The use of positive and negative penalty functions in solving constrained optimization problems and partial differential equations

The Rayleigh-Ritz Method together with the Penalty Function Method is used to investigate the use of different types of penalty parameters. The use of artificial springs as penalty parameters is a very well established procedure to model constraints in the Rayleigh-Ritz Method, the Finite Element Method and other numerical methods. Historically, large positive values were used to define the stiffness coefficient of artificial springs, until recent publications demonstrated that it is possible to use negative values to define the stiffness coefficients of the springs. Furthermore, recent publications show that constraints can be enforced using positive and negative mass or inertia in vibration problems and in a more generic sense using eigenpenalty parameters which are penalty parameters in the matrix associated with the eigenvalue. Before the commencement of this thesis, solutions using artificial inertia were published only for beams and simple spring-mass systems. In this thesis the use of all possible types of penalty parameters are investigated in vibration problems of Euler-Bernoulli beams, thin plates and shallow shells and in elastic stability analysis of Euler-Bernoulli beams, including penalty parameters associated with the geometrical stiffness matrix. The study includes the use of penalty parameters for both enforcing support boundary conditions and continuity conditions along structural joints. This investigation started with the selection of the set of admissible functions that would: (a) allow modelling of beams, plates and shells in completely free boundary conditions; (b) not present any limitation in the number of functions that can be used in the solution. This gives the possibility to converge to the constraint solution and to model any type of boundary conditions. The procedure proposed in this work combines several advantages: accuracy of the results, relative fast convergence, simplicity of the set of admissible functions and flexibility to define boundary conditions. While there are other procedures that may give better accuracy for specific cases, the proposed method is more widely applicable. The procedure used in this work also includes a way to check for round-off errors and ill-conditioning in the results; as well as a way to bracket the exact solution with upper and lower-bound results

Persistence of a pinch in a pipe

The response of low-dimensional solid objects combines geometry and physics in unusual ways, exemplified in structures of great utility such as a thin-walled tube that is ubiquitous in nature and technology. Here we provide a particularly surprising consequence of this confluence of geometry and physics in tubular structures: the anomalously large persistence of a localized pinch in an elastic pipe whose effect decays very slowly as an oscillatory exponential with a persistence length that diverges as the thickness of the tube vanishes, which we confirm experimentally. The result is more a consequence of geometry than material properties, and is thus equally applicable to carbon nanotubes as it is to oil pipelines.Comment: 6 pages, 3 figure

Hierarchic plate and shell models based on p-extension

Formulations of finite element models for beams, arches, plates and shells based on the principle of virtual work was studied. The focus is on computer implementation of hierarchic sequences of finite element models suitable for numerical solution of a large variety of practical problems which may concurrently contain thin and thick plates and shells, stiffeners, and regions where three dimensional representation is required. The approximate solutions corresponding to the hierarchic sequence of models converge to the exact solution of the fully three dimensional model. The stopping criterion is based on: (1) estimation of the relative error in energy norm; (2) equilibrium tests, and (3) observation of the convergence of quantities of interest

Optimization of plastic spherical shells

http://www.ester.ee/record=b1547873~S1*es

The analysis and behavior of thin-steel hyperbolic paraboloid shells

An extensive experimental and analytical investigation of thin-steel hyperbolic paraboloid (hypar) structures was carried out to provide design information. As a result of this work, empirical data is provided regarding the behavior of such structures and computer programs are presented for the analysis of thin steel hypar structures. Hyperbolic paraboloid structures possess a unique combination of structural and architectural properties; some of them are the following: 1) Due to the double curvature of the surface the internal stresses in the deck are generally low and the deflections are small. 2) Since a hypar surface can be generated by straight lines, thin-steel or light-gage panels may be used to form the shell; furthermore such panels are well suited to carry the in-plane shear forces in hypar shells. 3) Basic hypar units can be combined in a large variety of ways to produce attractive roofs (Fig. 1-2, page 212). 4) The dead load to live load ratio is very low in the case of thin-steel shell structures. A hypar unit is a warped surface bounded by straight lines (Fig. 1-1, page 211). The equation of the surface is z = Cxy/AB. According to the simple membrane theory, a uniform load p produces pure shear forces Nxy = ABF/2C. This membrane shear transmits uniform eccentric axial forces to the edge members. The following are the major problems associated with the design of thin-steel hypar structures: 1) The deflections, i stresses, and the stability of hypars depends greatly on the shear rigidity of the thin-steel deck. This property must be evaluated experimentally for each combination of decking, connections to edge members, and seam connections. Furthermore, in the case of hypars the deck is warped and thus the shear rigidity may be different from that of an equivalent flat diaphragm. 2) The deck may buckle due to the shear stresses~ and the buckling load must be evaluated for highly orthotropic shells. 3) The design of thin-steel hypar structures is generally governed by stiffness (deflections or buckling) requirements. The evaluation of the deflections is a very complex matter because it depends on the deck rigidity, the edge member axial and bending stiffnesses, and on the eccentricity of the deck-to-edge member connection. 4) If the curvature (or riseto- span ratio) of a hypar is small, the deflections may be large and a considerable portion of the load is carried by bending rather than by membrane shear. 5) Partial or concentrated loads may cause large local deflections, especially if singlelayer decks are used. The present investigation studied all the above-mentioned factors. The experimental and the analytical studies are summarized briefly in the following paragraphs. The ~~perimental investigatio~ consisted of four types of tests: a) Four medium-scale (12 ft by 12 ft in plan) inverted umbrella tests to study the stresses, deflections, and the deck buckling; b) Test on a small-scale (2 ft by 2 ft) inverted umbrella structure to study scaling effects and the overall i1 buckling of hypars; c) Sixteen flat shear tests to determine the shear rigidity of the decks used in the hypar tests; d) Twelve saddle-s~aped hypar tests (S ft by S ft in plan) with various rise-to-span ratios to evaluate the effect of rise or warping on the shear rigidity and to study other factors such as partial loading and single versus double layered decks. Photos of the various types of tests are shown in Figs. 7.1 to 7.S. The experimental program is described in detail in Chapter VII. Prior to the main test program, several small-scale (2 ft by 2 ft) four-quadrant tests and medium-scale single-quadrant tests were also conducted. These tests were however discontinued because of the severe scaling effects in the case of the small-scale models and the violation of the symmetry conditions in the case of single-quadrant experiments where the neighboring quadrants were missing. Nevertheless, these tests produced useful qualitative information and experience with manufacturing and testing thin-steel hypar structures. The edge members of the umbrella-type specimens were made of tubular members since this afforded easy connection of the warped surface to the straight edges. The decking consisted of single or double layers of standard corrugated panels. One layer was connected to the edge ~enbers with sheet metal screws at various spacings. The seam connections between the panels were also made by means of sheet metal screws. In the case of shells with two layers, the top layer was connected to the bottom layer in a similar manner. iii The medium-scale umbrella models were loaded using air bags under each of the four quadrants. The saddle-shaped hypars were loaded with sand, whereas the small-scale models were loaded through loading pads and suspended weights. The following are the principal conclusions of the experimental part of this investigation: The effective shear stiffness of the cold-formed deck and the rise (or curvature) of the structure are the most important factors influencing the behavior of hypars. For low shear stiffnesses and for small rise-to-span ratios the deflections may be large, the bending stresses tend to increase relative to the membrane stresses, and the possibility of deck buckling increases. As in the case of flat shear diaphragms, the shear stiffness depends strongly on the seam and edge connections. The increase in shear stiffness due to the addition of a second layer of deck was found to be only about 1/3 if the second layer was connected only to the first layer and not directly to the edge members. Similarly, the deflections of a double-layered shell are more than half of those of a corresponding single shell. If the two layers are interconnected with sheet metal screws (on an 8 in. grid in the present saddleshaped hypar tests), the deflections are further reduced by about 10 to 20%, depending on the rise ratio. A particular problem of certain types of hypar structures is the deflection of unsupported outside corners (see Fig. 1-2, page 212). The membrane shear cannot carry the load over such flat corners and thus considerable bending and deflections may develop. The tests showed that the bending stiffness of the iv edge members has a great effect on the corner deflections, in fact, they indicate that the design of the edge members in hypars with flat corners is usually governed by deflection limitations. The measured bending strains in flat saddle shells (riseto- span ratio of 1.8), was much greater than the bending in hypars with greater curvature (rise ratio of 1/3). The membrane theory is insufficient for the design of flat hypar structures. However, the design of the connections (seam or edge) may be based on the shear forces obtained from the simple membrane theory. Several single and double layered saddle-shaped models were tested under partial loading. Since such loads must be carried mainly by bending of beam strips along the deformations of single decks, relatively large deflections were noted. The deflections under the 8 in. by 8 in. loaded area were about three times greater in the single decks than on the doublelayered structures. Since the effective shear rigidity of the deck is of paramount importance, the effect of curvature (warping) on it is an important question. The effective shear rigidity of various deck, edge member, and connection configurations are determined by tests on flat diaphragms. The comparison of the measured deflections for saddle hypars with various rise-span ratios and the evaluation of the effective shear rigidities backwards from the measured deflections indicated that the shear rigidity is reduced by about 20% due to the warping effect. The buckling of the deck is one of the design factors. For small rise-span ratios and for low deck shear rigidities the deck may buckle. As an example, a 12 ft by 12 ft model v having a single layer 24 gage corrugated sheet deck buckled at a uniform load of 70 psf (see Fig. 6.14). This model had relatively stiff edge members (3 in. dia. tubular sections). The corner deflections remained linear with increasing load beyond the buckling load. The buckling load of double-layered structures is much larger than that for single deck shells. A model, similar to the above but with two layers of 28 gage standard corrugated decks, did not buckle up to a load of 145 psf, when the test was discontinued. The major part of the ~~lytical investigation consisted of two finite element approaches for the calculation of deflections, stresses, and instability. In addition, two simple methods were developed for estimating the deck buckling load and the buckling of the compression edge members, which would suffice in preliminary designs. Two types of finite elements were used: curved shallow shell elements and flat elements. The details of the analysis are described in Chapter III. Both approaches were verified by comparisons with existing experimental and analytical results. The stiffness of the eccentric edge members were properly accounted for in the mathematical representation of the structure. The connection of the decks to the edge members may allow rotation about the axis of the edge members and movement normal to the edges dUe to slip at the connections. These possibilities were also considered in the analysis. The instability of the decks was studied with the help of the incremental stiffness matrix approach. The effective stiffvi ness of the system is reduced due to the in-plane forces in the deck. The in-plane forces depend on the deflections of the shell and to obtain the buckling load, the eigenvalues of a large order system need to be evaluated. In the present study the load increment at ion method was used instead. The effect of the in-plane forces was evaluated iteratively at successive load increwents. The buckling load is obtained from the nonlinear load-deflection curve, (Fig. 6-6, Page 276). The comparison of the results of the flat element and the curved element approaches reveals that both give good results for shells supported around the perimeter. However, the flat element method gave better results in the neighborhood of unsupported flat corners. The analysis of the structures tested in this and in other studies confirmed the conclusions of the experimental part of the investigation. The stresses in most types of hypars are low and the design is usually controlled by deflection or buckling linitations. The relative stiffness of the deck and the edge members is an important factor. For stiff edge members the deck tends to bend between opposite edges, whereas in the case of flexible cantilevered edge members the shell partially supports the edge members. Analysis of a structure including the weight of the edge members indicated that this effect may have to be considered in the design of hypar structures. The analysis of buckling of hypar decks showed that the buckling load of double-layered shells is three to four times vii greater than that of single decks. The predicted buckling loads compared well with experimental or previous analytical evidence. The buckling load does not depend much on pre-buckling deflections, however it depends on the axial stiffness of the edge members. The finite element analysis was also used to calculate the deflection of an unsymmetrically loaded inverted umbrella structure. The results, which compared well with experimental data, showed that these deflections are about four times greater than those due to symmetric loading. This increase of deflections obviously depends on the type of structure; in this case much of the flexibility was due to the bending of the central column of the umbrella structure. Since the instability a~alysis of hypars by the finite element method involves considerable amount of computer capacity and expense, approximate methods were developed for the calculation of buckling loads. The buckling of the compression edge members was studied by isolating them from the structure. The instability of columns loaded by tangential axial forces that remain parallel to the member during deflection was evaluated. The results are tabulated in Fig. 6-13, page 284. The buckling of hypar decks was also investigated by the energy method (Section VI-7. The resulting equation has to be minimized to get the critical load; this can easily be done with the help of a computer. This approach is much simpler than the finite element instability analysis and is preferable in preliminary designs. viii A few buckling analyses of cold-formed hypar shells showed that the critical load for double-layers is about three to four times greater than a shell with a single deck. The finite element analysis computer program will be made available to designers by the A~erican Iron and Steel Institute

The analysis and behavior of thin-steel hyperbolic paraboloid shells

An extensive experimental and analytical investigation of thin-steel hyperbolic paraboloid (hypar) structures was carried out to provide design information. As a result of this work, empirical data is provided regarding the behavior of such structures and computer programs are presented for the analysis of thin steel hypar structures. Hyperbolic paraboloid structures possess a unique combination of structural and architectural properties, some of them are the following 1) Due to the double curvature of the surface the internal stresses in the deck are generally low and the deflections are small. 2) Since a hyper surface can be generated by straight lines, thin-steel or light-gage panels may be used to form the shell; furthermore such panels are well suited to carry the in-plane shear forces in hyper shells. 3) Basic hyper units can be combined in a large variety of ways to produce attractive roofs (Fig. 1-2, page 216). 4) The dead load to live load ratio is very low in the case of thin-steel shell structures. A hypar unit is a warped surface bounded by straight lines (Fig. I-I, page 215). The equation of the surface is z = cxy/ab. According to the simple membrane theory, a uniform load p produces pure shear forces N XY = abp/2c. This membrane shear transmits uniform eccentric axial forces to the edge members. The following are the major problems associated with the design of thin-steel hypar structures: 1) The deflections, stresses and the stability of hypars depends greatly on the shear rigidity of the thin-steel deck. This property must be evaluated experimentally for each combination of decking, connections to edge members, and seam connections. Furthermore, in the case of hypars the deck is warped and thus the shear rigidity may be different from that of an equivalent flat diaphragm. 2) The deck may buckle due to the shear stresses, and the buckling load must be evaluated for highly orthotropic shells. 3) The design of thin-steel hypar structures is generally governed by stiffness (deflections or buckling) requirements. The evaluation of the deflections is a very complex matter because it depends on the deck rigidity, the edge member axial and bending stiffnesses, and on the eccentricity of the deck-to-edge me.ber connection. 4) If the curvature (or rise-to-span ratio) of a hypar is small, the deflections may be very large and a considerable portion of the load is carried by bending nather than by membrane shear. 5) Partial or concentrated loads may cause large local deflections, especially if single-layer decks are used. The present investigation studied all the above mentioned factors. The experimental and the analytical studies are summarized briefly in the following paragraphs. The experimental investigation consisted of four types of tests: a) Four medium-scale (12 ft by 12 ft in plan) inverted umbrella tests to study the stresses, deflections, and the deck buckling, b) Test on a small-scale (2 ft by 2 ft) inverted umbrella structure to study scaling effects and the overall buckling of hypars, c) Sixteen flat shear tests to determine the shear rigidity of the decks used in the hyper tests, d) Twelve saddle-shaped hypar tests (5 ft by 5 ft in plan) with various rise-to-span ratios to evaluate the effect of rise or warping on the shear rigidity and to study other factors such as partial loading and single versus double layered decks. Photos of the various types of tests are shown in Figs. 7.1 to 7.5. The experimental program is described in detail in Chapter VII. Prior to the main test program, several small scale (2 ft by 2 ft) four-quadrant tests and medium-scale single-quadrant tests were also conducted. These tests were however discontinued because of the severe scaling effects in the case of the small-scale models and the violation of the symmetry conditions in the case of single-quadrant experiments where the neighboring quadrants were missing. Nevertheless, these tests produced useful qualitative information and experience with manuafacturing and testing thin-steel hypar structures. The edge members of the umbrella-type specimens were made of tubular members since this afforded easy connection of the warped surface to the straight edges. The decking consisted of single or double layers of standard corrugated panels. One layer was connected to the edge members with sheet metal screws at various spacings. The seam connection between the panels was also by means of sheet metal screws. In the case of shells with two layers, the top layer was connected to the bottom layer in a similar manner. The medium-scale umbrella models were loaded using air bags under each of the four quadrants. The saddle-shaped hypars were loaded with sand, whereas the small-scale models were loaded through loading pads and suspended weights. The following are the principal conclusions of the experimental part of this investigation. The effective shear stiffness of the cold-formed deck and the rise (or curvature) of the structure are the most important factors influencing the behavior of hypars. For low shear stiffnesses and for small rise-to-span ratioa the deflections may be large, the bending stresses tend to increase relative to the membrane stresses, and the possibility of deck buckling increases. As in the case of flat shear diaphragms, the shear stiffness depends strongly on the seam and edge connections. The increase in shear stiffness due to the addition of a second layer of deck was found to be only about 1/3 if the second layer was connected only to the first layer and not directly to the edge members. Similarly, the deflections of a double-layered shell are more than half of those of a corresponding single shell. If the two layers are interconnected with sheet metal screws (on an 8 in. grid in the present saddle-shaped hypar tests), the deflections are further reduced by about 10 to 20%, depending on the rise ratio. A particular problem of certain types of hypar structures is the deflection of unsupported outside corners (see Fig. 1-2, page 216). The membrane shear cannot carry the load over such flat corners and thus considerable bending and deflections may develop. The tests showed that the bending stiffness of the edge members has a great effect on the corner deflections, in fact, they indicate that the design of the edge members in hypars with flat corners is probably governed by deflection limitations. The measured bending strains in flat saddle shells (rise-to-span ratio of 1/8) was much greater than the bending in hypers with greater curvature (rise ratio of 1/3). The membrane theory is insufficient for the design of flat hypar structures. However, the design of the connections (seam or edge) may be based on the shear forces obtained from the simple membrane theory. Several single and double layered saddle-shaped models were tested under partial loading. Since such loads must be carried mainly by bending of beam strips along the deformations of single decks, relatively large deflections were noted. The deflections under the 8 in. by 8 in. loaded area were about three times greater in the single decks than on the double-layered structures. Since the effective shear rigidity of the deck is of paramount importance, the effect of curvature (warping) on it is an important question. The effective shear rigidity of various deck, edge member and connection configurations are determined by tests on flat diaphragms. The comparison of the measured deflections for saddle hypars with various rise-span ratios and the evaluation of the effective shear rigidities backwards from the measured deflections indicated that the shear rigidity is reduced by about 20% due to the warping effect. The buckling of the deck is one of the design factors. For small rise-span ratios and for low deck shear rigidities the deck may buckle. As an example, a 12 ft by 12 ft model having a single layer 24 gage corrugated sheet deck buckled at a uniform load of 70 psf (see Fig. ). This model had relatively stiff edge members (3 in. dia. tubular sections). The corner deflections remained linear with increasing load beyond the buckling load. The buckling load of double-layered structures is much larger than that for single deck shells. A model, similar to the above but with two layers of 28 gage standard corrugated decks, did not buckle up to a load of 145 psf, when the test was discontinued. The major part of the analytical investigation consisted of two finite element approaches for the calculation of deflections, stresses, and instability. In addition, two simple methods were developed for estimating the deck buckling load and the buckling of the compression edge members, which would suffice in preliminary designs. Two types of finite elements were used: curved shallow shell elements and flat elements. The details of the analysis are described in Chapter Ill. Both approaches were verified by comparisons with existing experimental and analytical results. The stiffness of the eccentric edge members were properly accounted for in the mathematical representation of the structure. The connection of the decks to the edge members may allow rotation about the axis of the edge members and movement normal to the edges due to slip at the connections. These possibilities were also considered in the analysis. The instability of the decks was studied with the help of the incremental stiffness matrix approach. The effective stiffness of the system is reduced due to the in-plane forces in the deck. The in-plane forces depend on the deflections of the shell and to obtain the buckling load, the eigenvalues of a large- order system needs to be evaluated. In the present study the load incrementation method was used instead. The effect of the in-plane forces was evaluated iteratively at successive load increments. The buckling load is obtained from the non-linear load deflection curve,(Fig. 6-6, Page 280). The comparison of the results of the flat-element and the curved-element approaches reveals that both give good results for shells supported around the perimetry. However, the flat element method gave better results in the neighborhood of unsupported flat corners. The analysis of the structures tested in this and in other studies confirmed the conclusions of the experimental part of the investigation. The stresses in most types of hypars are low and the design is usually controlled by deflection or buckling limitations. The relative stiffness of the deck and the edge members is an important factor. For stiff edge members the deck tends to bend between opposite edges, whereas in the case of flexible cantilevered edge members the shell partially supports the edge members. Analysis of a structure including the weight of the edge members indicated that this effect may have to be considered in the design of hyper structures. The analysis of buckling of hypar decks showed that the buckling load of double-layered shells is three to four times greater than that of single decks. The predicted buckling loads compared well with experimental or previous analytical evidence. The buckling load does not depend much on pre-buckling deflections, however it depends on the axial stiffness of the edge members. The finite element analysis was also used to calculate the deflection of an unsymmetrically loaded inverted umbrella structure. The results, which compared well with experimental data, showed that these deflections are about four times greater than those due to symmetric loading. This increase of deflections obviously depends on the type of structure. in this case much of the flexibility was due to the bending of the central column of the umbrella structure, Since the instability analysis of hypars by the finite element method involves considerable amount of computer capacity and expense, approximate methods were developed for the calculation of buckling loads. The buckling of the compression edge members was studied by isolating them from the structure. The instability of columns loaded by tangential axial forces that to the member remain parallel during deflection was evaluated. The results are tabulated in Fig. 6-13, page 287. The buckling of hypar decks was also investigated by the energy method (Section VI-7). The resulting equation has to be minimized to get the critical load. this can easily be done with the help of a computer. This approach is much simpler than the finite element instability analysis and is preferable in preliminary designs. A few buckling analyses of cold-formed hypar shells showed that the critical load for double-layers is about three to four times greater than a shell with a single deck

Recent advances in shell theory

The results reviewed are divided into two categories: those that relate two-dimensional shell theory to three-dimensional elasticity theory and those concerned with shell theory per se. In the second category results for general elastic systems that carry over, by specialization or analogy, to shells and results that are unique to shell theory itself are considered

Multistable Shell Structures

Multistable structures, which possess by definition more than one stable equilibrium configuration, are capable of adapting their shape to changing loading or environmental conditions and can further improve multi-purpose ultra-lightweight designs. Whilst multiple methods to create bistable shells have been proposed, most studies focussed on free-standing ones. Considering the strong influence of support conditions on related stability thresholds, surprisingly little is known about their influence on multistable behaviour. In fact, the lack of analytical models prevents a full understanding and constitutes a bottle-neck in the development process of novel shape-changing structures. The relevance becomes apparent in a simple example: whilst an unsupported sliced tennis ball can be stably inverted without experiencing a reversion, fixing its edge against rotation erodes bistability by causing an instantaneous snap-back to the initial configuration. This observation reveals the possibility to alter the structural response dramatically by a simple change of the support conditions. This dissertation explores the causes of this behaviour by gaining further insight into the promoting and eschewing factors of multistability and aims to point out methods to exploit this feature in optimised ways. The aforementioned seemingly simple example requires a geometrically nonlinear perspective on shells for which analytical solutions stay elusive unless simplifying assumptions are made. In order to captures relevant aspects in closed form, a novel semi-analytical Ritz approach with up to four degrees of freedom is derived, which enforces the boundary conditions strongly. In contrast to finite element simulations, it does not linearise the stiffness matrix and can thus explore the full solution space spanned by the assumed polynomial deflection field. In return, this limits the method to a few degrees of freedom, but a comparison to reference calculations demonstrated an excellent performance in most cases. First, the level of influence of the boundary conditions on the critical shape for enabling a bistable inversion is formally characterised in rotationally symmetric shells. Systematic insight is provided by connecting the rim to ground through sets of extensional and rotational linear springs, which allows use of the derived shell model as a macro-element that is connected to other structural elements. It is demonstrated that bistability is promoted by an increasing extensional stiffness, i.e. bistable roller-supported shells need to be at least twice as tall compared to their fixed-pinned counterparts. The effect of rotational springs is found to be multi-faceted: whilst preventing rotation has the tendency to hinder bistable inversions, freeing it can even allow for extra stable configurations; however, a certain case is emphasised in which an increasing rotational spring stiffness causes a mode transition that stabilises inversions. In a second step, a polar-orthotropic material law is employed to study variations of the directional stiffness of the shell itself. A careful choice of the basis functions is required to accurately capture stress singularities in bending that arise if the radial Young’s modulus is stiffer than its circumferential equivalent. A simple way to circumvent such singularities is to create a central hole, which is shown not to hamper bistable inversions. For significantly stiffer values of the radial stiffness, a strong coupling with the support conditions is revealed: whilst roller-supported shells do not show a bistable inversion at all for such materials, fixed-pinned ones feel the most disposed to accommodate an alternative equilibrium configuration. This behaviour is explained via simplified beam models that suggest a new perspective on the influence of the hoop stiffness: based on observations in free-standing shells, it was thought to promote bistability, but it is only insofar stabilising, as it evokes radial stresses; if these are afforded by immovable supports, it becomes redundant and even slightly hindering. Finally, combined actuation methods in stretching and bending that prescribe non-Euclidean target shapes are considered to emphasise the possibility of multifarious structural manipulations. When both methods are geared to each other, stress-free synclastic shape transformations in an over-constrained environment, or alternatively, anticlastic shape-changes with an arbitrary wave number, are achievable. Considering nonsymmetric deformations offers a richer buckling behaviour for certain in-plane actuated shells, where a secondary, approximately cylindrical buckling mode as well as a ‘hidden’ stable configuration of a higher wave number is revealed by the presented analytical model. Additionally, it is shown that the approximately mirror-symmetric inversion of cylindrical or deep spherical shells can be accurately described by employing a simpler, geometrically linear theory that focusses on small deviations from the mirrored shape. The results of this dissertation facilitate a versatile practical application of multistable structures via an analytical description of more realistic support conditions. The understanding of effects of the internal stiffness makes it possible to use this unique structural behaviour more efficiently by making simple cross-sectional adjustments, i.e. by adding appropriate stiffeners. Eventually, the provided theoretical framework of emerging actuation methods might inspire novel morphing structures.Friedrich Ebert Foundation (Friedrich-Ebert-Stiftung) Corpus Christi College, Cambridge Department of Engineering, Cambridg

The dynamics and control of large flexible space structures. Volume 3, part B: The modelling, dynamics, and stability of large Earth pointing orbiting structures

The dynamics and stability of large orbiting flexible beams, and platforms and dish type structures oriented along the local horizontal are treated both analytically and numerically. It is assumed that such structures could be gravitationally stabilized by attaching a rigid light-weight dumbbell at the center of mass by a spring loaded hinge which also could provide viscous damping. For the beam, the small amplitude inplane pitch motion, dumbbell librational motion, and the anti-symmetric elastic modes are all coupled. The three dimensional equations of motion for a circular flat plate and shallow spherical shell in orbit with a two-degree-of freedom gimballed dumbbell are also developed and show that only those elastic modes described by a single nodal diameter line are influenced by the dumbbell motion. Stability criteria are developed for all the examples and a sensitivity study of the system response characteristics to the key system parameters is carried out