Fundamental Frequency of Fourier Equation in Cosine Form

Problem
Given the Fourier equation:

f(t) = 5 cos (20πt) + 2 cos (40πt) + cos (80πt)

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.
 

2016-may-counter-doagonals.gif

 

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.
 

01-circular-gate-problem.png

 

  1. Find the horizontal reaction at the hinge.
    A.   412 kN
    B.   408 kN
    C.   410 kN
    D.   414 kN
  2. 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
  3. 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:

Pipe 1:   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 m3/s and f for all pipes is 0.018.
 

01-three-reservoir-problem-given-discharge.gif

 

  1. Find the elevation of the energy grade line at P in m.
    A.   75.512
    B.   73.805
    C.   72.021
    D.   74.173
  2. Determine the flow on pipe 2 in m3/s.
    A.   0.025
    B.   0.031
    C.   0.029
    D.   0.036
  3. 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:

Type of Pavement = Item 311 (Portland Cement Concrete Pavement)
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.
 

01-reversed-curve-problem.gif

 

  1. Find the radius of the first curve.
      A.   123 m
      B.   156 m
      C.   182 m
      D.   143 m
  2. Find the length of road from A to D. Use arc basis.
      A.   552 m
      B.   637 m
      C.   574 m
      D.   468 m
  3. Find the cost of the concrete pavement from A to D.
      A.   P2.81M
      B.   P5.54M
      C.   P3.42M
      D.   P4.89M

 

Influence Lines for Trusses

Example
For the Pratt truss shown below, draw the influence diagram for members JK, DK, and DE.
 

influence-line-pratt-truss-6-panels.jpg

 

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?

  1. 150 m
  2. 130 m
  3. 120 m
  4. 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-line-moment-at-c.jpg

 

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...

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

 

Value of the function for any type of load
 

influence-line-any-load.jpg

 

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

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