ACTIVE AND PASSIVE CONTROL SYSTEMS CONSIDERING EARTH QUAKE EXCITATION

     

The energy imparted to a structure from earth quakes is dissipated by damping, if the earth quake is large enough; it is accompanied by plastic yielding. The damping energy is associated with inherent structural damping of the conventional members, and the external damping contributed by the structural control systems, such as viscoelastic (VE) damper and active bracing system (ABS) .The strain energy is composed of recoverable elastic strain energy and irrecoverable hysteretic energy due to plastic yielding known as yield strain energy. Yielding is inelastic behaviour of the structures subjected to large earth quakes ,Theoretically, the inelastic behavior can be described by plastic zone approach or elastic plastic hinge model . The plastic zone approach approach considers explicitly the gradual spread of yielding throughout the volume of the structure, such residual stresses and geometric imperfections and the material strain hardening .The elastic plastic hinge model assumes that inelastic behavior is concentrated at the plastic hinge location.

In recent years, new ideas for the structural integrity and structural control motion have been developed. A procedure for designing the full scale ABS control system for an earth quake was  also presented .the stiffness and the damping ratio of the viscoelastic damper were estimated based on the energy based systems. The number, size, and the optimal location of the VE dampers for supporting the structure and their effects on stiffness and the damping ratio were     studied by the Zhang and Soong in 1992

THEORY

An active structure is a structure composed of two types of load resisting members:  Conventional static members designed to support static loads and active braces or active variable stiffness members designed to resist dynamic loads. The structure is optimally designed structure in terms of minimizing its cost while obtaining the optimal control forces for the active control systems. The cost of the structure is expressed in terms of structural volume, the structural design variables are the member areas and the control variables are the optimal control forces. 

 The design of active structures was also addressed by Cheng and Pantelides and Pantelides who formulated the active structure problem in two stages: first, the optimal control was expressed in the closed form as implicit functions of the design variables (moment of inertia of the structural members); and second, the design variables where modified iteratively in order to minimize the structural volume with constraints imposed on the structural response. The definition of the active structures can be extended to include. In addition to the active control system, not only purely static members, but passive control members such as viscoelastic dampers as well.  

In the present paper, the energy dissipation is expressed in terms of:

1. The inherent damping from the conventional structural and non structural members of the structure—the inherent structural damp of the structures is assumed to be present at fixed level;.
2. The yielding of the conventional members ;and
3. The external damping and the stiffness contribution of the passive structural control (VE dampers), and the active structural control (ABS) systems.

The optimal design of the conventional members of the active structures is investigated in terms of minimizing the structural volume while satisfying the structural response constraints during an earthquake. Structures equipped with VE dampers or ABS control systems are considered. The influence of inherent structural damping on the optimum designs is investigated.

The following optimal designs are done:

1. The optimal design of the 10 storey frame with different levels of inherent structural damping
2. The optimal design of the 10 storey frame with the two ABS controllers
3. The optimal design of the 10 storey frame with the two VE dampers
4. The optimal design of the 10 storey frame with the one ABS controller and one VE damper

A ten storey frame shown in Fig1 with different assumed levels of inherent structural damping is used to demonstrate the optimal design of the conventional members of the active structure .Three cases are discussed 

a) One active bracing system (ABS) installed on the first floor and one ABS on the second floor 

b) One viscoelastic (VE) damper on the first floor VE damper on the second floor 

1. One ABS on the first floor and one viscoelastic VE damper on the second floor.

The optimal design of conventional members of the structures is expressed as follows: minimize the structural volume subject to drift constraints between floors j and (j-1), and stress constraints for the conventional structure members. No constraints are imposed on the damping level to be achieved. It should be noted that for the optimization, linear behavior of the structural members is assumed. The objective of the optimization is to minimize the structural volume of the convention members of the structure. This measure represents the cost of the conventional structural system. The design variables are defined as the cross sectional areas of the structural members. The moment of inertia I, of the steel frame members can be obtained using well known relations as follows. (Haug .E.I and Arora ,J.S.)

                                        

where  Zb,  Si  ,Ai are the sectional modulus ,the least radius of  gyration and the cross sectional area of the ith  element; αi  βi  and  χi  are  constants and equal to 0.58,0.58and 0.67 respectively. The allowable combined bending and axial stress for each conventional member of the steel frame is assumed as 150MPa, and  the allowable inter storey drift for each floor is set at h/180. Where h=floor height, which is 2.5cms for the first floor, and 2.0cms for the second up to the 10th floor.

In cases   (1) and (3) the maximum control forces of the ABS controllers are limited to less than 20% of excitation lateral force. The next section describes how the optimal dimensions of the VE dampers can be obtained for an increased effective damping ratio. The strong column-weak beam philosophy of the design codes is also implemented. 

A modified iterated simulating and annealing (MISA) method (Tzan,S.R and Pantelides.C.P.) is used for optimal design. This method was found to be advantages for the optimization of structural systems with constrains arising from the dynamic excitation such as earthquakes.

         

Optimization techniques have for most part focused on structural optimization of systems with the static constrains. When the dynamic or time varying constraints are imposed, as in the case of the buildings in earth quakes, the feasible region becomes disjoint and the local minima exist A modified annealing method with sensitivity analysis and automatic reduction of the search range was recently presented by Tzan and Pantelides .The latter strategy was found to be advantageous for optimization of the structural systems with constraints arising from dynamic loading as compared to classical optimal design methods. 

The optimal configuration and dimension design of VE dampers in a building is also studied. The optimal number, location and cross-section area are found for increasing effective damping ratio for structures with different assumed inherent damping levels. The optimal design of constant number of VE dampers for different effective damping level is also examined.

Finally the elastic and inelastic analysis conventional and active structures equipped with ABS controls are performed. In these analysis the energy time histories and in particular the energy dissipation of conventional and active structures in term of the effective damping and yielding are contrasted. The number of yielding events in the structure elements of conventional and active structures undergoing inelastic deformation is examined for different earthquakes.

ENERGY DISSIPATION AND ITS DISTRIBUTION IN THE SEISMIC    STRUCTURE

The input energy imparted to the structure by the earth quake is dissipated by damping associated with the conventional members as well as any passive and/or active control systems, and by yielding of the structural members when the inelastic behaviors develops. .The energy terms can defined by integrating the equation of the motion of the structure as

Where     are the resisting forces in the yielding members of the system .(Chopra.A.K) In equation (1).m=mass matrix,  c*=damping matrix for the structure control system  which includes the effect of the inherent  structural damping  and that of the passive and /or active control systems xg(t) is the horizontal ground  acceleration, and x,x and x are the relative displacement ,velocity and the acceleration response of the dynamic degrees of  freedom. Equation (1) can be described in terms of the energy of the system as

                       

    are the kinetic energy associated with the motion of the structure relative to the ground, the damping energy, the yield strain energy, the elastic strain (recoverable strain)energy and the input energy.

These energies defined as

                  

Where k=elastic stiffness matrix 

Note that Equation (3d) equals zero when the linear elastic structural system is considered .The damping energy is provided by the inherent structural damping which is assumed to exist at the certain fixed level, as well as the additional external damping due to structural control systems.

In this paper, the passive system considered is viscoelastic (VE) damper and the active control system considered is active bracing system (ABS) .The instantaneous optimal control (Yang. J.N., Akbarpur, A and Ghammaghmi .P) algorithm is used for the seismic structures equipped with the ABS. A. time dependent performance index. J(t) was defined  

                                     

where ZT(t) is the transpose of the state vector; ZT(t)={x(t),x(t)}T and x(t), and x(t) are the relative displacement and the velocity responses of the structure and R are the weighing matrices chosen by the designer. The optimal control force ,can be obtained by minimizing J subject to the constraint of the equation of the motion .Matrix Q in the equation(4) is usually assigned to be an identity matrix and R is a diagonal matrix multiplied by a constant, r, at the non zero elements of the diagonal

The equation of the motion is expressed as 

                                      

Where m, c and k are the N*N mass, damping and stiffness matrices, N being the number be the number of degrees of freedom. Note that the c includes the contribution of inherent structural damping, which is assumed to exist at the certain fixed level, and that of the passive control structures such as viscoelastic dampers, if present is the N*m matrix which defines the ABS controller locations with respect to the structure topology ,m being the number of controllers; δ is effective loading vector which for the building under earth quake excitation   is   δT ={-m1, -m 2 ………--.mn }   Where mi the mass of the mass ith floor, CABS  is  the equivalent damping matrix due to the presence of the active structural control and ∆t is the time step at  which the control force is evaluated. Equation (5) can be used to obtain the response of the structure equipped with the ABS control system and /or viscoelastic damper during an earth quake.

 In the recent years, the material properties of the VE dampers have been studied by the Chang et al and Kasai et.al. In this paper, a frequency dependent calculation for the contribution of the VE dampers through the shear and the loss factors. G′ and G″, is considered. The VE damper contribution to the damping and stiffness matrices of the system is derived based on the parameters G′ and G″ evaluated at the natural frequencies of the structure(Tzan.S.R and Pantelides C.P.,)A frequent dependent effective damping ratio is implemented as

                          

where .G′  and G″ can be found by substituting ωo,j the jth   mode natural frequency of the structure ζo,j  is the jth    mode inherent  structural damping ratio ; the  η is the damper loss factor j is the   jth  modal shape function ;and K b  and  Ks are the stiffness matrices of the structure without  VE dampers and with the additional VE dampers, respectively .Thus the damping contribution of the first n nodes is uniquely determined through the damping ratio of the equation (6) and is considered invariant with respect  to the remaining excitation frequencies .The  stiffness of the VE damper ,K d can be expressed as 

                                    

where A  and th   are the cross sectional area  and the thickness of the VE damper β=cosθ 

and θ is the damper inclination with respect to the horizontal directions. It should be noted that only fundamental mode parameters   G′( ωo,1)  and  G″( ωo,1) are used to determine the VE damper stiffness.

OPTIMAL DESIGN OF ACTIVE STRUCTURES

a. OPTIMAL DESIGN OF THE CONVENTIONAL STRUCTURE.

The optimal design of the conventional structure with an assumed value of inherent structural damping level of 1 to 5% of critical is taken for all the modes in each of three cases and are  are fixed through the optimization. A time history approach is used for the analysis required for the optimization procedure. The 18 May 1940 El-Centro is used as earthquake excitation. The minimum volume of the optimal structure is decreased as inherent structural damping is increased. 

b. OPTIMAL DESIGN OF THE ACTIVE STRUCTURE.

The 10 storey frame shown in Fig 1 with the different assumed levels of the inherent structural damping is used to optimal design of the conventional members of the active structure 

Three cases are discussed 

1)  One active bracing system (ABS) installed on the first floor and one ABS on the second floor 

2) One viscoelastic (VE) damper on the first floor one viscoelastic (VE) damper second floor 

3) one active bracing system (ABS) installed on the first floor one viscoelastic (VE) damper second floor .

The inherent structural damping levels range from 1 to 5% of the critical for all modes in each of the three cases under fixed at these levels through the optimization. The 1940 El-Centro earthquake is used as the earthquake excitation. The minimum value of the active structure is compared with that of the conventional structures is shown fig 2-4. The volume of the optimal structures is decreased as the effective damping level is increased for all active structures. 

A relationship between the effective damping and optimal value of the active structures by comparing cases 1-3 is shown in fig 5. In general, the minimum volume of the active structure is inversely proportional to first mode effective damping ratio for all three structural control cases. The comparison of additional damping introduced in the first mode by the control systems in the three cases with respect to the assumed structural damping, is shown in fig 6

For the same level of inherent structural damping, the additional damping introduced by the VE dampers (Case 2) is on average 0.6% less than that of the active bracing system. (case1), and 0.9% less than that of combined system (case 3). A specific cross-sectional area of the VE dampers and the maximum control force of the ABS controller were chosen for the active structures to achieve a comparable level of performance. The aim of comparison is not to run a competition among the three cases of active cases, but to demonstrate that reduction of the structural volume (cost) can be achieved by more than one way. All three structural control systems reduce the optimal volume of the active structure by 23% on average as compared to volume of the optimal conventional structure. Fig 7 shows the ratio of the optimal volume of the active structure to that of conventional structures. For different levels of assumed inherent structural damping for the structure with an assumed inherent structural damping of 1% of critical the reduction of structural volume is more pronounced (35% on average). Note that the structural volume reduction achieved each of three active structures equipped with two ABS, two VE damper and one ABS and one VE damper on an average, for all inherent damping ratios 23,.21 and 25% less than the conventional structural optimal volume. This indicates that the cost of the conventional structural system could be reduced if the presence of active or passive structural control is taken into account at the design stage. 

c. OPTIMAL DESIGN OF VE DAMPER

An optimization method is used to design the minimum size of the viscoelastic damper for achieving an increased effective damping level of the structural system.

The modified method of feasible directions is used to optimize the required cross sectional area of the VE dampers on the different floors of the structure.

The parameters G′ and G″ of the VE dampers in equation (6) are determined at the natural frequency of each building. The damping ratios are used as a bench mark to find the optimum area required for the VE damper. 

The optimal design of the VE damper is expressed as follows: minimize the cross sectional area of the VE dampers subject to a constraint on the first mode effective damping level. The side constraint for the maximum and minimum sizes of the VE damper is done. It can be observed that for a certain number of dampers, the total VE damper cross sectional area obtained from the optimal design is less than the area required by using the same uniform size for each damper. For example, the required  VE damper area when the building is equipped with the damper at each floor obtained from the optimal design is about 38% less compared to the design using the same uniform m size for the each damper .

Thus the optimization can be obtained in both material (cross-sectional area of the VE damper) and in labour (installation of three verses five dampers).

The optimal designs using the constant number of the VE dampers but the different level of   the effective damping ratio are also investigated. By increasing the size of the VE damper one can increase the effective damping level for the same numbers of the VE dampers located in the structures. However for the same level of the effective damping, the structures equipped with the VE dampers on the first and the second floors only, requires the more material than that equipped with the VE dampers on the first, second, third and fourth floors .For the 10 storey frame,if only first and the second floors are equipped  with the VE dampers, the structure cannot even reach the effective damping level of 25%of critical

INELASTIC ANALYSIS OF CONVENTIONAL AND ACTIVE STRUCTURES. 

An inelastic analysis carried out in this section using finite element software DRAIN-2DX for both conventional and active structures. A ten story steel frame with an assumed inherent structural damping level of 5% of critical in all the modes is used to investigate the inelastic behavior of elastic plastic structures. The strain hardening ratio is assumed as 0.0588. The beam and the beam column conventional members of the frame are assumed to be structural steel W-shapes. The yield moments and the moment of inertia of the conventional structural elements are shown in above table

The yield moments for positive and negative bending moments are assumed to be same.

Three earthquake excitations are selected by scaling two earthquake records to have the same global energy

1. The1971 San Fernando earthquake.
2. The  scaled 1940 El- Centro earthquake and the 
3. The scaled 1994 Northridge earthquake.

The peak acceleration duration and global energy bound of the three earthquake excitations as shown in the below table.    

                      

 The maximum response in terms of displacement, drift velocity and acceleration at each floor is shown in figs for San Fernando earthquake. The displacement, drift, velocity, and acceleration response obtained by the elastic analysis are on average 27, 31, 33, and 20% larger than those obtained by the inelastic analysis. Similar results can be observed for the scaled records of 1940 El_centro earthquake and the 1994 Northridge earthquake. In general the maximum responses obtained from the elastic analysis were more conservative than those obtained from the inelastic analysis if significant yielding events had occurred. 

However, the permanent deformation can only be determined from the inelastic analysis.

After peak ground acceleration occurs, the structure maintains some permanent deformation. The permanent deformation for the each floor of the structure shown in the fig13. The permanent deformation for the structure subjected to scaled record of the 1994 Northridge earthquake is larger compared to the other earthquake records. 

The Fig 12 shows the time history displacement of the 10th floor displacement obtained from the three exications shown in the table 11 for both elastic and inelastic analysis

The energy time history of the elastic analysis for the structure subjected to 1991 San Fernando earthquake and the scaled records of the 1940 El-Centro and 1994 Northridge earthquakes are compared in the figs 14-16 .It can be observed that the damping energy reduces when the yielding occurs. This is because in the inelastic analysis the input energy is dissipated by the structural damping as well as yielding.

• 

Two active structures whose conventional members can develop elastic plastic behavior are studied in this section. In the first case the 10 story frame equipped with an ABS on the first and the second floor is used. The control force is constrained to a maximum of 20% of the excitation effective lateral force. The effective lateral force is determined by multiplying the total mass of the structure by the peak acceleration of the excitation. The effective damping for the first two modes of this active structure is equal to 10.6 and 16.8% of the critical, respectively. In the second case, the 10 storey frame is equipped with one ABS controller on each of first, second, third and fourth floors. The control force is also constrained to a maximum of 20% of the excitation effective lateral force. The effective damping level for this active structure equals 20.6 and 27.6% of the critical for the first second modes respectively.

The structure response for the cases stated above is compared to the structure without the ABS structural controls. The displacement, drift, velocity and acceleration response reduction obtained for the three earthquake records of table 11 is shown in the table 12.

In general the structures with 4 ABS structural controls can reduce the response even more compared to the structure with only 2 ABS. The displacement response for the structure with the two ABS structural controls was an average 9 to 16% less than that of the structure without control. The displacement response for the structure with 4 ABS structural controls was reduced by the 29 to 35% on average. 

More importantly, the additional structural control system not only reduces the dynamic response of the structure, but also reduces the yielding events which occur in the structure. 

Fig 17 shows the number of yielding events as a function of time for the three structures stated above in the 1971 San Fernando earthquake. The total number of yielding events is only two thirds of those of the uncontrolled systems, for the structure equipped with the two ABS and only one third for the structure with four ABS Fig 18 and 19 show a breakdown of yielding events with respect to beam and column members of the structure. Yielding in the column occurred only in the columns of the first floor. Similar results can also be observed from the scaled records of 1940 El-Centro earthquake and 1994 Northridge earthquake, the total number of yielding events for the three cases is shown in table 13.The total number of yielding for the active structures with two ABS is reduced to half compared to the conventional structure; the number of yielding events for the active structures with 4 ABS is much less.. 

CONCLUSIONS

The assumed inherent the damping level of the structure due to its structure and non structural members affects the optimal design of conventional and active structures for earthquake excitation. Larger inherent damping levels require a smaller structural volume. For the active structures, as the effective damping ratio increases, the structural volume decreases approximately in a linear fashion regardless of the source of the external damping i.e. active or passive. This has the implication for the design of economical structures equipped with active or passive control systems. 

The optimization of the cross sectional a dimension of the VE dampers has been presented for the structures under earthquake excitation. The procedure not only produces the dimensions of the dampers, but also their optimal number and the location required to achieve a target effective damping level. 

The inelastic analysis of the active structures with a number of ABS controllers have shown that the yielding of the conventional members reduces the peak structural response and the total input earthquake energy. The structural control systems in the active structures reduce the peak structural response as well as the number of the yielding events when compared to conventional structures.