# Shear, Torsion, Bond, Anchorage and Development Length

 Question 1
A reinforced concrete beam with rectangular cross section (width = 300 mm, effective depth = 580 mm) is made of $M30$ grade concrete. It has 1% longitudinal tension reinforcement of $Fe 415$ grade steel. The design shear strength for this beam is 0.66 $N/mm^2$. The beam has to resist a factored shear force of 440 kN. The spacing of two-legged, 10 mm diameter vertical stirrups of $Fe 415$ grade steel is ______mm. (round off to the nearest integer)
 A 127 B 101 C 254 D 331
GATE CE 2022 SET-1   RCC Structures
Question 1 Explanation:
$b=300mm$
$d=580 mm$
$V_u=440 kN$
Concrete used is M30
Raft steel is Fe415
$V_{cu}=\tau _c Bd=0.66 \times 300 \times \frac{580}{1000}=114.84 kN$
$V_{su}=V_u-V_{cu}=440-114.84=325.16 kN$
Spacing of 2-legged shear reinforcement
\begin{aligned} s_V&=\frac{A_{SV} \times 0.87 f_y \times d}{V_{su}}\\ &=\frac{2 \times \frac{\pi}{2} \times (10)^2 \times 0.87 \times 415 \times 580}{325.16 \times 1000}\\ &=101.16 mm \end{aligned}
 Question 2
A reinforcing steel bar, partially embedded in concrete, is subjected to a tensile force P. The figure that appropriately represents the distribution of the magnitude of bond stress (represented as hatched region), along the embedded length of the bar, is A A B B C C D D
GATE CE 2020 SET-1   RCC Structures
 Question 3
Tie bars of 12 mm diameter are to be provided in a concrete pavement slab. The working tensile stress of the tie bars is 230 MPa, the average bond strength between a tie bar and concrete is 2 MPa, and the joint gap between the slabs is 10 mm. Ignoring the loss of bond and the tolerance factor, the design length of the tie bars (in mm, round off to the nearest integer) is _____
 A 350 B 700 C 1050 D 850
GATE CE 2019 SET-1   RCC Structures
Question 3 Explanation: $L_d=\frac{\phi \sigma _{st}}{4\tau _{bd}}=\frac{12 \times 230}{4 \times 2}=345mm$
Length of tie bar $=L_d+10+L_d=345+10+345=700mm$
 Question 4
For a given loading on a rectangular plain concrete beam with an overall depth of 500 mm, the compressive strain and tensile strain developed at the extreme fibers are of the same magnitude of $2.5 \times 10^{-4}$. The curvature in the beam cross-section (in $m^{-1}$, round off to 3 decimal places), is ____
 A 0.008 B 0.006 C 0.004 D 0.001
GATE CE 2019 SET-1   RCC Structures
Question 4 Explanation:
Simple bending equation
\begin{aligned} \frac{m}{I}&=\frac{f}{y}=\frac{E}{R} \\ \frac{1}{R}&=\frac{f}{E}\cdot \frac{1}{y} \\ &=\frac{\varepsilon }{y}=\frac{2.5 \times 10^{-4}}{(0.5/2)}=0.001m^{-1} \end{aligned}
 Question 5
An RCC beam of rectangular cross section has factored shear of 200 kN at its critical section. Its width b is 250 mm and effective depth d is 350 mm. Assume design shear strength $\tau _{c}$ of concrete as 0.62 $N/mm^{2}$ and maximum allowable shear stress $\tau _{c,max}$ in concrete as 2.8 $N/mm^{2}$ . If two legged 10 mm diameter vertical stirrups of Fe250 grade steel are used, then the required spacing (in cm, up to one decimal place) as per limit state method will be ______
 A 5.5 B 8.2 C 15.6 D 19.4
GATE CE 2018 SET-1   RCC Structures
Question 5 Explanation:
Nominal shear stress,
\begin{aligned} \tau_{v}&=\frac{V_{u}}{b d}=\frac{200 \times 10^{3}}{250 \times 350}=2.286 \mathrm{N} / \mathrm{mm}^{2} \lt \tau_{c m a x} \\ &\text{(OK)}\\ SF&\text{ taken by stirrups}\\ &=\left(\tau_{v}-\tau_{c}\right) b d \\ &=(2.286-0.62) \times 250 \times 350 \\ &=145.75 \mathrm{kN} \\ \text{Now, }V_{u s}&=\frac{0.87 f_{y} A_{s v} d}{S_{v}} \\ \Rightarrow \quad S_{v}&=\frac{0.87 \times 250 \times 350 \times 2 \times \frac{\pi}{4} \times 10^{2}}{145.75 \times 10^{3}} \\ &=82 \mathrm{mm}=8.2 \mathrm{cm} \end{aligned}
 Question 6
As per IS 456-2000 for the design of reinforced concrete beam, the maximum allowable shear stress $\tau _{cmax}$ depends on the
GATE CE 2016 SET-2   RCC Structures
 Question 7
In shear design of an RC beam, other than the allowable shear strength of concrete $(\tau _{c})$ there is also an additional check suggested in IS 456-2000 with respect to the maximum permissible shear stress $(\tau _{cmax})$. The check for $(\tau _{cmax})$ is required to take care of
 A additional shear resistance from reinforcing steel B additional shear stress that comes from accidental loading C possibility of failure of concrete by diagonal tension D possibility of crushing of concrete by diagonal compression
GATE CE 2016 SET-1   RCC Structures
 Question 8
The development length of a deformed reinforncement bar can be expressed as (1/k) $(\phi \sigma _{s}/\tau _{bd})$. From the IS: 456-2000, the value of k can be calculated as________.
 A 6.4 B 9.6 C 3.4 D 8.3
GATE CE 2015 SET-1   RCC Structures
Question 8 Explanation:
Bond strength of concrete
\begin{aligned} &=\text{ Tensile force in steel}\\ \tau_{b d} \times\left(L_{d} \pi \phi\right)&=0.87 f_{y} \times\left(\frac{\pi}{4} \phi^{2}\right)\\ \Rightarrow \quad L_{d}&=\frac{0.87 f_{y} \phi}{4 \tau_{b d}} \\ \end{aligned}
For deformed bars $\tau_{\text {bd }}$ value is increased by 60 %
\begin{aligned} \therefore \quad L_{d}&=\frac{\phi \sigma_{s}}{4 \times\left(1.6 \tau_{b d}\right)}=\frac{\phi \sigma_{s}}{6.4 \tau_{b d}}\\ \therefore \quad k&=6.4 \end{aligned}
 Question 9
A rectangular beam of width (b) 230 mm and effective depth (d) 450 mm is reinforced with four bars of 12 mm diameter. The grade of concrete is M20 and grade of steel is Fe500. Given that for M20 grade of concrete the ultimate shear strength, $\tau _{uc} = 0.36 N/mm^{2}$ for steel percentage, p = 0.25, and $\tau _{uc}= 0.48 N/mm^{2}$ for p = 0.50. For a factored shear force of 45 kN, the diameter (in mm) of Fe500 steel two legged stirrups to be used at spacing of 375 mm, should be
 A 8 B 10 C 12 D 16
GATE CE 2014 SET-1   RCC Structures
Question 9 Explanation: \begin{aligned} \tau_{u c}=& 0.36 \mathrm{N} / \mathrm{mm}^{2} &[\text{For M20}]\\ & \text { for } \frac{A_{s t}}{b d} \times 100=0.25\\ \tau_{u c}=& 0.48 \mathrm{N} / \mathrm{mm}^{2} &[\text{For M20}]\\ & \text { for } \frac{A_{s t}}{b d} \times 100=0.5 \end{aligned}
Factored $S F=45 \mathrm{kN}=V_{u}$
We have to calculate the dia of Fe 500 2-Legged stirrup to be used at a spacing of 325 mm c/c
\begin{aligned} \tau_{v}&=\frac{V_{u}}{b d}=\frac{45 \times 1000}{230 \times 450}=0.4348 \mathrm{N} / \mathrm{mm}^{2}\\ \therefore \quad \%&\text{tensile steel}\\ &=\frac{4 \times \frac{\pi}{4}(12)^{2}}{230 \times 450} \times 100=0.437 \% \\ \tau_{c} &=0.36+\frac{0.12}{0.25} \times(0.437-0.25) \\ &=0.45 \mathrm{N} / \mathrm{mm}^{2} \end{aligned}
since $\tau_{v}-\tau_{c} \lt 0$
$\Rightarrow$ Min shear reinforcement is required
$\Rightarrow$ Min shear reinforcement is given by
\begin{aligned} \frac{A_{s v}}{b S_{v}} &=\frac{0.4}{0.87 f_{y}} \\ A_{s v} &=\frac{0.4 \times\left(S_{v}\right)(b)}{0.87 f_{y}} \end{aligned}
since we limit $f_{y}$ to $415 \mathrm{N} / \mathrm{mm}^{2}$ hence,
\begin{aligned} A_{s v}=2 \times \frac{\pi}{4}(\phi)^{2} &=\frac{0.4 \times 325 \times 230}{0.87 \times 415} \\ &= 82.814 \mathrm{mm}^{2} \\ \phi &=7.26 \mathrm{mm} \\ \text { adopt } \phi&=8 \mathrm{mm} \end{aligned}
 Question 10
As per IS 456:2000 for M20 grade concrete and plain barsin tension the design bond stress $\tau _{bd}=1.2 MPa$.Further, IS 456:2000 permits this design bond stress value to be increased by 60% for HSD bars. The stress in the HSD reinforcing steel barsin tension, $\sigma _{s}=360MPa$. Find the required development length, $L_{d}$, for HSD barsin terms of the bar diameter, $\phi$. __________
 A 34.54 B 46.87 C 89.25 D 93.47
GATE CE 2013   RCC Structures
Question 10 Explanation:
\begin{aligned} L_{d} &=\frac{\phi \sigma_{s t}}{4 \tau_{b d}}=\frac{\phi \times 360}{4 \times 1.2 \times 1.60} \\ &=46.875 \phi \end{aligned}
There are 10 questions to complete.