INTRODUCTION
According to Naiem (1989) in order to design a structure
to withstand an earthquake, the forces on the structure must be specified. The
exact forces that will occur during the life of the structure cannot, of course,
be known. Many methods are available for the structural analysis of buildings
and other civil engineering structures under seismic actions. The differences
between the methods lie in the way they incorporate the seismic input and in
the idealization of the structure. There are two commonly used procedures for
specifying seismic design forces. The equivalent static force procedure and
dynamic analysis. In the equivalent static force procedure the inertial forces
are specified as static forces using empirical formulas. The empirical formulas
do not explicitly account for the dynamic characteristics of the particular
structure being designed or analyzed. The formulas were, however, developed
to adequately represent the dynamic behavior of what are called regular structures
which have a reasonably uniform distribution of mass and stiffness. For such
structures the equivalent static force procedure is often adequate.
Structures that do not fit into this category are termed irregular. Common
irregularities include large floor to floor variation in mass or center of mass
and soft stories. Such structures violate the assumptions on which the empirical
formulas used in the equivalent static force procedures are based. Therefore,
its use may lead to erroneous results. In these cases a dynamic analysis should
be used to specify and distribute the seismic design forces. A dynamic analysis
can take a number of forms, but should take account of the irregularities of
the structure by modeling its dynamic characteristics including natural frequencies,
mode shapes and damping (Naiem, 1989).
According to Edward (2002) the mode displacement super-position
method provides an efficient means of evaluating the dynamic response of most
structures because the response analysis is performed only for a series of SDOF
systems. The response analysis for the individual modal equations requires very
little computational effort and in most cases only a relatively small number
of the lowest modes of vibration need to be included in the superposition. With
this regard, it is important to realize that the physical properties of the
structure and the characteristics of the dynamic loading generally are known
only approximately; hence, the structural idealization and the solution procedure
should be formulated to provide only a corresponding level of accuracy. Nevertheless,
the mathematical models developed to solve practical problems in structural
dynamics range from very simplified systems having only a few degrees of freedom
to highly sophisticated finite element models including hundreds or even thousands
of degrees of freedom in which as many as 50 to 100 modes may contribute significantly
to the response (Edward, 2002).
The basic mode superposition method, which is restricted to linearly
elastic analysis, produces the complete time history response of joint
displacements and member forces. The method involves the calculation of
only the maximum values of the displacements and member forces in each
mode using smooth design spectra that are the average of several earthquake
motions.
Generally, maximum total response cannot be obtained by merely adding the modal
maxima because these maxima usually do not occur at the same time. In most cases,
when one mode achieves its maximum response, the other modal responses are less
than their individual maxima. Therefore, although the superposition of the modal
spectral values obviously provides an upper limit to the total response, it
generally over estimates this maximum by a significant amount. A number of different
formulas have been proposed to obtain a more reasonable estimate of the maximum
response from the spectral values (Edward, 2002):
| • |
SRSS (square root of the sum of squares ) |
| • |
CQC (the complete quadratic combination) |
| • |
ASCE4-98 (ASCE) |
| • |
TEN the 10% |
| • |
ABS (the absolute) |
| • |
CSM (closely-spaced mode) |
This research studies the effect of these approaches on the analysis
and design of high rise reentrant steel buildings. It compares the resulting
steel sections weight using static and dynamic analysis utilizing mode
combination methods approaches to show the difference among these approaches,
to determine the most influenced structural members and to obtain a vertical
loads factor in order to get the required sections using common static
analysis for preliminary design purposes.
DYNAMIC EQUILIBRIUM
The force equilibrium of a multi-degree-of-freedom lumped mass system as a
function of time can be expressed by the following relationship (Edward,
2002; Ray and Penzien, 2003):
| F (t)I+F
(t)D+F (t)S = F (t) |
(1) |
in which the force vectors at time t are:
| F (t)I |
= |
Vector of inertia forces on the node masses |
| F (t)D |
= |
Vector of viscous damping or energy dissipation, forces |
| F (t)S |
= |
Vector of inernal forces carried by the structure |
| F (t) |
= |
Vector of externally applied loads |
Equation 1 is based on physical laws and is valid for
both linear and nonlinear systems if equilibrium is formulated with respect
to the deformed geometry of the structure. For many structural systems,
the approximation of linear structural behavior is made to convert the
physical equilibrium statement, Eq. 1, to the following
set of second-order, linear, differential equations:
where, M is the mass matrix (lumped or consistent), C is a viscous damping
matrix (which is normally selected to approximate energy dissipation in
the real structure) and K is the static stiffness matrix for the system
of structural elements. The time-dependent vectors 
and 
are the absolute node displacements, velocities and accelerations, respectively.
For seismic loading, the external loading F(t) is equal to zero. The
basic seismic motions are the three components of free-field ground displacements
u (t)ig that are known at some point below the foundation level
of the structure. Therefore, we can write Eq. 2 in terms
of the displacements 
,
velocities 
and accelerations
that are relative to the three components of free-field ground displacements.
Therefore, the absolute displacements, velocities and accelerations can
be eliminated from Eq. 2 by writing the following simple
equations:
where, Ii is a vector with ones in the i directional degrees-of-freedom
and zero in all other positions. The substitution of Eq.
3 into Eq. 2 allows the node point equilibrium equations
to be rewritten as:
where, Mi = MIi
The simplified form of Eq. 4 is possible since the rigid
body velocities and displacements associated with the base motions cause
no additional damping or structural forces to be developed.
There are several different classical methods that can be used for the
solution of Eq. 4. Each method has advantages and disadvantages
that depend on the type of structure and loading. The different numerical
solution methods are :
| • |
Step-by-step solution method |
| • |
Mode superposition method |
| • |
Response spectra analysis |
| • |
Solution in the frequency domain |
| • |
Solution of linear equations |
| • |
Undamped harmonic response |
| • |
Undamped free vibrations |
The most common and effective approach for seismic analysis of linear
structural systems is the mode superposition method. After a set of orthogonal
vectors have been evaluated, this method reduces the large set of global
equilibrium equations to a relatively small number of uncoupled second
order differential equations. The numerical solution of those equations
involves greatly reduced computational time. The dynamic force equilibrium,
Eq. 4 can be rewritten in the following form as a set
of Nd second order differential equations:
All possible types of time-dependent loading, including wind, wave and
seismic, can be represented by a sum of J space vectors fj,
which are not a function of time and J time functions g (t)j.
The fundamental mathematical method that is used to solve Eq.
5 is the separation of variables. This approach assumes the solution
can be expressed in the following form:
where, Ø is an Nd by N matrix containing N spatial vectors that
are not a function of time and Y(t) is a vector containing N functions
of time. From Eq. 6a, it follows that:
Before solution, we require that the space functions satisfy the following
mass and stiffness orthogonality conditions:
where, I is a diagonal unit matrix and Ω2 is a diagonal
matrix in which the diagonal terms are 
.
The term ωn has the units of radians per second and may
or may not be a free vibration frequencies. It should be noted that the
fundamentals of mathematics place no restrictions on those vectors, other
than the orthogonality properties. In this research each space function
vector, Φn, is always normalized so that 
.
The generalized mass is equal to one, or after substitution of Eq.
6 into Eq. 5 and the pre-multiplication by ΦT,
the following matrix of N equations is produced:
where, Pj = ΦT fj and are defined
as the modal participation factors for load function j. The term Pnj
is associated with the nth mode. Note that there is one set of N modal
participation factors for each spatial load condition fj. For
all real structures, the N by N matrix d is not diagonal; however, to
uncouple the modal equations, it is necessary to assume classical damping
where there is no coupling between modes. Therefore, the diagonal terms
of the modal damping are defined by:
where, ζn is defined as the ratio of the damping in mode
n to the critical damping of the mode. A typical uncoupled modal equation
for linear structural systems is of the following form:
For three-dimensional seismic motion, this equation can be written as:
where, the three-directional modal participation factors, or in this
case earthquake excitation factors, are defined by 
,
where, j is equal to x, y or z and n is the mode number. Note that all
mode shapes are normalized so that 
.
For three-dimensional seismic motion, the typical modal Eq.
11 is rewritten as:
where, the three mode participation factors are defined by 
,
which i is equal to x, y or z. Two major problems must be solved to obtain
an approximate response spectrum solution to this equation. First, for
each direction of ground motion, maximum peak forces and displacements
must be estimated. Second, after the response for the three orthogonal
directions has been solved, it is necessary to estimate the maximum response
from the three components of earthquake motion acting at the same time.
STRUCTURE ANALYSIS AND DESIGN
A reentrant steel building consisting of twenty stories, shown in Fig.
1 and 2, has been analyzed using STAAD PRO 2007
software for dynamic analysis under the following loads:
| • |
Self weight |
| • |
Finish loads = 4.9 kN m-2 |
| • |
Live loads = 2 kN m-2 |
| • |
UBC 1985 code according to STAAD PRO 2007 function |
| • |
Element load = DL+25% LL |
| • |
Element load = 4.9 (FL) +0.25x2.0 (LL) = 6.4 kN m-2 |
| • |
Response spectra according to STAAD PRO 2007 function |
Then combinations of loads including checker board pattern have been
done as shown in Table 1, taking into consideration
the effect of seismic forces (as equivalent static force and response
spectra forces for the only first five modes), the coefficients used in
seismic analysis are shown in Table 2-4
and 6, also P-delta analysis has been done.
|
| Fig. 2: |
Building plan and section |
| Table 1: |
Load cases used in analysis and design of building |
 |
 |
 |
| Table 2: |
UBC factors used in analysis |
 |
| Table 3: |
Response spectrum factors used in analysis |
 |
| Table 4: |
Response spectra used in analysis and design |
 |
Then the structural design of members was carried out to select members
sections, which were in turn organized in groups for each floor as shown
in Table 5 and Fig. 3, 4
to obtain the optimum steel sections weight. Six modal combination approaches
are used for analysis, namely (SRSS, CQC, ASCE-98, TEN, ABS, CSM) to predict
the difference in design among them. Also analysis was carried out for
the building in case of using equivalent static force only and for common
static analysis, to predict the difference in steel sections weights.
|
 |
| Fig. 3: |
Design group for first storey |
|
 |
| Fig. 4: |
Design group for 10th and 20th storey |
| Table 5: |
Members groups description |
 |
RESULTS AND DISCUSSION
The results are shown in Fig. 5. It is obvious that
ABS approaches differ from other approaches of modal combinations. While
SRSS and CQC provided values that are slightly differs and in good agreement
with Amr et al. (2008).
Table 6 shows the weight of steel sections and percentage
of steel weight relative to static method for the entire building according
to methods of structural analysis. In Table 7, the weight
of columns steel sections for selected stories of building according to
their locations and methods of analysis is shown. While in Table
8 the weight of beams steel sections for selected stories of building
according to their locations and methods of analysis is shown also.
Common static and UBC analysis under estimate the design. Figure
6 and 7 show that the columns, especially those
at the lower floors, are influenced with analysis type more than beams.
For this reason the building has been analyzed (common static analysis
only) and designed for several percentage increments in total vertical
loads to get the percentage at which the same weight of steel section
can be obtained, compared to dynamic analysis. In Fig. 8,
it is obvious that this percentage is (10.5% ).
This percentage (10.5%) has been used to analyze and design two other
buildings under the same conditions and load cases to show whether this
percentage is correct or not.
Figure 9 shows the plan and section of building No.
1. Total steel sections weight obtained under dynamic loads was (900.2
kN), while the total steel sections weight under common static analysis
with vertical loads multiplied by the factor (10.5%) was (906.4 kN). This
indicates that the factor (10.5%) is correct.
Figure 10 shows the plan and section of building No.
2. In this case, the total steel sections weight obtained under dynamic
loads was (912 kN), whereas, the total steel sections weight under common
static analysis with vertical loads multiplied by the factor (10.5%) was
(868 kN). This indicates that the factor (10.5%) is correct for this case
also and can be used for preliminary design purposes.
| Table 6: |
Weight of steel sections for the entire building according
to methods of structural analysis |
 |
| Table 7: |
Weight of columns steel sections for selected stories of building
according to their locations and methods of analysis |
 |
| Table 8: |
Weight of beams steel sections for selected stories of building
according to their locations and methods of analysis |
 |
|
| Fig. 5: |
Relation between type of structural analysis and total weight
of steel sections |
|
| Fig. 6: |
Effect of structural analysis type on columns steel sections
weight |
|
| Fig. 7: |
Effect of structural analysis type on beams steel sections
weight |
|
| Fig. 8: |
Percent of total vertical loads increment and designed steel
sections weight |
|
| Fig. 9: |
Building No. 1 plan, section and 3D view |
|
| Fig. 10: |
Building No. 2 plan, section and 3D view |
NOMENCLATURE
| F(t)I |
: |
A vector of inertia forces on the node masses |
| F(t)D |
: |
A vector of viscous damping, or energy dissipation, forces |
| F(t)S |
: |
A vector of inernal forces carried by the structure |
| F(t) |
: |
A vector of externally applied loads |
| M |
: |
The mass matrix (lumped or consistent) |
| C |
: |
A viscous damping matrix |
| K |
: |
The static stiffness matrix |
| u(t)a |
: |
Vector are the absolute node displacements |
|
: |
Vector are the absolute node velocities |
| ü(t)a |
: |
Vector are the absolute node accelerations |
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