Vol.8, No.4
EARTHQUAKE ENGINEERING AND ENGINEERING VIBRATION
Earthq Eng & Eng Vib (2009) 8: 469-479
December, 2009
DOI: 10.1007/s11803-009-9126-0
Design of controlled elastic and inelastic structures A. M. Reinhorn1†, O. Lavan2‡ and G.P. Cimellaro3§ 1. Dept. of Civil, Structural & Environmental Engineering, Univ. at Buffalo- The State University of NewYork, USA 2. Faculty of Civil and Environmental Engineering, Technion–Israel Institute of Technology, Haifa 32000, Israel 3. Dept. of Structural & Geotechnical Engineering DISTR, Politecnico di Torino, Turin 10129, Italy
Abstract: One of the founders of structural control theory and its application in civil engineering, Professor Emeritus Tsu T. Soong, envisioned the development of the integral design of structures protected by active control devices. Most of his disciples and colleagues continuously attempted to develop procedures to achieve such integral control. In his recent papers published jointly with some of the authors of this paper, Professor Soong developed design procedures for the entire structure using a design – redesign procedure applied to elastic systems. Such a procedure was developed as an extension of other work by his disciples. This paper summarizes some recent techniques that use traditional active control algorithms to derive the most suitable (optimal, stable) control force, which could then be implemented with a combination of active, passive and semi-active devices through a simple match or more sophisticated optimal procedures. Alternative design can address the behavior of structures using Liapunov stability criteria. This paper shows a unified procedure which can be applied to both elastic and inelastic structures. Although the implementation does not always preserve the optimal criteria, it is shown that the solutions are effective and practical for design of supplemental damping, stiffness enhancement or softening, and strengthening or weakening.
Keywords: active control; integral control; design – redesign procedure; inelastic structures;
1 Introduction In the last 30 years, the possibility of integrated design of structural/control systems in which both the structure and its vibration control system are optimized simultaneously has been extensively researched. Integrated design of optimal structural/control systems has been acknowledged as an advanced methodology for space structures, but not many applications can be found in civil engineering. Numerous researchers addressed the (i) topology; (ii) shape; and (iii) size optimization of structures using some form of control devices (see references provided by Cimellaro et al., 2009b which are not repeated here). The fundamental idea of redesign was proposed by Smith et al. (1992) and more recently, by Gluck et al. (1996) in a form close to the one presented in this paper. The idea of redesign is incorporated into the integrated design of structural/control systems as a second stage of a two stage procedure. (1) First stage: a desired structure is chosen based Correspondence to: A. M. Reinhorn, Dept. of Civil, Structural & Environmental Engineering, Univ. at Buffalo- The State Univ. of New York, 135 Ketter Hall, Buffalo, NY 14260, USA Tel: 716-645-2839 E-mail:
[email protected] † Clifford C Furnas Eminent Professor; ‡Senior Lecturer; §Assistant Professor Received October 11, 2009; Accepted October 31, 2009
viscoelastic braces
on best practice using engineering experience and is assumed to be fixed. An “active” often “adaptable” controller is designed to obtain a desired performance requirement, e.g., drift, absolute acceleration, base shear, etc. of the initial structure. The dynamic response of the initial structure in this stage is defined as the “ideal response.” (2) Second stage: the structure and the controller are redesigned by modifying the structural system to deliver part of the controller actions, and part is preserved to be delivered by active components to achieve a common goal prescribed by the performance obtained in the first step. The structure is therefore redesigned for better economy and controllability by modifying the structural system itself, i.e., changing stiffness, damping, and weights, and by reducing the amount of active control power needed to achieve the “ideal response.” In the following sections, this idea is developed progressively from simple elastic structures to inelastic structures.
2
Design of simple elastic structure with supplemental viscoelastic braces
The first development of this idea for simple frame structures and structural systems was suggested by Gluck et al. (1996), for a frame structure braced by control devices (active or passive) that control its vibration, for which the equation of motion may be written as:
470
EARTHQUAKE ENGINEERING AND ENGINEERING VIBRATION
Mx (t ) + Cx (t ) + Kx (t ) = Ef (t ) + Hu(t )
(1)
⎧ u1 ⎫ ⎡ g11, x g12, x ⎪u ⎪ ⎢ ⎪ 2⎪ ⎢ ⎨ ⎬=⎢ ⎪ ⎪ ⎢ ⎪⎩un ⎪⎭ ⎢⎣ g n1, x g n 2, x ⎡ g11, x g12, x ⎢ ⎢ ⎢ ⎢ ⎢⎣ g n1, x g n 2, x ...
where matrices M, C, K characterize mass, structural damping and stiffness related to the deformation x(t) at various degrees of freedom. (1) In the first stage, the brace forces are included in the system as a set of control forces u(t), at locations indicated by matrix D, designed to reduce the response due to excitation forces f(t) at locations indicated by E. The equation of motion can be easily compacted to a state space formulation (Soong, 1990): z (t ) = Az (t ) + Bu(t ) + Df (t )
(2)
where z(t) = {x(t), x(t ) }T, and the parameter matrices for the system, A, for the control location, B, and for force operation, H, are: ⎡ 0 A=⎢ -1 ⎣− M K
z (t ) = Ac z (t ) + Df (t )
with Ac = A + BG
(4)
(5)
The gain matrix G, (the matrix of controlled coefficients) is obtained by minimizing a performance index: J=
∫
tf
( z T (t ) Qz (t ) + uT (t ) Ru(t )) dt
0
J=
∫
tf
{zT(t) [Q + GT R G ] z(t)} dt,
or (6)
G = −1 / 2 R −1 B T P with P solved from AT P + PA − 1 / 2 PBR −1 B T P + 2Q = 0
(7)
The control forces are obtained from Eq. (4) or explicitly:
(8)
(9)
The coefficients of the passive formulations, kij, cij, as shown, do not correspond exactly to the gain factors in Eq. (8) which were derived from the gain coefficients gij,x, gij , x . (2) In the second stage, using several approximations, the stiffness coefficients ki and the damping coefficients ci are determined here using a least square approximation. For simplicity of further derivations, Eqs. (8) and (9) can be transformed using story-drift formulation obtained from the linear transformation: x (t) = T d (t)
0
which is constrained by the equilibrium equation, Eq. (5). The matrices Q and R are weighting matrices of factors for the optimization. The selection of matrices Q and R enable solutions within the structural or resource limitations as illustrated further in the numerical example. The gain matrix G is obtained from the minimization of the performance index J:
g1n, x ⎤ ⎧ x1 ⎫ ⎥⎪ ⎪ ⎥ ⎪⎨ x2 ⎪⎬ ⎥⎪ ⎪ ⎥ g nn, x ⎥⎦ ⎪⎩ xn ⎪⎭
− k2 ⎧u1* ⎫ ⎡ k1 ⎤ ⎧ x1 ⎫ ⎪ *⎪ ⎢ ⎥ ⎪x ⎪ ⎪u2 ⎪ ⎢ −k2 k1 + k2 −k3 ⎥ ⎪⎨ 2 ⎪⎬ + ⎨ ⎬= − kn ⎥ ⎪ ⎪ ⎪ ⎪ ⎢ ⎥ ⎪un* ⎪ ⎢⎣ −kn kn ⎦ ⎪⎩ xn ⎪⎭ ⎩ ⎭ −c2 ⎡ c1 ⎤ ⎧ x1 ⎫ ⎢ −c c + c −c ⎥ ⎪x ⎪ 3 ⎢ 2 1 2 ⎥ ⎪⎨ 2 ⎪⎬ ⎢ −cn ⎥ ⎪ ⎪ ⎢ ⎥ −cn cn ⎦ ⎪⎩ xu ⎪⎭ ⎣
Assume for simplicity that the control forces are of linear form:
in which the gain matrix, G includes the coefficients, Gx, Gx , for the structural control devices. The equation of motion reduces to:
g1n, x ⎤ ⎧ x1 ⎫ ⎥⎪ ⎪ ⎥ ⎪⎨ x2 ⎪⎬ + ⎥⎪ ⎪ ⎥ g nn, x ⎥⎦ ⎪⎩ xn ⎪⎭
The control forces can be supplied by passive diagonal braces with stiffness and damping properties such as viscoelastic, fluid, or hysteretic braces. These forces are dependent on their constant stiffness and damping coefficient as follows:
1 ⎤ ⎡ 0 ⎤ ⎡ 0 ⎤ ; B = ⎢ -1 ⎥ ; D = ⎢ -1 ⎥ -1 ⎥ −M C ⎦ ⎣M H ⎦ ⎣M E ⎦ (3)
u(t ) = Gz (t ) = [Gx Gx ] z (t ) = Gx x (t ) + Gx x (t )
Vol.8
where ⎡1 1 1 ⎢ 1 1 ⎢ T =⎢ 1 ⎢ ⎢ ⎢⎣0
1⎤ 1⎥⎥ 1⎥ ⎥ ⎥ 1⎥⎦
(10)
Simultaneously, the control forces in the diagonal braces u(t) in terms of story drifts and drift velocities are obtained from Eq. (8) with the transformation from Eq. (10): (11) u(t ) = Gd d (t ) + Gd d (t ) in which Gd = T T GxT ; and Gd = T T GxT
No.4
A. M. Reinhorn et al.: Design of controlled elastic and inelastic structures
Using the same transformation, the brace forces of Eq. (9) can be written as: u* (t ) = K d d (t ) + C d d (t )
with
T K d = T T K xT = diag (ki ) ; C d = T C xT = diag (ci ) (12)
where, ki, ci are the supplemental stiffness and damping properties from each brace at level i. To determine the individual components of matrices Kd and Cd in Eq. (12), a least squares approach is considered. Since the stiffness Kd and damping Cd can be assumed to be independent, the least squares will be applied independently for Kd and Cd: T
T
kk = ∫ ∑ g kj , d d j (t ) dt / ∫ d k (t ) dt 0
and
j
T
o
T
ck = ∫ ∑ g kj , d d j (t ) dt / ∫ d k (t ) dt 0
j
(13)
o
Other techniques can be used for this redesign procedure in the second stage as shown by Gluck et al. (1996). Numerical examples were presented in the reference mentioned above. The performance of such a re-designed structure with the additional braces subjected to several earthquakes was shown to be almost the same (within 2%) of the actively controlled solution.
3
Redesign of elastic structures using the optimal approach
Although alternative methods can be used for optimal design of structures with passive components considering Lyapunov stability criteria (Lavan and Levy, 2009), the method below allows design of both active and passive components. 3.1 Methodology, basic equations and expressions Cimellaro et al. (2009b) recently expanded the procedure to allow for adjustment of the initial basic structure in the second design stage, i.e., the redesign stage, altering the stiffness K, damping C and weight Mg matrices by subtraction, not only by additions, producing weakened and lighter structures which perform better than the original. In the first stage, the design follows the same procedure using the Linear Quadratic Regulator (LQR) as presented above, although more advanced control algorithms, such as poles assignments, H2 and H∞, Soong and Manolis (1987) that can be used to determine the control forces, u(t). It is important to note that LQR implies optimality for a white noise excitation, an assumption that leads to a Ricatti equation
471
and its solution. For any other motion, this is suboptimal (Yang et al., 1990). However, the controllers designed using LQR were proven efficient in practical applications for seismic protection (Reinhorn et al., 1993). Moreover, the active control forces obtained for each DOF considered in the design procedure can be easily converted to equivalent passive devices using a method described in Lavan et al. (2008) and Cimellaro et al. (2009a). In the second stage, the structure is redesigned in order to achieve the same performance, but the control force is resolved into an active part and a passive part depending on the constant coefficients which can be used as structure modifiers. During the redesign process, mass, stiffness and damping are therefore modified in order to achieve this goal. At the end of this step, the building will maintain the same performance, but with less amount of “active control forces.” If the control force is separated u(t)=ua(t)+up(t), where ua(t) is an active part and up(t) is a passive part, the equation of the redesigned structure including the change of weight (mass), stiffness and damping matrices, respectively, by ΔM, ΔK and ΔC, becomes:
( M + ΔM ) x ( t ) + ( C + ΔC ) x ( t ) + ( K + ΔK ) x ( t ) (14) = Hua ( t ) + Ef (t ) where similarly to Eq. (4) the active component of the control force u(t) is: ua ( t ) = Ga z ( t )
(15)
where Ga is the active part of the controller after redesign. Therefore, the control law can be written in the following form: ⎡ x ( t )⎤ Hu ( t ) = HG ⎢ ⎥ ⎣ x ( t )⎦ ⎡ x ( t )⎤ ⎡ x ( t )⎤ = HGa ⎢ ⎥ − [ ΔK ; ΔC ] ⎢ ⎥ − ΔMx ( t ) (16) ⎣ x ( t )⎦ ⎣ x ( t )⎦ and the closed-loop system after redesign is
( M + ΔM ) x ( t ) + ( C + ΔC ) x ( t ) + ( K + ΔK ) x ( t ) = HGa z ( t ) + Ef (t ) (17) where ua(t), which is given by the Eq. (14), is the active part of the controller in Eq. (1) and ΔMx ( t ) + ΔCx ( t ) + ΔKx ( t ) is the passive part, up(t). The objective of the redesign is to find the passive control modifiers (ΔM, ΔK, ΔC) in order to minimize the control power needed to satisfy Eq. (14) for any given G. Note that the closed-loop system response before and after redesign remains unchanged; therefore, all the designed closed-loop system properties remain unchanged. Let Bk, Bc and Bm be the stiffness, damping
472
EARTHQUAKE ENGINEERING AND ENGINEERING VIBRATION
and mass connectivity matrices of the structural system. The changes in the structural parameters can be expressed in the form: ΔK = Bk Gk BkT ΔC = BcGc BcT ΔM = BmGm BmT where
(18)
(19)
Substituting the solution of from Eq. (1), into Eq. (16) yields: HGz ( t ) = (Gactive + Gpassive ) z ( t )
Gactive = HGa
and
(20)
Gpassive = - I 0 BpGp BpT L
(21)
with ⎡ Bk Bp = ⎢⎢ 0 ⎢⎣ 0
0 Bc 0
I0 = [ I
and
⎡Gk Gp = ⎢⎢ 0 ⎢⎣ 0
0 ⎤ 0 ⎥⎥ , Bm ⎥⎦ I
I]
I ⎡ L = ⎢ −1 ⎣ M ( HG − [ K
0 Gc 0
0 ⎤ 0 ⎥⎥ , Gm ⎥⎦
(23)
The necessary and sufficient condition to resolve the control law into active and passive parts as in Eq. (20), it is given as follows: According to Smith et al. (1992) there exists an active controller Ga: HGa = HG + I 0 BpGp BpT L if and only if HH + I 0 BpGp BpT L = I 0 BpGp BpT L
(24)
(26)
An objective function representing the power of the active part of the control law is given by: (27)
where RXX is the covariance matrix of the response. A constraint optimization is formulated minimize
F ( Gp ) =trace (Ga RXXGaT R )
(28)
where Ga is given by Eq. (25), subjected to the equality constraints of Eq. (24) and inequality constraint in Eq. (26). An approach to numerically solve the constrained optimization problem is to use the “Exterior penalty function method” that is part of the Sequential Unconstrained Minimization Techniques (SUMT) (Vanderplaats, 2005). The approach consists of creating an unconstrained objective function of the form: (29)
where F(Gp) is the original objective function, P(Gp) is the penalty function and rp is a multiplier which determines the magnitude of the penalty and is held constant during a complete unconstraint minimization. The penalty function P(Gp) is given by the following expression in this case: T P ( Gp ) = trace ⎡ Z ( GP + S - C )(GP + S - C ) ⎤ + ⎣ ⎦
{
trace ( BB + I 0 BpGp BpT LHH + − I 0 BpGp BpT L ) ×
( BB
+
I 0 BpGp BpT LHH + + − I 0 BpGp BpT L )
T
}
(30)
where Z = diag(…zi…) is a diagonal matrix where the scalars zi are chosen such that zi = 1 if the corresponding inequality constraint Gpi+si–ci≥0 is active, and zi = 0 if the constraint is not active. So the new objective function is given by the following expression:
Φ (GP , rp ) = trace (Ga RXXGaT R ) +
Such that Ga is given by: Ga = G + I 0 BpGp BpT L
Gp + S ≥ C
Φ (Gp , rp ) = F (Gp ) + rp P (Gp ) (22)
⎤ C ]) ⎥⎦
S = diag[k0i,..c0i,…m0i,…] is the matrix of the initial parameters, then these constraints can be presented as:
F ( Gp ) = ∫ uaT ( t ) Rua ( t ) dt = trace (Ga RXXGaT R )
Gk = diag (… , Δki ,…) Gc = diag (… , Δci ,…) Gm = diag (… , Δmi ,…)
where
Vol.8
(25)
where ( )+ denotes the Moore-Penrose inverse of a matrix. The stiffness and the damping of any element of the system after redesign cannot be negative while the weight of any element cannot decrease lower than a specified bound, therefore imposing specific constraints. Therefore, if C = diag[ki,..ci,…mi,…] is a matrix with diagonal elements containing the specified lower bound values of the structural elements after redesign and
T rp trace ⎡ Z ( GP + S - C )(GP + S - C ) ⎤ + ⎣ ⎦
{
rp trace ( BB + I 0 BpGp BpT LHH + − I 0 BpGp BpT L )i
( BB
+
I 0 BpGp BpT LHH + − I 0 BpGp BpT L )
T
}
(31)
Minimization of Eq. (31) requires that the following first-order necessary condition is satisfied. P1 vec diag ( Gp ) = r1
(32)
No.4
A. M. Reinhorn et al.: Design of controlled elastic and inelastic structures
where vec diag(Gp) denotes a vector with diagonal elements of Gp as its components. We have
(B
P1 =
I B + T RB + I 0 Bp )i( BpT LH + HRXX H + HLT Bp ) +
T T p 0
rp ( BpT I 0T I 0 Bp )i( BpT LH + HLT Bp ) − rp ( BpT I 0T B + I 0 Bp )i
(B
T p
LH + HLT Bp ) + Z
(33)
and r1 = vec diag ( BpT I 0T B + T RGHRXX H + HLT Bp ) − rp Zvec diag ( S - C )
(34)
Therefore, the following algorithm can be used to find the optimal solution, where it is assumed that the matrix P1 is invertible. The general algorithm for the exterior penalty function approach is shown in Fig. 1. If a small value of rp is chosen, the resulting function Φ (GP , rp ) is easily minimized, but may yield large constraints violations. On the other hand, a large value of rp will ensure near satisfaction of all constraints but will create a very poorly conditioned optimization problem from a numerical standpoint. Therefore, the algorithm starts with a small value of rp and minimize Φ (GP , rp ) . Then rp is increased by a factor γ, say γ = 3, and Φ ( GP , rp ) is minimized again, each time beginning the optimization from the previous solution, until a satisfactory result is obtained. 3.2 Numerical example - MDOF 9-story shear-type building A nine-story structure considered in this example is 45.73 m (150 ft) by 45.73 m (150 ft) in plan, and
Start Given Gp0 rp, γ Minimize Φ (GP , rp ) as unconstrained function Eq. (33) Find Gp to min. Φ (GP , rp )
| Φ (G q ) − Φ (G q-1 ) |< ε Φ (G 0 )
No
rp = γ rp
Yes Exit Fig. 1 Algorithm for the exterior penalty function method
473
37.19 m (122 ft) in elevation. The bays are 9.15 m (30 ft) on center, in both directions, with five bays each in the North-South (N-S) and East-West (E-W) directions. The building’s lateral load-resisting system is comprised of steel perimeter moment-resisting frames (MRFs) with simple framing on the furthest south E-W frame. The interior bays of the structure contain simple framing with composite floors. Typical floor-to-floor heights (measured from center-of-beam to center-of-beam for analysis purposes) are 3.96 m (13 ft). The floor-tofloor height of the basement level is 3.65 m (12 ft) and for the first floor is 5.49 m (18 ft). The floor system is comprised of 248 MPa (36 ksi) steel wide-flange beams acting compositely with the floor slab, each frame resisting one-half of the seismic mass associated with the entire structure. The seismic mass at the ground level is 965 t (66.0 kip-sec2/ft), 1010 t (69.0 kips-sec2/ft) for the first level, 989 t (67.7 kip-sec2/ft) for the second through eighth levels and 1070 t (73.2 kip-sec2/ft) for the ninth level. The seismic mass of the above ground levels of the entire structure is 9000 t (616 kip-sec2/ft). More details about the model can be found in Cimellaro et al. (2009b). The lateral stiffness of the shear-type model are reported in column 4 of Table 1, and the first three frequencies of the shear-type model are 0.45, 1.28 and 1.99 Hz. Rayleigh proportional damping is considered, including 2% of damping ratio for the first two modes. The structure was subjected to the first 30 s of white noise with amplitude of 0.15 g and with a sampling frequency of 0.02 s. The drift and acceleration response during the first stage of the algorithm are shown in Table 3. Columns 2 and 3 show the drift and acceleration response of the structure with a hypothetical active control force which is defined here as the “Ideal Response.” Initially, the story lateral stiffness is reduced proportionally to 30% of the initial stiffness value in order to obtain a first natural period increment of 83%. Response of the lighter structure is shown in columns 4 and 5 of Table 3 . Then, a fully active brace is placed at each story level in order to achieve the same performance in terms of drift of the uncontrolled initial structure. Values of the maximum active control force at each story level are shown in column 8 of Table 3 . After the structure and controller were designed independently in first stage, the controller and the building are redesigned together in the second stage to achieve the same performance (Ideal Response) by reducing the amount of active control and changing the passive components as shown in Table 1 and Table 2. The initial total energy transferred to the structure from the controller is equal to 2623.0 N·m· and, after redesign, is equal to 1972.1 N·m, so the percentage of reduction of the total energy transferred is 24.81%. Results of the redesign procedure are shown in columns 4, 5 and 6 of Table 3. Comparisons between the fully controlled structure and the redesigned structure response are shown in Fig. 2.
474
EARTHQUAKE ENGINEERING AND ENGINEERING VIBRATION
Vol.8
Table 1 Structural parameters after redesign Original
Story level No.
M kN.s2/m
C kN.s/m
(1)
(2)
9 8
Redesign K 10 kN/m
Mopt kN.s2/m
Copt kN.s/m
10 kN/m
(3)
(4)
(5)
(6)
(7)
534.1
411.4
100.02
350.3
1352.5
18.21
494.7
1152.8
291.12
342.2
5843.8
62.79
7
494.7
390.9
71.52
336.0
434.9
15.68
6
494.7
1077.4
247.63
348.3
2284.2
56.02
5
494.7
487.5
75.03
361.2
289.9
16.89
4
494.7
877.3
170.08
370.0
596.6
39.98
3
494.7
1119.4
224.76
423.7
984.1
54.02
2
494.7
1301.4
263.02
474.1
1841.0
65.92
1
503.5
906.5
143.48
441.3
723.7
36.21
3
4 Redesign of inelastic structures using optimal approach
Kopt 3
Table 2 Percentage change in structural parameters (negative indicates removing) Story level
ΔM
ΔC
ΔK
Umax
Consider a multi-degree-of-freedom inelastic building structure subjected to a one-dimensional external excitation. The general equation of motion of the inelastic system with active control forces is given by
No.
(%)
(%)
(%)
(kN)
(1)
(2)
(3)
(4)
(5)
9
-34.4
228.8
-81.8
134.97
8
-30.8
406.9
-78.4
107.11
Mx (t ) + Cx (t ) + Kx (t )+Ts fs ⎡⎣ x ( t ) ⎤⎦ = Hu(t )+Ef (t ) (35)
7
-32.1
11.3
-78.1
110.81
6
-29.6
112.0
-77.4
114.80
where x(t) is the displacement vector, M and C are the mass and inherent damping matrices, respectively, K is the stiffness matrix of all linear elements, u(t) is the active control force vector; H is the location matrix for the active control forces; E is the excitation influence matrix; Ts is the location matrix of the restoring forces and fs[x(t)] is a vector of nonlinear restoring forces in the structural
5
-27.0
-40.5
-77.5
125.26
4
-25.2
-32.0
-76.5
113.00
3
-14.4
-12.1
-76.0
113.42
2
-4.2
41.5
-74.9
108.17
1
-12.3
-20.2
-74.8
57.02
4.1 Basic equations and expressions
Table 3 Drift and acceleration response for the two stages of the algorithm Story level No.
Drift (%)
(1)
(2)
(3)
(4)
(5)
(6)
(7)
(8)
9
0.31
3.61
0.80
3.09
0.31
2.03
173.39
8
0.18
2.97
0.47
2.64
0.15
1.85
159.72
7
0.94
2.71
2.55
2.46
0.77
1.77
6
0.27
2.71
0.76
2.86
0.22
5
0.90
2.64
2.74
2.99
0.86
4
0.42
3.74
1.39
2.64
3
0.38
3.50
1.04
2
0.38
2.95
1
0.79
2.79
Ideal response xa (m/s2)
Totally active [LQR]
T*1/T1=1.83# Drift (%)
Drift (%)
Redesigned structure Umax (kN)
(9)
(10)
(11)
0.23
1.87
134.97
0.10
1.67
107.11
153.86
0.67
1.89
110.81
1.87
163.49
0.19
1.70
114.80
1.88
165.26
0.73
1.80
125.26
0.43
2.04
151.52
0.34
1.90
113.00
2.67
0.37
2.10
127.90
0.31
1.98
113.42
0.92
2.62
0.38
1.99
97.91
0.31
1.96
108.17
1.87
2.47
0.76
1.98
66.67
0.71
1.93
57.02
xa (m/s2)
Note: #The stiffness is reduced proportionally to 30% of the initial lateral stiffness
Umax(kN)
Drift (%)
xa (m/s2)
xa (m/s2)
No.4
A. M. Reinhorn et al.: Design of controlled elastic and inelastic structures
Ideal response Active redesign
9
8
8
7
7
6
6
Story level
Story level
9
5
5
4
4
3
3
2
2
1
475
1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 Drift (%) (a)
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 Acceleration (m/s2) (b)
Fig. 2 Response comparison of “actively controlled” and “redesigned” system
elements. fs[x(t)] in Eq. (35) can be separated into two parts, one representing the elastic behavior and the other representing the nonlinear hysteretic behavior, i.e., f s ⎡⎣ x ( t ) ⎤⎦ = Kx ( t ) + Hfs ⎡⎣ x ( t ) ⎤⎦
(36)
in which K is a linear elastic stiffness matrix for the linear elastic parts of the nonlinear or hysteretic structural components; f s =[f1 f2 … fl]T is a vector representing the nonlinear or hysteretic parts of the restoring forces for nonlinear structural components; and H is a location matrix for the nonlinear elements. The component i of the vector of nonlinear forces, which is the nonlinear force in element i, is modeled using the continuous evolutionary Sivaselvan-Reinhorn model (Sivaselvan and Reinhorn, 2000) discretized as follows: fsi ⎡⎣ xi ( t + τ ) , xi ( t + τ )⎤⎦ = ai ki xi ( t + τ ) + (1 - α i ) ⋅ f yi ⋅ (ν i + τν i ) (37) i = 1,… , n where the normalized nonlinear force, νi=(fsi- αikixi) / (1-αi)fyi, is expressed by
ν i (t ) =
xi ( t ) ⎡ ni 1 − ν i ( t ) η1i + η 2i ⋅ sgn (ν i ( t ) xi ( t ) ) ⎤ ⎣ ⎦ xyi
i = 1,… , n
(
)
(38)
in which ki is the elastic stiffness; αi is the ratio of post-yielding to pre-yielding stiffness; ni is the power controlling the transition from the elastic to inelastic range; η1i and η2i are parameters controlling the shape of
the unloading curve (η2i=1– η1i, for compatibility with the plasticity theory); fsi is the portion of the applied force shared by the hysteretic spring; and fyi is the yielding force of the hysteretic spring. The procedure suggested herein can use any inelastic-hysteretic model without changing the procedure. Note that the ith DOF is linear elastic if αi=1.0. The term αikixi(t) is the post-yielding hardening linear elastic stiffness that will appear in the K matrix of Eq. (36). The design method presented further herein reverses the conventional procedure by designing the structure after the controller has been determined. The procedure is done in two stages as before: In the first stage, based on the structural parameters ξ (e.g., strength, damping and weight), a controller is designed to satisfy the desired performance requirements (e.g., drift, absolute acceleration, base shear, etc.) of the structure, which is assumed to be inelastic. An appropriate nonlinear control law that takes into account stability issues is adopted, since the structure is intended to yield, i.e., becoming inelastic. One of the methods that are able to accommodate the nonlinear behavior of the system and the uncertainties involved in the structural analysis is the sliding mode control (see Utkin, 1992 and Yang et al., 1995). The main advantages of sliding mode control are the: (i) possibility of stabilizing nonlinear systems, which cannot be stabilized by using continuous state feedback laws; (ii) robustness against a large class of perturbations, or model uncertainties; and (iii) need for a reduced amount of information in comparison to classical control techniques. In general, the design of the sliding mode control can be summarized in two fundamental steps as shown by Lavan et al. (2008). The first step is the selection of sliding surfaces such
476
EARTHQUAKE ENGINEERING AND ENGINEERING VIBRATION
that the system exhibits the desired behavior in the sliding mode. The second step is determining the control laws that guarantee that the structure reaches the sliding condition. Yang et al. (1995) presented details of this procedure. For the first step, a possible law for the continuous sliding mode control in the case of a saturated controller is given by ⎧α i*Gi − δ i λi ⎪ ui ( t ) = ⎨ * ⎪⎩ui max sgn (α i Gi − δ i λ ) i = 1,… , n
if α i*Gi − δ i λi ≤ ui max otherwise
;
(39)
in which 0 ≤ α i* ≤ 1, δ i ≥ 0 , is the sliding margin and ui max represents the upper bound of the ith control force. λi and Gi are r-vectors obtained using the procedure described in Yang et al. (1995). In the second stage, the redesign is accomplished by modifying the structural parameters ξ subjected to certain constraints by minimizing the amount of active control power needed to achieve the desired performance requirements. An objective function expressed by the sum of the power of the active control forces is minimized. The objective function is given by
( )
F1 ξ = ∫ u ( t ) Rua ( t ) dt T a
(40)
where ua is the active force vector of the controllers after redesign and is not known apriori, but can be determined by assuming that the building has the same structural response x, before and after redesign. Minimizing Eq. (40) can lead to spikes in the time history of the active control force ua that cannot be followed by the controller given in Eq.(39). Therefore, another objective function that minimizes the maximum of the active control force was suggested:
( )
F2 ξ = max ( uaT ( t ) Rua ( t ) )
(41)
each objective function, subjected to all constraints, independently. F1w( ξ ) and F2w( ξ ) are the values of the objective functions associated with the initial conditions of the structural parameters ξ. In order to determine the objective function in Eq. (42) it is necessary to determine the active force vector ua. Similar to the force derived in Eq. (14), the active component can be written as:
ua ( t ) = H −1 ⎡⎣ Hu ( t ) + ΔMx ( t ) + ΔCx ( t ) + Ts Δf s [ x ( t )]⎤⎦
(43)
If H is not invertible, ua(t) can be approximated by performing a pseudo-inverse of H or by employing a least square procedure. Substituting Eq. (38) into Eq. (37), the hysteretic force can be then written as follows
f si ( t ) = Bi ( t ) + Di ( t ) βi
( ) ( )
( ) ⎞⎟ ( ) ⎟⎠
( )
2
( ) ( )
( ) ⎞⎟ ( ) ⎟⎠
⎛ F2 ξ − F2 * ξ +⎜ w ⎜ F2 ξ − F2 * ξ ⎝
where W1 is the weighting factor; F1( ξ ) and F2( ξ ) are, respectively, the objective function given in Eqs. (40) and (41); F1*( ξ ) and F2*( ξ ) are the target objective functions; F1w( ξ ) and F2w( ξ ) are the worst known value of the two objective functions. F1*( ξ ) and F2*( ξ ) are the most difficult values to define in advance. In this paper, these values have been determined by optimizing
(44)
Bi ( t ) = α i ki xi ( t + τ ) + (1 − α i ) ki xi ( t ) ⋅τ ⋅
(
)
⎡1 − ν ( t ) ni η + η ⋅ sgn (ν ( t ) x ( t ) ) ⎤ i i 1i 2i i ⎣ ⎦ Di ( t ) = (1 − α i ) f yiν i ( t )
i = 1,…, n (45)
Assuming that the history of νi is known and it is the same for the structure before and after redesign, substituting Eq. (44) in Eq. (43) yields the following equation:
ua ( t ) = H −1 ⎡⎣ Hu ( t ) + ΔMx ( t ) + ΔCx ( t ) + Ts (1 − β ) D ( t )⎤⎦
(46) Finally, the optimization problem can be solved by substituting Eq.(46) in Eqs.(40) and (41). Define: ⎡ ΔM Ξ = ⎢⎢ 0 ⎢⎣ 0
2
(42)
i = 1,…, n
where βi is a multiplier of the yielding restoring force (1-αi)fyi such that the redesign yielding force in the element i is βi multiplied by its original yielding force, while
A multi-objective optimization problem is then formulated where the combined objective function is given by ⎛ F1 ξ − F1 * ξ F ξ = W1 ⎜ w ⎜ F1 ξ − F1 * ξ ⎝
Vol.8
0 ΔC 0
0⎤ 0 ⎥⎥ B ⎥⎦
(47)
where ΔM = diag (… , Δmi ,…) ΔC = diag (… , Δci ,…) B = diag (… , βi ,…)
i = 1,… , n
(48)
are diagonal matrices of the design variables, and βi are in fact weakening or strengthening factors. Then, if L = diag[…,miL,..ciL,…βiL,…] and U = diag[…, U mi ,..ciU,…βiU,…] are matrices with diagonal elements containing the specified lower and upper bound values
No.4
A. M. Reinhorn et al.: Design of controlled elastic and inelastic structures
of the structural elements, respectively, and S = diag[…, mio,..cio,…βio,…] is the matrix of the initial parameters, then these constraints can be presented in matrix format as L≤Ξ + S ≤U (49)
T Φ ( Ξ , ρ ) = F ( Ξ ) + ρ ⋅ trace ⎡ Z L ( Ξ + S - L )( Ξ + S - L ) ⎤ + ⎣ ⎦ T + ρ ⋅ trace ⎡ Z U ( −Ξ − S + U ) ( −Ξ − S + U ) ⎤ ⎣ ⎦ (53)
If a small value of ρ is chosen, the resulting function Φ ( Ξ , ρ ) is easily minimized, but may yield large constraints violations. On the other hand, a large value of ρ will ensure near satisfaction of all constraints but will create a very poorly conditioned optimization problem from a numerical standpoint. Therefore, the algorithm starts with a small value of ρ and minimize Φ ( Ξ , ρ ) . Then ρ is increased by a factor γ, say γ = 3, and Φ ( Ξ , ρ ) is minimized again, each time beginning the optimization from the previous solution, until a satisfactory result is obtained. Finally, at the end of the second stage, the building will maintain the same performance by transforming the control forces in equivalent changes of the mass, damping matrix and restoring force, respectively, by ΔM, ΔC and ΔfS.
Therefore, the optimization problem is defined as follows: Minimize: F ( Ξ ) as objective function
(50)
Subjected to: L ≤ Ξ + S ≤ U as side constraints The solution of the optimization problem can be determined numerically using the Exterior Penalty Function Method (Vanderplaats, 2005) described above and reformulated here. The approach consists of creating an unconstrained objective function of the form
Φ (Ξ , ρ ) = F (Ξ ) + ρ P (Ξ )
(51)
4.2 Numerical example - MDOF 8-story building
where F(Ξ) is the original objective function in Eq. (42), P(Ξ) is the penalty function and ρ is a multiplier which determines the magnitude of penalty and is held constant during a complete unconstraint minimization. The penalty function P(Ξ) is given by
An eight-story nonlinear structure is considered where the parameters follow the example of Yang et al. (1990), however, the stiffness at each story was scaled to result in a period of 0.81 s. The mass, damping, stiffness and yield strength of the shear-type model are reported in columns 2, 3, 4 and 5 of Table 4. The structure was subjected to the first 30 s of El Centro earthquake with a sampling frequency of 0.02 s. The drift and acceleration responses during first stage of the algorithm are shown in Table 5. The drift and the acceleration response of the initial building or, in other words, the performance level to be achieved (Target Response), are shown in columns 2 and 3. The initial story lateral stiffness of the elastic portion of the Sivaselvan-Reinhorn model is first reduced proportionally to 50% of the initial stiffness value in
T P ( Ξ ) = trace ⎡ Z L ( Ξ + S - L )( Ξ + S - L ) ⎤ + ⎣ ⎦ T trace ⎡ Z U ( −Ξ − S + U ) ( −Ξ − S + U ) ⎤ ⎣ ⎦
477
(52)
where ZL= diag(…,zi,…) is a diagonal matrix where the scalars zi are chosen such that zi=1 if the corresponding lower bound inequality constraint ξi+si-li≥0 is active and zi=0 if the constraint is non active. An analogous expression of ZU exists for the upper bound. Therefore, the new objective function is given by
Table 4 Optimal structural parameters after redesign using El Centro earthquake Story level No.
Original
Redesign
C (kN.s/m)
K
Fy
Mopt
(kN.s2/m)
(103kN.m)
(kN)
(kN.s2/m)
(2) 345.6
(3) 196
(4) 301.4
(5)
(6)
(7)
(8)
(9)
8
4521
197.6
768.3
5091.7
123.5
7
345.6
243
371.8
6321
223.2
486.0
4400.9
207.3
6
345.6
298
455.4
8653
261.1
745.2
5035.3
223.9
5
345.6
349
534.6
10692
273.3
1078.3
5905.5
226.2
4
345.6
386
591.8
12428
289.6
471.2
6495.6
239.7
3
345.6
410
627.0
13794
308.5
1159.7
6986.7
227.9
2
345.6
467
704.0
16192
312.6
1950.8
7986.9
218.1
1
345.6
490
748.0
17952
319.8
2632.1
8806.3
215.2
(1)
M
Copt (kN.s/m)
Fy,opt
Umax
(kN)
(kN)
478
EARTHQUAKE ENGINEERING AND ENGINEERING VIBRATION
Vol.8
Table 5 Drift and acceleration response Story level
Target response
T*1/T1=1.41#
Active response Umax=250 kN
Redesign approach
No.
Drift (%)
xa (m/s2)
Drift (%)
xa (m/s2)
Drift (%)
xa (m/s2)
Drift (%)
xa (m/s2)
(1) 8
(2) 0.36
(3) 12.28
(4) 0.45
(5) 6.67
(6) 0.43
(7) 7.37
(8) 0.20
(9) 9.86
(10) 123.5
7
0.65
11.69
1.06
6.76
0.80
7.12
0.41
8.58
207.3
6
0.81
9.62
0.97
5.63
0.75
4.99
0.65
8.04
223.9
5
0.71
9.44
0.83
5.91
0.56
5.31
0.70
5.85
226.2
4
0.61
8.21
0.61
6.10
0.50
5.94
0.62
6.03
239.7
3
0.71
6.77
0.71
5.83
0.49
5.58
0.81
6.38
227.9
2
0.58
6.68
0.58
4.63
0.48
4.43
0.68
3.90
218.1
1 0.58 4.35 0.58 3.56 0.48 3.87 # Note: The stiffness is reduced proportionally to 50% of the initial lateral stiffness
0.67
3.76
215.2
order to obtain a first natural period increment of 41%. The response of the lightweight structure is shown in columns 4 and 5 of Table 5. An active brace with the same maximum saturation control force (Umax = 250 kN) was introduced at each story level in order to achieve the same performance in terms of drift of the uncontrolled initial structure. After the structure and controller are designed independently in the first stage, the controller and the building are redesigned together in the second stage to achieve the same performance (Target Response) by reducing the objective function in Eq. (42). The values of the target objective functions selected are F1*(ξ)=1.02×107 and F2*(ξ)=0.80×103, while the value of the objective functions corresponding to the initial conditions are F1w(ξ)=6.62×107 and F2w(ξ)=7.72×104, while the value of the weight factor selected is W1=1.32×106. In this case, the total power of the active control force after redesign is reduced by 24.5%. The results of the optimal structural parameters (M, C, Fy and Umax) after redesign are shown in columns 6, 7 8 and 9 of Table 4 while the percentages of variation with respect to the initial structure are shown in Table 6. The performance in terms of drift and acceleration of the redesigned structure are shown in columns 8 and 9 of Table 4. Note that the weight is reduced, especially in the upper stories. Globally, a total weight reduction of 21% is obtained.
5 Concluding remarks An integrated design procedure for the design of elastic and inelastic structures equipped with control systems, either active, or passive, or a combination, was summarized in this paper. The method follows a two stage approach. First, an initial structure is assumed including its topology, stiffness and strength, and a control force is determined using a rigorous active control algorithm to ensure its controllability and stability. Then, the control
Umax(kN)
force is resolved in passive and active components. The passive components are implemented by modifying the structural system, i.e., weight, stiffness, damping and strength, while the remaining part is implemented with an active controller. In order to achieve an optimal structural configuration, optimization of the passive and active components is implemented in the second step. In comparison with a traditional design, the proposed integrated design procedure is able to achieve an overall weight saving, a stiffness reduction and a minimal active control component. Such reductions consequentially may lead to a reduction of initial investments costs. Note that the integral design leads to softer, weaker and highly damped stable structures ensured by the stability of the algorithms applied in the first stage of design. This paper is a summary of developments prepared for multiple journal archival papers reformulated to obtain a unified and unique approach. More details can be found in the References. The work is, in particular, an implementation of the vision of the mentor of the authors, Professor Emeritus Tsu T. Soong.
Acknowledgement The authors wish to acknowledge the guidance and direct contributions of Professor Soong to the integral design approach and to the inspiration and vision he always provided. The authors also acknowledge the financial support of the Multidisciplinary Center for Earthquake Engineering Research (MCEER) headquartered at the University at Buffalo, State University of New York, which excelled in the development of seismic protective systems for structures and equipment.
References Cimellaro GP, Lavan O and Reinhorn AM (2009a),
No.4
A. M. Reinhorn et al.: Design of controlled elastic and inelastic structures
“Design of Passive Systems for Controlled Inelastic Structures,” Earthquake Engineering and Structural Dyamics., 38(6): 783–804. Cimellaro GP, Soong TT and Reinhorn AM (2009b), “Integrated Design of Controlled Structural Systems,” ASCE/Journal of Structural Engineering, 135(7): 853–862. Gluck N, Reinhorn A and Levy R (1996), “Design of Supplemental Dampers for Control of Structures,” ASCE/Journal of Structural Engineering, 122(12): 1394–1399. Lavan O, Cimellaro GP and Reinhorn AM (2008), “A Noniterative Optimization Procedure for Seismic Weakening and Damping of Inelastic Structures,” ASCE/Journal of Structural Engineering, 134(10): 1638–1648. Lavan O and Levy R (2009), “Simple Iterative Use of Lyapunov’s Solution for the Linear Optimal Seismic Design of Passive Devices in Framed Buildings,” Journal of Earthquake Engineering, 13(5): 650–666. Reinhorn AM, Soong TT, Riley MA, Lin RC, Aizawa S and Higashino M (1993), “Full Scale Implementation of Active Control Part II: Installation and Performance,” ASCE/Journal of Structural Engineering, 119(6): 1935–1960. Sivaselvan M and Reinhorn A (2000), “Hysteretic
479
Models for Deteriorating Inelastic Structures,” ASCE/ Journal of Engineering Mechanics, 126(6): 633–640. Smith MJ, Grigoriadis KM and Skelton R (1992), “Optimal Mix of Passive and Active Control in Structures,” Journal of Guidance, Control and Dynamics, 15(4): 912–919. Soong TT (1990), Active Structural Control: Theory and Practice, Longman Scientific, London. Soong TT and Manolis GD (1987), “Active Structures,” ASCE/Journal of Structural Engineering, 113(11): 2290–2301. UtkinVI (1992), “Sliding Modes in Control Optimization,” Springer: New York. Vanderplaats GN (2005), Numerical Optimization Techniques for Engineering Design, Vanderplaats Research & Developments, Inc., Colorado Spring, CO 80906. Yang JN, Akbarpour A and Askar G (1990), “Effect of Time Delay on Control of Seismic-Excited Buildings,” ASCE/Journal of Structural Engineering, 116(10): 2801–2814. Yang JN, Wu JC and Agrawal AK (1995), “Sliding Mode Control of Nonlinear and Hysteretic Structures,” ASCE/Journal of Engineering Mechanics, 121(12): 1386–1390.