**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:**

δ_{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-m**^{2} are shown in the figures.

If the mid-span deflections of these beams are denoted by δ_{1} and δ_{2} (as indicated in the figures), the correct option is [2017 : 2 Marks, Set-II]

(a) δ_{1} = δ_{2}

(b) δ_{1} < δ_{2}

(c) δ_{1 }> δ_{2}

(d) δ_{1} >> δ_{2}

Ans. (A)

**Solution:**

∴ δ_{1 }= δ_{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 δ**_{max1} and δ_{max1}, respectively, in the case of beam PQ and the corresponding quantities for the beam XZ be δ_{max2} and δ_{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

⇒

∴ R = 0

In beam PQ also at support Q, vertical reaction in zero because of roller support.

So, beam PQ, xyand yz are same.

∴

**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 10^{6} N-m^{2}, 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**

Due to symmetrical loading,

According to Macaulay method,

After integrating once

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

∴

∴ 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,

Δ_{B} = 0 (Compatibility condition)

∴ R_{B} = 15 kN

BM at a distance x from free end,

BM_{x} = 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 36**^{0}C and 72^{0}C respectively. Considering coefficient of thermal expansion (α) as 1.50 x 10^{-5} per ^{0}C, 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 ‘δ’ very small so neglect δ^{2})

⇒

= 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:**

⇒ 2T cosθ = 120 ...(i)

Here,

⇒

**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.

∴ From equation (i),

⇒ V_{Q} = 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 δ**_{Q} at the loaded point Q is given by Find the value of β 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