two prismatic beams having the same flexural rigidity of 1000 kN-m2 are shown in the figures.
Past Year Questions: Deflection of Beams | Solid Mechanics - Mechanical EngineeringIf the mid-span deflections of these beams are denoted by δ1 and δ2 (as indicated in the figures), the correct option is [2017 : 2 Marks, Set-II]
(a) δ1 = δ2
(b) δ1 < δ2
(c) δ1 > δ2
(d) δ1 >> δ2
Ans: (a)

Past Year Questions: Deflection of Beams | Solid Mechanics - Mechanical Engineering

∴ δ1 = δ2


Q10: Two beams PQ (fixed at P with a roller support at Q, as shown in Figure I, which allows vertical movement) and XZ(with a hinge at Y) are shown in the Figures I and II respectively. The spans of PQ and XZ are L and 2L respectively. Both the beams are under the action of uniformly distributed load (w) and have the same flexural stiffness, EI (where, E and I respectively denote modulus of elasticity and moment of inertia about axis of bending). Let the maximum deflection and maximum rotation be δmax1 and δmax1, respectively, in the case of beam PQ and the corresponding quantities for the beam XZ be δmax2 and δmax2, respectively.

Past Year Questions: Deflection of Beams | Solid Mechanics - Mechanical Engineering

Past Year Questions: Deflection of Beams | Solid Mechanics - Mechanical EngineeringWhich one of the following relationship is true?    [2016 : 2 Marks, Set-I]
(a)Past Year Questions: Deflection of Beams | Solid Mechanics - Mechanical Engineering
(b) Past Year Questions: Deflection of Beams | Solid Mechanics - Mechanical Engineering
(c) Past Year Questions: Deflection of Beams | Solid Mechanics - Mechanical Engineering
(d) Past Year Questions: Deflection of Beams | Solid Mechanics - Mechanical Engineering
Ans: (d)

Past Year Questions: Deflection of Beams | Solid Mechanics - Mechanical Engineering

Deflection in beam xy at y = Deflection in beam yz at y
⇒ Past Year Questions: Deflection of Beams | Solid Mechanics - Mechanical Engineering

∴ R = 0
In beam PQ also at support Q, vertical reaction in zero because of roller support.
So, beam PQ, xyand yz are same.
∴ Past Year Questions: Deflection of Beams | Solid Mechanics - Mechanical Engineering

Q11: A 3 m long simply supported beam of uniform cross -section is subjected to a uniformly distributed load of w= 20 kN/m in the central 1 m as shown in the figure
Past Year Questions: Deflection of Beams | Solid Mechanics - Mechanical EngineeringIf the flexural rigidity (El) of the beam is 30 x 106 N-m2, the maximum slope (expressed in radians) of the deformed beam is [2016 : 2 Marks, Set-I]
(a) 0.681 x 10-7
(b) 0.361 x 10-3
(c) 4.310 x 10-7
(d) 5.910 x 10-7
Ans: (b)
Method-1

Past Year Questions: Deflection of Beams | Solid Mechanics - Mechanical Engineering

Due to symmetrical loading,

Past Year Questions: Deflection of Beams | Solid Mechanics - Mechanical Engineering

Past Year Questions: Deflection of Beams | Solid Mechanics - Mechanical Engineering

Past Year Questions: Deflection of Beams | Solid Mechanics - Mechanical Engineering

According to Macaulay method,

Past Year Questions: Deflection of Beams | Solid Mechanics - Mechanical Engineering

After integrating once

Past Year Questions: Deflection of Beams | Solid Mechanics - Mechanical Engineering

Due to symmetrical loading slope will be zero at mid section (x = 1.5 m),

Past Year Questions: Deflection of Beams | Solid Mechanics - Mechanical Engineering

Past Year Questions: Deflection of Beams | Solid Mechanics - Mechanical Engineering

∴ Past Year Questions: Deflection of Beams | Solid Mechanics - Mechanical Engineering

∴ Equation of slope,

Past Year Questions: Deflection of Beams | Solid Mechanics - Mechanical Engineering

The slope will be maximum at the support,

Past Year Questions: Deflection of Beams | Solid Mechanics - Mechanical Engineering

Method-ll
Moment Area Method,

Past Year Questions: Deflection of Beams | Solid Mechanics - Mechanical Engineering

Past Year Questions: Deflection of Beams | Solid Mechanics - Mechanical Engineering

Past Year Questions: Deflection of Beams | Solid Mechanics - Mechanical Engineering

Past Year Questions: Deflection of Beams | Solid Mechanics - Mechanical EngineeringArea of M/El diagram between points P and B.

Past Year Questions: Deflection of Beams | Solid Mechanics - Mechanical Engineering

Q12: Two beams are connected by a linear spring as shown in the following figure. For a load P as shown in the figure, the percentage of the applied load P carried by the spring is_____.    [2015 : 2 Marks, Set-I]

Past Year Questions: Deflection of Beams | Solid Mechanics - Mechanical Engineering

Ans:

Past Year Questions: Deflection of Beams | Solid Mechanics - Mechanical Engineering

Compression of spring

Past Year Questions: Deflection of Beams | Solid Mechanics - Mechanical Engineering

Past Year Questions: Deflection of Beams | Solid Mechanics - Mechanical Engineering

Past Year Questions: Deflection of Beams | Solid Mechanics - Mechanical Engineering

% force carried by spring = 25%

Q13: A steel strip of length, L = 200 mm is fixed at end A and rests at 6 on a vertical spring of stiffness, k = 2 N/mm. The steel strip is 5 mm wide and 10 mm thick. A vertical load, P= 50 N is applied at 6, as shown in the figure. Considering E = 200 GPa, the force (in N) developed in the spring is ________ .    [2015 : 2 Marks, Set-II]
Past Year Questions: Deflection of Beams | Solid Mechanics - Mechanical EngineeringAns:

Past Year Questions: Deflection of Beams | Solid Mechanics - Mechanical Engineering

Deflection of point B = Deflection of spring

Past Year Questions: Deflection of Beams | Solid Mechanics - Mechanical Engineering

Where, R= Force in the spring,

Past Year Questions: Deflection of Beams | Solid Mechanics - Mechanical Engineering

0.064(50 - R) = R
3.2= R+ 0.064 R
R = 3.0075 N

Q14: A horizontal beam ABC is loaded as shown in the figure below. The distance of the point of contraflexure from end A (in m) is _______ .    [2015 : 1 Mark, Set-II]
Past Year Questions: Deflection of Beams | Solid Mechanics - Mechanical Engineering

Ans:
Reaction at B,

Past Year Questions: Deflection of Beams | Solid Mechanics - Mechanical Engineering

Past Year Questions: Deflection of Beams | Solid Mechanics - Mechanical Engineering

Past Year Questions: Deflection of Beams | Solid Mechanics - Mechanical Engineering

ΔB = 0 (Compatibility condition)

Past Year Questions: Deflection of Beams | Solid Mechanics - Mechanical Engineering

Past Year Questions: Deflection of Beams | Solid Mechanics - Mechanical Engineering

∴ RB = 15 kN

BM at a distance x from free end,

Past Year Questions: Deflection of Beams | Solid Mechanics - Mechanical EngineeringBMx = 10 * x - 15 x (x - 0,25)= 0
⇒ 10x = 15x-3.75
⇒ 5x = 3.75
∴ x = 0.75m
∴ From end A, distance is 0.25 m.

Q15: The beam of an overall depth 250 mm (shown below) is used in a building subjected to two different thermal environments. The temperatures at the top and bottom surfaces of the beam are 360C and 720C respectively. Considering coefficient of thermal expansion (α) as 1.50 x 10-5 per 0C, the vertical deflection of the beam (in mm) at its midspan due to temperature gradient is _______ . [2014 : 2 Marks, Set-II]
Past Year Questions: Deflection of Beams | Solid Mechanics - Mechanical EngineeringAns:
Method-I

Past Year Questions: Deflection of Beams | Solid Mechanics - Mechanical Engineering

Past Year Questions: Deflection of Beams | Solid Mechanics - Mechanical Engineering

From properties of circle,

Past Year Questions: Deflection of Beams | Solid Mechanics - Mechanical Engineering

(Considering ‘δ’ very small so neglect δ2)

⇒ Past Year Questions: Deflection of Beams | Solid Mechanics - Mechanical Engineering

Past Year Questions: Deflection of Beams | Solid Mechanics - Mechanical Engineering

Past Year Questions: Deflection of Beams | Solid Mechanics - Mechanical Engineering

Past Year Questions: Deflection of Beams | Solid Mechanics - Mechanical Engineering

= 2.43

Method-ll

Past Year Questions: Deflection of Beams | Solid Mechanics - Mechanical Engineering

Past Year Questions: Deflection of Beams | Solid Mechanics - Mechanical Engineering

Past Year Questions: Deflection of Beams | Solid Mechanics - Mechanical Engineering

Past Year Questions: Deflection of Beams | Solid Mechanics - Mechanical Engineering


Q16: The tension (in kN) in a 10m long cable, shown in the figure, neglecting its self-weight is    [2014 : 2 Marks, Set-II]

Past Year Questions: Deflection of Beams | Solid Mechanics - Mechanical Engineering

(a) 120
(b) 75
(c) 60
(d) 45
Ans: (b)

Past Year Questions: Deflection of Beams | Solid Mechanics - Mechanical Engineering

Past Year Questions: Deflection of Beams | Solid Mechanics - Mechanical Engineering

Past Year Questions: Deflection of Beams | Solid Mechanics - Mechanical Engineering

⇒ 2T cosθ = 120 ...(i)

Here,

Past Year Questions: Deflection of Beams | Solid Mechanics - Mechanical Engineering

Past Year Questions: Deflection of Beams | Solid Mechanics - Mechanical Engineering

⇒ Past Year Questions: Deflection of Beams | Solid Mechanics - Mechanical Engineering

Q17: The axial load (in kN) in the member PQ for the arrangement/assembly shown in the figure given below is _________.    [2014 : 2 Marks, Set-II]
Past Year Questions: Deflection of Beams | Solid Mechanics - Mechanical EngineeringAns:
Free body diagram,
For principle of superposition,

Past Year Questions: Deflection of Beams | Solid Mechanics - Mechanical Engineering

Past Year Questions: Deflection of Beams | Solid Mechanics - Mechanical Engineering

Deflections due to axial forces will be very less as compared to bending forces.
So we can neglect the axial deformation.
∴ From equation (i),

Past Year Questions: Deflection of Beams | Solid Mechanics - Mechanical Engineering

⇒ VQ = 50 kN

Q18: For the cantilever beam of span 3 m (shown below), a concentrated load of 20 kN applied at the free end causes a vertical displacement of 2 mm at a section located at a distance of 1 m from the fixed end. If a concentrated vertically downward load of 10 kN is applied at the section located at a distance of 1 m from the fixed end (with no other load on the beam), the maximum vertical displacement in the same beam (in mm) is ____.    [2014 : 2 Marks, Set-I]
Past Year Questions: Deflection of Beams | Solid Mechanics - Mechanical EngineeringAns:

Past Year Questions: Deflection of Beams | Solid Mechanics - Mechanical Engineering

Past Year Questions: Deflection of Beams | Solid Mechanics - Mechanical Engineering

Past Year Questions: Deflection of Beams | Solid Mechanics - Mechanical Engineering

Q19: A uniform beam (EI= constant) PQ in the form of a quarter circle of radius R is fixed at end P and free at the end Q, where a load H/is applied as shown. The vertical downward displacement δQ at the loaded point Q is given by Past Year Questions: Deflection of Beams | Solid Mechanics - Mechanical Engineering Find the value of β correct to 4-decimal place. [2013 : 2 Marks]
Past Year Questions: Deflection of Beams | Solid Mechanics - Mechanical EngineeringAns:

Past Year Questions: Deflection of Beams | Solid Mechanics - Mechanical Engineering

Past Year Questions: Deflection of Beams | Solid Mechanics - Mechanical Engineering

Past Year Questions: Deflection of Beams | Solid Mechanics - Mechanical Engineering

Past Year Questions: Deflection of Beams | Solid Mechanics - Mechanical Engineering

Q20: A simply supported beam is subjected to a uniformly distributed load of intensity w per unit length, on half of the span from one end. The length of the span and the flexural stiffness are denoted as I and EI respectively. The deflection at mid-span of the beam is [2012 : 2 Marks]
Past Year Questions: Deflection of Beams | Solid Mechanics - Mechanical Engineering
Past Year Questions: Deflection of Beams | Solid Mechanics - Mechanical Engineering
Past Year Questions: Deflection of Beams | Solid Mechanics - Mechanical Engineering
Past Year Questions: Deflection of Beams | Solid Mechanics - Mechanical Engineering

Ans:
Method-I

Past Year Questions: Deflection of Beams | Solid Mechanics - Mechanical Engineering

Past Year Questions: Deflection of Beams | Solid Mechanics - Mechanical Engineering

Method-ll

Past Year Questions: Deflection of Beams | Solid Mechanics - Mechanical Engineering

Total deflection

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FAQs on Past Year Questions: Deflection of Beams - Solid Mechanics - Mechanical Engineering

1. What is beam deflection and why is it important in engineering?
Ans.Beam deflection refers to the displacement of a beam under load. It is important in engineering because excessive deflection can affect the structural integrity, safety, and functionality of a structure. Understanding deflection helps engineers design beams that can safely support loads without excessive bending or deformation.
2. What are the common methods used to calculate beam deflection?
Ans.Common methods for calculating beam deflection include the double integration method, the moment-area theorem, and the virtual work method. Each method has its advantages and is used based on the complexity of the beam and loading conditions.
3. How does the type of beam support influence deflection?
Ans.The type of beam support, such as simply supported, fixed, or cantilever, significantly influences deflection. Fixed supports typically restrict rotation and can reduce deflection compared to simply supported beams, while cantilever beams experience greater deflection due to their unsupported ends.
4. What factors affect the deflection of a beam?
Ans.Factors affecting beam deflection include the material properties (modulus of elasticity), the beam's moment of inertia, the length of the beam, the magnitude and type of load applied, and the support conditions. Each of these factors plays a crucial role in determining how much a beam will deflect under a given load.
5. How can engineers minimize beam deflection in their designs?
Ans. Engineers can minimize beam deflection by selecting materials with a higher modulus of elasticity, increasing the beam's moment of inertia through its cross-sectional design, limiting the length of the beam, or using additional supports to reduce the effective span. Proper design and analysis ensure that the deflection remains within acceptable limits for safety and performance.
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