ANALYSIS OF AXISYMMETRICAL SHELLS BY THE DIRECT STIFFNESS METHOD

The direct stiffness method is used to analyze the structural response of the cross section of an arteriole which is both heterogeneous and nonlinear in terms of structural response. A typical segment of the cross section is represented by an assembly of triangular elements. The stiffness matrix and strain-displacement relationships are then derived in terms of polar coordinates for a triangular plate element. Viscoelasticity is included in this structural analysis by using a piecewise linear approach. Stress-time and displacement-time data are presented for two typical cross sections of the arteriole. These results reveal special properties of a heterogeneous structure which is undergoing viscoelastic deformations.

Flexural analysis of concrete retaining walls on rough elastic half-space

Of the two main finite element approaches, the displacement method has progressed more rapidly than the force method in the area of element representation. Many authors have contributed to the displacement method with both static and dynamic stiffness matrices for beam elements, plane stress elements and plate bending elements. The plane stress and bending elements being rectangular, triangular or quadrilateral in form. A dynamic stiffness matrix consists of a static stiffness matrix and a mass matrix. In the force method, .static flexibility matrices have been developed for beam elements and plane stress elements. However, static flexibility matrices for plate bending elements have only recently been published (Kaufman and Hall, Morley, Robinson, Przemieniecki). For the past few years, the author has been investigating the rank force method for structural vibration analysis. In ref. 6 a dynamic flexibility matrix is presented for a beam element. This matrix consists of a static flexibility matrix and an inverse mass matrix. Ref. 5 contains the derivation of a dynamic flexibility matrix for a rectangular plate element in bending, twisting and shear. The author, in collaboration with Petyt, demonstrated how a dynamic stiffness and flexibility matrix can be extended to include material damping.

Introduction Carbon supported Pt shell-Pd core structured catalyst (Pt/Pd/C) is a promising candidate for decreasing Pt usage owing to high Pt utilization and ORR activity [1]. We found that ORR specific activity of Pt/Pd/C catalyst was drastically enhanced with accelerated durability test (ADT) performed at 80°C although electrochemical surface area (ECSA) of the catalyst largely decreased, which moderately enhanced ORR mass activity [2]. Thus, we developed high activation protocol (HAP) to mitigate ECSA decay and largely enhance ORR mass activity [3]. Furthermore, we developed H2-O2 chemical activation treatment which mimics HAP performed on glassy carbon (GC) electrode and is suitable for mass-production of highly active catalyst [3]. Besides ORR activity enhancement, synthetic method of Pt/Pd/C catalyst suitable for mass-production is highly important. Although we demonstrated that Pt/Pd/C catalyst could be mass-produced by modified Cu-UPD/Pt displacement method [2, 4], the Cu-UPD method was complicated. Therefore, in this study, we synthesized Pt/Pd/C catalyst via direct displacement (DD) method in which Pd core was directly displaced with Pt precursors, which is much simpler synthetic method compared with Cu-UPD one. Durability and electrochemical properties of Pt/Pd/C catalyst synthesized via DD method were investigated. Experimental 200 mL of H2SO4 aqueous solution (pH1) was deaerated with N2 gas and 300 mg of carbon supported Pd core (Pd/C, mean diameter: 4.7 nm, metal loading: 33 wt.%, ISHIFUKU Metal Industry) was dispersed with stirring, and the solution temperature was controlled in a range of 5-70°C. Then, Pt precursors (K2PtCl4 or H2PtCl6) corresponding to Pt monolayer shell were added at the controlled temperature and stirred for 3 h. The obtained Pt/Pd/C catalyst was activated by HAP in which rectangular potential cycling of 0.4 V (300 s)-1.0 V (300 s) was applied to GC electrode in Ar saturated 0.1 M HClO4 at 80°C for 60 cycles [3]. Pt/Pt/C catalyst was also activated with H2-O2 chemical treatment which mimics HAP performed on GC electrode and is suitable for mass-production. 300 mg of Pt/Pd/C catalyst was dispersed in 2 M H2SO4 aqueous solution at 80°C and H2 and O2 gasses were alternately introduced for every 6 minutes in which equilibrium potentials of H2 (ca. 0.0 V) and O2 (ca. 1.0 V) were alternately applied to the catalyst [3]. ADT was carried out by using rectangular potential cycling of 0.6 V (3 s)-1.0 V (3 s) in Ar saturated 0.1 M HClO4 at 80°C for 10,000 cycles. Characterization of Pt/Pt/C catalyst was conducted by using XRF, TG, XRD, TEM, LEIS and CV techniques. ORR activity of the catalyst was evaluated by RDE method performed at 1,600 rpm with positive scan rate of 10 mV/s in O2 saturated 0.1 M HClO4 at 25°C. Results and Discussion TEM images of Pt/Pd/C catalyst synthesized via modified Cu-UPD/Pt displacement method and DD method are shown in Fig. 1. In DD method (H2PtCl6@70°C), fine catalyst particles disappeared and particle density decreased compared with Cu-UPD method, which improved durability of the catalyst as depicted in Fig. 2. This tendency became evident when H2PtCl6 was used as Pt precursor, which was considered to be due to higher Cl- content in H2PtCl6 precursor than that in K2PtCl4 one. ORR mass activity enhancement of Pt/Pd/C catalyst with HAP is summarized in Fig. 3. The highest initial ORR mass activity was obtained when K2PtCl4 was used as Pt precursor and displacement was performed at 70°C, and the activity increased to 876 A/g-Pt with HAP, which was 2.7-fold of a reference Pt/C catalyst (TEC1050E, TKK). ORR activity was influenced by Pt shell coverage of as-synthesized Pt/Pd/C catalyst. LEIS spectra of the catalysts displaced at 70°C using K2PtCl4 and H2PtCl6 precursors are demonstrated in Fig. 4. Pt/Pd signal ratio obviously increased in Pt/Pd/C catalyst displaced with K2PtCl4, indicating that the catalyst has higher Pt shell coverage. It was considered that the higher Pt shell coverage was obtained through one to one displacement reaction, i.e., Pd + Pt2+ = Pd2+ + Pt, when Pt2+ precursor (K2PtCl4) was used. In the meeting, a revised DD method improving Pt shell coverage and activation by H2-O2 chemical treatment will be presented. Acknowledgement We would like to thank to KOBELCO Research Institute for LEIS measurement. This study was supported by NEDO, Japan.

Preliminary Considerations Nodal Variables - Stiffness and Flexibility Matrices of a Structure Element Variables - Total Stiffness Matrix of an Element Direct Stiffness Method Supplementary Procedures to the Stiffness Method Basic Element Variables The Equations of Equilibrium and the Equations of Compatibility of a Structure Computation of the Displacements of Statically determinate Framed Structures The Flexibility Method The Displacement or Stiffness Method Appendix A. The International System of Units Appendix B. End Actions in Fixed-At-Both-Ends Elements Appendix C. Vector Quantities Appendix D. Computation of the Total Stiffness Matrix of an Element and the Equivalent Actions to be Placed at the Nodes of a Structure Using Shape Functions.

Consistent formulations of direct and iterative methods of pile group analysis are presented. The iterative methods are derived from the same equilibrium and compatibility equations as the direct methods but are cast in different forms. These new iterative methods highlight the shortcomings of some existing iterative methods of analysis, although conceptually they are similar. Unlike the direct methods of analysis in which the stiffness matrix of the pile group system is generally fully coupled, the individual pile stiffness matrix can be uncoupled in the iterative methods and requires only a minimal amount of computer core storage. The accuracy of the iterative methods is compared with that of the direct method, and generally good agreement can be obtained. Field tests on the axial and flexural response of pile groups are studied and show that, in general, the computed response compares favourably with the field measurements. L'article présente des formulations logiques de méthodes directes et itératives pour l'analyse des groupes de pieux. Les méthodes itératives découlent des mêmes équations d'équilibres et de compatibilité que les méthodes directes, mais elles prennent des formes différentes. Ces nouvelles méthodes itératives accentuent les insuffisances de quelques méthodes itératives d'analyse à présent disponibles, malgré une certaine ressemblance de conception. En contraste avec la méthode directe d'analyse, dans laquelle les matrices de rigidité du système représenté par le groupe de pieux sont en général pleinement couplées, la matrice de rigidité du pieu individuel peut être découplée dans les méthodes itératives et n'a besoin que d'une quantité minimale de mémorisation par tores magnétiques dans l'ordinateur. Une comparaison de la précision des méthodes itératives avec celle de la méthode directe indique qu'en général l'accord est bon. Une étude des essais in situ sur la réponse axiale et de flexion des groupes de pieux démontre que généralement la réponse calculée s'accorde bien avec les mesures en place

This paper introduces the direct global stiffness matrix (DGSM) method for analyzing the dynamics of cylindrical shells with axial symmetry. The dynamic stiffness matrix of a straight, circular cross-section cylindrical element is formulated in terms of ring forces and displacements at each end on a per circumferential mode basis. The stiffness matrix is based on an exact wave-type solution of the thin shell Donnell equations. More complicated shell systems are analyzed by assembling any number of cylindrical elements. For example, a cylinder with a gradual variation in radius may be modeled by a series of shell elements welded together at their ends. Total system dynamics are found at any given frequency by inverting the global system matrix formed from the individual stiffness element matrices and summing over circumferential mode number. Results for a test case are presented to demonstrate and verify the implementation. [Research sponsored by ONR.]

The effects of prestressing forces on a reinforced concrete frame are shown in the paper. The structure is a statically indeterminate frame with two fixed supports. The load is due to tendon prestressing with Nk = 600 kN and due to self-weight of columns and beam. The columns height is 6.00 m and the beam span is 20.00 m. Effects of prestressing can be replaced by an external equivalent load. Calculations are performed by the force method, by the direct stiffness method and using the 'Tower' software. The rotational restraints are released at fixed supports and the restraint moments are calculated by the force method. The direct stiffness method uses the members' stiffness relations for computing member forces and displacements in structures. The unknown nodal rotations and translations are obtained by solving a system of equations. The considered tendon profile is without eccentricity above the columns. 'Tower 7' is software for static and dynamic structural analysis and for design of concrete elements. It enables overall structural analysis of plane and three-dimensional structures. Automated, integrated tools allow fast and reliable design or the verification of the results obtained by standard methods. Version 7 is a professional tool with high performance. Three different calculation methods have given close results, with minor differences.

Summary A problem of elastic structures stress-strain field determination is considered in this article. A theoretical background and computation algorithms of equilibrium finite element method are presented there. The different external effects are estimated, namely: load, prestressing, inicial strains and support settlements. The dual relationships (equilibrium and geometrical equations, stiffness and flexibility equations) of equilibrium element, also the expressions of stiffness and flexibility matrices are given. These relationships describe the stress-strain field of the finite element and allow to transforme the flexibility matrix to the stiffness matrix and on the contrary. There are presented direct and variational formulations of the problem. The algorithms of the forces method and displacements method are made for their solution. Easier is the algorithm of displacements method, because making equations of the forces method needs to solve the system of equilibrium equations. But the formulation and solving of the displacements equations can be changed by direct formulation of global flexibility matrix of the structure. In these cases, when the degree of static indeterminacy of the discrete model of structure is quite less than the number of freedom degrees, suth the change can be effective. The computational results illustrate the high accuracy of equilibrium elements not only with respect to forces, but also to displacements. The direct stiffness method in equilibrium elements makes it possible not only to simplify significantly implementation of the equilibrium element method, but also to include equilibrium elements in libraries of geometrically compatible elements.

This paper presents a modelling technique based on the direct stiffness method (DSM) to describe the behaviour in time of composite beams with partial shear interaction accounting for creep and shrinkage of the slab. The time‐dependent behaviour of the concrete is modelled using algebraic representations, such as the age‐adjusted effective modulus method and the mean stress method, while the steel joist, the reinforcement and the shear connection are assumed to behave in a linear‐elastic manner. Only one discretization (i.e. in the time domain) is required by the proposed stiffness formulations to perform a time analysis, while two discretizations (i.e. one in the time domain and the other in the spatial domain along the beam axis) are required by other modelling techniques available in the literature. The ability of the derived elements to overcome curvature locking problems observed to occur in some conventional displacement formulations is also highlighted. The proposed DSM approach is then validated against analytical solutions derived by the authors for simple structural systems. The applicability of this method for the time analysis of continuous composite beams is also illustrated. Copyright © 2008 John Wiley & Sons, Ltd.

Analysis of composite beams with partial shear interaction using available modelling techniques: A comparative study

Modeling procedures for the stress analysis of riveted lap joints in aircraft structure are proposed in this paper using both analytical and numerical methods. In the analytical method, a complex variational approach is employed to determine the stresses in joined plates containing single or multiple loaded holes. The effects of finite geometry are taken into account by the variational formulations. An iterative scheme is carried out to handle the deformation compatibility between all joined members, hi the numerical method, finite element analyses are conducted using the commercial packages MSC/Patran and MSC/Nastran. Gap elements are used to simulate the rivet-hole interactions. Both linear and nonlinear deformations are considered. The two modeling procedures are complementary to each other. While the analytical method is efficient for parametric studies, the numerical method is capable of dealing with relatively more complicated geometry and loading conditions. Results of several example problems are presented and reasonably good agreement between the two modeling procedures is demonstrated.

This paper presents the recursive algorithm of stiffness matrix method with improved efficiency for computing the total and surface stiffness matrices for a general multilayered anisotropic media. Based on the eigensolutions commonly available for analysis of such media, the recursive algorithm deals with eigen-submatrices directly and bypasses all intermediate layer stiffness submatrices. The improved algorithm obviates the need to compute certain inverse of the original scheme and makes the stiffness matrix recursion more robust. In situation where transfer matrix is numerically stable and easily accessible, an improved recursive algorithm is also given directly in terms of transfer submatrices without involving their explicit inverse.

An efficient hybrid method (HM) is proposed by combining the direct stiffness method (DSM) and the mode superposition method (MSM) for analyzing the train–track–bridge coupled system (TTBS). The train and the track are modeled by applying the multi-body dynamics and the DSM, respectively. The bridge is modeled by applying the MSM that is efficient in capturing the dynamic behavior with a small number of modes. The train–track subsystem and the bridge subsystem are coupled by the interaction forces between them. The computational efficiency is significantly improved because of the considerably reduced number of equations of motion of the TTBS. Numerical simulations of a train traversing an arch railway bridge are performed and the results are compared with the field test data and the data from other methods, demonstrating the efficiency and accuracy of the proposed method.

This chapter presents a methodology for designing a simple finite element solver. The implementation is carried out inside MATLAB™ and the simulation engine is driven by the Direct Stiffness Method (DSM) presented in Chap. 3. In order to illustrate this, the authors have presented an in-house finite element solver they created called: MATLAB Finite Element Simulation Engine (MATFESE™). This chapter presents the structure, implementation and execution of MATFESE™. The chapter concludes by illustrating the robustness of MATFESE™ in tackling a range of structural mechanics problems. It is expected that after reading through this chapter, the reader can start creating simple finite element solver by utilizing the principles of the DSM. MATFESE ™ can be downloaded from the textbook website.