INTRODUCTION
Background information
Inflatable structures, commonly utilized in entertainment and performance projects like children's play areas and welcome gates, represent a relatively new field in Vietnam Designing and analyzing these structures poses significant challenges due to their structural responses, which heavily rely on the air within and the material of the skin Among various types, inflatable beams are frequently employed in practical applications and serve as a fundamental model for understanding the mechanical behavior of inflatable structures While research has extensively documented the flexural performance and strength of inflatable beams through analytical and numerical modeling, studies focusing on their buckling and stability responses remain limited.
Orthotropic fabric materials are increasingly utilized across various industries due to advancements in weaving techniques that enhance their strength and resistance to diverse conditions Recently, the mechanical advantages of these materials have prompted interest in their application for inflating structures, yet research on their use as inflating beams remains limited in both experimental and numerical modeling contexts Economically, numerical modeling has gained preference in recent years, benefiting from advancements in computer science.
Isogeometric Analysis (IGA) is a cutting-edge numerical method that integrates Computer-Aided Design (CAD) tools with Finite Element Analysis (FEA), utilizing Non-uniform Rational B-spline (NURBS) functions for geometrical modeling This innovative approach streamlines the design process by eliminating the need for transformation between sketching and analysis modeling, allowing for precise representation of geometries with high smoothness The use of NURBS not only enhances the accuracy of geometric representation but also improves convergence rates due to their higher-order capabilities Additionally, IGA offers superior continuity among elements compared to traditional FEA methods, which typically rely on Lagrange interpolation functions with C0 continuity These advanced features make IGA an ideal choice for analyzing small-scale structures using high-order continuum theories.
The motivation of the thesis
The increasing popularity of woven fabric composite materials necessitates a thorough investigation into their applications in inflatable structures This study focuses on evaluating the structural performance of woven fabric beams subjected to compressive loads through both experimental and numerical modeling methods Additionally, the innovative application of Isogeometric Analysis (IGA) to assess the stability behavior of inflatable structures has not been previously explored, making this new numerical approach a significant contribution to the field.
The objectives and scope of the study
This study aims to explore the critical loads of inflated beams constructed from composite textiles through experimental and numerical modeling methods The key objectives of the research can be summarized as follows.
1) Develop an experimental program, in which:
1.1) Determine textile composite material constants These constants are used as input data for calculation programs
1.2) Set up experiments on inflatable beams in terms of equipment and supplies that can be found locally and purchased from abroad
1.3) Determine the critical load of steam beam structures of composite textile materials with different boundary conditions
1.4) Investigate the effect of initial internal pressure on the strength of the inflatable beam
1.5) Investigate the effect of initial internal pressure on critical load causing buckling of the inflatable beam structure
2) Apply the "Isogeometric Analysis - IGA" technique to develop a numerical program to determine the critical load for the composite woven fabric's inflating beam with different boundary conditions A piece of code in Matlab is developed
3) Compare the experimental results and those obtained from the numerical approach to validate the accuracy of the developed program.
Methodology
To achieve the study scopes, this thesis has used several methods as follows:
- Studying literature review related to the subjects of textile composite materials and inflating structures
This section focuses on reviewing, analyzing, and synthesizing critical load calculation models for inflatable beam structures made of composite textiles The aim is to select an appropriate model for both analytical equations and finite element analysis By enhancing the available analytical and traditional finite element tools, this research seeks to validate the results obtained from Isogeometric Analysis (IGA) and the experimental findings presented in the thesis.
- Review of NURBS-based geometry and isogeometric analysis (IGA)
- Derive theories for nonlinear buckling analysis of inflating composite structures under the IGA framework and investigate numerical models
- Construct analytical model and experiment program for verifying the proposed theory.
Outline of the thesis
The contents of this thesis are briefly organized as follows:
Chapter 1 provides an overview of the study's background, objectives, methodology, and thesis outline, highlighting its significance and contributions It emphasizes the importance of developing an effective analysis model for inflating structures.
- Chapter 2 gives a brief review of fibrous composite materials and their applications A literature review of previous studies on inflating structures is presented
Chapter 3 explores the fundamental features of Isogeometric Analysis (IGA) and the theoretical development of stability governing equations related to buckling problems It introduces essential concepts vital to the IGA approach, including knot vectors, B-splines, and NURBS basis functions Additionally, the chapter compares the advantages and disadvantages of IGA with traditional finite element methods The latter part delves into the theoretical framework for deriving the governing equations of stability problems, detailing the basic assumptions and the intricate derivation process.
Chapter 4 focuses on the advancements in the isogeometric analysis (IGA)-based numerical model, beginning with a brief overview of IGA It details the development procedure of the IGA-based model and outlines the general approach for solving the global equation This research emphasizes linear eigen analysis and nonlinear buckling analysis of inflating beams made from orthotropic materials using IGA The study evaluates the effects of geometric nonlinearities and inflation pressure on the stability response of inflating beams with various boundary conditions, highlighting how beam aspect ratios impact the buckling load coefficient Additionally, the results obtained are compared with existing experimental data from the literature.
Chapter 5 focuses on materials selection and the prototyping plan, emphasizing the relationship between load and buckling behavior through varying pressure It presents an experimental program investigating the buckling behavior of inflatable beams made from woven fabric composites The chapter begins with a review of thin-walled shell structures' buckling, followed by material testing of woven fabric composites It then details the fabrication process of the inflatable beams and the setup for buckling tests, concluding with a discussion of the results obtained.
Chapter 6 summarizes the research contributions and achievements of this work, highlighting key conclusions and findings Additionally, it offers suggestions for future studies to further explore the topic.
Original contributions of the thesis
In this study, the original contributions of the thesis are covered as follows:
This article explores an advanced IGA-based numerical method to investigate the nonlinear buckling behavior of inflated beams constructed from woven fabric composite materials The proposed approach utilizes a Higher Order Weak Formulation (HOWF) to analyze a 3D Timoshenko beam model A finite element model is developed with C1 continuity through quadratic NURBS-based Timoshenko elements Furthermore, the biaxial orthotropic mechanical properties of the composite material are identified as essential inputs for both the finite element model and the IGA analysis.
- Experimental investigation on determining the critical buckling load and load-carrying capacity of the inflating composite fabric beams
- Study on effects of different air pressures to determine the load-displacement relation of the inflating beam.
Significances of the thesis
In today's industrial landscape, various materials such as wood, metal, stone, and fabric are extensively utilized, with inflating structures increasingly replacing traditional materials like columns, beams, and arches Advances in weaving techniques have enabled the creation of closed tubes from construction fabrics, which can be inflated to support their own weight and additional loads Modern textile materials offer significant advantages over conventional options, including customization for specific applications, ease of deployment, lightweight properties, and compact storage These innovative inflating structures find applications across diverse fields, including aerospace, civil engineering, military, marine, agriculture, and entertainment, necessitating a thorough understanding of material behavior for effective structural design and optimization.
Research on the stability and bearing capacity of inflating structures in Vietnam remains limited, with a notable absence of scientific literature addressing the use of this innovative material in construction This thesis aims to fill this gap by exploring, developing, and modeling the mechanical properties of technical fabrics, as well as establishing calculation theories for inflating composite structures, ultimately enhancing their application in the construction industry.
OVERVIEW OF FIBROUS COMPOSITE MATERIALS AND
An overview of fibrous composite materials and practical applications of inflating
In addition to traditional construction materials like wood, stone, and metal, fabric materials are increasingly utilized in various industrial and construction applications due to advancements in weaving techniques that enhance their durability These construction fabrics are typically designed to form inflated structures, which can support critical loads and are applicable in diverse settings such as lunar bases, stadium domes, exhibition halls, and temporary outdoor structures The use of these lightweight materials allows for easy deployment and rearrangement, while also accommodating unique geometries and minimizing storage space Furthermore, their durability, low production costs, and reduced development expenses—eliminating the need for specialized deployment tools—provide significant advantages over conventional construction methods.
The study of composite materials has a rich history, as natural entities like the human body, plants, and animals are inherently composites In the early 1960s, the potential of engineered fibrous composite materials was recognized, leading to rapid advancements in their development and application The textile structural composite highlighted in this research exemplifies modern materials, defined as composites consisting of two or more distinct phases, making them heterogeneous In fibrous composites, one phase serves as reinforcement for another, enhancing their overall properties.
The second phase is called the matrix The challenge is to combine the fibers and the matrix to form the most efficient material for the extended application
Textile preforms represent an innovative material that consists of fibers arranged in a specific orientation and pre-impregnated with matrix materials The microstructural configuration of these fibers significantly influences various material characteristics, including pore geometry, pore distribution, and tortuosity within composites Consequently, textile preforms play a crucial role in not only conveying the properties of the fibers but also in impacting the overall properties of the matrix.
Flexible fibers such as glass, carbon, and aramid can be woven into textile fabrics and impregnated with matrix materials, offering a variety of weave patterns Among these, plain woven composites and homogeneous orthotropic woven fabric (HOWF) composites are notable for their orthotropic properties, featuring plain weaves where every fiber alternates over and under perpendicular fibers, and two-harness satin weaves that interlace fibers differently Woven fabrics exhibit superior in-plane transverse effective properties compared to unidirectional lamina, making them ideal for structural configurations with significant curvature and enhancing their durability during handling.
Continuous fiber composites utilize various matrix materials, including polymers, metals, and ceramics Among polymeric matrices, thermoplastics and thermosets are the two main categories Thermoplastics, such as polyphenylene sulfide (PPS) and polysulfone, can be reshaped when heated, making them suitable for high-toughness, low-cost processing, with a useful temperature range up to 225°C In contrast, thermosets like polyesters, epoxies, and polyimides are cross-linked during fabrication and do not soften upon reheating Polyesters are favored for their cost-effectiveness and lightweight properties, while epoxies offer superior moisture resistance despite being pricier Polyimides, although challenging to fabricate, can withstand temperatures up to 300°C High-performance laminated composite structures are increasingly utilized in modern aerospace designs to improve structural efficiency, prompting further research into advanced fiber materials such as glass, carbon, and aramid fibers for applications in membrane and thin shell structures.
Inflating structures are composed of elastic or plastic fabric textiles that utilize air pressure to maintain their shape and stiffness These inflatable beams offer several advantages, including impact load absorption, durability, lightweight design, easy assembly, and minimal storage space requirements Additionally, their low manufacturing cost makes them highly effective for industrial applications Examples of composite materials used in aerospace can be found in Figures 2.2 to 2.5.
Figure 2.2 30 meter ECHO I Balloon Satellite [8]
Figure 2.4 Inflating lunar habitat proposal [8]
Figure 2.5 Inflating aircraft [8] inflating composite structures
The first pneumatic building, designed by Frederick William Lanchester in 1917, utilized low air pressure for support, eliminating the need for poles In response to demands from the War Production Board in 1942, numerous inflatable structures were created, which are now commonly used as emergency shelters after natural disasters, decontamination zones, and tents for organizations like the Red Cross and military Additionally, inflatable places of worship, including churches and mosques, have been constructed This innovative technology has significantly reduced the transportable weight of tents by 66%, their volume by 75%, and setup time by 50%, making payload and optimal design crucial for engineers Inflatable structures have also made their mark in civil engineering, with notable examples such as the Carrier Dome in the USA (1980), BC Place Stadium in Canada (1983), and the Tokyo Dome in Japan (1988).
Figure 2.6 Inflating structures (source: internet) a Carrier Dome, USA b BC Place Stadium, Canada c Tokyo dome, Japan
Figure 2.7 Inflating stadiums (source: internet)
Inflating structures are increasingly utilized across various sectors, including marine, submarine, and agricultural applications, due to their lightweight and efficient design Understanding the stable behavior of these structures is crucial for users; however, there is a lack of comprehensive references on the buckling behavior of inflating structures made from plain woven composites A review of existing literature highlights the necessity of developing effective analytical models for these structures to ensure their reliability and performance.
Literature review on the analysis of inflatable beams
Research on inflatable beams in Vietnam remains limited Recently, Nguyen [9] undertook a project focused on the analysis and application of inflatable structures in construction This study involved the development of both analytical and numerical models to assess the structural response of inflatable beams Additionally, an experimental program was implemented to validate the accuracy of these numerical and analytical methods.
This subsection presents a literature review focused on the analysis of inflatable beams, which can be categorized into two primary approaches: analytical and numerical The subsequent sections will delve into these methodologies and their respective analyses.
Numerous researchers have extensively studied the behavior of inflating structures using analytical methods, with some applying Euler-Bernoulli kinematics to model inflating beams Notably, Comer and Levy derived load-deflection theory for an inflating isotropic beam, which was later expanded upon by Webber.
Recent studies have focused on predicting the behavior of inflating cantilever beams, with significant contributions from researchers such as Main et al., who enhanced existing theories through experimental work on isotropic beams Suhey et al analyzed pressurized tubes under uniform loads using Euler-Bernoulli kinematics, which assumes isotropic material properties and provides theoretical results for deflection However, Timoshenko's kinematics has been identified as more suitable for structures under pressure loads, as it addresses limitations in Euler-Bernoulli's approach Fichter derived nonlinear equations for bending and twisting in inflating cylindrical beams based on key assumptions about cross-sectional behavior and strain characteristics Topping and Douglas further explored the structural stiffness of inflating cantilever beams by integrating finite elasticity theory and considering geometric and material changes during inflation Additionally, Wielgosz and Thomas developed analytical solutions for inflating panels and tubes using Timoshenko beam theory, demonstrating how applied pressure affects limit loads and deflections Their experimental and analytical work on highly inflating fabric tubes confirmed the validity of beam theory in predicting deflections, while Le and Wielgosz formulated nonlinear equations for inflating isotropic beams using Lagrangian principles.
The literature review focuses on the linearization of nonlinear equilibrium equations around a pre-stressed reference configuration, rather than the natural state that needs to be defined This advancement has led to improvements in Fichter's theory through the application of these linearized equations.
Despite extensive research efforts over the years in developing analytical methods, most studies have concentrated on isotropic fabric materials To date, there has been limited research addressing orthotropic fabric materials.
Inflating beams present considerable challenges for analysts, particularly when analytical solutions are elusive due to complex loading and boundary conditions Significant prior research has focused on the numerical modeling of these beams Steeves formulated governing differential equations for lateral deformation based on minimal potential energy, simplifying the model to one dimension while assuming undeformed cross-sections The finite element approach was utilized by Quigley et al and Cavallaro et al to predict the linear load-deformation response, incorporating a pressure stiffening term that suggested an unbounded increase in beam stiffness with rising inflation pressure Wielgosz and Thomas, along with Thomas and Wielgosz, explored the load-deflection behavior of expanding fabric tubes and panels using Timoshenko beam theory, treating internal pressure as a follower force while neglecting fabric wrinkling Bouzidi et al reported theoretical and numerical advancements for axisymmetric bending problems in pressured isotropic membranes, focusing on normal pressure and significant displacements Suhey et al conducted a numerical simulation for an inflated aquaculture cage, employing nonlinear finite elements to enhance stability Le and Wielgosz discretized nonlinear equations for strongly inflated isotropic beams, achieving close numerical results to their 3D finite element models Davids and Zhang developed a Timoshenko beam finite element for nonlinear analysis, emphasizing the importance of pressure work in their formulations Malm et al validated standard beam theory against a finite element model for isotropic fabric beams, while research into orthotropic fabric materials has emerged, with Plaut et al investigating the effects of environmental stresses on inflated arches using Sanders' linear thin-shell theory.
LITERATURE REVIEW was considered to be linearly elastic, nonhomogeneous, and orthotropic The Rayleigh-Ritz technique was used to get approximate solutions Plagianakos et al
A study on low-pressure Tensairity explored its application potential, focusing on axial compressive stresses through experiments on a spindle-shaped column Displacements were measured along the span, while axial forces were determined using strain gauges, yielding results that aligned well with finite element and analytical predictions Additionally, Nguyen et al examined an analytical method to approximate the critical load for a 3D Timoshenko beam, finding that the inflatable beam model effectively matched existing literature The governing equations for the beam were formulated using the total Lagrangian form of Timoshenko kinematics and virtual work principles Mohebpour's research addressed the deflection analysis of inflatable beams and panels using the finite element method to assess flexural behavior under high pressure across various conditions Furthermore, Apedo et al conducted a free vibration analysis of inflatable beams made from woven fabric, employing the dynamic stiffness method alongside a 3D Timoshenko beam model to describe kinematic relations Eslabbagh implemented nonlinear analysis of axisymmetric inflatable beams in their work.
The proposed model effectively simulates the performance of inflatable beams, capturing the wrinkling phenomenon in a nonlinear framework Clapp et al investigated the bending response of reinforced inflatable beams using a 3D shell-based finite element model Experimental and numerical studies by Guo et al revealed that air-inflated arch frames demonstrate excellent structural integrity, with coupling beams crucial for spatial action Liu et al utilized a Fourier series method to explore the interactive bending wrinkling behavior of inflated beams, finding that buckling characteristics are sensitive to structural dimensions and boundary conditions Tao et al discussed the wrinkling and collapse behavior of mesh-reinforced membrane inflated beams, noting that the collapse load improves due to the interaction between mesh and membrane Young et al developed an efficient finite element model for inflatable members with axial reinforcement, achieving good agreement with shell-based models and component-level tests Chen et al conducted deformation analysis of tapered inflatable beams based on Timoshenko beam theory, using analytical and finite element methods to assess geometric influences and taper ratios Venter and Venter proposed a simulation study for dunnage bags made from plain woven polypropylene fabric, while Xue et al examined the buckling behavior of inflated arches, discovering that wrinkles can reduce failure loads significantly Wang et al created a model to investigate wrinkling and restabilization effects in hyperelastic sheets, and Vernarsky et al reported on numerical simulations predicting the response of Tensairity cylindrical beams under circumfluence wind pressure.
The literature review focuses on the tensairity cylindrical beam, examining vortex-shedding phenomena and galloping effects Davids et al conducted both experimental and numerical studies on the flexural responses of inflatable drop-stitch fabric panels and beams, revealing that their model effectively predicts pre- and post-wrinkling bending behavior across a broad range of deflections Their findings highlight the significant influence of pressure–volume work, shear deformations, and drop yarns on bending responses Additionally, Wei et al developed a quasi-static model to analyze the restoring moment concerning the deployment angle of an inflated beam, demonstrating that both the restoring force and moment increase with pressure at a defined angle, while the restoring moment exhibits a nonlinear relationship with the deployment angle.
Current literature primarily focuses on inflating tensile structures and their response to service loads, particularly lightweight designs Previous studies typically assumed homogeneous isotropic materials for beams and utilized membrane or thin shell theory to analyze structural responses Libai identified governing equations for the incremental stress state in membrane tube-shaped orthotropic circles, considering hyperelastic membranes under small perturbations from uniform internal pressure and longitudinal extension The approach relied on linearization around a known homogeneous reference state, employing Hermite cubic shape functions with variational principles Researchers like Wielgosz and Thomas developed inflating beam finite elements to compute deflections in hyperstatic beams, incorporating membrane elements Bouzidi introduced two finite elements for 2D inflating membrane problems, focusing on axisymmetric and cylindrical bending under large deflections and finite strains By solving a direct optimization problem using the minimum total potential energy theorem, numerical solutions were achieved Cavallaro's work highlighted that pressurized structural tubes differ fundamentally from conventional metal and fiber/matrix composite structures, noting that plain-woven fabric behaves as an orthotropic material rather than a continuum, with effective material qualities influenced by the internal pressure of individual tows and their weave geometry.
[13] used membrane elements to create a finite element model of an expanding open- ocean aquaculture cage, assuming the material is anisotropic
To analyze the deflection and stress at the onset of wrinkling in fabric materials, researchers have utilized nonlinear components to model tension-only behavior Main [42] applied a modified conventional beam theory to validate the results, while computational findings were derived from a beam element based on Fichter's earlier work [14] and the 3D isotropic fabric membrane finite element developed by Le van [19] and Le van [23] The governing equations were discretized using the virtual work principle, Timoshenko's kinematics, finite rotations, and modest stresses A linear eigen buckling study was performed using 3D membrane finite element calculations, alongside a mesh convergence test Fichter [14] also explored both linear and nonlinear finite element solutions in bending by discretizing nonlinear equilibrium equations from his previous analytical model, which considered a homogeneous orthotropic fabric.
Inflating structures face nonlinear challenges due to local buckling, which results in wrinkle formation The complexity of solving the associated nonlinear equations arises from geometric nonlinearity, where significant deformations necessitate equilibrium equations based on the deformed geometry Despite the importance of this topic, there have been limited studies on the buckling analysis of inflated structures Notably, Diaby et al conducted a numerical analysis of buckles and wrinkles in membrane structures using the complete Lagrangian formulation, as developed by Le van Their bifurcation analysis was notable for not making assumptions regarding structural defects Additionally, Davids contributed to this field with further developments.
This literature review discusses a quadratic Timoshenko beam element designed for an inflating beam, utilizing an incremental virtual work approach that addresses fabric wrinkling through moment-curvature nonlinearity It is important to note that the materials examined in these experiments were considered isotropic.
Finite element studies of expanding fabric constructions present challenges due to material and geometric nonlinearities, which arise from the fabric's nonlinear load/deflection behavior at low loads, pressure stiffening during inflation, fabric-to-fabric contact, and wrinkling on the structural surface To address these complexities, finite element models are employed to predict the fundamental mode of inflating fabric beams and assess the loads on fabric elements Notably, Aledo conducted a theoretical analysis using a homogeneous orthotropic fabric, while Apedo developed a three-dimensional Timoshenko beam model to explore the nonlinear equations associated with bending issues.
Conclusions
This chapter provides an overview of fibrous composite materials, which are renowned for their advanced properties and widespread application in both industrial and scientific fields Particularly in structural engineering, these materials are primarily utilized for enhancing the performance of inflatable structures.
The literature review indicates that the analytical approach is predominantly used to analyze the structural responses of inflating structures, with most previous research focusing solely on isotropic materials There is a noticeable lack of studies addressing orthotropic materials, and the application of numerical methods in this field remains limited This highlights the need for further investigation, motivating this research to explore the structural behavior of composite fabric inflating structures through both experimental and numerical methods.
THEORETICAL FORMULATIONS
Continuum-based governing equations of stability problems of inflating beams
The literature extensively explores analytical analyses of inflating beams and arches through both theoretical and experimental investigations Developing optimal analytical models for beam structures is crucial, with the Euler-Bernoulli and Timoshenko kinematics frequently employed to derive solutions and formulas for inflating woven fabric beams Comer introduced a load-deflection hypothesis for isotropic beams, while Main and Main proposed a shear-moment approach model for evaluating inflated fabric beams, utilizing the orthotropic membrane model Fichter applied an energy minimization technique in his analytical buckling analysis of Timoshenko cylindrical inflating beams made from elastic isotropic textile fabric, examining how air pressures impact the beam's load-carrying capacity.
The beam theoretical model is founded on several key assumptions: the cross-section of the inflating beam remains undeformed under load, translation and rotation are minimal, and circumferential strain is negligible Wielgosz presented analytical solutions for inflating plates and tubes using Timoshenko kinematics, addressing geometric stiffness and the residual force effect from internal pressure They asserted that the applied pressure sets the limit load, with deflections inversely related to the material properties of the fabric and the pressure applied To enhance Fichter’s theory, Le-Van and Wielgosz introduced a new formulation based on the virtual work principle in Lagrangian form and Kirchhoff's hypothesis, leading to the derivation of nonlinear equations for inflating beams.
[24] and Davids [25] presented the nonlinear load-deflection response of Timoshenko inflating beams Parametric studies have been also investigated in their work Malm
[26] used a 3D isotropic fabric membrane finite element model to predict the beam load-deformation response
This chapter utilizes theoretical formulations established by Nguyen and his colleagues to address the buckling issues of inflated composite beams The governing equations derived will be discretized using Isogeometric Analysis (IGA) in the subsequent chapter to obtain numerical solutions for these buckling problems It is important to highlight that Nguyen's earlier research employed the conventional finite element method to tackle similar challenges.
3.1.1 Mathematical description of inflating beams
This study investigates Timoshenko beams constructed from orthotropic materials, focusing on their behavior under a two-stage loading process for inflatable structures Initially, the beam is subjected to an internal pressure of zero, remaining in its original state before inflating to a pressure p Following this, additional external pressures are applied The reference configuration is established upon the completion of the first stage, as illustrated in Graph Figure 3.1a and Figure 3.1b Due to the presence of geometrical nonlinearities, the Green-Lagrange strain measure is employed to assess the deformation of the beams.
Figure 3.1 illustrates an inflating cylindrical beam constructed from a HOWF material The parameters L, R, t, A, and I denote the length, external radius, fabric thickness, cross-sectional area, and the second moment of inertia about the principal axes, respectively.
Y and Z of the beam in the reference configuration which is the inflating configuration A 0 and I 0 are given by
I = A R 3.2 where the reference dimensions l R 0 , 0 and t 0 depending on the inflation pressure and the mechanical properties of the fabric Apedo [44]:
= + 3.5 where l R , and t are length, fabric thickness, and external radius of the beam in the natural configuration
The internal pressure \( p \) is considered constant, streamlining the analysis and supporting experimental results and prior studies on inflated fabric beams and arches The structural analysis excludes the initial pressurization phase, which takes place prior to the introduction of concentrated and dispersed external stresses.
= where L= l 0 is the beam length and
= A is the beam radius of gyration The coefficient takes different values according to the boundary conditions of the beam
The centroid of the current cross-section is located on the X-axis, with point M situated on this cross-section When axial loading is applied to the beam, two of Fichter's simplifying assumptions come into play.
- The inflating beam under discussion is assumed to have a circular cross- section that maintains its shape after deformation, preventing distortion and local buckling;
- The rotations around the beam's primary inertia axes are minor, and the rotation around the beam axis is insignificant
In this section, the governing equations of inflatable beams are derived The general hypotheses and assumptions are based on the work of Nguyen and his colleagues ([3], [2], and [4])
The material is considered orthotropic, with the warp direction of the fabric aligned with the beam axis, resulting in a circumferential pattern for the weft yarn The model can be adapted for different orientations by applying an additional rotation to connect the orthotropic directions to the beam axes However, this article does not address scenarios where the orthotropic primary directions coincide with the longitudinal and circumferential directions of the cylinder.
With the hypotheses proposed by Fichter were applied, the displacement field of an arbitrary point M(X, Y, Z) are expressed as follows:
3.6 where u X ,u Y and u Z are the components of the displacement at the arbitrary point
M, whilst u X v X ( ) ( ) , and w X ( ) are the displacements of the centroid G 0 of the current cross-section at abscissa X, related to the base (X, Y, Z); Y ( ) X and Z ( ) X are the rotations of the current section at abscissa X around both principal axes of inertia of the beam, respectively By defining u as an arbitrary virtual displacement from the current position of the material point M:
The definition of the strain at an arbitrary point as a function of the displacements is: l nl
The equation E = E + E 3.8 defines the relationship between Green-Lagrange linear strain (E t) and nonlinear strain (E nl), with the latter incorporating geometrical nonlinearities Additionally, the strain fields are influenced by the displacement fields.
The higher-order nonlinear terms are the product of the vectors that are defined as follows
Herein, the Saint Venant-Kirchhoff orthotropic material is employed The energy function = E ( ) E now is known as the Helmholtz free-energy function
To analyze the behavior of the inflating beam, we establish two coordinate systems: one that aligns with the local warp and weft directions of the membrane at each point, corresponding to the fabric's principal directions, and another that is a Cartesian coordinate system fixed to the beam.
The components of the second Piola-Kirchhoff tensor S are given by the nonlinear Hookean stress-strain relationships o o
Figure 3.2 Transform of coordinate systems where, S o is the inflation pressure prestressing tensor
- The second Piola-Kirchhoff tensor is written in the beam coordinate system as
- C is the fourth-order elasticity tensor expressed in the beam axes
In general, the inflation pressure prestressing tensor is assumed spheric and isotropic Wielgosz [44] So, o o
S =S I 3.13 where I is the identity second-order tensor and o o o
The prestressing scalar, denoted as A, is integral to calculating the elasticity tensor in beam axes This tensor can be derived from the local orthotropic elasticity tensor through the application of the rotation matrix R, as referenced in Apedo [44].
The elasticity tensor in the beam axes then obtained as
3.17 where c=cos and s = sin with =( e n Z , ) being the angle between the Z-axis and the normal of the membrane The components of the elasticity tensor are given by
1 ; t l tl lt tl lt tl t l t lt lt tl lt tl
The balance equations of an inflating beam come from the Virtual Work Principle (VWP) The VWP applied to the beam in its pressurized state is int ext d ext p ,
In the context of the equations presented, the body forces per unit volume, denoted as f, and the traction forces, which are derived from the second Piola-Kirchhoff tensor S, are essential components of the formulation Specifically, the traction forces on the left-hand side of Eq 3.18 are expressed in relation to the virtual Green strain δE, highlighting the interplay between these physical quantities in the analysis.
The virtual Green strain tensor is written in the beam coordinate system as l nl
XX YY ZZ YZ ZX XY
= 3.21 nl nl nl nl nl nl T
XX YY ZZ YZ ZX XY
The generalized resultant forces and moments, and the quantities Q i i ( =1, ,10 ) acting over the reference cross-section A o can be related to the stresses in the beam by
XX y XY z A XZ o y XX z XX
XY XZ YY ZZ YZ
In the analysis of structural forces, N represents the axial force, while T_y and T_z denote the shear forces in the Y and Z directions, respectively Additionally, M_y and M_z refer to the bending moments around the Y and Z axes The quantities Q_i are influenced by the initial geometry of the cross-section.
Then the internal virtual work may be written as:
The external virtual work \( \delta W_{ext} \) arises from dead loads and pressure loads Body forces generated by dead loads encompass concentrated loads, moments, and distributed loads Additionally, the inflating pressure exerts traction forces on both ends and along the cylindrical surface The first term on the right side of Equation 3.19 can be reformulated accordingly.
In which f x , f y and f z are respectively the distributed loads along the X, Y, and Z axes, while F b a ( ) , and M a ( ) b (With a = X Y Z b , , ; = X 1 , , X n ) are the external support reactions and the external loads and moments
Conclusion
This chapter develops the governing equations for inflating beams using an analytical approach based on the HOWF 3D Timoshenko theory The theoretical framework incorporates the complete Lagrangian version of the virtual work principle and Timoshenko kinematics to construct stability-regulating equations In the following chapter, these equations are discretized to formulate the global buckling equations.
The study reveals that the mechanical characteristics E l and G lt significantly influence the critical load solution through C 11 and C 66, while E t affects it indirectly via the beam's reference dimensions, owing to the model's orthotropic nature Notably, the degree of orthotropy in the fabric leads to substantial variations in the buckling behavior of the inflating beam, attributed to the unequal mechanical properties of the yarn directions Additionally, the differences observed in the analyzed models stem from the formulation of the constitutive equations.
The material was initially considered to be hyperelastic isotropic, adhering to the Saint Venant-Kirchhoff law, which primarily focused on the S XX and S YY stress components In the Hookean stress-strain relationship, the Young modulus E is directly utilized However, this model incorporates all components of the second Piola-Kirchhoff tensor Instead of relying on the Young modulus E, the mechanical properties of the orthotropic material are defined using an elasticity tensor with specific tensor components.
IGA-BASED BUCKLING ANALYSIS OF INFLATING
A brief review on isogeometric analysis
4.1.1 The concept of isogeometric analysis
Since the 1960s, designers have utilized computers for calculations, primarily in the aerospace and automobile industries, marking the inception of Computer Aided Design and Drafting (CADD) By the late 1980s, Non-Uniform Rational B-Splines (NURBS) emerged as the most efficient method, capable of representing free-form curves, surfaces, and solids in three-dimensional space.
Designers create CAD files that need conversion into analysis-ready geometries, which are then meshed for use in large-scale finite element analysis software As engineering designs become increasingly complex, it is evident that design and analysis must be integrated rather than treated as separate processes Advanced engineering systems rely on various computer analysis and simulation methods, including structures, fluid dynamics, acoustics, electromagnetics, and heat transport This integration ensures continuous communication between design and analysis; however, it is important to note that CAD geometry does not automatically produce or mesh models suitable for analysis.
The integration of Computer-Aided Design (CAD) and Computer-Aided Analysis (CAA) is essential for addressing current engineering challenges However, achieving this integration has proven to be complex, necessitating significant adjustments to align engineering design with analysis Recent advancements in engineering analysis and high-performance computing demand greater accuracy in the modeling-analysis process Typically, a finite element mesh is used as an approximation of CAD geometry, but this often leads to errors in analytical outcomes.
This chapter provides an overview of NURBS theory, emphasizing the mathematical description of free-form curves For a comprehensive understanding of IGA and NURBS-based modeling, readers are referred to the works of Piehl, Hughes et al., and Conttrel et al NURBS, developed from Bézier curves and surfaces in the late 1960s and early 1970s, can accurately represent a wide variety of geometries, particularly conic sections Their flexibility and precision have made NURBS the standard for geometric modeling in computer-aided design The chapter begins with a brief review of Bézier curves, the precursor to B-Spline geometry, followed by a detailed explanation of B-Spline curves, as many definitions and properties apply to NURBS It concludes with a discussion of NURBS as a generalization of B-Splines.
The advantages of Isogeometric Analysis (IGA) over conventional Finite Element Method (FEM) include its ability to maintain the integrity of the computing domain regardless of the level of discretization, even with coarse meshes This characteristic simplifies the detection of connections at the interfaces of two surfaces, particularly in scenarios involving large deformations where relative positions frequently change IGA also enables precise and accurate representation of sliding joints between surfaces, making it advantageous for applications sensitive to geometric imperfections, such as shell buckling analysis, boundary layer phenomena, and fluid dynamics.
NURBS-based CAD models facilitate automatic mesh generation, eliminating the need for geometry clean-up or feature removal This efficiency significantly reduces the time spent on meshing and clean-up, which typically constitutes about 80% of the total analysis time in a given problem (Cottrell).
Effective communication with CAD geometry streamlines the mesh refinement process, making it quicker and more efficient This benefit stems from the use of identical basis functions for both modeling and analysis The separation of geometry positioning and mesh refinement is simplified through an automated knot insertion algorithm, which transforms partitioned segments into new elements, resulting in a precise and accurate mesh.
The use of inter-element regularity, particularly with a maximum of C p − 1 in the absence of repeated knots, proves to be an effective approach for solving mechanics problems This method leverages higher-order element derivatives in formulations such as the Kirchhoff-Love shell theory, gradient elasticity, and the Cahn-Hilliard equation for phase separation The outcomes derived from this technique, which utilizes B-spline/NURBS, are based on thorough analytical calculations.
Unlike the basic functions of Finite Element Method (FEM), which exhibit C^0 continuity and are defined locally within an element, Isogeometric Analysis (IGA) employs functions that span multiple contiguous elements, ensuring higher regularity and interconnectivity This results in a more continuous approximation, leading to improved convergence rates compared to traditional methods, particularly when using the innovative k-refinement technique Importantly, the extended support of IGA basis functions does not increase the bandwidth of the numerical approximation, allowing the sparse matrix's bandwidth to remain consistent with that of classical FEM functions.
Although the IGA has some advantages, it has still disadvantages This methodology presents some challenges that require special treatments
A key challenge in utilizing B-splines and NURBS in Isogeometric Analysis (IGA) is their tensor-product structure, which restricts true local refinement Specifically, any insertion of knots results in a global propagation effect throughout the computational domain.
Due to the lack of Kronecker delta property, in addition, the application of in- homogeneous Dirichlet boundary condition or forces/physical data exchange in a coupled analysis are highly involved
The IGA's enhanced support for fundamental functions leads to a denser matrix system, characterized by a higher number of non-zero entries, in comparison to both the Finite Element Method (FEM) and tri-diagonal banding structures.
Figure 4.1 An example of B-spline curve
B-Spline curves are constructed in a parametric space through a linear combination of control points and B-Spline basis functions This space is divided into intervals, with B-Splines defined piecewise according to specific continuity requirements Notably, the polynomial degree can be chosen independently from the number of control points, allowing for flexibility in the number of intervals Consequently, a large number of data points can be approximated using low-degree polynomials The structure of the parametric space is characterized by the knot vector.
The knot vector is a set of non-decreasing real numbers representing coordinates in parametric space:
In B-Spline theory, the notation = 4.1 indicates that the knots are indexed by i, ranging from 1 to n + p + 1, where p represents the polynomial degree and n is the number of basis functions The intervals [1, n + p + 1] are referred to as a patch, while the intervals [i, i + 1] are known as a knot span Notably, a B-Spline basis function is C∞ continuous within a knot span and exhibits C p − 1 continuity at individual knots.
In B-Splines, the BEAMS knot value can be repeated, leading to what are known as multiple knots When knots are evenly spaced in the parametric space, the knot vector is classified as uniform; otherwise, it is considered non-uniform An open knot vector is defined by the condition that the first and last knots have a multiplicity of p+1 In this configuration, the first and last control points of the B-Spline are interpolated, ensuring that the curve is tangential to the control polygon at both the beginning and the end.
B-Splines basis functions N i p , ( ) of degree p0 are defined by the Cox- deBoor [53] recursive formula as follows:
Important properties of B-Spline basis functions are:
Local support i.e N i p , ( ) is non-zero only in the interval i , i p + + 1
Examples of quadratic and cubic B-Spline basis functions for open, non- uniform knot vectors are presented in Figure 4.2 a Basis functions with knot {0 0 0 1 2 3 3 3} b Basis funtion with knot = {0 0 0 1 2 2 3 3 3}
Figure 4.2 Examples of Quadratic B-spline basis functions
The derivatives of the B-Spline basis functions are computed by the following formula:
A B-Spline curve of p order is defined by a tensor product of B-spline basis functions and control points, as follows:
Control points \( P_i \in \mathbb{R}^d \) for \( i=1, 2, ,n \) define a control polygon in d-dimensional space A quadratic B-Spline curve with an open knot vector is illustrated in Figure 4.3, where the first and last control points are interpolated, ensuring the curve is tangential to the control polygon at both its start and end Additionally, the derivative of a B-Spline curve is also a B-Spline curve, which can be calculated using a specific formula.
Some important characteristics of B-spline curves are:
• Convex hull property: the inside curve contained in the convex hull of controlling polygon
• The controlling points are generally not interpolated
• The controlling points influence maximum p+1 sections
IGA-based formulations for the buckling problems of inflating composite beams
In linear buckling analysis, the beam is subjected to an inflated prestressing pressure tensor \( S_0 \) The initial step involves applying an arbitrary reference level of external load \( \{ F_{ref} \} \) to the inflating beam and conducting a standard linear analysis to assess the finite element stresses Additionally, a general formula for the finite element stress stiffness matrix \( [k_{\sigma}] \) and the finite element elastic stiffness matrix \( [k] \) is sought The strain energy per unit volume of the beam is also a critical consideration in this analysis.
The governing equations for a finite inflating beam are derived from the principle of virtual work By integrating over the beam's volume, considering the cross-sectional area \( A_o \) and length \( l_o \), we obtain an expression for the virtual strain energy.
In beam analysis, the stiffness matrix is derived by interpolating a displacement field that includes parameters such as axial displacement, bending rotation, and transverse displacement Commonly used elements include the two-noded Euler-Bernoulli beam employing Hermite polynomials as shape functions, and higher-order elements like the three-node quadratic beam with reduced integration or the Timoshenko beam, which utilizes quadratic shape functions for transverse displacement alongside linear functions for bending rotation and axial displacement Additionally, quadratic NURBS basis functions are applied for enhanced interpolation accuracy.
There are five degrees of freedom (DOF) associated with a control point The displacement vector is defined as {d} defines DOF vector as
4.15 where index j defines the control point j, [N] the matrix of NURBS functions, which are discussed in the previous chapter, and ncp is the total number of control points
The strain energy component U m of the beam is associated with the stress stiffness matrix k and U b relates to the conventional elastic stiffness k of the beam, as
By applying the discretization procedure, the global equation is obtained as follows
The axial load, represented as F, is related to a proportionality coefficient λ, defined by the equation δ = δ + λ * 4.18 The matrix coefficients [k] and [k_ref] remain constant, influenced by the beam's geometry, material properties, and prestressing pressure conditions To evaluate the stiffness matrix, the Gauss numerical integration scheme is employed The assembly of the element stiffness matrices for the entire structure results in the equilibrium matrix equation in global coordinates The potential energy of the beam is the sum of the potential energies of its individual finite elements, and a complete structural matrix is generated by adhering to the standard finite element method (FEM) assembly procedure.
The structural equilibrium equations can be obtained by applying the principle of minimum potential energy This is expressed in the form of eigenvalue problem:
Equation 4.19 presents an eigenvalue problem where the eigenvalue \( \lambda_i \) corresponds to the first buckling mode The smallest root, \( \lambda_{cr} \), indicates the minimum external load level at which structural decomposition occurs.
The buckling mode represents the eigenvector linked to the critical load parameter ( cr) when a beam is subjected to a specific external load (F ref) In the context of linear buckling analysis, while the exact magnitude of the load remains indeterminate, it effectively defines the shape of the buckling mode without specifying its amplitude ( D).
Consider the geometrically nonlinear behavior of the HOWF inflating beam formed of a material that is assumed to be linear elastic The NLFEIB model is a
The BEAMS nonlinear finite element method is applied to model the inflation of beams, utilizing the complete Lagrangian technique to accurately describe geometric nonlinearity, where displacements are referenced to the original configuration This approach facilitates the calculation of a tangent stiffness matrix, essential for analyzing the structural response of inflated beams.
K T , includes the effect of changing geometry as well as the effect of inflated pressure The axial load i th is signified in the following formula:
With a known element, the nonlinear equilibrium equation can be formulated as
The element tangent stiffness matrix, denoted as \([k^T]\), relates to the external load increments vector \(\{f_i\}\) and the unknown displacement increment \(\{\Delta d\}\) that needs to be solved Once all elements are assembled within the model, the equilibrium equation can be established.
The incremental scheme based on the straightforward Newton method interprets Eq 4.23 by utilizing nodal load increments (ΔF) along with load correction terms and updates of the stiffness matrix (K) after each step In this approach, the model displacement vector is defined as D_i = D_{i-1} + ΔD, where ΔD represents the unknown displacement increment at step i, and D_{i-1} is the displacement vector from the previous solution step The equilibrium solution tolerance is established to ensure accuracy in the results.
R i = ( R T i R i ) 1 2 0.0001 4.25 with R i = R D ( ) i − 1 = K T D i being the globally unbalanced residual force vector from the previous increment As a limit point is approached, displacement increments D become very large Either at a limited point or bifurcation point,
The outline of the algorithm at the element level developed by Nguyen et al
[4] is employed in this study, (numerical integration procedure for calculating the element stiffness matrix at the jth element) The algorithm is described as follows:
Require Nodal unknown displacements D i , element number jth, model description
Ensure Element stiffness matrix K T e , element load vectors int
Loop on 1D Gauss integration m point(s) in the ξ direction: for m = 1 to 3 do
Set sampling point location ξ = ξ m and associated weight factor W m ,
Call shape function subroutine to calculate element matrix B and Jacobian operator J, all at point ξ m
Calculate product [ ]B T ( int − ext )[ ]B W m and add it to an array K T e Calculate element internal load factor and T int e W k add it to F int e
Calculate the element external load factor ( T ext d + T ext p ) W k and add it to the array F ext e end for
The matrices concerning internal and external forces for calculating the tangent stiffness, respectively, are
The strain–displacement matrix is given by
4.2.3 Implementation of an iterative algorithm in solving nonlinear model
The iterative method for determining nodal displacement increment solutions employs the fundamental Newton-Raphson iteration combined with adaptive load stepping This process assumes that a solution approximation has been achieved at the current increment, despite the residual not equaling zero.
At increment step i, one seeks an approximation D i of the solution such that:
R D ( ) i = R D ( i − 1+ D i ) 0 4.27 The algorithm is obtained by using the first-order Taylor series in the vicinity of D i
MATLAB, a powerful numerical computing software, is utilized to develop models through linearizable and incremental iterative techniques, incorporating an iterative equation solution This process involves analyzing each material (Gaussian) point within the structural loop to ensure accurate results.
BEAMS incremental-iterative algorithm will be invoked The beam parameter is set in every loop during an incremental loading step
Table 4.5 outlines the prescribed boundary conditions along with the input variables for the global level routine The output results from this routine are derived from Equation Eq 4.23, which is solved iteratively within the structural level In the element level sub-routine, each element is analyzed to compute the tangent stiffness matrix \(\left[K^T_e\right]\) and the internal loading vectors \(\{\text{int}\}\).
The article discusses the process of incremental loading in a computational framework, where the global counter (i, k, m) indicates the current loading step, the number of elements, and the number of Gauss integration points, respectively After each loading step, the converged displacement solution \(\{\Delta D_i\}\) at the current load \(\Delta F\) is used to determine the incremental displacement for the subsequent loading step Additionally, the convergence criterion at the material level can be defined using equations 4.24 or 4.25, which are formulated in terms of displacement vectors.
The nonlinear solutions for tracing the load-deflection response of the model is presented as follows:
Require Beam geometry, material properties, external loads, and model description
Ensure Displacement incrementation solutions D i for tracing load- deflection response
Loop over load increments: for i = 1 to n inc do
=n in which i is the current load increment
Call global level routine for computing K T , F ext and F int
Solve the nonlinear equation K T D i −( F ext − F int )=0 for D i
Convergence check for stopping the iteration loop: D i 10 − 6
Save the current solution D i in the global solution vector D end for
Numerical examples
This section presents numerical examples and their results, emphasizing that a convergence analysis is conducted for all cases before extracting findings The study investigates the effects of compressive focused force on cantilever and simply-supported inflated composite beams, characterized by the slenderness ratio \( \lambda_s = \frac{L}{\rho} \), where \( L = \mu l_o \) represents the effective length of the beam A comparison of numerical results obtained from the traditional finite element method and the Isogeometric Analysis (IGA) approach highlights the latter's accuracy and efficiency Notably, the C1 continuity of IGA elements is inherently achieved due to the method's fundamental characteristics, offering a distinct advantage over traditional finite element methods in numerical applications.
This study conducts a linear buckling analysis of inflatable beams subjected to concentrated compressive loads to determine critical load parameters By pressurizing the beam, we evaluate the effects of inflation pressure on its stability The analysis focuses on the normalized linear buckling load coefficient to investigate the eigen buckling behavior of the inflated structure.
The formula K c l = 10^5 × σ cr / E eq, proposed by Ovesy, relates to the linear buckling critical stress (σ cr) of a beam, with E eq representing the equivalent Young’s modulus of the material (Paschero) The specific material, geometric parameters, and pressure values utilized in the LFEIB model are detailed in Table 4.3.
Table 4.3 Input parameters for modeling LFEIB model
Boundary condition Simply-supported Fixed-free
Figure 4.8 illustrates a cylindrical inflating composite beam under simply- supported constrains and subjected to axial compression load
Figure 4.8 Model of a simply-supported inflating beam subjected to axial compression load
The input parameters are presented in Table 4.3 The simply-supported boundary condition is assigned by u= =v 0 at X =0 , and v=0 at
The convergence studies on the normalized buckling coefficient \( K_{cl} \) of the LFEIB model indicate that approximately four quadratic NURBS-based Timoshenko elements are adequate for achieving converged results, aligning closely with those obtained from the standard 3-node Timoshenko element utilized by Nguyen [2] Comparative analysis in Table 4.4 highlights the error of numerical solutions against closed-form solutions derived from Nguyen's analytical approach, demonstrating that IGA models yield superior results compared to FEM models Notably, the implementation of four quadratic NURBS-based elements requires fewer degrees of freedom (30 DOFs with 6 control points) than the standard finite element mesh (45 DOFs with 9 nodes), leading to significant reductions in computational effort Additionally, the IGA model consistently produces stable solutions, particularly under high inflation pressure scenarios (case p4@kPa) Furthermore, the buckling load coefficient \( K_{cl} \) gradient is influenced by the normalized pressure \( p_n \), with a noticeable increase in the gradient at elevated \( p_n \) levels.
Figure 4.9 Linear Eigen buckling: mesh convergence test of normalized linear buckling load coefficient ( K c l = 10 5 cr / E eq ) for a simply-supported LFEIB model
Table 4.4 Normalized critical loads K c l of simply-supported LFEIB inflating beam
*(2) & (1) denotes the differences between FEM and closed-form solutions,
(3) & (1) denotes the differences between IGA and closed-form solution
A cantilever LFEIB model is illustrated in Figure 4.10
Figure 4.10 Model of a cantilever inflating beam under axial compression load
Material and geometric properties are assumed in Table 4.3 The clamped boundary condition is assigned by u = = = v w x = y = 0 at X =0
The buckling load of the cantilever inflating beam with different inflation pressures based on isogeometric analysis is plotted in Figure 4.11
Figure 4.11 Linear eigen buckling: mesh convergence test of normalized linear buckling load coefficient ( K c l =10 5 cr / E eq ) for a cantilever LFEIB model
The results align closely with those obtained using standard finite element methods, as demonstrated by Nguyen [2] Notably, isogeometric analysis exhibits rapid convergence due to the high continuity of the finite element mesh Moreover, it necessitates fewer total degrees of freedom (DOFs) compared to traditional FEM, leading to significant computational savings, particularly in the nonlinear analysis of inflating composite beams.
The linear eigen buckling of inflating composite beams has been effectively analyzed using NURBS-based isogeometric analysis, demonstrating high accuracy through various numerical tests across different boundary conditions and geometric configurations This method exhibits rapid convergence and requires fewer degrees of freedom (DOFs), highlighting the robustness of the isogeometric analysis models for inflating composite beams, which shows promise for future investigations into geometric and material nonlinearity.
In linear buckling analysis, the critical load is only applicable when there is minimal or no interaction between membrane deformation and bending Figure 4.12 illustrates this concept, showcasing a small initial imperfection that can influence the results.
Initial curvature or slight eccentricity in the compressive load F of beams can lead to significant displacements instead of buckling as original flaws develop Consequently, linear bifurcation analysis may underestimate the actual collapse load.
The normalized nonlinear load parameter at i th an increment of axial load is defined by,
Figure 4.12 Model for nonlinear analysis (Nguyen [2])
The model utilizes materials 1 and 2 as outlined in Table 4.6 The deflection solutions \( D_v \) along the Y-axis from the NLFEIB model indicate a change in the flexion-to-radius ratio \( \left( \frac{R_{fr} D_v}{R_0} \right) \), while the axial displacement solutions \( D_u \) along the X-axis reflect a change in the length-to-radius ratio \( \left( \frac{R_{lr} D_u}{R_0} \right) \) For consistent normalized pressure and material properties, lower values of \( R_{lr} \) and \( R_{fr} \) signify a more stable beam.
Table 4.5 Input parameters for modeling models
Parameter type Input Physical interpretation Value
E l Young modulus in the warp direction See
E t Young modulus in the weft direction
G lt In-plane shear modulus v lt
Poisson ratio due to the loading in l direction and contraction in the t direction v tl
Poisson ratio due to the loading in t direction and contraction in the l direction
Beam geometry l Length of the inflating beam See
Table 4.6 (in the natural state)
R The external radius of the inflating beam t The thickness of the inflating beam
External load p Inflation pressure 10-200 (kPa)
n inc Number of load increments 10
The model consists of 4 elements, each having 3 control points In total, there are 12 control points globally Each node features 5 degrees of freedom, while each element has its own set of degrees of freedom, contributing to the overall functionality of the model.
BEAMS g dof Number of global degrees of freedom n dof n n m Number of Gauss integration points 3
Table 4.6 Data set for inflating beam
Orthotropic fabric's mechanical properties: Material 1 Material 2 Young modulus in the warp direction, E l
Young modulus in weft direction, E t (MPa) 2994 14240
In-plane shear modulus, G lt (MPa) 1171 6450
Table 4.7 Normalized pressure ( ) p n for different values of internal pressure ( ) p used in the study p (kPa) p n
This article explores the nonlinear buckling behavior of a simply supported inflated beam under an axial compressive force F, utilizing the method outlined in Section § 4.2.2 Through numerical examples, the study demonstrates the effectiveness of the proposed method in analyzing significant deformations within the NLFEIB model Additionally, a parametric analysis is conducted to assess the influence of normalized pressure on the NLFEIB model's performance.
At each level of normalized pressure, the corresponding crushing load
( F crush = F p ) is the upper bound of the axial load applied to the beam The displacements at the middle span of the beam are extracted from the global solution
Figures 4.13 and 4.14 illustrate how the flexion-to-radius and length-to-radius ratios change with increasing normalized load parameter K c nl across two material cases The linear buckling analysis indicates that four elements are adequate for achieving converged results At low pressures, the model exhibits instability and is likely to fail first In contrast, at elevated pressures, the R fr ratio responses display a quasi-linear trend for lower increments of K c nl, gradually transitioning to a nonlinear behavior as K c nl increases.
The study emphasizes the importance of fabric qualities and their interaction with normalized pressure in a parametric investigation Two materials are utilized to construct HOWF inflating beams, and as detailed in Section § 4.3, nonlinear iterative solutions are obtained using normalized pressure inputs alongside two aspect ratios for normalization (R lr and R fr).
The responses of simply-supported (SS) inflating beams highlight the significant impact of boundary conditions and material properties Specifically, when comparing two materials, material 1, with a lower elastic modulus than material 2, exhibits a heightened sensitivity to buckling under high internal pressure Additionally, as the axial compression loads increase, a mode jump behavior is observed in the SS beam In contrast, clamped inflating beams do not demonstrate any distortion in load-deflection under similar conditions.
Figure 4.13 Nonlinear buckling: variation of flexion-to-radius ratio ( R fr =D v / R o ) with increasing normalized nonlinear load parameter ( K c nl = 10 6 F i / ( E A eq 0 ) ) for a simply supported NLFEIB model
Figure 4.14 Nonlinear buckling: variation of length-to-radius ratio ( R lr =D u / R o ) with increasing normalized nonlinear load parameter K c nl for a simply supported
In this example, the nonlinear buckling of a cantilever inflating beam subjected to an axial compressive load F is investigated The discrepancy due to the normalized
The results demonstrate a clear relationship between pressure and performance in beams, as illustrated in Figure 4.15, which depicts the variation of the flexion-to-radius ratio with increments of the normalized load parameter K c nl for two material cases Furthermore, Figure 4.16 shows the length-to-radius ratio R lr in relation to the incremental load ratio K c nl Notably, beams subjected to higher pressures exhibit enhanced load-carrying capacity and stability.
Conclusions
This chapter introduces the fundamental concepts of Isogeometric Analysis (IGA) and its implementation as an alternative finite element method It covers both linear and nonlinear buckling analyses of an inflating beam, deriving governing equations through an energy approach that considers changes in membrane and bending strain energy The equations are discretized using the IGA approach, employing NURBS basis functions to accurately represent geometry and serve as interpolation functions.
The linear buckling analysis demonstrated a significant enhancement of the proposed numerical model over the standard finite element method, as evidenced by a mesh convergence test on the beam's critical force Additionally, the buckling coefficient results aligned well with existing literature In the nonlinear buckling analysis, the method effectively captured the load-deflection response of inflating beams.
Two numerical methods, Finite Element Method (FEM) and Isogeometric Analysis (IGA), were utilized to validate the numerical approach for the inflating beam model A straightforward beam model was simulated and analyzed, revealing that the IGA method provides a higher level of accuracy in constructing numerical models for this problem.