# Footing, Columns, Beams and Slabs

 Question 1
A rectangular cross-section of a reinforced concrete beam is shown in the figure. The diameter of each reinforcing bar is 16 mm. The values of modulus of elasticity of concrete and steel are $2.0 \times 10^{4} \mathrm{MPa} \text { and } 2.1 \times 10^{5} \mathrm{MPa}$, respectively.

The distance of the centroidal axis from the centerline of the reinforcement (x) for the uncracked section (in mm,round off to one decimal place) is ____________
 A 129.4 B 178.6 C 145.6 D 98.2
GATE CE 2021 SET-2   RCC Structures
Question 1 Explanation:

\begin{aligned} m&=\frac{E_{s}}{E_{C}}=\frac{2.1 \times 10^{5}}{2 \times 10^{4}}=10.5 \\ A_{s t}&=3 \times \frac{\pi}{4}(16)^{2}=603.20 \mathrm{~mm}^{2} \\ \bar{y}&=\frac{\left(B \cdot D \cdot \frac{D}{2}+(m-1) \times A_{s t} \times d\right)}{B \cdot D+(m-1) \cdot A_{s t}} \\ &=\frac{\left(200 \times \frac{350^{2}}{2}+(10.5-1) \times 603.2 \times 315\right)}{200 \times 350+(10.5-1) \times 603.2}=185.59 \mathrm{~mm}\\ & \text{Distance of N-A from reinforcement}\\ y_{2} &=d-\bar{y} \\ &=315-185.59=129.41 \mathrm{~mm} \end{aligned}
 Question 2
A combined trapezoidal footing of length L supports two identical square columns ($P_{1}$ and $P_{2}$) of size 0.5m x 0.5m, as shown in the figure. The columns $P_{1}$ and $P_{2}$ carry loads of 2000 kN and 1500 kN, respectively.

If the stress beneath the footing is uniform, the length of the combined footing L (in m,round off to two decimal places) is _____
 A 4.52 B 2.78 C 5.83 D 2.45
GATE CE 2021 SET-1   RCC Structures
Question 2 Explanation:
C.G. of load from $P_{1}$

\begin{aligned} P_{R} \bar{x} &=P_{1} \times 0+P_{2} \times 5 \\ \bar{x} &=\frac{1500 \times 5}{3500}=2.143 \mathrm{~m} \end{aligned}
Distance of C.G. of footing from face of $P_{1}$
$\bar{y}=\bar{x}+0.25=2.393 \mathrm{~m}$

C.G of footing \begin{aligned} \bar{y} &=\left(\frac{B_{1}+2 B_{2}}{B_{1}+B_{2}}\right) \times \frac{L}{3} \\ 2.393 &=\left(\frac{5+2 \times 1.5}{5+1.5}\right) \times \frac{L}{3} \\ L &=5.833 \mathrm{~m} \text { say } 5.83 \mathrm{~m} \end{aligned}
 Question 3
The cross-section of the reinforced concrete beam having an effective depth of 500 mm is shown in the figure (not drawn to the scale). The grades of concrete and steel used are M35 and Fe550, respectively. The area of tension reinforcement is 400 $mm^2$. It is given that corresponding to 0.2% proof stress, the material safety factor is 1.15 and the yield strain of Fe550 steel is 0.0044.

As per IS 456:2000, the limiting depth (in mm, round off to the nearest integer) of the neutral axis measured from the extreme compression fiber, is ________.
 A 125.14 B 235.21 C 365.47 D 221.52
GATE CE 2020 SET-2   RCC Structures
Question 3 Explanation:

For a RCC T-Beam
(For limiting depth of neutral axis)
Considering d=500mm
\begin{aligned} \frac{0.0035}{x_{u,lim}}&=\frac{0.0044}{d-x_{u,lim}}\\ d-x_{u,lim}&=\frac{0.0044}{0.0035} \times x_{u,lim}\\ 35 \times 500&=35x_{u,lim}+44x_{u,lim}\\ &=79x_{u,lim}\\ x_{u,lim}&=\frac{35 \times 500}{79}\\ &=221.52 mm \end{aligned}
Limiting depth of neutral axis
$x_{0,lim}=221.52mm$
 Question 4
The singly reinforced concrete beam section shown in the figure (not drawn to the scale) is made of M25 grade concrete and Fe500 grade reinforcing steel. The total crosssectional area of the tension steel is 942 $mm^2$.

As per Limit State Design of IS 456 : 2000, the design moment capacity (in kNm round off to two decimal places) of the beam section, is __________
 A 151.77 B 252.36 C 121.12 D 158.28
GATE CE 2020 SET-1   RCC Structures
Question 4 Explanation:
\begin{aligned} B&=300mm\\ d&=450mm\\ A_{st}&=942 mm^2\\ M_u&=?\\ (i)\; x_{ulim}&=0.46 \times d\\ &=0.46 \times 450=207mm\\ (ii)\;\; x_u&=\frac{0.87f_yA_{st}}{0.36 f_{ck}B}\\ &=\frac{0.87 \times 500 \times 942}{0.36 \times 25 \times 300}\\ &=151.77mm\\ (iii)\;\; x_y &\lt x_{ulim}\; \text{It is under reinforcement section}\\ (iv)\;\; M_u&=0.36f_{ck}B x_u(d-0.42x_u)\\ &=0.36 \times 25 \times 300 \times 151.77 \\ &\times (450-0.42 \times 151.77)/10^6\\ &=158.28kN-m \end{aligned}
 Question 5
A singly-reinforced rectangular concrete beam of width 300 mm and effective depth 400 mm is to be designed using M25 grade concrete and Fe500 grade reinforcing steel. For the beam to be under-reinforced, the maximum number of 16 mm diameter reinforcing bars that can be provided is
 A 3 B 4 C 5 D 6
GATE CE 2018 SET-2   RCC Structures
Question 5 Explanation:
\begin{aligned} B&=300 \mathrm{mm}\\ d&=400 \mathrm{mm} \text{ (effective depth)} \\ &\mathrm{M} 25\text{ and }\mathrm{Fe} 500 \\ A_{s t,} \lim &=0.414\left(\frac{f_{c k}}{f_{y}}\right) x_{u, l \operatorname{irn}} b \\ &=0.414\left(\frac{25}{500}\right) 0.46 \times 400 \times 300 \\ &=1142.64 \mathrm{mm}^{2}\\ \text{Number of }&16 \mathrm{mm} \phi \\ &=\frac{1142.64}{\frac{\pi}{4}(16)^{2}}=5.68\\ \end{aligned}
For $A_{s t} \lt A_{s t \text { lim' }},$ maximum number of bars to be provided is 5.
 Question 6
A reinforced-concrete slab with effective depth of 80 mm is simply supported at two opposite ends on 230 mm thick masonry walls. The centre-to-centre distance between the walls is 3.3 m. As per IS 456 : 2000, the effective span of the slab (in m, up to two decimal places) is ______
 A 3.15 B 2.25 C 1.15 D 2
GATE CE 2018 SET-2   RCC Structures
Question 6 Explanation:
Effective depth
$d=80 \mathrm{mm}$
Width of support =230 mm
c/c distance between walls =3.30 m
Clear span of slab =3.30-0.23=3.07 m
Effective span
= Minimum $\left\{\begin{array}{l}\left(L_{\text {clear }}+d\right) \\ c / c \text { distance between supports }\end{array}\right.$
= Minimum $\left\{\begin{array}{l}(3.07+0.08=3.15 \mathrm{m}) \\ 3.3 \mathrm{m}\end{array}\right.$
$So, \quad L_{\mathrm{eft}}=3.15 \mathrm{m}$
 Question 7
A structural member subjected to compression, has both translation and rotation restrained at one end, while only translation is restrained at the other end. As per IS 456 : 2000, the effective length factor recommended for design is
 A 0.5 B 0.65 C 0.7 D 0.8
GATE CE 2018 SET-2   RCC Structures
Question 7 Explanation:
One end is fixed

Other end is pin jointed
Effective length of column (as per IS:456-2000)$=0.08L$
 Question 8
An RCC short column (with lateral ties) of rectangular cross section of 250 mm x 300 mm is reinforced with four numbers of 16 mm diameter longitudinal bars. The grades of steel and concrete are Fe415 and M20, respectively. Neglect eccentricity effect. Considering limit state of collapse in compression (IS 456 : 2000), the axial load carrying capacity of the column (in kN, up to one decimal place), is ______
 A 918.1 B 120.5 C 540.4 D 650.6
GATE CE 2018 SET-1   RCC Structures
Question 8 Explanation:
since eccentricity effect is being neglected so column can be considered as concentrically loaded.
Ultimate axial load carrying capacity of column.
\begin{aligned} P_{u} &=0.45 \mathrm{f}_{c k} A_{g}+\left(0.75 \mathrm{f}_{y}-0.45 \mathrm{f}_{c k}\right) A_{\mathrm{sc}} \\ &=0.45 \times 20 \times 250 \times 300 \\ &+(0.75 \times 415-0.45 \times 20) 4 \times \frac{\pi}{4} \times 16^{2} \\ &=918.1 \mathrm{kN} \end{aligned}
 Question 9
Two rectangular under-reinforced concrete beam sections X and Y are similar in all aspects except that the longitudinal compression reinforcement in section Y is 10% more. Which one of the following is the correct statement?
 A Section X has less flexural strength and is less ductile than section Y B Section X has less flexural strength but is more ductile than section Y C Sections X and Y have equal flexural strength but different ductility D Sections X and Y have equal flexural strength and ductility
GATE CE 2018 SET-1   RCC Structures
Question 9 Explanation:

Due to presence of more compression steel in section Y, NA of section of Y is above than as of X. It means Y is more under-reinforced than X so ductility of Y is more.
Since compression steel of Y is more so flexure resistance of X is less than as of Y.
 Question 10
A column of height h with a rectangular cross-section of size ax2a has a buckling load of P. If the cross-section is changed to 0.5a x 3a and its height changed to 1.5h, the buckling load of the redesigned column will be
 A P/12 B P/4 C P/2 D 3P/4
GATE CE 2018 SET-1   RCC Structures
Question 10 Explanation:
\begin{aligned} \text { For column, } & P=\frac{\pi^{2} E I_{\min }}{L^{2}} \\ &=\frac{\pi^{2} E\left(\frac{2 a \times a^{3}}{12}\right)}{h^{2}}=\frac{\pi^{2} E a^{4}}{6 h^{2}}\\ \text{For new column, }P&=\frac{\pi^{2} E\left[\frac{3 a \times(0.5 a)^{3}}{12}\right]}{(1.5 h)^{2}} \\ &=\frac{1}{12} \times \frac{\pi^{2} E a^{4}}{6 h^{2}}=\frac{P}{12} \end{aligned}
There are 10 questions to complete.