Average acceleration discrete algorithm for force identification in state space Y. Ding a,⇑ , S.S. Law b , B. Wu a , G.S. Xu a , Q. Lin c , H.B. Jiang a , Q.S. Miao d a School of Civil Engineering, Harbin Institute of Technology, Harbin 150090, People’s Republic of China b Civil and Structural Engineering Department, The Hong Kong Polytechnic University, Hunghom, Kowloon, Hong Kong, China c China Southwest Architectural Design and Research Institute CORP.LTD, Chengdu 610041, People’s Republic of China d Beijing Institute of Architecture Design, Beijing, People’s Republic of China article info Article history: Received 25 June 2012 Revised 28 July 2013 Accepted 3 August 2013 Available online 14 September 2013 Keywords: Force identification State space Average acceleration Regularization First-Order-Hold Zeroth-Order-Hold abstract A discrete force identification method based on average acceleration discrete algorithm is proposed in this paper. The method is formulated in state space and the external excitation acting on a structure is estimated with regularization method. A three-dimensional three-storey frame structure subject to an impact force and random excitations is studied respectively with numerical simulations. Uncertainties such as measurement noise, model error and unexpected environmental disturbances are included in the investigation of the accuracy and robustness of the proposed method. Experimental results from a seven-storey planar frame structure in laboratory are also used for the validation. The above results are also compared with those from two existing force identification methods, which are based on the Zer- oth-Order-Hold (ZOH) discrete algorithm and the First-Order-Hold (FOH) discrete algorithm. Model of a fourteen-storey concrete shear wall building is studied experimentally with shaking table tests to further validate the proposed method. The shear wall structure has a two-storey steel frame on top with base isolation. The interface force in the isolation at the bottom of the steel frame during the seismic excitation is estimated with the proposed force identification method. Results from both numerical simulations and laboratory tests indicate that the proposed method can be used to identify external excitations and interface forces effectively based on the structural acceleration responses from only a few accelerometers with accurate results. The proposed method is capable to iden- tify the dynamic load fairly accurately with measurement noise, model error and environmental disturbances. Ó 2013 Elsevier Ltd. All rights reserved. 1. Introduction The dynamic load environment is an important component of design, condition assessment and health monitoring of structure. The external excitation estimation methods can be classified into two categories of direct methods and indirect methods. Force transducers are installed at locations where the forces are applied in the direct method. Though numerous types of force transducers have been developed, it is impossible to measure all the excitations on a structure directly due to the lack of access to the loading posi- tion. The number of required sensors may also be very large. The indirect method is an alternative tool for the evaluation of external forces acting on a structure [1,2]. The load environment assessment is an inverse problem with ill-conditioning, and regularization method is usually adopted [3] for a solution. A lot of indirect force re-construction or force identification methods have been developed [3–6]. They are usually based on the finite element model of the structure which is often inaccurate [4]. One of the most common errors is with the properties of materials which affect the accuracy of both the forward and backward analysis. The force identification algorithm is usually formulated in the state space. The discretization of the continuous state space equa- tion will, however, affect the accuracy and stability of computa- tion. It would subsequently affect the estimation result in the inverse analysis. The formulation for force identification in state space has been solved directly with regularization method based on the Zeroth-Order-Hold (ZOH) Discrete method [7–10]. Some researchers employed the ZOH discrete method for the first-order regularization [9,10] in force identification based on the dynamic programming method, but the computational is expensive. Many methods have also been proposed for the identification of external excitations, including deterministic forces [11], stochastic forces [9] and methods based on artificial intelligence [12]. However, the accuracy of methods based on different algorithm of 0141-0296/$ - see front matter Ó 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.engstruct.2013.08.004 ⇑ Corresponding author. Tel.: +86 15846532083. E-mail address: [email protected](Y. Ding). Engineering Structures 56 (2013) 1880–1892 Contents lists available at ScienceDirect Engineering Structures journal homepage: www.elsevier.com/locate/engstruct
Y. Ding a,⇑, S.S. Law b, B. Wu a, G.S. Xu a, Q. Lin c, H.B. Jiang a, Q.S. Miao d
a School of Civil Engineering, Harbin Institute of Technology, Harbin 150090, People’s Republic of Chinab Civil and Structural Engineering Department, The Hong Kong Polytechnic University, Hunghom, Kowloon, Hong Kong, Chinac China Southwest Architectural Design and Research Institute CORP.LTD, Chengdu 610041, People’s Republic of Chinad Beijing Institute of Architecture Design, Beijing, People’s Republic of China
a r t i c l e i n f o a b s t r a c t
Article history:Received 25 June 2012Revised 28 July 2013Accepted 3 August 2013Available online 14 September 2013
A discrete force identification method based on average acceleration discrete algorithm is proposed inthis paper. The method is formulated in state space and the external excitation acting on a structure isestimated with regularization method. A three-dimensional three-storey frame structure subject to animpact force and random excitations is studied respectively with numerical simulations. Uncertaintiessuch as measurement noise, model error and unexpected environmental disturbances are included inthe investigation of the accuracy and robustness of the proposed method. Experimental results from aseven-storey planar frame structure in laboratory are also used for the validation. The above resultsare also compared with those from two existing force identification methods, which are based on the Zer-oth-Order-Hold (ZOH) discrete algorithm and the First-Order-Hold (FOH) discrete algorithm. Model of afourteen-storey concrete shear wall building is studied experimentally with shaking table tests to furthervalidate the proposed method. The shear wall structure has a two-storey steel frame on top with baseisolation. The interface force in the isolation at the bottom of the steel frame during the seismic excitationis estimated with the proposed force identification method.
Results from both numerical simulations and laboratory tests indicate that the proposed method can beused to identify external excitations and interface forces effectively based on the structural accelerationresponses from only a few accelerometers with accurate results. The proposed method is capable to iden-tify the dynamic load fairly accurately with measurement noise, model error and environmentaldisturbances.
� 2013 Elsevier Ltd. All rights reserved.
1. Introduction
The dynamic load environment is an important component ofdesign, condition assessment and health monitoring of structure.The external excitation estimation methods can be classified intotwo categories of direct methods and indirect methods. Forcetransducers are installed at locations where the forces are appliedin the direct method. Though numerous types of force transducershave been developed, it is impossible to measure all the excitationson a structure directly due to the lack of access to the loading posi-tion. The number of required sensors may also be very large. Theindirect method is an alternative tool for the evaluation of externalforces acting on a structure [1,2].
The load environment assessment is an inverse problem withill-conditioning, and regularization method is usually adopted [3]for a solution. A lot of indirect force re-construction or force
identification methods have been developed [3–6]. They areusually based on the finite element model of the structure whichis often inaccurate [4]. One of the most common errors is withthe properties of materials which affect the accuracy of both theforward and backward analysis.
The force identification algorithm is usually formulated in thestate space. The discretization of the continuous state space equa-tion will, however, affect the accuracy and stability of computa-tion. It would subsequently affect the estimation result in theinverse analysis. The formulation for force identification in statespace has been solved directly with regularization method basedon the Zeroth-Order-Hold (ZOH) Discrete method [7–10]. Someresearchers employed the ZOH discrete method for the first-orderregularization [9,10] in force identification based on the dynamicprogramming method, but the computational is expensive. Manymethods have also been proposed for the identification of externalexcitations, including deterministic forces [11], stochastic forces[9] and methods based on artificial intelligence [12]. However,the accuracy of methods based on different algorithm of
Y. Ding et al. / Engineering Structures 56 (2013) 1880–1892 1881
discretizing the continuous function, such as ZOH discrete methodand First-Order-Hold (FOH) discrete method is rarely comparedand investigated.
In this paper, a new force identification method based on aver-age acceleration discrete algorithm is proposed in state space. Fewliterature reports on this kind of discrete method for the forceidentification. The method is formulated recursively in state space.A three-dimensional three-storey steel frame is firstly studiednumerically with single excitation. In addition to measurementnoise in the responses, model error in the structural material andunexpected environmental excitation are included in the study toinvestigate the robustness of the proposed method. Results ob-tained are compared with those from the force identification meth-ods based on ZOH discrete method and FOH discrete method. Forpractical application purpose [13], a scenario of multiple randomexcitations identification is also numerically investigated withthe three-dimensional steel frame structure.
A planar seven-storey steel frame constructed in the laboratoryof the Hong Kong Polytechnic University was experimentally inves-tigated to validate the proposed method. The impact force on the se-ven-storey is identified with the ZOH discrete method, FOH discretemethod and the proposed average acceleration discrete method.Scaled model of a 14-storey concrete shear wall building was teston a shaking table in the laboratory of the Institute of EngineeringMechanics, China Earthquake Administration, for further investiga-tion of the proposed method with substructural external force iden-tification. The shear wall building made of reinforced concrete has atwo-storey steel frame fixed at the roof with rubber base isolation.The property of this isolation could not be evaluated directly duringthe shaking table test. However, the interface shear force in the iso-lator between the steel frame and the shear wall building can beidentified satisfactorily with the proposed method.
2. Dynamic responses of a structural system
The equation of motion of a N DOFs damped structural systemsubjected to external excitation can be represented as
M€xþ C _xþ Kx ¼ LF ð1Þ
where M, C and K are the mass, damping and stiffness matrices ofthe structural system respectively. F is the vector of external excita-tion forces on the structure and L is the mapping matrix for the in-put forces. €x, _x and x are vectors of acceleration, velocity anddisplacement of the structural system respectively. Rayleigh damp-ing model is assumed with
C ¼ a1 �Mþ a2 � K ð2Þ
where a1 and a2 are the Rayleigh damping coefficients.When the structure is subjected to seismic excitation, Eq. (1)
becomes
M€xþ C _xþ Kx ¼ �M€xg ð3Þ
where €xg is the acceleration at ground level.
3. Force identification methods in time domain
3.1. Discrete equation in state space
The equation of motion of the structural system shown in Eq.(1) can be expressed continuously in the state space as
_z ¼ ACzþ BCL � F ð4Þ
where z ¼x_x
� �; AC ¼
0 I�M�1K �M�1C
� �and BC ¼
0M�1
� �:
The superscript C denotes matrices for the continuous system.Vector y(t) is assumed to represent the output of the structural sys-tem and it is assembled from the measurements with
y ¼ Ra€xþ Rv _xþ Rdx ð5Þ
where Ra, Rv and Rd e Rm�ndof are the output influence matrices forthe measured acceleration, velocity and displacement respectively,m is the dimension of the measured responses and ndof is the num-ber of DOFs of the structure. Eq. (5) can be rewritten as
y ¼ RCzþ DC � L � F ð6Þ
where RC = [Rd � RaM�1KRv � RaM�1C] and DC = RaM�1. When theexternal force is known or measured, the value of state variable zand y can be calculated accurately. However, in practice, the mea-surement data is discrete and the continuous state equation is re-quired to be transformed into discrete equation. With the discretemethod, Eqs. (4) and (6) can be converted into the following dis-crete equations as
zðjþ 1Þ ¼ ADzðjÞ þ BD � L � FðjÞ ð7Þ
yðjÞ ¼ RCzðjÞ þ DC � L � FðjÞ ðj ¼ 1;2; . . . ;NÞ ð8Þ
where superscript D denotes the matrices for the discrete structural sys-tem and N is the total number of sampling points. The output y(j) issolved from Eqs. (7) and (8) with zero initial conditions of responsesin terms of the previous input F(k), (k = 0, 1, . . . , j) and we have
yðjÞ ¼Xj
k¼0
HDk � L � Fðj� kÞ ð9Þ
where the expressions of HD0 and HD
k vary with different discretealgorithms. The ZOH discrete method and FOH discrete methodfor the force identification are referred to Appendices A and Brespectively. The following paragraphs will present the averageacceleration discrete algorithm for the force identification.
3.2. Average acceleration discrete method in force identification
The Newmark-b is generally used in civil engineering to evalu-ate the dynamic response of a structural system. Average acceler-ation step-by-step integration method is a variant of theNewmark-b method with the following assumptions
_xkþ1 ¼ _xk þ€xk þ €xkþ1
2
� �Dt
xkþ1 ¼ xk þ _xkDt þ€xk þ €xkþ1
4
� �Dt2
ð10Þ
Rearranging, the incremental acceleration and velocity can berepresented as
D _x ¼ _xkþ1 � _xk ¼ 2DxDt2 � 2 _xk
D€x ¼ €xkþ1 � €xk ¼ 4DxDt� 4
_xk
Dt� 2€xk
ð11Þ
Substituting Eq. (11) into Eq. (1) at the (k + 1)th time instant,the following equation can be obtained
4Dt2 Mþ 2
DtCþ K
� �Dx ¼ Fkþ1 þM€xk þ Cþ 2
DtM
� �_xk � Kxk ð12Þ
The incremental displacement can also be represented as
Dx ¼ 4Dt2 Mþ 2
DtCþ K
� ��1
Fkþ1 þM€xk þ Cþ 2Dt
M� �
_xk � Kxk
� �
¼ 4Dt2 Mþ 2
DtCþ K
� ��1
Fkþ1 þ Fk þ2Dt
M _xk � 2Kxk
� �
ð13Þ
1882 Y. Ding et al. / Engineering Structures 56 (2013) 1880–1892
Substituting Eq. (13) into Eq. (11), the incremental velocity canbe represented as:
D _x ¼ _xkþ1 � _xk ¼ 2DxDt2 � 2 _xk ¼ 2
DxDt2
4Dt2 Mþ 2
DtCþ K
� ��1
� Fkþ1 þ Fk þ2Dt
M _xk � 2Kxk
� �ð14Þ
The displacement and velocity at the (k + 1)th time instant can berepresented as the summation of those at the previous time instantand the increments of displacement and velocity as
xkþ1
_xkþ1
� �¼
xk þ Dx_xk þ D _x
� �ð15Þ
Eq. (15) can be written in state space as
zkþ1 ¼ ANDzk þ BNDL � Fk ð16Þ
where
zkþ1 ¼xkþ1
_xkþ1
� �; BND ¼ 4
Dt2 Mþ 2Dt Cþ K
� ��12Dt
4Dt2 Mþ 2
Dt Cþ K� ��1
� �
AND ¼ I� 2 4Dt2 Mþ 2
Dt Cþ K� ��1
K �4Dt
4Dt2 Mþ 2
Dt Cþ K� ��1
M �4Dt
4Dt2 Mþ 2
Dt Cþ K� ��1
K �4Dt2
4Dt2 Mþ 2
Dt Cþ K� ��1
M� I� �
With the assumption of the average acceleration discrete algo-rithm, output y(j) can also be written similar to Eq. (9) as
yðjÞ ¼Xj
k¼0
HNDk � L � Fðj� kÞ ð17Þ
where
HND0 ¼ RBND þ DC and HND
k ¼ RCANDBND þ RCðANDÞk�1
BND:
It is noted from Eq. (17) that the matrix Hk from the average accel-eration discrete algorithm is different from those in Eqs. (A.3) and(B.15) in Appendixes A and B with the assumption of ZOH discretemethod and the FOH discrete method respectively. The externalforce vector F can be identified from Eq. (17) as follows similar tothe force identification with ZOH discrete method from Eq. (A.3)and the FOH discrete method from Eq. (B.15).
4. Force identification with iterative regularization method
The force identification based on the ZOH discrete method, FOHdiscrete method or the average acceleration algorithm as shown inEqs. (A.3), (B.15), and (17) respectively can all be written in a gen-eral form asY ¼ HLF ð18Þ
It is noted that Eq. (18) is ill-posed for the identification whenthere is measurement noise in the system. A straightforward least-squares solution will produce unbounded solution. Regularizationmethod would provide an improved solution to the ill-posed prob-lem. The damped least-squares method proposed by Tikhonov[14] is adopted to give bounds to this force identification problem as
HTLY ¼ HT
LHL þ kI� �
F ð19Þ
where k is the non-negative damping coefficient governing the par-ticipation of the least-squares error in the solution. Solving Eq. (19)is equivalent to minimizing the function
JðF; kÞ ¼ kHLF� Yk2 þ kkFk2 ð20Þ
The L-curve method proposed by Hansen [15] is adopted to find theoptimal regularization parameter k. The relative percentage error inthe identified forces can be calculated as
errorF ¼kFid � FtruekkFtruek
� 100% ð21Þ
where Ftrue is the real force acting on the structure and Fid is theidentified force. The exact value of external force Ftrue is difficultto obtain in practice. Therefore, an alternative form of error of iden-tification from measurement can be represented as
erroracc ¼k€xm � €xrekk€xmk
� 100% ð22Þ
where the erroracc denotes the relative error between the measuredacceleration and re-constructed acceleration on application of theidentified forces.
5. Implementation procedure
Step 1: Obtain the mass, damping and stiffness matrices of the
target structure.Step 2: Conduct dynamic measurement on the structure. In thecase of simulation studies, compute the responses of the struc-ture under excitation using the state space method as the ‘‘mea-sured’’ responses.Step 3: Obtain the matrix of system Markov parameters fromthe finite element model of the structure based on the corre-sponding discrete method from Eqs. (17).Step 4: Solve Eq. (18) and identify the forces acting on the struc-ture through the iterative regularization method in Eq. (19)with the finite element model of the structure. It should benoted that this approach of force identification only needs thefinite element model of the structure and a few accelerometers.Step 5: Convergence is considered achieved if the following cri-teria is met
DFkþ1 � DFk
DFkþ1
��������
�������� 6 Tol ð23Þ
where k denotes the number of iteration and Tol is a small pre-scribed value which is taken as 10�4 in this work.
6. Numerical studies
A numerical model of a three-dimensional steel frame structureas shown in Fig. 1 is investigated to validate the proposed forceidentification method with the average acceleration algorithm.The midpoints and two ends of the beams and columns are mod-eled as nodes in the finite element model. Nodes 1–16 and someimportant nodes useful for the simulation are shown in the figure.Dimensions and properties of the frame members are shown in Ta-ble 1 and the material properties are shown in Table 2. The damp-ing ratios of the first two modes are taken to be 0.02 and 0.01respectively. Only three accelerometers are used for the force iden-tification, and they are placed at Nodes 17 in the x-direction, Node33 in the y-direction and Node 49 in the y-direction. The sampling
Position of the single external
force in Scenario 1
Six random forces in Scenario 2
Fig. 1. Three-storey steel frame.
Y. Ding et al. / Engineering Structures 56 (2013) 1880–1892 1883
rate is 1000 Hz and 0.1 s of measured data with1000 � 0.1 � 3 = 300 sampling points is used for the force identifi-cation. Two scenarios are studied in this paper to validate the pro-posed force identification method. In the first scenario a singleforce is applied at one corner of the top floor of the structure asshown in Fig. 1. The excitation force is modeled as
In the second scenario, multiple forces identification is investi-gated. Six random forces are applied at six joints of the beams andcolumns in x-direction as shown in Fig. 1. When there is noise in
Table 1Section shape and properties of the 3-storey steel frame.
Beam and girder Column
A (cm2) IX–X (cm4) IY–Y (cm4) J (cm4) q (kg/m) A (cm2) IX–X (cm4) IY–Y (cm
12.74 198.3 25.6 223.9 10.0 4.44 10.2 10.2
the ‘‘measured’’ response, the polluted response is simulated byadding a normal random component to the unpolluted structuralresponses as
€xm ¼ €xþ EPNnoiserð€xÞ ð25Þ
where EP is the percentage noise level, Nnoise is a standard normaldistribution with zero mean and unit standard deviation, rð€xÞ isthe standard deviation of the ‘‘measured’’ response.
Seven force identification cases are studied in Scenario 1 withdifferent effects from model error, measurement noise and envi-ronmental excitation and they are shown in Table 3. The identifiedforce time histories are plotted in Figs. 2–8. The force for the case
Fig. 6. Force identification result (with unexpected random base excitation only).
Table 2Parameter of steel.
E(GPa)
G(GPa)
t Post-yieldingstiffness ratio
Yielding stress(MPa)
Density (g/cm3)
206 78.63 0.31 0.02 235 7.85
1884 Y. Ding et al. / Engineering Structures 56 (2013) 1880–1892
without the model error, environmental excitation and measure-ment noise can be identified very accurately with the proposedmethod (Fig. 2). The effect of 10% measurement noise is noted tocause slight changes in the amplitude of the identified force asshown in Figs. 3, 5 and 8. When only 5% model error is considered,the identified force time histories shown in Figs. 4, 5 and 7 exhibita slight shift. This could be explained that the model error willintroduce an error in the stiffness matrix of the structure whichcould affect the amplitude and modal frequency of the identifiedforce as noted in the shift of the identified force time history.
Fig. 5 shows the identified results considering the model errorand measurement noise effect. Both fluctuations and shift existin the identified force time history. It can be concluded that themeasurement noise would cause fluctuation while the model errorin structural model would lead to a shift of amplitude and fre-quency in the force time history.
The effect of environmental excitation exists in all laboratoryand field measurements which may be due to the operation ofthe mechanical system in the laboratory or the nearby traffic loadexcitation. The slight environmental base disturbance may affectthe force identification result. It is investigated in this paper withthe inclusion of a white noise excitation with zero mean and0.001 gal standard deviation at the base of the three-storey steelframe. The external excitations are at the same positions as in pre-vious studies. The results for the cases including the environmentalexcitation are shown in Figs. 6–8. It is noted that there is some
Fig. 8. Force identification result (with random base excitation, 5% model error and10% noise).
-20
-15
-10
-5
0
5
10
15
20
25
Forc
e (N
)Fo
rce
(N)
Forc
e (N
)
Real forceIdentified force
(a) Identified force on Node 19
-30
-20
-10
0
10
20
30Real forceIdentified force
(b) Identified force on Node 35
-50
-40
-30
-20
-10
0
10
20
30
40
50Real forceIdentified force
(c) Identified force on Node 49
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
Time (s)
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
Time (s)
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
Time (s)
Fig. 9. Random excitations identification results (without noise).
Y. Ding et al. / Engineering Structures 56 (2013) 1880–1892 1885
error in the curve between 0.08 s and 0.1 s towards the end of thetime history. The effect of model error dominates compared tothose of noise and environmental excitation as shown in Fig. 8.The identification result illustrates that the external force couldalso be identified fairly accurately with environmental randomexcitation, measurement noise and model error.
In Scenario 2 with multiple random external excitations, theexternal excitations are acting at joints 17, 19, 33, 35, 49 and 51respectively. An extra acceleration response on Node 40 in x-direc-tion is measured due to the poor identification results with theacceleration response measurements on only three nodes as theScenario 1. In the first study, the mean of the six random excita-tions are all zero while the standard deviations are 7, 11, 10, 8,15, and 6 respectively. Three of the identified excitation time his-tories at Nodes 19, 35 and 51 are shown in Fig. 9 without measure-ment noise and in Fig. 10 with 10% measurement noise. Theidentification results for excitations without measurement noiseare very accurate. Though there are some errors in the forces withmeasurement noise, the identification results are still acceptable.This conclusion on multiple random excitations identification isconsistent with that for the single excitation identification in Sce-nario 1. It is noted in Figs. 9 and 10 that the peak value of the ran-dom force time histories are close to each other. In the secondstudy, only the standard deviation of the random excitation onNode 19 is changed as unity and the force time histories on otherNodes are the same as in the first study in Scenario 2. The excita-tion identification result on Node 19 is shown in Fig. 11. The exci-tation on Node 19 cannot be identified with 10% measurementnoise. However, the relative error as calculated from Eq. (22) isonly 5.07%. This indicates that when the excitation generates struc-tural response in the same order as the measurement noise orsmaller, they are too small to be identified. The measurement noise
will adversely affect the identification of multiple excitations caus-ing small amplitude structural responses.
7. Laboratory validation
The proposed method with average acceleration discrete algo-rithm will be validated with the experimental studies of a two-dimensional planar frame and a scaled model of a concrete shearwall building as follows.
7.1. Experimental study with a seven-storey planar steel frame
The two-dimensional seven-story steel frame is shown inFig. 12 and the node and element numbering systems are shownin Fig. 13. It was fabricated and test in the laboratory of The HongKong Polytechnic University. Dynamic responses were recorded
-30
-20
-10
0
10
20
30
Forc
e (N
)
Real forceIdentified force
(a) Identified force on Node 19
-30
-20
-10
0
10
20
30
40
Forc
e (N
)
Real forceIdentified force
(b) Identified force on Node 35
-50
-40
-30
-20
-10
0
10
20
30
40
50
Forc
e (N
)
Real forceIdentified force
(c) Identified force on Node 49
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
Time (s)
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
Time (s)
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
Time (s)
Fig. 10. Random excitations identification results (with 10% noise).
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
Time (s)
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
Time (s)
Forc
e (N
)
(a) Without measurement noise
-15
-10
-5
0
5
10
15
Forc
e (N
)
Real forceIdentified force
(b) With 10% measurement noise
Real forceIdentified force
Fig. 11. Identified random excitation on Node 19 with zero mean and unitestandard deviation.
Fig. 12. Photograph of the two-dimensional seven-storey frame structure.
1886 Y. Ding et al. / Engineering Structures 56 (2013) 1880–1892
with DEWESoft software and NI data acquisition equipment. Thephysical properties of the 7-storey steel frame are listed in Table4. Two lumped mass were placed on each floor of the frame struc-ture to simulate the mass effect of the floor slab, and the weightand their locations are listed in Table 5. The bottom of the framewas welded to a thick steel base plate, and the plate was connectedfirmly to the ground simulating rigidly fixity. The finite elementmodel consists of 56 elements and 51 nodes as shown in Fig. 13.
Before the force identification, the stiffness of the structure wasupdated based on the first seven modal frequencies and modeshapes with the optimization function ‘fmincon’ of MATLAB. Theinitial Young’s modulus is set as 2.0 � 1011 N/m2 for all compo-nents of the frame structure. The updated Young’ modulus of beamis 2.2 � 1011 N/m2 and that of column is 1.9 � 1011 N/m2. Compar-ison of the updated modal frequencies and those from the struc-ture is shown in Table 6. The updated finite element model isnoted very similar to the experimental structure.
Measured acceleration responses in the x-direction at Nodes 4,25, 32 and 39 of the structure were used to estimate the externalexcitation with the force identification method based on ZOH,
Fig. 13. Ketch of the two-dimensional seven-storey frame structure.
Table 4Properties of the seven-storey frame.
Properties Member
Sectional area of the beam (mm2) 49.98 � 8.92Sectional area of the column (mm2) 49.89 � 4.85Density of the beam (kg/m3) 7850Density of the column (kg/m3) 7734Poisson ratio 0.3Moment of area Iz (m4) 3.645 � 10�9
Torsional rigidity J (m4) 7.290 � 10�9
Table 5The weight and location of the lumped masses.
Storey number Node number Weight (kg) Node number Weight (kg)
Measured forceIdentified force with average acceleration discrete algorithmIdentified force with ZOH discrete algorithmIdentified force with FOH discrete algorithm
Fig. 14. Impact force identification results.
Y. Ding et al. / Engineering Structures 56 (2013) 1880–1892 1887
FOH and the proposed average acceleration discrete algorithm. Theimpact force is applied horizontally at Node 50 with a SINOCERALC-04A force hammer. The sampling rate is 1000 Hz and only thefirst 150 sampling data from each accelerometer are used for theimpact force estimation.
Fig. 14 shows the identified hammer impact force comparingwith the measured force from the hammer. The identification re-sults from different discrete methods are very close to the mea-sured impact force. There is slight fluctuation in the identifiedimpact force from the ZOH discrete algorithm and the identifiedforce time history from the FOH discrete algorithm is relativelysmooth. It is noted that the fluctuations are nearly absent withthe average acceleration discrete algorithm. The manual hammer-ing may not be absolutely horizontal, which may explain whythe identified force is a little different from the measured forceat the peak. Comparison of the measured responses and the
0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16
-3
-2
-1
0
1
2
Time (s)
0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16
Time (s)
0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16
Time (s)
0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16
Time (s)
Acce
lera
tion
(m/s
2 )Ac
cele
ratio
n (m
/s2 )
Acce
lera
tion
(m/s
2 )Ac
cele
ratio
n (m
/s2 )
Measured accelerationRe-constructed acceleration with average acceleration methodRe-constructed acceleration with ZOH discrete methodRe-constructed acceleration with FOH discrete method
(a) Acceleration at Node 4
-2
-1
0
1
2
3
4Measured accelerationRe-constructed acceleration with average acceleration methodRe-constructed acceleration with ZOH discrete methodRe-constructed acceleration with FOH discrete method
(b) Acceleration at Node 25
-3
-2
-1
0
1
2
Measured accelerationReconstructed acceleration with average accleration methodReconstructed acceleration with ZOH discrete methodReconstructed acceleration with FOH discrete method
(c) Acceleration at Node 32
-4
-3
-2
-1
0
1
2
Measured accelerationRe-constructed acceleration with average acceleration methodRe-constructed acceleration with ZOH discrete methodRe-constructed acceleration with FOH discrete method
(d) Acceleration at Node 39
Fig. 15. Comparison of accelerations.
1888 Y. Ding et al. / Engineering Structures 56 (2013) 1880–1892
re-constructed responses with the identified external force isshown in Fig. 15. The errors between the re-constructed responsesand measured acceleration responses calculated from Eq. (22) are
9.82% with the ZOH discrete method, 7.29% with the FOH discretemethod and 6.94% with the average acceleration discrete algorithm.
7.2. Experimental study with a fourteen-storey shear wall building
A scaled 14-storey concrete shear wall building under seismicexcitation was studied for further validation of the proposed forceidentification method. The scaled model is constructed on theshaking table of the Institute of Engineering Mechanics, ChinaEarthquake Administration, as shown in Fig. 16(a). The structurehas a scaled ratio of 1/6. The floor plan of the model is shown inFig. 17. The excitation for the test is in y-direction. The shear wallbuilding model is made from grade M7.5 mortar with an averagestrength of 9.148 MPa. The external wall of the structure was con-structed with reinforced mortar of grade M15 with an averagestrength of 14.878 MPa and steel wire mesh with average yieldstrength of 852.22 MPa. The steel in the shear wall was in two lay-ers at 1.9@25 � 25 in the weak direction (x-direction) and0.8@13 � 13 in the strong direction (y-direction). Two layers ofsteel wire mesh were also provided with 1.5@25 � 25 in both thex- and y-directions of the floor slab.
The two-storey steel frame structure is fabricated from40 mm � 60 mm � 2 mm Q235 rectangular steel tube, and is fixedto the top of the 14-storey shear wall building with rubber isola-tion. The photograph of the steel frame can be seen in Fig. 16(b)and the floor plan of the steel frame is shown in Fig. 18. Each storeyof the steel frame is 483.3 mm high. Each column of the frame iswelded at the bottom to a base plate which has four U20 mm boltholes for connecting to the concrete roof of the building as shownin Fig. 19(a). A sketch of the base isolation is also shown inFig. 19(b).
The weight of the whole scaled model is 7.62 tonne and addi-tional mass of 678 kg is placed at each floor level to simulate thefloor mass effect, i.e. the mass of the structure is mainly distributedat each floor level. Considering this distribution of mass, the 14-storey shear wall building and the additional two-storey steelframe could be simplified into a lumped mass cantilever structurewith 16 equivalent beam elements. Comparison of the modal fre-quency and damping ratio of the first two modes from the cantile-ver model and the real shear wall building structure is shown inTable 7, and they are very close together indicating the cantilevermodel could suitably model the dynamic behavior of the structure.
Accelerometer model 941B manufactured by the Institute ofEngineering Mechanics, China Earthquake Administration and6000DAS Data Acquisition System are used. The sampling rate is200 Hz. The horizontal accelerations in the y-direction at the 8th,10th, and 13th floors are measured for the force identification.
The structure is divided into two substructures, which are theshear wall building plus the foundation and the additional steelframe plus the inter-storey isolation. Substructure methods [16–20] have been developed for force estimation or damage detection.This paper used the substructural force identification method [17]with the proposed average acceleration discrete algorithm to iden-tify the horizontal interaction between the shear wall building andthe two-storey steel frame. The shear force of the isolation couldalso be evaluated experimentally as
Fis ¼ � msf €xsf þ csf _xsf þ ksf xsfð Þ ð26Þ
where sf denotes the DOFs of the steel frame and Fis is the shearforce in the base isolation.
In the experiment, the relative displacement between the four-teenth floor and the fifteenth floor and the horizontal accelerationon the fourteenth and fifteenth floors are measured. The displace-ment transducer model SW-5 is also made by Institute of Engineer-ing Mechanics, China Earthquake Administration with a responsefrequency range of 0–10 Hz. However the velocity at these floors
(a) Shear wall building (b) Shear wall building withadditional steel frame
Fig. 16. Shear wall building model on shaking table.
(a) Floor Plan for the lowest five storeys (b) Floor Plan for the sixth to fourteenth storeys
y
x
Fig. 17. Floor plan for the concrete structure.
Y. Ding et al. / Engineering Structures 56 (2013) 1880–1892 1889
Fig. 18. Floor plan for the two-storey steel frame.
(a) Connection plan (b) Base Isolation
Fig. 19. Details of the base isolation and connection plate.
Table 7Comparison of natural frequencies of the shear wall building.
Order Experimental model Numerical cantilever model
1 Modal frequency 4.59 4.58Damping ratio 1.2% 1.2%
2 Modal frequency 9.159 9.157Damping ratio 1.14% 1.14%
1890 Y. Ding et al. / Engineering Structures 56 (2013) 1880–1892
is difficult to obtain in the test and it is numerically integratedfrom the measured acceleration. The calculated shear force fromEq. (26) is compared with the identified force in Fig. 20. The
identified force is close to and has the same pattern as thecalculated shear force in the isolation with some losses of higherfrequency components in the first half of the duration. This maybe due to the poor frequency response of the displacementtransducer which has not captured the higher frequency compo-nents in the displacement responses. The relative errors calculatedbetween the re-constructed response and measured response asEq. (22) are 11.45% with the ZOH discrete method, 9.77% withthe FOH discrete method and 8.93% with the average accelerationdiscrete algorithm. This indicates that the average accelerationdiscrete algorithm could be used as an alternative accurate toolfor the discrete force identification method.
0 2 4 6 8 10 12-3000
-2000
-1000
0
1000
2000
3000
Forc
e (N
)
Time (s)
Calculated force From Eq.(26)Identified force with average acceleration discrete algorithm
Fig. 20. Comparison of identified force and calculated force.
-2T 2T-T T
1.0
Fig. B-I. Impulse response of the extrapolation filter [21].
eTs-2+eTs 1/s 1/Ts H(s)
uu w v y
Fig. B-II. Block diagram of the triangle-hold equivalent [14].
Y. Ding et al. / Engineering Structures 56 (2013) 1880–1892 1891
It has been demonstrated above that the proposed force identi-fication method could also be used for the substructural identifica-tion and the identification result is fairly accurate. With the aboveexperiences, the proposed method could also be used to evaluatethe output force of innovative energy dissipation devices whichdo not have a distinct analytical model. It could also be used asan alternative tool to evaluate the performance of the innovativeenergy dissipation devices.
8. Conclusions
A new method based on the average acceleration discrete algo-rithm is proposed for the inverse identification of external forceacting on a structure. The force identification methods based onthe equation in state space with ZOH and FOH discrete methodsare also studied and compared with the proposed method. IterativeTikhonov regularization is adopted for the force identification.Numerical studies with a steel frame and experimental validationwith a steel frame and a scaled model of a reinforced concretebuilding are conducted. The external force can be identified fairlyaccurately with polluted measurements, model error and environ-mental base excitation. However, the proposed method has thelimitation that the number of sensors should not be less than halfnumber of the external excitations. Also when the external forcecontributes little to the structural response in a noisy environment,the external force cannot be identified accurately.
With experiences from studies in this paper, it may be con-cluded that the average discrete method could be used for the force
identification and model updating simultaneously in future re-search. The recursive computational technique can be used forthe response calculation in state space with the proposed method.The proposed method could be developed further for the simulta-neous identification or two-stage identification of excitation andstructural damage with less numerical error than the ZOH discreteand the FOH discrete methods.
Acknowledgements
The work described in this paper was supported by a grant fromthe Niche Area Research Funding of the Hong Kong PolytechnicUniversity Project No. 1-BB6F, Project No. 01319406 Supportedby Natural Scientific Research Innovation Foundation in HarbinInstitute of Technology, Projects No. 51161120360, No. 51308160and No. 91315301 of National Natural Science Foundation of Chinaand Beijing Institute of Architecture Design. Also thank for thesupport of Prof. Li Hui in Harbin Institute of Technology.
Appendix A. Zeroth-Order-Hold discrete
With the ZOH discrete method [8,13], the discrete equation instate space can be converted from Eqs. (4) and (6) as
zðjþ 1Þ ¼ AZDzðjÞ þ BZD � L � FðjÞ ðA:1Þ
yðjÞ ¼ RCzðjÞ þ DC � L � FðjÞ ðj ¼ 1;2; . . . ;NÞ ðA:2Þ
where AZD = exp(AC � dt), BZD = (AC)�1(AZD � I)BC. The output y(j)can be represented as
yðjÞ ¼Xj
k¼0
HZDk � L � Fðj� kÞ ðA:3Þ
where
HZ0 ¼ DC and HZ
k ¼ RCðAZDÞk�1
BZD
.
Appendix B. First-Order-Hold discrete
When the number of external force increases, the influence ofthese forces on the response of the structural system increases cor-respondingly, and an inaccurate matrix BZD will result in large er-ror in the state variable. This is because the force in a samplingperiod has been assumed to be constant. This discretization withina sampling period is treated differently in the FOH discrete meth-od, namely the triangle hold discrete method, where the discretedata is interpolated as
uðtÞ ¼ uðiÞ þ uðiþ 1Þ � uðiÞT
ðt � iTÞ ðiT 6 t 6 ðiþ 1ÞT i
¼ 1;2 � � � N � 1Þ ðB:1Þ
where T is the sampling period. The continuous input u can be rep-resented as
u ¼ LF ðB:2Þ
Define the unit impulse function d as
dðtÞ ¼þ1 t ¼ 00 t–0
8><>: ðB:3Þ
whereRþ1�1 dðtÞdt ¼ 1. The impulse response and block diagram of
the FOH are shown Fig. B-I.The Laplace transformation of the extrapolation filter [21] that
follows the impulse sampling is
1892 Y. Ding et al. / Engineering Structures 56 (2013) 1880–1892
HtriðsÞ ¼eTs � 2þ e�Ts
Ts2 ðB:4Þ
Based on the block diagram shown as Fig. B-II, the state vari-ables v and w are defined as
where d(t) is the unit impulse shown in Eq. (B.3).It can be shown from the integration of Eqs. (B.5) and (B.6) that
v(i) = u(i) and w(i) = u(i + 1) � u(i), and a new state space equationcan be derived as
_z_v_w
264
375 ¼
AC BC 00 0 1=T
0 0 0
264
375
zvw
264
375þ
001
264
375�u ðB:7Þ
where �u represents the input impulse function as shown in Fig. B-I.The matrix on the right-hand-side of the Eq. (B.7) is defined as
FT ¼AC BC 00 0 1=T
0 0 0
264
375 ðB:8Þ
If the one step solution to Eq. (B.7) is written as
fðiT þ 1Þ ¼ eFT TfðiTÞ ðB:9Þ
Then
expðFT TÞ ¼U C1 C2
0 1 00 0 1
264
375 ðB:10Þ
The iterative equation in variable x can be written as
xðiþ 1Þ ¼ UxðiÞ þ C1vðiÞ þ C2wðiÞ ðB:11Þ
If a new state is defined as z(i) = x(i) � C2u(i), the equation ofmotion in state space for the triangle FOH can be rewritten as
zðiþ 1Þ ¼ AFDzðiÞ þ BFDuðiÞ ðB:12Þ
The output equation becomes
yðiÞ ¼ CFDzðiÞ þ DFDuðiÞ ðB:13Þ
The parameter for the state equation can then be represented as
AFD ¼ U;
BFD ¼ C1 þUC2 � C2;
CFD ¼ RC;
DFD ¼ DC þHC2:
ðB:14Þ
Based on the above triangle FOH discrete method, the forceidentification can be conducted following Eqs. (B.12), (B.13),(B.14) with more accurate results than the previous ZOH discretemethod. The output y(j) is solved from Eqs. (B.12) and (B.13) andwe have
yðjÞ ¼Xj
k¼0
HFDk � L � Fðj� kÞ ðB:15Þ
where
HF0 ¼ DC and HF
k ¼ RCðAFDÞk�1
BFD
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