Page 177 - Bridge and Highway Structure Rehabilitation and Repair

152 SECTION 2 STRENGTHENING AND REPAIR WORK

1. Matrix inversion.
2. Matrix partitioning and solving banded matrices.
3. Gaussian elimination method.
4. Gauss-Seidel iteration methods.

The deflection function in response to applied load can be expressed in the form of algebraic

polynomial function, such as R. H. Wood’s series for slab and beam systems.
Harmonic analysis such as the Fourier series in terms of assumed deflection function with

unknown coefficients: Well-known Navier series, Ritz series, Galerkin series, and Allen and

Severn series have been used for thin plates subjected to vertical deflection and for stress func-
tion representing membrane stress.

The finite difference method derived from a Taylor series is based on a differential equations
2
2
approach in which slopes (dw/dx) and (d w/dx ) and other derivatives of defl ections are replaced

by algebraic equations, assuming successive differences in deflections at selected nodes.
1. Deflection function is assumed as y 3 f(x).

A change in assumed deflection curve over a small distance h can be expressed as:

n
2
n
f(x 4 h) 3 f(x) 4 h f1(x) 4 h /2! f2(x) 4 ………. 4 h /n! f (x) (4.13)
Neglecting higher order terms, the first derivative represents slope as a rate of change of

defl ection. f1(x) 3 [f(x 4 h) 6 f(x)] / h
2. Successive derivation will give expressions for bending curvature, vertical shear, and rate
of loading at selected nodes.
3. The resulting system of algebraic equations can then be solved in the form of matrices.
4.9 ANALYSIS OF APPROACH SLAB RESTING ON GRADE
4.9.1 Alternate Approaches Using Differential Equations or Finite Element Methods
Approach slabs are required when ADTT is high (greater than 500 trucks). The new method
is to support the end of the approach slab on a groove located behind the backwall so that settle-
ment will not take place. The factors affecting stresses are:
1. Thickness of approach slab—Bridge and highway design manuals for each state have de-
veloped thicknesses varying from 12 to 18 inches. However, the thickness of the slab needs
to be developed from a finite element analysis.

2. End conditions or method of support of approach slab—15 to 30 feet length in direction of

traffic is generally used.
Three types are possible:
• Approach slabs on grade
• Approach slab at bridge end simply supported on a ledge or groove at the top of the back-
wall and on grade at highway end.
• Approach slab at integral abutment bridge end cast in place (integral) with concrete of
backwall and deck slab. On highway end simply supported by an ell beam resting on grade.
Eighteen-inch-thick slab is generally used.
3. Subgrade reaction of the soil-supporting slab—Usually subgrade material is selected by the
highway engineer and an estimate of quantities is included in the highway costs. However,
for integral abutments, selected fill material is used. The material is compacted in layers.

The coefficient of subgrade reaction “k” values varies between 200 and 800 lbs/sq. inch/
inch.
When the bridge is located on a stream, the rise and subsequent fall of the water table may
cause the subgrade to settle. Small diameter perforated pipes for drainage are commonly
placed at a level where buildup of water will not take place.