Layup optimization against buckling of shear panels

The object of the study was to optimize the shear buckling load of laminated composite plates. The laminates lacked coupling between bending and extension (B _ij=0) but had otherwise arbitrary selection of the ply angle variation through the thickness. The plates were rectangular and either simply supported or clamped on all edges. For orthotropic plates, it was seen that there is only one parameter necessary for finding the optimal design for different materials and plate aspect ratios. This parameter can be interpreted as the layup angle in a (+/–) orthotropic laminate. When bendingtwisting coupling is present, the buckling strength depends on the direction of the applied load. A laminate with non-zero bending-twisting coupling stiffnesses can be described with four lamination parameters. The allowable region of these parameters was investigated, and an optimization of the buckling load within this region was performed. It was seen that even this is a one parameter problem. This parameter can be interpreted as the layup anlge in an off-axis unidirectional laminate ().Notations A _ij in-plane stiffnesses of anisotropic plates, Tsai and Hahn (1980) - B _ij coupling stiffnesses of anisotropic plates - D _ij bending stiffnesses of anisotropic plates - D _ij ^* normalized bending stiffnesses - a, b, h length, width and thickness of the plate - x, y in-plane coordinates - z through-the-thickness coordinate - z ^* normalized through-the-thickness coordinate - w (x, y) out-of-plane deformation - N _xy shear buckling load - W ₁ ^* toW ₄ ^* lamination parameters - U ₁ toU ₅ linear combinations of the on-axis moduli - (z) layup angle - f _k functional of(z)     

Composite plate optimization only requires one parameter

The object of the study was to optimize composite plates concerning free vibration frequencies, buckling loads, and deflections under constant pressure. Layups without coupling between bending and extension but otherwise arbitrary selection of the ply angle variation through the thickness of the laminate were included. For these plates, four different boundary conditions were studied. The number of relevant parameters was successively reduced from the initial six bending stiffnesses that any laminate has. Bending-twisting coupling has only negative influence on fundamental eigenfrequency, buckling load and average deflection under a constant pressure, so the number of parameters could be reduced by two. The remaining four parameters are not independent, but are functions of only two independent parameters, the flexural lamination parameters. It was further seen that the optimal designs always were found on the boundary of the allowable region of the flexural lamination parameters, i.e. there is only one relevant parameter for the optimization problems. This parameter can be interpreted as the layup angle () in an orthotropic (+/ – ) laminate.Notation D _ij bending stiffnesses - D _ij ^* normalized bending stiffnesses - a, b, h length, width, and thickness of the plate - x, y inplane coordinates - w (x, y) out-of-plane deformation - area density of the laminate - eigenfrequency - N buckling load - _m average deflection - p applied pressure normal to the plane - U potential energy - W ₁ ^* ,W ₂ ^* lamination parameters - U ₁ –U ₅ linear combinations of the on-axis moduli     

Variable stiffness composite material design by using support vector regression assisted efficient global optimization method

In this work, a surrogate assisted optimization method is utilized to optimize buckling loads of variable stiffness composites made by fiber steering. To improve the efficiency of optimization procedure, an expected improvement criterion is employed. Moreover, considering uncertainties of the fiber placement, a robust surrogate, least square support vector regression (LSSVR) considering empirical and structural risks is integrated with the expected improvement (EI) criterion and applied to two applications. The first case is the fiber path design of a variable stiffness plate under the compression load. The second one is the fiber path design of a variable stiffness cylinder under the bending load. According to results of the optimization, the buckling load of the variable stiffness plate has 52.63% improvement than the constant stiffness plate and 24.3% improvement than the quasi-isotropic plate. The buckling load of the variable stiffness cylinder has 40.22% improvement than the constant stiffness cylinder and 31.25% improvement than the quasi-isotropic cylinder. Furthermore, to verify the robustness of optimal design variables for the variable stiffness cylinder, the perturbed optimum design is presented and demonstrates that the results are reliable.     

Size of the Largest Antichain in a Partition Poset

Partitions {(k ₁,..., k )} of a given set are considered as a partially ordered set (poset) with a natural partial ordering with respect to inclusion. Asymptotics for the size of the largest antichain in this poset is found for fixed.     

Genetic algorithms with local improvement for composite laminate design

This paper describes the application of a genetic algorithm to the stacking sequence optimization of a laminated composite plate for buckling load maximization. Two approaches for reducing the number of analyses required by the genetic algorithm are described. First, a binary tree is used to store designs, affording an efficient way to retrieve them and thereby avoid repeated analyses of designs that appeared in previous generations. Second, a local improvement scheme based on approximations in terms of lamination parameters is introduced. Two lamination parameters are sufficient to define the flexural stiffness and hence the buckling load of a balanced, symmetrically laminated plate. Results were obtained for rectangular graphite-epoxy plates under biaxial in-plane loading. The proposed improvements are shown to reduce significantly the number of analyses required for the genetic optimization.Presented at the ASME Winter Annual Meeting Structures and Controls Optimization, pp. 13–28. Printed with permission from ASME.     

Application of optimization methods to helicopter rotor blade design

The minimum weight design of helicopter rotor blades with constraints on multiple coupled flap-lag natural frequencies is studied in this paper. A constraint is also imposed on the minimum value of the autorotational inertia of the blade to ensure sufficient autorotational inertia to autorotate in case of an engine failure. A stress constraint is used to guard against structural failure due to blade centrifugal forces. Design variables include blade taper ratio, dimensions of the box beam located inside the airfoil and magnitudes of the nonstructural weights. The program CAMRAD is used for the blade modal analysis and the program CONMIN for the optimization. In addition, a linear approximation involving Taylor series expansion is used to reduce the analysis effort. The procedure contains a sensitivity analysis which consists of analytical derivatives of the objective function, the autorotational inertia constraint and the stress constraints. A central finite difference scheme is used for the derivatives of the frequency constraints. Optimum designs are obtained for both rectangular and tapered blades. The paper also discusses the effect of adding constraints on higher frequencies and stresses on the optimum designs. b box beam width - c chord - f ₁,f ₃,f ₄ first three lead-lag dominated frequencies (elastic modes) - f ₂,f ₅ first two flapping dominated frequencies (elastic modes) - g constraint function - h box beam height - h(z) box beam height variation along blade span - n number of blades - r _j distance from the root to the center of thej-th segment - t ₁,t ₂,t ₃ box beam wall thicknesses - x, y, z reference axes - A box beam cross-sectional area - AI autorotational inertia - E Young's modulus - F objective function - FS factor of safety - GJ torsional stiffness - I _x ,I _y total principal area moments of inertia about reference axes - L _j length ofj-th segment - M _j total mass ofj-th segment - N total number of blade segments - NDV number of design variables - R blade radius - W total blade weight - W() blade weight as a function of design variable - W _b box beam weight - W _o nonstructural blade weight (weight of skin, honeycomb, etc. along with tuning/lumped weights) - prescribed autorotational inertia - design variable increment - _h taper ratio inz direction - _i i-th design variable - _j mass density of thej-th segment - _j stress inj-th segment - _max maximum allowable stress - blade RPM - r root value - t tip value - L lower bound - U upper bound - ^ approximate value This paper is declared a work of the U.S. Government and is not subject to copyright protection in the United States     

A Novel Class of Symmetric and Nonsymmetric Periodizing Variable Transformations for Numerical Integration

Variable transformations for numerical integration have been used for improving the accuracy of the trapezoidal rule. Specifically, one first transforms the integral via a variable transformation that maps [0,1] to itself, and then approximates the resulting transformed integral by the trapezoidal rule. In this work, we propose a new class of symmetric and nonsymmetric variable transformations which we denote , where r and s are positive scalars assigned by the user. A simple representative of this class is . We show that, in case , or but has algebraic (endpoint) singularities at x = 0 and/or x = 1, the trapezoidal rule on the transformed integral produces exceptionally high accuracies for special values of r and s. In particular, when and we employ , the error in the approximation is (i) O(h ^r ) for arbitrary r and (ii) O(h ^2r ) if r is a positive odd integer at least 3, h being the integration step. We illustrate the use of these transformations and the accompanying theory with numerical examples.      

Optimization and experimental investigations of stiffened, axially compressed CFRP-panels

Thin-walled, unstiffened and stiffened shell structures made of fibre composite materials are frequently applied due to their high stiffness/strength to weight ratios in all fields of lightweight constructions. One major design criterion of these structures is their sensitivity with respect to buckling failure when subjected to inplane compression and shear loads. This paper describes how the structural analysis program BEOS (Buckling of Eccentrically Orthotropic Sandwich shells) is combined with the optimization procedure SAPOP (Structural Analysis Program and Optimization Procedure) to produce a tool for designing optimum CFRP-panels against buckling. Experimental investigations are used to justify the described procedures.Nomenclature C, C _b material stiffness matrices (shell, beam stiffener) - f objective function - F _cr buckling load - g vector of inequality constraints - K,K _g , stiffness matrix, geometrical stiffness matrix, condensed stiffness matrix - n _s ,n _b vector of stress resultants (shell, beam stiffener) - N _x ,N _y ,N _xy membrane forces of the shell - P _x ,P _y ,P _xy membrane forces of the stiffener - r _x ,r _y ,r _xy radii of curvature - ⁿ n-dimensional Euclidean space - W strain energy - u, v, w deformations inx, y, z direction - x, y, z global coordinate system - x vector of design variables - y, z right-hand side, left-hand side eigenvector - variational symbol - _i ^k Kronecker's delta - e _s ,e _b strain vectors (shell, beam stiffener) - , _cr load factor, buckling load factor - , e total energy, external potential energy A (^) sign above a variable points out that this variable belongs to the prebuckling state.     

Analysis of optimized plates for buckling

This paper is concerned with the analysis of optimized plates for buckling. The Rayleigh-Ritz technique is used to solve variable thickness rectangular plates. A double sine series is used to represent the lateral displacements and convergence is attained for a six term series. The result yields a buckling load which is 44% higher than that of a uniform thickness plate having the same volume. The plate itself was optimized using variational calculus to obtain the optimality condition which states that the thickness is proportional to the strain energy density and a truncated Fourier series solution (one term) was used to obtain an optimal shape having a critical load 112% higher than the uniform thickness plate.     

Optimum design of Beck's column with a constraint on the static buckling load

The paper is concerned with optimization of a damped column subjected to a follower load. The aim is to determine the colum of least volume which has the same critical load as a uniform reference column. The stability analysis is based on the finite element method. The optimization problem is solved by sequential linear programming. By only including a constraint on the flutter load in the volume minimization, a very large volume reduction is possible but the static buckling load (by a pure conservative loading) becomes very small.In applications, it may be important that the optimal column also is capable of supporting a conservative load. Consequently, the volume is minimized with constraints on both the flutter load and the static buckling load. The constraint on the buckling loadp _b has the formp _b ^opt cp _b ⁰ , 0c1, where the upper index opt refers to the optimal design while the upper index 0 refers to the uniform initial design. It is found that, as the constantc approaches 1, the optimal column approaches the optimal Euler column of Tadjbakhsh and Keller (1962).Notation c slack parameter on the constraint on the static buckling load; defined by (9) - c _int,c _ext dimensionless internal and external damping parameters defined by (3) - d _j eigenvalue margin defined by (9) - d vector of time-independent nodal displacements and rotations - _e length of thee-th finite element - L total length of the column - vector of element lengths defined by (11) - m, m(x) mass distribution function - m _i design variables; the mass distribution function evaluated at the nodal points - upper and lower bounds on the design parameters - m design vector with elementsm _i - M mass matrix - N _e the number of finite elements used - p load parameter - Q load matrix - S stiffness matrix - t time - x distance along the column, measured from the clamped end - y lateral deflection of the column - y vector of nodal displacements and rotations - complex eigenvalue - b refers to buckling (static instability by conservative loading) - d refers to divergence (static instability by nonconservative loading) - f refers to flutter (dynamic instability by nonconservative loading)     

Vibration and local instability of thermally stressed plates

The vibration and buckling of a double wedge square cantilever plate has been investigated. It is shown that the free vibration modes, which occur at ΔT_ref = 0, transition into the buckled modes which occur at ΔT_ref = ΔT_refcr for the respective mode. ΔT_refcr for a particular mode is defined as the magnitude of thermal load at which the frequency of the particular mode vanishes. The analysis, in which no assumption whatsoever is made about the shape of the vibration modes, about the vibration frequencies, about the shape of the buckled modes, or about the magnitude of the critical loads, yields the same number of buckling eigenvalues and buckling modes as there are vibration eigenvalues and vibration modes. Gradual application of the load in the analysis permits the change in each vibration frequency of interest and its associated mode to be followed up to the load at which the frequency of the mode becomes zero. This constitutes the limit of linear theory. Only linear theory is used in this paper; thus, no post buckled behavior is considered. As the load is increased, the thin edges of the plate begin to duform during vibration. This local deformation, which begins in the vibration mode, is shown to transition into the phenomena of local edge buckling at ΔT_refcr for the mode.     

Analysis of high-dimensional repeated measures designs: The one sample case

A one sample statistic is derived for the analysis of repeated measures design when the data are multivariate normal and the dimension, d, can be large compared to the sample size, n, i.e. d>n. Quadratic and bilinear forms are used to define the statistic based on Box’s approximation [Box, G.E.P., 1954. Some theorems on quadratic forms applied in the study of analysis of variance problems I: Effect of inequality of variance in the one-way classification. Annals of Mathematical Statistics 25 (2), 290-302]. The statistic has an approximate distribution, even for moderately large n. One of the main advantages of the statistic is that it can be used both for unstructured and factorially structured repeated measures designs. In the asymptotic derivations, it is assumed that n→∞ while d remains finite and fixed. However, it is demonstrated through simulations that for n as small as 10, the new statistic very closely approximates the target distribution, unaffected by even large values of . The application is illustrated using a sleep lab example with .     

An Efficient Fixed Parameter Tractable Algorithm for 1-Sided Crossing Minimization

We give an O(^k · n²) fixed parameter tractable algorithm for the 1-Sided Crossing Minimization. The constant in the running time is the golden ratio = (1+5)/2 1.618. The constant k is the parameter of the problem: the number of allowed edge crossings.     

Optimal design of rigid-plastic annular plates with piecewise constant thickness

Rigid-plastic stepped annular plates under uniform pressure load are considered. Both plate edges are supported. Four types of boundary conditions are studied. Tresca's yield condition is used. Such plate dimensions are sought for which the plate of constant volume has the maximal load carrying capacity.Notation a, d, R, h ₁, h₂ plate dimensions (Fig. 1) - Q ^* shear force - Q dimensionless shear force - M _r ^* radial bending moment - M ₁ dimensionless radial bending moment - M _t ^* circumferential bending moment - M ₂ dimensionless circumferential bending moment - M _k maximum value of bending moments in the rigid region - p ^* uniform pressure load - p dimensionless uniform pressure load - r radial coordinate - x dimensionless radial coordinate - , , dimensionless parameters for plate (5) - V plate volume - dimensionless plate volume - M ₀ ^* yield moment - ₀ yield stress - p ₀ load carrying capacity - p ₀ ^m maximum value of the load carrying capacity - p ₀ ^u load carrying capacity for uniform plate - ^m, ^m optimal parameters, which correspond to the maximum value ofp ₀ - s _i radius of circle between different plastic stages - () /x     

A Fat Boundary Method for the Poisson Problem in a Domain with Holes

We consider the Poisson equation with Dirichlet boundary conditions, in a domain , where ⁿ , and B is a collection of smooth open subsets (typically balls). The objective is to split the initial problem into two parts: a problem set in the whole domain , for which fast solvers can be used, and local subproblems set in narrow domains around the connected components of B, which can be solved in a fully parallel way. We shall present here a method based on a multi-domain formulation of the initial problem, which leads to a fixed point algorithm. The convergence of the algorithm is established, under some conditions on a relaxation parameter . The dependence of the convergence interval for upon the geometry is investigated. Some 2D computations based on a finite element discretization of both global and local problems are presented.     

A phase II nonparametric control chart based on precedence statistics with runs-type signaling rules

Nonparametric control charts do not require knowledge about the shape of the underlying distribution and can thus be attractive in certain situations. Two new Shewhart-type nonparametric control charts are proposed for monitoring the unknown location parameter of a continuous population in Phase II (prospective) applications. The charts are based on control limits given by two specified order statistics from a reference sample, obtained from a Phase I (retrospective) analysis, and using some runs-type signaling rules. The plotting statistic can be any order statistic in a Phase II sample; the median is used here for simplicity and robustness. Exact run length distributions of the proposed charts are derived using conditioning and some results from the theory of runs. Tables are provided for practical implementation of the charts for a given in-control average run length between 300 and 500. Comparisons of the average run length ARL, the standard deviation of run length (SDRL) and some run length percentiles show that the charts have robust in-control performance and are more efficient when the underlying distribution is t (symmetric with heavier tails than the normal) or gamma (1, 1) (right-skewed). Even for the normal distribution, the new charts are quite competitive. An illustrative numerical example is given. An added advantage of these charts is that they can be applied before all the data are collected which might lead to savings in time and resources in certain applications.     

C1 plate and shell finite elements for geometrically nonlinear analysis of multilayered structures

The aim of this work is to analyze the geometrically nonlinear mechanical behaviour of multilayered structures by a high order plate/shell finite element in order to predict displacements and stresses of such composite structures for design applications. Based on a conforming finite element method, a C¹ triangular six node finite element is developed using trigonometric functions for the transverse shear stresses. The geometric nonlinearity is based on von-Karmann assumptions and only five generalized displacements are used to ensure:

• •a cosine distribution for the transverse shear stresses with respect to the thickness co-ordinate, avoiding shear correction factors;
• •the continuity conditions between layers of the laminate for both displacements and transverse shear stresses;
• •the satisfaction of the boundary conditions at the top and bottom surfaces of the shells.

     

Diagrammatic Analysis of Interval Linear Equations - Part II: The Two-Dimensional Case and Generalization to n Dimensions

Using the results obtained for the one-dimensional case in Part I (Reliable Computing 9(1) (2003), pp. 1–20) of the paper, an analysis of the two-dimensional relational expression a ₁ x ₁ + a ₂ x ₂ b, where {, , , =}, is conducted with the help of a midpoint-radius diagram and other auxiliary diagrams. The solution sets are obtained with a simple boundary-line selection rule derived using these tools, and are characterized by types of one-dimensional cuts through the solution space. A classification of basic possible solution types is provided in detail. The generalization of the approach for n-dimensional interval systems and avenues for further research are also outlined.     

On coupling of probability distributions and estimating the divergence through variation

Let X be a discrete random variable with a given probability distribution. For any α, 0 ≤ α ≤ 1, we obtain precise values for both the maximum and minimum variational distance between X and another random variable Y under which an α-coupling of these random variables is possible. We also give the maximum and minimum values for couplings of X and Y provided that the variational distance between these random variables is fixed. As a consequence, we obtain a new lower bound on the divergence through variational distance.