Page 167 - Bridge and Highway Structure Rehabilitation and Repair

142 SECTION 2 STRENGTHENING AND REPAIR WORK

2
2
2
2
6M 3 D (; w/; y ) 4 ' D (; w/; x ) (4.6b)
y
2
6M 3 M 3 (1 6 ') D (; w/;x ;y) (4.6c)
xy yx
For load intensity resulting from one-directional beam bending can be generalized for two-
directional plate bending as
4
2
4
4
4
4
2
q (x, y) 3 D [(; w/; x ) 4 2 (; w/; x ;y ) 4 (; w/; y )] (4.7)
The above equation is known as Lagrange Biharmonic Equation for plate bending and as-
sociated twisting.
4.5.5 Application of Finite Element Method

Modeling of concrete decks and piers is based on finite elements using computer methods.

FEM is a powerful computational method. Like the finite differences method, defl ections are
calculated for elements of deck supported by grid beams or the wall of a pier by an indetermi-
nate analysis for each position of the unit load. While the finite differences method uses line

elements in two directions, the finite element method uses surface elements and therefore has

a higher degree of accuracy.
Using Energy Principle:
Work Done 3 (A) 8 (C) 3 (B) 8 (D)
(B) and (C) are moving unit load quantities at two points X and Y. A and D are the recipro-

cal deflections at Y and X.
Finite Element Modeling:
Stiffness matrix [k] 3 < [B] [D] [B] { } (4.8)
T
vol
Plate elements with 3, 4, and 8 nodes or beam elements are used. Energy theorems and
Castigliano’s Principles are applied. The well-known method was originally developed by Clough
and Wilson at the University of California Berkeley.
Due to availability of high speed and large storage computers, FEM has been successfully
applied to the modeling of bridges.
Recent developments in FEM:
1. Boundary Elements Method (BEM)—Developed by Carlos Brebbia and Hugh Tottenham,
it uses the variational principle, which is transformed into a sequence of FEM equations.
The number of equations is smaller than for FEM.
Accurate predictions of stress concentrations near skew boundaries are possible. Presently
approximate shear correction factors for skew angles range between 60 and 90 degrees.
2. Probabilistic Finite Element Method (PFEM)—Developed by T. Belytschko at Northwestern
University, this method eliminates various spurious modes that are associated with FEM
modeling of nodal forces. The Lagrange Variational Principle is used. Probabilistic char-
acteristics of stress-strain, strain-displacement, and random character of applied loads and
outer boundary are accounted for, resulting in more accurate solutions.

4.5.6 Substructure Stiffness Methods
Piers are usually made of reinforced concrete. Hammerhead, multi-frame, and pile bent
methods are in use. Each type uses the stiffness method of analysis.
Abutments are designed as retaining walls subjected to vertical loads from the bridge and
lateral loads from braking forces, active and passive earth pressures, longitudinal and transverse
wind forces, and thermal and seismic forces. Settlement due to erosion is a possibility for scour
critical bridges.