Static Determinacy Research Articles

On the computation of in vivo transmural mean stress of patient-specific aortic wall

It is well known that residual deformations/stresses alter the mechanical behavior of arteries, e.g., the pressure-diameter curves. In an effort to enable personalized analysis of the aortic wall stress, approaches have been developed to incorporate experimentally derived residual deformations into in vivo loaded geometries in finite element simulations using thick-walled models. Solid elements are typically used to account for "bending-like" residual deformations. Yet, the difficulty in obtaining patient-specific residual deformations and material properties has become one of the biggest challenges of these thick-walled models. In thin-walled models, fortunately, static determinacy offers an appealing prospect that allows for the calculation of the thin-walled membrane stress without patient-specific material properties. The membrane stress can be computed using forward analysis by enforcing an extremely stiff material property as penalty treatment, which is referred to as the forward penalty approach. However, thin-walled membrane elements, which have zero bending stiffness, are incompatible with the residual deformations, and therefore, it is often stated as a limitation of thin-walled models. In this paper, by comparing the predicted stresses from thin-walled models and thick-walled models, we demonstrate that the transmural mean stress is approximately the same for the two models and can be readily obtained from in vivo clinical images without knowing the patient-specific material properties and residual deformations. Computation of patient-specific mean stress can be greatly simplified by using the forward penalty approach, which may be clinically valuable.

Three-dimensional DEM investigation of critical state and dilatancy behaviors of granular materials

The critical state is significant to the mechanical behaviors of granular materials and the foundation of the constitutive relations. Using the discrete element method (DEM), the mechanical behaviors of granular materials can be investigated on both the macroscopic and microscopic levels. A series of DEM simulations under true triaxial conditions have been performed to explore the critical state and dilatancy behavior of granular materials, which show the qualitatively similar macroscopic responses as the experimental results. The critical void ratio and stress ratio under different stress paths are presented. A unique critical state line (CSL) is shown to indicate that the intermediate principal stress ratio does not influence the CSL. Within the framework of the unique critical state, the stress–dilatancy relation of DEM simulations is found to fulfill the state-dependent dilatancy equations. As a microscopic parameter to evaluate the static determinacy of the granular system, the redundancy ratio is defined and investigated. The results show that the critical state is very close to the statically determinate state. Other particle-level indexes, including the distribution of the contact forces and the anisotropies, are carefully investigated to analyze the microstructural evolution and the underlying mechanism. The microscopic analysis shows that both the contact orientations and contact forces influence the mechanical behaviors of granular materials.

Solving membrane stress on deformed configuration using inverse elastostatic and forward penalty methods

Wall stress in a curved membrane depends on the surface geometry and applied load, not the constitutive law of the wall material (static determinacy). This remarkable property suggests that the membrane stress can be determined without knowing of the material property. Being able to determine the wall stress without material knowledge is practically significant in numerous applications, particularly in in vivo biomedical stress analysis wherein patient-specific tissue properties are extremely difficult to obtain. In this article, we present two non-traditional methods that utilize the static determinacy to compute the membrane stress. A common theme is to start with a deformed configuration and use artificial material models. In one of these approaches, the equilibrium problem is formulated as an inverse elastostatic problem whereby the stress is determined by way of finding a (fictitious) reference configuration. The other approach utilizes the standard forward analysis, but in a penalty setting using artificially stiff material models. A continuum sensitivity analysis is implemented to quantitatively assess the influence of material parameters. These developments open new pathways of membrane analysis enabling stress prediction without using realistic material parameters. Numerical examples are provided to demonstrate and compare these two methods.

A miniaturized, high frequency mechanical scanner for high speed atomic force microscope using suspension on dynamically determined points.

One of the major limitations in the speed of the atomic force microscope (AFM) is the bandwidth of the mechanical scanning stage, especially in the vertical (z) direction. According to the design principles of "light and stiff" and "static determinacy," the bandwidth of the mechanical scanner is limited by the first eigenfrequency of the AFM head in case of tip scanning and by the sample stage in terms of sample scanning. Due to stringent requirements of the system, simply pushing the first eigenfrequency to an ever higher value has reached its limitation. We have developed a miniaturized, high speed AFM scanner in which the dynamics of the z-scanning stage are made insensitive to its surrounding dynamics via suspension of it on specific dynamically determined points. This resulted in a mechanical bandwidth as high as that of the z-actuator (50 kHz) while remaining insensitive to the dynamics of its base and surroundings. The scanner allows a practical z scan range of 2.1 μm. We have demonstrated the applicability of the scanner to the high speed scanning of nanostructures.

Stiffness threshold of randomly distributed carbon nanotube networks

For carbon nanotube (CNT) networks, with increasing network density, there may be sudden changes in the properties, such as the sudden change in electrical conductivity at the electrical percolation threshold. In this paper, the change in stiffness of the CNT networks is studied and especially the existence of stiffness threshold is revealed. Two critical network densities are found to divide the stiffness behavior into three stages: zero stiffness, bending dominated and stretching dominated stages. The first critical network density is a criterion to judge whether or not the network is capable of carrying load, defined as the stiffness threshold. The second critical network density is a criterion to measure whether or not most of the CNTs in network are utilized effectively to carry load, defined as bending–stretching transitional threshold. Based on the geometric probability analysis, a theoretical methodology is set up to predict the two thresholds and explain their underlying mechanisms. The stiffness threshold is revealed to be determined by the statical determinacy of CNTs in the network, and can be estimated quantitatively by the stabilization fraction of network, a newly proposed parameter in this paper. The other threshold, bending–stretching transitional threshold, which signs the conversion of dominant deformation mode, is verified to be well evaluated by the proposed defect fraction of network. According to the theoretical analysis as well as the numerical simulation, the average intersection number on each CNT is revealed as the only dominant factor for the electrical percolation and the stiffness thresholds, it is approximately 3.7 for electrical percolation threshold, and 5.2 for the stiffness threshold of 2D networks. For 3D networks, they are 1.4 and 4.4. And it also affects the bending–stretching transitional threshold, together with the CNT aspect ratio. The average intersection number divided by the fourth root of CNT aspect ratio is found to be an invariant at the bending–stretching transitional threshold, which is 6.7 and 6.3 for 2D and 3D networks, respectively. Based on this study, a simple piecewise expression is summarized to describe the relative stiffness of CNT networks, in which the relative stiffness of networks depends on the relative network density as well as the CNT aspect ratio. This formula provides a solid theoretical foundation for the design optimization and property prediction of CNT networks.

Algorithms for Solution of Direct and Inverse Geometric Problems Involving Determination of Angular Platform Orientation

A six-DOF mechanism of a Steward (or Gough–Steward) type of platform is isolated among parallel-structure mechanisms. Steward [1] and Gough and Whitehall [2] propose a specific application of platforms in special test benches. The general theory of multiple-DOF platforms was developed in the 1980s–1990s [3–9]; however, since then serious investigations have been performed on this subject [10–21]. Analytical methods of solving analytical problems of the statics, kinematics, and dynamics of six-DOF platforms make it possible to establish qualitative properties. Computer programs are currently employed for practical solution of mechanics problems of these platforms. The direct geometry problem for the parallel-structure mechanisms under consideration is formulated in the following manner: the position of the platform (a free solid body with six degrees of freedom in space) is assigned by six geometric parameters, and the platform is hinge-connected to a stationary base with rigid driving limbs equipped with linear drive for varying their lengths (within certain limits). In classical studies involving the mechanics of six-DOF platforms, it is assumed that all degrees of freedom on the stationary base are situated in the same plane, and the hinges on the moving platform are arranged in a single moving plane. In certain structures (for example, [18]), these restrictions are removed, and the hinges on the stationary base and moving platform may be arranged at the whim of the investigator. Problems involving the selection of the position of the spherical hinges of the driving limbs on the stationary base and platform are solved separately on the basis of different requirements (for example, provision for maximum stiffness of the structure on the whole, leaving zones free, etc.); it is then considered that arrangement of the hinges is assigned. Basic Geometric Relationships. It is necessary and sufficient that there be six driving limbs for single-valued assignment of the position of the moving platform on the one hand, and for static determinacy on the other. In the direct geometry (or kinematics) problem, values of the lengths li (distance between centers of spherical hinges) of the six limbs of variChemical and Petroleum Engineering, Vol. 50, Nos. 9–10, January, 2015 (Russian Original Nos. 9–10, Sept.–Oct., 2014)

Structural Topology Optimization with Stress Constraint Considering Loading Uncertainties

This paper deals with the consideration of loading uncertainties in topology optimization via a fundamental optimization problem setting. Variability of loading in engineering design is realized e.g. in the action of various load combinations. In this study this phenomenon is modelled by the application of two mutually excluding (i.e. alternating) forces such that the magnitudes and directions are varied parametrically in a range. The optimization problem is stated as to find the minimum volume (i.e. the minimum weight) load-bearing elastic truss structure that transfers such loads acting at a fix point of application to a given line of support provided that stress limits are set. The aim of this paper is to numerically determine the layout, size, and volume of the optimal truss and to support the numerical results by appropriate analytical derivations. We also show that the optimum solution is non-unique, which aects the static determinacy of the structure as well. In this paper we also create a truss-like structure with rigid connections based on the results of the truss optimization and analyse it both as a bar structure (frame model) and a planar continuum (disk) structure to compare with the truss model. The comparative investigation assesses the validity of computational models and proves that the choice aects design negatively since rigidity of connections resulted by usual construction technologies involve extra stresses leading to significant undersizing.

Method for Generating Statically Determinate Cable Net Topology Configurations of Deployable Mesh Antennas

AbstractLarge deployable high-frequency mesh antennas are being envisioned for future space missions. The static determinacy is required to ensure the long-term dimensional stability of cable net reflectors of mesh antennas under extreme space environments. In this paper, the design methodologies for creating statically determinate cable net topology configurations of AstroMesh and umbrella-type antennas are proposed on the basis of the generalized Maxwell’s rule, respectively. To generate these configurations, the number of subdivisions of mesh reflectors is first determined by the allowable surface accuracy. Then, by introducing the concept of nodal connectivity, the cable net topology configurations of AstroMesh antennas are deduced from a general analytical solution of linear algebraic theory. The desired topology configurations of umbrella-type antennas are obtained by adjusting the number of nodes, cables, and kinematic constraints under uncertain supporting boundary conditions. Some novel cable net...

Geometric sensitivity of statically determinate planar truss families

We define a truss family Ti by a statically determinate truss T0 and a recursive step Ti+1=f(Ti), such that Ti+1Ti and that step f(Ti) keeps static determinacy. Such recursive algorithms have been broadly discussed in the literature, e.g. the Henneberg operation 1 is a well-known example. Earlier we introduced the concept of geometric sensitivity index rg of trusses, here we investigate the sensitivity of truss families, in particular, the limit sensitivity (Ti).

Geometric sensitivity of statically determinate planar truss families

We define a truss family Ti by a statically determinate truss T0 and a recursive step Ti+1=f(Ti), such that Ti+1Ti and that step f(Ti) keeps static determinacy. Such recursive algorithms have been broadly discussed in the literature, e.g. the Henneberg operation 1 is a well-known example. Earlier we introduced the concept of geometric sensitivity index rg of trusses, here we investigate the sensitivity of truss families, in particular, the limit sensitivity (Ti).

Indeterminacy of the Offset Mechanism and Microscale Static Determinacy of Roll System in Four-high Mill

In the four-high mill used for rolling plate,an offset is generally set to keep the rolls parallel and prevent the cross between the rolls,and which becomes the parallel positioning mechanism of roll system in four-high mill entered into the textbook of rolling mills.In all contemporary four-high mills,there is an offset between the rolls.But the offset cannot ensure the certainty of its mechanism,and the dynamic cross between the rolls happens frequently,which induces ultra-large axial force to damage the thrust rolling bearing or generate support reaction deviation in rolling to break the stability of plate shape control.Online inspection test of the axial force of a high-speed four-high mill for rolling aluminum foil and with a 3 mm offset distance is carried out,which reveals the occurrence of regular dynamic cross between work roll and backup roll during the rolling process.The dynamic cross behavior between the rolls is corresponding to the size of small clearance between bearing seat and housing window column,limited only in the range of β≤0.05° cross angle of rolling mill,and far lower than β=1°～1.5° cross angle of PC mill.But the derivative axial force of the dynamic cross exceeds three times of the rated value The static acyclic test of the offset distance of four-high mill roll system proves that with constant original installation position between the rolls there is no horizontal displacement of rolls.The dynamic cross phenomenon and zero static horizontal displacement reveal the indeterminacy of the mechanism of setting offset distance,so it cannot be used for parallel positioning and stability design of four-high mill.

The Stringer System — A Truss Model of Membrane Shells for Analysis and Design of Boundary Conditions

This paper deals with the problem: What sort of boundary conditions are to be prescribed on the edge of a shell that is to be in a statically determinate membrane state? Triangulated single layer trusses have been suggested as a model for membrane shells by (Calladine 1983) and have been investigated by (Szabo and Tarnai 1992). However, these efforts have not solved the problem completely. The paper presents a triangular single layer truss system – the stringer system – which differs from the above mentioned models in two ways: the systematic topology makes it relatively simple to control whether or not the system is statically and geometrically/kinematically determinate, the geometry corresponds to the curvature of the considered surface and ensures that the system can be applied as a valid static and geometric model for the membrane shell. In most cases, the stringer system regarded as a static model of membrane shells leads to the same results as the theory of partial differential equations, as investigated by (Tarnai 1980a, 1981, 1983). However, some new boundary conditions for shells with positive and zero Gaussian curvature have been found and the types of supports and the connection between statical and geometrical determinacy have been further generalised. The paper is based on parts of the Ph.D. Thesis by (Almegaard 2003).

A New Design Conception for Large Span Deployable Flat Grid Structures

The deployable flat grids, consisting of only hinged pantographic units, usually have low structural stiffness, and the bending moment constitutes a great part of internal forces. This results in structural inefficiency, especially for large span grids. Diagonals (web struts) and lower chords are added herein to enhance the structural stiffness. The newly added web struts must be compatible with the deployable geometry. Cables or foldable bars can be used as lower chords. The statical determinacy characteristics are analysed for the newly proposed grid system. On the basis of this concept, several flat grids are firstly developed for potential applications. And a joint conception also developed. The structural performances are investigated for a series of deployable flat grids. Steel quantities and deflections are compared for four different grids. The results indicate that the newly proposed conception has fairly high economic scale, structural efficiency and promising potential application field.

Buckling of thin cylindrical shells: an attempt to resolve a paradox

The classical theory of buckling of axially loaded thin cylindrical shells predicts that the buckling stress is directly proportional to the thickness t, other things being equal. But empirical data show clearly that the buckling stress is actually proportional to t 1.5, other things being equal. As is well known, there is wide scatter in the buckling-stress data, going from one half to twice the mean value for a given ratio R/ t. Current theories of shell buckling explain the low buckling stress—in comparison with the classical—and the experimental scatter in terms of “imperfection-sensitive”, non-linear behaviour. But those theories always take the classical analysis of an ideal, perfect shell as their point of reference. Our present principal aim is to explain the observed t 1.5 law. So far as we know, no previous attack has been made on this particular aspect of thin-shell buckling. Our work is thus breaking new ground, and we shall deliberately avoid taking the classical analysis as our starting point. We first point out that experiments on self-weight buckling of open-topped cylindrical shells agree well with the mean experimental data mentioned above; and then we associate those results with a well-defined post-buckling “plateau” in load/deflection space, that is revealed by finite-element studies. This plateau is linked with the appearance of a characteristic “dimple” of a mainly inextensional character in the deformed shell wall. A somewhat similar post-buckling dimple is also found by quite separate finite-element studies when a thin cylindrical shell is loaded axially at an edge by a localised force; and it turns out that such a dimple grows under a more-or-less constant force that is proportional to t 2.5, other things being equal. This 2.5-power law can be explained by analogy with the inversion of a thin spherical shell by an inward-directed force. Thus, the deformation of such a shell is generally inextensional except for a narrow “knuckle” or boundary layer in which the combined local elastic energy of bending and stretching is proportional to t 2.5, other things being equal. Similarly, the modes of deformation in the post-buckling dimples in a cylindrical shell are practically independent of thickness, except in the highly deformed boundary-layer regions which separate the inextensionally distorted portions of the shell. These ideas lead in turn to an explanation of the t 1.5 law for the post-buckling stress of open-topped cylindrical shells loaded by their own weight. We attribute the absence of experimental scatter in the self-weight buckling of open-topped cylindrical shells to the statical determinacy of the situation, which allows a post-buckling dimple to grow at a well-defined “plateau load”. Conversely, the large experimental scatter in tests on cylinders with closed ends may be attributed to the lack of statical determinacy there. Our paper contains several arguments that are not mathematically water-tight, in contrast to many reports in the field of mechanics of structures. We plead that the problem which we have tackled is so difficult that the only way forward is one of “over-simplification”. We hope that our work will be judged not with respect to its absence of mathematical precision, but by the light which it sheds upon the problem under investigation.

Engineering Mechanics: Statics

TO THE INSTRUCTOR / TO THE STUDENT 1. REVIEW OF CONCEPTS IN MECHANICS Brief History of Mechanics / Newtonian Mechanics, Engineering Mechanics / Units of Measure and Physical Dimensions / Numerical Computations in Engineering / Suggestions for Problem Solving / Problems 2. FORCES AND OTHER VECTOR QUANTITIES Characteristics of Forces / Newton's Third Law / Force Resultants and Force Components / Vectors and Vector Quantities / Rectangular Components of a Vector / Unit Vectors / Problems 3. EQUILIBRIUM OF A PARTICLE Concept of Particle Equilibrium / Free-Body Diagrams / Types of Forces / Concept of a Rigid Body / Equilibrium of a Two-Force Member / Transmissibility of Forces That Act on a Rigid Body / Equilibrium of a Rigid Body Subjected to Concurrent Forces / Problems 4. TWO-DIMENSIONAL FORCES, COUPLES, AND RIGID-BODY EQUILIBRIUM Moment of Coplanar Forces with Respect to an Axis / Resultant of Coplanar Forces That Act on a Rigid Body / Parallel Coplanar Forces and Couples / Moment of a Couple / Lateral Displacement of Forces / Equilibrium of a Rigid Body Subjected to Coplanar Forces / Alternative Formulations of Equilibrium for Coplanar Forces / General Theory of Parallel Forces / Problems 5. THREE-DIMENSIONAL FORCES, COUPLES, AND RIGID-BODY EQUILIBRIUM Vector Algebra / Moment of a Force about a Point / Moment of Force about an Axis / Vector Representation of Couples / Representation of a Couple as a Vector Product /Composition of Forces That Act on a Rigid Body / Resultant Axis and the Wrench / The Fundamental Principle of Equilibrium of a Rigid Body / Problems 6. TRUSSES The Truss-A System of Two-Force Members / Stability and Statical Determinacy / Methods of Truss Analysis / The Method of Joints / The Method of Sections / Compound and Complex Trusses / Space Trusses / Problems 7. SIMPLE STRUCTURES AND MACHINES The Lever / The Pulley / Simple Plane Frames / Mechanisms / Parabolic Cables and the Suspension Bridge / The Catenary-A Freely Hanging Cable / Problems 8. CENTER OF GRAVITY, CENTROIDS, AND DISTRIBUTED FORCES Gravity Axis of a Body / Center of Gravity in Cartesian Coordinates / Center of Gravity by Integration / Centroids of Plane Areas and Lines / Composite Bodies / Theorems of Pappus-Guldinus / Resultants of Distributed Parallel Forces / Problems 9. FLUID STATICS Definitions of Stress, Fluid, and Pressure / The Concept of Pressure at a Point / Pressure Variation with Depth in a Fluid / Normal Force on a Flat Surface. (Part contents)

Compatibility requirements for yield-line mechanisms

Yield-line analysis proves a powerful and convenient method for establishing the peak load capacity of reinforced concrete slabs. Its application requires compatible mechanisms to be postulated, comprising rigid regions intersecting at yield-lines where relative rotations may occur. A systematic procedure is described for checking the compatibility of postulated yield-line mechanisms. The similarity between this method and equilibrium requirements in a plane pin-jointed truss is highlighted, and an analogy between yield-line mechanism compatibility and statical determinacy in pin-jointed trusses is established.

Transit Motion Analysis of Enveloping Grasp Based on the Linear Complementarity Problem

This paper discusses a transit motion of a planar enveloping grasp with friction point contacts. We first formulate the kinematics and dynamics of the enveloping grasp on the assumption that Coulomb friction is applied at each contact point. The conditions for static determinancy of the enveloping grasp are derived based on the linear complementarity problem. We also show the relation between the static determinacy of the enveloping grasp and kinetic friction. Finally the conditions for the feasibility of a continuous motion by the enveloping grasp are presented.

Reaction forces on a ladder leaning on a rough wall

The determination of the reactions on a ladder is discussed including friction against the wall. In a recent article, this problem was studied considering elastic compression and it was argued that an additional condition holds that allows one to find all the reactions. We show that this condition is incorrect for typical ladders, when flexion is important. We clarify the issues of static determinacy by means of an analysis that takes into account both compression and flexion. We find that when the climber arrives at the top, the result depends on which deformation prevails. We conclude that the reactions can never be determined using only statics, because there is no way to ascertain which kind of deformation dominates, and so which limiting condition will be attained, if any.

Static determinacy of the limiting state of a rigid body subject to separation

Static determinacy of the limiting state of a rigid body subject to separation

Linear programming in prestressed trusses

This paper discusses the effect of prestress on the minimum weight design of singly loaded trusses of fixed geometry which are required to satisfy stress constraints. Both general trusses and trusses whose design requires a predetermined ratio for the cross-sectional areas (same for simplicity) are considered. Formally synthesized as nonlinear programming problems, they are later reduced to linear programming problems by removing the constitutive equations from the formulation by utilizing the results of theorems relating to realizability through prestressing. With equilibrium only left to be considered the general truss problem reduces identically to that of optimal design without prestress having the well-known properties of fully stressedness and statical determinacy. Prestressing, therefore, has no meaning as far as weight is concerned. However, for trusses in the special category, prestressing can reduce the weight as the structure is understressed because usually one member governs the design. Posterior considerations are given for attaining the prestress and an example that shows the weight advantages of prestressing is presented.