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T.h.g. Megson - One of the best experts on this subject based on the ideXlab platform.
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Analysis of Pin-Jointed Trusses
Structural and Stress Analysis, 2014Co-Authors: T.h.g. MegsonAbstract:This chapter discusses the types of trusses, their functions, and the idealization of a truss into a form amenable to analysis. In practice trusses are not pin-jointed but are constructed, such as in the case of steel trusses, by bolting, riveting, or welding the ends of the members to gusset plates. In a timber roof truss, the members are connected using spiked plates driven into their vertical surfaces on each side of a joint. The joints in trusses are therefore semi-rigid and can transmit moments, unlike a frictionless pinned joint. Furthermore, if the loads are applied at points on a member away from its ends, that member behaves as a fixed or built-in beam with unknown moments and shear forces as well as axial loads at its ends. Such a truss would possess a high degree of Statical Indeterminacy and would require a computer-based analysis. The chapter also investigates the criterion, which indicates the degree of their Statical determinacy, the action of the members of a truss in supporting loads and, finally, methods of analysis of both plane and space trusses.
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Analysis of Statically Indeterminate Structures
Structural and Stress Analysis, 2014Co-Authors: T.h.g. MegsonAbstract:This chapter examines methods of analysis of different forms of Statically indeterminate structures. In the analysis of Statically indeterminate structures two basic methods are employed. In the first method—flexibility or force method—the structure is reduced to a Statically determinate state by employing releases. The alternative procedure, known as the stiffness or displacement method, is analogous to the flexibility method, the major difference being that the unknowns are the displacements at specific points in the structure. Generally the procedure requires a structure to be divided into a number of elements for each of which load-displacement relationships are known. Equations of equilibrium are then written in terms of the displacements at the element junctions and are solved for the required displacements. Both the flexibility and stiffness methods, for practical structures having a high degree of Statical Indeterminacy, generally result in a large number of simultaneous equations that are most readily solved by computer-based techniques. However, the flexibility method requires the structure to be reduced to a Statically determinate state by inserting releases, a procedure requiring some judgment on the part of the analyst. But, the stiffness method requires no such judgment to be made and is therefore particularly suitable for automatic computation.
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Chapter 15 – Virtual Work and Energy Methods
Structural and Stress Analysis, 2014Co-Authors: T.h.g. MegsonAbstract:The majority of the structural problems we have encountered so far have involved structures in which the support reactions and the internal force systems are Statically determinate. These include beams, trusses, cables and three-pinned arches and, in the case of beams, we have calculated displacements. Some Statically indeterminate structures have also been investigated. These include the composite structural members in Section 7.10 and the circular section beams subjected to torsion and supported at each end in Section 11.1. These relatively simple problems were solved using a combination of Statical equilibrium and compatibility of displacements. Further, in Section 13.6, a Statically indeterminate propped cantilever was analysed using the principle of superposition (Section 3.7) while the support reactions for some cases of fixed beams were determined by combining the conditions of Statical equilibrium with the moment-area method (Section 13.3). These methods are perfectly adequate for the comparatively simple problems to which they have been applied. However, other more powerful methods of analysis are required for more complex structures which may possess a high degree of Statical Indeterminacy. These methods will, in addition, be capable of providing rapid solutions for some Statically determinate problems, particularly those involving the calculation of displacements.
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Chapter 4 – Analysis of Pin-jointed Trusses
Structural and Stress Analysis, 2005Co-Authors: T.h.g. MegsonAbstract:Publisher Summary This chapter discusses the types of trusses, their functions, and the idealization of a truss into a form amenable to analysis. In practice trusses are not pin-jointed but are constructed, such as in the case of steel trusses, by bolting, riveting, or welding the ends of the members to gusset plates. In a timber roof truss, the members are connected using spiked plates driven into their vertical surfaces on each side of a joint. The joints in trusses are therefore semi-rigid and can transmit moments, unlike a frictionless pinned joint. Furthermore, if the loads are applied at points on a member away from its ends, that member behaves as a fixed or built-in beam with unknown moments and shear forces as well as axial loads at its ends. Such a truss would possess a high degree of Statical Indeterminacy and would require a computer-based analysis. The chapter also investigates the criterion, which indicates the degree of their Statical determinacy, the action of the members of a truss in supporting loads and, finally, methods of analysis of both plane and space trusses.
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Chapter 16 – Analysis of Statically Indeterminate Structures
Structural and Stress Analysis, 2005Co-Authors: T.h.g. MegsonAbstract:Publisher Summary This chapter examines methods of analysis of different forms of Statically indeterminate structures. In the analysis of Statically indeterminate structures two basic methods are employed. In the first method—flexibility or force method—the structure is reduced to a Statically determinate state by employing releases. The alternative procedure, known as the stiffness or displacement method, is analogous to the flexibility method, the major difference being that the unknowns are the displacements at specific points in the structure. Generally the procedure requires a structure to be divided into a number of elements for each of which load-displacement relationships are known. Equations of equilibrium are then written in terms of the displacements at the element junctions and are solved for the required displacements. Both the flexibility and stiffness methods, for practical structures having a high degree of Statical Indeterminacy, generally result in a large number of simultaneous equations that are most readily solved by computer-based techniques. However, the flexibility method requires the structure to be reduced to a Statically determinate state by inserting releases, a procedure requiring some judgment on the part of the analyst. But, the stiffness method requires no such judgment to be made and is therefore particularly suitable for automatic computation.
Taiji Adachi - One of the best experts on this subject based on the ideXlab platform.
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Residual Stress in Bone Structure: Experimental Observation and Model Study with Uniform Stress Hypothesis
Biomechanics, 1996Co-Authors: M. Tanaka, Taiji AdachiAbstract:In soft tissues, residual stress/strain have been observed experimentally and their role has been investigated from the point of view of stress/strain regulation or mechanical remodeling. In this chapter, the residual stress of bone structure is examined for the leporine tibiofibula bone and the bovine coccygeal vertebra. The fibula-cutting experiment for the leporine tibiofibula demonstrated that the axial strain induced in the tibia is distributed in the circumferential direction, suggesting that the residual stress is caused by Statical Indeterminacy. Numerical fibula cutting was studied for the remodeling equilibrium, which was simulated by the remodeling model considering residual stress and in which stress distribution is uniform under the working load. It was found that the strain induced by the real fibula cutting experiment coincides with that by the numerical experiments. For the bovine coccygeal vertebra, the end-plates and cancellous bone were removed sequentially, and positive strains were observed in the cephalocaudal and circumferential directions. The strain induced in the cephalocaudal direction was examined for the uniform stress state of the model of two-bar elements of the cortical and cancellous bone under working load, and the strain induced in the circumferential direction was for that of the model of two-cylinder elements as well. From these considerations, it was confirmed that the residual stress of bone structure suggested experimentally can be understood in the context of model studies based on the uniform stress hypothesis at remodeling equilibrium.
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Model and Simulation of Bone Remodeling Considering Residual Stress
Computational Biomechanics, 1996Co-Authors: Masao Tanaka, Taiji AdachiAbstract:In bone remodeling, some bone material is resorbed and new bone material is then synthesized. In this context, the new bone material may have its own natural state which is different from that of the old bone materials in its neighbors, and this nonuniform distribution of the natural state is suggested to cause the stress/strain that remains even when all external loads are removed. Our preliminary observations have also suggested the existence of residual stress/strain in the bone structure, and the model of bone remodeling is expected to take these into account. This chapter describes a mathematical model of the bone remodeling that considers residual stress from the Statical Indeterminacy of the bone structure. The basic idea is discussed by using a lumped parameter system and is then extended to distributed parameter systems of the conventional continuum. The idea is also combined with the lattice continuum, that is, a continuum consisting of the rigidly interconnected elastic members as the microstructure, and this is used as a mechanical model of cancellous bone with trabecular architecture. The fundamental characteristics of the model are examined through remodeling simulations of the diaphysis bone under axial load by using a cylindrical model. The proposed model for the conventional continuum and the lattice continuum is applied to remodeling simulations of leporine tibiofibula bone and bovine vertebral bone. The capability of the remodeling simulation is demonstrated by the morphology of the bone structure as well as residual stress in comparison with experimental observations.
Alan Williams - One of the best experts on this subject based on the ideXlab platform.
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Chapter 1 – Statical Indeterminacy
Structural Analysis, 2009Co-Authors: Alan WilliamsAbstract:Publisher Summary A Statically determinate structure is one in which all the member forces and external reactions can be determined by applying the equations of equilibrium. An indeterminate or redundant structure is one that possesses more unknown member forces and reactions than available equations of equilibrium. To determine the member forces and reactions, additional equations must be obtained from conditions of geometrical compatibility. The number of unknowns, in excess of the available equations of equilibrium, is the degree of Indeterminacy, and the unknown forces and reactions are the redundants. The redundants can be removed from the two-dimensional plane structure, leaving a stable, determinate structure known as the cut-back structure. In a pin-jointed frame, external reactions are provided by either roller supports or hinge supports. The roller support provides only one degree of restraint in the vertical direction, and both horizontal and rotational displacements can occur. The hinge support provides two degrees of restraint in the vertical and horizontal directions, and only rotational displacement can occur.
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chapter 1 Statical Indeterminacy
Structural Analysis#R##N#In Theory and Practice, 2009Co-Authors: Alan WilliamsAbstract:Publisher Summary A Statically determinate structure is one in which all the member forces and external reactions can be determined by applying the equations of equilibrium. An indeterminate or redundant structure is one that possesses more unknown member forces and reactions than available equations of equilibrium. To determine the member forces and reactions, additional equations must be obtained from conditions of geometrical compatibility. The number of unknowns, in excess of the available equations of equilibrium, is the degree of Indeterminacy, and the unknown forces and reactions are the redundants. The redundants can be removed from the two-dimensional plane structure, leaving a stable, determinate structure known as the cut-back structure. In a pin-jointed frame, external reactions are provided by either roller supports or hinge supports. The roller support provides only one degree of restraint in the vertical direction, and both horizontal and rotational displacements can occur. The hinge support provides two degrees of restraint in the vertical and horizontal directions, and only rotational displacement can occur.
Alan J. Levy - One of the best experts on this subject based on the ideXlab platform.
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On the Treatment of Statical Indeterminacy in the Elementary Statics of Particles and Rigid Bodies
International Journal of Mechanical Engineering Education, 2012Co-Authors: Alan J. LevyAbstract:The inability to determine reactions beyond a maximum of six for a three-dimensional rigid body (or three for a planar rigid body) is termed Statical Indeterminacy and represents a flaw in the elementary statics of particles and rigid bodies because the indeterminate reactions cannot be obtained by methods developed from within the subject. As noted in virtually every modern textbook on statics, Statical Indeterminacy may be resolved by relaxing the assumption of structural rigidity. This theory forms the content of the mechanics of deformable solids, which considers the stretching, twisting and bending of elastic bars. The Indeterminacy may also be resolved by maintaining structural rigidity but relaxing the assumption of support rigidity. Because deformable supports may be modeled by lumped linear and torsional springs (which are traditionally part of the statics of particles and rigid bodies), analysis of Statically indeterminate reactions in such systems utilizes no new concepts except that of rigid b...
M. Bill Wong - One of the best experts on this subject based on the ideXlab platform.
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Structural Analysis—Stiffness Method
Plastic Analysis and Design of Steel Structures, 2008Co-Authors: M. Bill WongAbstract:Publisher Summary This chapter introduces general plastic analysis methods, which take advantage of the availability of modern computational tools, such as linear elastic analysis programs and spreadsheet applications. The powerful number crunching capability of these tools enables plastic analysis and design to be performed for structures of virtually any size. The amount of computation required for structural analysis is largely dependent on the degree of Statical Indeterminacy of the structure. Structural analysis, whether linear or nonlinear, is mostly based on matrix formulations to handle the enormous amount of numerical data and computations. Matrix formulations are suitable for computer implementation and can be applied to two major methods of structural analysis: the flexibility (or force) method and the stiffness (or displacement) method. This chapter focuses on the stiffness method. Formulation of the matrix equations for the stiffness method is done routinely and the solution procedure is systematic. Therefore, the stiffness method is adopted in most structural analysis computer programs. This method is particularly useful for structures with a high degree of Statical Indeterminacy, although it can be used for both determinate and indeterminate structures. Furthermore, the stiffness method is used in the elastoplastic analysis described in this chapter. It concludes by stating that in structural analysis, the degree of Statical Indeterminacy is important, as its value may determine whether the structure is globally unstable or stable.