# Properties of Metals, Stress and Strain

 Question 1
Strain hardening of structural steel means
 A experiencing higher stress than yield stress with increased deformation B strengthening steel member externally for reducing strain experienced C strain occurring before plastic flow of steel material D decrease in the stress experienced with increasing strain
GATE CE 2021 SET-2   Solid Mechanics
Question 1 Explanation:
Strain hardening is experiencing higher stress than yield stress with increased deformation
In the figure AB = Strain hardening zone
OA = Linear elastic zone
Stress corresponding to point 'A' is yield stress.

 Question 2
A square plate O-P-Q-R of a linear elastic material with sides 1.0 m is loaded in a state of plane stress. Under a given stress condition, the plate deforms to a new configuration O-P'-Q'-R' as shown in the figure (not to scale). Under the given deformation, the edges of the plate remain straight.

The horizontal displacement of the point (0.5 m, 0.5 m) in the plate O-P-Q-R (in mm,round off to one decimal place) is ________
 A 1.2 B 6.3 C 5.2 D 2.5
GATE CE 2021 SET-1   Solid Mechanics
Question 2 Explanation:

So horizontal displacement of the point (0.5 m, 0.5 m)
$=-2.5 \mathrm{~mm}+5 \mathrm{~mm}=2.5 \mathrm{~mm}$
 Question 3
The state of stress represented by Mohr's circle shown in the figure is
 A uniaxial tension B biaxial tension of equal magnitude C hydrostatic stress D pure shear
GATE CE 2020 SET-2   Solid Mechanics
Question 3 Explanation:
In pure shear condition, Mohr's circle has its center at origin.
 Question 4
A rigid, uniform, weightless, horizontal bar is connected to three vertical members P, Q and R as shown in the figure. All three members have identical axial stiffness of 10 kN/mm. The lower ends of bars P and R rest on a rigid horizontal surface. When NO laod is applied, a gap of 2 mm exist between the lower end of the bar Q and the rigid horizontal surface. When a vertical load W is placed on the horizontal bar in the downward direction, the bar still remains horizontal and gets displayed by 5 mm in the vertically downward direction.

The magnitude of the load W (in kN, round off to the nearst integer), is ______
 A 110 B 150 C 130 D 160
GATE CE 2020 SET-1   Solid Mechanics
Question 4 Explanation:

\begin{aligned} P_1 +P_1+P_2&=W \\ P_1&=P_3 \\ \frac{AE}{L}&=10kN/mm\\ \delta _1&=5mm=\frac{P_1L}{AE} \\ \delta _2&=3 mm = \frac{P_2L}{AE} \\ P_1 &=10 \times 5 =50 kN \\ P_2 &=10 \times 3 =30kN \\ W&=2(20)+30=130 kN \end{aligned}
 Question 5
The total stress paths corresponding to different loading conditions, for a soil specimen under the isotropically consolidated stress state (O), are shown below:

The correct match between the stress paths and the listed loading conditions, is
 A OP-I, OQ-II, OR-IV, OS-III B OP-IV, OQ-III, OR-I, OS-II C OP-III, OQ-II, OR-I, OS-IV D OP-I, OQ-III, OR-II, OS-IV
GATE CE 2020 SET-1   Solid Mechanics
Question 5 Explanation:

 Question 6
An isolated concrete pavement slab of length L is resting on a frictionless base. The temperature of the top and bottom fibre of the slab are $T_t \; and \; T_b$, respectively. Given: the coefficient of thermal expansion $=\alpha$ and the elastic modulus =E. Assuming $T_t \gt T_b$ and the unit weight of concrete as zero, the maximum thermal stress is calculated as
 A $L \alpha (T_t-T_b)$ B $E \alpha (T_t-T_b)$ C $\frac{E \alpha (T_t-T_b)}{2}$ D Zero
GATE CE 2019 SET-1   Solid Mechanics
Question 6 Explanation:
Due to frictionless, thermal stress developed in concrete pavement slab is zero
$\sigma _{th}=0$
 Question 7
An element is subjected to biaxial normal tensile strains of 0.0030 and 0.0020. The normal strain in the plane of maximum shear strain is
 A Zero B 0.001 C 0.0025 D 0.005
GATE CE 2019 SET-1   Solid Mechanics
Question 7 Explanation:
$\varepsilon _x=0.0030$
$\varepsilon _y=0.0020$
Normal strain in the plane of maximum shear strain
$\varepsilon _{avg}=\frac{\varepsilon _x+\varepsilon _y}{2}=\frac{0.0030+0.0020}{2}=0.0025$
 Question 8
A plate in equilibrium is subjected to uniform stresses along its edges with magnitude $\sigma_{xx}$ = 30 MPa and $\sigma_{yy}$ = 50 MPa as shown in the figure.

The Young's modulus of the material is $2 \times 10^{11} N/m^{2}$ and the Poisson's ratio is 0.3. If $\sigma_{zz}$ is negligibly small and assumed to be zero, then the strain $\varepsilon _{zz}$ is
 A $-120\times 10^{-6}$ B $-60\times 10^{-6}$ C 0 D $120\times 10^{-6}$
GATE CE 2018 SET-1   Solid Mechanics
Question 8 Explanation:
\begin{aligned} \sigma _{xx} &=30MPa \\ \sigma _{yy} &=50MPa \\ \sigma _{zz} &=0 \\ \varepsilon _{zz} &=\frac{\sigma _{zz}}{E}-\mu \frac{\sigma _{xx}}{E}-\mu \frac{\sigma _{yy}}{E} \\ &= -\frac{\mu }{E} (\sigma _{xx}+\sigma _{yy})\\ &= -\frac{0.3}{2 \times 10^5}(30+50)\\ &=-120 \times 10^{-6} \end{aligned}
 Question 9
A 2 m long, axially loaded mild steel rod of 8 mm diameter exhibits the load-displacement $(P-\delta )$ behavior as shown in the figure.

Assume the yield stress of steel as 250 Mpa. The complementary energy (in N-mm) stored in the bar up to its linear elastic behavior will be____
 A 7500 B 15000 C 15707.9 D 15007.9
GATE CE 2017 SET-2   Solid Mechanics
Question 9 Explanation:

Elastic strain, $\epsilon_{E}=\frac{\Delta L}{L}=\frac{2.5}{2000}=1.25 \times 10^{-2}$
Elastic strain energy $=\frac{1}{2} \sigma_{y} \in_{E} A L$
$=\frac{1}{2} \times 250 \times 1.25 \times 10^{-3} \times \frac{\pi}{4} \times 8^{2} \times 2000$
$=15707.96 \mathrm{Nmm}$
Note: For linear elastic material both complementary energy and strain energy is same.
 Question 10
In a material under a state of plane strain, a 10x10 mm square centered at a point gets deformed as shown in the figure.

If the Shear strain $\gamma _{xy}$ at this point is expressed as 0.001k(in rad),the value of k is
 A 0.5 B 0.25 C -0.25 D -0.5
GATE CE 2017 SET-2   Solid Mechanics
Question 10 Explanation:
According to the sign convention.

In question since angle has been increase
Therefore, shear strain should be negative.
\begin{aligned} \therefore \; \gamma_{x y} &=-0.0005 \mathrm{rad} \\ &=0.001 k \\ -0.0005 &=0.001 k \\ \Rightarrow \;\; k &=-0.50 \end{aligned}
There are 10 questions to complete.