A study on the stability of metal arches as a membrane support Vitória Marques Alves Departamento de Engenharia Civil, Arquitectura e Georrecursos Instituto Superior Técnico, Universidade de Lisboa, Lisboa, Portugal Abstract This work presents and discusses the results of a study on metal arches stability behavior as a support of a membrane structure. Arches simply supported with circular tubular sections as support elements of a tensioned membrane structure were considered in the study. A research about membrane structures in constructions is presented which also includes general information on the structural design of this type of structures. A preliminary stability study of arched structural elements under vertical loads is presented, in order to understand the overall respective behavior. This work aim is to quantify the stabilizing effect of membranes in support elements, by comparing the elastic ultimate loading obtained with and without the influence of the membrane. To that end, a stability analysis in a model membrane-shaped umbrella has been performed . This model is used as reference by the community of membrane researchers and designers. Analyses are performed in Ansys APDL program that uses the finite element method with polynomial approximation functions to solve structural problems. The geometrically non-linear stability analysis are incremental-iterative by the Newton Rapshon method. Finally, the results and conclusions about considering the membrane influence on the supporting arch elements elastic stability behavior are presented. Comparisons have also been made with known analytical solutions and results in parabolic hinged arches under vertical distributed loads. Keywords: Tension membrane structures, Finite element models of membrane structures, Support elements of membrane structures, Arches elastic stability, Membrane stabilizing effect
Transcript
A study on the stability of metal arches as a membrane
support
Vitória Marques Alves
Departamento de Engenharia Civil, Arquitectura e Georrecursos
Instituto Superior Técnico, Universidade de Lisboa, Lisboa, Portugal
Abstract
This work presents and discusses the results of a study on metal arches stability behavior
as a support of a membrane structure. Arches simply supported with circular tubular sections as
support elements of a tensioned membrane structure were considered in the study.
A research about membrane structures in constructions is presented which also includes
general information on the structural design of this type of structures.
A preliminary stability study of arched structural elements under vertical loads is
presented, in order to understand the overall respective behavior.
This work aim is to quantify the stabilizing effect of membranes in support elements, by
comparing the elastic ultimate loading obtained with and without the influence of the membrane.
To that end, a stability analysis in a model membrane-shaped umbrella has been performed .
This model is used as reference by the community of membrane researchers and designers.
Analyses are performed in Ansys APDL program that uses the finite element method with
polynomial approximation functions to solve structural problems. The geometrically non-linear
stability analysis are incremental-iterative by the Newton Rapshon method.
Finally, the results and conclusions about considering the membrane influence on the
supporting arch elements elastic stability behavior are presented. Comparisons have also been
made with known analytical solutions and results in parabolic hinged arches under vertical
distributed loads.
Keywords: Tension membrane structures, Finite element models of membrane structures,
Support elements of membrane structures, Arches elastic stability, Membrane stabilizing effect
1 Introduction Membrane structures are tensioned structures used to cover large spans with high
interest space and wide variety of geometric forms. From an engineering point of view this
structures are slender membrane with uniform thickness which, by virtue of its shape and high
deformations inherent in their behavior, are able to support regulatory actions [1].
Tensioned membranes structures represents constructions where structure and form are
closely linked. This characterization makes them very specific structures, which require a total
balance between its component elements. [2]
In general, the tensioned membrane structures are designed with mixed forms derived
from one or more typical structural conceptions. In cases where the orientation of the curvature
of the membrane is in the opposite direction of the initial stresses applied to the edges it is required
the same action in another direction to ensure equilibrium. The tension relationships are linked to
the curvature and for these cases the membrane support structures require a support construction
in addition to the tensioned textile coating. In this case, it should be noted that the vertical external
forces can not be leveled without the existence of elements that are resistant to compression
which makes the support elements have the load leveling function in addition to offer geometric
boundary conditions for the support surfaces of the membranes [3].
Arch structures are frequently used for the support of tensile membrane structures. This
arches can be quite slender since the prestress in the membranes have a stabilizing effect [3].
For this assumption it was proposed to study the stabilizing effect of tensile membrane structures
on its arch supporting elements.
In this study, it was taken into account a tensile membrane structure with and umbrella
geometric form. The umbrella rods are paradigmatic in this sense.
For a perfect understanding on the matter, it was studied general functions and
constructions which included arch beam supports and also structural concepts for the design of
tensile membranes [4]. It was also studied general concepts of structures stability and its
application on arches beams [1,5].
Because of the specific configuration of tensile membranes and the propose of this work,
it was studied and designed a tensile membrane structure. The shape finding of the membrane
was made with a finite element formulation with polynomial approximation functions implemented
in Ansys apdl. To find an equilibrium membrane shape it was adopted the Newton- Rapshon
method in the solution of non-linear problem [6].
It was created a model with the membrane and an isolated arches model with the same
configurations and conditions.
The geometric non-linear stability analyses were also performed with the Newton-
Rapshon method and in the case of isolated support elements it was also taken into account the
membrane prestress and the transmission loadind configuration of the membrane.
These results were compared with analytical solutions with an equivalent distributed
vertical loading as the force resultant in the models are vertical and this loading configuration is
the one that makes compression axial forces critical in the arches. In addition and by simplicity,
this loading configuration in parabolic arches is the one more explicit in buckling analytical solution
under vertical loads [1,5].
As the analysis in the two models are under the same considerations, except for the
membrane integration on the system, the results were compared. In this way the stabilizing effect
of the membrane could be quantified in the results.
2-Tensile membrane structures
Tensile membrane structures are already being applied in a wide set of constructions.
Due to the ability of wining large spans it is used mostly in large spaces covers. These structures
are a high level architecture solutions with the capacity to resist to regular actions. Most of the
tensile membrane solutions have curved elements supports [2].
The characteristics design and analysis stages of tensile membrane structures are as
follows i) the shape finding of the membrane ii) the determination of the cutting patterns, linked
to the construction of the membrane itself and iii) the analysis and verification of the structural
loading response [4].
The shape finding is due the fact membrane structures only can be utilized as as a
construction coverage when pretensioned. For this specific characteristic, the first design
procedure is to find a membrane shape that corresponds to an equilibrium tension field.
3- Usual arches stability behavior under vertical distributed loads
For the understanding of arches stability behavior, were studied some articles and
studies. What is considered relevant of the research is the existing approaches calculation in
obtaining loads associated with hinged parabolic arches under distributed vertical loads instability.
These results are also compared with numerical linear stability analyses, performed in circular
tubular sections for a given range of diameter and thickness. Two configurations of hinged
parabolic arches were considered. One with 10m rise to 50m span and another with 20m rise to
a 100m span. Both with a rise-to-span 0.2 relation value. The loading configuration is vertical and
distributed along the arch span, being the analysis performed with Ansys program. The linear
analysis performed had the aim to understand the general simplified behavior of metal arches
under vertical distributed loads. The figure 1 presents the buckling loads results, figure 2 presents
the four first buckling modes considering an in plane linear behavior and the figure 3 illustrates
the first buckling mode when the three-dimension linear behavior is considered in the linear
buckling analysis. The axial arches rotation is restrained.
0
10
20
30
40
50
60
70
0 500 1000 1500
pcr A.L. f=10 l=50
pcr T.G. f=10 l=100
pcr A.L. f=20 l=100
pcr T.G f=20 l=100
0
5
10
15
20
0 500 1000 1500
pcr A.L. f=10 l=50
pcr S.K. f=10 l=50
Figure 1- Buckling loads from analysis and studied aproches – a) in plane b) out of plane
𝑝𝑐𝑟 (𝐾𝑁 𝑚⁄ ) 𝑝
𝑐𝑟 (𝐾𝑁 𝑚⁄ )
𝐿 𝑖⁄ 𝐿 𝑖⁄
.
Figure 3- First linear buckling mode considering 3D behavior
4- Membrane Shape- Finding method
The model is composed for 1728 triangular tridimensional membrane elements (Shell 41),
four arches, each one composed with 24 beam elements, and border cables with 12 link elements
formulated to be tension resistant as the Shell membrane elements. The membrane has a 1mm
thickness, the arches are HCS508 with 12,5 mm thickness and cables with 20cm2 area. A
geometric nonlinear large displacements analysis was performed, being used an incremental-
iterative Newton’s method
The initial geometry to start the cables and membrane form finding procedure may be
quite arbitrary. In some cases, however, Ansys pre-processing routines enable to begin analysis
with forms that already constitute equilibrium configurations under prestressing loads [6]. The
figure 3 represents the initial configuration considered.
Further complexity arises when an elastic boundary is added to the structure. That is the
case of the model. The membrane is connected to border cables anchored at membrane edges
[6]. If the boundary is excessively flexible, the model tends to relax the prestressing field, making
convergence harder to achieve. To maintain the arches with the wanted geometry in the final
Figure 4- Initial condition
Figure 2- In plane linear buckling modes
shape finding process, it was considered that the arches were fixed along the incremental
procedure.
A uniform 2MPa prestress field in the membrane and a 40kN presstress force was applied
to the membrane model. Since the initial geometry is not a double curvature surface, it obviously
can not constitute an equilibrium shape. However, different equilibrium shapes can be obtained
by varying the membrane elastic module, the shape is more peaked when the material is more
flexible.
In the shape finding procedure the membrane material characteristics has no specific
meaning. The aim is to find a membrane configuration that corresponds to an equilibrium imposed
tension field. For this reason, the elastic material modulus can be changed during the iterative
procedure to facilitate the convergence method. Reducing the material’s elastic modulus, more
uniform stress fields are obtained. However, prescription of lower elastic modulus allows larger
geometric variations, possibly degrading numerical convergence [6]. A valid alternative was
made, it was adopted a moderate elastic modulus and then proceed to successive updates of the
nodal coordinates. Nevertheless, the process may still lead to excessive element distortion [6].
Mesh adaptation is a last resource which was not tested insofar [6]. In the final iteration, it was
adopt the real elastic material modulus and the arches were considered simply supported at its
ends. Figure 4 shows the final configuration with a quite slender geometry, with a stress field
varying from 1,72MPa to 2,34 MPa , (figure 6), and the border cables with tensions from 21.5Mpa
to 22.5Mpa resulting on force variation between 43kN and 45kN (figure 7) with very small
displacements as shown in Figure 5 a valid and realistic shape form of a tensile membrane
structure. The figure 8 shows the resulted displacements and tensions on the arched beams from
de shape finding procedure.
Figure 5- Final Shape Finding membrane configuration
Figure 7- Effective membrane tensions
Figure 6- displacement modulos of the model
Figure 8- Effective border cable tensions
5- Elastic non-linear stability Analyses
5.1- Comparison of behavior due to membrane consideration
The structure stability analysis were performed for a hydrostatic downwards pressure
applied to the membrane. Taking advantage of the system symmetry, it is clear that the resulting
load is only vertical. Thus, the load applied on the arches associated to a vertical distributed load
becomes simple, which enhances the comparison with the results existing approaches for arches
ultimate elastic loading. (3).The analysis was made by an incremental-iterative procedure by
Newton-Rapshon method with Ansys Apdl program and the ultimate elastic loading were
validated with the arc-length method.
In the case of the isolated arches model, the loading adopted in the analyses were by the
load transmission of the membrane under a 1KN/m2 hydrostatic pressure. The tensions and
displacements resultants of the form-finding procedure were also taken into account on this
model. In the figures 11 and 13 is represented the displacement behavior of one of the support
arches integrated on the two models, with and without considering the membrane influence, figure
10. In the figures 12 and 14 is represented the equilibrium path of the arch in and out-of- plane.
Figure 9- (a) displacement modulos (b) normal tension on arches beams due to the membrane prestress
Figure 10- Study Models
5.2- Comparison of ultimate elastic loading with analytical solutions
Based on the models presented in figure 10, a geometric non-linear stability analysis in
the support arches with the variation of rise-to-span ratios from 0.1 to 0.3, considering and not
the influence of the membrane was performed.
Figure 11- In plane equilibrium paths (a) with membrane influence (b) without membrane influence
Figure 12- in plane displacement of a support arch (a) with membrane influence (b) without membrane influence
Figure 13- Out-of-plane equilibrium paths (a) with membrane influence (b) without membrane influence
Figure 14- out-of-plane displacement of a support arch (a) with membrane influence (b) without membrane influence
5.2.1- Membrane consideration with in plane ultimate elastic loading of support arches.
As presented in 4.1, in this situation the arches tend to instabilize on the plane. This is
also assumed by membrane designers in the predesign of support elements and it was also
verified in the analysis made. For that reason, the ultimate elastic loading was compared with the
results obtained from Austin and Timoshenko table [1] based on elastic stability analyses of
arches under vertical distributed loads [1].
The plots presented in figure 15 are associated with an equivalent ultimate elastic
uniformed distributed vertical loading obtained with the analysis vertical reactions on the arch
supports and the solution from Austin and Timoshenko table for hinged arches with HCS508x12.5
section for a length range.
5.2.2- Isolated support arches with ultimate elastic loading
In the case of the analysis made in arch support elements not considering the influence
of the membrane, the arches tend to instabilize out-of-plane as predicted by Sakimoto and
Kumatsi on the studies made about ultimate strength formula for steel arches under vertical
distributed loads [5].
To verify the accuracy of the assumption made on the association of elastic ultimate loads
on the model to an equivalent uniform vertical distributed load, the results were compared with
the solutions by Sakimoto and Kumatsi formula to elastic ultimate vertical load in arches braced
at mid span to consider the bracing effect of the arches connection on each other. The results
obtained are represented in the figure 17.
Figure 15- Ultimate elastic loading with membrane influence
