## Fundamental Frequency of Fourier Equation in Cosine Form

**Problem**

Given the Fourier equation:

*f*(

*t*) = 5 cos (20π

*t*) + 2 cos (40π

*t*) + cos (80π

*t*)

What is the fundamental frequency?

A. 10 | C. 40 |

B. 20 | D. 30 |

## Truss With Flexible Cable Diagonals

**Situation**

Flexible cables *BE* and *CD* are used to brace the truss shown below.

1. Determine the load *W* to cause a compression force of 8.9 kN to member *BD*.

A. 7.80 kN | C. 26.70 kN |

B. 35.64 kN | D. 13.35 kN |

2. Which cable is in tension and what is the tensile reaction?

A. BE = 12.58 kN |
C. BE = 6.29 kN |

B. CD = 6.29 kN |
D. CD = 12.58 kN |

3. If *W* = 20 kN, what will be the tensile reaction of member *CE*?

A. 6.67 kN | C. 0 |

B. 13.33 kN | D. 10 kN |

## Find the Integral of dx / sqrt(1 + sqrt(x))

**Problem**

Evaluate $\displaystyle \int_0^9 \dfrac{1}{\sqrt{1 + \sqrt{x}}}$

A. 4.667 | C. 5.333 |

B. 3.227 | D. 6.333 |

## Circular Gate with Water on One Side and Air on the Other Side

**Situation**

The figure below shows a vertical circular gate in a 3-m diameter tunnel with water on one side and air on the other side.

- Find the horizontal reaction at the hinge.

A. 412 kN

B. 408 kN

C. 410 kN

D. 414 kN - How far from the invert of the tunnel is the hydrostatic force acting on the gate?

A. 1.45 m

B. 1.43 m

C. 1.47 m

D. 1.41 m - Where will the hinge support be located (measured from the invert) to hold the gate in position?

A. 1.42 m

B. 1.46 m

C. 1.44 m

D. 1.40 m

## Three Reservoirs Connected by Pipes at a Common Junction

**Situation**

Three reservoirs *A*, *B*, and *C* are connected respectively with pipes 1, 2, and 3 joining at a common junction *P*. Reservoir *A* is at elevation 80 m, reservoir *B* at elevation 70 m and reservoir *C* is at elevation 60 m. The properties of each pipe are as follows:

*L*= 5000 m,

*D*= 300 mm

Pipe 2:

*L*= 4000 m,

*D*= 250 mm

Pipe 3:

*L*= 3500 m

The flow from reservoir *A* to junction *P* is 0.045 m^{3}/s and *f* for all pipes is 0.018.

- Find the elevation of the energy grade line at
*P*in m.

A. 75.512

B. 73.805

C. 72.021

D. 74.173 - Determine the flow on pipe 2 in m
^{3}/s.

A. 0.025

B. 0.031

C. 0.029

D. 0.036 - Compute the diameter appropriate for pipe 3 in mm.

A. 175

B. 170

C. 178

D. 172

## Reversed Curve to Connect Three Traversed Lines

**Situation**

A reversed curve with diverging tangent is to be designed to connect to three traversed lines for the portion of the proposed highway. The lines *AB* is 185 m, *BC* is 122.40 m, and *CD* is 285 m. The azimuth are Due East, 242°, and 302° respectively. The following are the cost index and specification:

Number of Lanes = Two Lanes

Width of Pavement = 3.05 m per lane

Thickness of Pavement = 280 mm

Unit Cost = P1,800 per square meter

It is necessary that the *PRC* (Point of Reversed Curvature) must be one-fourth the distance *BC* from *B*.

- Find the radius of the first curve.

A. 123 m

B. 156 m

C. 182 m

D. 143 m - Find the length of road from
*A*to*D*. Use arc basis.

A. 552 m

B. 637 m

C. 574 m

D. 468 m - Find the cost of the concrete pavement from
*A*to*D*.

A. P2.81M

B. P5.54M

C. P3.42M

D. P4.89M

## Problem 04 - Symmetrical Parabolic Curve

**Problem**

A highway engineer must stake a symmetrical vertical curve where an entering grade of +0.80% meets an existing grade of -0.40% at station 10 + 100 which has an elevation of 140.36 m. If the maximum allowable change in grade per 20 m station is -0.20%, what is the length of the vertical curve?

- 150 m
- 130 m
- 120 m
- 140 m

## Influence Lines for Beams

A downward concentrated load of magnitude 1 unit moves across the simply supported beam *AB* from *A* to *B*. We wish to determine the following functions:

- reaction at
*A* - reaction at
*B* - shear at
*C*and - moment at
*C*

when the unit load is at a distance *x* from support *A*. Since the value of the above functions will vary according to the location of the unit load, the best way to represent these functions is by influence diagram.

## Influence Lines

Influence line is the graphical representation of the response function of the structure as the downward unit load moves across the structure. The ordinate of the influence line show the magnitude and character of the function.

The most common response functions of our interest are *support reaction*, *shear at a section*, *bending moment at a section*, and *force in truss member*.

With the aid of influence diagram, we can...

- determine the position of the load to cause maximum value to the function.
- calculate the maximum value of the function.

Value of the function for any type of load

$\displaystyle \text{Function} = \int_a^b y_i (y \, dx)$