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



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

Reversed Curve to Connect Three Traversed Lines

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.



  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


Problem 04 - Symmetrical Parabolic Curve

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

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.



Problem 02 - Symmetrical Parabolic Curve

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.



Problem 01 - Symmetrical Parabolic Curve

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?




Problem 03 - Simple Curve

Given the following elements of a circular curve: middle ordinate = 2 m; length of long chord = 70 m. Find its degree of curve, use arc basis.



Problem 02 - Simple Curve

The angle of intersection of a circular curve is 36° 30'. Compute the radius if the external distance is 12.02 m.



Problem 01 - Simple Curve

The angle of intersection of a circular curve is 45° 30' and its radius is 198.17 m. PC is at Sta. 0 + 700. Compute the right angle offset from Sta. 0 + 736.58 on the curve to tangent through PC.



Parabolic Curve

Vertical Parabolic Curve
Vertical curves are used to provide gradual change between two adjacent vertical grade lines. The curve used to connect the two adjacent grades is parabola. Parabola offers smooth transition because its second derivative is constant. For a downward parabola with vertex at the origin, the standard equation is

$x^2 = -4ay$   or   $y = -\dfrac{x^2}{4a}$.





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