BAZI DÜZENSİZLİKLER İÇEREN BÜYÜK YAPI SİSTEMLERİNİN
YATAY YÜKLERE GÖRE HESABI İÇİN BİR YÖNTEM
Z.Canan
GİRGİN, Doktora Tezi, Mayıs 1996
Anahtar Kelimeler: Üç
boyutlu ortogonal yapı sistemi, düzlem içi şekildeğiştirmeler, düzensizlikler, düzlem
alt sistem, çubuk alt sistem, indirgenmiş
rijitlik matrisi, ikinci mertebe teorisi, guseli çubuk, elastik zemin, eylemsizlik
kuvvetleri, malzeme bakımından lineer olmayan sistem, sınır koşulları bakımından
lineer olmayan sistem, kazık, lineer olmayan yayılı şekildeğiştirme, göçme yükü
Özet: Bu çalışmada, bazı
yapısal düzensizlikler içeren, ortogonal, üç boyutlu büyük yapı sistemlerinin
yatay yüklere göre hesabı için bir yöntem geliştirilmiştir. Sistem, düşey ve
yatay düzlem alt sistemler ile bunları oluşturan özel eleman tipi alt
sistemlerden oluşmaktadır ve döşeme düzlem içi şekildeğiştirmeleri önemli olan,
büyük döşeme ve perde boşlukları içeren sistemler az sayıda bilinmeyenle
sistematik olarak hesaplanmaktadır. Zemine tümü veya bir bölümü ile oturma,
eksenel kuvvetlerin ikinci mertebe etkileri, eylemsizlik kuvvet ve kuvvet
çiftleri, uzama ve kayma şekildeğiştirmeleri, değişken kesitli elemanlar,
malzeme, zemin özellikleri veya sınır koşulları bakımından lineer olmayan
davranış gibi düzensizliklerin tümü Mohr yöntemi esas alınarak geliştirilen bir
algoritma ile dikkate alınır. Yöntem ve algoritmanın kullanıldığı
örneklerde yüksek bir doğruluk
düzeyi elde edilmiştir.
A METHOD FOR THE
LATERAL LOAD ANALYSIS
OF
IRREGULAR FRAMED STRUCTURES
Ph.D.
Dissertation by Z. Canan GİRGİN, May 1996
Keywords : three
dimensional orthogonal framed structures, inplane deformations, irregularities,
condensed stiffness matrix, planar substructures, element type substructures, second
order theory (P-D effects), non-prismatic element, elastic
foundation, inertia forces, material non-linearity, boundary non-linearity, pile,
distributed plastic bending deformations, collapse load
Abstract: In this study, a
method for the analysis of three dimensional orthogonal irregular framed
structures subjected mainly to lateral loads has been developed. Vertical
planar substructures, horizontal planar substructures and one dimensional element
type substructures for which an algorithm based on conjugate beam method is developed are utilized in this method.
Thus, the elements can be partially or completely in contact with soil which is
linear or non-linear. Any kind of
irregular cross-section, inertia and rotatory inertia forces, flexural, shear
and axial deformations, geometrical non-linearity due to P-D effects, distributed material and boundary non-linearity are taken
into account by means of this algorithm systematically. As seen the comparison
of numerical examples, the accuracy level obtained by this method and algorithm
is high.
A
METHOD FOR THE LATERAL LOAD ANALYSIS OF IRREGULAR FRAMED STRUCTURES
SUMMARY
Three dimensional earthquake analysis of regular framed
structures can be simplified so that the total number of unknowns is reduced to
a level that the problem can be solved either by the help of simple computers
with low memory capacity or even manually. If the slabs are infinitely rigid in
their own planes, if there are no big openings in slabs and/or in the
structural walls, if most of the elements are prismatic elements with constant
rigidities in their whole lengths, if the members are not very slender elements
in other words if there is no need to adopt the second order theory in the
design, if the members are not fully or partially supported by linear or
non-linear soil media, if the inertia and/or rotatory inertia forces are
negligible and if the material is elastic, the structure can be considered as a
regular structure. Otherwise, the
structure can be considered as an irregular
structure.
In order to achieve an equivalent simplicity in the
design of irregular structures which are supposed to have some of the above
mentioned irregularities, new and efficient methods are necessary.
In this study, a method for the analysis of three
dimensional orthogonal irregular framed structures subjected mainly to lateral
loads has been developed. The structure is supposed composed of substructures.
Three different type of substructures have been
identified in this study. Namely, vertical elements resisting lateral loads are
considered as vertical planar
substructures such as frames-shear walls or shear walls with big openings,
horizontal elements distributing lateral loads are considered as horizontal planar substructures such as
continuous slabs or slabs with openings and these two planar substructures are
supposed to be composed of one
dimensional element type substructures.
All kinds of irregularities mentioned above can be
taken into consideration in these substructures automatically. The geometrical
differences among planar substructures are taken into account by defining
fictitious elements.
At the beginning of analysis, only one of the similar
planar substructures has been utilized and its stiffness matrix is eliminated
up to lateral displacements independently by means of any well- known procedure.
For this purpose, any available simple computer program can be employed, also
parallel programming techniques can be used as well. The reduced matrix which
is a part of this eliminated matrix and composed of stiffness terms related to
only absolute lateral displacements of nodes moving together in one direction
is defined as condensed lateral stiffness
matrix. The condensed stiffness matrices corresponding to the other similar
planar substructures have also been
determined automatically.
After having obtained the condensed lateral stiffness
matrices of planar substructures, [S] defined as system stiffness matrix of three dimensional irregular framed
structure, is composed of these condensed matrices. The set of lateral equilibrium
equations of the related system where [q] indicates the vector of nodal
external loads is stated as below,
[S][d]=[q]
The number of equations is equal to the number of
unknowns that are taken as the lateral displacements of nodes moving together
in one direction, in other words lateral story displacements. The lateral
displacement vector [d] is obtained by solving equilibrium equations stated
above.
Using the lateral story displacements in the planar
substructures, other unknown rotational or axial displacement components of
nodes in each substructure are obtained by back-substitution in the related
eliminated stiffness matrices. Unknown end forces of elements are determined by
means of all these nodal displacement components in each planar substructure.
The details of substructuring, the composition of stiffness matrix and the
assumptions imposed have been represented in the second chapter of this study.
The third chapter outlines an algorithm developed for
one dimensional element type substructures consisting of one element or the
string of elements. Following aspects
are taken into consideration by the supposed simple algorithm based on conjugate beam method;
i- Not only an element, but also the string of
different elements can be considered as an
equivalent element,
ii- Any kind of irregular cross-section in plane or in
view along the span of elements can be taken into account,
iii- Flexural, shear and axial deformations can be
included in the element level analysis keeping the simplicity of the algorithm,
iv- Irregular elements can be partially or completely
in contact with ground and soil which can be linear or non-linear,
v- Geometrical non-linearity caused by second order
effects of irregularly distributed axial forces along the element can be taken into consideration,
vi-The accumulated or distributed material
non-linearity is included at the element level calculations after having defined the
corresponding softened rigidities for certain
parts of element and for a certain
load level,
vii- Boundary
non-linearity can also be considered within the framework of the
proposed algorithm depending on a
set of successive iterations.
All kinds of irregularities given above can be defined
by means of the same algorithm systematically.
One dimensional element and string of elements which
are theoretically divided to small parts or some additional intermediate points
are taken into account to obtain the mechanical properties such as flexibility,
stiffness coefficients and fixed end forces as well. The deformed shape of
element axis is obtained in a discreatized manner. It has to be kept in mind
that successive iterations on the deformed shape are needed in order to
determine the flexibility coefficients f11, f12, f22 and the loading
coefficients f10, f10 of the element type substructures with enough accuracy.
The fictitious forces determined according to second
order effects of axial forces depending on the deformed shape of the element
axis, the fictitious forces which simulate soil reactions, inertia and rotatory
inertia forces caused by the mass on or out of the neutral axis can be defined
and be taken into consideration.
Not only the fictitious
forces related to the flexural deformations, but also the fictitious moments related to the shear
and axial deformations can be included in all intermediate steps of analysis.
According to all these fictitious forces and moments,
there will be no difficulty to obtain flexibility and loading coefficients.
First, three independent flexibility coefficients f11, f12, f22 of flexibility
matrix [f] are determined using a simply supported element in the proposed
algorithm. Secondly, the inverse of this matrix which is the rigidity matrix
[k] has been achieved in closed form. Any kind of fictitious force defined
using the deformed shape of element axes are considered in the determination of
the rest of the dependent stiffness coefficients.
If the element type substructures are elastically
supported at the ends, it is possible to obtain the flexibility coefficients of
these special elements including the work done at the elastic supports either
against rotation or settlements.
Non-linear behavior of planar structures or
substructures which distributed plastic bending deformations are employed has
also been taken into consideration by using this algorithm. For this reason,
first the bending-curvature relationship is defined by utilizing the
stress-strain relationship of material. Secondly, materially non-linear
behavior is linearized in each load level by using a fictitious
moment-curvature relationship. Thus, new fictitious rigidities in each element
and section corresponding to the related load level are determined according to
this fictitious moment-curvature relationship defined as linearization technique.
In this study, the non-linear behavior of the structure
can be determined under,
i- Proportionally increasing gravity and lateral loads,
ii- Factored constant gravity loads and proportionally
increasing lateral loads
The structures that nonlinearities caused by geometrical changes are not taken into
account collapse due to large deflections, excessive plastic deformations or
the rupture of the critical sections specially in reinforced concrete
structures. This load parameter is defined as collapse load parameter.
Otherwise, the structure collapses through the loss of
stability and this load parameter is referred to as buckling load parameter. However, in certain cases, the structure
may collapse before buckling caused by reasons stated above.
Non-linearity due to the boundary conditions in the
case of a continuous foundation uplifting from the soil, can easily be taken
into account by setting up an iterative procedure where the supposed algorithm
is used. In this way, this non-linear problem is linearized. The contact
portion and new stiffness matrix of the element partially supported by soil
media can be identified without having a serious difficulty in each step of the
iteration procedure.
The fourth chapter of this study mainly outlines two
computer programs which are developed called as LLA-3D & IBS-3M. Both of
them are executed in a batch program structure and coded in QBASIC programming
language. Either elastic or any kind of linearized inelastic analysis can be
performed by means of these computer programs.
LLA-3D consists of five independent programs called as
LPRO, STIF, RIMAT, LSOLVE, FORCE. Mainly, this program collects the condensed
lateral stiffness matrices performed by RIMAT in order to obtain the system
stiffness matrix of the irregular three dimensional structure. Then, the set of
lateral equilibrium equations is solved by LSOLVE and unknown lateral story
displacements are obtained. Using these displacements, the rest of the unknowns
such as rotational and axial displacements is computed by the common routine
FORCE.
IBS-3M consists of six independent programs called as
PRO, STIF, SOLVE, DYNAPRO, FORCE, IFDR. Non-linear analysis of a planar
substructure or a planar structural system which represents a three dimensional
structure is carried out this program.
Iterative calculations in the element level according
to proposed algorithm are done in the program of STIFF commonly used by IBS-3M
and LLA-3D.
The period values and mode shapes of structure are
determined the program DYNAPRO.
The special routine IFDR deals with all kinds of
nonlinearities at element level. Each section of element type substructures is
checked to get the new fictitious rigidities in each step.
Special attention has been given to save the memory
capacity of computer. Namely, all two dimensional symmetric matrices of the
formulation are replaced by one dimensional arrays so that maximum benefit of
memory usage has been achieved. The general algorithm presented here can easily
be applied to parallel or distributed computational techniques.
Several kinds of numerical examples have been given in
the fifth chapter of this study.
The first example indicates the effectiveness of the
proposed method developed for three dimensional irregular framed structures
subjected to lateral loads. For this purpose, a four-story, four-bay in X
direction, three-bay in Y direction three dimensional structural system has
been chosen and this system has been analyzed according to two different
lateral loading case. In order to research the effectiveness of the
irregularities such as in plane and view irregularities and the in-plane
deformations in slabs, on the end forces and displacements , six different
cases are considered and analyzed. The results have been checked by the
structural analysis program called as SAP90. The accuracy achieved by means of
this method is very high.
The effectiveness of supposed algorithm for element
type structures has also been checked in the numerical examples. For this
purpose, the examples in which the element stiffness and loading matrices can
be compared by closed solutions in the literature and also the examples which
the effectiveness of the algorithm has been shown in the cases that closed
solutions are not existing are examined. Moreover, there are the examples in
which eigenvalue problems are also solved and compared by the closed solutions.
The numerical examples related to planar structures or
substructures consist of two groups. In the first group of examples, the planar
systems which behave linearly due to material, soil properties, geometrical
changes and boundary conditions have been solved. In the second group, a
five-story steel frame non-linear due to material, a pile example non-linear
caused by material and geometrical changes and a non-linear pile example due to
soil properties are solved.
The sixth chapter covers the conclusions. The basic
features of the proposed method and algorithm, the some properties of developed
programs and the evaluation of numerical investigations are presented in the
following steps:
i- Computers having low memory and simple computer
programs for planar structures can effectively be utilized to analyze the three
dimensional irregular framed structures subjected to mainly lateral static
loads. Furthermore, using of substructuring technique and only one of the
similar substructures reduces the required memory and computation time.
The number of unknowns of a standard three dimensional
analysis is 6xr where r indicates the
number of nodes. Using proposed method, this number can be reduced to lx (m+n) where l, m, n are the number of stories, the number of vertical planar
substructures in X direction and the number of vertical planar substructures in
Y direction, respectively.
ii- All the two dimensional nxn symmetric matrices are substituted by one dimensional arrays
containing only the elements from upper triangular portion. Namely the
dimensions of these arrays are only (n+1)xn/2
instead of nxn. Thus, the main
memory has effectively been utilized. In other words, relatively bigger
structures can also be analyzed in small computers.
iii- Openings in the slabs and / or in the shear walls
can be considered easily. Sometimes their effects on the overall structural
behavior and on the fields of internal forces are not negligible.
The achieved accuracy by means of the proposed
procedure is very high. The well-known computer program SAP90 has been used for
having reliable comparisons.
iv- All kinds of irregularities on the element level
are satisfactorily taken into account by means of the proposed algorithm based
on conjugate beam theorem. Very few iterations are needed to achieve a
reasonable accuracy. From engineering point of view, if the intervals are
chosen as 1 meter then a relative accuracy at the order of 1E-3 approximately can be obtained and
if the intervals reduced to 1/4 of it then the relative accuracy rises up to 1E-7 approximately. Shortly
it can be specified that the required accuracy is not strongly dependent on the
number of section and interval. The algorithm is a stable procedure as well
even though the successive iteration procedure performed is very close to the
stability load of the element or when it is nearby the resonance frequencies.
But it has to be noticed that in several cases such as the elements supported
by elastic soil media have the characteristic length of gl=2.3, the number
of iterations necessary for a specified accuracy becomes very high. This case
occurs in the regions where discontinuities are expected. The easiest way of
overcoming this difficulty, is to divide the element to independent elements.
v- Boundary non-linearity can easily be taken into
account setting up an successive iterative procedure. Specified accuracy is
obtained after 3-4 iterations.
vi- Distributed plastic deformations are easily taken
into account in the supposed algorithm. A relative accuracy at the order of 1E-5 in the element level can be
achieved after 6-7 successive
iterations at a certain load level which is even very close to the collapse
load parameter of the structure.
vii- As it is expected, ultimate load capacities
determined according to the assumptions that plastic deformations are
distributed or accumulated are very close to each other in the case of steel
structures. The structure has collapsed due to excessive displacements and
reducing rigidities in a high load level.
viii- Plastic deformations are getting more distributed
as it is achieved in the case of laterally loaded reinforced concrete pile
or short column with single curvature.
ix- The non-linearity due to geometrical changes is
also taken into account in the analysis of the pile. The pile has been
collapsed by rupturing in a section before buckling load parameter.
x- The non-linear behavior of soil and the group effect
among the piles in a pile group are also taken into consideration by proposed
algorithm.
After having had all the explanations above, one can
easily conclude that the following contributions to the present knowledge have
been done by this research work:
The method presented is utilized to compose the results
of the individual analyses of two dimensional substructures which can also be
executed by different computers with parallel processor as well. The proposed
method is a versatile tool for the lateral load analysis of three dimensional
framed structures with several kinds of irregularities. Among those
irregularities, the in-plane deformations of slabs with or without big openings
and any kind of cross-sectional variations can be stated.
The number of unknowns is not increased by this method.
All the irregularities associated with element are taken into account on the
element level preparatory works referring to a unified single algorithm.
Utilizing that algorithm, it has been proven that tapered beams partially or
fully in contact with soil can be handled according to the first and second
order theories. All the numerical examples indicate a good convergence rate
which is achieved both in the element and system level on the iterations for
any kind of non-linearity.