# Recent questions and answers in Mechanics of Materials

The process, that uses a tapered horn to amplify and focus the mechanical energy for machining of glass, is electrochemical machining electrical discharge machining ultrasonic machining abrasive jet machining
Bars of square and circular cross-section with $0.5\: m$ length are made of a material with shear strength of $20\: MPa$. The square bar cross-section dimension is $4\:cm \times$ $4\:cm$ and the cylindrical bar cross-section diameter ... to the applied load as per maximum shear stress theory? Tensile and compressive load specimens Torsional load specimen Bending load specimen None of the specimens
A cam with a translating flat-face follower is desired to have the follower motion $y\left ( \theta \right )=4\left [ 2\pi \theta -\theta ^{2} \right ],\:\:\:0\leq \theta \leq 2\pi .$ Contact stress considerations dictate that the radius of curvature ... profile should not be less than $40\:mm$ anywhere. The minimum permissible base circle radius is _________$mm$ (round off to one decimal place).
Consider a linear elastic rectangular thin sheet of metal, subjected to uniform uniaxial tensile stress of $100$ MPa along the length direction. Assume plane stress conditions in the plane normal to the thickness. The Young’s modulus $E=200$ MPa and Poisson’s ratio $v=0.3$ are given. The principal strains in the plane of the sheet are $(0.35, – 0.15)$ $(0.5, 0.0)$ $(0.5, – 0.15)$ $(0.5, – 0.5)$
Endurance limit of a beam subjected to pure bending decreases with decrease in the surface roughness and decrease in the size of the beam increase in the surface roughness and decrease in the size of the beam increase in the surface roughness and increase in the size of the beam decrease in the surface roughness and increase in the size of the beam
Consider two concentric circular cylinders of different materials $M$ and $N$ in contact with each other at $r=b$, as shown below. The interface at $r=b$ is frictionless. The composite cylinder is subjected to internal pressure $P$. Let $(u_r^M, u_{\theta}^M)$ ... $\sigma_{rr}^M = \sigma_{rr}^N \text{ and } \sigma_{\theta \theta}^M = \sigma_{\theta \theta}^N$
A prismatic, straight, elastic, cantilever beam is subjected to a linearly distributed transverse load as shown below. If the beam length is $L$, Young’s modulus $E$, and area moment of inertia $I$, the magnitude of the maximum deflection is $\dfrac{qL^4}{15EI} \\$ $\dfrac{qL^4}{30EI} \\$ $\dfrac{qL^4}{10EI} \\$ $\dfrac{qL^4}{60EI}$
A hollow circular shaft of inner radius $10 \: mm$ outer radius $20 \: mm$ length $1 \: m$ is to be used as a torsional spring. If the shear modulus of the material of the shaft is $150 \: GPa$, the torsional stiffness of the shaft (in $kN-m/rad$) is ________ (correct to two decimal places).
Fatigue life of a material for a fully reversed loading condition is estimated from $\sigma_a=1100N^{-0.15}$, where $\sigma_a$ is teh stress amplitude in $MPa$ and $N$ is the failure life in cycles. The maximum allowable stress amplitude (in $MPa$) for a life of $1 \times 10^5$ cycles under the same loading condition is ________ (correct to two decimal places).
The true stress (in MPa) versus true strain relationship for a metal is given by $\sigma=1020 \varepsilon^{0.4}$. The cross sectional-area at the start of the test (when the stress and strain values are equal to zero) is $100 \: mm^2$. The cross-sectional area at the time of necking ( in $mm^2$) is ______________ correct to two decimal palces.
A thin-walled cylindrical can with rigid end caps has a mean radius $R=100\:mm$ and a wall thickness of $t=5 \: mm$. The can is pressurized and an additional tensile stress of a $50 \: MPa$ is imposed along the axial direction as shown in the ... and circumferential components of stress in the can are equal, the pressure (in $MPa$) inside the can is ___________ (correct to two decimal places)
A bimetallic cylindrical bar cross sectional area $1 \: m^2$ is made by bonding Steel (Young's modulus $= 210 \: GPa$) and Aluminium (Young's modulus =$70 \: GPa$) as shown in the figure. To maintain tensile axial strain of magnitude $10^{-6}$ in Steel bar and ... $P$ (in $kN$) along the indicated direction is $70$ $140$ $210$ $280$
A bar of circular cross section is clamped at ends P and Q as shown in the figure. A torsional moment $T=150 \: Nm$ is applied at a distance of $100 \: mm$ from end $P$. The torsional reactions $(T_P, T_Q)$ in $Nm$ at ends $P$ and $Q$ respectively are $(50, 100)$ $(75, 75)$ $(100, 75)$ $(120, 30)$
The minimum axial compressive load, $P$, required to initiate buckling for a pinned-pinned slender column with bending stiffness $EI$ and length $L$ is $P=\dfrac{\pi^2 EI}{4L^2} \\$ $P=\dfrac{\pi^2 EI}{L^2} \\$ $P=\dfrac{3\pi^2 EI}{4L^2} \\$ $P=\dfrac{4\pi^2 EI}{L^2}$
A bar is compressed to half to its original length. The magnitude of true strain produced in the deformed bar is __________ (correct to two decimal places)
The true stress $(\sigma)$ - true strain $(\varepsilon)$ diagram of a strain hardening material is shown in figure. First, there is loading up to point A, i.e., up to stress of $500$ MPa and strain of $0.5$. Then from point A, there is unloading up to ... $E=200$ GPa, te natural strain at point B$(\varepsilon_B)$ is _____ (correct to three decimal places)
A simply supported beam of width $100$ mm, height $200$ mm and length 4 m is carrying a uniformly distributed load of intensity $10 \: kN/m$. The maximum bending stress (in MPa) in the beam is _________ (correct to one decimal place)
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In a linearly hardening plastic material, the true stress beyond initial yielding increases linearly with the true strain decreases linearly with the true strain first increases linearly and then decreases linearly with the true strain remains constant
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A rod of length $20$ mm is stretched to make a rod of length $40$ mm. Subsequently, it is compressed to make a rod of final length $10$ mm. Consider the longitudinal tensile strain as positive and compressive strain as negative. The total true longitudinal strain in the rod is $-0.5$ $-0.69$ $-0.75$ $-1.0$
For a loaded cantilever beam of uniform cross-section, the bending moment (in N.mm) along the length is $M(x)=5x^{2}+10x$, where $x$ is the distance (in mm) measured from the free end of the beam. The magnitude of shear force (in N) in the cross-section at $x=10$ mm is __________.
An initially stress-free massless elastic beam of length $L$ and circular cross-section with diameter $d$ $(d << L)$ is held fixed between two walls as shown. The beam material has Young's modulus $E$ and coefficient of thermal expansion $a$. If the beam is slowly and uniformly heated, the temperature rise required to cause the beam to buckle is proportional to $d$ $d^{2}$ $d^{3}$ $d^{4}$
The value of true strain produced in compressing a cylinder to half its original length is $0.69$ $-0.69$ $0.5$ $-0.5$
A cylindrical job with diameter of $200$ $mm$ and height of $100$ $mm$ is to be cast using modulus method of riser design. Assume that the bottom surface of cylindrical riser does not contribute as cooling surface. If the diameter of the riser is equal to its height, then the height of the riser (in $mm$) is $150$ $200$ $100$ $125$
A cantilever beam having square cross-section of side $a$ is subjected to an end load. If $a$ is increased by $19\%$, the tip deflection decreases approximately by $19\%$ $29\%$ $41\%$ $50\%$
The dimensions of a cylindrical side riser (height = diameter) for a $25 \: cm \times 15 \: cm \times 5 \: cm$ steel casting are to be determined. For the tabulated shape factor values given below, the diameter of the riser (in $cm$ ...
The strain hardening exponent $n$ of stainless steel SS $304$ with distinct yield and UTS values undergoing plastic deformation is $n < 0$ $n = 0$ $0 < n < 1$ $n = 1$
A hollow shaft ($d_o$ = $2d_i$ where $d_o$ and $d_i$ are the outer and inner diameters respectively) needs to transmit $20$ $kW$ power at $3000$ $RPM$. If the maximum permissible shear stress is $30$ $MPa$, do is $11.29$ $mm$ $22.58$ $mm$ $33.87$ $mm$ $45.16$ $mm$
A hollow shaft of $1 \: m$ length is designed to transmit a power of $30\: kW$ at $700 \text{rpm}$. The maximum permissible angle of twist in the shaft is $1^{\circ}$. The inner diameter of the shaft is $0.7$ times the outer diameter. The modulus of rigidity is $80 \: GPa$. The outside diameter (in $mm$) of the shaft is__________
In a plane stress condition, the components of stress at a point are $\sigma_x =20$ $MPa$, $\sigma _y=80$ $MPa$ and $\tau_ {xy}$=$40$ $MPa$ The maximum shear stress (in $MPa$) at the point is $20$ $25$ $50$ $100$
A cantilever beam with square cross-section of $6 \: mm$ side is subjected to a load of $2 \: kN$ normal to the top surface as shown in the figure. The Young’s modulus of elasticity of the material of the beam is $210 \: GPa$. The magnitude of slope (in radian) at $Q$ ($20 \: mm$ from the fixed end) is ________
A cantilever beam OP is connected to another beam PQ with a pin joint as shown in the figure. A load of $10$ $kN$ is applied at the mid-point of PQ. The magnitude of bending moment (in $kN$-$m$) at fixed end O is $2.5$ $5$ $10$ $25$
A rope-brake dynamometer attached to the crank shaft of an $I.C$. engine measures a brake power of $10$ $kW$ when the speed of rotation of the shaft is $400$ $rad/s$. The shaft torque (in $N$-$m$) sensed by the dynamometer is ___________
The uniaxial yield stress of a material is $300$ $MPa$. According to von Mises criterion, the shear yield stress (in $MPa$) of the material is _______
A gas is stored in a cylindrical tank of inner radius $7$ $m$ and wall thickness $50$ $mm$. The gage pressure of the gas is $2$ $MPa$. The maximum shear stress (in $MPa$) in the wall is $35$ $70$ $140$ $280$
A rod is subjected to a uni-axial load within linear elastic limit. When the change in the stress is $200$ $MPa$, the change in the strain is $0.001$. If the Poisson’s ratio of the rod is $0.3$, the modulus of rigidity (in $GPa$) is ________________
The diameter of a recessed ring was measured by using two spherical balls of diameter $d_2$ = $60$ $mm$ and $d_1$ = $40$ $mm$ as shown in the figure. The distance $H_2$ = $35.55$ $mm$ and $H_1$ = $20.55$ $mm$. The diameter ($D$, in $mm$) of the ring gauge is _______
Consider a long cylindrical tube of inner and outer radii, $r_i$ and $r_o$ , respectively, length, $L$ and thermal conductivity, $k$. Its inner and outer surfaces are maintained at $T_i$ and $T_o$ , respectively ( $T_i > T_o$ ). Assuming one-dimensional steady state heat conduction in the radial ... $\dfrac{1}{4\pi kL}\ln \bigg (\dfrac{r_o}{r_i} \bigg )$
Consider the two states of stress as shown in configurations I and II in the figure below. From the standpoint of distortion energy (von-Mises) criterion, which one of the following statements is true? $I$ yields after $II$ $II$ yields after $I$ Both yield simultaneously Nothing can be said about their relative yielding
A component can be produced by any of the four processes $I$, $II$, $III$ and $IV$. The fixed cost and the variable cost for each of the processes are listed below. The most economical process for producing a batch of $100$ ... $I$ $II$ $III$ $IV$