2.1. The Timoshenko Beam with De Saint-Venant Torsional Behavior (e-TBM)
The equivalent beam model presented in [
8] to estimate the three-dimensional response of asymmetric tall buildings was based on the Timoshenko beam with coupled De Saint-Venant torsional behavior (e-TBM). As it is well known, the Timoshenko beam model enhances the Euler–Bernoulli beam model by considering the shear stiffness, the cross-section rotation inertia and the possibility to have deformation where the beam axis can also be not normal to the cross section.
In this Section the ruling equations are briefly recalled and additional details can be found in [
8]. Assuming a reference system where
z is the beam axis,
x and
y are the cross-section principal axis, the problem in the planar case,
, can be described by the following equations:
the balance equations
where
and
,
and
are the distributed load and moment, the shear and the bending moment, respectively.
the constitutive equations
where
,
,
and
are curvature, shear deformation, bending stiffness and shear stiffness, respectively.
the compatibility equation, assumed as in Timoshenko (1921)
Assuming constant stifnesses along the beam axis, the following equations are obtained
which can be extended to the equations of motion assuming inertial forces and moments per unit length and D’Alembert’s principle
where
is the mass density, assumed to be uniform over both the beam cross-section area and beam length.
The previous planar model can be easily extended to the 3D case writing the equations of motion in the
x–
z plane and describing the torsional deformation by the classical De-Saint Venant’s model where, assuming the torsional moment,
, stiffness,
, and distributed moment,
, and using the constitutive equation
and the balance equation
one obtains
Finally, assuming uniform mass density
and using D’Alembert’s principle, the equation of motion becomes
where
is the polar moment of inertia and
is the cross section surface.
The equations ruling the 3D dynamic problem can be obtained by coupling the Timoshenko beam model in the
x and
y directions with the torsional model in (
13) and assuming a reference system with the origin in the cross-section centroid
where
with
,
,
,
and
being the classical first and second inertia moments and
and
are the coordinates of the shear centre. Equations (
14)–(
18) can be used to compute natural frequencies and the corresponding mode shapes [
8].
2.2. The Sub-Structure Approach (e-SBM)
The equivalent beam model proposed by Potzta and Kollar [
11] is based on the sub-structure approach, which uses an equivalent cantilever sandwich beam to model the lateral load-resisting system of a tall building starting from the stiffnesses of the different structural sub-systems (e-SBM). The equivalent sandwich beam is given by coupling one Timoshenko beam with bending stiffness,
, and shear stiffness,
S, and one Euler–Bernoulli beam with bending stiffness,
, assuming to have the same horizontal displacements at all levels. The parameter
is also referred to as “local" stiffness since it is obtained from each of the subsystem columns and beams. Moreover, the 3D behavior is also described using the Vlasov non-uniform torsion theory. In this Section, the main features of the model described in [
11] that will be used to develop the proposed novel equivalent beam model are recalled.
In the sub-structure approach, each lateral load-resisting sub-system of the building (i.e., wall, truss, frame, coupled shear wall) is replaced by a sandwich beam or a Timoshenko beam with suitable stiffness parameters. In this paper, a tall building with two sub-systems, frame and truss, will be considered. For these two cases, Table 1 in [
11] gives the following stifness values:
k-th frame with
columns, having cross section area,
, and inertia,
,
, and
n beams having cross section inertia,
,
:
where
is the distance of the
i-th column from the frame axis,
is the span of the
i-th beam (i.e., distance between columns
i and
) and
h is the constant inter-storey height. It is worth noting that these values do not depend on the number of floors.
k-th truss with columns and bracing having cross section area,
and
, respectively:
where
L is the span of the truss,
h is the inter-storey height and
d is the length of each of the two cross-bracing at each storey, with
. In this case, a pure Timoshenko beam is assumed, since
is null.
In the general 3D case, the tall building equivalent sandwich beam parameters can be found by using the approach based on the deformation energy and can be collected into three matrices,
,
and
. In the following, adopting the assumptions in [
11], the reference system
is used with the beam axis in the
x direction;
v and
w are the displacements of the beam axis in the
y and
z directions, while
,
,
are the rotations about the
x,
y, and
z axis, respectively. Finally,
and
are the shear deformations, while
is the torsional deformation with the two components,
and
, due to bending and shear of the single sub-structures, respectively. In the approach proposed in [
12,
13] the following equations hold
It is worth noting that neglecting the shear deformation, i.e., one obtains the standard Vlasov model with , and .
The values of
,
and
are obtained assuming that the deformation energy of the tall building equivalent sandwich beam is the same of the sum of the deformation energy of all the
N sub-structures
with
and where
,
and
are the stiffness parameters of each equivalent beam sub-structure in the global reference system
;
and
are the torsional stiffness of the equivalent beam and the
k-th sub-structure, respectively. These matrices are computed starting from the stiffness values in the local reference system
(
horizontal axis and
vertical axis)
and then changing the orientation and location of the lateral load-resisting sub-system by the transformation
where
X can be
,
or
S,
is the angle between
and
y, and
and
are the coordinates of the origin of the system
in the reference system
.
Solving Equation (
31), the following relations are obtained
where
Approximate values for are suggested when vibration analyses are of interest. For example, , , can be assumed for modeling the first, second and third vibration mode, respectively.
2.2.1. Modal Analysis
Natural frequencies and associated mode shapes can be found solving the following eigenvalues and eigenvectors problem [
13,
14]
where
where
H is the equivalent beam height,
m is the mass per unit length,
is the polar moment of mass,
and
are the coordinates of the centroid. The values of
and
are respectively 3.52 and 0.5
for the first vibration mode, 22.03 and 1.5
for the second vibration mode and 61.7 and 1.25
for the third vibration mode. The polar moment of mass is given by
where
A is the region occupied by the generic building plan and
is the mass per unit area. Assuming uniform mass distribution
is constant, giving
where
and
are moments of inertia about
A principal axes.
2.2.2. Torsional Stiffness
When dealing with torsion, it is well-known that two stiffnesses may be considered: the classical De Saitnt-Venant torsion stiffness and the one associated with the section warping. The latest is implicit in the contribution given by
and
, while the first one, represented by
in Equation (
42) need to be estimated. The following espression for
is proposed in [
15] when
w walls and
f frames and/or coupled beams are the considered sub-structures
where
is the torsional constant of the
k-th wall and
and
are the shear stiffnesses of the
h-th frame or coupled-beam having
y-
z centroind coordinates equal to
and
, with
and
being the coordinates of the stiffness center. Furthermore, the finding in [
15] suggest using the effective shear stiffness,
, in place of the shear stiffness,
S, as follows
The values of
and
can be estimated by
where
and
are the frequencies in
y and
z direction of the wall and frame, that can be estimated with
where
,
and
are the frequencies considering only shear stiffness, global bending stiffness and local bending stiffness, respectively, given by
with
a factor related to the mass distribution.
2.3. Estimation of Parameters and Genetic Algorithms
The mechanical characteristics of the e-TBM described in
Section 2.1 are calibrated minimizing a suitable function of
natural frequencies and
static displacements of the e-TBM and the e-SBM in
Section 2.2 that has general expression:
where
,
, are the
e-TBM design variables (e.g., mechanical parameters such as cross-section area, second moments, bending, shear and torsional stifnesses);
and
,
, are the e-TBM and e-SBM natural frequencies;
and
,
, are the e-TBM and e-SBM static response displacements;
and
are suitable weights to drive the solution on the desired target response parameters. At each step of the minimization procedure, a check on the modal shapes is performed in order to match both natural frequencies and mode shapes of the e-SBM.
Genetic algorithms [
16] are chosen to minimize the function in Equation (
53). This choice is usually done when the optimization problem cannot be solved using classical algorithms, for example because of the discontinuous, not derivable or stochastic objective functions. Classical minimizing algorithms were not successful in the case study that will be presented in this paper because of the significant influence of the initial design variable values. Sensitivity analysis was also performed to have information both on the influence of the variation of the design parameters on the quantities of interest and on the presence of local minima of Equation (
53). The genetic algorithms were successful to overcome the problem due to their inherent feature to randomly search in the solution domain and to easily follow decreasing values of the objective function. The classic genetic algorithm can be summarized as follows [
17]:
generation of a random set of values for the design variables (initial population);
evaluation of the objective function and selection of the best solutions;
generation of a new population of hybrid solutions by means of genetic operators (cross-over and mutation);
iterative repetition of points 2. and 3. with this population;
termination of the algorithm when a specified condition is reached (e.g., convergence criteria, maximum number of generations, time limit).
A crucial point to find the optimal solution is to give a wide range for the initial population, which is related to the initial domain of the design variables.