2. 同济大学 土木工程防灾国家重点实验室，上海 200092;
3. School of Eng., Univ. of British Columbia, British Columbia, Canada V1V1V7
2. State Key Lab. of Disaster Reduction in Civil Eng., Tongji Univ., Shanghai 200092, China;
3. School of Eng., Univ. of British Columbia, British Columbia, Canada V1V1V7
Thermal power plant is an important life line structure^{[1]}. Maintaining its workability during and after earthquakes can be essential for people’s life and recovery^{[1]}. However, due to the functional requirements, thermal power plants are usually designed as complex structures with various irregularities^{[2–6]}. Besides, typically thermal power plants consists heavy coal scuttles at relatively high floors, which may generate significant inertial force and be detrimental to the structural seismic performance^{[4]}. An effective strategy to solve this problem is to convert the scuttles to suboscillators and tuning to the main structure, i.e.nonconventional multiple tuned mass damper (NCMTMD)^{[5–7]}.
During the real operational process, the coal storage in the scuttles may change^{[6–7]}, which leads to random mass of the tuned mass dampers of NCMTMD system. This can be classified as a mass uncertain NCMTMD (MUNCMTMD) system. Jensen et al^{[8]}. first investigated TMD system with uncertain mass and proofed the necessities to consider uncertainty during design for moderate (coefficient of variation,
In this paper, a reliabilitybased optimization design framework for MUNCMTMD systems was proposed. Several parameters of hazard, main structures and scuttles were considered to be random. Unconditional failure probability considering multiple limit state bounds was adopted as objective function. A thermal power plant with multiple scuttles was used as a case to illustrate the application of the framework. Latin hypercube sampling (LHS) method was implemented to reduce the sample size needed and sampling size study was performed to determine the appropriate sampling size. Optimum design was obtained and discussed. A parametric study was further performed to study the influence of isolation mechanism and seismic gap. Pendulum MUNCMTMD and failure probability considering collision were investigated.
1 Structural modelFor a NCMTMD with a n_{M} DOF main structure (mass matrix
$\begin{aligned}[b]\left[ {\begin{array}{*{20}{c}}{{{{M}}_{{n_M} \times {n_M}}}} & {{O}}\\{{O}}& {{{{m}}_{{n_m} \times {n_m}}}}\end{array}} \right]\left[ {\begin{array}{*{20}{c}}{{{\ddot X}}}\\{{{\ddot x}}}\end{array}} \right] + \left[ {\begin{array}{*{20}{c}}{{{{C}}_{{n_M} \times {n_M}}} + {{{l}}^{\, \rm{T}}}{{{c}}_{{n_m} \times {n_m}}}{{l}}}& {{{{l}}^{\, \rm{T}}}{{{c}}_{{n_m} \times {n_m}}}}\\{{{{c}}_{{n_m} \times {n_m}}}{{l}}}& {{{{c}}_{{n_m} \times {n_m}}}}\end{array}} \right]\left[ {\begin{array}{*{20}{c}}{{{\dot X}}}\\{{{\dot x}}}\end{array}} \right] + \\\left[ {\begin{array}{*{20}{c}}{{{{K}}_{{n_M} \times {n_M}}} + {{{l}}^{\, \rm{T}}}{{{k}}_{{n_M} \times {n_M}}}{{l}}}& {{{{l}}^{\, \rm{T}}}{{{k}}_{{n_m} \times {n_m}}}}\\{{{{k}}_{{n_m} \times {n_m}}}{{l}}}& {{{{k}}_{{n_m} \times {n_m}}}}\end{array}} \right]\left[ {\begin{array}{*{20}{c}}{{X}}\\{{x}}\end{array}} \right] = \left[ {\begin{array}{*{20}{c}}{{{{M}}_{{n_M} \times {n_M}}}}& {{O}}\\{{O}}& {{{{m}}_{{n_m} \times {n_m}}}}\end{array}} \right]{ l}{{a}}\end{aligned}$  (1) 
where
By transforming only the main structure to its modal space, i.e., using the transformation in Eq. (2), and performing mode truncation^{[12]}, one has the dynamic equation for the simplified NCMTMD system (Fig. 1) as Eq. (3).
$\left[ {\begin{aligned}{{X}}\\{{x}}\end{aligned}} \right] = {{\hat \varPhi }}\left[ {\begin{aligned}{{q}}\\{{x}}\end{aligned}} \right] = \left[ {\begin{aligned}& {{{{\varPhi }}_{{n_M} \times {n_{\overline M}}}}} & {{O}}\\& {{O}} & {{{{I}}_{{n_m} \times {n_m}}}}\end{aligned}} \right]\left[ {\begin{aligned}{{q}}\\{{x}}\end{aligned}} \right] $  (2) 
where
$\begin{aligned}[b]\left[ {\begin{array}{*{20}{c}}{{{{{\overline M}}}_{{n_{\bar M}} \times {n_{\bar M}}}}}& {{O}}\\{{O}}& {{m}}\end{array}} \right]\left[ {\begin{array}{*{20}{c}}{{{\ddot q}}}\\{{{\ddot x}}}\end{array}} \right] + \left[ {\begin{array}{*{20}{c}}{{{{{\overline { C}}}}_{{n_{\bar M}} \times {n_{\bar M}}}} + {{{\varPhi }}^{\text{T}}}{{{l}}^{\text{T}}}{{cl\varPhi }}}& {  {{{\varPhi }}^{\text{T}}}{{{l}}^{\text{T}}}{{c}}}\\{  {{cl\varPhi }}}& {{c}}\end{array}} \right]\left[ {\begin{array}{*{20}{c}}{{{\dot q}}}\\{{{\dot x}}}\end{array}} \right] + \\ \left[ {\begin{array}{*{20}{c}}{{{{{\overline K}}}_{{n_{\bar M}} \times {n_{\bar M}}}} + {{{\varPhi }}^{\text{T}}}{{{l}}^{\text{T}}}{{kl\varPhi }}}& {  {{{\varPhi }}^{\text{T}}}{{{l}}^{\text{T}}}{{k}}}\\{  {{kl\varPhi }}}& {{k}}\end{array}} \right]\left[ {\begin{array}{*{20}{c}}{{q}}\\{{x}}\end{array}} \right] = {{{{\hat \Phi }}}^{\text{T}}}\left[ {\begin{array}{*{20}{c}}{{M}}& {{O}}\\{{O}}& {{m}}\end{array}} \right]{{\iota a}}\end{aligned}$  (3) 
where
The uncertainty ubiquitously exists in the structural system and therefore the structural random variable set can be expressed as Eq. (4).
${{{\psi }}_S} = \left\{ {{{M}}, {{C}}, {{K}}, {{m}}, {{c}}, {{k}}} \right\}$  (4) 
Stationary KanaiTajimi (KT) model^{[13]} was adopted to model the ground motion excitation. A threedimensional filter, with one DOF in each direction, was assembled to the NCMTMD system. For simplification, the filter was assumed to be isotropic and expressed as Eq. (5),
$\left\{\begin{aligned}& {{{I}}_{3 \times 3}}{{{\ddot x}}_{\text{f}}} + 2{\xi _{\text{f}}}{\omega _{\text{f}}}{{{I}}_{3 \times 3}}{{{\dot x}}_{\text{f}}} + \omega _{\text{f}}^2{{{I}}_{3 \times 3}}{{{x}}_{\text{f}}} =  {{w}}\\& {{a}} = {{{\ddot x}}_{\text{f}}}{\text{ + }}{{w}} =  \left( {2{\xi _{\text{f}}}{\omega _{\text{f}}}{{{{\dot x}}}_{\text{f}}} + \omega _{\text{f}}^2{{{x}}_{\text{f}}}} \right)\end{aligned}\right.$  (5) 
where
Substitute Eq. (5) into Eq. (3), one has Eq. (6),
${{{M}}_{{\text{ft}}}}\left[ {\begin{array}{*{20}{c}}{{{\ddot q}}}\\{{{\ddot x}}}\\{{{{{\ddot x}}}_{\text{f}}}}\end{array}} \right] + {{{C}}_{{\text{ft}}}}\left[ {\begin{array}{*{20}{c}}{{{\dot q}}}\\{{{\dot x}}}\\{{{{{\dot x}}}_{\text{f}}}}\end{array}} \right] + {{{K}}_{{\text{ft}}}}\left[ {\begin{array}{*{20}{c}}{{q}}\\{{x}}\\{{{{x}}_{\text{f}}}}\end{array}} \right] = {{{f}}_{{\text{bw}}}}$  (6) 
where
${{{M}}_{{\text{ft}}}} = \left[ {\begin{array}{*{20}{c}}{{{\overline M}}}& {{O}}& {{O}}\\{{O}}& {{m}}& {{O}}\\{{O}}& {{O}}& {{{{I}}_{3 \times 3}}}\end{array}} \right]$  (7) 
${{{C}}_{{\text{ft}}}} = \left[ {\begin{array}{*{20}{c}}{{{\overline{ C}}} + {{{\varPhi }}^{\text{T}}}{{{l}}^{\text{T}}}{{cl{ \varPhi} }}}& {  {{{\Phi }}^{\text{T}}}{{{l}}^{\text{T}}}{{c}}}& {  2{{{\varPhi }}^{\text{T}}}{{M}}{{{\iota }}_{n \times 3}}{\xi _{\text{f}}}{\omega _{\text{f}}}}\\{  {{cl{ \varPhi}}}}& {{c}}& {  2{{m}}{{{\iota }}_{{n_m} \times 3}}{\xi _{\text{f}}}{\omega _{\text{f}}}}\\{{O}}& {{O}}& {2{\xi _{\text{f}}}{\omega _{\text{f}}}{{{I}}_{3 \times 3}}}\end{array}} \right]$  (8) 
${{{K}}_{{\text{ft}}}} = \left[ {\begin{array}{*{20}{c}}{{{\overline K}} + {{{\varPhi }}^{\text{T}}}{{{l}}^{\text{T}}}{{kl\varPhi }}}& {  {{{\varPhi }}^{\text{T}}}{{{l}}^{\text{T}}}{{k}}}& {  {{{\varPhi }}^{\text{T}}}{{M}}{{{\iota }}_{n \times 3}}\omega _{\text{f}}^2}\\{  {{kl\varPhi}}}& {{k}}& {  {{m}}{{{\iota }}_{{n_m} \times 3}}\omega _{\text{f}}^2}\\{{O}}& {{O}}& {\omega _{\text{f}}^2{{{I}}_{3 \times 3}}}\end{array}} \right]$  (9) 
${{{f}}_{bw}}{\text{ = }}{\left[{\begin{array}{*{20}{c}}{{{{O}}_{1 \times \left( {n + {n_i}} \right)}}}& {  {{{w}}^{\text{T}}}}\end{array}} \right]^{\text{T}}}$  (10) 
Three components of earthquake excitation are not necessarily correlated in its principle directions. With
${{S}}{\text{ = }}E \left( {{{w}}{{{w}}^{\text{T}}}} \right){\text{ = }}\left[ {\begin{array}{*{20}{c}}{{S\!\!_0}}& 0& 0\\0& {0.75{S\!\!_0}}& 0\\0& 0& {0.5{S\!\!_0}}\end{array}} \right]$  (11) 
where E(•) is expectation operator.
Amplitude and duration are important characters of ground motion.In this paper, the amplitude was expressed as a function of peak ground acceleration (
${S_0} = \frac{{2{\xi _{\text{f}}}{{\left( {PGA} \right)}^2}}}{{{3^2}{{\pi }}\left( {1 + 4\xi _{\text{f}}^2} \right){\omega _{\text{f}}}}}$  (12) 
$PGA{\text{ = }}{b_1}{{\rm{e}}^{{b_2}M_{\rm s}}}{\left( {R{\text{ + }}{R_0}} \right)^{  {b_3}}}$  (13) 
where
The probability model of moment magnitude
$p\left( {M_{\rm s}} \right){\text{ = }}\frac{{{\beta _{M_{\rm s}}}{{\rm{e}}^{  {\beta _{M_{\rm s}}}M_{\rm s}}}}}{{{{\rm{e}}^{  {\beta _{M_{\rm s}}}{{M_{\rm s}}_{{\min} }}}}  {{\rm{e}}^{  {\beta _{M_{\rm s}}}{{M_{\rm s}}_{{\max} }}}}}}$  (14) 
where
The groundmotion duration
${t_{\text{d}}}{\text{ = }}{T_{\text{p}}}{\text{ + }}{{0.5} /{{f_{\text{a}}}}}$  (15) 
where
Considering the uncertainty that associated in
${{{\psi }}_H} = \left\{ {{\omega _{\text{f}}}, {\xi _{\text{f}}}, M_{\rm s}, R} \right\}$  (16) 
Strictly speaking, the random variables should also include the bed rock white noise process. However, the structural response under random process can be solved by stochastic dynamics and eventually be expressed as a deterministic response variance. Therefore, it was not included in the random variable set.
3 Reliability AnalysesThe state vector for the NCMTMD system with the KT filter is
${{\dot {{y}}}} = {{Ay}} + {{B}}{{{f}}_{{\text{bw}}}}$  (17) 
where A and B are state matrices given by
${{A}} = \left[ {\begin{array}{*{20}{c}}{{O}}& {{I}}\\{  {{M}}_{{\text{ft}}}^{  1}{{{K}}_{{\text{ft}}}}}& {  {{M}}_{{\text{ft}}}^{  1}{{{C}}_{{\text{ft}}}}}\end{array}} \right], {{B}} = \left[ {\begin{array}{*{20}{c}}{{O}}\\{{{M}}_{{\text{ft}}}^{  1}}\end{array}} \right]$  (18) 
The stationary response can be obtained by solving the Lyapunov equation, as Eq. (19).
${{AR}} + {{R}}{{{A}}^{\rm{T}}} + 2{{\pi }}\left[ {\begin{array}{*{20}{c}}{{O}}& {{O}}\\{{O}}& {{S}}\end{array}} \right] = {{O}}$  (19) 
where
The response corresponding to certain state variable
$\left\{\begin{array}{l}\sigma _{{{\textit{z}}_i}}^2 = {{n}}_i^{\text{T}}{{GR}}{{{G}}^{\text{T}}}{{{n}}_i}, \$6pt]\sigma _{{{\dot {\textit{z}}}_i}}^2 = {{n}}_i^{\text{T}}{{GAR}}{{{A}}^{\text{T}}}{{{G}}^{\text{T}}}{{{n}}_i}\end{array}\right. $  (20) 
where
Take structure performance space
${\Pi _{\text{s}}} = \left\{ {{{\textit{z}}} \in R: \left {{{\textit{z}}_i}} \right < {\beta _i}} \right\}, i = 1, 2, \cdots, {n_{\textit{z}}}$  (21) 
The first passage failure probability was expressed as Eq. (22).
${P_{\rm{f}}}\left( {{t_{\rm{d}}}} \right) = \int_0^T {P\left[ {{\textit{z}}\left( \tau \right) \notin {\Pi _{\text{s}}}} \right]} {\rm{d}}\tau {\text{ = }}1  \exp \left( {  v_{\textit{z}}^ + {t_{\rm{d}}}} \right)$  (22) 
where
$\left\{\begin{aligned}& v_z^ + \approx \sum\limits_{i = 1}^{{n_z}} {{w_{{{\textit{z}}_i}}}\left( {{\lambda _{{{\textit{z}}_i}}}r_{{{\textit{z}}_i}}^ + } \right)}, \\ & {w_{{{\textit{z}}_i}}} = \int_{{{\text{B}}_i} \cap {\text{F}}} {p\left( {{{{z}}_ \bot }{{\textit{z}}_i} = {\beta _i}} \right)} {\rm{d}}{{{z}}_ \bot }, \\ & r_{{{\textit{z}}_i}}^ + = \frac{{{\sigma _{{{\dot {\textit{z}}}_i}}}}}{{\pi {\sigma _{{{\textit{z}}_i}}}}}\exp \left\{ {  \frac{{\beta _i^2}}{{2\sigma _{{{\textit{z}}_i}}^2}}} \right\}, \\ & {\lambda _{{{\textit{z}}_i}}} \approx \frac{{1  \exp \left\{ {  {q^{0.6}}{{\left( {\displaystyle\frac{2}{{\sqrt \pi }}} \right)}^{0.1}}\displaystyle\frac{{2\sqrt 2 }}{{{n_b}}}\displaystyle\frac{{{\beta _i}}}{{{\sigma _{{{\textit{z}}_i}}}}}} \right\}}}{{1  \exp \left( {  {{\beta _i^2} / {2\sigma _{{{\textit{z}}_i}}^2}}} \right)}}, \\ & q = \frac{{\sigma _{{{\textit{z}}_i}}^5}}{{4\pi \int_{  \infty }^{ + \infty } {\left \omega \right{S_{{{\textit{z}}_i}{{\textit{z}}_i}}}\left( \omega \right){\rm{d}}\omega \int_{  \infty }^{ + \infty } {S_{{{\textit{z}}_i}{{\textit{z}}_i}}^2\left( \omega \right){\rm{d}}\omega } } }}\end{aligned}\right.$  (23) 
where
Noting that the duration of earthquake is a function of hazard variables^{[20]}, the unconditional failure probability can be obtained by the integration as Eq. (24).
${P_{\rm{f}}} = \int {\int {{p_{\rm{f}}}\left( {{t_{\rm{d}}}\left( {{\psi _{\rm{H}}}} \right){\rm{ /}}{t_d}\left( {{\psi _{\rm{H}}}} \right)\left\{ {{\psi _{\rm{S}}},{\psi _{\rm{H}}}} \right\}{\rm{ }}} \right)p\left( {{\psi _{\rm{S}}}} \right)p\left( {{\psi _{\rm{S}}}} \right){\rm{d}}{\psi _{\rm{H}}}{\rm{d}}{\psi _{\rm{H}}}} } $  (24) 
Assuming failure event follows engineering Poisson distribution of independent occurrences^{[25]}, the failure probability after a time period
$\left\{\begin{array}{l}{P_{\rm F}}\left( {{t_{\text{l}}}} \right) = 1  {{\rm e}^{  {t_{\text{l}}}{v_{\rm f}}{P_{\rm f}}}}, \\{v_{\rm f}} = {{\rm e}^{{\alpha _{M_{\rm s}}}  {\beta _{M_{\rm s}}}M_{\rm s}{_{{\min} }}}}  {{\rm e}^{{\alpha _{M_{\rm s}}}  {\beta _{M_{\rm s}}}M_{\rm s}{_{{\max} }}}}\end{array}\right.$  (25) 
where
With given limit state bounds, the objective function that adopted in this paper was defined as
$\left\{\begin{array}{l}{\rm min}\;{P_F({\theta})}, \\{{\theta }}{\text{ = }}\left\{ {\begin{array}{*{20}{c}}{{\omega _1}, {\omega _2}, \cdots , {\omega _{{n_m}}}}& {{\xi _1}, {\xi _2}, \cdots , {\xi _{{n_m}}}}\end{array}} \right\}, \\{{\theta }} \in \mathop {I}\limits_{j = 1}^{{n_m}} {\left\{ {\left\{ {{\omega _j}: {\omega _j} \in \left[ {{\omega _{j{\min} }}, {\omega _{j{\max} }}} \right]} \right\} \cap \left\{ {{\xi _j}: {\xi _j} \in \left[ {{\xi _{j{\min} }}, {\xi _{j{\max} }}} \right]} \right\}} \right\}} \end{array}\right.$  (26) 
where
The case considered here is a concentrically braced steel thermal power plant building (Fig. 2). The structure consists of boiler frames, air heater houses and a scuttle bay. In its scuttle bay, 7 scuttles with each weight 1 098.3 tons locate at 32.2 m height of the structure.
5.1 Deterministic model
The structure was located at a Chinese site with relatively high hazard level. The expected bounds of earthquake magnitude are
The main structure was modeled by its first 12 modes, at which the cumulated participation factor at two horizontal directions is larger than 90% (Tab. 1). Rayleigh damping was adopted to model the inherent damping of the main structure, with 2% at 0.1 s and 1 s^{[29]}, respectively. Scuttles were modeled as a 2DOF lumped mass each, with a DOF at each horizontal direction.
5.2 Probabilistic modelParameters in random variable sets
Latin hypercube sampling (LHS) strategy was adopted in this paper to reduce the needed sample size. Appropriate sample size investigation was further carried out to determine the necessary sample size.With design variables
5.3 Statement of optimization problem
Considering that a too high dimensional structural output may cause a prohibitive computational burden, only eight structural outputs that expected to be the most critical, i.e., drifts of corner columns at the 1st and 3th floor (shown in Fig. 5), were considered, which are expected to have higher value^{[30]}.
Denoting
$\left\{\begin{array}{l}\!\!\!\! {\min}\;{P_{\text{F}}({{\theta}})}\\\!\!\!\! {{\theta }}{\text{ = }}\left\{ {\begin{array}{*{20}{c}}{{\omega _{{A_ x}}}, {\omega _{{ A_y}}}, \cdots , {\omega _{G_y}}}& {{\xi _{{A_x}}}, {\xi _{{A_y}}}, \cdots , {\xi _{{G_y}}}}\end{array}} \right\},\\\!\!\!\! {{\theta }} \in \!\!\!\!\!\!\!\!\! \mathop {I}\limits_{j = \{ {{\text{A}}_x}, {\text{ }}{{\text{A}}_y}, \cdots , {\text{ }}{{\text{G}}_y}\} }\!\!\!\!\! {\left\{ {\left\{ {{\omega _j}\!: \!{\omega _j} \!\in \!\left[ {{\omega _{j{\min} }}, {\omega _{j{\max} }}} \right]} \right\}\! \cap \!\left\{ {{\xi _j}\!: \!{\xi _j} \!\in\! \left[ {{\xi _{j{\min} }}, {\xi _{j{\max} }}} \right]} \right\}} \right\}} \end{array} \right.$  (27) 
where
The problem contains uncertainty and deterministic algorithms (e.g., the gradientbased algorithms and direct search algorithms) are not fit for this situation. The genetic algorithm, which is one of the most popular stochastic optimization algorithms, was adopted in this paper to solve the optimization problem.
The optimum design obtained by the optimization process was summarized in Tab. 3.
It can be seen that
Uncertain mass of the scuttles caused by changing coal storage cause large variation on oscillator frequencies.The variation can further cause problem on tuning. Pendulum system has an independent period with system mass, because of the perfect correlation between stiffness and mass. Consequently, the pendulum strategy is potentially a solution for the mass uncertainty. Another important practical problem is collision between coal scuttles and its surrounding structural members. These two important practical aspects were therefore investigated in this section.
6.1 Influence of oscillator typeFor translational MUNCMTMD system (Fig. 6), stiffness of suboscillators was induced by elastic potential energy of isolators. As an alternative, the pendulum MUNCMTMD, in which stiffness of suboscillators is the function of gravity potential energy of oscillators, can possibly be effective to enhance the system robustness when mass uncertainty presents.
For pendulum MUNCMTMD, all modeling methods are same as translational MUNCMTMD, but stiffness. For pendulum MUNCMTMD, stiffness was expressed as Eq. (28).
$k = {\rm{diag}}\left\{ {{m_1}g/{\rho _1},{m_2}g/{\rho _2}, \cdots ,{m_{{n_m}}}g/{\rho _{{n_m}}}} \right\}$  (28) 
where
1/
It can be observed from Tab. 4 that the translational system has slightly smaller
The gap between the coal scuttle and the surrounding structural members is 130 mm. Considering possible collision in {A_{x}, A_{y},
For the optimum design in Tab. 3, probability of failure with the consideration of collision
The author adopted multiobjective method^{[5]} in a previous engineering problem. The multiple output unconditional PF integrates multiple objectives into one failure probability and therefore avoids cumbersome multiobjective optimization.
7 ConclusionIn this paper, a multiple output unconditional reliabilitybased design framework for nonconventional multiple tuned mass damper for a complex structure was proposed. With the description of simplified structure model and earthquake hazard model, a multiple output reliability method was presented. Optimization problem was then formulated. A case of coal scuttles isolation for thermal power plant was solved to illustrate the method. From the study, conclusions as follows can be drawn:
1) The framework adopts failure probability with multiple limit state bounds as the objective function and therefore avoids the cumbersome multiobjective optimization. Multiple responses are integrated into one failure probability, which is clearer.
2) The pendulum system does not perform better than translational system in the case with equal uncertainty. Performance of pendulum system is not sensitive to the degree of its curvature uncertainty.
3) Collision problem is not critical for this case but it does affect the failure probability. Failure probability with the consideration of collision can be further optimized to search a design solution that mitigates both structural failure probability and collision problem.
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