# GATE Civil Engineering 2023 SET-1

 Question 1
For the integral

$\mathrm{I}=\int_{-1}^{1} \frac{1}{\mathrm{x}^{2}} \mathrm{dx}$

which of the following statements is TRUE?
 A $\quad \mathrm{I}=0$ B $\quad \mathrm{I}=2$ C $\quad \mathrm{I}=-2$ D The integral does not converge
Engineering Mathematics   Calculus
Question 1 Explanation:
\begin{aligned} I & =\int_{-1}^{1} \frac{1}{x^{2}} d x \\ & =2 \int_{0}^{1} \frac{1}{x^{2}} d x \quad(\because \quad f(-x)=f(x)) \\ & =2 \lim _{t \rightarrow 0^{+}} \int_{t}^{1} \frac{d x}{x^{2}} \\ & =2 \lim _{t \rightarrow 0^{+}}\left(\frac{-1}{x}\right)_{t}^{1} \\ & =-2 \lim _{t \rightarrow 0^{+}}\left(1-\frac{1}{t}\right) \\ & =2 \lim _{t \rightarrow 0^{+}}\left(\frac{1}{t}-1\right) \\ & =2 \lim _{h \rightarrow 0^{+}}\left(\frac{1}{0+h}-1\right)\\ & =2(\infty-1) \\ & =\infty \;\;\; (Divergent) \end{aligned}
 Question 2
A hanger is made of two bars of different sizes. Each bar has a square cross-section. The hanger is loaded by three-point loads in the mid vertical plane as shown in the figure. Ignore the self-weight of the hanger. What is the maximum tensile stress in $\mathrm{N} / \mathrm{mm}^{2}$ anywhere in the hanger without considering stress concentration effects? A 15 B 25 C 35 D 45
Solid Mechanics   Principal Stress and Principal Strain
Question 2 Explanation: $\sigma_{A B}=\frac{P_{A B}}{A_{A B}}=\frac{250 \times 10^{3}}{100 \times 100}=25 \mathrm{~N} / \mathrm{mm}^{2}$
$\sigma_{B C}=\frac{P_{B C}}{A_{B C}}=\frac{50 \times 10^3}{50 \times 50}=20 \mathrm{N} / \mathrm{mm}^{2}$
$\sigma_{\max }=\sigma_{\mathrm{AB}}=25 \mathrm{~N} / \mathrm{mm}^{2}$

 Question 3
Creep of concrete under compression is defined as the
 A increase in the magnitude of strain under constant stress B increase in the magnitude of stress under constant strain C decrease in the magnitude of strain under constant stress D decrease in the magnitude of stress under constant strain
RCC Structures   Working Stress and Limit State Method
Question 3 Explanation:
Under sustained compressive loading, deformation in concrete increases with time even through the applied stress level is not changed. The time dependent component of strain is called creep.
 Question 4
A singly reinforced concrete beam of balanced section is made of M20 grade concrete and $\mathrm{Fe} 415$ grade steel bars. The magnitudes of the maximum compressive strain in concrete and the tensile strain in the bars at ultimate state under flexure, as per IS 456: 2000 are _______ respectively. (round off to four decimal places)
 A 0.0035 and 0.0038 B 0.0020 and 0.0018 C 0.0035 and 0.0041 D 0.0020 and 0.0031
RCC Structures   Prestressed Concrete Beams
Question 4 Explanation:
Given data,
Balanced section, singly reinforced beam.
As per Clause No. 38.1, IS $456: 2000$,
Maximum strain in concrete at the outermost compression fibre $=0.0035$
and strain in the tension reinforcement for balanced section at ultimate state under flexure
\begin{aligned} & =0.002+\frac{f_{y}}{1.15 E_{s}} \\ & =0.002+\frac{415}{1.15 \times 2 \times 10^{5}}=0.0038 \end{aligned}