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

## 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

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

Method-ll
Moment area method:

Î´Q/P = Defleciton of point Q wrt to tangent at point P

Q.2 Two prismatic beams having the same flexural rigidity of 1000 kN-m2 are shown in the figures.

If the mid-span deflections of these beams are denoted by Î´1 and Î´2 (as indicated in the figures), the correct option is    [2017 : 2 Marks, Set-II]
(a) Î´1 = Î´2
(b) Î´1 < Î´2
(c) Î´> Î´2
(d) Î´1 >> Î´2
Ans.
(A)
Solution:

âˆ´ Î´= Î´2

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 Î´max1 and Î´max1, respectively, in the case of beam PQ and the corresponding quantities for the beam XZ be Î´max2 and Î´max2, respectively.

Which one of the following relationship is true?    [2016 : 2 Marks, Set-I]
(a)
(b)
(c)
(d)
Ans. (D)
Solution:

Deflection in beam xy at y = Deflection in beam yz at y
â‡’
âˆ´ R = 0
In beam PQ also at support Q, vertical reaction in zero because of roller support.
So, beam PQ, xyand yz are same.
âˆ´

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

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

According to Macaulay method,

After integrating once

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

âˆ´
âˆ´ Equation of slope,

The slope will be maximum at the support,

Method-ll
Moment Area Method,

Area of M/El diagram between points P and B.

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]

Solution:

Compression of spring

% 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]

Solution:

Deflection of point B = Deflection of spring

Where, R= Force in the spring,

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]

Solution:

Reaction at B,

Î”B = 0 (Compatibility condition)

âˆ´ RB = 15 kN
BM at a distance x from free end,

BMx = 10 * x - 15 x (x - 0,25)= 0
â‡’ 10x = 15x-3.75
â‡’ 5x = 3.75
âˆ´ x = 0.75m
âˆ´ 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 (Î±) 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]

Solution:
Method-I

From properties of circle,

(Considering â€˜Î´â€™ very small so neglect Î´2)
â‡’

= 2.43
Method-ll

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]

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

â‡’ 2T cosÎ¸ = 120 ...(i)
Here,

â‡’

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]

Solution:

Free body diagram,
For principle of superposition,

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

â‡’ 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]

Solution:

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 Î´Q at the loaded point Q is given by  Find the value of Î² correct to 4-decimal place.    [2013 : 2 Marks]

Solution:

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]

Ans.
(B)
Solution:
Method-I

Method-ll

Total deflection

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