$.$ A simply supported beam of length $8$ meters carries a uniformly distributed load of $10$ kN$/$m and a point load of $20$ kN at the center. Calculate the reactions at the supports and draw the shear force and bending moment diagrams.

Geotechnical Engineering:

A soil sample has a void ratio of $0\mathrm{.}65$ and a specific gravity of solids of $2\mathrm{.}70.$ If the sample is fully saturated, calculate the moisture content and the bulk unit weight of the soil.

Hydraulics:

Water is flowing through a circular pipe of diameter $1$ meter with a velocity of $3$ m$/$s$.$ Calculate the Reynolds number and determine whether the flow is laminar or turbulent if the kinematic viscosity of water is

$1$

$$

$1$

$0$

$$

$6$

$$

$$

$2$

$/$

$$

$110$

$6$

m

$2$

$/$s$.$

Environmental Engineering:

A wastewater treatment plant treats $10,000$ m$^3/$day of sewage. The influent has a BOD $($Biochemical Oxygen Demand$)$ of $300$ mg$/$L$.$ Calculate the amount of oxygen required per day to treat the sewage if the treatment efficiency is $90\%.$

Transportation Engineering:

A highway curve with a radius of $400$ meters is designed for a speed of $100$ km$/$h$.$ Calculate the required super$-$elevation if the coefficient of lateral friction is $0\mathrm{.}15.$

Structural Dynamics:

A single$-$degree$-$of$-$freedom system has a mass of $500$ kg and a stiffness of $2000$ N$/$m$.$ Calculate the natural frequency and the period of oscillation of the system.

Fluid Mechanics:

A rectangular channel of width $3$ meters carries water with a depth of $2$ meters and a velocity of $1\mathrm{.}5$ m$/$s$.$ Calculate the flow rate and the Froude number. Determine whether the flow is subcritical or supercritical.

Concrete Technology:

A concrete mix design requires a water$-$cement ratio of $0\mathrm{.}45.$ If $500$ kg of cement is used, calculate the amount of water required. Also, if the mix design specifies a sand$-$to$-$aggregate ratio of $1$:$2\mathrm{.}5,$ calculate the quantities of sand and aggregate needed.

Foundation Engineering:

A square footing of $2$m x $2$m is to be constructed on a sandy soil with an allowable bearing capacity of $200$ kN$/$m$^2.$ Calculate the maximum load the footing can support.

Water Resources Engineering:

A reservoir has an inflow rate of $10$ m$^3/$s and a constant outflow rate of $7$ m$^3/$s$.$ Calculate the time required to fill the reservoir if its capacity is $2$ million cubic meters..

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