Influence of wind angle incidence and architectural elements on the

external pressure coefficient of hyperbolic paraboloid roofs

GUILHERME S. TEIXEIRA, MARCO D. DE CAMPOS

Institute of Exact and Earth Sciences

Federal University of Mato Grosso,

Av. Valdon Varjão, 6390, Barra do Garças, 78605-091, Mato Grosso

BRAZIL

Abstract: In the study of wind loads in buildings, the aerodynamics of roofs with parabolic shapes, which cause

complex pressure distributions due to their sensitivity to wind, are often omitted and neglected by several codes

and norms. In this way, computer simulations are a viable and reliable alternative. Here, wind action was

considered in an innovative project composed of parabolic and circumferential generatrices: the Church of

Saint Francis of Assisi. Designed by Brazilian architect Oscar Niemeyer in Belo Horizonte, Brazil, two

paraboloid vaults and three circular arches of reinforced concrete composed its structure. This work generated

great international recognition for the architect after 1943, as the design of the roofs did not require walls. For

geometry modeling, Autodesk AutoCAD software was adopted, and the models were considered in a control

volume. The simulations were performed using Ansys Workbench software and the RNG K-Epsilon turbulence

model. The wind speed at different heights was calculated using the Power-law approximation. A basic wind

speed of 30 m/s was adopted, and the mesh used was composed of tetrahedrons. To validate the methodology,

different models with hyperbolic-paraboloid roofs from the literature were considered. In addition, the

visualization of the flow around the geometry from the streamlines, the wind profile, and the analysis of the

isobaric lines of the external pressure coefficients for different directions of incidence and architectural

elements that make up the building were presented.

Key-Words: Wind action, parabolic roof, Ansys, pressure coefficients, computational method.

Received: July 28, 2021. Revised: April 22, 2022. Accepted: May 17, 2022. Published: June 30, 2022.

1 Introduction

Widely used in engineering and architecture are

hyperbolic paraboloid roofs, especially in buildings

with large spans in which the rooftop and wall

constitute a single element. The arch axis is

composed of a parabola, allowing for the

construction of large spaces. This type of building is

a specific class of laminar structures with double

curvature that allows the design to carry forces

either solely or predominantly through membrane

forces such as uniform tension, compression, and

shear through the thickness of the shell [1]. Arched

roof structures often have a low permanent load and

are wind sensitive with a small payload capacity [2].

Using a parabolic roof as a roof has the advantages

of a harmonious shape, satisfactory strength

performance, and easy construction.

However, some restrictions limit the use of these

structures: the complexity of this type of project and

the absence of mandatory norms for the structural

project in the most different international codes.

Thus, numerical and experimental experiments are

the only way to study and investigate the wind-

structure interaction in this geometry [3].

Among these, Rizzo and Sepe [4] explored the

possibility of defining static pressure fields capable

of reproducing the dynamic displacements of the

cable network based on experimental wind tunnel

tests on hyperbolic paraboloid roofs [5] and

simplified pressure maps evaluated in Rizzo et al.

[6]. The tests were carried out in a CRIACIV

boundary layer wind tunnel in Prato, Italy, to

determine the pressure fields for various angles of

attack of the incident wind in different models of

hyperbolic paraboloid roofs. Xu et al. [7]

investigated the mechanical characteristics of the

open hyperbolic-parabolic membrane structure

under a wind load and the influences of wind

directions and speeds on the mean wind pressure

distribution. The wind load shape coefficients of

constructions with an open hyperbolic-parabolic

membrane were obtained from a series of numerical

calculations and compared with values in the

literature. Rizzo and Demartino [8] proposed a

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modification of mean pressure coefficients on

hyperbolic paraboloid roofs. Singular value

decomposition (SVD) on data from wind tunnel

tests applied to eight different geometries of

buildings covered was used to obtain the pressure

coefficients. Three wind incidence angles, 45° and

parallel to downward and upward cables, namely, 0°

and 90°, respectively, were investigated.

The mean pressure coefficients were modified using

the superpositions of pressure for all eight

geometries. To estimate the wind action and the

vertical displacements of a cable net by FEM

analyses, symmetrical and asymmetrical pressure

coefficient modes were adopted. The results indicate

that these load combinations allow for capturing

large downward and upward displacements not

accurately predicted using mean experimental

pressure coefficients. Rong et al. [9] numerically

simulated the wind pressure distribution on the roof

of Wanda Ski Park, located in Harbin, China,

comparing the mean wind pressure distribution data

with the results of existing tests in wind tunnels.

Based on the total pressure coefficients, they

proposed an evaluation method where only the roof

geometry and the wind incidence angle need to be

known.

Many architects have used this type of

construction, for example, in the restaurant of Ciutat

de Les Arts I Les Ciències in Valencia, Spain, and

the Restaurante Los Manantiales in Xochimilco,

Mexico, designed by Felix Candela. Additionally,

Bosjes Chapel by architect Coetzee Steyn in South

Africa. In Brazil, one of the most representative

examples is the Church of Saint Francis of Assisi by

Oscar Niemeyer in Pampulha Modern Ensemble,

Belo Horizonte, Brazil. The inclination of the roof

and the repetition of a uniform motif on the main

floor integrate itself magnificently with the

simplicity and lack of pretension of already known

buildings. Far from being a pastiche, the building

maintains its contemporaneity and offers a plastic

form. Its structure is composed of reinforced

concrete arches forming two parabolic domes (Fig.

1). The use of this form allowed a single element to

form the roof and walls. The bell tower and the

marquee at the entrance appear as independent

structures (Fig. 1).

In this work, the pressure coefficient with

numerical tests on different geometries of

hyperbolic paraboloid roofs was investigated

considering different wind angles of attack as well

architectural element influences (the bell tower and

marquee).

Fig. 1 Aerial photo of the Church of Saint Francis of

Assisi [10]

2 Methodology

In this work, meshes and postprocessing were

performed with Ansys Workbench software, and the

simulations took place with the CFX solver. For

geometry modeling, Autodesk AutoCAD software

with dimensions proposed by Macedo [11] was

used, as shown in Fig. 2a. According to Franke et al.

[12], a control volume was used for low-rise

buildings (i.e., H~B~L), whose adopted dimensions

were as follows: length of 5H+L+15H, width of

5H+B+5H, and height of H+5H, dependent on the

building height of (H = 9.16 m), length L of the

building in the flow direction and width B (Fig 2c).

For the local refinement of the mesh, an

influencing body with the following dimensions was

used (Fig 2b): L for the length of the downstream

region and half of this length upstream and, in

addition, twice this dimension for the width and

height [13].

The wind speed was estimated using the Power-

law approximation with a height of 10 m.

Here, an unstructured mesh composed of

tetrahedrons with curvature and capture proximities

and three levels of refinement was used. The first

level defines the dimensions of the elements in the

fluid domain. The second and third levels,

respectively, characterize the refinement of local

(with the body of influence) and geometric faces

that intercept the flow. In terms of advection, a

High-resolution scheme was chosen in all

applications [14].

(a)

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(b)

(c)

Fig. 2 (a) Dimensions and architectural components

of the Church of Saint Francis of Assisi, (b)

geometry and control volume, (c) geometry and

body of influence

Using the residual root mean square (RMS)

between subsequent iterations of a variable over all

the domain's volumes, it is possible to estimate the

iteration error present in codes that use iterative

solvers. The convergence criterion for the

simulations, in double precision, was defined as a

root mean square residual target value of 1×10-4.

The maximum and minimum numbers of iterations

were 400 and 500, respectively.

Using tetrahedral volumes, discretization was

performed. In this case an unstructured mesh was

used, considering its easy construction and that CFX

in Ansys Workbench employs an unstructured code.

Furthermore, the "capture curvature and proximity"

tool, activated during all simulations, generated an

adaptive process of the mesh elements close to the

edges and curves of the adopted geometries. As a

result, the stretching and compression of the

tetrahedral mesh were avoided for high gradient

regions. Finally, this resulted in a small truncation

error [14].

To solve the governing equations of problems

involving Computational Fluid Dynamics (CFD)

and transform complex equations into algebraic

equations, discretization is used. Faced with this,

Franke et al. [14] recommend high-order schemes

for discretizing advective terms. In this work, the

advection scheme and numerical turbulence were

high resolution. Table 1 depicts the boundary

condition details.

The convergence analysis was performed

according to the following criteria. The first is

related to the residual RMS of the energy, mass,

momentum, and additional turbulence equations due

to the RNG K-Epsilon model. The second was the

monitoring of mass conservation, given by the

IMBALANCE monitor, which showed consistent

values (<1%) according to [15].

Table 1. Boundary conditions and nondimensional

parameters.

Condition Parameters

Method of mesh Tetrahedron

Capture curvature and proximity On

Reference pressure 101325 [Pa]

Air temperature 25º [C]

Turbulence intensity Medium (5%)

Flow regime Subsonic

Inlet U/U

ref

= (Z/Z

ref

)

α

Relative pressure of outlet 0 [Pa]

Wall (terrain) Rough wall

Model wall roughness Smooth wall

Roughness Application 1 0.0025 [m]

Applications 2 and 3 0.01 [m]

Reference

height

Application 1 0.1 [m]

Applications 2 and 3 10 [m]

Wind

speed

Application 1 16.7 [m/s]

Applications 2 and 3 30 [m/s]

Advection scheme High resolution

Turbulence numeric High resolution

3 Numerical applications

Application 1 (Validation): For the validation of

the methodology, the external wind pressure results

on hyperbolic paraboloid roofs for three wind

directions were calculated. Hereafter, the results

were compared with [4].

The model dimensions are deﬁned in Fig. 3 and

reported in Table 2 according to [4]; f1 and f2 denote

the sag of the cable (upward curvature) and of the

stabilising cable (downward curvature),

respectively, H and h are the maximum and

minimum height of the roof on the ground,

respectively, and L is the dimension of the plan

sides.

Table 3 shows the computational mesh applied to

the domain.

Parameters such as the element quality,

skewness, and orthogonal quality were estimated

with special attention. The element quality describes

the relationship between element area and border

length (with recommended values close to 1).

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Table 2. Dimensions of sample cases.

Sample H

[m]

h

[m]

f1

[m]

f2

[m]

L

[m]

1 0.2133 0.1333 0.0267 0.0533 0.8000

2 0.2666 0.1333 0.4440 0.0890 0.8000

Table 3. Results for Models 1, 2, 3, 4, 5 and 6 with their respective quality metrics.

Model 1 2 3 4 5 6

Geometric configuration 1 1 1 2 2 2

Wind direction 0º 45º 90º 0º 45º 90º

Element size in the fluid domain

(m)

0.07 0.1 0.08 0.1 0.1 0.1

Element size in the body of

influence (m)

0.035 0.05 0.04 0.05 0.05 0.05

Element size on geometry faces

(m)

0.005 0.01 0.005 0.01 0.01 0.02

Nodes 465186 190999 402417 177443 242653 126447

Elements 2571471

1066180

2207163

984627 1363700

712401

Element quality (average) 0.84401 0.84735 0.84260 0.84623 0.84851 0.85038

Skewness (average) 0.21802 0.21293 0.22013 0.21448 0.21095 0.20805

Orthogonal quality (average) 0.78080 0.78593 0.77867 0.78435 0.78792 0.79083

Fig. 3 Geometric characteristics of the model.

In turn, the skewness characterizes the proximity

of the ideal geometry (in this case, the tetrahedron),

the mesh cells, or faces (with recommended values

between 0 and 0.5). Finally, orthogonal quality

defines the element's orthogonality (with

recommended values close to 1) [16].

To model the physical conditions, the wind

profile was defined using the Power-law

approximation for a reference height equal to 0.1 m,

an exponent equal to 0.233, and a wind speed of

16.7 m/s, and for the terrain, the roughness adopted

a value of 0.0025 m.

In general, the simulations showed good

agreement in the distribution of the isobaric lines

with the experimental samples. Considering a

precision house, the best concordance, in absolute

values, occurred in Model 1 (Fig. 4). The highest

discrepancy occurred at the midpoint of the left edge

in Model 2, with a value of 0.4. The color hue

represents the pressures that act on the surfaces

corresponding to the pressure coefficient ranges:

warm colors represent the overpressure regions, and

the cold colors represent the suction regions

indicating the detachment of the flow in the corners

of the lateral faces of the geometry before reaching

the cover. The results are presented in Table 4.

Then, to determine significant differences

between the present work results and the literature,

the T-test was used, considering a null hypothesis

that the means are not different. Thus, considering a

one-tailed distribution, p-value=0.42 was obtained

for a critical t=1.72. As 0.42<1.72, it was possible to

conclude that the difference between the mean

values of Cpe is insignificant.

Application 2 (External pressure coefficients

for Church of Saint Francis of Assisi): Here, the

external pressure coefficients of the Church of Saint

Francis of Assisi were acquired considering eight

wind angles.

According to [17], a wind speed equal to 30 m/s

was adopted. Thus, the directions of 0°, 45°, 90°,

135°, 180°, 225°, 270°, and 315° (Fig. 5) were

simulated in different domains, and the data are

presented in Table 5.

Considering an average height of the surrounding

obstacles of 10 m, the exponent is equal to 0.25 for

the power-law parameter [18]. The results showed

how the pressures act on the structure, emphasizing

paraboloid concrete covers (Fig. 6).

Directions 0° and 180° (Models 7 and 11) were

those in which the fluid showed detachment from

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the building at the corners of the main facades (Fig.

7), generating mostly suction zones on the roofs.

The case at 0° presented a more intense peak in the

main canopy bark (Cpemin= -0.72).

(a) [4]

(b) Present work

Fig. 4 Cpe contours for Model 1.

Table 4. Pressure coefficients comparison.

Model 1 2 3

Cpe

max

Cpe

min

Cpe

max

Cpe

min

Cpe

max

Cpe

min

[4] -0.2 -1.3 -0.2 -1.6 -0.8 -1.4

Present

work

-0.2 -1.3 -0.3 -2.0 -0.7 -1.4

Difference

0.0 0.0 0.1 0.4 0.1 0.0

Model 4 5 6

Cpe

max

Cpe

min

Cpe

max

Cpe

min

Cpe

max

Cpe

min

[4] -0.1 -0.9 -0.2 -1.8 -0.1 -1.2

Present

work

-0.1 -1.0 0.0 -2.1 -0.1 -1.3

Difference

0.0 0.1 0.2 0.3 0.0 0.1

Fig. 5 Different angles of incidence of the wind in

the Church of Saint Francis of Assisi.

For cases in which the wind directly or partially

impinges on the parabolic roofs, the building

presented suction peaks in the inflection region of

the concrete shell and a clear separation between the

suction and overpressure regions. In terms of higher

suctions, Model 4 (90°) stood out with Cpemin= -1.86

(Fig. 8).

Fig. 6 Maximum and minimum Cpe for different

angles of incidence of the wind (in absolute values).

Despite the significant similarity in the isobaric

lines, no axes of symmetry were observed in the

inclined roofs for cases of direct or partial incidence

of the fluid.

(a)

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(b)

Fig. 7 Streamlines in a longitudinal plane for (a) Model 7 and (b) Model 11.

Table 5. Results for Models 7, 8, 9, 10, 11 and 12 with their respective quality metrics.

Model Wind

direction

Nodes Elements Element

quality

(average)

Skewness

(average)

Orthogonal

quality

(average)

7 0º 118465 666741 0.82183 0.24691 0.75202

8 45º 147195 834265 0.81955 0.24946 0.74948

9 90º 109654 615181 0.82166 0.24747 0.75144

10 135º 140317 793734 0.82035 0.24854 0.75039

11 180º 119801 673748 0.82199 0.24660 0.75232

12 225º 144692 819670 0.81924 0.24989 0.74905

13 270º 123124 694732 0.82107 0.24804 0.75089

14 315º 138645 783688 0.82025 0.24876 0.75018

Application 3 (Architectural elements

influence): Architectural and constructive elements,

such as sun visor devices and projection in the

structure, are relevant factors in the thermal comfort

study and natural ventilation in buildings and their

influence in this type of analysis. However, they can

be neglected during the pressure coefficient

analysis, going through simplifications.

This application analyzed the bell tower and

marquee influence (E and F in Fig. 2a) present in

the church architecture. The bell tower has the shape

of an inverted pyramid trunk and contributes to

supporting the marquee, which has a slight

inclination. The elements are the only ones that have

straight lines in their design, creating contrast

between the circular and parabolic curves of the

domes and sideward arches (Fig. 1). The 315°

direction was chosen (Fig. 5) because, in this case,

the fluid intersects two architectural elements before

the building. These elements were disregarded in the

simulation, and the results were compared with

Model 14. The meshes were composed of 4.0 m

tetrahedrons in the fluid domain, 2.0 m in the body

of influence, and 0.5 m in the building faces. For

this application 0.25 for the power-law parameter

[18] was also adopted, and the terrain was defined

as a rough wall with a roughness of 0.01 m. Table 6

presents the mesh results for these models with their

respective quality metrics.

Model 15 presented a distribution of pressure

coefficients in the coverage similar to Model 14; for

the case of the complete model, the range was from

-0.03 to -1.68, equivalent to -0.07 to -1.77 (Fig.

8(h)-(i)). The distinction of the main dome's

overpressure and suction regions (A in Fig. 1) was

conserved, and the suction peak occurred in the

parabola inflection region for both cases.

The analysis of the interferences of the marquee

and bell towers alone in the fluid flow allows the

study of the slight difference in the results between

these two simulations. Figure 9 presents the velocity

contours parallel to the terrain at different levels (2

m, 3.5 m, and 5 m) intersecting the architectural

elements (Model 15). On the first level (Fig. 9a),

where the contour intersects the bell tower below

the marquee, a large part of the fluid, including that

near the bell tower, was at low speeds

(approximately 14 m/s). This behavior is expected,

given that for a height of 2 m, the power-law will

return low values for the velocity component.

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(a) Model 7 (b) Model 8 (c) Model 9

(d) Model 10 (e) Model 11 (f) Model 12

(g) Model 13 (h) Model 14 (i) Model 15

Fig. 8 Separation of overpressure and suction zones.

Table 6. Results of Model 15 with their respective quality metrics.

Model

Direction Nodes Elements Element quality

(average)

Skewness

(average)

Orthogonal quality

(average)

15 315° 115680 656071 0.81684 0.25258 0.74638

However, a small amount of fluid accelerates

past the tower bell, reaching 25 m/s. On the other

hand, downstream of the architectural element, the

color scale indicated values close to zero. The

disturbed flow quickly returns to 14 m/s, returning

to the initial configuration, before reaching the main

dome shell (A in Fig. 1). This behavior was repeated

for the 3.5 m and 5 m levels (Fig. 9b-c), with their

respective speed variations. Thus, these architectural

elements caused a small local change in the flow.

Their positions and geometric configurations

allowed the fluid to regain its initial characteristics

before generating effects on the parabolic coverage.

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(a)

(b)

(c)

Fig. 9 Ground-parallel velocity contours for Model

15, assuming heights of (a) 2 m, (b) 3.5 m, and

(c) 5 m.

4 Conclusions

This paper used computational studies in wind

action analyses in an innovative project composed

of parabolic and circumferential generatrices: the

Church of Saint Francis of Assisi, designed by

Brazilian architect Oscar Niemeyer in Belo

Horizonte, Brazil.

The numerical simulations were carried out on

signiﬁcant regions of the paraboloid roof using

Ansys Workbench software.

This work investigated the pressure coefficient

with numerical tests on seven different geometries

of hyperbolic paraboloid roofs.

In the flow, eight wind angles of attack (0°,

45°, 90°, 135°, 180°, 225°, 270°, and 315°) and

architectural elements (bell tower and marquee)

were considered.

The validation methodology considered two

hyperbolic paraboloid roofs with square plan

models. The maximum difference of 0.4 occurred in

the suction of the hyperbolic paraboloid.

For the various directions of wind incidence

under the Church of Saint Francis of Assisi

simulated, the parabolic cover presented similar

behavior in the two situations.

In the first one, the wind orthogonally reached

the main facades (front and back). The pressure

coefficient contours were distributed uniformly,

without great values for peaks (in the absolute

values).

In the second, the wind flowed directly or

partially on the sloping concrete roofs, showing

significant detachment points and a clear separation

between the suction and overpressure regions.

In general, the directions 45°, 135°, 225°, 270°,

and 315° were considered unfavorable.

The results showed that the marquee and bell

tower caused a low local disturbance in the flow,

which was not enough to generate high effects on

the structure.

In this case, with or without the presence of

architectural elements, the maximum difference in

the external pressure coefficients was 0.1.

Finally, these results can motivate the

elaboration of a roadmap to reduce the accidents in

buildings due to wind. Furthermore, to fill the

material gap for scholars in the area, the effects of

wind on paraboloid roofs should be reduced.

Other works may study the remaining wind

directions and the internal pressure coefficients. In the

critical regions, one could analyze the existence of

crack openings and their connection with the action of

the winds. Finally, the effects on the structure due to

the vortices caused by the steeple interception in the

flow are determined.

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Volume 2, 2022

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Doctoral dissertation, Federal University of Rio

Grande do Sul, 2019 (in Portuguese).

[18] J. Blessmann, O vento na Engenharia

Estrutural, UFRGS, 2013 (in Portuguese).

Contribution of individual authors to

the creation of a scientific article

Guilherme Teixeira was responsible for the

methodology, carrying out the simulation, and

writing the results. Marco Campos carried out the

conceptualization, review, and editing.

Creative Commons Attribution

License 4.0 (Attribution 4.0

International , CC BY 4.0)

This article is published under the terms of the

Creative Commons Attribution License 4.0

https://creativecommons.org/licenses/by/4.0/deed.en

_US

DESIGN, CONSTRUCTION, MAINTENANCE

DOI: 10.37394/232022.2022.2.27

Guilherme S. Teixeira, Marco D. De Campos

E-ISSN: 2732-9984

216

Volume 2, 2022