Keywords

1 Introduction

With the use of new materials and technologies, large-span structures are becoming slenderer and lighter, thus tend to have lower fundamental frequencies. If the structural frequency becomes close to the pacing frequency, human-induced loads may cause excessive structural vibration that may exert physiological or psychological discomfort to the exposed personnel, causing the so-called vibration serviceability problem [1]. If large vibration occurs during the service life of the structure, huge efforts are usually necessary to reduce the unpleasant large vibration. Such efforts sometimes can be highly time and cost consuming. Therefore, it is important to carefully address the vibration serviceability problem in the design stage, especially for those light-weight large-span structures whose design is usually dominated by such vibration.

In most current design codes, the vibration serviceability assessment is usually addressed by either setting a lower limit of structural fundamental frequency, or by setting an upper limit of the dynamic response acceleration [2]. Because the calculation of fundamental frequency and acceleration requires an available analytical structural model, the vibration serviceability assessment is usually carried out after the structural safety design is finished. However, once the serviceability requirement is unsatisfied, the original design needs to be modified, and the vibration serviceability check is repeated. Such a trial and error verification process makes the design process inefficient, especially for those large-span structures where vibration serviceability plays a controlling role. Moreover, current design process does not guarantee that the design solution obtained is theoretically optimal. An effective method that is capable of obtaining the optimal design for structural vibration serviceability is thus strongly desired.

In recent years, the topology optimization method has been shown to actively accomplish generative design of various types of structures. It can seek the optimal layout form of structural materials in the design domain according to certain design criteria, so that the optimized design of a structure under various constraints can be achieved. Structural topology optimization is divided into two main categories, one considering topology optimization under static load and one under dynamic loads. For the structural design under static loads, topology optimization has been adopted for the tensegrity structures for the Grote Marktstraat in The Hague, Netherlands [3], the Qatar National Convention Center in Doha [4], and the Los Angeles International Airport long-span pedestrian bridges [5]. For dynamic loads, Pedersen [6] used topology optimization in designing the collision-resistant structures. Fuchs and Moses [7] achieved good results in topology optimization of hydrostatic pressure dams. Yan et al. [8] presented an optimal topology design of damped vibrating plate structures subject to initial excitations.

Inspired by the widespread success and effectiveness of topology optimization, a novel method for structural vibration serviceability using topology optimization is proposed in this paper. Corresponding to the serviceability requirement in current design codes, three different design objectives are considered, namely minimum displacement under static load, maximum fundamental frequency, and minimum acceleration under dynamic load. This paper is organized as follows: In Sect. 2, the basic theory of topology optimization is briefly introduced, which is then implemented in the design of structural design of a large-span truss structure as described in Sect. 3. The optimized results are then compared and discussed. Finally, some concluding remarks are made in Sect. 4.

2 Basic Theory of Topology Optimization

2.1 Variable Density Method

Currently, the mainstream topology optimization methods include the Homogenization Method, the Variable Density Method, and the Evolution Structure Optimization Method. In this paper, the Variable Density Method is adopted because of its high optimization efficiency, low number of reanalysis, and simple program implementation. In this method, a design domain is firstly required to be given according to the constraints. The design domain is divided into a finite number of grid units. For each unit, material property is assumed constant and the design variable is the unit relative densities ρe, which represents whether there the material exists in this unit. The variation range of ρe is (0, 1), in which ρe = 0 means no material in the unit and ρe = 1 means material is present in the unit. The topological form of the structure is changed according to the presence or absence of the unit material. The variable density method is widely applied to continuum topology optimization problems such as stress, frequency, displacement, and dynamic response. The common interpolation models of this method include Solid Isotropic Microstructures with Penalization Model (SIMP) [9, 10] and Rational Approximation of Material properties Model (RAMP) [11].

2.2 SIMP Model for Topology

In this paper, the SIMP model is used. The idea of the SIMP method is to penalize the intermediate density val-ues of the design variables between (0, 1) by a penalty fac-tor to make the density converge to 0 or 1. The above pe-nalized relaxation model is implemented by the following equation

$$ E_e \left( {\rho_e } \right) = \rho_e^p E_0 $$
(1)

where ρe is the element density, Ee is element Young’s modulus of solid material, and p is penalization parameter, whose value is usually taken as 3.

In general, the topology optimization mathematical model of SIMP [11] is

$$ {\mathop {\min }\limits_{x_{e,e} = 1, \ldots ,N_e }} f $$
$$ s.t. \left\{ {\begin{array}{*{20}c} {Ku + C\dot{u} + M\ddot{u} = P} & \quad {(a)} \\ {\mathop \sum \nolimits_{e = 1}^{N_e } \rho_e V_e - V^* \le 0} & \quad {(b)} \\ {0 < \rho_{min} \le \rho_e \le 1} & \quad {(c)} \\ \end{array} } \right. $$
(2)

in which Eqs. (2a), (2b), and (2c) indicates the mechanical constraint, volume constraint, and element density constraint, respectively. In these equations, f represents the objective function, K, C and M are the global stiffness, damping, and mass matrix of the structure, respectively, P is the external load, Ve and V* are element volume and design volume, respectively, and ρmin is the lower limit of density set to prevent the overall stiffness matrix singularity due to zero density, which is empirically taken as 0.0001.

The SIMP model is a nonlinear programming problem with inequality constraints, whose characteristics include multiple design variables, large analytical solution scale, and structural analysis at each iteration step. The mainstream algorithms currently used to solve SIMP model are Mathematic Programming Method, Heuristic Algorithms, and Optimality Criterion Method [13]. In this paper, the optimization criterion method is used, and the specific solution process is shown in later sections.

3 Vibration Serviceability Design of a Truss Structure Using Topology Optimization

In this section, the topology optimization method is implemented to the vibration serviceability design of a simply-supported pedestrian bridge with a span of 54 m. To serve as the main supporting system of this pedestrian bridge, a Q355C steel plane truss is to be designed. Three different objective functions corresponding to different vibration serviceability requirements are separately adopted in the optimization process.

In topology optimization with three different objective functions, the SIMP model and constraints are similar, as shown in Eq. (2). Three different objective functions are denoted as f1, f2, and f3. The mechanical constraints of Eq. (2a) change according to the static and dynamic conditions, and the volume constraints and density ranges of Eqs. (2b), (2c) remain the same for different objective functions.

To solve the SIMP model, the Optimality Criterion Method is adopted in this paper for all the three objective functions. Following Bendsøe [15], a heuristic updating scheme for the design variables can be formulated as

$$ \rho_e^{(k + 1)} = \left\{ {\begin{array}{*{20}c} {\left. {\left( {B_e^{\left( k \right)} } \right.} \right)^\eta \rho_e^{\left( k \right)} } & {if\;{\text{max}}\left. {\left( {\rho_{min} ,\left( {1 - m} \right)\rho_e^{\left( k \right)} } \right.} \right) < \left. {\left( {B_e^{\left( k \right)} } \right.} \right)^\eta \rho_e^{\left( k \right)} < {\text{min}}\left. {\left( {1,\left( {1 + m} \right)\rho_e^{\left( k \right)} } \right.} \right)} \\ {{\text{max}}\left. {\left( {\rho_{min} ,\left( {1 - m} \right)\rho_e^{\left( k \right)} } \right.} \right)} & {if \left. {\left( {B_e^{\left( k \right)} } \right.} \right)^\eta \rho_e^{\left( k \right)} \le {\text{max}}\left. {\left( {\rho_{min} ,\left( {1 - m} \right)\rho_e^{\left( k \right)} } \right.} \right)} \\ {{\text{min}}\left. {\left( {1,\left( {1 + m} \right)\rho_e^{\left( k \right)} } \right.} \right)} & {if \left. {\left( {B_e^{\left( k \right)} } \right.} \right)^\eta \rho_e^{\left( k \right)} \ge {\text{min}}\left. {\left( {1,\left( {1 + m} \right)\rho_e^{\left( k \right)} } \right.} \right)} \\ \end{array} } \right. $$
(3)

where m (move) is a positive move-limit, η is a numerical damping coefficient which is usually taken as 1/2, and Be is found from the optimality condition as

$$ B_e = \frac{{ - \frac{\partial f}{{\partial {\uprho }_e }}}}{{{\uplambda }\frac{\partial V_e }{{\partial {\uprho }_e }}}} $$
(4)

where λ is a Lagrangian multiplier that can be found by a bi-sectioning algorithm.

3.1 Minimum Static Displacement as the Objective Function

In some design codes, the structural static displacement under external load is limited within an acceptable threshold, which might be the simplest requirement for structural vibration serviceability. This sub-section aims to provide an optimized design for the plane truss that exhibits minimum static displacement under external load. The objective function f1 and mechanical constraints can be formulated as:

$$ f_1 = {\mathop {\min }\limits_{\uprho }} C\left( \rho \right) = U^T KU = \mathop \sum \limits_{e = 1}^N \left( {{\uprho }_e } \right)^p u_e^T k_e u_e $$
$$ K\left( \rho \right)U\left( \rho \right) = P $$
(5)

where ue and ke are the element displacement vector and stiffness matrix, respectively.

The update direction of the design variables is obtained through the partial derivatives (i.e., sensitivity) of the objective and constraints with respect to the design variables. The sensitivity of the objective can be formulated as:

$$ \frac{\partial C}{{\partial {\uprho }_e }} = u_e^T \frac{\partial k_e }{{\partial {\uprho }_e }}u_e = - p\left( {{\uprho }_e } \right)^{p - 1} u_e^T k_e u_e $$
(6)

Sensitivity of volume constraint can be formulated as:

$$ \frac{{\partial \left( {\sum_{e = 1}^n V_e {\uprho }_e } \right)}}{{\partial {\uprho }_e }} = V_e $$
(7)

Directly using Eq. (6) to update the density will cause checkerboard phenomenon. In this regard, Sigmund [16] proposes a filtering method based on image filtering technology and the filter works by modifying the element sensitivities as follows:

$$ \frac{\partial C}{{\partial {\uprho }_e }} = \frac{1}{{{\uprho }_e \sum_{f = 1}^n \widehat{H_f }}}\mathop \sum \limits_{f = 1}^n \widehat{H_f }{\uprho }_f \frac{\partial C}{{\partial {\uprho }_f }} $$
$$ \widehat{H_f } = r_{{\text{min}}} - dist\left( {e,f} \right),\left\{ {f \in N|dist\left( {e,f} \right) \le r_{{\text{min}}} } \right\},e = 1, \ldots ,N $$
(8)

where the operator dist(e, f) is defined as the distance between center of element e and element f. Instead of the original sensitivities expressed by Eq. (6), the modified sensitivities in Eq. (8) are used in the Optimality Criteria update shown by Eq. (3).

For complex topology optimization problems, such as multi-objective problems and dynamic problems, there are often gray-scale units whose density does not converge. This paper introduces the gray filter function [17] for improvement. The gray filter function is formulated as:

$$ {\Phi }\left( {{\uprho }_e } \right) = \frac{{\tan^{ - 1} \left[ {\frac{{\left( {\rho_e - 0.5} \right)}}{\mu }} \right]}}{{2\tan^{ - 1} \left( {\frac{0.5}{\mu }} \right)}} + \frac{1}{2} $$
$$ \mu^{(k)} = \left\{ {\begin{array}{*{20}c} {0.5,} & {k = 1,2, \ldots ,5} \\ {\left( {1 - \xi } \right)\mu^{(k - 1) } ,} & {k \ge 6} \\ \end{array} } \right. $$
(9)

where k is the number of iteration step and ξ is taken as 0.05. In the optimization process, the gray-scale filtering is embedded in the dichotomy loop.

This paper uses numerical examples to verify the calculation process. The target is to design a pedestrian bridge that satisfies vibration serviceability requirement. Thus, assuming that the optimization target structure is a simply-supported steel beam. The stiffer material is Q355C steel with Young’s modulus E = 210 GPa, Poisson’s ratio υ = 0.3 and the mass density ρ = 7850 kg/m3. Constraints and loads are shown in Fig. 1. The structure is discretized into 300 × 50 four-node rectangular elements. The penalty factor is taken as p = 3 and the minimum filter radius rmin is 2. The design volume V* is set as 0.3 V0 (i.e. the original volume of design domain). The external load is taken as 7200 N.

Fig. 1.
figure 1

Constraints and loads of the numerical example

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The calculation process of the numerical examples presented in this paper is realized by MATLAB code. The result is shown in Fig. 2.

Fig. 2.
figure 2

The result of topology optimization with objective function f1

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3.2 Maximum Fundamental Frequency as the Objective Function

The second part of our work is topology optimization with maximum fundamental frequencies as objective function, which aims at avoiding resonance under external excitation frequencies. This objective can be achieved by maximizing the structural fundamental frequency [18].

Assuming that damping can be neglected, the objective function f2 and mechanical constraints can be formulated as

$$ f_2 = {\mathop {\max }\limits_{\rho_1 , \ldots ,\rho_{N_e } }} \left\{ {{\mathop {\min }\limits_{j = 1, \ldots ,J}} \{ \omega_j^2 \} } \right\} $$
$$ \left\{ {\begin{array}{*{20}c} {K{\Phi }_j = \omega_j^2 M{\Phi },} & \quad {j = 1, \ldots ,J} & \quad {(a)} \\ {{\Phi }_j^T M{\Phi }_k = \delta_{jk} ,} & \quad {j \ge k, k,j = 1, \ldots ,J} & \quad {(b)} \\ \end{array} } \right. $$
(10)

where ωj is the jth eigenfrequency, Φj is the corresponding eigenvector, and δjk is Kronecker’s delta.

Similar to the sensitivity analysis in Sect. 3.1, the sensitivity analysis in this section can be divided into two parts: acquisition of sensitivity resolution expressions and sensitivity filtering.

First, to get the sensitivity of the eigenvalue λj = ωj2 of design variable ρe, the vibration Eq. (10a) is differentiated with respect to ρe.

$$ \left( {K - \lambda_j M} \right)\left( {{\Phi }_j } \right)_{\rho_e }^{\prime} + \left( {K_{\rho_e }^{\prime} - \lambda_j M_{\rho_e }^{\prime} - \left( {\lambda_j } \right)_{\rho_e }^{\prime} M} \right){\Phi }_j = 0 $$
(11)

Thus, the sensitivity of the eigenvalue λj with respect to the design variable ρe becomes

$$ \left( {\lambda_j } \right)_{\rho_e }^{\prime} = {\Phi }_j^T \left( {p\rho_e^{\left( {p - 1} \right)} K_e^* - \lambda_j q\rho_e^{\left( {q - 1} \right)} M_e^* } \right){\Phi }_j $$
(12)

Second, the density ρe is iteratively updated by ρe(k+1) = ρe(k) + Δρ. This iteration process is terminated when ||Δρ|| <ε, where ε is a predetermined small value.

The constraints and parameters of the numerical example are set as in Sect. 3.1. The theoretical optimal result is shown in Fig. 3.

Fig. 3.
figure 3

The result of topology optimization with objective function f2

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3.3 Minimum Acceleration as the Objective Function

The third part of our work is dynamic topology optimization with minimum acceleration as objective function. A numerical method is proposed to calculate the sensitivity for the acceleration optimization scheme, and other processes are the same as in Sect. 3.1. Assuming that damping can be neglected, the objective function f3 and the corresponding mechanical constraints are formulated as:

$$ f_3 = {\mathop {\min }\limits_{\rho_1 , \ldots ,\rho_{N_e } }} \ddot{u} $$
$$ M\ddot{u} + Ku = P $$
(13)

where ωn is the frequency of nth mode; ü is the acceleration of each mode.

Only the first, second, and third mode shapes of the structure are considered. For each mode shape, Eq. (14) is used to calculate the corresponding modal mass:

$$ m_n = \mathop \int \nolimits_0^L m\left( x \right)\phi_n \left( x \right)^2 dx $$
(14)

where m(x) is mass per unit length and ϕn(x) is the nth mode shape. For simply-supported beam structures, the nth mode shape is composed of n half-sine waves.

The external force of the corresponding mode shape can be formulated as:

$$ P_n \left( t \right) = P\left( t \right)\phi_{n,input} $$
(15)

where P(t) is the external force, and ϕn,input is the nth mode shape at the excitation point of the external force.

The calculation of the frequency is the same as described in Sect. 3.2. The stiffness corresponding to the first three modes are obtained from k = mω2. Combining Eq. (13), (14), (15) and the Newmark-beta method, the corresponding modal acceleration response ün can be solved. The acceleration is then calculated through the following formula

$$ \ddot{u} = \mathop \sum \limits_{n = 1}^3 \ddot{u}_n \phi_{n,input} $$
(16)

Using the analytical method to obtain the sensitivity expression of the acceleration requires a lot of mathematical and mechanical analysis. Thus, a numerical method is introduced to calculate the sensitivity. Considering that the physical meaning of sensitivity is the opposite number of the partial derivative of the objective function, the calculation process of the numerical method is as follows. For any grid unit (elx, ely) in the design domain, it has a unit density ρe (i.e. design variable), and the acceleration a1 corresponding to ρe can be calculated through Eq. (16). Then, a small increment ∆ is added to the unit density ρe and a new density \(\rho_e^{\rm{\prime}}\) is obtained. The acceleration a2 corresponding to the new density ρe’ is then calculated. The sensitivity can be formulated as:

$$ dc\left( {elx,ely} \right) = \frac{a_1 - a_2 }{{\Delta }} $$
(17)

The constraints and parameters of the numerical example are set as in Sect. 3.1. A half-cycle sinusoidal force P = Psinωt is used to represent the human-induced load to excite the structure at mid-span, and ω is taken as 2 Hz, P is taken as 720 N. The theoretical optimal result is shown in Fig. 4.

Fig. 4.
figure 4

The result of topology optimization with objective function f3

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3.4 Comparison of Optimization Results with Different Objective Functions

Comparing the results of topology optimization with displacement and frequency as the objective function (corresponding to Fig. 2 and Fig. 3, respectively). It can be found that the fundamental frequency of the latter structure has a substantial increase, and its dynamic stiffness performance is better. However, from the structure shape of the optimization results, the displacement of the latter result in the span will be much higher than the former result when the concentrated load is applied in the span, which may produce structural instability, indicating that the static stiffness performance of the latter structure is relatively weak. From Fig. 5, it can be found that the mass is mainly concentrated near the point of load action, so that the vibration acceleration near this point can be reduced. Thus, the global acceleration of the structure can be controlled. The maximum displacements, fundamental frequencies, and maximum accelerations, are calculated for the above three structures. The results are shown in Table 1 by percentages according to the optimized values for comparison.

Table 1. Comparison of optimized results
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Compared with the analytical method used in Sect. 3.1 and 3.2, the numerical method used in acceleration problem takes more time and calculation for each iteration, sacrificing a certain efficiency.

4 Conclusion

This paper uses topology optimization to find the optimal structure of large-span pedestrian bridge under the constraints of displacement, fundamental frequency, and acceleration. A gray-scale filter function is adopted to avoid gray-scale non-convergence. A numerical method is used to realize the sensitivity calculation of acceleration, instead of complex mathematics and mechanical derivation analysis. Specific numerical examples and results are also given, and qualitative analysis is performed.