Civil Engineering Reference
In-Depth Information
4
Static Equilibrium of Structures
4.1
Introduction
Practical finite element analysis had as its starting point matrix analysis of “structures”,
by which engineers usually mean assemblages of elastic, line elements. The matrix dis-
placement (stiffness) method is a special case of finite element analysis, and since many
engineers still begin their acquaintance with the finite element method in this way, the
opening applications chapter of this topic is devoted to “structural” analysis.
The first program, Program 4.1, permits the analysis of a rod subjected to combinations
of axial loads and displacements at various points along its length. Each 1D rod element can
have a different length and axial stiffness but the element stiffness matrices, being simple
functions of these two quantities are easily formed by a subroutine. Indeed, in nearly
all the programs in this chapter, the element stiffness matrices consist of simple explicit
expressions which are conveniently provided by subroutines. Program 4.2 introduces a
more general treatment involving rod elements, allowing analyses to be performed of 2D
or 3D pin-jointed frames.
Program 4.3 permits the analysis of slender beams subjected to combinations of trans-
verse and moment loading. Optionally, the program allows the inclusion of an elastic foun-
dation enabling analysis of problems generally classified as “beams on elastic foundations”.
When 1D beam and rod elements are superposed, the result is a “beam-rod” ele-
ment, which is a powerful general element that can sustain axial, transverse, and moment
loading. This element is able to analyse all conventional structural frames. Program 4.4
implements the “beam-rod” elements in the analysis of two- or three-dimensional framed
structures.
Program 4.5 introduces material non-linearity in the form of an elastic-perfectly plastic
moment/curvature relationship for beams. The program can compute plastic collapse of
one-, two-, or three-dimensional structures when subjected to incrementally changing loads.
The non-linearity is dealt with using an iterative, constant stiffness (modified Newton-
Raphson) approach. Unlike more traditional approaches, at each iteration the internal loads
on the structure are altered rather than the stiffness matrix itself. This approach will be
