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

 

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   $y = -\dfrac{x^2}{4a}$.

 

004-verical-symmetrical-parabolic-curve.gif

 

Spiral Curve

Spirals are used to overcome the abrupt change in curvature and superelevation that occurs between tangent and circular curve. The spiral curve is used to gradually change the curvature and superelevation of the road, thus called transition curve.
 

003-spiral-curve-transition-curve.gif

Simple Curves

Formulas for Circular Curves
The formulas we are about to present need not be memorized. All we need is geometry plus names of all elements in simple curve. Note that we are only dealing with circular arc, it is in our great advantage if we deal it at geometry level rather than memorize these formulas.

001-circular-simple-curve.gif

 

Example 03: Finding the Number of 32-mm Steel Bars for Doubly-Reinforced Concrete Propped Beam

Problem
A propped beam 8 m long is to support a total load of 28.8 kN/m. It is desired to find the steel reinforcements at the most critical section in bending. The cross section of the concrete beam is 400 mm by 600 mm with an effective cover of 60 mm for the reinforcements. f’c = 21 MPa, fs = 140 MPa, n = 9. Determine the required number of 32 mm ø tension bars and the required number of 32 mm ø compression bars.
 

wsd-example-03-propped-beam.jpg

 

Example 02: Finding the Number of 28-mm Steel Bars of Singly-Reinforced Concrete Cantilever Beam

Problem
A reinforced concrete cantilever beam 4 m long has a cross-sectional dimensions of 400 mm by 750 mm. The steel reinforcement has an effective depth of 685 mm. The beam is to support a superimposed load of 29.05 kN/m including its own weight. Use f’c = 21 MPa, fs = 165 MPa, and n = 9. Determine the required number of 28 mm ø reinforcing bars using Working Stress Design method.
 

wsd-example-02-cantilever-beam.jpg

 

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