Given matrix $$\left[ A \right] = \left[ {\matrix{ 4 & 2 & 1 & 3 \cr 6 & 3 & 4 & 7 \cr 2 & 1 & 0 & 1 \cr } } \right],$$ the rank of the matrix is

If $$L$$ denotes the laplace transform of a function, $$L\left\{ {\sin \,\,at} \right\}$$ will be equal to

A box contains $$10$$ screws, $$3$$ of which are defective. Two screws are drawn at random with replacement. The probability that none of the two screws is defective will be

The vector field $$\,F = x\widehat i - y\widehat j\,\,$$ (where $$\widehat i$$ and $$\widehat j$$ are unit vectors) is

A concrete column carries an axial load of $$450$$ $$kN$$ and a bending moment of $$60$$ $$kN$$-$$m$$ at its base. An isolated footing of size $$2m \times 3m$$ side along the plane of the bending moment is provided under the column. Centers of gravity of column and footing coincide. The net maximum and the minimum pressures in $$kN/{m^2}$$ on soil under the footing are respectively.

A reinforced concrete beam, size $$200$$ $$mm$$ wide and $$300$$ $$mm$$ deep overall is simply supported over a span of $$3$$ $$m.$$ It is subjected to two point loads $$P$$ of equal magnitude placed at middle third points. The two loads are gradually increased simultaneously. Beam is reinforced with $$2$$ $$HYSD$$ $$415$$ bars of $$16$$ $$mm$$ diameter placed at an effective cover of $$40$$ $$mm$$ on bottom face and nominal shear reinforcement. The characteristic compressive strength and the bending tensile strength of the concrete are $$20.0\,\,N/m{m^2}$$ and $$2.2\,\,N/m{m^2}$$ respectively.

Ignoring the presence of tension reinforcement, the value of load $$P$$ in $$kN$$ when the first flexure crack will develop in the beam is

The effective length of a column in a reinforced concrete building frame, as per $$IS:$$ $$456$$-$$2000,$$ is independent of the

A reinforced concrete beam, size $$200$$ $$mm$$ wide and $$300$$ $$mm$$ deep overall is simply supported over a span of $$3$$ $$m.$$ It is subjected to two point loads $$P$$ of equal magnitude placed at middle third points. The two loads are gradually increased simultaneously. Beam is reinforced with $$2$$ $$HYSD$$ $$415$$ bars of $$16$$ $$mm$$ diameter placed at an effective cover of $$40$$ $$mm$$ on bottom face and nominal shear reinforcement. The characteristic compressive strength and the bending tensile strength of the concrete are $$20.0\,\,N/m{m^2}$$ and $$2.2\,\,N/m{m^2}$$ respectively.

The theoretical failure load of the beam for attainment of limit state of collapse in flexure is

Maximum strains in an extreme fibre in concrete and in the tension reinforcement (Fe-$$415$$ grade and $${E_s} = 200\,\,kN/m{m^2}$$) in a balanced section at limit state of flexure are respectively

In the design of lacing system for a, built-up steel column, the maximum allowable slenderness ratio of lacing bar is

A truss tie consisting of $$2$$ $$ISA$$ $$75 \times 75 \times 8\,\,mm$$ carries a pull of $$150$$ $$kN.$$ At ends the two angles are connected, one each on either side of a $$10$$ $$mm$$ thick gusset plate, by $$18$$ $$mm$$ diameter rivets arranged in one row. The allowable stresses in rivet are $${f_s} = 90.0\,\,N/m{m^2}$$ and $${f_{br}} = 250\,\,N/m{m^2}$$.

Maximum tensile stress in the tie in $$N/m{m^2}$$ is

A truss tie consisting of $$2$$ $$ISA$$ $$75 \times 75 \times 8\,\,mm$$ carries a pull of $$150$$ $$kN.$$ At ends the two angles are connected, one each on either side of a $$10$$ $$mm$$ thick gusset plate, by $$18$$ $$mm$$ diameter rivets arranged in one row. The allowable stresses in rivet are $${f_s} = 90.0\,\,N/m{m^2}$$ and $${f_{br}} = 250\,\,N/m{m^2}$$.

Minimum number of rivets required at each end is

Group-$${\rm I}$$ contains some elements in design of a simply supported plate girder and Group-$${\rm II}$$ gives some qualitative locations on the girder. Match the items of two lists as per good design practice and relevant codal provisions.

Group $${\rm I}$$
$$P.$$$$\,\,\,\,\,\,$$ flange splice
$$Q.$$$$\,\,\,\,\,\,$$ web splice
$$R.$$$$\,\,\,\,\,\,$$ bearing stiffeners
$$S.$$$$\,\,\,\,\,\,$$ horizontal stiffener

Group $${\rm II}$$
$$1.$$$$\,\,\,\,\,\,$$ at supports (minimum)
$$2.$$ $$\,\,\,\,\,\,$$ away from center of span
$$3.$$$$\,\,\,\,\,\,$$ away from support
$$4.$$$$\,\,\,\,\,\,$$ in the middle of span
$$5.$$$$\,\,\,\,\,\,$$ longitudinally some where in the compression flange

A $$''H''$$ Shaped frame of uniform flexural rigidity $$EI$$ is loaded as shown in the figure. The relative outward displacement between points $$K$$ and $$O$$ is

For linear elastic systems, the type of displacement function for the strain energy is

List - $${\rm I}$$ shows different loads acting on a beam and list - $${\rm II}$$ shows different bending moment distributions. Match the load with the corresponding bending moment diagram.

A curved number with a straight vertical leg is carrying a vertical load at $$Z,$$ as shown in the figure. The stress resultants in the $$XY$$ segment are. Bending movement, shear force and axial force

A long structural column (length $$-L$$) with both ends hinged is acted upon by an axial compressive load $$P.$$ The differential equation governing the bending of column is given by: $$$EI{{{d^2}y} \over {d{x^2}}} = - Py$$$

Where $$y$$ is the structural lateral deflection and $$EI$$ is the flexural rigidity. The first critical load on column responsible for its buckling is given by

A simply supported beam of uniform rectangular cross-section of width $$b$$ and depth $$h$$ is subjected to linear temperature gradient, $${0^ \circ }$$ at the top $${T^ \circ }$$ at the bottom, as shown in the figure. The coefficient of linear expansion of the beam material is $$\alpha .$$ This resulting vertical deflection at the mid-span of the beam is

A stepped steel shaft shown below is subjected to $$10$$ $$N$$-$$m$$ torque. If the modulus of rigidity is $$80$$ $$GPa,$$ the strain energy in the shaft in $$N$$-$$mm$$ is

A beams $$PQRS$$ is $$18$$ $$m$$ long and is simply supported at points $$Q$$ and $$R$$ $$10$$ $$m$$ apart. overhangs $$PQ$$ and $$RS$$ are $$3$$ $$m$$ and $$5$$ $$m$$ respectively. A train of two point loads of $$150$$ $$kN$$ and $$100$$ $$kN$$, $$5$$ $$m$$ apart, crosses this beam from left to right with $$100$$ $$kN$$ load leading.

The maximum hogging moment in the beam anywhere is

A truss, as shown in the figure, is carrying $$180$$ $$kN$$ load at node $${L_2}.$$ The force in the diagonal member $${M_2}{U_4}$$ will be

A steel portal frame has dimensions, plastic moment capacitance and applied loads as shown in the figure. The vertical load is always twice of the horizontal load. The collapse load $$P$$ required for the development of a beam mechanism is

In a redundant joint model, three bar members are pin connected at $$Q$$ as shown in the figure. Under some load placed at $$Q$$, the elongation of the members $$MQ$$ and $$OQ$$ are found to be $$48$$ $$mm$$ and $$35$$ $$mm$$ respectively. Then the horizontal displacement $$'u'$$ and the vertical displacement $$'v'$$ of the node $$Q,$$ in $$mm,$$ will be respectively,

The stiffness $$K$$ of a beam deflecting in a symmetric mode, as shown in the figure, is _______

A beams $$PQRS$$ is $$18$$ $$m$$ long and is simply supported at points $$Q$$ and $$R$$ $$10$$ $$m$$ apart. overhangs $$PQ$$ and $$RS$$ are $$3$$ $$m$$ and $$5$$ $$m$$ respectively. A train of two point loads of $$150$$ $$kN$$ and $$100$$ $$kN$$, $$5$$ $$m$$ apart, crosses this beam from left to right with $$100$$ $$kN$$ load leading.

During the passage of the loads, the maximum and the minimum reactions at supports $$'R'$$ in $$kN,$$ are respectively

Muller Breslau Principle in structural analysis is used for

A beams $$PQRS$$ is $$18$$ $$m$$ long and is simply supported at points $$Q$$ and $$R$$ $$10$$ $$m$$ apart. overhangs $$PQ$$ and $$RS$$ are $$3$$ $$m$$ and $$5$$ $$m$$ respectively. A train of two point loads of $$150$$ $$kN$$ and $$100$$ $$kN$$, $$5$$ $$m$$ apart, crosses this beam from left to right with $$100$$ $$kN$$ load leading.

The maximum sagging moment under the $$150$$ $$kN$$ load anywhere is