TECHNICAL NOTES
Tension and Compression Stability and Second-Order
Analyses of Three-Dimensional Multicolumn Systems:
Effects of Shear Deformations
J. Dario Aristizabal-Ochoa
1
Abstract: The stability and second-order analyses of three-dimensional 3D multicolumn systems including the effects of shear defor-
mations along the span of each column are presented in a condensed manner. This formulation is an extension to an algorithm presented
recently by the writer in 2002 and 2003 by which the critical load of each column, the total critical load, and the second-order response
of a 3D multicolumn system with semirigid connections can be determined directly. The proposed solution includes not only the combined
effects of flexural deformations and shear distortions along the columns in their two principal transverse axes, but also the effect of the
shear forces along each member induced by the applied end axial force as the columns deform and deflect as suggested by Haringx in
1947 and explained by Timoshenko and Gere in 1961 in their two principal transverse axes. The extended characteristic transcendental
equations corresponding to multicolumn systems with sidesway and twist uninhibited, partially inhibited, and totally inhibited that are
derived and discussed in this publication find great applications in the stability and second-order analyses of 3D multicolumn systems
made of materials with relatively low shear stiffness such as orthotropic composite materials fiber reinforced plastic and multilayer
elastomeric bearings used for seismic isolation of buildings. The phenomenon of buckling under axial tension in members with relatively
low shear stiffness observed by Kelly in 2003 in multilayer elastomeric bearings, and recently discussed by the writer in 2005 is captured
by the proposed method. Tension buckling must not be ignored in the stability analysis of multicolumn systems made of columns in which
the shear stiffness GA
s
is of the same order of magnitude as
2
EI / h
2
.
DOI: 10.1061/ASCE0733-93992007133:1106
CE Database subject headings: Buckling; Building codes; Columns; Computer applications; Frames; Shear deformation;
Compression.
Introduction
The elastic stability and second-order analyses of multicolumn
systems become complicated when shear deformations must be
included, such as in the case of buildings supported by multilayer
elastomeric bearings that are utilized in seismic isolation in which
even buckling under tension forces may occur Kelly 2003. Gen-
erally, such analyses require the use of special algorithms i.e.,
nonlinear finite element analysis programs that are not generally
available in most engineering offices. In addition, the design of
orthotropic-column systems made of composites fiber reinforced
plastic FRP may be governed by deflection and elastic buckling
limitations because of their low shear moduli Roberts 2002.
The main objectives of this publication are 1 to present a
complete classical formulation an extension to that presented re-
cently by Aristizabal-Ochoa 2003 by including the effects of
shear deformations in their two principal transverse axes that can
be used to determine the critical load and the second-order re-
sponse of a three-dimensional 3D multicolumn system without
the use of complex models and cumbersome nonlinear numerical
procedures; 2 to show the importance of shear deformations and
their detrimental effects on the lateral stability and stiffness of 3D
multicolumn systems; and 3 to describe the phenomenon of
buckling under axial tension in members with relatively low shear
stiffness observed by Kelly 2003 in multilayer elastomeric
bearings, and discussed recently by the writer Aristizabal-Ochoa
2005.
Structural Model
Consider the 3D multicolumn system shown in Fig. 1b. It is
assumed that the floor diaphragm is rigid in its own plane and lies
on the XY plane with the origin O located at a convenient point
generally, at the centroid of the multicolumn system. This is a
linear elastic model consisting of n prismatic orthotropic col-
umns, with the centroid of Column i located at Point X
i
, Y
i
on
the XY plane, under axial load P
i
with individual properties in-
cluding: gross cross-section area A
i
; effective shear-areas A
sxi
and A
syi
and the principal second moments of area I
xi
and I
yi
with the major x
i
axis making an angle
i
with the global X axis;
1
125-Year Generation Professor, National Univ., School of Mines, A.
A. 75267, Medellín, Colombia.
Note. Associate Editor: Hayder A. Rasheed. Discussion open until
June 1, 2007. Separate discussions must be submitted for individual pa-
pers. To extend the closing date by one month, a written request must be
filed with the ASCE Managing Editor. The manuscript for this technical
note was submitted for review and possible publication on February 23,
2005; approved on June 5, 2006. This technical note is part of the
Journal of Engineering Mechanics, Vol. 133, No. 1, January 1, 2007.
©ASCE, ISSN 0733-9399/2007/1-106–116/$25.00.
106 / JOURNAL OF ENGINEERING MECHANICS © ASCE / JANUARY 2007