A UNIFIED ANALYSIS OF STIFFENER REINFORCED COMPOSITE BEAMS WITH ARBITRARY CROSS-SECTION by GIANFRANCO RIOS Presented to the Faculty of the Graduate School of The University of Texas at Arlington in Partial Fulfillment of the Requirements for the Degree of DOCTOR OF PHILOSOPHY THE UNIVERSITY OF TEXAS AT ARLINGTON December 2009 Copyright © by Gianfranco Rios 2009 All Rights Reserved ACKNOWLEDGEMENTS I would like to thank my beloved parents, Luis Enrique Rios and Marghe Lombardi de Rios, for their support and guidance; and my brother Giancarlo Rios. Also, my girlfriend Nhat Linh Do for all her love, support, and never ending encouragement. And my special dog Chewy for his companionship and unconditional love. I would like to express my sincere appreciation and gratitude to my supervising professor, Dr. Wen S. Chan for his guidance, patience, support, and encouragement. I would also like to thank the committee members: Dr. Lawrence, Dr. Nomura, Dr. Wang, and Dr. Huang for their valuable advice and assistant. November 19, 2009 iii ABSTRACT A UNIFIED ANALYSIS OF STIFFENER REINFORCED COMPOSITE BEAMS WITH ARBITRARY CROSS-SECTION Gianfranco Rios, PhD The University of Texas at Arlington, 2009 Supervising Professor: Wen S. Chan A number of methods are available to analyze the laminated composite beams using smear property of laminate stiffness ignoring unsymmetrical behavior of laminates. Some include their effect of laminate but did not include the unsymmetrical effect of the cross-section of the beam. On the other hand, using finite element method to analyze the beam is dependent on the structural configurations including the laminate lay-up sequence. An analytical method was developed from lamination theory in order to study the structural response of composite laminated beams. The present method is capable of predicting the axial and bending stiffnesses, the centroid location, and the stresses in each ply of the whole structure. The results were compared with finite element methods. iv Laminated beams studied in this research include the beams with a rectangular cross-section bonded with a stiffener. The stiffener placed in both aligned and non-aligned position with respect to the centerline of the beam is considered. Thin-walled beams with circular and airfoil sections are also studied. A parametrization method to define the median of the cross-section is developed. Then the stress analysis is conducted. For two laminates aligned bonded together top and bottom, the present method results agree very well with the finite element results for all the cases less the unsymmetrical case. For the non-aligned case, all the results agree. On the other hand, the results also are close to each other for the laminates bonded side by side less the un-balanced case. Finally, for the circular and airfoil cross-sections the results agree with the finite element results. v TABLE OF CONTENTS ACKNOWLEDGEMENTS ................................................................................................................ iii ABSTRACT ...................................................................................................................................... iv LIST OF ILLUSTRATIONS ............................................................................................................ xiii LIST OF TABLES ......................................................................................................................... xviii LIST OF SYMBOLS ....................................................................................................................... xxi Chapter Page 1. INTRODUCTION…………..……………………………………..………..….. ..................... 1 1.1 Composite Materials Overview ......................................................................... 1 1.1.1 Definition ........................................................................................... 1 1.1.2 Ancient History ................................................................................. 1 1.1.3 Classification .................................................................................... 3 1.1.4 Advantages and Disadvantages....................................................... 3 1.2 Literature Review .............................................................................................. 4 1.2.1 Analytical Studies ............................................................................. 4 1.2.2 Experiment Studies .......................................................................... 8 1.2.3 Finite Element Analysis .................................................................... 9 1.3 Objective of this Research ............................................................................. 12 1.4 Outline of the Dissertation .............................................................................. 13 2. BASIC PROPERTIES OF COMPOSITE LAMINATED BEAMS .................................. 14 2.1 Brief Review of Lamination Theory ................................................................ 14 2.2 Special Cases of Laminates ........................................................................... 20 2.2.1 Balanced vs. Un-balanced Laminate ............................................. 20 2.2.2 Symmetric, Un-symmetric, and Anti-symmetric Laminate ............. 20 vi 2.3 Evaluation of Stiffness of Composite Beam ................................................... 21 2.3.1 Axial Stiffness ................................................................................. 21 2.3.1.1 Isotropic Material ............................................................ 21 2.3.1.2 Composite Laminate ...................................................... 22 2.3.2 Bending Stiffness............................................................................ 23 2.3.2.1 Isotropic Material ............................................................ 23 2.3.2.2 Composite Materials (Smeared Property Approach) ...................................................................... 24 2.4 Parallel Axis Theorem .................................................................................... 24 2.5 Axial and Bending Stiffness in Laminated Rectangular cross-section Beams ...................................................................................... 25 2.5.1 Narrow Beams ................................................................................ 26 2.5.2 Wide Beams ................................................................................... 28 2.5.3 General Beam ................................................................................ 30 2.6 Centroid .......................................................................................................... 30 2.6.1 Isotropic Material ............................................................................ 30 2.6.2 Composite Laminate ...................................................................... 30 2.6.3 Smeared Property Approach .......................................................... 31 3. STIFFENER REINFORCED LAMINATED BEAMS ..................................................... 32 3.1 Constitutive Equations of Composite Beam ................................................... 32 3.2 Aligned Rectangular Strip Stiffener Reinforcement ....................................... 34 3.2.1 Description of Geometry ................................................................. 34 3.2.2 Equivalent Axial Stiffness ............................................................... 35 3.2.3 Equivalent Bending Stiffness .......................................................... 36 3.2.4 Centroid .......................................................................................... 37 3.3 Results Comparison of Centroid Calculations ............................................... 39 vii 3.3.1 Isotropic Material ............................................................................ 39 3.3.2 Symmetric Laminate ....................................................................... 40 3.3.3 Un-symmetric and Balanced Laminate .......................................... 41 3.3.4 Un-symmetric and Un-balanced Laminate ..................................... 43 3.4 Ply Stress Calculations ................................................................................... 44 3.5 Finite Element Model ...................................................................................... 45 3.6 Finite Element Results.................................................................................... 47 3.6.1 Axial and Bending Stiffnesses obtained from FEM ....................... 47 3.6.2 Centroid Locations.......................................................................... 48 3.6.3 Comparison of Laminate Stiffnesses ............................................. 49 3.6.4 Ply Stresses of Isotropic Material ................................................... 50 c 3.6.4.1 Beam Laminate under Axial Load, N x .......................... 50 c 3.6.4.2 Beam Laminate under Bending Moment, M x .............. 54 3.6.5 Ply Stresses of 0° Laminate ................. .......................................... 57 3.6.5.1 N c acting on [0 ] and [0 ] of parent and x 4 S 2 S stiffener .......................................................................... 57 3.6.5.2 M c acting on [0 ] and [0 ] of parent and x 4 S 2 S stiffener .......................................................................... 60 3.6.6 Ply Stresses of [+45 2/0 2]s Symmetric Laminate .............................. 63 c 3.6.6.1