Tuesday, September 10, 2013

Westergaard's theory for rigid pavements

Hi, 

Rigid Pavements are constructed with some rigid materials like Cement Concrete(Plain, reinforced or prestressed).

Here the load is transferred  through the slab action not like in the flexible pavements. Westergaard's theory is considered good to design the rigid pavements.

He considered rigid pavement slab as a thin elastic plate resting on soil sub-grade, which is assumed to be a dense liquid. So, here the upward reaction is assumed to be proportional to the deflection, i.e. p = K.d, where K is a constant defined as modulus of subgrade reaction. Units of K are kg/cm^3.




  • Westergaard's modulus of sub-grade reaction:

Modulus of sub-grade reaction is proportional to amount of deflection d. Displacement level is taken as 0.125 cm in calculating K i.e. d = 0.125 cm, so modulus of sub-grade reaction
K = p/d = p/0.125 kg/cm^2


  • Radius of relative stiffness of slab to sub-grade:

Amount of deflection which will occur on the pavement surface depends on the stiffness of the slab and also on the stiffness of the sub-grade. Same amount of deflection will occur on the top surface of the sub-grade.

This means that the amount of deflection which is going to occur in the rigid pavement pavement layer depends both on relative stiffness of the pavement slab with respect to that of sub-grade.

Westergaard defined this by a term "Radius of relative stiffness" which, can be written numerically as below:
            l = [Eh^3/ (12K(1-U^2)]^(1/4)

Where,  l = radius of relative stiffness, cm
         E = Modulus of elasticity of cement concrete kg/cm^2
         U = Poisson's ratio for concrete = 0.15
         K = Modulus of Sub-grade reaction in kg/cm^2

Traffic Parameters:
(1) Design Wheel Load
(2) Traffic Intensity



  • Critical Load Positions:
    Rigid Pavement - a 3D view

When the wheel load is applied on the pavement surface, flexural stresses are induced in the pavement. There are three critical positions which are to be checked for maximum stresses.


  1. Interior loading
  2. Edge Loading
  3. Corner loading
Whenever loading is applied at the interior of the slab, remote than the edges and corner, this is called interior loading.
When loading is applied on the edges, remote than the corners is called edge loading.
When the loading is applied on the corner angle bisector and loading is touching the corner the edges.

  • Equivalent Radius of Resisting section:
When the loading is at the interiors there is a particular area which will resist the bending moment. Westergaard assumed that the area will be circular in plan and its radius is called as Equivalent radius of Resisting section.
Numerically,
     b= (1.6.a^2 + h^2)^(1/2)  - 0.675.h
Here,
      b = equivalent radius of resisting section, cm when 'a' is less than 1.724.h
      a = radios of wheel load distribution, cm
      h = slab thickness, cm
When 'a' is greater than 1.724.h, b =a.

  • In case of corner loading, maximum stresses are not produced at corner but they are produced at a certain distance X along the corner bisector. This is given by the relation:
     X = 2.58.(a.l)^1/2

 Here,    X = distance from apex of the slab corner to section of maximum stress along the corner bisector, cm.
   a= Radius of wheel load distribution, cm
    l = Radius of relative stiffness, cm.
Here is an image which shows you the formulas used to calculate the amount of stresses developed at the three critical positions due to the given wheel load P.


Rigid Pavement- Stresses at interior, edges and corners - Westergaard's theory


References:
 
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