Questions & Answers of Solid Mechanics

A column of height h with a rectangular cross-section of size $\style{font-family:'Times New Roman'}{a\times2a}$ has a buckling load of $P$. If the cross-section is changed to $\style{font-family:'Times New Roman'}{0.5a\;\times\;3a}$ and its height changed to 1.5h, the buckling load of the redesigned column will be

A solid circular beam with radius of 0.25 m and length of 2 m is subjected to a twisting moment of 20 kNm about the z-axis at the free end, which is the only load acting as shown in the figure. The shear stress component $\style{font-family:'Times New Roman'}{\tau_{xy}}$ at Point ‘M’ in the cross-section of the beam at a distance of 1 m from the fixed end is

A cantilever beam of length 2 m with a square section of side length 0.1 m is loaded vertically at the free end. The vertical displacement at the free end is 5 mm. The beam is made of steel with Young’s modulus of 2.0×1011 N/m2. The maximum bending stress at the fixed end of the cantilever is

A plate in equilibrium is subjected to uniform stresses along its edges with magnitude $ \sigma_{xx}=30 $ MPa and $ \sigma_{xx}=50 $ MPa as shown in the figure.

The Young’s modulus of the material is 2×1011 N/m2 and the Poisson’s ratio is 0.3. If $\style{font-family:'Times New Roman'}{\sigma_{zz}}$ is negligibly small and assumed to be zero, then the strain $\style{font-family:'Times New Roman'}{\varepsilon_{zz}}$ is

A structural member subjected to compression, has both translation and rotation restrained at one end, while only translation is restrained at the other end. As per IS 456 : 2000, the effective length factor recommended for design is

A vertical load of 10 kN acts on a hinge located at a distance of $L/4$ from the roller support Q of a beam of length $L$ (see figure).

 

 

The vertical reaction at support Q is

An elastic bar of lenght L, uniform cross sectional area A, coefficient of thermal expansion $\style{font-family:'Times New Roman'}\alpha$, and Young's modulus E is fixed at the two ends. The temperature of the bar is increased by T, resulting in an axial strees $\style{font-family:'Times New Roman'}\sigma$. Keeping all other parameter unchanged if the length of the bar is doubled, the axial stress would be

A simply supported beam is subjected to a uniformly distributed load. Which one of the following statements is True?

Consider the stepped bar made with a linear elastic material and subjected to an axial load of 1 kN, as shown in the figure.

Segment 1 and 2 have cross-sectional area of 100 mm2 and 60 mm2, Young's modulus of $\style{font-family:'Times New Roman'}{2\times10^5\mathrm{MPa}\;\mathrm{and}\;3\times10^{5\;}\mathrm{MPa}}$, and length of 400 mm and 900 mm, respectively. The strain energy (in N-mm, up to one decimal place) in the bar due to the axial load is ___________

The value of M in the beam ABC shown in the figure is such that the joint B does not rotate.

The value of support reaction (in kN) at B should be equal to __________________

Consider two axially loaded columns, namely, 1 and 2, made of a linear elastic material with Young's modulus 2×105 MPa, square cross-section with side 10 mm, and length 1 m. For coloumn 1 one end is fixed and is free Column 2, one end is fixed and the other end is pinned. Based on the Euler's theory the ratio (up to one decimal place) of the buckling load of column 2 to the buckling load of column 1 is _______________

In a material under a state of plane strain, a 10 × 10 mm square centered at a point gets deformed as shown in the figure.

If the shear strain $\style{font-family:'Times New Roman'}{\gamma_{xy}}$ at this point is expressed as $ 0.001k $ (in rad), the value of $ k $ is

Two prismatic having the same flexural rigidity of 1000 kN-m2 are shown in the figures.

If the mid-span deflections of these beams are denoted by $\style{font-family:'Times New Roman'}{\delta_1\;\mathrm{and}\;\delta_2}$ (as indicated in the figures), the correct option is

Consider the three prismatic beams with the clamped supports P, Q and R as shown in the figures.

Given that the modulus of elasticity, E is 2.5 × 10MPa; and the moment of interta I is 8 × 108 mm4, the correct comparison of the magnitude of the shear force S and the bending moment M developed at the support is

A hollow circular shaft has an outer diameter of  100 mm and inner diameter of 50 mm. If the allowable shear stress is 125 MPa. The maximum torque (in kN-m) that the shaft can resist is ______ 

 A 2 m long, axially loaded mild steel rod of 8 mm diameter exhibits the load-displacement $ \style{font-family:'Times New Roman'}{(P-\delta)} $  behavior as shown in the figure.

Assume the yield stress of steel as 250 MPa. The complementary strain energy (in N-mm) stored in the bar up to its linear elastic behavior will be ___________

A rigid member ACB is shown in the figure. The member is supported at A and B by pinned and guided roller supports, respectively. A force P acts at C as shown. Let RAh and RBh be the horizontal reactions at supports A and B, respectively, and RAv be the vertical reaction at support A. Selfweight of the member may be ignored.

Which one of the following sets gives the correct magnitudes of RAv, RBh and RAh ?

An elastic isotropic body is in a hydrostatic state of stress as shown in the figure. For no change in the volume to occur, what should be its Poisson's ratio?

For the stress state (in MPa) shown in the figure, the major principal stress is 10 MPa.

The shear stress $ \style{font-family:'Times New Roman'}\tau $ is

Consider the structural system shown in the figure under the action of weight W. All the joints are hinged. The properties of the members in terms of length (L), area (A) and the modulus of elasticity (E) are also given in the figure. Let L, A and E be 1 m, 0.05 m2 and 30 × 106 N/m2, respectively, and W be 100 kN.

Which one of the following sets gives the correct values of the force, stress and change in length of the horizontal member QR?

A haunched (varying depth) reinforced concrete beam is simply supported at both ends, as shown in the figure. The beam is subjected to a uniformly distributed factored load of intensity 10 kN/m.  The design shear force (expressed in kN) at the section X-X of the beam is ______

A 450 mm long plain concrete prism is subjected to the concentrated vertical loads as shown in the figure. Cross section of the prism is given as 150 mm × 150 mm. Considering linear stress distribution across the cross-section, the modulus of rupture (expressed in MPa) is ________

For the beam shown below, the value of the support moment M is _____ kN-m.

A tapered circular rod of diameter varying from 20 mm to 10 mm is connected to another uniform circular rod of diameter 10 mm as shown in the following figure. Both bars are made of same material with the modulus of elasticity, E = 2 × 105 MPa. When subjected to a load = 30π kN, the deflection at point A is ________ mm.

 

Two beams are connected by a linear spring as shown in the following figure. For a load P as shown in the figure, the percentage of the applied load P carried by the spring is ________.

A horizontal beam ABC is loaded as shown in the figure below. The distance of the point of contraflexure from end A (in m) is ____

In a system, two connected rigid bars $ AC $ and $ BC $ are of identical length $ L $ with pin supports at $ A $ and $ B $. The bars are interconnected at $ C $ by a frictionless hinge. The rotation of the hinge is restrained by a rotational spring of stiffness, $ k $. The system initially assumes a straight line configuration, $ ACB $. Assuming both the bars as weightless, the rotation at supports, $ A $ and $ B $, due to a transverse load, $ P $ applied at $ C $ is:

A steel strip of length, L = 200 mm is fixed at end A and rests at B on a vertical spring of stiffness, k = 2 N/mm. The steel strip is 5 mm wide and 10 mm thick. A vertical load, P = 50 N is applied at B, as shown in the figure. Considering E = 200 GPa, the force (in N) developed in the spring is________.

A fixed end beam is subjected to a load, W at 1/ 3rd span from the left support as shown in the figure. The collapse load of the beam is

The possible location of shear centre of the channel section, shown below, is

A box of weight 100 kN shown in the figure is to be lifted without swinging. If all forces are coplanar, the magnitude and direction (θ) of the force (F) with respect to x-axis should be

Mathematical idealization of a crane has three bars with their vertices arranged as shown in the figure with a load of 80 kN hanging vertically. The coordinates of the vertices are given in parentheses. The force in the member QR, FQR will be

The values of axial stress (σ) in kN/m2, bending moment (M) in kNm, and shear force (V) in kN acting at point P for the arrangement shown in the figure are respectively

The beam of an overall depth 250 mm (shown below) is used in a building subjected to two different thermal environments. The temperatures at the top and bottom surfaces of the beam are 36°C and 72°C respectively. Considering coefficient of thermal expansion (α) as 1.50×10−5 per °C, the vertical deflection of the beam (in mm) at its mid-span due to temperature gradient is ________

The axial load (in kN) in the member PQ for the arrangement/assembly shown in the figure given below is _______________

The tension (in kN) in a 10 m long cable, shown in the figure, neglecting its self-weight is

For the state of stresses (in MPa) shown in the figure below, the maximum shear stress (in MPa) is _______________

The ‘plane section remains plane’ assumption in bending theory implies:

A symmetric I-section (with width of each flange 50 mm,thickness of each flange = 10 mm,depth of web = 100 mm, and thickness of web =10 mm) of steel is subjected to a shear force of 100 kN. Find the magnitude of the shear stress(in N/mm2 the web at its junction with the top flange. __________

The Poisson’s ratio is defined as

The following statements are related to bending of beams:

I   The slope of the bending moment diagram is equal to the shear force.
II  The slope of the shear force diagram is equal to the load intensity.
III The slope of the curvature is equal to the flexural rotation.
IV The second derivative of the deflection is equal to the curvature.

The only FALSE statement is

The ratio of the theoretical critical buckling load for a column with fixed ends to that of another column with the same dimensions and material, but with pinned ends, is equal to

A simply supported beam is subjected to a uniformly distributed load of intensity w per unit length, on half of the span from one end. The length of the span and the flexural stiffness are denoted as l and EI, respectively. The deflection at mid-span of the beam is

The sketch shows a column with a pin at the base and rollers at the top. It is subjected to an axial force P and a moment M at mid-height. The reaction(s) at R is/are

A symmetric frame PQR consists of two inclined members PQ and QR, connected at ‘Q’ with a rigid joint, and hinged at ‘P’ and ‘R’. The horizontal length PR is l. If a weight W is suspended at ‘Q’, the bending moment at ‘Q’ is

For the cantilever bracket, PQRS, loaded as shown in the adjoining figure(PQ=RS=L, and, QR=2L),which of the following statements is FALSE?

A rigid beam is hinged at one end and supported on linear elastic spring(both having a stiffness of (‘k’) at points "1" and "2", and an inclined load acts at "2", as shown

Which of the following options represents the deflections δ1 and δ2 at point "1" and "2"?

A rigid beam is hinged at one end and supported on linear elastic spring(both having a stiffness of (‘k’) at points "1" and "2", and an inclined load acts at "2", as shown

If the load P equals 100 kN, which of the following options represents forces R1 and R2 in the springs at points "1" and "2"?

Two people weighing W each are sitting on a plank of length L floating on water at L4 from either end. Neglecting the weight of the plank, the bending moment at the centre of the plank is

The effective length of a column of length L fixed against rotation and translation at one end and free at the other end is

A solid circular shaft of diameter d and length L is fixed at one end and free at the other end. A torque T is applied at the free end. The shear modulus of the material is G. The angle of twist at three free ends is

For the simply supported beam of length L, subjected to a uniformly distributed moment M kN-m per unit length as shown in the figure, the bending moment (in kN-m) at the mid-span of the beam is

The point within the cross sectional plane of a beam through which the resultant of the external loading on the beam has to pass through to ensure pure bending without twisting of the cross-section of the beam is called

A hollow circular shaft has an outer diameter of 100mm and a wall thickness of 25mm. The allowable shear stress in the shaft is 125MPa. The maximum torque the shaft can transmit is

Group I gives the shear force diagrams and Group II gives the diagrams of beams with supports and loading. Match the Group I with Group II

 

Cross-section of a column consisting of two steel strips, each of thickness t and width b is shown in the figure below. The critical loads of the column with perfect bond and without bond between the strips are P and Po respectively. The ration P/Po is

The maximum shear stress in a solid shaft of circular cross-section having diameter subjected to a torque T is τ . If the torque is increased by four times and the diameter of the shaft is increased by two times, the maximum shear stress in the shaft will be

The maximum tensile stress at the section X - X shown in the figure below is

A steel column, pinned at both ends, has a buckling load of 200kN. If the column is restrained against lateral movement at its mid-height, its buckling load will be

The maximum and minimum shear stresses in a hollow circular shaft of outer diameter 20 mm and thickness 2 mm, subjected to a torque of 92.7 N.m will be

The shear stress at the neutral axis in a beam of triangular section with a base of 40 mm and height 20 mm, subjected to a shear force of 3 kN is