Civil Engineering Reference
In-Depth Information
9.1
Introduction
In current design practice, structural analysis for reinforced concrete frames
is generally based on the assumption that plane sections remain plane after
loading and the material is homogeneous and elastic. Therefore, linear
elastic methods of analysis are normally adopted for the design of simple
reinforced concrete beams and frames to obtain the member forces and
bending moments that will enable the design and detailing of the sections to
be carried out, despite the fact that reinforced concrete is not a
homogeneous and elastic material (British Standard BS 8110:1985).
However, the elementary theory of bending for simple beams may not be
applicable to deep beams even under the linear elastic assumption. A deep
beam is in fact a vertical plate subjected to loading in its own plane. The strain
or stress distribution across the depth is no longer a straight line, and the
variation is mainly dependent on the aspect ratio of the beam. (
Figure 9.1
)
.
The analysis of a deep beam should therefore be treated as a two-
dimensional plane stress problem, and two-dimensional stress analysis
methods should be used in order to obtain a realistic stress distribution in deep
beams even for a linear elastic solution. There are several methods available
for the analysis of deep beams that are either simply supported or continuous.
The classifical analytical method is based on the classical theory of
elasticity and it relies on finding a solution for the biharmonic differential
equation of AiryÓs stress function satisfying all boundary conditions. But in
the practical situation, a mathematical solution is not always possible.
The finite difference technique may be used to solve the differential
equation to obtain a numerical solution if the analytical solution is not readily
available. Both methods are more suitable for deep beams with rectangular
shapes, straight top and bottom soffits, prismatic constant cross-section and
with uniform material properties (Timoshenko and Goodier, 1951)
The finite element method is a much more versatile tool compared with
the former methods. It can be used to analyse variable thickness deep beams
with curved, stepped or inclined edges. Edge stiffening, openings and
loading at any location of the beam can be easily dealt with; and the
different properties of the constituent materials, concrete and steel, can be
separately represented. By incorporating a known constitutive law and an
iterative procedure, the non-homogeneous and non-linear nature of the
composite construction can be accounted for (Zienkiewicz, 1971).
9.2
Concept of finite element method
The finite element method can be regarded as an extension of the
displacement method for beams and frames to two and three dimensional
continuum problems, such as plates, shells and solid bodies. The actual
continuum is replaced by an equivalent idealised structure composed of
discretised elements connected together at a finite number of nodes.
