No / This paper considers both the linear elastic and creep buckling of polymeric pipe linings used for the rehabilitation of gravity pipes, for which external groundwater pressure has been identified as the prime source of loading. Theoretically perfect and imperfect conditions are considered, with the imperfections taken to be in the form of a concentric or eccentric annulus between the rigid host pipe (cylindrical constraint) and polymeric lining. Under these conditions two recently obtained mathematical procedures for the prediction of linearly and non-linearly elastic buckling are compared with the results of complementary laboratory testing. Linear elastic conditions are shown to be well approximated by undertaking short-term (¿30 min) testing under increasing pressure to failure. Controlled imperfections are introduced into the laboratory tests and excellent correlation with the theoretical predictions is obtained. In particular, the dominant geometrical imperfections are shown to be major influences on the obtained buckling pressure. The mathematical models are then adapted to simulate the creep buckling process under long-term constant pressure. The results obtained are again compared with those provided by complementary physical testing, and appropriate conclusions are made.

No / This paper considers both the linear elastic and creep buckling of polymeric pipe linings used for the rehabilitation of gravity pipes, for which external groundwater pressure has been identified as the prime source of loading. Theoretically perfect and imperfect conditions are considered, with the imperfections taken to be in the form of a concentric or eccentric annulus between the rigid host pipe (cylindrical constraint) and polymeric lining. Under these conditions two recently obtained mathematical procedures for the prediction of linearly and non-linearly elastic buckling are compared with the results of complementary laboratory testing. Linear elastic conditions are shown to be well approximated by undertaking short-term (¿30 min) testing under increasing pressure to failure. Controlled imperfections are introduced into the laboratory tests and excellent correlation with the theoretical predictions is obtained. In particular, the dominant geometrical imperfections are shown to be major influences on the obtained buckling pressure. The mathematical models are then adapted to simulate the creep buckling process under long-term constant pressure. The results obtained are again compared with those provided by complementary physical testing, and appropriate conclusions are made.

The considerable progress in the research and development of thin-walled beam structures responds to their growing use in engineering construction and to their increased need for efficiency in strength and cost. The result is a structure that exhibits large shear strains and important non uniform warping under different loadings, such as non uniform torsion, shear bending and distortion... A unified approach is formulated in this thesis for 3D thin walled beam structures with arbitrary profile geometries, loading cases and boundary conditions. A single warping function, defined by a linear combination of longitudinal displacements at cross sectional nodes (derived from Prokic work), is enhanced and adapted in order to qualitatively and quantitatively reflect and capture the nature of a widest possible range of behaviors. Constraints are prescribed at the kinematics level in order to enable the study of arbitrary cross sections for general loading. This approach, differing from most published theories, has the advantage of enabling the study of arbitrary cross sections (closed/opened or mixed) without any restrictions or distinctions related to the geometry of the profile. It generates automatic data and characteristic computations from a kinematical discretization prescribed by the profile geometry. The amount of shear bending, torsional and distortional warping and the magnitude of the shear correction factor is computed for arbitrary profile geometries with this single formulation. The proposed formulation is compared to existing theories with respect to the main assumptions and restrictions. The variation of the location of the torsional center, distortional centers and distortional rotational ratio of a profile is discussed in terms of their dependency on the loading cases and on the boundary conditions. A 3D beam finite element model is developed and validated with several numerical applications. The displacements, rotations, amount of warping, normal and shear stresses are compared with reference solutions for general loading cases involving stretching, bending, torsion and/or distortion. Some examples concern the case of beam assemblies with different shaped profiles where the connection type determines the nature of the warping transmission. Other analyses –for which the straightness assumption of Timoshenko theory is relaxed– investigate shear deformation effects on the deflection of short and thin beams by varying the aspect ratio of the beam. Further applications identify the cross sectional distortion and highlight the importance of the distortion on the stresses when compared to bending and torsion even in simple loading cases. Finally, a non linear finite element based on the updated lagrangian formulation is developed by including torsional warping degrees of freedom. An incremental iterative method using the arc length and the Newton-Raphson methods is used to solve the non linear problem. Examples are given to study the flexural, torsional, flexural torsional and lateral torsional buckling problems for which a coupling between the variables describing the flexural and the torsional degrees of freedom occurs. The finite element results are compared to analytical solutions based on different warping functions and commonly used in linear stability for elastic structures having insufficient lateral or torsional stiffnesses that cause an out of plane buckling.

Civil Engineering thin cylindrical shells such as silos and tanks are normally subjected to axial compression that arises from a stored solid, wind, earthquake, self-weight or roof loads. The walls of these shells are very thin, generally of the order of 6 to 25 mm, and massively less than the radius, which is typically 5 to 30 m. They are thus very thin shell structures, like those of rockets, spacecraft, motor vehicles and aircraft. The commonest failure mode is elastic buckling under axial compression. It has long been known that the buckling strength of a thin cylindrical shell under axial compression is very sensitive to tiny deviations of geometry, reducing the buckling strength to perhaps 10 or 20% of the value for the perfect structure. A normal internal pressure usually accompanies the axial compression, caused by stored granular solids or fluids. At relatively low pressures, the elastic buckling strength under axial compression rises, but an elastic-plastic buckling phenomenon intervenes at higher pressures, causing a dramatic decrease in buckling resistance associated with an elephant’s foot collapse mode. To construct such large shells, the fabrication technique is generally the assembly of many rolled plates or panels, joined by short longitudinal welds and continuous circumferential welds. The process of welding produces a distinctive geometric imperfection form at each weld joint, which in turn is extremely detrimental to the shell axial buckling carrying capacity. The strength may be further reduced by slight misalignments between adjacent panels, or in bolted construction, by vertical and horizontal lap splices. Due to the pattern of loading, both the axial compression and internal pressure increase progressively down the wall. Accordingly, practical construction usually uses a stepped wall, formed from panels of uniform thickness, but with larger thicknesses at lower levels. Since the loading varies smoothly, but each panel has constant thickness, the critical location for buckling lies at the base of a panel. But the greater thickness of the lower panel can usefully enhance the buckling strength of the critical panel above it. This thesis presents an extensive computational study that examines all the above influences, divided into chapters that are outlined here. A full exploration of the effect of the cylinder length on the perfect and imperfect elastic buckling strength is presented in Chapter 3. In Chapter 4, the elastic-plastic buckling resistance of imperfect cylinders is described, including strain hardening. These lead to many capacity curves, for which the key parameters are extracted. The effect of internal pressure on the buckling resistance of imperfect elastic cylinders is explored in Chapter 5. Chapter 6 studies the effect of high pressures that produce elastic-plastic elephant’s foot buckling at circumferential welds in imperfect shells. Next, a step change in plate thickness is studied in Chapter 7 for imperfect butt jointed cylinders with and without the internal pressure. Chapter 8 presents an exploration of the effect of plate misalignments at a circumferential joint, as well as the full misalignment of a circumferential lap joint in bolted construction. These are investigated in both the elastic and elastic-plastic domains. The entire thesis is conceived in the context of EN 1993-1-6 (2007) and the ECCS Recommendations on Shell Buckling (2008). This research has shown significant weaknesses in both the concepts and the detailed rules of these standards. Many conditions are found where either the standard is unnecessarily conservative, or its safety margin may be too low. Thus, some new provisions are proposed for each of the above practical problems. These are expected to provide useful knowledge for the design of such structures against buckling in the future.

Presented are approximate finite strip methods for use in predicting elastic buckling strength of cold-formed steel columns. These methods were developed by examining elemental behavior of cross-sections in eigen-buckling analyses and validated using a large database of finite element rack-type columns with perforation patterns. The influence of perforations is accounted by reduced thicknesses related to the plate buckling coefficient and transverse web rotational stiffness in the prediction of local and distortional buckling respectively. Global buckling prediction including the influence of perforations uses critical elastic loads of an unperforated section multiplied by the ratio of weighted to gross cross-sectional moment of inertia for flexural buckling and the ratios of weighted to gross cross-sectional warping torsion constant and weighted to gross St. Venant torsional constant for flexural-torsional buckling. Concern for end-user was given and methods are presented in a way for incorporation into governing design standards. Data to support these findings are available at http://hdl.handle.net/10919/23797 / Master of Science

A thesis submitted to the School of Graduate Studies in partial fulfilment of the requirements of the Master of Applied Science degree / This thesis investigated structure-function relationships of materials designed to mimic the plant cell wall by comparing their mechanical properties measured using strain-induced buckling methods. The plant cell wall mimics are submicrometer-thick films composed of cellulose nanocrystals (CNCs) and various types of xyloglucan (XG), a common plant hemicellulose. Our goal was to establish links between film composition/architecture and elastic modulus, to better understand the interactions between plant cell wall components and their influence on mechanical properties. Three buckling methods for measuring mechanical properties of supported films were compared. All methods involved compressing a thin film deposited onto a shape memory polymer or an elastomeric substrate, through thermal shrinking or mechanical compression, respectively. Two thermal shrinking methods (constrained in one axis or unconstrained) and one compression method (using a mechanical strain stage) were used. Based on the mismatch of mechanical properties between the film and the substrate, the rigid thin film “buckles” upon compression to dissipate strain. The resulting surface wrinkle sizes are characteristic of the mechanical properties of the thin film. A Fourier analysis algorithm with Gaussian curve fitting was optimized to extract wrinkle sizes accurately and reproducibly from microscopy images to reliably quantify the elastic moduli of thin films. To select the most precise strain-induced buckling method, model layer-by-layer (LbL) thin films composed of CNCs and polyethylene imine were tested. All three buckling methods precisely quantified the elastic moduli of the films and helped us build connections between the mechanical properties and the film composition. Elastic moduli determined were 15-44 GPa (depending on composition) and films up to 350 nm-thick were tested. Based on sensitivity analyses, however, unconstrained thermal shrinking proved to be the most robust method for calculating the elastic modulus. We believe these buckling methods may find widespread use in the characterization and surface structuring of thin films for applications in biosensors, flexible electronics, point-of-care diagnostics, and for studying plant cell wall mimics. Using the unconstrained thermal shrinking method, plant cell wall mimics were constructed using LbL thin film assembly with various concentrations of CNCs and XG. Three types of XG were compared: (1) unmodified XG, (2) XG with a fraction of the galactosyl residues removed (degalactosylated), and (3) a fragmented lower molecular weight XG. It was inferred that molecular weight impacts the stiffness of XG-CNC based on adsorption conformation of XG onto CNCs, where lower molecular weight XG results in a higher modulus film (27 ± 1 vs. 19 ± 2 GPa). As well, saccharide residues of XG, specifically galactosyl, impact XG’s capacity for self-association and interaction with CNCs, because saccharide residues hinder association through their glucan backbone. This is evidenced by the higher elastic moduli calculated for degalactosylated XG-CNC films (75 ± 6, GPa), compared to native XG-CNC films (19 ± 2 GPa). This work highlights the importance of material structure as it relates to overall performance and therefore function in natural systems, such as the plant cell wall. These studies contribute to a greater understanding of the mechanical properties of the plant cell wall and serve as a basis to extend bio-based and biomimetic materials to applications such as drug delivery, packaging, and coatings. / Thesis / Master of Applied Science (MASc) / The plant cell wall boasts impressive mechanical properties, balancing seemingly opposing properties of structural strength with flexibility. These natural materials have been a source of inspiration for new material design, but the phenomena that govern interactions between components and how their structures translate into function, have yet to be fully understood. In this work, we have constructed thin multilayered films to mimic the plant cell wall, composed of cellulose nanocrystals (rod-shaped nanoparticles derived from plant cellulose) and xyloglucan (a common hemicellulose “glue”). When the films on flexible supports are compressed, they buckle into wrinkled surface patterns that can be used to calculate their mechanical properties. This investigation compares three buckling methods and supports the notion that the mechanical performance of the plant cell wall is strongly dependent on the structure of the different components and the way they interact.

Experimental testing and elastic buckling studies were performed on 68 C-section cold-formed steel joists with unstiffened rectangular web holes. Four Steel Stud Manufacturers Association (SSMA) cross-section types; 800S200-33, 800S200-43, 1000S162-54, and 1200S162-97, were evaluated to explore the influence of holes on local, distortional, and global bucking failure modes. Hole depth was varied in the tests to identify trends in ultimate strength. Ultimate strength was observed to decrease with increasing hole depth for 800S200-33, 1200S162-97 cross-sections. Due to small number of specimen and unidentified behavior of the beams, a more in depth study of the behavior of 800S200-43 and 1000S162-54 beams are necessary. Local buckling of the unstiffened strip above the hole was observed to accompany distortional buckling at the hole for the locally slender 800S200-33 and 1000S162-54 cross-sections. Thin shell finite element eigen-buckling analysis of each joist specimen, including measured cross-section dimensions and tested boundary and loading conditions, were conducted in parallel with the experiments to identify those elastic buckling mode shapes which influence load-deformation response. The distortional and lateral-torsional buckling moments were observed to decrease with increasing hole depth while a contrasting behavior was captured for local buckling modes. A modification to the AISI Direct Strength Method equations for beams with slotted web-holes was compared against the experimental results with predictions lower than tested strength. Initial cross-section imperfections led to inclined webs which decreased the capacity of the beams. The use of a water-jet cutting process was employed successfully to produce accurate holes sizes and locations in each joist specimen and is recommended for researchers and manufacturers as a method for custom fabrication of cold-formed steel members. / Master of Science

Most thin-walled metallic structural members experience some extent of interactive buckling that corrodes the load carrying capacity. Current design methods predict the strength of thin-walled metallic structural members based on individual buckling limit-states and limited case of interactive buckling limit state. In order to develop design methods for most coupled buckling limit states, the interaction of buckling modes needs to be studied. This dissertation first introduces a generally applicable methodology for Generalized Beam Theory (GBT) elastic buckling analysis on members with holes, where the buckling modes of gross cross-section interact with those of net cross-section. The approach treats member with holes as a structural system consisting of prismatic sub-members. These sub-members are connected by enforcing nodal compatibility conditions for the GBT discretization points at the interfaces. To represent the shear lag effect and nonlinear normal stress distribution in the vicinity of a hole, GBT shear modes with nonlinear warping are included. Modifications are made to the GBT geometric stiffness because of the influence from shear lag effect caused by holes. In the following sections, the GBT formulation for a prismatic bar is reviewed and the GBT formulation for members with holes is introduced. Special aspects of analyzing members with holes are defined, namely the compatibility conditions to connect sub-members and the geometric stiffness for members with holes. Validation and three examples are provided. The second topic of this dissertation involves a buckling mode decomposition method of normalized displacement field, bending stresses and strain energy for thin-walled member displacement field (point clouds or finite element results) based on generalized beam theory (GBT). The method provides quantitative modal participation information regarding eigen-buckling displacement fields, stress components and elastic strain energy, that can be used to inform future design approaches. In the method, GBT modal amplitudes are retrieved at discrete cross-sections, and the modal amplitude field is reconstructed assuming it can be piece-wisely approximated by polynomials. The unit displacement field, stress components and strain energy are all retrieved by using reconstructed GBT modal amplitude field and GBT constitutive laws. Theory and examples are provided, and potential applications are discussed including cold-formed steel member design and post-disaster evaluation of thin-walled structural members. In the third part, post-buckling modal decomposition is made possible by development of a geometrically nonlinear GBT software. This tool can be used to assist understanding couple-buckling limit-states. Lastly, the load-deformation response considering any one GBT mode is derived analytically for fast computation and interpretation of structural post-buckling behavior. / Ph. D.

Master of Science / Department of Mechanical and Nuclear Engineering / Kevin B. Lease / Xiao J. Xin / The study of buckling behavior of tubular and cellular structures has been an intriguing area of research in the field of solid mechanics. Unlike the global Euler buckling of slender structures under compressive loads, tubular and cellular structures deform with their walls buckling as individual supported plates. The aspect ratio and the dimensional characteristics of the tube define the buckling behavior of any tube structure. In this thesis, a thorough study on the buckling of polygon tubular structures with different cross sections is discussed. In the first study, the theoretical buckling formulation of a square tube using the energy method is reviewed from existing solutions in literature. The elastic critical load of a square tube derived from the theoretical solution is then compared with results of finite element elastic buckling simulations. The formation of lobes along the height of the walls at different aspect ratios of the tube is investigated and compared to theory. Also, the buckling behavior of multi-wall structures is studied and the relationship between these structures and a rectangular simply supported plate is established. A brief study on the buckling behavior of rhombic tubes is also performed. The results of the simulation match closely with the theoretical predictions. The study is then extended to quadrilateral tubes with cross-sections in the shape of square, rectangle, rhombus and parallelogram. The theory of buckling of these tubes is explicitly defined using classical plate mechanics based on the previous works presented in literature. Also, the possibility of global Euler buckling in the tubular structures after a certain critical height is discussed. The prediction from the theory is validated using extensive finite element elastic buckling simulations and experimental tests on square and rhombic tube specimens. The results of the simulations and experiments are observed to be consistent with the theory. Using the formulation of plate buckling under different boundary conditions, the buckling behavior of triangular tubes is also determined. A theoretical formulation for calculating the critical load of triangular tubes is derived. The theoretical critical loads for a range of aspect ratios are compared with corresponding finite element simulation results. The comparisons reveal high degree of similarity of the theoretical predictions with the simulations.

Thesis (MScEng (Process Engineering))--University of Stellenbosch, 2010. / ENGLISH ABSTRACT: The goal of this study is to produce cellulose acetate (CA) hollow-fine-fibre membranes with good water flux performance in the 95 – 96% salt retention range for brackish water desalination from first principles. First, the acceptable range of fibre dimensions was determined by means of a collapse pressure calculation using the elastic buckling pressure equation (thin shell assumption). Second, the pressure drop across the fibre wall in the hollow-fine fibre was determined by using the Hagen-Poiseuille equation, in order to determine how this would affect the chosen fibre dimensions. It was determined that the acceptable range of fibre dimensions was 222 – 247 m, and the wall thickness was 50 m. Fibres with these dimensions exhibited a high resistance to brackish water operating pressure of 20 – 25 bar, without collapse. The pressure drop calculations of these dimensions showed a sufficiently low pressure drop across the fibres. A dry-wet spinning technique was used for the preparation of the hollow-fine-fibre membranes. Hollow-fine fibres were spun using CA dissolved in a suitable solvent and non-solvent mixture comprising acetone and formamide. The effects of the dope composition and spinning parameters such as solvent to non-solvent ratio, bore fluid ratio, take-up speed, dope extrusion rate and heat treatment on the membrane morphology and performance were investigated. The spun fibres showed a good morphological structure, with no macrovoids (sponge-like structure), which is favourable for reverse osmosis (RO) applications. The hollow-fine-fibre membranes showed a good brackish water desalination performance within brackish water operating conditions. Statistical analysis was used to generate a fabrication formulation for producing cellulose acetate hollow-fine-fibre membrane for brackish water desalination with improved salt retention and flux. A three-level three-factor factorial was used to the study of the effect of spinning parameters (solvent to non-solvent ratio, bore fluid ratio and air gap distance). A regression equation was successfully established and was used to predictably produce membranes with good performance within the limits of the factors studied. RO performance of these hollow-fine-fibre membranes was good: The salt retention ranged from 96 to 98% and the permeate flux ranged from 60 to 46 L/m2.d (2 000 ppm, NaCl, 20 bar, 24 oC). / AFRIKAANSE OPSOMMING: Die studie het ten doel gehad om selluloseasetaat holveselmembrane vanaf eerste beginsels vir brakwaterontsouting te ontwikkel. Die ontsoutingsvlakke van die membrane moet tussen 95 en 96% lê met ’n aanvaarbare waterproduksievermoë. Aanvaarbare deursneë vir die holvesels is eerstens bepaal deur platval-berekeninge met behulp van die inmekaarvouvergelyking uit te voer (dunwand aanname). Hierna is drukval oor die wand van die holvesel met behulp van die Hagen-Poiseuille vergelyking bepaal ten einde vas te stel hoe dit die gekose dimensies sal beïnvloed. Daar is vasgestel dat vesel deursneë tussen 222 en 247 um met ’n 50 um wand aanvaarbaar is. Vesels met hierdie dimensies het ’n hoë weerstand teen inval getoon by brakwater opereringsdrukke tussen 20 en 25 bar. ’n Droë-nat spintegniek is in die voorbereiding van die holveselmembrane gebruik. Holvesel membrane is met ’n selluloseasetaat stroop gespin wat uit ’n oplosmiddel (asetoon) en nieoplossmiddel (formamied) bestaan het. Die effek van die spinstroop samestelling en spinparameters soos die oplosmiddel tot nieoplosmiddel verhouding, lumen-vloeistof verhouding, opneemspoed, spinstroop ekstrusie tempo en hittebehandeling op membraan morfologie en werkverrigting is ondersoek. Die gespinde vesels toon ’n sponsagtige struktuur sonder die teenwoordigheid van enige mikroleemtes wat voordelig is vir tru-osmose toepassings. Die holvesel membrane het aanvaarbare brakwater ontsoutings werkverrigting. Statistiese analise is gebruik in die generasie van produksieformulasies vir die produksie van brakwater ontsoutingsmembrane met verbeterde retensie en vloed. ’n Drie-vlak driefaktoriaal ontwerp is tydens die studie gebruik om die effek van spinparameters (oplosmiddel tot nie-oplosmiddel verhouding, lumen vloeistof verhouding, en lug-gaping) te ondersoek. ’n Regressie vergelyking is suksesvol daargestel en gebruik om voorspelbaar membrane met goeie werkverrigting binne die limiete van die studie te produseer. Die tru-osmose werkverrigting van die membrane was goed: die sout retensie het tussen 96 en 98% gelê en die permeaatvloed tussen 60 en 46 L/m2.d (2 000 ppm NaCl, 20 bar, 24oC).