Page 178 - Bridge and Highway Structure Rehabilitation and Repair

CHAPTER 4 AN ANALYTICAL APPROACH TO FRACTURE AND FAILURE 153

4. Properties of concrete—Usually the same concrete is used as for deck slabs, with fc1 values
between 4.0 and 5.0 ksi. This helps in rapid construction.
5. Thermal expansion and contraction—Longitudinal and transverse joints are provided at
regular intervals. The edges of slab joints are armored.
The S. P. Timoshenko and Woinowsky-Krieger Theory of Plates and Shells proposes the
following mathematical method for elastic subgrade reaction for soil spring stiffness k:
Lagrange general biharmonic equation for plate bending may be written as:
4
4
2
2
4
4
4
D (; w/;x ) 4 2D (; w/; x ;y ) 4 D(; w/;y ) 3 (q 6 wk) (4.14a)
Dividing by D,
4
2
(; w/;x ) 4 (; w/; x ;y ) 4 (; w/;y ) 3 (q/D 6 wk/D) (4.14b)
4
4
2
4
4
where q is intensity of vertical load.
w 3 Defl ection of edges of bottom plate, intensity of foundation reaction 3 k (w 6 w)/D
0
0
k 3 modulus of foundation in lb/sq in/in. k varies between 100 and 800 based on Casgrade’s

soil classification but needs to be verified by borehole tests.

4.9.2 Closed Form Solutions
The format can be developed in terms of fourth order differential equations. The following
expressions for deflections of slab on grade are based on the Navier solution and are of practical

®
interest in application to design of the approach slab. They can be programmed using Excel
spreadsheets:
1. For an approach slab of width a, length b simply supported on all four edges,
Apply the Navier solution to obtain the expression of deflection of plate as:

For rigid foundations, w 3 $ $ A (sin m!x/a) (Sin n!y/b)
m n
2 2
m n
where 3 4
a 2 b 2
Apply the Navier solution to obtain the expression of distribution of load acting on the plate
as:
q 3 $ m3 $ 1 n3 a (sin m!x/a) (Sin n!y/b) ..... (4.14c)
1
m n
Similarly, for reaction of subgrade:
p 3 kw 3 $ $kA (sin m!x/a) (Sin n!y/b)

m n

4
2
2
2
2
A 3 a /[! D (m /a 4 n /b ) 4 k]
m n
m n
For uniformly distributed load of intensity q:
w 3 (16 q / ! $ m3 m31, 3, 5 $ n3 31 (sin m!x/a) (Sin n!y/b)/
2
2
mn[! D (m /a 4 n /b ) 4 k] .... (4.14d)
2
2
4
2
When k 3 0, deflection reduces to that given by Navier equation for deflection of a uniformly

loaded plate. Refer to the theory of plates and shells by Timoshenko, Chapter 8.
4.9.3 HL-93 Trucks versus Heavier Cooper Railway Train Live Load
Shear forces transmitted by wheels of railway wagons are very high and punching shear
failure is possible. In railway tracks, sleepers and ballasts are used to distribute wheel loads at
45 degrees to subgrade.