Using Laplace transforms, solve $${a \over {{s^2} - {a^2}}}\,\,\left( {{d^2}y/d{t^2}} \right) + 4y = 12t\,\,$$
given that $$y=0$$ and $$dy/dt=9$$ at $$t=0$$

The Laplace transform of the following function is $$$f\left( t \right) = \left\{ {\matrix{ {\sin t} & {for\,\,0 \le t \le \pi } \cr 0 & {for\,\,t > \pi } \cr } } \right.$$$

The directional derivative of the following function at $$(1, 2)$$ in the direction of $$(4i+3j)$$ is : $$f\left( {x,y} \right) = {x^2} + {y^2}$$

The value of the following definite integral in $$\int\limits_{{\raise0.5ex\hbox{$\scriptstyle { - \pi }$} \kern-0.1em/\kern-0.15em \lower0.25ex\hbox{$\scriptstyle 2$}}}^{{\raise0.5ex\hbox{$\scriptstyle \pi $} \kern-0.1em/\kern-0.15em \lower0.25ex\hbox{$\scriptstyle 2$}}} {{{Sin2x} \over {1 + \cos x}}dx = \_\_\_\_\_\_\_\_.} $$

The value of the following improper integral is $$\,\int\limits_0^1 {x\,\log \,x\,dx} = \_\_\_\_\_.$$

Limit of the following sequence as $$n \to \infty $$ $$\,\,\,$$ is $$\,\,\,$$ $${x_n} = {n^{{1 \over n}}}$$

The following function has local minima at which value of $$x,$$ $$f\left( x \right) = x\sqrt {5 - {x^2}} $$

Eigen values of the following matrix are $$\left[ {\matrix{ { - 1} & 4 \cr 4 & { - 1} \cr } } \right]$$

In natural water, hardness is mainly caused by

During a 6-hour storm the rainfall intensity was 0.8 cm/hr on a catchment of area 8.6 km². The measured runoff volume during this period was 2,56,000 m³. The total rainfall was lost due to infiltration, evaporation and transpiration in cm/hr is

A $$10m$$ long prestressing bed is used to cast $$4$$ (pretensioned) prestressed concrete beams of $$2.3$$ $$m$$ each. A schematic representation of the bed is given in the following figure. The continuous prestressing reinforced is pulled at the end $$'Y'$$ of the bed through a distance of $$20$$ $$mm$$ to introduce the required prestress, before the concrete is cast. After the concrete has hardened, the prestressing reinforcement is cut at points $$A,B,C,D$$ and $$E.$$ Assume that the prestress is introduced without eccentricity, what is the loss in prestress on account of elastic deformation of concrete. Assume $${E_s} = 200000\,\,N/m{m^2},$$ reinforcement is $$500\,\,m{m^2},$$ size of beams $$ = 200\,\,mm\,\left( b \right) \times 400\,\,mm\,\left( h \right).$$

As per provision of $$IS$$ $$456$$-$$2000$$, in the limit state method for design of beams, the limiting value of the depth of neutral axis in a reinforced concrete beam of effective depth $$'d'$$ is given as

Read the following two statements

$${\rm I}.\,\,\,\,\,\,\,\,\,\,\,\,$$ Maximum strain in concrete at the outermost compression fibre is taken to be $$0.0035$$ in bending
$${\rm II}.\,\,\,\,\,\,\,\,\,\,\,\,$$ The maximum compressive strain in concrete in axial compression is taken as $$0.002$$ Keeping the provisions of $$IS$$ $$456$$-$$2000$$ on limit state design in mind, which of the following is true?

Modulus of elasticity of M25 grade concrete in MPa is

An $$ISMB$$ $$500$$ is used as a beam in a multi-storey construction. From the view point of structural design, it can be considered to be 'laterally restrained' when,

When designing steel structures, one must ensure that local buckling in webs does not take place. This check may not be very critical when using rolled steel sections because

Given reason for the following is not more than $$20$$ words

(a) A maximum permissible distance between lacing and battens in steel columns is
$$\,\,\,\,\,\,\,$$ usually specified.
(b) It is sometimes preferable to have unequal flange angle with the longer legs
$$\,\,\,\,\,\,\,$$ horizontal in plate girder.
(c) If two channel sections need to be used as a steel column, they may be connected
$$\,\,\,\,\,\,\,$$ 'face-to-face' rather than 'back-to-back'
(d) It is sometimes preferred to have a small gap between the web and the flange
$$\,\,\,\,\,\,\,$$ plate in a plate girder.
(e) A maximum permissible 'outstand may be specific for flange in built-up sections.

$$ISA$$ $$100 \times 100 \times 10mm$$ (cross sectional area $$ = 1903\,\,m{m^2}$$) is welded along $$A$$ and $$B$$ (Refer to figure in the below question) such that the lengths of the weld along $$A$$ and $$B$$ are $${l_1}$$ and $${l_2}$$ respectively. Which of the following is a possibly acceptable combination of $${l_1}$$ and $${l_2}$$

$$ISA$$ $$100 \times 100 \times 10\,\,mm$$ (Cross sectional area $$ = 1903\,\,m{m^2})$$ serves as tensile member. This angle is welded to a gusset plate along $$A$$ and $$B$$ appropriately as shown. Assuming the yield strength of the steel to be $$260$$ $$N/m{m^2}$$ the tensile strength of the member can be taken to be approximately

For the loading given in the figure below, two statements ( $${\rm I}$$ and $${\rm II}$$ ) are made.

$${\rm I}.$$ $$\,\,\,\,\,\,\,\,\,$$Member $$AB$$ carries shear force and bending moment
$${\rm II}.$$ $$\,\,\,\,\,\,\,$$Member $$BC$$ carries axial load and shear force

Which of the following is true ?

In the propped cantilever beam carrying a uniformly distributed load of $$w$$-$$kN/m,$$ shown in the following figure, the reaction at the support $$'B'$$ is

The shear modulus (G), modulus of elasticity (E) and the Poisson's ratio ($$\mu$$) of a material are related as

Calculate the shape factor for the $$T$$-section shown in the following figure made up of two plates $$100\,\,mm \times 10\,\,mm.$$ What will be the load factor if the permissible stress in bending is only $$2/3$$ of the yield stress $$\left( {{\sigma _y}} \right)$$

A steel beam (with a constant $$EI,$$ and span $$L$$) is fixed at both ends and carries a uniformly distributed load ($$w$$ $$kN/m$$), which is gradually increased till the beam reaches the stage of plastic collapse (refer to the following figure). Assuming $$'B'$$ to be at mid-span, which of the following is true.

A propped cantilever beam of span $$'L'$$ is loaded with $$u.d.l$$ of intensity $$w$$/unit length, all through the span. Bending Moment at the fixed end is

SSD and Friction coefficients are