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.

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

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

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

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

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 $${f_{br}} = 250\,\,N/m{m^2}$$.

Minimum number of rivets required at each end 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 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 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

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

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

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

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 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

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

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

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