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Global Local Stress Analysis NASA
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GLOBAL LOCAL STRESS ANALYSIS OF COMPOSITE PANELSt,Jonathan B Ransom and Norman F Knight Jr. NASA Langley Research Center,Hampton Virginia, A method for pcrforming a global local stress analysis is described and its capabilities are. demonstrated The method employs spline interpolation functions which satisfy the linear plate. bending equation to determine displacements and rotations from a global model which are used. as boundary conditions for the local model Then the local model is analyzed independent. of the global model of the structure This approach can be used to determine local detailed. stress states for specific structural regions using independent refined local models which exploit. information from less refined global models The method presented is not restricted to having. a priori knowledge of the location of the regions requiring local detailed stress analysis This. approach also reduces the computational effort necessary to obtain the detailed stress state. Criteria for applying the method are developed The effectiveness of the method is demonstrated. using a classical stress concentration problem and a graphite epoxy blade stiffened panel with a. discontinuous stiffener,Nomenclature, Vector of unknown spline coefficients of the interpolation function. Polynomial coefficients of the interpolation function i 0 1 9. Cross scctional area,Net cross sectional area W 2ro h. Stiffener spacing,Flexural rigidity,Young s modulus of elasticity.
Spline interpolation function, Natural logarithm coefficients of the interpolation function i 1 2 n. Panel thickness,Blade stiffener height,Stress colicexitration factor. Number of points on the local model boundary,Panel length. Number of points in the interpolation region, t Invited Talk at the Third Joint ASCE ASME Mechanics Conference University of California. at San Diego July 9 12 1989, Aerospace Engineer Structural Mechanics Branch Structural Mechanics Division.
Longitudinal stress resultant,Average running load per inch. Maximum longitudinal stress resultant j u z m a z h. Nominal longitudinal stress resultant u nomh,Applied load. Pressure loading on panel,Radius of the panel cutout. Radial coordinate of the i th node in the interpolation region or. along the local model boundary,Radius of the interpolation region. Radius of the circular local model,Spline matrix,Element strain energy per unit area.
Maximum element strain energy per unit area,Panel width. x coordinate of the i th node in the iiiterpolation region or. along the local model boundary, y coordinate of the i th node in the interpolation region or. along the local model boundary, Parameter used to facilitate spline matrix computation. Change in stress,Longitudinal stress,Maximum longitudinal stress. Nominal longitudinal stress A, Natural logarithm terms of the spline matrix i j 1 2 n.
Biharmonic operator,Introduction, Discontinuities and eccentricities which are coinmon in practical structures increase. the difficulty in predicting accurately detailed local stress distributions especially when the. component is built of a composite material such as a graphite epoxy material The use of. composite materials in the design of aircraft structures introduces added complexity due to. the nature of the material systems and the complexity of the failure modes The design and. certification process for aerospace structures requires an accurate stress analysis capability. Detailed stress analyses of complex aircraft structures and their subcomponents are required. and can severely tax even today s computing resources Embedding detailed local finite. element models within a single global finite element model of an entire airframe structure is. usually impractical due to the computational cost associated with the large number of degrees of. freedom required for such a global detailed model If the design load envelope of the structural. component is extended new regions with high stress gradients may be discovered In that case. the entire analysis with embedded local refinements may have to be repeated and thereby further. reducing the practicality of this brute force approach for obtaining the detailed stress state. The phrase global local analysis has a myriad of definitions among analysts e g refs 1 2. The concept of global and local may change with every analysis level and also from one analyst. to another An analyst may consider the entire aircraft structure to be the global model and a. fuselage section to be the local model At another level the fuselage or wing is the global model. and a stiffened panel is the local model Laminate theory is used by some analysts to represent. the global model and micromechanics models are used for the local model The global local. stress analysis methodology herein is defined as a procedure to determine local detailed stress. states for specific structural regions using information obtained from an independent global stress. analysis A separate refined local model is used for the detailed analysis Global local analysis. research areas include such methods as substructuring submodeling exact zooming and hybrid. techniques, The substructuring technique is perhaps the most common technique for global local. analysis in that it reduces a complex structure to smaller more manageable components and. simplifies the structural modeling e g refs 3 51 A specific region of the structure may be. modeled by a substructure or multiple levels of substructures to determine the detailed response. Submodeling refers to any method that uses a node by node correspondence for the displacement. field at the global local interface boundary e g refs 6 91 A form of submodeling is used in. reference 8 to perform a two dimensional to three dimensional global local analysis Recently. the specified boundary stiffness force SBSF method 9 has been proposed as a global local. analysis procedure This approach uses an independent subregion model with stiffnesses and. forces as boundary terms These stiffnesses and forces represent the effect of the rest of the. structure upon the subregion The stiffness terms are incorporated in the stiffness of the. subregion model and the forces are applied on the boundary of the local model. An efficient zooming technique as described in reference lo employs static condensation. and exact structural reanalysis methods Although this efficient zooming technique involves. the solution of a system of equations of small order all the previous refinement processes are. needed to proceed to a new refinement level An exact zooming technique ll employs an. expanded stiffness matrix approach rather than the reanalysis method described in reference. lo The exact zooming technique utilizes results of only the previous level of refinement. For both methods separate locally refined subregion models are used to determine the detailed. stress distribution in a known critical region The subregion boundary is coincident with nodes. in the global model or the previously refined subregion model which is akin to the submodeling. technique discussed earlier, I Hybrid techniques e g refs 12 13 make use of two or more methods in different domains. of the structure In the global variational methods the domain of the governing equation is. treated as a whole and an approximate solution is constructed from a sequence of linearly. independent functions i e Fourier series that satisfy the geometric boundary conditions. In the finite element method the domain is subdivided into small regions or elements within. which approximating functions usually low order polynomials are used t o describe the. continuum behavior Global local finite element analysis may refer to an analysis technique that. simultaneously utilizes conventional finite element modeling around a local discontinuity with. classical Rayleigh Ritz approximations for the remainder of the structure The computational. effort is reduced as a result of the use of the limited number of finite element clegrees of freedom. however this approach presupposes that the analyst can identify an approxhstion sequence for. The aforementioned global local methods with the exception of the subn lodeling technique. require that the analyst know where the critical region is located before perfolming the global. analysis However a global local methodology which does not require a priori knowledge of. the location of the local region s requiring special modeling could offer advantages in many. situations by providing the modeling flexibility required to address detailed local models as their. need is identified, The overall objective of the present study is to develop such a computational strategy for. obtaining detailed stress states of composite structures Specific objectives are. 1 To develop a method for performing global local stress analysis of composite structures. 2 To develop criteria for defining the global local interface region and local modeling. requirements, 3 To demonstrate the computational strategy on representative structural analysis.
The scope of the present study includes the global local linear two dimensional stress analysis. of finite element structural models The method developed is not restricted to having a priori. knowledge of the location of the regions requiring detailed stress analysis The guidelines for. developing the computational strategy include the requirement that it be compatible with. general purpose finite element computer codes valid for a wide range of elemcmts extendible. to geometrically nonlinear analysis and cost effective In addition the computational strategy. should include a procedure for automatically identifying the critical region and defining. the global local interface region Satisfying these guidelines will provide a general purpose. global local computational strategy for use by the aerospace structural analysis community. Global Local Methodology, I Global local stress analysis methodology is defined as a procedure to detcrnine local. detailed stress states for specific structural regions using information obtained from an. independent global stress analysis The local model refers to any structural subregion within. the defined global model The global stress analysis is performed independent of the local stress. analysis The interpolation region encompassing the critical region is specified A surface spline. interpolation function is evaluated at every point in the interpolation region yielding a spline. matrix S z y and unknown coefficients a The global field is used to compute the unknown. coefficients An independent more refined local model is generated within the previously defined. interpolation region The global displacement field is interpolated producing a local displacement. field which is applied as a boundary condition on the boundary of the local model Then a. complete two dimensional local finite element analysis is performed. The development of a global local stress analysis capability for structures generally involves. four key components The first component is an adequate global analysis In this context. adequate implies that the global structural behavior is accurately determined and that local. structural details are at least grossly incorporated The second component is a strategy for. identifying in the global model regions requiring further study The third component is a. procedure for defining the boundary conditions along the global locd interface boundary. Finally the fourth component is an adequate local analysis In this context adequate. implies that the local detailed stress state is accurately determined and that compatibility. requirements along the global local interface are satisfied The development of a global local. stress analysis methodology requires an understanding of each key component and insight into. their interaction, In practice the global analysis model is adequate for the specified design load cases. However these load cases frequently change in order to extend the operating region of the. structure or to account for previously unknown affects In these incidents the global analysis. may identify new hot spots that require further study The methodology presented herein. provides an analysis tool for these local analyses. I Terminology, I The terminology of the global local methodology presented herein is depicted in figure 1. to illustrate the components of the analysis procedure The global model figure la is a finite. element model of an entire structure or a subcomponent of a structure A region requiring a. more detailed interrogation is subsequently identified by the structural analyst This region. may be obvious such as a region around a cutout in a panel or not so obvious such as a. local buckled region of a curved panel loaded in compression Because the location of these. regions are usually unknown prior to performing the global analysis the structural analyst. must develop a global model with sufficient detail to represent the global behavior of the. structure An interpolation region is then identified around the critical region as indicated in. figure lb An interpolation procedure is used to determine the displacements and rotations used. as boundary conditions for the local model The interpolation region is the region within. which the generalized displacement solution will be used to define the interpolation matrix The. global local interface boundary indicated in figure IC coincides with the intersection of the. boundary of the local model with the global model The definition of the interface boundary. may affect the accuracy of the interpolation and thus the local stress state The local model. lies within the interpolation region as shown in figure ICand is generally more refined than. the global model in order to predict more accurately the detailed state of stress in the critical. region However the local model is independent of the global finite element model as indicated. in figure Id The coordinates or nodes of the local model need not be coincident with any of the. coordinates or nodes of the global model, The global local interpolation procedure consists of generating a matrix based on the. global solution and a local interpolated field The local interpolated field is that field which is. interpolated from the global analysis and is valid over the domain of the local model Local. stress analysis involves the generation of the local finite element model use of the interpolated. field to impose boundary conditions on the local model and the detailed stress analysis. Global Modeling and Analysis, The development of a global finite element moclel of an aerospace structure for accuratc.
stress predictions near local discontinuities is often too time consuming to impact the design. and certification process Predicting the global structural response of these structures often has. many objectives including overall structural response stress analysis and determining internal. load distributions Frequently structural discontinuities such as cutouts are only accounted for in. the overall sense Any local behavior is then obtained by a local analysis possibly by another. analyst The load distribution for the local region is obtained from the global analysis The. local model is then used to obtain the structural behavior in the specified region For example. the global response of an aircraft wing is obtained by a coarse finite element analysis A typical. subcomponent of the wing is a stiffened panel with a cutout Since cutouts are known to produce. high stress gradients the load distributions from the global analysis of the wing are applied to. the stiffened panel to obtain the local detailed stress state One difficulty in modeling cutouts. is the need for the finite element mesh to transition from a circular pattern near the cutout. to a rectangular pattern away from the cutout This transition region is indicated in figure 2. and will be referred to as a transition square i e a square region around the cutout used to. transition from rings of elements to a rectangular mesh This transition modeling requirement. impacts both the region near the cutout and the region away from the cutout Near the cutout. quadrilateral elements may be skewed tapered and perhaps have an undesirable aspect ratio. In addition as the mesh near the cutout is refined by adding radial spokes of nodes and. rings of elements the mesh away from the cutout also becomes refined For example adding. radial spokes of nodes near the cutout also adds nodes and elements in the shaded regions see. figure 2 away from the cutout This approach may dramatically increase the computational. requirements necessary to obtain the detailed stress state Alternate mesh generation techniques. using transition zones of triangular elements or multipoint constraints may be used however the. time spent by the structural analyst will increase substantially. The global modeling herein although coarse is sufficient to represent the global structural. behavior The critical regions have been modeled with only enough detail to represent their. effect on the global solution This modeling step is one of the key components of the global local. methodology since it provides an adequate global analysis Although the critical regions are. known for the numerical studies discussed in a subsequent section this a priori knowledge is not. required but may be exploited by the analyst,Local Modeling and Analysis. The local finite element modeling and analysis is performed to obtain a dctailed analysis. of the local structural region s The local model accurately represents the geometry of. the structure necessary to provide the local behavior and stress state The discretization. requirements for the analysis are governed by the accuracy of the solution desired The. discretization of the local model is influenced by its proximity to a high stress gradient. One approach for obtaining the detailed stress state is to model the local region with an. arbitrarily large number of finite elements Higher order elements may be usell to reduce the. number of elements requircd Detailed refinement is much more advantageous for use in the. local model than in the global model The local refinement affects only the lo al model unlike. embedding the same refinement in the global model which would propagate to regions not. requiring such a level of detailed refinement A second approach is to refine the model based. on engineering judgement Mesh grading in which smaller elements are used near the gradient. may be used An error measure based on the change in stresses from element to element may. be used to determine the accuracy in the stress state obtained by the initial l c d finite element. mesh If the accuracy of the solution is not satisfactory additional refinement j are required The. additional refinements may be based on the coarse global model or the displacement field in the. local model which suggests a third approach The third approach is a multi level global local. analysis At the second local model level any of the three approaches discussc d may be used to. obtain the desired local detailed stress state Detailed refinement is used for local modeling in. this study,Global Local Interface Boundary Definition. The definition of the global local interface boundary is problem dependent Herein the. location of the nodes on the interface boundary need not be coincident with any of the nodes. in the coarse global model Reference 7 concluded that the distance that the local model must. extend away from a discontinuity is highly dependent upon the coarse model used The more. accurate the coarse model displacement field is the closer the local model boundary may be to. the discontinuity This conclusion is based on the results of a study of a flat isotropic panel with. a central cutout subjected to uniform tension but it extends to other structures with high stress. Stresses are generally obtained from a displacement based finite element analysis by. differentiation of the displacement field For problem with stress gradients the element. stresses vary from element to element and in some cases this change Au may be substantial. The change in stresses Au may be used as a measure of the adequacy of the finite element. discretization Large Au values indicate structural regions where more modeling refinement is. needed Based on this method structural regions with small values of Au have obtained uniform. stress states away from any gradients Therefore the global local interface boundary should. be defined in regions with small values of A a ic away from a stress gradient Exploratory. studies to define an automated procedure for selecting the global local interface boundary. have been performed using a measure of the strain energy The strain energy per unit area is. selected since it represents a combination of all the stress components instead of a single stress. component Regions with high stress gradients will also have changes in this measure of strain. energy from element to element,Global Lo cal Interpolation Procedure. The global local analysis method is used to determine local detailed stress states using. independent refined local models which exploit information from less refined global models A. two dimensional finite element analysis of the global structure is performed to obtain its overall. behavior A critical region may be identified from the results of the global analysis The global. solution may be used to obtain an applied displacement field along the boundary e boundary. conditions of an independent local model of the critical region This step is one of the key. components of the global local methodology namely interpolation of the global solution to. obtain boundary conditions for the local model, Many interpolation methods are used to approximate functions e g ref 14 The. interpolation problem may be stated as follows given a set of function values fj at n coordinates. z y determine a best fit surface for these data Mathematically this problem can be stated. where S ti yi is a matrix of interpolated functions evaluated at n points the array a defines. the unknown coefficients of the interpolation functions and the array f consists of known values. of the field being interpolated based on n points in the global model Common interpolation. methods include linear interpolation Lagraiigiaii iiileryolatioii and least squares techniques for. polynomial interpolation Elementary linear interpolation is perhaps the simplest method and. is an often used interpolation method in trigonometric and logarithmic tables Another method. is Lagrangian interpolation which is an extension of linear interpolation For this method data. for n points are specified and a unique polynomial of degree n 1 passing through the points. can be determined However a more common method involving a least squares polynomial fit. minimizes the the sum of the square of the residuals The drawbacks of least squares polynomial. fitting include the requirement for repeated solutions to minimize the sum of the residual and. the development of an extremely ill conditioned matrix of coefficients when t l edegree of the. approximating polynomial is large A major limitation of the approximating 1 olynomials which. fit a given set of function values is that they may be excessively oscillatory between the given. points or nodes,Mat hematical Formulation of Spline Interpolation.
Spline interpolation is a numerical analysis tool used to obtain the best local fit through a. set of points Spline functions are piecewise polynomials of degree m that are connected together. at points called knots so as to have m 1 continuous derivatives The mathematical spline. is analogous to the draftman s spline used to draw a smooth curve through a itumber of given. points The spline may be considered to be a perfectly elastic thin beam resti ig on simple. supports at given points A surface spline is used to interpolate a function of wo variables. and removes the restriction of single variable schemes which require a rectangitlar array of. grid points The derivation of the surface spline interpolation function used herein is based. on the principle of minimum potential energy for linear plate bending theory This approach. incorporates a classical structural mechanics formulation into the spline interpolation procedure. in a general sense Using an interpolation function which also satisfies the linear plate bending. equation provides inherent physical significance to a numerical analysis technique The spline. interpolation is used to interpolate the displacements and rotations from a global analysis. and thereby provides a functional description of each field over the domain The fields are. interp Jlated separately which provides a consistent basis for interpolating glol la1 solutions based. on a plate theory with shear flexibility effects incorporated That is the out of plane deflections. and the bending rotations are interpolated independently rather than calculat tng the bending. rotations by differentiating the interpolated out of plane deflection field. The spline interpolation used herein is derived following the approach described by Harder. and Desmairis in reference 15 A spline surface is generated based on the solution to the linear. isotropic plate bending equation,DV4w I q 2, Extensions have been made to the formulation presented in reference 15 to include higher. order polynomial terms underlined terms in equation 3 The extended Cai tesisn form which. satisfies equation 2 is written as,Tf x X i 2 y y 2 4. and z i y i are the coordinates of the i th node in the interpolation region The higher order. polynomial terms were added to help represent a higher order bending response than was being. approximated by the natural logarithm term in the earlier formulation The additional terms. increase the number of unknown coefficients and constraint equations to n 10 The n 10. unkiiowns a0 a1 a2 as F are found from equation 3 and by solving the set of equations. The constraint equations given in equation 5 are used to prevent equation 2 from becoming. unbounded when expressed in Cartesian coordinates The modified matrix equations still of the. form Sa f are,0 0 0 0 1 1 1 0,0 0 0 0 21 22 X n 0,0 0 0 0 y1 Y2 Yn 0. 0 0 0 0 xq 2 x 0,0 0 0 0 xly1 2,e 2 xnyn 0,0 0 0 0 y y Yn. 0 0 0 0 z x x 0,0 0 0 0 xqy1 x y2 2,0 0 0 0 zly X I Y 2.
0 0 0 0 y y 3,1 21 y1 y QII 012 Qln fl,1 2 2 y2 y 021 Q22 R2n f2. 3 Qn1 Qn2 Qnn,1 2n Yn fn,where R i j In T j,T E for i j. 1 2 n and E is a parameter used to insure numerical. stability for the case when i vanishes, j The extended local interpolation function is similar to. equation 3 except that it is evaluated at points along the global local interface boundary That. fi a0 a1xi a2yi a3xi2 a 4 i y i a5yi2 agxi3 a7si2yi. a8ziyi2 a9yi3 F T ln r j i 1 2 1 7, Upon solving equation 6 for the coefficients ao al a 2 as Fj equation 7 is used to. compute the interpolated data at the required local model nodes. Computational Strategy, A schematic which describes the overall solution strategy is shown in figure 3 The.
computational strategy described herein is implemented through the use of the Computational. Structural Mechanics CSM Testbed see refs 16 and 17 The CSM Testbed is used. I to model and analyze both the global and local finite element models of a structure Two. computational modules or processors were developed to perform the global local interpolation. procedure Processor SPLN evaluates the spline coefficient matrix S z i yi Processor INTS. solves for the interpolation coefficients a a Fi and performs the local interpolation to obtain. the boundary conditions for the local model Various other Testbed processors are used in the. stress analysis procedure The overall computational strategy for the global local stress analysis. methodology is controlled by a high level procedure written using the command language of. the Testbed see ref 17 The command language provides a flexible tool for performing. computational structural mechanics research,Numerical Results. The effectiveness of the computational strategy for the global local stress analysis outlined. in the previous sections is demonstrated by obtaining the detailed stress states for an isotropic. panel with a cutout and a blade stiffened graphite epoxy panel with a discontinuous stiffener. The first problem was selected to verify the global local analysis capabilities while the second. problem was selected to demonstrate its use on a representative aircraft subcomponent The. objectives of these numerical studies are, 1 To demonstrate the global local stress analysis methodology and. 2 To obtain and interrogate the detailed stress states of representative subcomponents of. complete aerospace structures, All numerical studies were performed on a Convex C220 minisupercomputer The. computational effort of each analysis is quantified by the number of degrees of freedom used in. the finite element model the computational time required to perform a stress analysis and the. amount of auxiliary storage required The computational time is measured in central processing. unit CPU time The amount of auxiliary storage required is measured by the size of the data. library used for the input output of information to a disk during a Testbed execution. Isotropic Panel with Circular Cutout, An isotropic panel with a cutout is an ideal structure to verify the global local. computational strategy since closed form elasticity solutions axe available Elasticity solutions. for an infinite isotropic panel with a circular cutout e g ref IS predict a stress concentration. factor of three at the edge of the cutout The influence of finite width effects on the stress. concentration factors for isotropic panels with cutouts are reported in reference 19 By. including finite width effects the stress concentration factor is reduced from 1he value of three. for an infinite panel, The global local linear stress analysis of the isotropic panel with a circular cutout shown.
in figure 4 has been performed The overall panel length L is 20 in the overall width W is. 10 in the thickness h is 1 in and the cutout radius TO is 0 25 in This geometry gives a cutout. diameter to panel width ratio of 0 05 which corresponds to a stress concentration factor of 2 85. see ref 19 The loading is uniform axial tension with the loaded ends of the panel clamped. and the sides free The material system for the panel is aluminum with a Young s modulus of. 10 000 ksi and Poisson s ratio of 0 3,Global Analysis. The finite element model shown in figure 4 of the isotropic panel with a circular cutout is a. representative finite element model for representing the global behavior of the panel as well as. a good approximation to the local behavior The finite element model shown In figure 4 will be. referred to as the coarse global model or Model G1 in Table 1 The finite element model has a. total of 256 4 node quadrilateral elements 296 nodes and 1644 active degrees of freedom for the. linear stress analysis This quadrilateral element corresponds to a flat C shell element which. is based on a displacement formulation and includes rotation about the outward normal axis. Originally developed for the computer code STAGS see refs 20 21 this element has been. installed in the CSM Testbed software system and denoted ES5 E410 see ref IS. The longitudinal stress resultant N distribution shown in figure 5 reveals several features of. the global structural behavior of this panel First away from the cutout the Y distribution in. the panel is uniform Secondly the N load near the center of the panel is much greater than the. N load in other portions of the panel due to the redistribution of the N loall as a result of the. cutout Thirdly the N load at the edge of the cutout at the points ninety dc grees away from. the stress concentration is small relative to the uniform far field stress state. The distribution of the longitudinal stress resultant N at the panel midlength normalized. by the nominal stress resultant is shown in figure 6 as a function of the distance from the cutout. normalized by the cutout radius The results indicate that high inplane stresses and a high. gradient exist near the cutout However a stress concentration factor of 2 06 is obtained from. a linear stress analysis using the coarse finite element model see figure 4 This value is 28. lower than the theoretical value of 2 85 reported in reference 19 Therefore even though the. overall global response of the panel is qualitatively correct as indicated by the stress resultant. contour in figures 5 the detailed stress state near the discontinuity is inaccurate. Accurate detailed stress distributions require a finite element mesh that is substantially more. refined near the cutout Adding only rings of elements Model G 2 in Table 1 does not affect the. discretization away from the cutout however the stress concentration factor is still 22 lower.

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