Systematic and Quantitative Assessment of Reduced Model Resolution on the Transient Structural Response Under Wind Load

Abstract

The wind-induced response of structures is typically studied in wind tunnels either on scaled models or using numerical approaches under similar transient load conditions. In early design phases—where the potential for impactful change is most significant—information is often limited. As a result, studies are frequently conducted on simplified or reduced-resolution structural models. Typical applications for dimensionally reduced engineering models include early design phases, deciding on the need for high-fidelity analyses, and verifying wind tunnel models, which are often constructed using beams with lumped masses. In this contribution, the validity of these approaches is tested. Various limitations intrinsically arising from such modeling assumptions, showcased on a generic high-rise under dynamic wind load conditions, are highlighted. The systematic parametric analysis focuses on the variations in transient structural responses, particularly displacement and accelerations at the top of a building. Various wind loading cases are studied, with the reduction of the resolution taking place either in the original or in modal space. Results indicate that a considerable reduction is possible, but characteristic design values tend to deteriorate in cases of a high reduction, particularly when higher mode contributions are truncated. It is observed that the top-floor acceleration and displacement can be captured with considerable accuracy with three lumped masses for tall buildings. It is critical to study the impact of simplifying models starting at the highest level of detail possible. Here, a three-DoF model was able to capture the displacement up to a deviation of 11% and accelerations up to 20%. These approximate models are useful for initial design stages, optimization, uncertainty quantification, etc., where fast, cheap, and moderately accurate model evaluations are necessary.

1. Introduction

Structural wind engineering is the art and science of modeling and analyzing structures under natural wind flows. It aims at estimating the accurate responses of these structures under natural wind loads. Wind tunnel measurements traditionally represent the established method for quantifying wind loads [1]. These typically result from high-resolution pressure time histories (in the case of high-frequency pressure integration—HFPI) or global structural forces (from high-frequency force balance—HFFB). Additionally, codes and standards provide further insights into the quantification of the effects of wind flow on structures. These, however, have severe restrictions and limitations for special constructions (such as high-rise, large span, lightweight, etc.) and inherently cannot provide sufficiently detailed information on local/global loads and respective responses in high resolution (in time and space). In recent decades, computational wind engineering (CWE) has gained a lot of attention from the structural wind engineering practice, where wind loads are estimated by modeling the wind flows around the structure using computational methods like computational fluid dynamics (CFD).

To obtain structural responses, like displacement and accelerations, an aeroelastic model that incorporates the structural properties of the system is used in wind tunnels. The structures are equivalently represented in these scaled-down aeroelastic models. The sizing effect needs to be taken into account during such modeling. For tall buildings, a commonly used simplification for the structural system is to use lumped mass models. The number of degrees of freedom in such lumped mass models is chosen by the researcher/modeler. Wind tunnel testing with aeroelastic models commonly uses lumped mass models for tall buildings. Newer techniques like hybrid aeroelastic pressure balance (HYPB) use a single degree of freedom (SDoF) [2] for the structural system in a wind tunnel, capturing the two-way coupling of structure and wind. In order to capture higher vibration modes in these aeroelastic tests, MDoF is used in previous studies. Vortex-induced vibrations (VIVs) and resonance were investigated with aeroelastic models in a wind tunnel to model super high-rise buildings using MDoF in [3,4]. A series of wind tunnel tests with the MDoF model of Shanghai World Financial Centre is carried out in [5] for vibration measurements.

In computational wind engineering, this is captured by coupling the CFD model developed for wind with a structural model. This is referred to as a typical fluid–structure interaction problem. Recent advancements in algorithms and computational resources have made it possible to perform fully coupled simulations in complex wind flows and complex structures. Large Eddy simulation (LESs) of wind effects on tall buildings with and without nearby buildings are presented in [6], and wind effects on tall steel buildings are presented in [7]. Multiple turbulence models were numerically investigated for predicting wind effects on tall buildings in [8,9]. The wind effects on structures may be captured by a two-way coupled or one-way coupled simulation. The accuracy of these modeling choices depends on the flexibility of the structure. In tall building simulations, it is observed that a one-way coupled simulation is capable of capturing important characteristics sufficiently. In CWE, the wind loads evaluated from the CFD may be imposed on a finite element (FE) structural model. It is common practice to model the structure in FE software tools for evaluating the building response and, hence, for design. The FE model of a tall building comprises multiple structural elements and is an accurate representation of the structure. These complex FEM models are expensive to evaluate in many cases.

In a design scenario, it is not possible for the designer to know exact structural properties like the mass, stiffness, and damping of the structural system accurately in the early stages. It is a common norm to start with simplified models in such scenarios. Also, in wind tunnel testing, it is not possible to capture all the structural behaviors, and a common approach is to use MDoF systems. An MDoF system is the best representation of a tall building and is widely used in computational studies for evaluating the wind effects on structures. A more complex and accurate structural model may be built at a later time as more information and decisions on the structural system are made in a design scenario. However, the accuracy of these simplified models has not been not well studied in the literature. Here, in this contribution, a systematic study is carried out to quantify the effects of simplified modeling in structures under transient loads with a focus on the wind.

The current study aims to assess (dynamic) wind load in various scenarios (angle of attack, degree of turbulence) obtained from CFD simulations, which represent the input load case in the form of force time history on the structural model. A generic highrise structure (CAARC B—a well-established model in the wind engineering community proposed by the International Association of Wind Engineering—IAWE [10,11]) is chosen due to the vast literature available related to the airflow around it as well as necessary information regarding the choice and set up of the structural model. Originally, the structural model is only prescribed a certain geometry (uniform rectangular along height), mass distribution (uniformly along height), eigenfrequencies of the first three modes (weak and strong bending, torsion), and type of mode shape (linear, as the initial study way intended for HFFB in experimental wind tunnels). Additional remarks on structural damping are given for the assessment of responses. This study takes these existing prescriptions and uses them as a base model: continuous and uniform mass and stiffness distribution along the height chosen so that target parameters (eigenfrequencies) are met. The study of slight variations of this model, aiming to assess other scenarios typical for the engineering practices for such constructions, such as three intervals along the height with varying mass and stiffness distribution, is included. A model of the building with a full FEM model is presented, comparing the responses with approximate MDoF systems.

Wind loading is considered as a given input, with the variations described previously. Structural models and respective modifications will be analyzed as briefly outlined. The systematic parametric study is further enhanced with the investigation of the spatial resolution of the structural model itself. This last step aims to quantify the effect of structural detailing through the usage of various (decreasing) numbers of structural elements. It is aimed to highlight what minimum requirements related to proper analysis are needed, such that certain target response parameters (e.g., time and magnitude of maximum displacements and accelerations) are captured with sufficient accuracy. The insights are linked to modal analysis and a subsequent reduction of the number of modes in order to identify minimally relevant ones. The numerical toolchain that is developed and used permits a flexible and thorough parametric assessment of these aspects. The accuracy assessment of these simplified models can be useful for both simplified numerical modeling and the use of simplified models in wind tunnel tests.

The rest of this paper is organized as follows: Section 2 describes the methodology used for modeling and analyzing fluids and structures. Section 3 describes the details of the numerical study and results, and discusses the obtained results. Section 4 presents the conclusions and outlooks.

2. Methodology

This section focuses on the detailed methodological description and underlying assumptions for each of those modules.

2.1. Wind Load Modeling and Simulation

Wind load is modeled using CFD analysis with the open-source Kratos Multiphysics tool [12,13,14]. This involves a finite element method (FEM) formulation for flow problems based upon a variational multi scale (VMS) formulation [15]. The computational domain setup is the one presented in ref. [16], which was set up for the CAARC B building [10,11]. Tetrahedral elements are employed to discretize the fluid domain. The computational domain setup for the current simulation is illustrated in Figure 1. The figure also outlines the boundaries of the fluid domain and illustrates the applied boundary conditions.

Inlet Boundary Condition

The wind characteristics are taken care of by the inlet boundary condition. The wind effects in the atmospheric boundary layer (ABL) are modeled by decomposing the wind velocity field as a Reynolds decomposition, $V=\overline{V}+V^{\prime }$, where $\overline{V}$ is the steady mean profile component and $V^{\prime }$ is the unsteady turbulent fluctuations component. The mean wind profile, denoted as $\overline{V}$, represents the averaged contribution to the overall wind field V. This profile undergoes changes over the time scale of several hours to days, as indicated in [17]. Therefore, for the numerical simulation, the mean profile is assumed to remain constant. The turbulent fluctuations of wind $V^{\prime }$ represent the wind gusts with a time span of seconds. The gust wind speed triggers transient conditions of peak wind loading. If the structural eigenfrequencies of large structures align with frequencies in the gust-induced wind load pattern, resonant effects may occur. Separate models are employed for each term, $\overline{V}$ and $V^{\prime }$.

An exponential profile is adopted for the study for the mean wind profile. The wind velocity at any height z is given by

$$\overline{V}\left(z\right)=V_{\mathrm{ref}}{\left({\displaystyle \frac{z}{z_{\mathrm{ref}}}}\right)}^{\alpha }$$

(1)

where $z_{\mathrm{ref}}$ is the reference height and $\alpha$ is the exponent which depends on the terrain. To capture wind statistics at the location of interest, wind fluctuations $V^{\prime }(t,y,z)$ are modeled. Various numerical models are available to address this issue [18,19,20]. In this study, the well-established Mann model is utilized to simulate wind gust fluctuations [21,22]. This model, widely accepted in structural wind engineering [23,24,25,26], is known for its spectral approach in wind generation.

2.2. Structural Modeling and Analysis

The prescribed structural model of the CAARC B building is a typical cantilever with continuous mass and stiffness, constant along its height. It cannot be modeled as a full cross-section; as with such an assumption, the prescribed modal frequencies cannot all be fulfilled at the same time. These target modes and frequencies can only be achieved by a detailed structural model, typically set up for extensive analysis during design. For the current investigation, a custom parametrizable Timoshenko beam with an FEM formulation is developed and used, referred to as ParOptBeam. Two structural models were used in this study: the high-fidelity model (HFM), which is a detailed FEM model, and the ParOptBeam model, which is a low-fidelity model (LFM).

A simplified structural model for the dynamic time history analysis of tall buildings with different structural systems is presented here. The three-dimensional (3D) structure is replaced by a simplified structural model that considers both bending and shear deformations. The 3D Timoshenko beam theory is therefore used to model the equivalent structural system. Ref. [27] presented a simplified methodology for the analysis of the framed structure shear wall interaction problem. It can be seen that the Timoshenko beam model can effectively model various structural systems used for tall buildings in practice.

2.2.1. Timoshenko Beam Model

The Timoshenko beam theory is adopted to model the simplified structural system to analyze the actual 3D building. A 3D prismatic homogeneous isotropic beam element is used for the element formulation. The element considers both the shear deformation and rotational inertia (related to bending). Since the shear deformations are taken into account, the planes that were initially perpendicular to the neutral axis are no longer perpendicular to it after deformation.

A consistent mass formulation is used. Hence, both the stiffness and mass matrix depend on the relative importance of the shear deformations (relative to the contribution from bending). This ratio is given by

$$\mathrm{\Phi }={\displaystyle \frac{12EI}{G(A/\alpha )L^2}}$$

(2)

where E and G are the moduli of elasticity and shear of the material, respectively, I is the moment of inertia of the cross-section, $A/\alpha$ is the effective shear area, and L is the length of the discrete beam element. In 3D, the properties of cross-sections can change in the two principle directions; hence, two relative shear importance factors may be defined. The torsional stiffness is computed from the polar inertia of the equivalent cross-section. A Rayleigh damping model is assumed. For a dynamic problem, the governing equation of motion (EOM) is shown in (3).

The numerical structural models of the version described are set up such that the eigenfrequencies can be met. Furthermore, in the initial description of CAARC B, linear mode shapes (with linear referring to the distribution along the height and not referring to geometric/material linearity) are mentioned, as these would be typically achieved by the target study at that time involving a HFFB measurement. These distributions along the height are only possible if the model is an infinitely rigid beam along the height, and deformations are only possible at the mount (clamped edge). This could be seen as a rigid body deformation along the height permitted by springs at the fixity. These mode shapes are, however, not representative (nor realistic) of typical aeroelastic models used in wind tunnels, nor of high-rise structures modeled using computational methods. Realistically, these have mode shapes and sway deformations that lie somewhere between the theoretical limits of pure shear buildings and pure bending cantilevers and are definitely not linear. The latter would correspond to the Bernoulli/Timoshenko beam formulation, whereas the former can only be achieved with a dedicated MDoF shear model. Current modeling techniques and implementations permit the modification of the relative importance of shear. Also, the addition of point mass and stiffness entries to model the possible effect of outriggers is enabled.

2.2.2. Implementation Details of the Structural Model in ParOptBeam

The Timoshenko beam model realized in the current work (ParOptBeam [28]) is capable of carrying out a 3D analysis of tall buildings subjected to wind forces. A linear (geometric and material) model is used for the structure, and the time domain dynamic problem is solved with a generalized – α time integration scheme. Mode shapes (two sway modes and torsion) are uncoupled.

Analysis in the Modal Coordinates in the Framework

The structural solver is capable of computing time-dependent responses in the modal coordinates as well. Such computations help to reduce the time required for analysis, as one needs to solve a smaller set of equations (which comprise a series of scalar solves). This can be seen as useful if the number of DoFs of the system is very large, and if the system is mainly influenced by a few of the modes. The capabilities are implemented for the structural model. The responses computed in transformed coordinates are thereafter transferred back to the initial system of reference.

The Capability of Stepping

In many designs in practice, it so happens that the stiffness values are reduced along the height of the structure towards the top. This is realized by different zones (intervals) having changing prescribed stiffness and mass values. Constant stiffness (material and geometric) and mass (density) are assumed within an interval.

Modeling of Outriggers

In practice, outriggers are generally designed in-between changes of mass and stiffness. These are mostly used to help reduce the bending moment at the core. The mass and stiffness contributions of outriggers are added to the corresponding matrices.

Variations in the model regarding mass and stiffness considered are shown in Figure 2 and are as follows:

• Continuous—constant distribution over height;
• With intervals—varying (but constant on intervals) distribution of these over height (three intervals considered);
• With outriggers—additional inclusion of outriggers (at the two changes between intervals).

Figure 2. Main assumptions and parameters of the structural model.

Figure 2. Main assumptions and parameters of the structural model.

2.3. Coupling of Fluid and Structure

The wind flow results in nodal forces on the surface mesh of the structure, typical for the numerical approach (CFD) at hand. These fluid forces need to be transferred to the structural model (modeled using computational structural mechanics—CSM methods) in a conservative way, referring to the conservation of forces in an integral sense. Thus, the resulting forces on the surface mesh on the fluid solver side need to be the same as the acting forces on the structure during the CSM simulation. These data transfer operations can be generically seen as a mapping operation. It needs to be noted that the deformations of the structural model are not sent back to the fluid solver; as such, the wind flow CWE model acts as if we were only able to see the rigid structure throughout the total duration of the simulation.

This manner of assessing and transferring fluid forces is typical to either HFPI techniques (on a rigid model) or CFD simulations and leads to a one-way coupled fluid–structure interaction (FSI) approach, which means that only the wind flow affects the structure (due to arising forces), but structural deformations (kinematics) do not have an influence on the geometry of the flow domain considered.

The main goal is to identify the minimum amount of considered structural detailing (represented here by the minimum amount of finite elements in discretization and/or number of mode shapes used for modal reduction), which leads only to a minimal degree of variation from the full resolution model. The minimum (or acceptable) level of refinement is established based on intended applications. The EOM in spatial coordinates is displayed in Equation (3), where $x\left(t\right)$ denotes the vector of the DoFs of the generalized displacement, with time derivatives also being marked. The matrices $\left[M\right]$, $\left[B\right]$, and $\left[K\right]$ represent mass (consistent), stiffness, and damping (Rayleigh), respectively. The corresponding vector of external forces is represented by $\left\{F\right(t\left)\right\}$. The sizes of the matrices and vectors are shown using the number of DoFs n.

Generalized discrete EOM.

$${\left[M\right]}_{n\times n}{\left\{\ddot{x}\left(t\right)\right\}}_{n\times 1}+{\left[B\right]}_{n\times n}{\left\{\dot{x}\left(t\right)\right\}}_{n\times 1}+{\left[K\right]}_{n\times n}{\left\{x\left(t\right)\right\}}_{n\times 1}={\left\{F\left(t\right)\right\}}_{n\times 1}$$

(3)

Transformation into modal coordinates. The eigenvalue problem for the conservative (undamped) structure is

$${\left[K\right]}_{n\times n}{\left[\mathrm{\Phi }\right]}_{n\times n}={\left[M\right]}_{n\times n}{\left[\mathrm{\Phi }\right]}_{n\times n}{\left[\mathrm{\Lambda }\right]}_{n\times n}$$

(4)

where ${\left[\mathrm{\Phi }\right]}_{n\times n}={\left[{\left\{{\varphi }_1\right\}}_{n\times 1},{\left\{{\varphi }_2\right\}}_{n\times 1},\dots ,{\left\{{\varphi }_n\right\}}_{n\times 1}\right]}_n^T$ is the modal matrix, consisting of n modal vectors; ${\left[\mathrm{\Lambda }\right]}_{n\times n}={\left[diag\left\{{\lambda }_1,{\lambda }_2,\dots ,{\lambda }_n\right\}\right]}_{n\times n}$ is the modal stiffness matrix, with ${\lambda }_i={\omega }_i^2$ for $i=1,n$, consisting of n eigenvalues on the main diagonal. This assumption should be well accepted for civil engineering structures, which are typically highly underdamped.

Mass-normalization is used for obtaining the modal matrix ${\left[\mathrm{\Phi }\right]}_{n\times n}$, such that

$$\begin{array}{cc}\hfill {\left[\mathrm{\Phi }\right]}_{n\times n}^T{\left[M\right]}_{n\times n}{\left[\mathrm{\Phi }\right]}_{n\times n}& =\left[I\right]={\left[\overline{M}\right]}_{n\times n}\hfill \\ \hfill {\left[\mathrm{\Phi }\right]}_{n\times n}^T{\left[K\right]}_{n\times n}{\left[\mathrm{\Phi }\right]}_{n\times n}& =\left[\mathrm{\Lambda }\right]={\left[\overline{K}\right]}_{n\times n}\hfill \\ \hfill {\left[\mathrm{\Phi }\right]}_{n\times n}^T{\left[B\right]}_{n\times n}{\left[\mathrm{\Phi }\right]}_{n\times n}& =\left[\mathrm{B}\right]={\left[\overline{B}\right]}_{n\times n}\hfill \end{array}$$

(5)

where ${\left[\mathrm{B}\right]}_{n\times n}={\left[diag\left\{{\beta }_1,{\beta }_2,\dots ,{\beta }_n\right\}\right]}_{n\times n}$ is the modal damping matrix, with ${\beta }_i=2{\zeta }_i{\omega }_i$ for $i=1,n$, ${\zeta }_i$ being the damping ratio for the i-th mode.

The kinematics spatial coordinates and respective time derivatives are transformed into modal coordinates using

$${\left\{x\left(t\right)\right\}}_{n\times 1}={\left[\mathrm{\Phi }\right]}_{n\times n}{\left\{\overline{x}\left(t\right)\right\}}_{n\times 1}$$

(6)

and pre-mutiplying the resulting equation with ${\left[\mathrm{\Phi }\right]}_{n\times n}^T$, leading to

$${\left[\overline{M}\right]}_{n\times n}{\left\{\ddot{\overline{x}}\left(t\right)\right\}}_{n\times 1}+{\left[\overline{B}\right]}_{n\times n}{\left\{\dot{\overline{x}}\left(t\right)\right\}}_{n\times 1}+{\left[\overline{K}\right]}_{n\times n}{\left\{\overline{x}\left(t\right)\right\}}_{n\times 1}={\left\{\overline{F}\left(t\right)\right\}}_{n\times 1}$$

(7)

where ${\left[\overline{M}\right]}_{n\times n}$$,{\left[\overline{B}\right]}_{n\times n}$ and ${\left[\overline{K}\right]}_{n\times n}$ are the modal mass, stiffness and damping matrices (all diagonal), respectively.

It will be assumed that the following condition holds:

$${\left[\overline{B}\right]}_{n\times n}{\left[\overline{M}\right]}_{n\times n}^{-1}{\left[\overline{K}\right]}_{n\times n}={\left[\overline{K}\right]}_{n\times n}{\left[\overline{M}\right]}_{n\times n}^{-1}{\left[\overline{B}\right]}_{n\times n}$$

(8)

ensuring the fact that the damped system will possess the same mode shapes as its undamped counterpart. Consequently, the eigenvectors will be real and the equation will contain n uncoupled systems of equations, representing an SDoF-analogy for each modal component and leading to a scalar system for each, individually.

Modal reduction.

Systematically decreasing the involved modal components, one chooses $m\le n$ modes to represent a subset ${\left[\mathrm{\Phi }\right]}_{n\times m}={\left[{\left\{{\varphi }_1\right\}}_{n\times 1},{\left\{{\varphi }_2\right\}}_{n\times 1},\dots ,{\left\{{\varphi }_n\right\}}_{n\times 1}\right]}_m^T$ for $i=1,m$ of the full modal matrix. Carrying out the transformation considering only this selection of modes, the following equation holds:

$${\left[\overline{M}\right]}_{m\times m}{\left\{\ddot{\overline{x}}\left(t\right)\right\}}_{m\times 1}+{\left[\overline{B}\right]}_{m\times m}{\left\{\dot{\overline{x}}\left(t\right)\right\}}_{m\times 1}+{\left[\overline{K}\right]}_{m\times m}{\left\{\overline{x}\left(t\right)\right\}}_{m\times 1}={\left\{\overline{F}\left(t\right)\right\}}_{m\times 1}$$

(9)

representing the EOM in reduced modal space. Consequently, the response in the initial coordinate system is made up only by the contribution (mathematically a linear superposition) of a reduced subset of (decoupled) modes.

$$\left\{x(s,t)\right\}=\sum _{j=1}^{\infty }{\mathrm{\Phi }}_j\left(s\right){\overline{x}}_j\left(t\right)\to \dots \to {\left\{x\left(t\right)\right\}}_{n\times 1}={\left[\mathrm{\Phi }\right]}_{n\times m}{\left\{\overline{x}\left(t\right)\right\}}_{m\times 1}$$

(10)

It should be noted that transferring from continuous space to finite space is implicitly already a model-order reduction, where the number of modes is reduced from ∞ to n, further reduced here to m.

For large systems, determining all mode shapes and frequencies is practically prohibitive. Analytical solutions are not viable for such cases. Thus, numerical methods (such as the Rayleigh–Ritz) exist, and they approximate a certain subset of eigenforms and values, mostly starting at the lowest frequency (in civil/structural engineering applications). Similarly, experimental mode shape analysis only enables the determination of a limited number of lower-frequency modes. The result is, from either procedure, that one only has a limited subset of eigenforms corresponding to the lower eigenfrequencies. The current study uses a parametric FEM formulation of the Timoshenko beam, such that all eigenforms and values are assessed for a (relatively) highly detailed spatial model. The spatial and modal resolution is systematically reduced such that the effect of a lower number of FE elements or the mode truncation can be properly measured.

3. Numerical Study

The time history of the forces acting on the structure is recorded level-wise. Figure 3 represents how the structure model is split into 60 regions along the height, such that a series of force components (six per floor: three forces and three moments in a body-attached coordinate system) are made available with the intention of being transferable to the structural model for structural analysis. This defines the highest spatial resolution, meaning that at the highest detailing level, 60 floors along the height are taken into account, with six DoFs at each floor and corresponding force/moment components. Due to the time settings (total time, time step) of the computational wind flow simulation, the time resolution is set to 50 steps/s, leading to 30,000 of total time instances for 10 min, which defines the temporal resolution. The total simulation time of 10 min is deemed sufficient to capture naturally turbulent wind loading effects on structures at full scale.

The reduction of the forces from each point on the surface mesh of the structure is similar in manner to what would be available from HFPI experimental wind tunnel techniques. Forces and moments are reduced to a center point, which represents the geometrical center of each fictitious floor of the considered structural model. Thus, the resulting nodal forces (on mesh nodes in flow-attached coordinates) are part of the solution of the CFD solving procedure. These values are transferred onto the structural model, which is reduced to the centers of floors and transformed into the body-attached coordinate system (corresponding to the principal axes of inertia) of the building.

3.1. Numerical Study I: Comparison of a Detailed FEM Model for Structures with Low-Fidelity ParOptBeam Model

In this study, a detailed CFD model for wind is coupled to a detailed FEM model and compared with the parametric optimizable beam (ParOptBeam) model. The same wind forces from the CFD model are used for both models.

The high-fidelity model (HFM) for the case study was developed to conduct a detailed time series analysis, focusing on the wind effects on the CAARC building. GiD was used during the preprocessing stage to set up the model, which included defining the geometry, assigning structural elements, and generating a finite element mesh. The material properties were considered linear-elastic, as non-linear behavior was deemed excessive for assessing operational performance. The modeling of the structural elements is summarized as slabs and walls, which were modeled as thick shell elements, beams and columns as Timoshenko beam elements, and supports, which were modeled as pinned fixities at the base.

Geometrical non-linearities, which impact the amplification of structural responses in tall and slender buildings, were also incorporated into the HFM. Although these effects can be ignored when displacements are minimal, it is advisable to account for them.

To account for geometrical non-linearities, the beam and shell elements were modeled using the co-rotating theory [29], and the Newton–Raphson method was applied in the analysis. The co-rotating approach references elements relative to a local coordinate system rather than a global one, where total displacements and rotations are considered. This method decomposes the element’s overall motion into two components: rigid-body motion and deformable motion. The inclusion of shear deformations is particularly significant for deeper beams and thicker shell elements. Thick shells may be ideal for applications like foundation plates, while thin elements may be more suitable for slabs. Incorporating shear deformation provides a more realistic approach to the HFM.

To assess wind effects on a tall building, it is crucial to examine the first three modes, as illustrated in Figure 4. This focus is due to the fact that wind forces in tall buildings are not commonly excited by higher-order modes since wind forces do not exert force in one direction up to a certain height and then shift to another direction for the remainder of the structure.

A modal analysis was conducted on the HFM to explore the building’s dynamic behavior. The results of the eigenvalue analysis are depicted in Figure 4. The first and second eigen-mode is characterized mainly by a dominant bending effect, with minimal torsional impact, representing a nearly pure bending scenario. In contrast, the third eigen-modes exhibit translation in the y-direction combined with torsional rotation along the building’s Z-axis.

Low-Fidelity Model

After creating the HFM of the building, a low-fidelity model (LFM) consisting of beam elements can be established by considering the characteristics of the building. Dynamic simulations with the LFM require less computational time to complete. To create the LFM, it is necessary to ensure that this model has attributes similar to those of the detailed building model, such as eigen-frequencies and mode shapes. Timoshenko beam theory was used, as elaborated in Section 3.

The top-floor accelerations are presented for two angles of the wind in

Table 1. Comparison of HFM and LFM—acceleration at the top for 2.5% damping and 0° angle of wind.

	Mean	Std	Mean + 3Std	Max	99th Percentile
HFM
$A_x$ [m/s2]	$1.67\times {10}^{-4}$	$1.32\times {10}^{-1}$	$3.96\times {10}^{-1}$	$4.34\times {10}^{-1}$	$3.04\times {10}^{-1}$
$A_y$ [m/s2]	$-1.42\times {10}^{-4}$	$1.17\times {10}^{-1}$	$3.52\times {10}^{-1}$	$3.73\times {10}^{-1}$	$2.73\times {10}^{-1}$
$A_{r}z$ [rad/s2]	$-2.34\times {10}^{-7}$	$1.30\times {10}^{-3}$	$3.90\times {10}^{-3}$	$4.96\times {10}^{-3}$	$3.22\times {10}^{-3}$
LFM
$A_x$ [m/s2]	$1.60\times {10}^{-4}$	$1.41\times {10}^{-1}$	$4.25\times {10}^{-1}$	$4.21\times {10}^{-1}$	$3.32\times {10}^{-1}$
$A_y$ [m/s2]	$-2.04\times {10}^{-5}$	$1.14\times {10}^{-1}$	$3.43\times {10}^{-1}$	$3.65\times {10}^{-1}$	$2.69\times {10}^{-1}$
$A_{r}z$ [rad/s2]	$-5.15\times {10}^{-7}$	$1.30\times {10}^{-3}$	$3.90\times {10}^{-3}$	$4.39\times {10}^{-3}$	$3.23\times {10}^{-3}$

and

Table 2. Comparison of HFM and LFM—acceleration at the top for 2.5% damping and 90° angle of wind.

	Mean	Std	Mean + 3Std	Max	99th Percentile
HFM
Ax [m/s2]	$-1.75\times {10}^{-4}$	$1.84\times {10}^{-1}$	$5.51\times {10}^{-1}$	$5.41\times {10}^{-1}$	$4.22\times {10}^{-1}$
Ay [m/s2]	$3.39\times {10}^{-5}$	$6.40\times {10}^{-2}$	$1.92\times {10}^{-1}$	$2.24\times {10}^{-1}$	$1.53\times {10}^{-1}$
Arz [rad/s2]	$-4.28\times {10}^{-7}$	$2.44\times {10}^{-3}$	$7.33\times {10}^{-3}$	$9.06\times {10}^{-3}$	$5.92\times {10}^{-3}$
LFM
Ax [m/s2]	$-1.17\times {10}^{-4}$	$1.85\times {10}^{-1}$	$5.56\times {10}^{-1}$	$5.31\times {10}^{-1}$	$4.10\times {10}^{-1}$
Ay [m/s2]	$2.40\times {10}^{-5}$	$5.17\times {10}^{-2}$	$1.55\times {10}^{-1}$	$1.50\times {10}^{-1}$	$1.20\times {10}^{-1}$
Arz [rad/s2]	$-8.56\times {10}^{-7}$	$2.45\times {10}^{-3}$	$7.35\times {10}^{-3}$	$8.66\times {10}^{-3}$	$5.95\times {10}^{-3}$

. Time statistics of the two models are compared in the table. Mean, standard deviation (std), mean + 3 sd, maximum, and the 99% values are compared for the two models for two angles of the incoming wind. The base reactions are presented in

Table 3. Comparison of HFM and LFM—base reactions for 2.5% damping and 0° angle of wind.

	Mean	Std	Mean + 3Std	Max	99th Percentile
HFM
Fx [MN]	$8.94\times {10}^0$	$4.11\times {10}^0$	$2.13\times {10}^1$	$2.49\times {10}^1$	$1.92\times {10}^1$
Fy [MN]	$-5.46\times {10}^{-2}$	$3.67\times {10}^0$	$1.09\times {10}^1$	$1.41\times {10}^1$	$8.78\times {10}^0$
Fz [MN]	$-3.71\times {10}^2$	$4.59\times {10}^1$	$-2.34\times {10}^2$	$-1.73\times {10}^{-1}$	$8.47\times {10}^1$
Mx [MNm]	$3.25\times {10}^{-1}$	$3.26\times {10}^2$	$9.77\times {10}^2$	$9.70\times {10}^2$	$8.14\times {10}^2$
My [MNm]	$8.72\times {10}^2$	$3.60\times {10}^2$	$1.95\times {10}^3$	$1.85\times {10}^3$	$1.66\times {10}^3$
Mz [MNm]	$2.84\times {10}^{-1}$	$1.62\times {10}^1$	$4.90\times {10}^1$	$5.63\times {10}^1$	$4.01\times {10}^1$
LFM
Fx [MN]	$8.85\times {10}^0$	$3.21\times {10}^0$	$1.85\times {10}^1$	$1.79\times {10}^1$	$1.57\times {10}^1$
Fy [MN]	$1.21\times {10}^{-2}$	$2.81\times {10}^0$	$8.45\times {10}^0$	$8.12\times {10}^0$	$6.59\times {10}^0$
Fz [MN]	$-3.67\times {10}^2$	$4.60\times {10}^1$	$-2.29\times {10}^2$	$0.00\times {10}^0$	$-8.09\times {10}^1$
Mx [MNm]	$9.93\times {10}^{-1}$	$3.11\times {10}^2$	$9.34\times {10}^2$	$9.70\times {10}^2$	$7.71\times {10}^2$
My [MNm]	$8.53\times {10}^2$	$3.60\times {10}^2$	$1.93\times {10}^3$	$1.93\times {10}^3$	$1.64\times {10}^3$
Mz [MNm]	$3.27\times {10}^{-1}$	$1.34\times {10}^1$	$4.06\times {10}^1$	$4.55\times {10}^1$	$3.24\times {10}^1$

and

Table 4. Comparison of HFM and LFM—base reaction for for 2.5% damping 90° wind.

	Mean	Std	Mean + 3Std	Max	99th Percentile
HFM
Fx [MN]	$3.04\times {10}^{-2}$	$4.01\times {10}^0$	$1.21\times {10}^1$	$1.73\times {10}^1$	$5.04\times {10}^0$
Fy [MN]	$-4.66\times {10}^0$	$1.80\times {10}^0$	$7.25\times {10}^{-1}$	$6.88\times {10}^{-1}$	$-2.46\times {10}^0$
Fz [MN]	$-3.71\times {10}^2$	$4.59\times {10}^1$	$-2.34\times {10}^2$	$-1.73\times {10}^{-1}$	$-3.78\times {10}^2$
Mx [MNm]	$4.70\times {10}^2$	$1.83\times {10}^2$	$1.02\times {10}^3$	$9.73\times {10}^2$	$7.16\times {10}^2$
My [MNm]	$1.96\times {10}^0$	$4.22\times {10}^2$	$1.27\times {10}^3$	$1.34\times {10}^3$	$5.52\times {10}^2$
Mz [MNm]	$7.72\times {10}^{-1}$	$2.60\times {10}^1$	$7.88\times {10}^1$	$1.10\times {10}^2$	$3.37\times {10}^1$
LFM
Fx [MN]	$9.17\times {10}^{-4}$	$3.45\times {10}^0$	$1.03\times {10}^1$	$1.08\times {10}^1$	$4.55\times {10}^0$
Fy [MN]	$-4.64\times {10}^0$	$1.63\times {10}^0$	$2.50\times {10}^{-1}$	$2.95\times {10}^{-2}$	$-2.64\times {10}^0$
Fz [MN]	$-3.67\times {10}^2$	$4.60\times {10}^1$	$-2.29\times {10}^2$	$0.00\times {10}^0$	$-3.76\times {10}^2$
Mx [MNm]	$4.58\times {10}^2$	$1.77\times {10}^2$	$9.88\times {10}^2$	$9.59\times {10}^2$	$6.97\times {10}^2$
My [MNm]	$1.88\times {10}^0$	$4.16\times {10}^2$	$1.25\times {10}^3$	$1.35\times {10}^3$	$5.44\times {10}^2$
Mz [MNm]	$8.72\times {10}^{-1}$	$2.42\times {10}^1$	$7.36\times {10}^1$	$9.54\times {10}^1$	$3.15\times {10}^1$

. It can be seen that the approximate model can capture the base reactions and the top-floor acceleration with reasonable accuracy. The mean of the top-floor acceleration is almost zero, and extreme statistics are of importance. Further, the time series of accelerations and base reactions are presented in Figure 5, Figure 6 and Figure 7. The computations were performed on the SuperMUC-NG system equipped with an Intel Xeon Platinum 8174 processor using 8 nodes and each node has 48 cores per node (Intel Corporation, Santa Clara, CA, USA). The computational time for both analyses was compared, and the high-fidelity model required approximately 13,824 CPU hours, whereas the low-fidelity model completed the analysis in approximately 144 CPU hours. The LF model demonstrated a significant computational saving, using only 1% of the resources required by the HF model.

Table 5. Comparison of the HFM and LFM—summary.

Case	Quantity	HFM	LFM	Difference in % from HFM
Table 1	Max $A_y$ [m/s2]	0.373	0.365	2.14
Table 2	Max $A_x$ [m/s2]	0.541	0.531	1.84
Table 3	Mean $F_x$ [MN]	8.94	8.85	1.01
Table 3	Mean $M_y$ [MNm]	872	853	2.18
Table 4	Mean $F_x$ [MN]	−4.66	−4.64	0.43
Table 2	Mean $M_y$ [MNm]	470	458	2.55

compiles a few of the QoIs that are interesting to each load case, and the % of the difference is presented from HFM. In both acceleration and base reactions, the low-fidelity model was able to capture the responses accurately to less than 3%. In almost all cases, the LFM underpredicts the responses. However, this difference is expected and is considerably justified considering the computational saving it presents.

3.2. Numerical Study II: Comparison of Responses with Reduced Degrees of Freedom

The low-fidelity model, which was elaborated and proven to be useful in the numerical Study I, is used here in this study for further parametric studies. A systematic parametric study is carried out in order to evaluate various modeling choices for the low-fidelity model elaborated above. One of the major points looked into was how many DoFs need to be present for the accurate prediction of the structural responses. The number of nodes or elements is systematically reduced in this study. The top-floor displacements, acceleration, and base reactions are the quantities of interest. Sixty elements are considered to comprise the highest resolution model. This resolution was used in the previous study to validate with a high-fidelity model. The major focus of the section is to identify the effect of the number of elements used for FEM modeling and the number of mode shapes considered in modal analysis for a reduction in various quantities of interest.

Outline of the Parametric Study

The parametric study is thought out such that the effect of variations in

• Load scenario:

  -

  Wind flow angle (0, 90)

  -

  Presence of turbulence (smooth exponential inlet profile or with fluctuations in time and space)

• Structural modeling: type of considered models

  -

  Damping characteristics

  -

  Mass and stiffness distribution along the height

  -

  Detailing the level of the model represented by spatial resolution, achieved by

  *

  Varying the number of finite elements used (in physical space - spatial reduction)

  *

  Systematic modal reduction (in modal space)

are considered.

The measured quantities of interest are

• Kinematics

  -

  At the top: considering the occurrence (time) and magnitude of maximum

  *

  Displacement

  *

  Acceleration

• Base moments: at the bottom (fixity) considering the occurrence (time) and magnitude of maximum
• Base shear at the bottom (fixity).

Numerous structural parameters and modeling parameters (like the number of elements considered) are varied, and the results are analyzed and discussed. The simulations are carried out for 10-min winds, and the structural response time history is evaluated. In many cases, the chosen representative values for structural responses are those of the absolute maximum. These are reported for comparative purposes.

The time series of responses, either displacement or base reactions for the base scenario, are presented in Figure 8 and Figure 9 for cases with and without incoming turbulence in the wind. The following figures compare the top-floor accelerations and base reactions with and without damping. Also, the variation in damping is presented. It can be seen that the response reduction is prominent in the case of a structure with damping. It is uncommon for the structures to have zero damping, as shown in the figure. However, this is presented for a reference.

For the spatial evaluation from 15 elements onward (for some cases even starting from 3 elements), maximum convergency is observed for top-floor displacements and accelerations. This is tabulated in

Table 6. Spatial reduction summary for no turbulence, damping ratio $\zeta$ = 0.01, continuous, 0° angle of wind.

No. of Elements	${\mathit{Disp}}_{\mathit{max}}$	${\mathit{Disp}}_{\mathit{RMS}}$	${\mathit{Acc}}_{\mathit{max}}$	${\mathit{Acc}}_{\mathit{RMS}}$
1	0.1565	0.1081	0.2323	0.0210
2	0.2056	0.1452	0.0745	0.0278
3	0.2364	0.1572	0.1576	0.0524
15	0.2238	0.1588	0.1319	0.0429
30	0.2265	0.1593	0.1386	0.0463
60	0.2234	0.1591	0.1307	0.0446

. Similar observation can be made for

Table 7. Spatial reduction summary for with turbulence, damping $\zeta$ = 0.01, continuous, 0° angle of wind.

No. of Elements	${\mathit{Disp}}_{\mathit{max}}$	${\mathit{Disp}}_{\mathit{rms}}$	${\mathit{Acc}}_{\mathit{max}}$	${\mathit{Acc}}_{\mathit{rms}}$
1	0.5884	0.2233	0.4389	0.1855
2	0.6804	0.2922	0.5340	0.2292
3	0.8332	0.3266	0.8698	0.3220
15	0.6785	0.3099	0.5926	0.2549
30	0.8567	0.3201	0.9044	0.3015
60	0.7511	0.3061	0.6936	0.2497

. The tabulated and compared quantities are the maximum and the RMS of displacements/accelerations at the top.

This part of this study focuses on the evaluation of an extensive parametric study, showing a clear summary of the comprehensive data available. The values on the vertical (y) axis are normalized with the values of the highest level of spatial resolution. The effects of

• Wind flow conditions

  -

  Angle of attack

  -

  Presence of natural incoming turbulence

• Structural model

are plotted for the top-floor displacement in all the above cases in Figure 10, Figure 11, Figure 12 and Figure 13. It can be observed that for the cases with and without turbulence, the quantity of interest shows similar characteristics. This may be due to the normalization with respect to the highest resolution. As commented earlier, the resolution of 15 spatial elements is accurate in representing the maximum and RMS of displacement at the top. The amount of structural damping plays an insignificant role compared to the spatial resolution of modeling.

A similar study was conducted on the model results and is plotted in Figure 14, Figure 15, Figure 16 and Figure 17. Interestingly, it can be seen that in the modal domain, the number of modes required to capture the mean and RMS of acceleration are very low. As seen in

Table 8. Model reduction summary for no turbulence, damping $\zeta$ = 0.025, continuous, 90° angle of wind.

No. of Modes	${\mathit{Disp}}_{\mathit{max}}$	${\mathit{Disp}}_{\mathit{RMS}}$	${\mathit{Acc}}_{\mathit{max}}$	${\mathit{Disp}}_{\mathit{RMS}}$
3	0.1997	0.1249	0.1722	0.0549
6	0.1978	0.1232	0.1723	0.0549
12	0.1979	0.1233	0.1724	0.0549
18	0.1979	0.1233	0.1724	0.0549
90	0.1979	0.1233	0.1724	0.0549
180	0.1979	0.1233	0.1724	0.0549

even with three modes, the responses can be captured with a 99% accuracy. This is similar to what is being used for wind tunnels with approximate modeling. However, it is important to note that the accuracy of the results depends on the accuracy of these mode shapes. It is important to obtain these shapes and eigen-frequencies correctly, as seen in Study I. It is also seen that from the table, the mean can be approximated by the lower mode considerations somewhat accurately. However, the RMS needs more modes. This observation is valid for more accelerations compared to displacements. Some additional figures that support this observation is shown in additional figures in Appendix A.

4. Conclusions

This study emphasizes the significance of approximate structural modeling in the early design phases of high-rise buildings with a focus on natural wind load conditions. The specific conclusions from the work are

• These early-phase structural models often exhibit inherent limitations due to simplifying assumptions. These limitations can greatly influence the accuracy of the measured responses to dynamic loads, especially wind loads. As a designer, one needs to be aware of the limitations of respective modeling assumptions.
• The parametric study reveals that low-fidelity models are a good representation of high-fidelity models, provided that the frequencies of the modes are matched.
• This study demonstrates that reduced structural models, specifically lumped-mass models with three lumped masses, can effectively capture the behavior of high-rise structures under wind loads, provided that the frequencies and mode shapes are matched.
• These models can approximate the behavior of complete systems with a certain degree of accuracy, quantified as around 11% for displacements and 20% for accelerations. Moreover, these LF models are fast and easy to evaluate.

These models are particularly beneficial in the initial design stages, where rapid, cost-effective, and moderately accurate evaluations are required. They are also useful for computationally intensive tasks, like optimization, uncertainty quantification, optimization under uncertainties, and the design of damping devices to reduce the structural acceleration responses with supplemental damping.