Christian Mittelstedt

Interlaminar Stress Concentrations in Layered Structures: Part I - A Selective Literature Survey on the Free-Edge Effect since 1967

Stress concentration phenomena in composite laminates are technically important situations. A well-known problem of this class is the free-edge effect in composite laminates or as a superordinated example the stress concentrations in the vicinity of free laminate corners (so-called free-corner effect). The present work is split into two parts. In the present contribution, after a short introduction to the given stress concentration problems in general we will survey relevant selected literature on the classical free-edge effect dating from 1967 until today. Beside accentuation on approximate closed-form analytic methods for the stress analysis in the free-edge effect situation, numerous references on numerical methods and investigations on the occurring stress singularities are also cited. In a subsequent paper we will present a simple closed-form method for the analysis of the stress fields in the vicinity of free laminate corners with arbitrary layup. The method is based on adequate stress shape assumptions and a variational principle. The present article contains 136 references.

Free-Edge Effects in Composite Laminates

There are many technical applications in the field of lightweight construction as, for example, in aerospace engineering, where stress concentration phenomena play an important role in the design of layered structural elements (so-called laminates) consisting of plies of fiber reinforced plastics or other materials. A well known stress concentration problem rich in research tradition is the so-called free-edge effect. Mainly explained by the mismatch of the elastic material properties between two adjacent dissimilar laminate layers, the free-edge effect is characterized by the concentrated occurrence of three-dimensional and singular stress fields at the free edges in the interfaces between two layers of composite laminates. In the present contribution, a survey on relevant literature from more than three decades of scientific research on free-edge effects is given. The cited references date back to 1967 and deal with approximate closed-form analyses, as well as numerical investigations by the finite element method, the finite difference method, and several other numerical techniques. The progress in research on the stress singularities which arise is also reviewed, and references on experimental investigations are cited. Related problems are also briefly addressed. The paper closes with concluding remarks and an outlook on future investigations. In all, 292 references are included.

Interlaminar Stress Concentrations in Layered Structures: Part II - Closed-Form Analysis of Stresses at Laminated Rectangular Wedges with Arbitrary Non-Orthotropic Layup

In the second of a series of two papers, a refined closed-form analysis method for the calculation of interlaminar stress concentrations in the vicinity of rectangular wedges of thermally loaded composite laminates with arbitrary layup is presented. Based on adequate layerwise shape assumptions for the in-plane components of Cauchy’s stress tensor that automatically fulfill the conditions of traction free edges, the interlaminar stresses are derived from the three-dimensional equilibrium conditions in combination with the exact fulfillment of the given homogeneous boundary conditions of traction-free laminate facings and the requirement of continuity of the interlaminar stresses at the ply interfaces. The far field conditions of recovery of the stress results by classical laminate plate theory in the inner laminate regions with increasing distance from the laminate corner are accounted for. Free constants in the stress shape functions are determined by the minimization of the laminate’s complementary potential energy which can be accomplished in an iterative manner. The stress shape functions are assumed as simple exponential terms with respect to the in-plane coordinates, whereas polynomials are applied as thickness functions. The present analysis methodology is found to be in good agreement with finite-element computations and yields reasonably accurate results with little computational effort.

Free-corner effects in cross-ply laminates: An approximate higher-order theory solution

Like in the well-known free-edge effect situation, stress fields in the vicinity of free corners of layered plates exhibit a distinct three-dimensional and singular behavior and thus represent an important technical situation. Nevertheless, there are only few thorough investigations available concerning stress concentrations near free laminate corners. Since numerical analyses of stress concentration phenomena in composite laminates are computationally expensive, the present contribution is devoted to a simple closed-form higher-order theory approach for the calculation of displacements, strains, and stresses in the vicinity of a rectangular corner of a symmetric cross-ply laminate under thermal load. Appropriate representations for the displacement field in the manner of a single-layer theory with unknown inplane components and assumed trigonometric functions through the thickness yield closed-form expressions for the strains and stresses throughout the whole laminate. The inplane displacement functions are determined by equilibrium considerations in an integrated form with the solution of some resultant characteristic equations. Boundary conditions are fulfilled in an integral sense. The present approach can be applied easily, requires little computational effort and is in good conformity with comparative finite-element calculations and other closed-form analyses.

A VARIATIONAL FINITE LAYER TECHNIQUE FOR THE INVESTIGATION OF THERMALLY INDUCED STRESS CONCENTRATIONS IN COMPOSITE STRUCTURES

The technical relevance of stress fields near free laminate edges under mechanical and/or hygrothermal loads (“free-edge effect”) has long been recognized. However, the state of stress near free laminate corners (i.e., at corners that are generated by two merging straight free laminate edges) has gone nearly unnoticed in the open literature. To gain further insight into the mechanics of free-corner stress fields (“free-corner effect”), the present contribution is devoted to the closed-form analysis of displacements, strains and stresses in the vicinity of free rectangular corners of symmetric crossply laminates under uniform thermal load by means of a layerwise C0-continuous displacement approach. The laminate is discretized into an arbitrary number of mathematical layers through the thickness. However, concerning the two in-plane directions, no discretization is employed, but on the contrary, unknown in-plane functions are assumed that are then determined by application of the principle of minimum potential energy of the laminate. Due to some simplifying prerequisite assumptions concerning the utilized displacement approach and performing a separation of the in-plane variables, the resultant governing Euler–Lagrange equations are ordinary second-order differential equations that can be solved in a closed–form way. Hence, all state variables of the given thermoelastic free-corner problem can be written in a closed-form manner, which makes the present method easily applicable and allows a good insight into the underlying mechanics. Given boundary conditions of traction-free laminate edges are satisfied in an average sense. The present method is easily applicable, requires little computational effort, and is in excellent conformity with accompanying finite element computations. Because the presented approach enables a closed-form analytic formulation with respect to the in-plane coordinates, it is appropriate to designate the methodology as a finite layer technique.

POSTBUCKLING OF COMPRESSIVELY LOADED IMPERFECT COMPOSITE PLATES: CLOSED-FORM APPROXIMATE SOLUTIONS

In this paper, closed-form approximate solutions for the geometrically nonlinear behaviour of rectangular laminated plates with flexural orthotropy under longitudinal compression are presented. Based on the governing Marguerre-type differential equations postulated for imperfect plates, two plate configurations are discussed in detail, representing important application cases in practical engineering work. The first configuration is a laminated plate that is simply supported at all four edges (the so-called SSSS plate), while for the second configuration clamped unloaded longitudinal edges are considered (denoted as the SSCC plate). For both plate configurations, rather simple closed-form approximations in the form of trigonometric shape functions are employed for the description of the out-of-plane postbuckling plate deflections. Based on the chosen shape functions, the compatibility condition with respect to the in-plane strains is fulfilled exactly, while the out-of-plane equilibrium condition for a deflected plate element is not, but is solved using a Galerkin-type formulation instead. Eventually, very simple closed-form solutions for all postbuckling state variables (deflections, in-plane edge displacements, and effective widths) are derived that can be used very conveniently in engineering practice. The high accuracy of the presented analysis methods is established by comparison with the results of other authors.

Thermoelastic Fields in Boundary Layers of Isotropic Laminates

An approximate approach to the calculation of displacements, strains, and stresses near edges and corners in symmetric rectangular layered plates of dissimilar isotropic materials under thermal load is presented. In the thickness direction the plate is discretized into an arbitrary number of sublayers/mathematical layers. A layerwise linear displacement field is formulated such that the terms according to classical laminate plate theory are upgraded with unknown in-plane functions and a linear interpolation scheme through the layer thickness in order to describe edge and corner perturbations. By virtue of the principle of minimum potential energy the governing coupled Euler-Lagrange differential equations are derived, which in the case of free-edge effects allow a closed-form solution for the unknown inplane functions. Free-corner effects are investigated by combining the displacement formulations of the two interacting free-edge effects. Hence, all state variables in the plate are obtained in a closed-form manner. Boundary conditions of traction free plate edges are satisfied in an integral sense. The present methodology is easily applied and requires only reasonable computational expenses.

Closed-Form Approximate Solutions for the Local Buckling Behavior of Composite Laminated Beams Based on Third-Order Shear Deformation Theory

This paper presents approximate closed-form analytical methods for the assessment of the local buckling behaviour (i.e. buckling of flanges and webs) of shear-deformable composite laminated beams. The analysis approach relies on the discrete plate method, i.e. the buckling analysis of flanges and webs modelled as separate plates that are elastically clamped at those edges where the adjacent section members are located. We perform these local buckling analyses within the framework of third-order shear deformation theory (short: TSDT) by Reddy (Mechanics of Laminated Composite Plates and Shells, 2004, [1]) and compare the results to analysis approaches that we established using classical laminated plate theory (CLPT) and first-order shear deformation theory (FSDT). The buckling analysis uses simple shape functions for the displacements and rotation angles and enables closed-form solutions based on the principle of minimum elastic potential of the buckled plates. The results show that the approximate analysis approach work with very satisfying accuracy despite its rather simple and straightforward nature and thus is especially suited for all practical applications where computational time and effort are crucial factors.

Closed-Form Approximate Solution for the Postbuckling Behavior of Orthotropic Shallow Shells Under Axial Compression

The current paper deals with a closed-form approximate solution for the postbuckling behavior of an unstiffened, singly-curved, orthotropic shell. As loading condition the case of uniform axial compression is treated. Concerning the boundary conditions all edges are supposed to be simply supported. Additionally, geometrical imperfections in form of an initial deflection of the shell can be accounted for. Choosing rather simple shape functions for the deflection a closed-form expression for the Airy stress function is obtained from the compatibility condition. As the equilibrium condition cannot be satisfied exactly the solution procedures of Galerkin as well as Ritz are employed to obtain an approximate solution. The resulting expressions from these procedures again allow for a closed-form solution of the load-deflection-relationship. After the force and the amplitude are known all other state variables such as stresses and displacements can be evaluated in a closed-form manner. Due to the rather simple formulation of the deflection shape the algorithm is limited to cases where the qualitative shape of the deflection does not change significantly. On the other hand the very high computational efficiency of the described solution procedure makes it ideally suited for use in the field of optimization and preliminary design, if the applied load does not exceed the linear buckling load too much.

On the Determination of Edge Reinforcement Properties for Optimum Lightweight Design of Composite Stiffeners

In this work the determination of the properties of an edge reinforcement of a composite plate stiffener is treated. A minimum stiffness criterion for the edge reinforcement on the basis of a closed-form buckling analysis of a composite plate with edge reinforcement and elastic clamping is given. The minimum stiffness criterion is given in explicit form and in a fully dimensionless representation. A composite stiffener designed by this criterion will exhibit a local buckling mode of the web, rather than a global simultaneous buckling of both the web and the edge reinforcement. The determination of an optimum lightweight design is discussed. In an example the criterion is applied to the dimensioning of a stiffener design.