A general solution for stress resultants and moments around holes in unsymmetric laminates V.G. Ukadgaonker * , D.K.N. Rao Department of Mechanical Engineering, Indian Institute of Technology, Powai, Mumbai 400 076, India Abstract The complex variable method of Lekhnitskii and Savin can be applied for solving stress concentration problems in symmetric laminates. Although signi®cant amount of literature is available on heterogeneous aelotropic plates, the stress concentration problems have not been addressed in it until Becker gave his complex potential method for plate problems with bending extension coupling. The present work is an extension of Becker's solution for elliptical hole problem for unsymmetric laminates. In this paper, it is aimed to bring out a general solution to determine the stress resultants and moments around holes of any shape with a simple mapping function under arbitrary biaxial loading condition. Ó 2000 Elsevier Science Ltd. All rights reserved. Keywords: Complex potential method; Unsymmetric laminates; Stress resultants and moments around holes www.elsevier.com/locate/compstruc Composite Structures 49 (2000) 27±39 Notation A j complex constants for ®rst stage potentials A ij , B ij , D ij extensional, coupling and bending stinesses a j , b j constants related with anisotropy a 5 ±a 8 , b 5 ±b 8 complex constants c j , d j , e j , f j , g j , h j complex constants associated with stress resultants and plate moments f x , f y membrane force components f(s) integrated loading function k, N number associated with terms of the mapping function K 11j ±K 18j complex constants K 11j ± K 18j complex conjugates of K 11j ±K 18j M x , M y , M xy moment resultants in the plate M n moment in the normal direction on the hole contour M nt moment in the tangential direction on the hole contour m k constants of the mapping function m(s) normal bending moment on the hole contour N x , N y , N xy stress resultants in the plate P integrated transversal force p j , q j complex constants for the laminate p(s) transversal force on the hole contour Q x , Q y transversal plate forces Q x ; Q y Kircho 's substitute transversal forces Q n transversal force normal to the hole boundary s j complex parameters of anisotropy t boundary value of f on unit circle, f  e ih u, v, w inplane and normal displacements z complex coordinate, z  x  iy z j anisotropic complex coordinate, z j  x  s j y b orientation angle of loading e x , e y , c xy longitudinal and shear strains v x , v y , v xy curvatures k biaxial loading factor U 00 j z j  ®nal potentials for a given plate problem U 00 j z j ; U 000 j z j  second and third derivatives of the potentials U 00 j1 z j  second derivative of ®rst stage potentials U 00 j2 f second derivative of second stage potentials in f-plane Uf potentials in f-plane Ut potentials on the unit circle * Corresponding author. Tel.: +91-22-576-7543; fax: +91-22-578- 3480. E-mail address: vgu@me.iitb.ernet.in (V.G. Ukadgaonker). 0263-8223/00/$ - see front matter Ó 2000 Elsevier Science Ltd. All rights reserved. PII: S 0 2 6 3 - 8 2 2 3 ( 9 9 ) 0 0 1 2 3 - 3