March 2017

Problem 01 - Symmetrical Parabolic Curve

Problem
A grade of -4.2% grade intersects a grade of +3.0% at Station 11 + 488.00 of elevations 20.80 meters. These two center gradelines are to be connected by a 260 meter vertical parabolic curve.

  1. At what station is the cross-drainage pipes be situated?
  2. If the overall outside dimensions of the reinforced concrete pipe to be installed is 95 cm, and the top of the culvert is 30 cm below the subgrade, what will be the invert elevation at the center?

 

01-004-problem-parabolic-sag-curve.gif

 

Problem 02 - Symmetrical Parabolic Curve

Problem
A descending grade of 6% and an ascending grade of 2% intersect at Sta 12 + 200 km whose elevation is at 14.375 m. The two grades are to be connected by a parabolic curve, 160 m long. Find the elevation of the first quarter point on the curve.
 

01-005-first-quater-point-sag-curve.gif

 

Problem 03 - Symmetrical Parabolic Curve

Board Problem
A grade line AB having a slope of +5% intersect another grade line BC having a slope of –3% at B. The elevations of points A, B and C are 95 m, 100 m and 97 m respectively. Determine the elevation of the summit of the 100 m parabolic vertical curve to connect the grade lines.
 

01-006-elevation-of-summit-parabolic-curve.gif

 

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?
A.   150 m
B.   130 m
C.   120 m
D.   140 m
 

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.
 

01-007-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

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 response in the function.
  2. calculate the maximum value of the function.

 

Value of the function for any type of load
 

if-beam-any-type-of-load.gif

 

$\displaystyle \text{Function} = \int_{x_1}^{x_2} y_i (y \, dx)$
 

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.
 

il-beam-moment-at-c.gif

 

Influence Lines for Trusses

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

il-truss-given-pratt-6panels.gif

 

Integration problem

Submitted by vipa2000 on March 16, 2017 - 4:28pm

∈=p/Ao((1-(x/2L) *E^(-1)

I am trying to integrate the above. For clarity p is over Ao((1-(x/2L) and then all multiplied by E^(-1). Do I need to deal with the Ao((1-(x/2L) first?

Thanks in advance Paul

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

A.   4.667 C.   5.333
B.   3.227 D.   6.333

 

Using Simpson’s One-third Rule for Integration Problems (Tagalog)

Problem
Find the distance from the point A(1, 5, -3) to the plane 4x + y + 8z + 33 = 0.

A.   1/2 C.   2/3
B.   2 D.   1.5

 

Situation
An open cylindrical vessel 1.3 m in diameter and 2.1 m high is 2/3 full of water. If rotated about the vertical axis at a constant angular speed of 90 rpm,
1.   Determine how high is the paraboloid formed of the water surface.

A.   1.26 m C.   2.46 m
B.   1.91 m D.   1.35 m

2.   Determine the amount of water that will be spilled out.

A.   140 L C.   341 L
B.   152 L D.   146 L

3.   What should have been the least height of the vessel so that no water is spilled out?

A.   2.87 m C.   3.15 m
B.   2.55 m D.   2.36 m

 

Problem
A 523.6 cm3 solid spherical steel ball was melted and remolded into a hollow steel ball so that the hollow diameter is equal to the diameter of the original steel ball. Find the thickness of the hollow steel ball.

A.   1.3 cm C.   1.2 cm
B.   1.5 cm D.   1.6 cm

 

Problem
A point moves in the plane according to equations x = t2 + 2t and y = 2t3 - 6t. Find dy/dx when t = 0, 2, 5.

A.   -3, -3, -12 C.   3, 3, 12
B.   3, -3, 12 D.   -3, 3, 12

 

Problem
Chords AB and CD intersect each other at E inside the circle. AE = 8 cm, CE = 12 cm, and DE = 20 cm. If AB is the diameter of the circle, compute the area of AEC.

A.   61.04 cm2 C.   39.84 cm2
B.   52.05 cm2 D.   48.62 cm2

 

Problem
A salesperson earns P60,000 per month plus a commission of 20% of sales. Find the minimum amount of sales needed to receive a total income of at least P150,000 per month.

A.   P150,000 C.   P450,000
B.   P350,000 D.   P250,000

 

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

 

Problem
Determine the absolute pressure in a vessel of mercury at a point 200 mm below its surface.

A.   126 kPa C.   128 kPa
B.   130 kPa D.   132 kPa

 

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

2016-may-design-3panel-truss-counter-diagonals.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

 

Situation
Diagonals BG, CF, CH, and DG of the truss shown can resist tension only.
 

2016-may-design-truss-with-tension-diagonals.gif

 

If W = 3 kN and P = 0, find the following:
1.   the force in member CF.

A.   4.76 kN C.   4.67 kN
B.   4.32 kN D.   4.23 kN

2.   the force in member BF.

A.   3.2 kN C.   3.4 kN
B.   3.3 kN D.   3.5 kN

3.   the force in member DH.

A.   2.8 kN A.   2.5 kN
B.   2.8 kN D.   2.7 kN

 

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