# Footing, Columns, Beams and Slabs

 Question 1
With regard to the shear design of RCC beams, which of the following statements is/are TRUE?
 A Excessive shear reinforcement can lead to compression failure in concrete B Beams without shear reinforcement, even if adequately designed for flexure, can have brittle failure C The main (longitudinal) reinforcement plays no role in the shear resistance of beam D As per IS456:2000, the nominal shear stress in the beams of varying depth depends on both the design shear force as well as the design bending moment
GATE CE 2023 SET-2   RCC Structures
Question 1 Explanation:
Option A is true, because when the area of shear reinforcement is large i.e. in case of excessive shear reinforcement, concrete becomes stronger is diagonal tension on failure compared to diagonal compression failure and compression failure may occur before the shear reinforcement has yielded.
Option '$B$' is true because for beams without shear reinforcement, once the flexural crack crosses the longitudinal reinforcement, the propagation of crack will be sudden and there can be brittle failure.
Option '$C$' is not true, as main reinforcement increases the shear resitance by providing dowel action, limiting crack width and by increasing the depth of concrete. Design shear strength of concrete is a function of grade of concree and percentage tensile reinforcement.
Option 'D' is true, as for beams with varying depth, nominal shear stress,
$\tau_{v}=\frac{V_{u} \pm \frac{M_{u} \tan \beta}{d}}{b d}($ clause 40.1.1, IS456: 2000)
it is dependent on both the design shear force and design bending moment.
Hence, options (A), (B) and (D) are true.
 Question 2
Read the following statements relating to flexure of reinforced concrete beams:

I. In over-reinforced sections, the failure strain in concrete reaches earlier than the yield strain in steel.
II. In under-reinforced sections, steel reaches yielding at a load lower than the load at which the concrete reaches failure strain.
III. Over-reinforced beams are recommended in practice as compared to the under-reinforced beams.
IV. In balanced sections, the concrete reaches failure strain earlier than the yield strain in tensile steel.

Each of the above statements is either True or False.
Which one of the following combinations is correct?
 A I (True), II (True), III (False), IV (False) B I (True), II (True), III (False), IV (True) C I (False), II (False), III (True), IV (False) D I (False), II (True), III (True), IV (False)
GATE CE 2022 SET-2   RCC Structures
Question 2 Explanation:
The question is based on LSM design principle as it is describing different conditions related to strain
Depending on amount of reinforcement in a cross- section, here ca be three types of sections viz. balanced, under reinforced and over reinforced.
Balanced section is a section that is expected to result in a balanced failure. It means at the ultimate limit state in flexure, the concrete will attain a limiting compressive strain of 0.0035 and steel will attain minimum specified tensile strain of $0.002+\frac{0.87f_y}{E_s}$
Under reinforced section is a section in which steel yield before collapse. Over reinforced section is a section in which crushing of concrete in compression i.e. attainment of compressive strain of 0.0035 occurs prior to yielding of steel.
In case of over reinforced section the deflection, crack width remain relatively low and failure occurs without any sign of warning and hence over reinforced flexural members are not recommended by IS code.
Based on the above information:
Statement I is true.
Statement II is true.
Statement III is false.
Statement IV is false.

 Question 3
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 3 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 4
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 4 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 5
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 5 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
$x_{0,lim}=221.52mm$

There are 5 questions to complete.