A PSEUDO STRENGTH FUNCTION FOR THE GENERATION OF LOCAL LAMINATE - PDF document

18 TH INTERNATIONAL CONFERENCE ON COMPOSITE MATERIALS A PSEUDO STRENGTH FUNCTION FOR THE GENERATION OF LOCAL LAMINATE REINFORCEMENT DOUBLERS B. Schläpfer 1* , G. Kress 1 1 Centre of Structure Technologies, ETH Zurich, Zurich, Switzerland * Corresponding author (bschlaepfer@mavt.ethz.ch) Keywords : Local Reinforcement, Pseudo Strength Function, Sensitivity Analysis, Structural Optimization failure indices. The heuristic approach is later enhanced with a genetic algorithm [12]. The 1 Introduction generation of reinforcement doublers for laminated Due to their superior mass-specific mechanical plates with holes using this method is presented in properties, laminated composite structures are today [13]. The parameterization scheme ensures that the well established for aircraft and spacecraft obtained designs are adapted for production. applications. The possibility to tailor the properties In this paper, a differentiable pseudo objective of laminates to local requirements at different function is introduced that unites the failure indices regions of the structure is an additional advantage. of the structure. This enables the determination of a The large number of design variables as well as the gradient field that expresses the influence of a layer anisotropic behavior of the layers result in a thickness change on the strength. It is later used to complex and time-consuming design process. The generate reinforcement doublers starting from an usage of an automated design procedure including unreinforced initial design. Material is added step- an optimization can reduce these costs and lead to wise based on gradient information until a required better solutions. strength is reached. Since weight is linearly Stress concentrations caused by cut-outs or load depending on the layer thicknesses, the placement of introduction points may reduce the strength of a the reinforcements is very efficient in terms of mass. structure. These areas can be reinforced with local laminate doublers. Finding efficient doubler 2 Pseudo Strength Function geometries is not trivial and hard to be done intuitively. Automated design processes can help The proposed pseudo strength function is based on improving solutions quality and reduce time-costs. the well-known Tsai-Hill criterion [14,15] for first In contrast to eigenfrequencies, buckling or ply failure in laminated composite structures. The compliance, strength is a local phenomenon. The Failure Index FI can be determined with equation problem of finding optimal, homogeneous laminates (1), where σ 1 , σ 2 and τ 12 stand for the plane-stress for a given loadcase has been considered by many components in material principal coordinates and X , researchers [1-7] (and references therein). Y and S are the corresponding strength values. Considering non homogeneous stress states, the Having different strength values for tension and strength of the structure is restricted by the compression, the choice of X , Y and S is depending maximum stresses. Allowing locally varying on the stress state. laminates, a design change may cause a sudden 2 2 2                (1) FI  1  1 2  2   12  relocation of the critically stressed region what leads       X X 2 Y S to a non-differentiable objective function. Gradient- By rearranging the stress components into a stress based optimization algorithms cannot be applied and vector, the equation can be transferred to matrix stochastic algorithms are often used instead [8-10]. notation (Eq. (2)). Hansel and Becker [11] address the problem of finding weight-minimal solutions considering strength constraints by taking advantage of local reinforcement doublers. Material is removed layer- wise depending on the principal stresses and the

    2  2 T ( ) X ( 2 ) X 0         1 1 Thus the Failure Indices FI can be expressed by            2 2  (2) FI ( 2 ) X ( ) Y 0     FI   T  S . (10) 2 2   TH xy xy        2   0 0 ( ) S       Also the transformed strength matrix has to be 12 12 determined only once for every single layer since Q , This equation again can also be written in a short R and T are element independent and thus unique for form. a single layer. FI    T S (3) 12 TH 12 Based on the strain-based matrix formulation of the failure indices FI , a pseudo strain function is The strength matrix S TH differs for the single layers defined. It is inspired by the optimality criterion that but not for the elements containing a specific layer. a structure has an optimal material distribution when Equation (3) can also be expressed with element the stress distribution is homogeneous. Kress [17,18] strains in global coordinates instead of the principal takes the stress standard deviation as objective stress components. This enables an efficient function to find homogeneous designs considering evaluation since global element strains arise directly different side constraints. Here, the objective from the displacement vector obtained in the finite function is formulated by summarizing the squares element analysis. For that, strains are mapped to of the failure indices of every ply of every finite stresses with the material stiffness matrix Q which is element, also called element-layers, subtracted by a also unique for a single layer. For simplicity, only given value β . plane stresses are considered so that Q becomes the nel     2 membrane stiffness matrix.      f T S (11) TH xy i , TH xy i ,   Q ε (4)  i 1 12 12 The minus sign in front converts the minimization Subsequently, the element strains in global problem to a maximization problem, so that areas coordinates are rotated to local coordinates by an where the structure should be reinforced show high angle φ with a transformation matrix T and a so gradient values. This function becomes maximal (or called Reuter-matrix R [16]. equal to 0), if all failure indices match the required   ε RTR ε 1 (5) value β . The choice of β has to be done manually by 12 xy the designer. Choosing a value of 1 implies that the with strength of the optimal design is critical in every part of the structure. To obtain non-critical designs, β   2  2    cos sin 2sin cos should be set to a low value (e.g. equal zero). It is    2  2     (6) T sin cos 2sin cos clear that the optimal value of the function may     never be reached in a real optimization process.      2   2  sin cos sin cos cos sin   Moreover, it is not guaranteed that all failure indices fall below the critical threshold of 1. However, this and formulation unifies the strength which is, as   1 0 0 mentioned before, a local phenomenon, into a global   expression that can be used as an objective function R  . (7) 0 1 0   for optimization.     0 0 2 Referring to equation (10), the pseudo strain function can be expressed in terms of element Combining equation (4) and (5) leads to strains.  σ QRTR ε . (8)   nel 2   12 xy   T    f S (12) TH TH xy i , xy i , Equation (8) can then be used to transform the  i 1 To be able to perform an optimization of the stress-based strength matrix to a strain-based laminate, sensitivities with respect to the thicknesses strength matrix applying a both-sided multiplication. of the element-layers will be determined. T       Thicknesses are chosen as design parameters to have 1 1 (9) S QRTR S QRTR  TH TH