Safety and serviceability assessment for high-rise tower crane to turbulent winds

Abstract

Tower cranes are commonly used facilities for the construction of high-rise structures. To ensure their workability, it is very important to analyze their response and evaluate their condition under extreme conditions. This paper proposes a general scheme for safety and serviceability assessment of high-rise tower crane to turbulent winds based on time domain buffeting response analysis. Spatially correlated wind velocity field at the location of the tower crane was first simulated using an algorithm for generating the time domain samples of a stationary, multivariate stochastic process according to some prescribed spectral density matrix. The buffeting forces applied to the structure were computed according to the above-simulated wind velocity fluctuations and the lift, drag, and moment coefficients obtained from a CFD computation. Those spatially correlated loads were then fed into a well calibrated finite element model and the nonlinear time history analysis was conducted to compute structural buffeting response. Compared with structural on-site response measurement, the computed response using the proposed method has good precision. The proposed method is then adopted for analyzing the buffeting response of an in-use tower crane under the design wind speed and the maximum operational wind speed for safety and serviceability assessment.

Keywords

tower crane / buffeting response / wind velocity / modeling

Introduction

Tower cranes are widely used for the construction of high-rise structures, where these cranes may rise hundreds of meters into the air with a required reach to cover the working range. Studies related to the dynamics of cranes were mostly focused on the vibration of the crane system due to the payload motions [1-4]. Since high-rise tower cranes normally operate under relatively severe conditions of high wind speed, its dynamic responses under extreme wind loads are also of great interest both for engineers and researchers. The recent collapsing accidents of several tower cranes under extreme wind loads addressed the importance of the related researches once more [5].

Previous studies related to wind resistance analysis for tower cranes were generally focused on the structural static analysis to constant wind velocity. Eden et al. [6] developed the empirical methods for calculating wind forces on latticed tower-like structures at any angle to the wind. Under the assumption that the crane is a system of a single degree of freedom and that the wind applies a constant overturning moment supplemented by a short duration gust moment, it is concluded that the crane design approach which is based on the structural static response analysis under the assumption of constant wind velocity is acceptable for most cranes. This conclusion is acceptable if the crane is an isolated structure without any wind disturbances due to surrounding buildings. However, due to its functions, it is impossible to erect an isolated crane with no other structure around. Because of the surrounding existing structures, the wind flow around a tower crane may exhibit unexpected and dangerous aerodynamic phenomena. Voisin et al. [7] studied the behavior of tower cranes in strong winds exposed to the disturbed shear flow induced by the surrounding existing structures. Based on results from the experiment, they concluded that the overturning moment of the crane in this heterogeneous wind field is strongly fluctuating and its maximum value is more than twice higher than in the case without any existing structures.

In this paper, a time domain buffeting response analysis scheme was proposed for workability assessment of high-rise tower cranes to turbulent wind. According to some prescribed spectral density matrix defined according to the location of the tower crane, the spatially correlated wind velocity field was first simulated using an algorithm for generating time domain samples of a stationary, multivariate stochastic process. Structural buffeting loads were computed according to these simulated wind velocity fluctuations and the lift, drag, and moment coefficients obtained from a CFD computation. Structural buffeting response was computed based on a nonlinear time history analysis on a well calibrated finite element model. Structural safety and serviceability could thus be assessed using the obtained responses.

Structure description and modeling

The Sutong Bridge is a cable-stayed bridge. Its main span is 1088 meters long and its cable pylons are 300.6 m high. During the construction of the cable pylons, a 315.7-meter-high tower crane was employed. Figure 1 shows the layout of the cable pylon and tower crane system. The tower crane is a beam-like lattice type structure of 51 repeated standard segments. Figure 2 shows the dimension of a standard segment of the tower crane. As shown, the standard segment is composed of four main chords and eight  type mast section panels. The main chords are steel bars, 200 mm in diameter. As a crane, there are some lifting devices, such as jib, trolley and counter-weights, at its top. For the most critical loading cases, the total weight of these devices along with the payloads is 126.9 tons. To enhance the safety and stability of the crane, six braces were respectively installed along the pylon to support the crane during different construction stages. For the full height construction stage studied in this paper, the braces at the heights of 118.7 m, 211.2 m and 269.0 m were bolted.

According to the design diagrams, a three-dimensional finite element model of the cable pylon-tower crane system was set up. Taking the eigenvalue analysis on the established finite element model, the natural frequencies and mode shape vectors of the structural system were computed. Figure 3 shows the first four mode shapes and natural frequencies when the jib is parallel to the cable pylon plane. It can be seen that the first mode corresponds to the out-of-plane bending of the cable pylon and the tower crane; the second and the fourth modes are the cantilever bending mode of the jib; the third mode is a torsion dominated vibration mode of the tower crane. Rayleigh damping is assumed for the upcoming time history integration procedure. It is determined according to the damping ratios of the fundamental out-of-plane bending mode and the fundamental in-plane bending mode, which are estimated from the dynamic response measurement collected from an on-site vibration test. According to the result of the vibration test, the damping ratio is 1.9% for the fundamental out-of-plane bending mode and is 2.2% for the fundamental in-plane bending mode [8].

Wind field simulation

For the time domain buffeting response analysis, the wind velocity fluctuation should first be digitally simulated. The wind velocity field where a high-rise tower crane stands can decompose into three fluctuation components: the horizontal along-wind fluctuation, the horizontal cross-wind fluctuation, and the vertical cross-wind fluctuation. Since the vertical cross-wind fluctuation will only induce structural axial tension and compression, which will not govern the failure of the masts, only the horizontal along-wind and cross-wind components are taken into account in this study. If the correlation between those components is further omitted, the wind field can decompose into two independent one-dimensional, n-variable stochastic vector processes. In this study, since 62 input points, equally spaced by 5.71 m along the tower crane which cover the entire height of the crane, are selected for wind fluctuation simulation, n is set to be 62. For the tower crane of two to three hundred meters high, the simulated sample functions should accurately describe the probabilistic characteristics not only in terms of temporal variation but also in terms of spatial distribution.

To simulate the structural along-wind wind velocity fluctuation, the spectral density matrix for along-wind wind velocity fluctuations, BoldItalic(ω), should first be determined:

(1)\mathit{S}(\omega )=\left[\begin{array}{llll}{s}_{\mathrm{11}}(\omega )& s_{\mathrm{12}}(\omega )& \cdots & s_{\mathrm{1}n}(\omega )\\ s_{\mathrm{21}}(\omega )& s_{\mathrm{22}}(\omega )& \cdots & s_{\mathrm{2}n}(\omega )\\ ⋮& ⋮& ⋮& ⋮\\ s_{n\mathrm{1}}(\omega )& s_{n\mathrm{2}}(\omega )& \cdots & s_{nn}(\omega )\end{array}\right].

The diagonal terms of BoldItalic(ω) are the auto-spectral density function for along-wind velocity fluctuation. In this study, the following Kaimal spectrum, which can account for the variations in vertical direction, is adopted:

(2)S_{ii}^u(\omega )=U_{*}^{\mathrm{2}}{\displaystyle \frac{\mathrm{200}f}{\omega {(\mathrm{1}+\mathrm{50}f)}^{\mathrm{5}/\mathrm{3}}}},f={\displaystyle \frac{\omega z_i}{U_{z_i}}},

where <InlineEquation></InlineEquation>S^u_ii(ω) is the power spectral density for the along-wind wind velocity fluctuation at the height of the i-th point z(i), and ω is the frequency of the wind fluctuation in Hz; U_{*}={\displaystyle \frac{K{U}_{z_i}}{\ln z_i-\ln z_{\mathrm{0}}}},Kis a unit-less constant set at 0.4 in this study, z₀ is the ground roughness height set at 0.05 meters in this study according to the site condition of the tower crane; U_zi is the mean wind speed at the height z_i determined from the following equation:

(3)U_{z_i}=U_{\mathrm{10}}{\left({\displaystyle \frac{z_i}{\mathrm{10}}}\right)}^a,

where U₁₀ is the mean wind speed at the gradient height of 10 m; α is a unit-less constant related to ground roughness. It is set at 0.12.

The non-diagonal terms of BoldItalic(ω) are the cross-spectral density function describing the spatial correlation of the wind fluctuation at two points, which can be determined by the auto-spectral density functions at these two points and the following coherence function:

(4){\rho }_{ij}(\omega )=\exp \left[-{\displaystyle \frac{\omega }{U}}\sqrt{C_x^{\mathrm{2}}{(x_i-x_j)}^{\mathrm{2}}+C_z^{\mathrm{2}}{(z_i-z_j)}^{\mathrm{2}}}\right],

where {\rho }_{ij}^{\mathrm{2}}(\omega )={\displaystyle \frac{S_{ij}^{\mathrm{2}}(\omega )}{S_{jj}(\omega )S_{ii}(\omega )}} is the spatial coherence function for the wind fluctuation of the two points i and j, x_i, x_j, z_i, z_j are the horizontal and vertical coordinates of the points i and j, U is the average wind velocity of the two points and C_x and C_z are the coefficients determining the decay of spatial correlation. In this study, C_x=8 and C_z=7 are adopted for along-wind fluctuation wind simulation.

Obtaining the spectral density matrix for the one-dimensional, 62-variable zero-mean stationary stochastic vector process {u_j(t)} (j=1,2,…,62) , an algorithm for generating time domain samples of a stationary, multivariate stochastic process according to the prescribed spectral density matrix was employed. The basic idea for the simulation of correlated time series from the power spectra and root-coherence function is that a time series can be represented by a sum of cosine waves with different phase, frequency and amplitude. The frequency and amplitude is obtained from discretization of the power spectra into segments (usually equally spaced), where the frequency associated with each segment is a typical frequency of that segment (for instance the central frequency) and the squared amplitude is proportional to the area of the segment. According to Shinozuka et al. [9], the simulated wind velocity fluctuation has the following form for a multivariate random process of the power spectral density matrix BoldItalic(ω)

(5)u_j(t)={\displaystyle \sum _{m=\mathrm{1}}^j{\displaystyle \sum _{l=\mathrm{1}}^N\left|{\mathit{H}}_{jm}({\omega }_l)\right|}\sqrt{\mathrm{2}\mathrm{\Delta }\omega }\cos [{\omega }_{mt}t+{\psi }_{jm}({\omega }_l)+{\theta }_{ml}],j=\mathrm{1},\mathrm{2},\mathrm{3},\cdots ,n,}

where the wind spectrum is divided into N equal portions in the frequency domain, Δω=ω/N is the frequency increment; H_jm(ω) is a lower-triangular matrix obtained via the Cholesky decomposition of BoldItalic(ω); |•| denotes the operation of taking modulus; ψ_jm(ω_l) is the phase angle between two points; θ_ml is the evenly distributed random number among 0 and 2π; and ω_ml is the double-indexed frequency defined as follows:

(6){\omega }_{ml}=(l-\mathrm{1})\mathrm{\Delta }\omega +{\displaystyle \frac{m}{n}}\mathrm{\Delta }\omega ,l=\mathrm{1},\mathrm{2},\cdots ,N.

During the simulation, the following relationship must be satisfied to avoid the distortion of simulation:Δt≤2π/2ΔωN.

Following the above-presented procedure, a set of 62 time records for the along-wind wind fluctuation process under the mean wind speed of 35.4 m/s was generated. Each record is a 100000-point along-wind wind velocity fluctuation time history. To check the accuracy of the presented simulation algorithm, the auto-/cross-power spectrum function of the generated sample sets was compared with the target auto-/cross-power spectrum function. For this comparison, two points were selected: the first point is at the bottom of the crane structure and the second point is at the top of the tower crane (just at the height of central line of the jib). Figure 4 shows the simulated time records at these two points. The temporal auto-/cross-power spectrum functions for the generated sample time records were then computed. As shown in Fig. 5, the auto-spectrum functions of the generated sample show very good agreement with the target functions. It is therefore concluded that the applied algorithm is accurate enough to simulate wind velocity fluctuations that are spatially correlated with the prescribed coherence function.

Similarly, the cross-wind velocity fluctuation can be simulated from the following determined spectral density function as well as the coherence function shown in Eq. (4).

(7)S_{ii}^v(\omega )=U_{*}^{\mathrm{2}}{\displaystyle \frac{\mathrm{15}f}{\omega {(\mathrm{1}+\mathrm{9.5}f)}^{\mathrm{5}/\mathrm{3}}}},f={\displaystyle \frac{\omega z_i}{U_{z_i}}}.

For cross-wind simulation, C_x=4.5 and C_z=5.5 are adopted in this study.

Buffeting load generation

Wind loads acting on a structure consist of self-excited aerodynamic loads due to structural motion and buffeting loads independent of structural motion. Concerning the lattice shape of the tower crane, the self-excited aerodynamic force is of low magnitude and is omitted in this study. For the buffeting loads, according to the quasi-steady theory, the lift, moment, and drag buffeting forces per unit length can be expressed as follows:

(8){\hat{F}}_{\mathrm{L}}(t)=\frac{\rho U^{\mathrm{2}}}{\mathrm{2}}H\left[\mathrm{2}{C}_{\mathrm{L}}(\alpha ){\displaystyle \frac{u(t)}{U}}+\left({\displaystyle \frac{\mathrm{d}{C}_{\mathrm{L}}}{\mathrm{d}\alpha }}+C_{\mathrm{D}}(\alpha )\right){\displaystyle \frac{v(t)}{U}}\right],

(9){\hat{F}}_{\mathrm{D}}(t)=\rho U^{\mathrm{2}}/\mathrm{2}H\left[\mathrm{2}{C}_{\mathrm{D}}(\alpha ){\displaystyle \frac{u(t)}{U}}+{\displaystyle \frac{\mathrm{d}{C}_{\mathrm{D}}}{\mathrm{d}\alpha }}{\displaystyle \frac{v(t)}{U}}\right],

(10){\hat{F}}_M(t)=\rho U^{\mathrm{2}}/\mathrm{2}H\left[\mathrm{2}{C}_{\mathrm{M}}(\alpha ){\displaystyle \frac{u(t)}{U}}+{\displaystyle \frac{\mathrm{d}{C}_{\mathrm{M}}}{\mathrm{d}\alpha }}{\displaystyle \frac{v(t)}{U}}\right],

where C_L(α), C_D(α), and C_M(α)are the dimensionless lift, drag, and moment coefficients at the wind angle of α; ρ is the density of the air (kg/m³); H is the projection width cross the structure; u(t), v(t) is the along-wind and cross-wind wind velocity fluctuation; and α is the wind attack angle. The static aerodynamic coefficients of the critical sections of the pylon-crane system were computed using the computational fluid dynamics program developed by the State Key Laboratory for Disaster Reduction in Civil Engineering of Tongji University [10]. Table 1 shows the computed static aerodynamic coefficients for the upper single-pylon-section pylon-crane structure system.

Nonlinear response analysis

To account for the geometrical nonlinearity effect for the time history analysis for the cable pylon-tower crane system, the following step-by-step integration method is employed to compute the structural time domain buffeting response under turbulent wind. For a time interval [t, t+Δt], structural stiffness matrix, BoldItalic_t, is assumed to be invariant over the full interval and the governing equations of motion can be expressed in the incremental form as

(11)\mathit{M}\mathrm{\Delta }\ddot{y}(t)+\mathit{C}\mathrm{\Delta }\dot{y}(t)+{\mathit{K}}_t\mathrm{\Delta }y(t)=\mathrm{\Delta }\hat{\mathit{F}}(t),

where BoldItalic, and BoldItalic are the mass and damping matrix of the system, \mathrm{\Delta }\ddot{y}(t)=\ddot{y}(t+\mathrm{\Delta }t)-\ddot{y}(t),\mathrm{\Delta }\dot{y}(t)=\dot{y}(t+\mathrm{\Delta }t)-\dot{y}(t),\mathrm{\Delta }y(t)=y(t+\mathrm{\Delta }t)-y(t); and\mathrm{\Delta }\hat{\mathit{F}}\mathit{t}\mathit{=}\hat{\mathit{F}}\mathit{t}\mathit{+}\mathrm{\Delta }\mathit{t}\mathit{-}\hat{\mathit{F}}\mathit{t}.

Based on the following assumptions for nodal velocities and displacements,

(12)\{\begin{array}{c}\dot{y}(t+\mathrm{\Delta }t)=\dot{y}(t)+[(\mathrm{1}-\gamma )\mathrm{\Delta }t]\ddot{y}(t)+(\gamma \mathrm{\Delta }t)\ddot{y}(t+\mathrm{\Delta }t),\\ y(t+\mathrm{\Delta }t)=y(t)+\mathrm{\Delta }t\dot{y}(t)+\left[\left({\displaystyle \frac{\mathrm{1}}{\mathrm{2}}}-\beta \right)\mathrm{\Delta }{t}^{\mathrm{2}}\right]\ddot{y}(t)+(\beta \mathrm{\Delta }{t}^{\mathrm{2}})\ddot{y}(t+\mathrm{\Delta }t),\end{array}

where the parameters γ and β are the required integration stability and accuracy, the following equation can be obtained:

(13){\hat{K}}_t\mathrm{\Delta }y(t)=\mathrm{\Delta }{\hat{F}}_i,

where

(14){\hat{K}}_t=K_t+{\displaystyle \frac{\gamma }{\beta \mathrm{\Delta }t}}C+{\displaystyle \frac{\mathrm{1}}{\beta \mathrm{\Delta }{t}^{\mathrm{2}}}}M,

(15)\mathrm{\Delta }{\hat{F}}_i=\mathrm{\Delta }{F}_i+\left({\displaystyle \frac{M}{\beta \mathrm{\Delta }t}}+{\displaystyle \frac{\gamma C}{\beta }}\right)\dot{y}(t)+\left[{\displaystyle \frac{M}{\mathrm{2}\beta }}+\mathrm{\Delta }t\left({\displaystyle \frac{\gamma }{\mathrm{2}\beta }}-\mathrm{1}\right)c\right]\ddot{y}(t).

Solving Eq. (13) and taking into consideration the relationship between Eq. (11) and Eq. (15), the structural responses at the time instant t+Δt can be computed. During each step of the computation, one should keep on updating the total tangential stiffness matrix according to the response obtained in the last step. For more details on formulating the total tangential stiffness matrix for nonlinear analysis, please reference the related textbooks (such as Ref. [10]).

Following the above procedure, the buffeting response analysis of the cable pylon-tower crane system was conducted at the deformed aerostatic equilibrium state by applying the fluctuating components of the lift, drag, and moment both to the pylon and the crane. To examine the difference between the linear and nonlinear analysis due to the wind fluctuation, the dynamic response computation was conducted using both methods. Figure 6 shows the out-of-plane displacements at the top of the tower crane for one sample set of wind velocity fluctuations. It can be seen that the differences between the linear and nonlinear analyses are obvious. The peak response of the nonlinear analysis is 28% higher than the peak response of the linear analysis. The lumped mass at the top of the tower crane, such as jib, trolley, counter-weights and payloads, is the contributing factor to the observed difference between the linear and nonlinear analyses.

To verify the accuracy of the proposed method, some on-site strain monitoring records were collected and the corresponding numerical computation was performed. Figure 7 shows the comparison of the computed and the measured results [11]. As shown in the figure, the computed maximum wind-induced response in the wind velocity field simulated according to the mean wind speed (U=6.1 m/s) measured during the test matches quite well with the strain response measurement: the tested and computed results have a similar trend and the computed peak values along the tower crane are bigger than the tested peak values except for point A, where the analyzed result (1.034 MN) is 4.15% smaller than the test result (1.079 MN). This means that the proposed method intends to give a more conservative analyzed result.

Safety and serviceability assessment

To evaluate the safety of the tower crane under the design wind speed (U=35.4 m/s), both static and a set of 600-second buffeting wind loads were simulated and then fed into a commercial FEM program for static and buffeting response analysis. Since the mast of the crane is mainly in compression, the axial forces of the mast are checked for safety assessment. Figure 8 shows the peak axial force of the mast along the tower crane only considering the static load and considering both static and buffeting loads. As shown in this figure, accounting for the buffeting vibration, the peak axial force (3.51 MN) at the top of the tower crane is 140.4% higher than the corresponding axial force (1.46 MN) to static wind load. This means the buffeting vibration analysis should be taken into account even when it is in an open terrain like the construction site of the Sutong Bridge. In this case, compared with the allowable internal forces (18.6 MN) of the mast components, the results show that the tower crane has enough safety redundancy under the design wind speed.

For serviceability assessment under operational wind speed during the construction stage (U=15 m/s), the peak accelerations along the tower crane (as shown in Fig. 9) under this wind velocity fluctuation are computed. Since the allowable maximum serviceability acceleration is 30 gal according to the Chinese Code of JGJ99-98 [12], the results verified that the vibration of the crane under the operational wind speed will not induce the crane operators uncomfortable.

Concluding remarks

This paper proposed a general scheme for safety and serviceability assessment of high-rise tower cranes to turbulent wind based on time domain buffeting response analysis. The geometrical nonlinearity effect on the buffeting response analysis of the tower crane was discussed. The results of the numerical study show that the geometric nonlinearity effect for a tower crane, whose top is lumped with jib, trolley, counterweights and weighting loads, contributes much for the structural dynamic response analysis to turbulent wind. The comparison of the numerical results obtained from the above-proposed nonlinear buffeting response analysis and the response measurement obtained from an on-site test verified the proposed time domain buffeting analysis method is efficient and accurate.

The verified nonlinear time domain buffeting response analysis method is adopted for safety and serviceability assessment of an in-use tower crane. Based on the result of the study, it is concluded that the buffeting response of the tower crane accounts much for the wind-induced response under design wind speed and thus not only the static response analysis to constant wind velocity but also the buffeting response analysis due to the wind fluctuation should be conducted for the safety assessment of the high-rise tower crane.

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