IN
DEGREE PROJECT ,
SECOND CYCLE, 30 CREDITS ,
Structural design and performance of tube mega frame in arch-shaped high-rise buildings
MATISS SAKNE
KTH ROYAL INSTITUTE OF TECHNOLOGY
SCHOOL OF ARCHITECTURE AND THE BUILT ENVIRONMENT
DEGREE PROJECT IN CIVIL ENGINEERING, SECOND LEVEL STOCKHOLM, SWEDEN 2017
TRITA-BKN Examensarbete 505, Brobyggnad 2017 ISSN 1103-4297
ISRN KTH/BKN /EX-505-SE
KTH ROYAL INSTITUTE OF TECHNOLOGY
KTH ARCHITECTURE AND THE BUILT ENVIRONMENT
AF223X DEGREE PPROJECT IN STRUCTURAL ENGINEERING AND BRIDGES 2017
Structural design and performance of tube mega frame
in arch-shaped high-rise buildings.
Abstract
A recent development and innovation in elevator technologies have sprawled interest in how these technologies would affect the forms and shapes of future high-rise buildings. The elevator that uses linear motors instead of ropes and can thus travel horizontally and on inclines is of particular interest.
Once the vertical cores are no longer needed for the elevators, new and radical building forms and shapes are anticipated. It is expected that the buildings will have bridges and/or the buildings themselves will structurally perform more like bridges than buildings, therefore this study addresses the following topic - structural design and performance of tube mega frame in arch-shaped high-rise buildings.
Evidently, for a structure of an arched shape, the conventional structural system used in high-rise buildings does not address the structural challenges. On the other hand, The Tubed Mega Frame system developed by Tyréns is designed to support a structural system for high-rise building without the central core, in which the purpose is to transfer all the loads to the ground via the perimeter of the building, making the structure more stable by maximizing the lever arm for the structure. The system has not yet been realized nor tested in realistic circumstances. This thesis aims at evaluating the efficiency of the Tubed Mega Frame system in arched shaped tall buildings. Multiple shapes and type of arches are evaluated to find the best possible selection.
Structural behavior of different arch structures is studied using analytical tools and also finite element method in software SAP2000. The most efficient arch shape is sought to distribute the self-weight of the structure. The analysis shows that it is possible to accurately determine efficient arch shape based on a specific load distribution.
Furthermore, continuing with the arch shape found in previous steps, a 3D finite element model is built and analyzed for linear static, geometric non-linearity (P-Delta) and linear dynamic cases in the ETABS software. For the given scope, the results of the analysis show that the Tubed Mega Frame structural system is potentially feasible and has relatively high lateral stiffness in the plane of the arch, while the out-of-plane lateral stiffness is comparatively smaller. For the service limit state, the maximum story drift ratio is within the limitation of 1/400 for in-plane deformations, while for out-of-plane the comfort criteria limit is exceeded.
Preface
This master thesis has been written at the division of structural design and bridges, department of the Civil and Architectural Engineering, at the Royal Institute of Technology (KTH). The writing process took place in Tyréns, in Stockholm and has been a very rewarding experience. This study concludes two-year Mater’s program at KTH.
I would like to express large appreciation and gratitude to Tyréns and particularly my supervisor Fritz King for providing this great opportunity to conduct my master thesis within an engineering field that I am very much interested in and also for offering his guidance and engineering judgment throughout the whole process.
I also thank my supervisor and examiner Professor Raid Karoumi, for his guidance and inspiration throughout my time in KTH and also for his help, support and feedback throughout the writing process.
Thanks to Mahir Ülker, for his suggestions and knowledge concerning my questions during the process.
Last but not least, I would like to thank the whole working group in Tyréns studying the tube mega frame concept for new building shapes and forms. Thanks to David Desimons, Matea Bradaric for helping with the wind load estimation, also, Grigoris Tsamis, Hamzah Maitham, Levi Grennvall, Sujan Rimal, Lydia Marantou and Paulina Chojnicka for great discussions throughout the process.
Stockholm, June 2017 Matiss Sakne
Notations
𝐴 Cross-section area
𝐵 Width of a structural section
𝑏 Width of a cross- section of mega tube 𝑐𝑝𝑤 Windward Coefficient
𝑐𝑝𝑙 Leeward Coefficient 𝐷 Arch bending stiffness
𝐷0 The arch stiffness at the crown 𝐷𝑎 The stiffness at the springing
𝑑𝑥 Horizontal projection of arch element 𝑑𝑦 Vertical projection of arch element 𝐸 Elastic modulus (Young’s modulus)
𝑒1 Eccentricity ratio for wind load (ASCE 7-10 Fig. 27.4-8) 𝑒2 Eccentricity ratio for wind load (ASCE 7-10 Fig. 27.4-8)
𝐹 Transverse load
𝑓𝑐 The cyclic frequency
𝑓 Rise of an arch
𝐺(𝑟) The geometric (P-Delta) stiffness due to the load vector 𝑟 𝐻 Height of a structural section
𝐻𝑞 Horizontal force in the arch
ℎ Height of a cross- section of mega tube
𝐼 The moment of inertia
𝐾 The stiffness matrix 𝐾𝑧𝑡 Topographical Factor 𝐾𝑑 Directionality Factor 𝐿 Theoretical span of an arch 𝑙 Stretched length of a string 𝑙1 Natural length of a string 𝑀𝑠 Moment at the support 𝑀 The diagonal mass matrix
𝑃 Axial load
𝑞(𝑥) Basic load
𝑅 A vector of forces acting at the direction of the structure global displacement
𝑟 The load vector
𝑠 Arc length
𝑇 Period of a Mode
𝑇1 Force at each end of a string
𝑡 Thickness of a cross- section of mega tube 𝑈 A vector of the system global displacements
𝑢 Poison’s Ratio
𝑉 Vertical for in the arch 𝑤 Weight of structural material 𝑥 Horizontal coordinate for the arch 𝑦 Vertical coordinate for the arch 𝑦′ Slope of the tangent line for the arch
𝑦′′ The change of the slope of the tangent line for the arch
𝛼 Thermal expansion coefficient 𝜆 The diagonal matrix of eigenvalues
𝜌(𝑠) Density function for the weight of the chain
𝛷 The matrix of the corresponding eigenvectors (mode Shapes) 𝜑 The angle which tangent to the arc makes with the horizontal axis 𝛹 The matrix of corresponding eigenvectors (Mode shapes)
𝜔 The circular frequency
𝛺2 The diagonal matrix of eigenvalues
Content
1 Introduction ... 1
1.1 Background ... 1
1.2 Aim and scope ... 1
1.3 Assumptions and limitations ... 2
1.4 Method ... 2
2 Arch shape and structural behavior... 5
2.1 Arch type ... 8
2.2 Arch shape... 9
2.2.1 Analytical calculation of the funicular line ... 11
2.2.2 Numerical calculation of the funicular line ... 13
2.3 Cross-section variation ... 14
2.4 Construction technique and sequence... 16
2.4.1 Temporary supports ... 17
2.4.2 Prefabrication ... 17
3 High-rise structures and Tube mega frame ... 19
3.1 High-rise structures ... 19
3.2 Tube mega frame ... 20
4 Finite Element Method ... 21
4.1 Linear elasticity... 21
4.2 Structural dynamics ... 23
4.3 Discrepancies in FEM application ... 23
4.3.1 Material ... 24
4.3.2 Load ... 24
4.3.3 Discretization ... 25
4.4 Finite element types ... 25
4.4.1 Beam elements ... 25
4.4.2 Shell elements ... 26
4.5 Construction sequence analysis ... 26
5 Arch Mega Structure ... 27
5.1 Introduction ... 27
5.1.1 Rules and Regulations ... 27
5.2 Geometry ... 28
5.2.1 Cross-sections ... 28
5.2.2 Support conditions - Foundation ... 29
5.2.3 Interactions ... 29
5.3 Material and Loads ... 30
5.3.1 Materials ... 30
5.3.2 Loads ... 30
5.4 Load Combinations ... 31
5.5 Quality assurance ... 32
5.5.1 Model mass ... 32
5.5.2 Convergence ... 33
6 Results and discussions ... 35
6.1 Preliminary arch shape evaluation ... 35
6.2 Arch shape and characteristics ... 40
6.3 Bracing system evaluation ... 50
6.4 Arch Mega Structure ... 53
6.4.1 Deformation ... 53
6.4.2 Story drift ... 56
6.4.3 Mode shapes and frequencies ... 57
6.4.4 Reaction forces ... 59
6.4.5 Buckling ... 60
7 Conclusion ... 61
8 Further studies ... 63
9 References ... 65
10 Annex ... 67
10.1 Annex A ... 67
10.2 Annex C ... 68
10.3 Annex D ... 72
10.4 Annex E ... 73
1 Introduction
1.1 Background
Company Tyrens, the Council on Tall Buildings and Urban Habitat and ThyssenKrupp have formed an architectural and structural engineering group to study the effects of shared-shaft elevator technologies on the forms and shapes of high-rise buildings.
ThyssenKrupp is developing an elevator that uses linear motors instead of ropes and can thus travel horizontally and on inclines. Once the vertical cores are no longer needed for the elevators new and radical building forms and shapes are anticipated. These new buildings will begin a new era in building forms. It is expected that the buildings will have bridges and/or the buildings themselves will structurally perform more like bridges than buildings.
Based on these novel ideas it is of high interest to evaluate the structural performance of these new building shapes and forms. By eliminating the need for vertical elevator shafts, which also functioned as the main structural stability elements for conventional high-rise structures, it is possible to consider different and new novel ways of stabilizing the structure. One such concept is the tube mega frame (TMF), which essentially provides the structural integrity of the structure on building perimeter and eliminates the central core.
The tube mega frame concept has been studied for more conventional high-rise shapes and proven to be by all means a structurally sound concept. Therefore, as further development of the TMF, it would be of interest to evaluate, whether or to what extent this concept would perform well with the new building shapes and forms. With initial structural restrictions limiting architectural designs, now it is of interest to evaluate where the new limitation lies.
For the thesis at hand, an evaluation of the structural performance of a novel architectural design of high-rise structure is performed. The structural shape takes a form of an arch. Arched megastructures have been commonly used in the construction of bridges and stadium roofs, where the span of the structure is usually much longer than the height. For this particular thesis, the other end of the spectrum is studied, where the height of the structure is greater than the span, allowing for the structure to enter the realms of high-rise structure.
Arched structures can take various forms and shapes, therefore, a thorough investigation of the optimal geometry is performed, which essentially leads to reduced material consumption and preferable construction sequence and, most importantly, demonstrate that such a structure can be realized under the right circumstances.
1.2 Aim and scope
The aim of the study is to evaluate an arch structure and its suitability for tall building application, in particular, using tube mega frame concept. In order to demonstrate the structural performance the study focuses on indicating relevant parameters that are vital for arched structures and play a major role in the efficiency of the structural system. Parameters influencing the shape of the structure, force distribution within the structure as well as the effect of different bracing systems are to be evaluated.
Furthermore, the emphasis is put on demonstrating the continuity between the results of analytical and numerical calculation with finite element models.
1.3 Assumptions and limitations
The study only considers one prototype of an arch structure. Different parameters are studied on various levels of complexity in order to define the structural elements of the structural system. Material nonlinearity is not considered. The effects of creep, shrinkage or temperature have not been analyzed for concrete. A limited amount of the loads are included in the study. Loads such as installation loads and façade loads among others are not considered. The assessment of the structure system using finite element method is limited to linear static load conditions, geometric non-linear conditions (P-Delta) and linear dynamic load conditions.
1.4 Method
LITERATURE STUDY
Literature study serves the purpose of acquisition of knowledge. As a part of the revision, brief history and structural principles of arch structures are studied, which essentially set the basis for the particular arch structure under consideration. Also, structural design principles of Tubed Mega Frame concept are investigated through previous publications, allowing for a qualitative estimation of the preferable structural system. Finite Element Method is briefly included in the literature study as it is selected as the main tool for the analysis. Through the literature review the strengths, but most importantly sources of error and limitations of FEM are highlighted to provide an understanding of potential drawbacks of more complex models.
FINITE ELEMENT METHOD
The study primarily uses finite element method to model and analyses the structure. The analysis is performed on multiple scales of complexity in ascending manner. This particular setting allows for better understanding and overall control of the models. For simpler models, equivalent analytical solution procedures are available, therefore, allowing for quality verification tools. Furthermore, results in more complex models can be traced back and must be comparable to ones in simpler models.
PRELIMINARY ARCH SHAPE EVALUATION
An evaluation of common arch shapes based on presented functions is performed. For constant material and cross-sectional geometry properties as well as a constant span to rise ratio - linear, parabolic, catenary and arc shapes are evaluated in terms of support reactions and also relative buckling safety.
The evaluation is performed using FEM software SAP2000, structures are analyzed using self-weight of the structure.
ARCH SHAPE CALCULATION
The arch shape is calculated by numerical iterative procedure, where the self-weight distribution within the structure is accounted for. Cross-sectional variation is applied by enforcing initial assumption upon the structural elements, particularly, varying cross-section height. For the acquired arch shape, few options are examined relating to mega tube location with respect to one another and the centerline of the arch shape. The evaluation is performed using FEM software SAP2000, structures are analyzed using self-weight of the structure. Available quantities from the numerical iterative procedure are evaluated with the results of FEM. Elastic modulus for the megastructure is kept constant as well as the thickness of the tube section. For more detailed information on the calculation procedure see section 2.
BRACING SYSTEM EVALUATION
An evaluation of different bracing systems is performed for specified tube mega frame arrangement.
2D FEM in SAP2000 is used to assess the deformation of the structure, buckling safety as well as mode shapes and frequencies. Three different bracing arrangements are evaluated to one another by having equivalent weight, which include some of the most common principles for typical bracing systems. X- BRACING system consist of common steel braces arranged in cross formation, F-BRACING utilizes the principles of steel moment frames, and B-BRACING includes steel truss principles at certain heights along the building. The evaluation more thoroughly examines three types of bracing systems for in- plane performance.
STATIC ANALYSIS
The 3D arch mega structure is evaluated using static analysis, which is implemented using finite element analysis software ETABS considering both linear static and non-linear static cases. The geometry of the finite element model is based on the results of the preliminary studies. Estimated loads are applied to the model and linear static analysis is performed. For geometric non-linearity analysis, P-delta effects are considered. The P-delta effects are considered as a separate load case in ETABS and analyzed as the initial load case. Upon converged results, for P-delta effects the stiffness of the model is further used for the analysis of remaining linear static cases. The assessment of the results includes self-weight of the whole structure, reaction force at the base, story drift ratios, and the deflections of the structure.
Also, forces in structural elements are evaluated including the effects of P-delta. Buckling safety of megastructure is evaluated.
DYNAMIC ANALYSIS
Dynamic analysis is performed within the same finite element model and includes Modal analysis. From the modal analysis, the natural frequencies and periods of the building are obtained which allow for the evaluation of the overall stiffness of the structure.
2 Arch shape and structural behavior
Arched structures have been used for various purposes and have been around for a long time. These structures provide very efficient transfer of forces if designed accordingly. The very first structures that resemble an arch were built in form of so-called “corbelled arch” (1). The structural behavior of these structures produce no horizontal force at the supports, however, for larger spans, extensive height is required.
Shortly after the “corbelled arch”, the use of “vault” structures emerged. These structures were built in a way to only support compressive forces, mainly, due to construction technique and material availability at the time. These type of structures were constructed using bricks or stones, with very weak or no binding material in between. The main and only difference between an arch, as we know it today, and a vault is that arches can withstand tensile forces within the structure, however, arches at times are designed to resemble vaults and the difference between the two often are somewhat blurred. (1) Structures which employ vaults are known to be
preserved from around 1000 BC (1), however, the Romans were the first to construct major structures utilizing the principles of an arch in the form of barrel vault, as shown in Figure 2.1. At the time structures like bridges, aqueducts and roof structures were constructed in this way (2).
Gothic arches emerged through Middle Ages, characterized by the pointed apex, and were extensively used in the construction of European cathedrals. These types of arches still functioned in compression, however, they had the tendency to produce large horizontal forces at the supports, also called springing. For the structure to work it was necessary to construct heavy structures that would withstand the horizontal force, without a collapse, therefore frequently flying buttresses were constructed to fulfill this function. (2)
In the 18th century, many bridges were designed and built utilizing masonry vault structures, as shown in Figure 2.2. Many of these bridges have been conserved to this day and carry the load that is vastly greater that the ones, that these structures were intended for at the time. (2) Substantial research has been conducted on these types of bridges and recently it has been possible to show that the interaction with the filling material greatly increases the strength of the bridge, hence, clarifying the strength and durability phenomena of these structures. (1)
When an unrestricted approach is taken when designing arched structures the word “arch” is used. For such structures materials with sufficient tensile capacity are required. Most common arches are of reinforced concrete, steel or laminated timber (Glulam), although in theory many materials could be used. (1)
Figure 2.1 – Roman barrel vault arch structure
Figure 2.2 – Masonry bridge vault structure (1)
Among the evident economic advantages, these structures give unique visual experience. They have been used in various shapes to get across the intended aesthetic impression for specific buildings.
Nowadays most new built arched mega structures can be observed in many bridge structures. As per (1) the development of the arch spans has increased considerably over past century, due to technological advances and increased understanding and knowledge of structural behavior. Nowadays the longest spans reach 552 m for steel structures and 445 m for concrete structures. See Annex D for a full list of largest arch bridges. Arch mega structures are common to use in stadium roof construction and massive dam constructions as well.
Another great structure, where the principles of the concept of the ideal arch shape, or at least the quest in pursuit of the ideal aesthetic and
structurally honest arch shape is present is the Gateway Arch – The Jefferson Memorial dedicated to Thomas Jefferson – to his vision of an America stretching across the continent, and his Louisiana Purchase, which roughly doubled the size of America at the time. This magnificent structure constructed in the shape of an arch is among the highest arch structures ever built, considering the rise of an arch.
In the context of the present study this monumental structure draws undeniable similarities, therefore, a more thorough look is taken to examine the origins of the shape and design principles of the Gateway Arch.
Although there has been some initial confusion about the shape of the arch, it has since been determined and proved that the arch takes the shape of a so-called weighted catenary. Very simply put, the arch resembles the shape of a hanging chain with varying size of the chain links.
More thorough discussion on arch shapes is presented further on in this study.
The size of an arch is 630 feet (192 m) high and 630 feet (192 m) wide forming the aspect ratio of 1:1. The term aspect ratio in this context describes no more that the width to height ratio of an arch structure. These aspect ratios are not typically used for arch bridges, dams or stadium roof trusses, instead, these structures strive for more shallow arch shapes with aspect ratios of around 5:1. This notion by itself arises design challenges that are quite different from the ones in more shallow arch megastructures, however, in general “it may be worth noting that the 1:1 aspect ratio is a very old tradition in architecture. In medieval times, it was known as ad quadratum. The original 1386 specification for the cathedral of Milan is an example”. (3)
Chaotianmen Lupo Huangpu New River Gorge Bayonne Kill van Kull
Wupper Bridge Eads Bridge
0 100 200 300 400 500 600 700
1700 1800 1900 2000
Max. SPAN [m]
Figure 2.3 - Largest arch bridges from 18th century to nowadays
Figure 2.4 - Eero Saarinen’s Gateway Arch, St. Louis.
Image reproduced from (3)
R. Osserman has studied the trade-offs between the mathematical and esthetical reasoning behind the description of the shape of the Gateway Arch and in his work, has provided some general notions concerning the relation between the formulation of hanging chain and that of an arch.
“It was Robert Hooke who in 1675 made the connection between the ideal shape of an arch and that of a hanging chain in an aphorism that says, in abbreviated form, “As hangs the chain, so stands the arch.”
In other words, the geometry of a standing arch should mirror that of a hanging chain.” (4)
He has decisively concluded that the aesthetic and structural decisions that have to be made when designing an arch are “inextricably entwined”. This statement, however, is not surprising and is repeatedly addressed in this study as well.
Once the fact that the hanging chain and an arch being similar, is established, it is evident that analogous issues have to be addressed. The quest for finding the ideal shape has to be commenced. The history of the problem has been clearly presented in R. Osserman’s research, therefore it is recommended to see (3) and (4) for further insight.
In general, the term catenary is used when describing the shape of a hanging cable under its own weight.
Evidently, the function describing the cable is derived based on the weight density1 of the cable. In structural mechanics2, the catenary can be described using the theory of elastic strings, which is based on the Hook’s law3. This theory describes the elongation of a string under tensile forces and only considers axial deformations. (5)
𝑙 − 𝑙1= 𝑙1𝑇𝐸1 ( 2-1 )
Where l is the stretched length of a string, l1 natural length, 𝑇1 the force at each end of a string, E elastic modulus (Young’s modulus).
“This law governs the extension of other substances besides elastic strings. It applies also to the compression and elongation of elastic roads.” (5)
“This law is true only when the extension does not exceed certain limits, called the limits of elasticity.
When stretching is too great the body either breaks or receives such a permanent change of structure that it does not return to its original length when the stretching force is removed.” (Routh Art. 490) Using theory of elastic strings, it is possible to calculate force equilibrium when the cable is subjected to any load and also to its own weight.
“It is shown that the equilibrium of an arch is achieved by combining forces at the points of an arch and, similarly to hanging chain, the horizontal component of the force is constant and simply transmitted along the arch or chain, while the vertical forces are mirror images.” (4)
An important aspect when using elastic string theory is to understand that the basis of this theory originates around tensile axial deformation and in no way, consider bending stiffness of a string,
1 Weight density - the weight per unit volume of a substance or object. (www.dictionary.com 28.03.2017)
2 Structural mechanics or Mechanics of structures is the computation of deformations, deflections, and internal forces or stresses (stress equivalents) within structures, either for design or for performance evaluation of existing structures. (Wikipedia 28.03.2017)
3 Hook’s law - is a principle of physics that states that the force (F) needed to extend or compress a spring by some distance X is proportional to that distance. That is: F = kX, where k is a constant factor characteristic of the spring:
its stiffness, and X is small compared to the total possible deformation of the spring. The law is named after 17th- century British physicist Robert Hooke. (Wikipedia 28.03.2017)
therefore, it is reasonable to believe that the accuracy of this theory, when used for arch structure, will not reflect the real behavior, especially, when structure is designed to have significant bending stiffness.
The procedure of finding the shape of the structure, however, is a sound concept, and similarly, for an arch structure, the shape can be derived based on the weight density of the structure.
2.1 Arch type
From the perspective of structural definition, arches often are categorized by their structural system. In (1) H. Sundquist uses terms “single-hinge”, “two- hinge”, “three-hinge” and “zero-hinge” arch referring to statical determinacy of the system. In Figure 2.5, it is possible to observe the notations used when describing an arch from a structural perspective. The respective arch lines represent the center-of-gravity lines for the arches. Each of these types has their own advantages and, therefore, the choice of a system for a particular case is based on the actual conditions. The major contributing factors are foundation condition and construction method, which will eventually determine the structural design.
Zero-hinge arches require good foundation conditions, also for most cases reinforced concrete material would be the most economic choice,
although steel has the advantage in cases where complex construction sequences have to be resorted to. Under sufficient conditions, especially for large spans, it is appropriate to choose zero-hinge arch, since this leads to reduced consumption of material for the megastructure, provided that the cross- sectional variation is resorted to. (1)
“The relationship is illustrated in Table 2.1, which applies for an evenly distributed variable load p and which indicates the maximum moment, the average value of the maximum moment curve and the relative buckling safety for a given section at the crown and a given load. The Table 2.1 shows that the zero-hinge arch has the lowest average moment and thus the lowest material consumption. At the same time, it has the greatest buckling safety, which permits slender dimensions and less weight for large spans.” (1)
Table 2.1 – Comparison of different arch types (1)
ARCH TYPE CROSS-SECTION VARIATION MAX. MOMENT AVERAGE MAX. MOMENT RELATIVE BUCKLING SAFETY
3-HINGE - ± 0,0188pL² ± 0,0125pL² 1,00
2-HINGE D = EI cos φ = D0 ± 0,0166pL² ± 0,0116pL² 1,33
0-HINGE Kararnowsky-type ± 0,0218pL² ± 0,0071pL² 3,28
Based on the knowledge of the structural definition, the study will focus on the zero-hinge arch, as it is expected that the sheer scale of the structure will require efficient design and solid foundation, therefore, an assumption is made that at this instance foundation is in good enough condition to support a zero-hinge structure.
Figure 2.5 - Arch types with respect to the number of hinges together with notations. (1)
2.2 Arch shape
As mentioned before the shape of the Gateway Arch is a weighted catenary, although it has been confused with catenary and parabola instead. It is not, however, something that visually is easily distinguishable, although the mathematical formulation is quite different. Parabola is a function of the single-variable quadratic polynomial ( 2-2 ), the catenary is a function of hyperbolic cosine ( 2-3 ), while weighted catenary is a function of hyperbolic cosine ( 2-4 ) but includes constants that allow for shrinking of the curve.
𝑓(𝑥) = 𝑎𝑥2+ 𝑏𝑥 + 𝑐 ( 2-2 )
𝑓(𝑥) = 𝑎 ∙ cosh (𝑥𝑎) = 𝑎(𝑒
𝑥 𝑎+𝑒−𝑎𝑥)
2 ( 2-3 )
𝑓(𝑥) = 𝑏 ∙ cosh (𝑥
𝑎) = 𝑏(𝑒
𝑥 𝑎+𝑒−𝑎𝑥)
2 ( 2-4 )
“The term “weighted catenary” conveys essentially no useful information; virtually any curve that one can picture as a possible candidate for weighted catenary can actually be reproduced by a suitably weighted chain. Furthermore, it is not descriptive, in that it says nothing about the actual shape of the arch, but only about the method used to produce it.” (3) Rather than using the term weighted catenary, R. Osserman uses the term flattened catenary. “In short, a “pure catenary” is the ideal shape for an arch of uniform thickness, and a flattened catenary is an ideal shape for an arch that is tapered in certain precise (and elementary) manner.” (3) Further, the term flattened catenary is adopted and resorted to whenever describing a tapered catenary.
In the case of weighted chain, R. Osserman puts forward equation for expressing the amount of the weight of the chain by a density function ρ(s) , where the weight of any arc of the chain is the integral of ρ(s) with respect to arc lengths over the arc. Osserman shows that a given flattened catenary of the form
𝑓(𝑥) = 𝐴 ∙ 𝑐𝑜𝑠ℎ(𝐵 ∙ 𝑥) + 𝐶 ( 2-5 )
Is obtained from the density function 𝜌 which, when expressed in terms of 𝑥, takes the form:
𝜌(𝑠(𝑥)) = (𝑎𝑦+𝑏)𝑑𝑠 𝑑𝑥
= (𝑎𝑓(𝑥)+𝑏)
√1+(𝑓′(𝑥))2 ( 2-6 )
Where the coefficients a and b are determined by the coefficients A, B, C of 𝑓(𝑥), up to an arbitrary multiple since multiplying the density by an arbitrary positive constant just changes the total weight, but not the shape of a hanging chain. The physical meaning of this is that the weight of any portion of the chain laying over a segment of the x-axis, given by ∫ 𝜌(𝑠)𝑑𝑠, is equal to ∫ 𝜌(𝑎𝑦 + 𝑏)𝑑𝑥 over the given x-interval, hence providing a method for estimation of arch shape based on its weight density.
The weight of the structure defines the most efficient shape of transferring the weight to the foundation, however, another important aspect that greatly affects the shape is the aspect ratio. By changing the aspect ratio, it is possible to either minimize the effects of weight distribution in defining the shape of the structure or highly increase these effects.
Also, a great discussion is brought up by R. Osserman about the influence of aspect ratio on the perception of the structure but more importantly differences between previously mentioned functions that are likely to mathematically describe the ideal shape for an arch. As he points out and also as can be observed in Figure 2.6 and Figure 2.7 the difference between the catenary and the parabola becomes indistinguishable as the aspect ratio increases, on the other hand, the differences are undeniable for smaller aspect ratios. The takeaway from these observations is that for shallower arches one can get away with using almost any of previously mentioned functions to describe the shape of the arch and still be relatively close to the ideal shape for the arch,
however, if one wants to be more accurate in finding the ideal shape for an arch with a small aspect ratio, it will be of utmost importance to resort to the function of flattened catenary to mathematically describe the shape.
Figure 2.7 - Different functions (aspect ratio 4:1)
An aspect ratio is somewhat more an aesthetic tool that will affect the visual perception of an arch. As discussed before, the structural design of an arch could be seen through for various aspect ratios.
Further in this study, the parameters affected by aspect ratio are discussed, however, the choice of aspect ratio for the arch evaluated will be based on more of an aesthetic motivation.
The ideal shape of an arch is a key concept and, therefore perhaps a more thorough explanation should be given to grasp the context. H. Sundquist has used term “funicular line” to describe the ideal shape.
The funicular line also known as a compression line is a factious curve which connects load position so that only compression forces occur in the arch. When the arch shape coincides with the funicular line – no moments in the arch arise. It must be reminded that the funicular line is determined for one specific load situation and for all other loading combinations bending moments are likely to occur in the arch.
For example, for a point load at the crown, the funicular line would be triangular, while for a uniformly
Numerical Parabola Catenary Flattened catenary
Numerical
Parabola
Catenary
Flattened catenary
Figure 2.6 - Different functions (aspect ratio 3:4)
distributed load (second-degree parabola) the funicular line would be parabolic. R. Osserman points out an interesting phenomenon, that if one would design an arch to take parabolic shape, since following Hooke’s dictum, in that case, it would require the greatest weight at the crown, and therefore an arch that was thickest at the top and slimming down toward the bottom, which is counterintuitive with what would be required from structural point of view.
It is also noted that the process of calculating the fully three-dimensional shape of the Gateway Arch was somewhat different from the more common approach of starting with a hanging chain and adopting the form it takes in order to build an arch. In this particular case, the desired shape was settled upon first and thereafter the degree of tapering was determined that would fit the shape from a structural perspective. Although different, the process still involved the same parameters and concepts. The relation between the weight distribution in the arch and the ideal shape are directly correlated.
2.2.1 Analytical calculation of the funicular line
H. Sundquist in (1) provides a method based on similar principles. Analytical procedure for determining arch shape when the distribution of the load is known is described by considering an arch element with horizontal projection dx loaded by basic load q(x) as in Figure 2.8
Figure 2.8 - Arch element (1)
Force equilibrium for vertical projection and bending moment give:
𝑞(𝑥)𝑑𝑥 − 𝑑𝑉 = 0 ( 2-7 )
𝑉 ∙ 𝑑𝑥 + 𝐻𝑞∙ 𝑑𝑦 = 0 ( 2-8 )
Which lead to
𝑉 = −𝐻𝑞∙𝑑𝑦𝑑𝑥= −𝐻𝑞∙ 𝑦′ ( 2-9 )
𝑑𝑉 = 𝑞(𝑥)𝑑𝑥 = −𝐻𝑞∙ 𝑦′′𝑑𝑥 ( 2-10 )
So that 𝑦′′= −𝑞(𝑥)𝐻
𝑞 ( 2-11 )
The condition that the funicular line must pass through both the crown and springing has to be met to determine the magnitude of the horizontal thrust.
The analytical funicular line in Figure 2.10 is derived for parabolic load approximation according to (1) illustrated in Figure 2.9. One can observe that the funicular line “raises its back” (1) the more the basic load increases towards the springing. It is also noted that in many cases, the basic load can be approximated by a parabolic function as shown in Figure 2.9.
Figure 2.9 – Parabolically varying basic load 𝑞(𝑥) = 𝑞(𝑢) = 𝑞0+ ∆𝑞(1 − 𝑢)2. (1)
Although the parabolic function is a good approximation for the basic load, it is not the exact solution. A different procedure is required in order to more accurately determine the funicular curve for the basic load when the load variation is more complex and cannot be directly estimated by known mathematical function.
The closer the structure resembles the funicular curve for the specific loading case the smaller the effect of bending moments in the structure, which in general leads to more optimized structures and reduced material consumption.
When evaluating the influence of span/raise ratios on parabolic load approximation of a basic load distribution it can be observed in
Figure 2.11, that the load magnitude increases rapidly towards the springing as the span/raise ratio decreases. Such relationship can be expected based on the load estimation principles mentioned previously.
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
y/f
x/L 1 2 4 9
∆𝑞/𝑞0
Figure 2.10 - The shape of the funicular curve for different distributions of the load.
Figure 2.11 - Basic load distribution for different span/raise ratios (constant cross-section area) If the ratio of load increase at springing with respect to the load at the crown is considered, it is possible to observe in Figure 2.12 that the ratio decreases exponentially as the span to raise ratio increases.
Figure 2.12 - Δq/q0 at springing for different span/raise ratios (constant cross-section area)
2.2.2 Numerical calculation of the funicular line
The geometric shape of an arch greatly affects the functionality, design, and performance of the structure. The key to quality design is to define a shape that closely follows the natural path of forces through the structure, in this way leading to the fundamental basic concept of an arch in that it shall primarily function as a structure under pure axial pressure.
Any arch can be designed to function purely under compressive forces for a particular basic load q.
Usually, the basic load is preferred to be a self-weight of the structure as this type of action is permanent on any structure.
The process of defining the structural shape, however, takes thorough consideration in early stages in the design process. It is clear, that it will not be possible to design a structure to act purely in compression for all loading situations, and since the self-weight of the megastructure accounts for large part of the force exerted on the structure, a sensible approach is to design the structure close to the funicular shape for the self-weigh of the structure. To compute the funicular line for self-weight (1) provides a methodology in which an iterative procedure is used. “Numerical determination of the funicular line where the shape of this line is dependent on the load, and where the load, in turn, is
0.0 2.0 4.0 6.0 8.0 10.0 12.0 14.0 16.0
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
q(x)/q0
x/L 0,50 0,75 1,00 1,50 5,00
𝐿/𝑓
1.0 2.0 4.0 8.0 16.0 32.0 64.0 128.0
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0
Δq/q0
L/f
dependent on the arch shape.” (1) At times, multiple iteration steps are required to acquire converged result. The amount of these steps depends on the quality of the first estimate of the form of the funicular line.
Rather than attempting to find an analytical solution to define the shape of an arch, it is possible to resort to the numerical calculation to determine the funicular line. A common way of implementing this procedure is by using numerical integration of ( 2-11 ). The accuracy of this solution can be adjusted by the amount of discretization of the calculation model. Trapezoidal rule is used for the integration. This procedure allows for easy incorporation computer software.
To construct a funicular curve, the load arrangement of the self-weight of the structure must be estimated. One advantage of this procedure is that the first estimation, even if not very accurate, will only increase the iteration steps required to arrive at desired accuracy for funicular line, therefore one can start by assuming a parabolic shape, which, if we consider the previous discussion, will certainly not be close to funicular line for this structure, but gives a rather easy way to define a shape which fits the overall rise and span set out for the arch.
This study resort to the numerical calculation procedure in order to determine the shape of the arch.
This procedure leads to the development of practical tools in MATLAB in order to make the calculation quick and also the further implementation in the FEM software is made more fluent in this way.
2.3 Cross-section variation
The cross-sectional variation of the arch is influenced by various variables. The choice of the material, the span, construction techniques and maintenance consideration play a significant role in the design process. Also, the way in which the load is transferred into the structure is vital. The choices for cross- sectional design must take into consideration economic aspects of the design and assembly. In general, it is beneficial to maintain a lower weight for the structure, however, the design must not lead to a complex design, which would rapidly outweigh the benefits of the lower weight. (1)
For long spans, it is of interest to have more sophisticated cross-section variation. As the forces increase towards the springing it should be reasonable to increase the cross-sectional dimensions in regions with higher force concentrations, as can be observed in Figure 2.13.
Figure 2.13 - Notations for bending stiffness of zero-hinge arch (1) Arch bending stiffness is expressed as
𝐷 = 𝐸𝐼 cos(𝜑) ( 2-12 )
Where E is the modulus of elasticity, 𝐼 is the moment of inertia and 𝜑 is the angle which tangent to the arc makes with the horizontal axis.
Cross-sectional variation methods mentioned by (1) provides few different possibilities of describing the variation. Strassner variation presents a case wherein ( 2-13 ) k = 1 while Ritter using the same ( 2-13 ) presents a case with k = 2, both are given for different values of n = D0/Da.
𝐷(𝑥)
𝐷0 =𝐸(𝑥)𝐼(𝑥) cos(𝜑(𝑥))
𝐸𝐼0 = 1
1−(1−𝐷𝑎
𝐷0)(1−2𝑥𝐿)𝑘 ( 2-13 )
Kasarnowsky has produced the case for variation in stiffness which is described by:
𝐷(𝑥) 𝐷0 =𝐷𝑎
𝐷0(1 −2𝑥
𝐿)4 𝑓𝑜𝑟 0 < 𝑥 <𝐿
2(1 − √𝐷𝐷𝑎
0
4 ) ( 2-14 )
𝐷(𝑥)
𝐷0 = 1 ,00 𝑓𝑜𝑟 𝐿2(1 − √𝐷𝐷𝑎
0
4 ) < 𝑥 <𝐿2 ( 2-15 )
Where 𝐷0 is the arch stiffness at the crown and 𝐷𝑎 is the stiffness at the springing.
The advantages of using one or the other variation strongly depend on the limitations that occur on the site, especially the condition of the foundation, other than that the variation type will determine the force distribution within the structure.
In Figure 2.14 one can observe an example of the comparison of the cross-sectional variation methods.
Figure 2.14 – Examples of cross-section variation when the stiffness ratio between the crown and springing is 𝐷𝑎/𝐷0= 3 (1)
The advantage of using predefined cross-section variation is that for these particular cases structural engineers have developed analytical solutions for force distribution within the structure, which can serve as a beneficial tool for verification of calculation models.
0.00 0.50 1.00 1.50 2.00 2.50 3.00
0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50
Dx/D0
x/L
Strassner Ritter Kasarowsky
2.4 Construction technique and sequence
It is expensive and complicated to construct the formwork and scaffolding for casting large concrete arch structures. Formwork for arch bridges is costly, various methods for rational formwork and assembly of arches have, therefore, been developed, see Figure 2.15. (1)
Types of formwork
Conventional formwork with shoring
Formwork suspended in hangers
Simply supported formwork Statically interacting formwork
Cantilever building procedures
The “Freivoorbau” method with suspended formwork
The “Freivoorbau” method with combined shoring suspended formwork
The “Freivoorbau” method with a truss supporting system
Suspended formwork approach
Figure 2.15 – Examples of construction procedures for arch bridges. (1)
The construction method, temporary support system and formwork largely affect the economic feasibility of building an arch structure. More so, the choice will affect the design procedure and principles. Stresses that accumulate during construction stages largely affect the structure and eventually the performance of the structure in exploitation. Without knowing exact steps in the construction sequence, it will be difficult to complete a thorough structural assessment analysis and provide accurately design structural components.
2.4.1 Temporary supports
Figure 2.16 illustrate a method which has been implemented in a construction process of an arch structure. The procedure follows the sequence where the foundation for the cast stages nearest to the supports is papered and the formwork is built with help of conventional form towers and a special truss system. Followed by temporary formwork with caissons is prepared and towers are erected on these.
The subsequent construction is done with cables as support. (1)
Figure 2.16 - Example of construction sequence of an arch bridge. (1)
This procedure might be difficult to apply for arches having lower aspect ratios, therefore, adequate methods likely have to be considered.
2.4.2 Prefabrication
Assembly of the arch structure is expected to be a lengthy process, more so, if concrete formwork is of varying nature so that it has to be adjusted for every consecutive stage. Evidently, this would largely affect the feasibility of the whole construction. A prefabrication process in these instances would be beneficial. A great example addressing this issue was used in The Gateway Arch construction. It was done by prefabricating stainless steel outer shells, which then would be assembled on site and filled with concrete, in this way avoiding time-consuming framework issues. While addressing some issues, other might have arisen, therefore, it is of utmost importance to evaluate all aspects of such a construction project to find the middle ground.
3 High-rise structures and Tube mega frame
3.1 High-rise structures
There is no absolute definition of what constitutes a “tall building” or a high-rise structure according to the Council of Tall Buildings and Urban Habitat (CTBUH), however, several criteria can be evaluated to establish whether a structure would qualify as one. Amongst the important aspects are height relative to context, proportions of the structure and tall building technologies. In more general terms, “The CTBUH defines “supertall” as a building over 300 meters (984 feet) in height, and a “mega tall” as a building over 600 meters (1,968 feet) in height. As of June 2015, there was 91 super tall and 2 mega tall buildings fully completed and occupied globally.” (6)
“In 2016, 128 buildings of 200 meters’ height or greater were completed, setting a new record for annual tall building completions and marking the third consecutive record-breaking year.” (7) Figure 3.1 illustrate the tendency of increasing height of high-rise structures. Clearly, it can be observed that the demand for building higher is genuine, which in turn sets the excellent environment for innovation to address new challenges arising as the height of the structures increase.
Figure 3.1 - The Average Height of the Tallest Buildings (7)
Conventional tall structures mostly employ common structural systems that have been used on various occasions and that have been thoroughly studied for their structural performance. Amongst the common systems are moment frames, tubes, core systems, tubed moment frames, trussed tubes, tube in a tube, outrigger systems and combinations of these systems as well. For a more thorough description of these systems, it is recommended to read (8).
In the urge of addressing the ever important aspect of usable floor area in high-rise structures, “Tyréns has developed a new structural system for super tall buildings called the Tubed Mega Frame (TMF). The main purpose of this system is to transfer all loads to the perimeter of the building and thereby achieve higher stability since the lever arm between the load bearing components will be longer than in a core system. With this structural system, there will be no central core. “ (8)
3.2 Tube mega frame
The structural principles of tube mega frame have been developed to increase the efficiency of the load bearing structure of high-rise buildings. The key features allowing this concept to accelerate ahead of the more conventional design techniques are the increase of moment of inertia at the base of the structure, leading to increased moment resistance, as well as the increase of a rentable floor space due to the elimination of central core (9).
The increase in the moment resistance at the base when using TMF leads to very appealing design possibilities. This aspect directly can contribute to increased slenderness of the structural system beyond what has been conceivable in the past (9).
A. Partovi and J. Svärd in (8) have comprehensively described and evaluated few different possibilities regarding structural design of TMF. The range of options varies from multiple frame members on the perimeter to few hollow mega columns. The following types of tubes mega frame layouts have been evaluated shown in Figure 3.2:
a) Perimeter frame with belt walls on three levels b) Perimeter frame with cross walls on three levels c) Mega columns with belt walls on three levels d) Mega columns with cross walls on three levels
a) b) c) d)
Figure 3.2 – Tube mega frame and bracing system arrangement (8)
The bracing system plays a major role in the force distribution within a structural system. In order to effectively transfer the forces between the mega tubes bracing system has to be designed accordingly.
In the study by A. Partovi and J. Svärd in (8) tube mega frame has been evaluated using belt walls for bracing purposes, while theoretically this function can be fulfilled by truss-like bracing, moment frames amongst others.
All of the options in Figure 3.2 cover the majority of different layout principles that can be engaged when using tube mega frame concept. While it has been shown that the TMF concepts perform better than more conventional structural systems, there is no clear favorite type of TMF that would be the most efficient across all of the loading situations. The structural system preference is strongly related to the actual condition at the location of the building site and can be tailored to the specific circumstances.
4 Finite Element Method
“Finite element method is a way of numerically solving field problems which are described by differential equations. There are many different areas where this method can be useful besides structural systems; heat transfer and magnetic fields among others. When using this method, the structure is divided into small elements which are assigned chosen geometry and material properties.
The division of a model into elements is called discretization. These elements are, as the name of the method implies, not infinitesimal but finitely small. The elements are connected to each other at nodes.
The nodes are assigned boundary values and restraints. It is at these nodes the analysis yields the results, and the values are then interpolated between the nodes to get results there as well. The network of elements attached to each other is called mesh. “ (10)
The outcome of the finite element analysis is only an approximation, considering the fact that the elements used for creation of the model are not infinitely small, therefore it is of utmost importance to be aware of the element type and size used for the analysis, as these factors greatly contribute to the accuracy of the desired results. In order to achieve more accurate results, either more elements or higher order elements have to be used, thus providing a solution that mathematically converges toward the real solution. The advantage of the finite element method is that it is possible to model any structure regardless of the complexity of the geometry and acquire convergent results.
The finite element method is applied by using computer software – SAP2000. It is very important to be familiar with the software and be aware of its shortcomings. There is various software available to solve finite element problems and most of them have developed unique algorithms to solve differential equations, therefore an important aspect of improving the quality of the results and procedures, is to be accustomed to the particular software at use.
4.1 Linear elasticity
The linear static analysis of a structure consists of solving the system of linear equation defined by:
𝐾 ∙ 𝑈 = 𝑅 ( 4-1 )
Where K is structure stiffness matrix, U is a vector of the system global displacements and R is a vector of forces acting at the direction of the structure global displacement. (11)
NONLINEAR STATIC ANALYSIS
The nonlinear static analysis is used for a wide variety of purposes; however, the subject of interest is to analyze the structure for geometric nonlinearity to form the P-Delta stiffness for subsequent linear analyses, also to investigate staged (incremental) construction sequence.
Geometric nonlinearity is considered in the form of both P-Delta effects and large- displacement/rotation effects. Strains within the elements are assumed to be small. The implementation is based on a step-by-step basis in nonlinear static analyses and incorporated in the stiffness matrix for linear analyses.
THE P-DELTA EFFECTS
“The P-delta effect refers specifically to the nonlinear geometric effect of a large tensile or compressive direct stress upon transverse bending and shear behavior. A compressive stress tends to make the structural member more flexible in transverse bending and shear, whereas a tensile stress tends to stiffen the member against transverse deformation.” (12)
The P-Delta effects are particularly of interest in structures where the effects significant gravity loads affect the lateral stiffness of the structure.
One can observe P-Delta effects in Figure 4.1, where cantilever beam is subjected to axial load P and a transverse load F. If equilibrium conditions are enforced the internal axial load in the element is equal to P. In the original configuration, the moment at the support is
𝑀𝑠= 𝐹𝐿 ( 4-2 )
And decreases linearly to zero at the loaded end. On the other hand, if deformed configuration is considered, the moment at the support becomes
𝑀𝑠= 𝐹𝐿 + 𝑃∆ ( 4-3 )
Where axial load causes additional moment due to the eccentric action on the beam and obliterates the linear moment variation along the beam. When the beam is loaded by compressive force, the moment throughout the beam are increased resulting in increased transverse deflection.
Once the compressive force reaches a certain magnitude, the transverse stiffness goes to zero forcing deflection to tend to infinity – the structure has buckled. (12)
LINEAR BUCKLING ANALYSIS
Linear buckling analysis estimates the instability modes of the structure due to the P-Delta effects under a specified set of loads. The analysis considers the solution of the generalized eigenvalue problem:
[𝐾 − 𝜆𝐺(𝑟)]𝛹 = 0 ( 4-4 )
Where K is the stiffness matrix, 𝐺(𝑟) is the geometric (P-Delta) stiffness due to the load vector 𝑟, 𝜆 is the diagonal matrix of eigenvalues, and 𝛹 is the matrix of corresponding eigenvectors (Mode shapes).
Each eigenvalue-eigenvector pair is referred to as a buckling mode of the structure. The mods are identified by numbers from 1 to 𝑛. The eigenvalue 𝜆 is referred to as the buckling factor. It is a scale factor that must multiply loads in 𝑟 to cause buckling in the given mode. If the buckling factor is greater than one, the load must be decreased to prevent buckling. Buckling modes depend on the load, therefore there are multiple sets of buckling modes for a structure, hence the structure must be explicitly evaluated for buckling for each set of loads of concern. (12)
For each buckling case either the stiffness matrix of the full structure in its unstressed state, or the stiffness of the structure at converged nonlinear state, to include P-Delta effects, can be applied.
Figure 4.1 – Geometry and moment diagrams for cantilever beam (12)
4.2 Structural dynamics
Modal analysis is used in order to determine the vibration modes of the structure. The results of the modal analysis provide useful tools to increase the understanding of the behavior of the structure.
Modal analysis is implemented as linear and is either based on the stiffness of the full unstressed structure or upon the stiffness at the end of a nonlinear load case. By using the stiffness at the end of a nonlinear case, one can evaluate the modes under P-delta conditions and at different stages if construction. (12)
Eigenvector analysis determines the undamped free vibration mode shapes and frequencies of the structural. These natural Modes provide insight into the behavior of the structural system. Eigenvector analysis consists of solving the generalized eigenvalue problem:
[𝐾 − 𝛺2𝑀]𝛷 = 0 ( 4-5 )
Where K is the stiffness matrix, M is the diagonal mass matrix, 𝛺2 is the diagonal matrix of eigenvalues, and 𝛷 is the matrix of the corresponding eigenvectors (mode Shapes). Each eigenvalue-eigenvector pair is referred to as a natural Vibration Mode of the structural system. The Modes are identified by numbers from 1 to n in the order in which the modes are found in the solution. (12)
The eigenvalue is the square of the circular frequency 𝜔 for that Mode
The cyclic frequency 𝑓𝑐 and period 𝑇 of the Mode are related to circular frequency by:
𝑇 =1𝑓 ( 4-6 )
𝑓𝑐 =2𝜋𝜔 ( 4-7 )
4.3 Discrepancies in FEM application
Finite Element Modelling of arched structures requires a thorough understanding of the FE method.
“One has to be careful when performing a finite element analysis since there are different kinds of errors that can be introduced that can affect the accuracy of the results. First of all, modeling error can arise if the model is wrongly computed or too many or too incorrect simplifications are made. Another problem can be discretization error which can occur if the elements are too large and can yield a less accurate result since the distance between the nodes becomes greater. This can be improved by dividing the structure into more elements. Furthermore, numerical error is introduced when the computer makes calculations with a finite number of decimals. These are just a few possible errors that can occur, there are others besides from these that one has to be aware of” (10)
The ability to predict and control the errors in finite element analysis, provide useful modeling tool.
Where possible, analytical or numerical hand calculation techniques will be used to grant reassurance of the results. This is common way handling complex models, as due to their complexity it is very difficult to recognize some design flaws or errors. In some sense, this might be considered as an improvement of the construct validity of the procedure, specifically, convergent validity, as the results of the two approaches taken should lead to the same result – converge.
4.3.1 Material
Applying accurate material properties in a Finite Element Model is often a difficult task. Frequently common building materials exhibit highly nonlinear material behavior, including concrete, which is largely modeled having linear elastic material properties, when analyzing the response of a structure and determining internal forces and moments. Implementing nonlinear material properties in large models require a large amount of work in defining the material accurately, as well as the computational requirements, increase drastically due to the complexity of the model. Linear material assumptions, however, are sufficient on many occasions. Although, certain cases explicitly require nonlinear material behavior. Nonlinear material behavior should be considered when designing slender columns and thin slab structures, where members undergo large deformations. Also, optimization of the load redistribution and reduction of member forces in case of restraints can be considered. Nonlinear material properties also must be used for comparison of test results to acquire exact deformation of the structure and simulate accurate failure conditions and damages.
4.3.2 Load
The applied load in a finite element model is always distributed to the nodes, regardless of the type or nature of the action.
Self-weight of the structure is integrated over the volume of a finite element and distributed to the connecting nodes in the model.
Live load is applied as a distributed area load on floor diaphragms and distributed to surrounding support structures.
Wind loads are automatically calculated by exposure to floor diaphragms using the available tools in ETABS. A separate lateral load is created for each diaphragm present at a story level. The wind loads at any story level are based on the story level elevation, the story height above and below that level, the assumed width of the diaphragm at the story level, and various code-dependent wind coefficients. (13) Figure 4.2 shows the principal lateral wind load distribution to the floor diaphragms.
Figure 4.2 - Example extent of wind loading (13)
