Deflection of Beams Civil Engineering (CE) Notes | EduRev

Topic wise GATE Past Year Papers for Civil Engineering

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Civil Engineering (CE) : Deflection of Beams Civil Engineering (CE) Notes | EduRev

The document Deflection of Beams Civil Engineering (CE) Notes | EduRev is a part of the Civil Engineering (CE) Course Topic wise GATE Past Year Papers for Civil Engineering.
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Q.1 The figure shows a simply supported beam PQ of uniform flexural rigidity El carrying two moments M and 2M
Deflection of Beams Civil Engineering (CE) Notes | EduRev
The slope at P will be    [2018 : 2 Marks, Set-I]
(a) 0
(b) ML/(9EI)
(c) ML/(6EI)
(d) ML/(3EI)
Ans. 
(C)
Solution:
Method-l
Deflection of Beams Civil Engineering (CE) Notes | EduRev
Deflection of Beams Civil Engineering (CE) Notes | EduRev
Method-ll
Moment area method:
Deflection of Beams Civil Engineering (CE) Notes | EduRev
Deflection of Beams Civil Engineering (CE) Notes | EduRev
δQ/P = Defleciton of point Q wrt to tangent at point P
Deflection of Beams Civil Engineering (CE) Notes | EduRev
Deflection of Beams Civil Engineering (CE) Notes | EduRev

Q.2 Two prismatic beams having the same flexural rigidity of 1000 kN-m2 are shown in the figures.
Deflection of Beams Civil Engineering (CE) Notes | EduRev
If 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) δ> δ2
(d) δ1 >> δ2
Ans. 
(A)
Solution:
Deflection of Beams Civil Engineering (CE) Notes | EduRev
∴ δ= δ2

Q.3 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.
Deflection of Beams Civil Engineering (CE) Notes | EduRev
Deflection of Beams Civil Engineering (CE) Notes | EduRev
Which one of the following relationship is true?    [2016 : 2 Marks, Set-I]
(a) Deflection of Beams Civil Engineering (CE) Notes | EduRev
(b) Deflection of Beams Civil Engineering (CE) Notes | EduRev
(c) Deflection of Beams Civil Engineering (CE) Notes | EduRev
(d) Deflection of Beams Civil Engineering (CE) Notes | EduRev
Ans. (D)
Solution:
Deflection of Beams Civil Engineering (CE) Notes | EduRev
Deflection in beam xy at y = Deflection in beam yz at y
Deflection of Beams Civil Engineering (CE) Notes | EduRev
∴ R = 0
In beam PQ also at support Q, vertical reaction in zero because of roller support.
So, beam PQ, xyand yz are same.
Deflection of Beams Civil Engineering (CE) Notes | EduRev

Q.4 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
Deflection of Beams Civil Engineering (CE) Notes | EduRev
If 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)
Solution:
Method-1
Deflection of Beams Civil Engineering (CE) Notes | EduRev
Due to symmetrical loading,
Deflection of Beams Civil Engineering (CE) Notes | EduRev
Deflection of Beams Civil Engineering (CE) Notes | EduRev
Deflection of Beams Civil Engineering (CE) Notes | EduRev
According to Macaulay method,
Deflection of Beams Civil Engineering (CE) Notes | EduRev
After integrating once
Deflection of Beams Civil Engineering (CE) Notes | EduRev
Due to symmetrical loading slope will be zero at mid section (x = 1.5 m),
Deflection of Beams Civil Engineering (CE) Notes | EduRev
Deflection of Beams Civil Engineering (CE) Notes | EduRev
Deflection of Beams Civil Engineering (CE) Notes | EduRev
∴ Equation of slope,
Deflection of Beams Civil Engineering (CE) Notes | EduRev
The slope will be maximum at the support,
Deflection of Beams Civil Engineering (CE) Notes | EduRev
Method-ll
Moment Area Method,
Deflection of Beams Civil Engineering (CE) Notes | EduRev
Deflection of Beams Civil Engineering (CE) Notes | EduRev
Deflection of Beams Civil Engineering (CE) Notes | EduRev Area of M/El diagram between points P and B.
Deflection of Beams Civil Engineering (CE) Notes | EduRev

Q.5 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]
Deflection of Beams Civil Engineering (CE) Notes | EduRev
Solution:
Deflection of Beams Civil Engineering (CE) Notes | EduRev
Compression of spring
Deflection of Beams Civil Engineering (CE) Notes | EduRev
Deflection of Beams Civil Engineering (CE) Notes | EduRev
Deflection of Beams Civil Engineering (CE) Notes | EduRev
% force carried by spring = 25%

Q.6 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]
Deflection of Beams Civil Engineering (CE) Notes | EduRev
Solution:

Deflection of Beams Civil Engineering (CE) Notes | EduRev
Deflection of point B = Deflection of spring
Deflection of Beams Civil Engineering (CE) Notes | EduRev
Where, R= Force in the spring,
Deflection of Beams Civil Engineering (CE) Notes | EduRev
0.064(50 - R) = R
3.2= R+ 0.064 R
R = 3.0075 N

Q.7 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]
Deflection of Beams Civil Engineering (CE) Notes | EduRev
Solution:

Reaction at B,
Deflection of Beams Civil Engineering (CE) Notes | EduRev
Deflection of Beams Civil Engineering (CE) Notes | EduRev
Deflection of Beams Civil Engineering (CE) Notes | EduRev
ΔB = 0 (Compatibility condition)
Deflection of Beams Civil Engineering (CE) Notes | EduRev
Deflection of Beams Civil Engineering (CE) Notes | EduRev
∴ RB = 15 kN
BM at a distance x from free end,
Deflection of Beams Civil Engineering (CE) Notes | EduRev
BMx = 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.

Q.8 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]
Deflection of Beams Civil Engineering (CE) Notes | EduRev
Solution:
Method-I
Deflection of Beams Civil Engineering (CE) Notes | EduRev
Deflection of Beams Civil Engineering (CE) Notes | EduRev
From properties of circle,
Deflection of Beams Civil Engineering (CE) Notes | EduRev
(Considering ‘δ’ very small so neglect δ2)
Deflection of Beams Civil Engineering (CE) Notes | EduRev
Deflection of Beams Civil Engineering (CE) Notes | EduRev
Deflection of Beams Civil Engineering (CE) Notes | EduRev
Deflection of Beams Civil Engineering (CE) Notes | EduRev
= 2.43
Method-ll
Deflection of Beams Civil Engineering (CE) Notes | EduRev
Deflection of Beams Civil Engineering (CE) Notes | EduRev
Deflection of Beams Civil Engineering (CE) Notes | EduRev
Deflection of Beams Civil Engineering (CE) Notes | EduRev

Q.9 The tension (in kN) in a 10m long cable, shown in the figure, neglecting its self-weight is    [2014 : 2 Marks, Set-II]
Deflection of Beams Civil Engineering (CE) Notes | EduRev
(a) 120 
(b) 75 
(c) 60 
(d) 45
Ans. (B)
Solution:
Deflection of Beams Civil Engineering (CE) Notes | EduRev
Deflection of Beams Civil Engineering (CE) Notes | EduRev
Deflection of Beams Civil Engineering (CE) Notes | EduRev
⇒ 2T cosθ = 120 ...(i)
Here,
Deflection of Beams Civil Engineering (CE) Notes | EduRev
Deflection of Beams Civil Engineering (CE) Notes | EduRev
Deflection of Beams Civil Engineering (CE) Notes | EduRev

Q.10 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]
Deflection of Beams Civil Engineering (CE) Notes | EduRev
Solution:

Free body diagram,
For principle of superposition,
Deflection of Beams Civil Engineering (CE) Notes | EduRev
Deflection of Beams Civil Engineering (CE) Notes | EduRev
Deflections due to axial forces will be very less as compared to bending forces.
So we can neglect the axial deformation.
∴ From equation (i),
Deflection of Beams Civil Engineering (CE) Notes | EduRev
⇒ VQ = 50 kN

Q.11 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]
Deflection of Beams Civil Engineering (CE) Notes | EduRev
Solution:

  Deflection of Beams Civil Engineering (CE) Notes | EduRev
Deflection of Beams Civil Engineering (CE) Notes | EduRev
Deflection of Beams Civil Engineering (CE) Notes | EduRev

Q.12 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 Deflection of Beams Civil Engineering (CE) Notes | EduRev Find the value of β correct to 4-decimal place.    [2013 : 2 Marks]
Deflection of Beams Civil Engineering (CE) Notes | EduRev
Solution:

Deflection of Beams Civil Engineering (CE) Notes | EduRev
Deflection of Beams Civil Engineering (CE) Notes | EduRev
Deflection of Beams Civil Engineering (CE) Notes | EduRev
Deflection of Beams Civil Engineering (CE) Notes | EduRev

Q.13 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]
Deflection of Beams Civil Engineering (CE) Notes | EduRev
Deflection of Beams Civil Engineering (CE) Notes | EduRev
Deflection of Beams Civil Engineering (CE) Notes | EduRev
Deflection of Beams Civil Engineering (CE) Notes | EduRev
Ans. 
(B)
Solution:
Method-I
Deflection of Beams Civil Engineering (CE) Notes | EduRev
Deflection of Beams Civil Engineering (CE) Notes | EduRev
Method-ll
Deflection of Beams Civil Engineering (CE) Notes | EduRev
Total deflection
Deflection of Beams Civil Engineering (CE) Notes | EduRev 

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