MATHalino

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
 

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.
 

Compound Curves
A compound curve consists of two (or more) circular curves between two main tangents joined at point of compound curve (PCC). Curve at PC is designated as 1 (R₁, L₁, T₁, etc) and curve at PT is designated as 2 (R₂, L₂, T₂, etc).
 

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.

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, f_s = 140 MPa, n = 9. Determine the required number of 32 mm ø tension bars and the required number of 32 mm ø compression bars.
 

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, f_s = 165 MPa, and n = 9. Determine the required number of 28 mm ø reinforcing bars using Working Stress Design method.
 

Problem
A 300 mm × 600 mm reinforced concrete beam section is reinforced with 4 - 28-mm-diameter tension steel at d = 536 mm and 2 - 28-mm-diameter compression steel at d' = 64 mm. The section is subjected to a bending moment of 150 kN·m. Use n = 9.

1. Find the maximum stress in concrete.
2. Determine the stress in the compression steel.
3. Calculate the stress in the tension steel.
 

Problem
The following are the dimensions of a concrete T-beam section

The beam is reinforced with 3-32 mm diameter bars in tension and is carrying a moment of 100 kN·m. Find the total compressive force in the concrete. Use n = 9.
 

Problem
A reinforced concrete beam 300 mm wide has an effective depth of 600 mm. It is reinforced with 4-32 mm diameter bars for tension. f’_c = 21 MPa and f_y = 275 MPa. Find the moment capacity of the beam.
 

Problem
A rectangular reinforced concrete beam with width of 250 mm and effective depth of 500 mm is subjected to 150 kN·m bending moment. The beam is reinforced with 4 – 25 mm ø bars. Use alternate design method and modular ratio n = 9.

1. What is the maximum stress of concrete?
2. What is the maximum stress of steel?
3. What is the total compressive force in concrete?