investigation of isotruss® structures in compression using

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2020-08-04

Investigation of IsoTruss® Structures in Compression Using Investigation of IsoTruss® Structures in Compression Using

Numerical, Dimensional, and Optimization Methods Numerical, Dimensional, and Optimization Methods

Hanna Belle Opdahl Brigham Young University

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Investigation of IsoTruss® Structures in Compression Using

Numerical, Dimensional, and Optimization Methods

Hanna Belle Opdahl

A thesis submitted to the faculty ofBrigham Young University

in partial fulfillment of the requirements for the degree of

Master of Science

David W. Jensen, ChairMichael A. ScottS. Andrew Ning

Department of Civil and Environmental Engineering

Brigham Young University

Copyright © 2020 Hanna Belle Opdahl

All Rights Reserved

ABSTRACT

Investigation of IsoTruss® Structures in Compression UsingNumerical, Dimensional, and Optimization Methods

Hanna Belle OpdahlDepartment of Civil and Environmental Engineering, BYU

Master of Science

The purpose of this research is to investigate the structural efficiency of 8-node IsoTruss®

structures subject to uniaxial compression using numerical, dimensional, and optimization meth-ods. The structures analyzed herein are based on graphite/epoxy specimens that were designed forlight-weight space applications, and are approximately 10 ft. (3 m) long and 0.3 lb. (0.14 kg). Theprincipal failure modes considered are material failure, global buckling, local buckling at the baylevel, and longitudinal strut buckling. Studies were performed with the following objectives: tocorrelate finite element predictions with experimental and analytical methods; to derive analyticalexpressions to predict bay-level buckling; to characterize interrelations between design parametersand buckling behavior; to develop efficient optimization methods; and, to compare the structuralefficiency of outer longitudinal configurations with inner longitudinal configurations.

Finite element models were developed in ANSYS, validated with experimental data, andverified with traditional mechanics. Data produced from the finite element models were used toidentify trends between non-dimensional Π variables, derived with Buckingham’s Pi Theorem.Analytical expressions were derived to predict bay-level buckling loads, and verified with dimen-sional analyses. Numerical and dimensional analyses were performed on IsoTruss structures withouter longitudinal members to compare the structural performance with inner longitudinal config-urations. Analytical expressions were implemented in optimization studies to determine efficientand robust optimization techniques and optimize the inner and outer longitudinal configurationswith respect to mass.

Results indicate that the finite element predictions of axial stiffness and global bucklingloads correlate with traditional mechanics equations, but overestimate the capacity demonstratedin previously published experimental results. The buckling modes predicted by finite element pre-dictions correlate with traditional mechanics and experimental results, except when the local andglobal buckling loads coincide. The analytical expressions derived from mechanics to predict localbuckling underestimate the constraining influence of the helical members, and therefore underes-timate the local buckling capacity. The optimization analysis indicates that, in the specified designspace, the structure with outer longitudinal members demonstrates a greater strength-to-weightratio than the corresponding structure with inner longitudinal members by sustaining the sameloading criteria with 10% less mass.

Keywords: IsoTruss, finite elements, Buckingham’s Pi Theorem, optimization, light-weight com-posite structures

ACKNOWLEDGMENTS

This research was funded, in part, by the Utah NASA Space Grant Consortium. Their

contribution is deeply appreciated.

I extend my sincere gratitude to my advisor, Dr. Jensen. Dr. Jensen has spent countless

hours with me to provide instruction, discuss new ideas, and coach me in pedagogy. I appreciate

every opportunity that he has extended to me, especially the experience of assistant teaching. I

have grown in many ways under his tutelage, and am grateful that he has seen potential in me, and

challenged me to give my best effort.

I would also like to recognize my committee members, Dr. Scott and Dr. Ning. It has been

a privilege to attend their lectures and learn from their expertise. Both demonstrate sincere interest

in the education of their students, dedication to progress knowledge and understanding, and deep

respect for the harmony of spiritual and intellectual truth.

To Kim and Jolene, staff of the BYU Civil Engineering Department: Thank you for pro-

viding assistance and coaching in the logistical details of this thesis. Your willingness to help

has eased much of my anxiety and made the experience enriching and seamless. The Department

would be a very different place without you.

To my friends at BYU: Thank you for offering diverse perspectives, providing a backboard

for new ideas, and knowing when it’s time to leave and go for a walk.

To my wonderful family: Thank you for keeping everything in perspective and cheering

me through till the end. Your prayers and encouragement are a force of faith that helped move this

mountain forward.

“And now, if the Lord has such great power, and has wrought so many miracles among the

children of men, how is it that He cannot instruct me, that I should build a ship?” -1 Nephi 17:51

TABLE OF CONTENTS

LIST OF TABLES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viii

LIST OF FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix

Chapter 1 General Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Nomenclature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.3 Research Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.4 Research Limitations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.5 IsoTruss Structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.5.1 Geometric Configuration . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.5.2 Failure Modes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

Chapter 2 Derivation of the shell-like buckling load of 8-node IsoTruss® structures . 132.1 Synopsis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132.2 Introduction and Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142.3 Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

2.3.1 Strain Energy Derivation . . . . . . . . . . . . . . . . . . . . . . . . . . . 162.3.2 Critical Shell-like Buckling Load . . . . . . . . . . . . . . . . . . . . . . . 21

2.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222.4.1 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222.4.2 Implications for Further Investigation . . . . . . . . . . . . . . . . . . . . 22

Chapter 3 Validation of a finite element model in ANSYS WorkBench for IsoTruss®

structures in uniaxial compression . . . . . . . . . . . . . . . . . . . . . . 233.1 Synopsis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233.2 Nomenclature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233.3 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243.4 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253.5 Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

3.5.1 Composite Material Properties . . . . . . . . . . . . . . . . . . . . . . . . 263.5.2 Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273.5.3 Finite Element Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . 283.5.4 Verification of FE Model using Mechanics . . . . . . . . . . . . . . . . . . 313.5.5 Validation of FE Model using Experimental Data . . . . . . . . . . . . . . 32

3.6 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 333.6.1 Axial Stiffness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 343.6.2 Eigenvalue Buckling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38


iv

Chapter 4 Dimensional analysis of shell-like buckling in IsoTruss® structures usingBuckingham’s Pi Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

4.1 Synopsis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 434.2 Nomenclature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 434.3 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 444.4 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 454.5 Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

4.5.1 Selecting Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 484.5.2 Dimensional Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 484.5.3 Finite Element Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

4.6 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 524.6.1 Shell-like Buckling Interrelations . . . . . . . . . . . . . . . . . . . . . . . 534.6.2 Global and Local Buckling Curves . . . . . . . . . . . . . . . . . . . . . . 564.6.3 Analytical Predictions vs. FE Predictions . . . . . . . . . . . . . . . . . . 57

4.7 Significance of Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 614.7.1 Shell-like Buckling Interrelations . . . . . . . . . . . . . . . . . . . . . . . 614.7.2 Global and Local Buckling Curves . . . . . . . . . . . . . . . . . . . . . . 614.7.3 Analytical Predictions vs. FE Predictions . . . . . . . . . . . . . . . . . . 62


Chapter 5 Gradient-based versus gradient-free optimization of IsoTruss® structuresin uniaxial compression . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

5.1 Synopsis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 655.2 Nomenclature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 655.3 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 665.4 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 675.5 Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

5.5.1 Resources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 685.5.2 Design Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 695.5.3 Design Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 705.5.4 Constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 705.5.5 Problem Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

5.6 Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 745.7 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 755.8 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

5.8.1 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 775.8.2 Implications for Further Investigation . . . . . . . . . . . . . . . . . . . . 77

Chapter 6 Gradient-based optimization of IsoTruss® structures in uniaxial com-pression . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

6.1 Synopsis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 786.2 Nomenclature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

v

6.3 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 806.4 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 806.5 Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

6.5.1 Analytical Expressions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 816.5.2 Algorithmic Differentiation . . . . . . . . . . . . . . . . . . . . . . . . . . 836.5.3 Design Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 836.5.4 Multi-modal Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 846.5.5 Problem Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

6.6 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 866.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

6.7.1 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 886.7.2 Implications for Further Investigation . . . . . . . . . . . . . . . . . . . . 89

Chapter 7 Dimensional analysis of buckling in IsoTruss® structures with outer lon-gitudinal members . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

7.1 Synopsis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 907.2 Nomenclature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 907.3 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 917.4 Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

7.4.1 Dimensional Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 947.4.2 Finite Element Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . 957.4.3 Optimization Techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . 97

7.5 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 997.5.1 Trend Analyses of OLC . . . . . . . . . . . . . . . . . . . . . . . . . . . . 997.5.2 Analytical vs. FE Predictions of OLC . . . . . . . . . . . . . . . . . . . . 1027.5.3 Trend Analyses of OLC vs. ILC . . . . . . . . . . . . . . . . . . . . . . . 1067.5.4 Optimization of OLC vs. ILC . . . . . . . . . . . . . . . . . . . . . . . . 108

7.6 Significance of Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1107.6.1 Influence of Helical Members . . . . . . . . . . . . . . . . . . . . . . . . 1107.6.2 OLC vs. ILC Performance . . . . . . . . . . . . . . . . . . . . . . . . . . 113


Chapter 8 Conclusions and Recommendations . . . . . . . . . . . . . . . . . . . . . . 1168.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116

8.1.1 Validity of Finite Element Modeling . . . . . . . . . . . . . . . . . . . . . 1168.1.2 Analytical Predictions of Bay-level Buckling . . . . . . . . . . . . . . . . 1178.1.3 Design Parameter Interrelations with Buckling Capacity . . . . . . . . . . 1178.1.4 Preferred Optimization Techniques . . . . . . . . . . . . . . . . . . . . . . 1188.1.5 Relative Buckling Capacity of Outer vs. Inner Longitudinal Members . . . 118

8.2 Recommendations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118

REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121

vi

Appendix A Composite Micromechanics Equations . . . . . . . . . . . . . . . . . . . . 123

Appendix B Geometric Correlations of IsoTruss® Structures . . . . . . . . . . . . . . . 126B.1 Nomenclature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126B.2 Sample Values of Rackliffe4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127B.3 Geometric Correlations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128

Appendix C Gradient-based Optimization Code . . . . . . . . . . . . . . . . . . . . . . 132C.1 Multi-start Optimization Framework . . . . . . . . . . . . . . . . . . . . . . . . . 133C.2 Single Iteration Optimization Framework . . . . . . . . . . . . . . . . . . . . . . 135C.3 Composite Material Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137C.4 Objective Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140C.5 Constraint Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143C.6 Failure Modes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146C.7 Supplementary Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151

vii

LIST OF TABLES

3.1 Material Properties of Fiber and Resin (McCune [1]) . . . . . . . . . . . . . . . . . . 273.2 Material Properties of Fiber and Resin (Rackliffe et al. [2]) . . . . . . . . . . . . . . . 273.3 Nominal Properties of Composite Materials . . . . . . . . . . . . . . . . . . . . . . . 283.4 Geometric Parameters of Experimental Specimens . . . . . . . . . . . . . . . . . . . . 283.5 Moment of Inertia Coefficients for IsoTruss Structures [3] . . . . . . . . . . . . . . . . 313.6 Previous Experimental Results [1] [2] . . . . . . . . . . . . . . . . . . . . . . . . . . 333.7 FE Models and Corresponding Parameters . . . . . . . . . . . . . . . . . . . . . . . . 343.8 Axial Stiffness from Mechanics and FE Model A . . . . . . . . . . . . . . . . . . . . 353.9 Axial Stiffness from Mechanics and FE Model C . . . . . . . . . . . . . . . . . . . . 383.10 Critical Buckling Loads from Mechanics and FE Model C . . . . . . . . . . . . . . . 393.11 Critical Buckling Loads from Mechanics and FE Model D . . . . . . . . . . . . . . . 403.12 Critical Buckling Loads from Mechanics and FE Model E . . . . . . . . . . . . . . . . 40

4.1 Parametric Studies on Open-lattice Composite Structures . . . . . . . . . . . . . . . . 464.2 Parameters and Corresponding Dimensions . . . . . . . . . . . . . . . . . . . . . . . 494.3 Fixed Π Variables of Dimensional Analyses . . . . . . . . . . . . . . . . . . . . . . . 514.4 Fixed Design Parameters of FE Analyses . . . . . . . . . . . . . . . . . . . . . . . . . 524.5 Coefficients relating Π0 to Π1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 534.6 Coefficients relating Π0 to Π2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 544.7 Coefficients relating Π0 to Π3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

5.1 Material and Geometric Properties Implemented in Optimization Analysis . . . . . . . 705.2 Optimized Results Compared to Rackliffe Specimen [2] . . . . . . . . . . . . . . . . . 75

6.1 Material and Geometric Properties Implemented in Optimization Analysis . . . . . . . 866.2 Mass and Dimensions of Optimized IsoTruss Structures . . . . . . . . . . . . . . . . . 876.3 Mass and Dimensions of Rackliffe4 Specimen . . . . . . . . . . . . . . . . . . . . . . 87

7.1 Fixed Π Variables of Dimensional Analyses . . . . . . . . . . . . . . . . . . . . . . . 957.2 Fixed Design Parameters of FE Analyses . . . . . . . . . . . . . . . . . . . . . . . . . 967.3 Coefficients relating Π0 to Π1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1007.4 Coefficients relating Π0 to Π2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1017.5 Coefficients relating Π0 to Π3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1027.6 Results of OLC and ILC Multi-modal Optimization Analysis . . . . . . . . . . . . . . 109

B.1 Varying Nomenclature of IsoTruss Structures . . . . . . . . . . . . . . . . . . . . . . 127

viii

LIST OF FIGURES

1.1 Reference coordinate system of IsoTruss structures. . . . . . . . . . . . . . . . . . . . 41.2 Longitudinal (blue) and helical (orange) members. . . . . . . . . . . . . . . . . . . . . 51.3 Helical segments differentiated by color. . . . . . . . . . . . . . . . . . . . . . . . . . 51.4 Longitudinal (blue) and helical (orange) struts. . . . . . . . . . . . . . . . . . . . . . . 61.5 Nodal connectivity: (a) interwoven; and, (b) bonded [2]. . . . . . . . . . . . . . . . . 61.6 IsoTruss nodal terminology [4]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71.7 Pyramidal helical trusses (blue) and planar X-style helical trusses (orange). . . . . . . 81.8 A finite element mesh generated in ANSYS Mechanical. . . . . . . . . . . . . . . . . 81.9 Global buckling as exhibited in FE analysis. . . . . . . . . . . . . . . . . . . . . . . . 101.10 Shell-like buckling as exhibited in FE analysis (side). . . . . . . . . . . . . . . . . . . 111.11 Shell-like buckling as exhibited in FE analysis (rotated side). . . . . . . . . . . . . . . 11

2.1 Wavelength of shell-like buckling mode. . . . . . . . . . . . . . . . . . . . . . . . . . 152.2 Transverse deflection of a longitudinal member during shell-like buckling as calculated

by Eq. 2.1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162.3 Potential planes of shell-like buckling: (a) one of the two planes of the planar X-style

truss; and, (b) the plane tangent to the inner diameter. . . . . . . . . . . . . . . . . . . 19

3.1 Percent deviation vs. FE mesh density. . . . . . . . . . . . . . . . . . . . . . . . . . . 303.2 Potential rotation of the longitudinal strut within each bay (NTS). . . . . . . . . . . . . 363.3 Effective stiffness vs. nominal stiffness of McCune specimens. . . . . . . . . . . . . . 37

4.1 Π0 vs. Π1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 534.2 Π0 vs. Π2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 544.3 Π0 vs. Π3. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 554.4 Π0 vs. Π1 Global and local buckling curves. . . . . . . . . . . . . . . . . . . . . . . . 564.5 Π0 vs. Π2 Global and local buckling curves. . . . . . . . . . . . . . . . . . . . . . . . 574.6 Π0 vs. Π1 Analytical and FE predictions. . . . . . . . . . . . . . . . . . . . . . . . . 584.7 Percent deviation of analytical predictions from FE predictions vs. Π1. . . . . . . . . . 584.8 Π0 vs. Π2 Analytical and FE predictions. . . . . . . . . . . . . . . . . . . . . . . . . 594.9 Percent deviation of analytical predictions from FE predictions vs. Π2. . . . . . . . . . 594.10 Π0 vs. Π3 Analytical and FE predictions. . . . . . . . . . . . . . . . . . . . . . . . . 604.11 Percent deviation of analytical predictions from FE predictions vs. Π3. . . . . . . . . . 60

5.1 Effective length of shell-like buckling (simplified method) [4]. . . . . . . . . . . . . . 725.2 Process of optimization analysis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 745.3 Convergence of gradient-based and gradient-free optimization algorithms. . . . . . . . 76

7.1 Location of longitudinal members in the: a) inner longitudinal configuration (green);and, b) outer longitudinal configuration (blue). . . . . . . . . . . . . . . . . . . . . . . 92

7.2 IsoTruss structure with outer longitudinal members (i.e., OLC). . . . . . . . . . . . . . 937.3 Local buckling of OLC (side view). . . . . . . . . . . . . . . . . . . . . . . . . . . . 967.4 Local buckling of OLC (top view). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 977.5 OLC Π0 vs. Π1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

ix

7.6 OLC Π0 vs. Π2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1017.7 OLC Π0 vs. Π3. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1027.8 OLC Π0 vs. Π1 Analytical and FE predictions. . . . . . . . . . . . . . . . . . . . . . 1037.9 OLC Π0 vs. Π1 Percent deviation of analytical predictions from FE predictions. . . . . 1037.10 OLC Π0 vs. Π2 Analytical and FE predictions. . . . . . . . . . . . . . . . . . . . . . 1047.11 OLC Π0 vs. Π2 Percent deviation of analytical predictions from FE predictions. . . . . 1047.12 OLC Π0 vs. Π3 Analytical and FE predictions. . . . . . . . . . . . . . . . . . . . . . 1057.13 OLC Π0 vs. Π3 Percent deviation of analytical predictions from FE predictions. . . . . 1057.14 Π0 vs. Π1 of OLC and ILC. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1067.15 Π0 vs. Π2 of OLC and ILC. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1077.16 Π0 vs. Π3 of OLC and ILC. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1087.17 Local buckling of Set 2 OLC with two carbon tows in helical members. . . . . . . . . 1127.18 Local buckling of Set 2 OLC with eleven carbon tows in helical members. . . . . . . . 1127.19 Rotation at the nodes of helical members with: a) two carbon tows; and, b) eleven

carbon tows. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112

x

CHAPTER 1. GENERAL OVERVIEW

1.1 Nomenclature

A Cross-sectional area of longitudinal (L) or helical (H) member

Ez Young’s modulus of composite in the z-direction

Ig Moment of inertia of global IsoTruss

IL Moment of inertia of longitudinal member

L Length of global IsoTruss

N Number of nodes

P Actual applied axial load

Pg Buckling load of global buckling mode

Pl Buckling load of longitudinal strut buckling mode

R Outer radius of IsoTruss

c Moment of inertia coefficient [3]

kg Boundary condition factor of global buckling (converts length to effective length)

l1 Short span of longitudinal member between transition nodes

l2 Long span of longitudinal member between transition nodes

α Stiffness factor of longitudinal member in bending

µl Boundary condition factor of longitudinal strut buckling (converts length to effective length)

σact Actual applied axial stress

1

1.2 Introduction

IsoTruss® structures are open-lattice composite structures that have been designed and

manufactured for diverse applications from civil infrastructure to aerospace components. With

a high strength-to-weight ratio, they exhibit excellent resistance to various loading scenarios, in-

cluding tension, compression, torsion, flexure, and shear. To design an IsoTruss structure for a

specific application and loading scenario, the geometric and material properties are reconfigured

multiple times to minimize the mass and maximize the structural efficiency.

Various configurations of IsoTruss structures have been designed in previous research by

manufacturing physical specimens for testing. Experimental methods are reliable in demonstrating

the actual failure modes and behavior of structures, but are expensive and time consuming, only

producing one set of data that is specific to the geometric configuration and parameters of that

specimen. Supplementary research methods have been developed to facilitate more efficient design

processes through numerical analysis, dimensional analysis, and optimization. These techniques

allow novel configurations to be tested quickly and inexpensively, without the need to manufacture

and physically load each specimen.

1.3 Research Objectives

The purpose of this research is to apply numerical, dimensional, and optimization methods

to the preliminary design of 8-node IsoTruss structures subject to uniaxial compression.

First, finite element (FE) methods are validated by modeling configurations that have been

tested experimentally, and verified with analytical expressions derived from traditional mechanics.

Next, FE models are used to gather data for dimensional analysis. The critical design pa-

rameters are reduced to a smaller set of non-dimensional Π variables via Buckingham’s Pi Theorem

(BPT). The data are plotted to characterize interrelations between design parameters and various

failure loads and modes. Trend analyses are performed with the Π variables to verify newly de-

rived analytical expressions with the validated FE predictions. The trend analyses are also used to

assess the performance of novel configurations of IsoTruss structures with the performance of the

experimentally-validated specimens.

2

Finally, the analytical expressions are implemented in optimization studies to select effi-

cient optimization techniques and minimize the mass of traditional and novel configurations. The

gradient-based optimization techniques also facilitate additional analysis of the modality of the

design space and the sensitivity derivatives at the minima. These methods are implemented in this

research to achieve the following objectives:

• Determine the extent to which FE predictions of axial stiffness and buckling loads correlate

with experimental data and traditional mechanics;

• Derive and verify analytical expressions to predict bay-level buckling loads in inner longitu-

dinal configurations (ILC) and outer longitudinal configurations (OLC);

• Characterize the interrelations between ILC and OLC design parameters and the buckling

behavior;

• Select efficient and robust optimization techniques to implement in the preliminary design

process of IsoTruss structures; and,

• Identify the relative advantages and disadvantages of the ILC and OLC with respect to

strength-to-weight efficiency.

The numerical, dimensional, and optimization studies that investigate these topics are pre-

sented in this thesis with individual synopses and conclusions. The general conclusions and rec-

ommendations for further research are summarized at the end of this document.

1.4 Research Limitations

This research focuses on 8-node IsoTruss structures subject to uniaxial compression. The

configurations analyzed herein are based on the extremely light-weight graphite/epoxy specimens

that were manufactured and tested by Rackliffe et al. [2], and are characterized by small-diameter

members and long global lengths. These specimens were designed for light-weight space applica-

tions, and are approximately 10 ft. (3 m) long and 0.3 lb. (0.14 kg).

The failure modes that are considered in this research are based on those typically exhibited

by IsoTruss structures in uniaxial compression. These include material failure, global column

3

buckling, shell-like buckling, and longitudinal strut buckling. While the analyses performed herein

are mainly validated with the experimental data of the Rackliffe et al. specimens, the methods are

transferable to other design spaces that are subject to similar failure criteria.

1.5 IsoTruss Structures

1.5.1 Geometric Configuration

The orientation of IsoTruss structures is referenced from the polar coordinate system, r-θ -

z, shown in Fig. 1.1. The r-axis is perpendicular to the tangent of the outer radius; the θ -axis is

in the plane perpendicular to the z-axis and lies tangent to the outer radius; and, the z-axis runs

parallel with the length of the IsoTruss structure.

Figure 1.1: Reference coordinate system of IsoTruss structures.

For the purpose of this thesis, IsoTruss structures are comprised of structural components

referred to (in decreasing size) as members, segments, struts, and elements. A ‘member’ is a com-

ponent that is continuous from the beginning to the end of the structure. Members are categorized

as longitudinal or helical members, shown in Fig. 1.2. Longitudinal members are straight and

continuous for the entire length of the structure, primarily providing flexural and axial strength.

Helical members are piece-wise linear members that wind around the structure and contribute to

4

the shear resistance and flexural rigidity of the structure. They also provide lateral stability to the

longitudinal members.

Figure 1.2: Longitudinal (blue) and helical (orange) members.

The term ‘segment’ is used to refer to the straight portion of a helical member as shown in

Fig. 1.3.

Figure 1.3: Helical segments differentiated by color.

‘Struts’ are the unconstrained portions of the members that span between joints (see Fig. 1.4).

5

Figure 1.4: Longitudinal (blue) and helical (orange) struts.

‘Strut connectivity’ refers to the boundary conditions, including the degrees of freedom,

at the joints of the struts. The joints are manufactured by interweaving or bonding, depicted in

Fig. 1.5, which is taken from Fig. 3.11 of Rackliffe [5]. Both interweaving and bonding permit the

fibers to be continuous through the joint. Joints that are bonded rather than interwoven are easier to

manufacture, but result in a weaker joint. Interweaving the joints increases the rigidity of the nodal

connectivity, but the fibers can separate as tension loads increase unless the joint is adequately

consolidated. Strut connectivity is not directly analyzed in the scope of this research, however, the

strut connectivity does affect the boundary constraints imposed on local strut buckling, which is

discussed herein.

Figure 1.5: Nodal connectivity: (a) interwoven; and, (b) bonded [2].

6

While the term ‘node’ is used within FE literature to refer to the FE mesh, it is also used

within IsoTruss nomenclature to refer to strut endpoints. There are three types of nodes in an

IsoTruss: nodes, anti-nodes, and transition nodes (see Fig. 1.6). Nodes are the intersections of two

helical members at the outer diameter of the structure; anti-nodes are the intersections of helical

members at the inner diameter; and, transition nodes are the intersections of helical members with


Figure 1.6: IsoTruss nodal terminology [4].

Sui et al. [6] introduced a nomenclature for the truss-like configurations formed by helical

struts that intersect at nodes and anti-nodes. The truss-like configuration that is formed by the heli-

cal struts that intersect at the nodes is referred to as a pyramidal truss. The truss-like configuration

that is formed by helical struts that intersect at the anti-nodes is referred to as a planar X-style truss.

Pyramidal and planar X-style helical trusses are depicted in Fig. 1.7.

7

Figure 1.7: Pyramidal helical trusses (blue) and planar X-style helical trusses (orange).

‘Elements’ are the fundamental units of an FE mesh, and discretize the global structure for

numerical analysis (see Fig. 1.8).

Figure 1.8: A finite element mesh generated in ANSYS Mechanical.

1.5.2 Failure Modes

The loading criteria of this research is limited to uniaxial compression. Typical failure

modes include material failure, global buckling, shell-like buckling, and local strut buckling of the

longitudinal members. When the overall structure is loaded in uniaxial compression, the helical

members do not typically buckle, but rather behave as a spring. Another potential failure mode

is radial crushing of the cross-section due to excessive bending of the struts induced by locally

8

applied transverse loads. Radial crushing is based on an empirical formula for short spans and is

not implemented in the current study.

Material Failure

Material failure occurs in the longitudinal members as the applied stress surpasses the ul-

timate strength of the composite material. The applied load is distributed among the longitudinal

members such that the effective cross-sectional area is the cumulative cross-sectional area of the

longitudinal members. Equation 1.1 is used herein to calculate the actual compression stress caused

by the axial load, P.

σact =P

N ·AL(1.1)

The ultimate strength of the composite material is calculated from the micromechanics

equations provided in Appendix A from Kollar and Springer [7]. The ultimate strength of the

composite is calculated from the strength of the fiber, the strength of the matrix, and the fiber

volume ratio.

Global Buckling

Global buckling typically governs failure when the bay length is short enough to restrict

bay-level buckling. It is approximated as Euler-column buckling. The longitudinal and helical

members are fixed at the base of the overall structure, restricting rotation and translation of the

nodes. The compression load is applied as an axial follower load at the free end. Fig. 1.9 demon-

strates global buckling as exhibited in an FE analysis performed in ANSYS WorkBench.

9

Figure 1.9: Global buckling as exhibited in FE analysis.

Equation 1.2 is used to predict the global buckling load. The inertia coefficient shown

in Eq. 1.2, c, is derived by Winkel [3] for 8-node IsoTruss structures with inner longitudinal

members. The derivation uses the parallel axis theorem and only includes the contribution from

the longitudinal members.

Pg =π2 ·Ez · Ig

(kg ·L)2

kg =2.0

Ig = c ·AL ·R2

c =2.343

(1.2)

Shell-like Buckling

Shell-like buckling is exhibited in the longitudinal members and has a wavelength of ap-

proximately two bay lengths, beginning and ending at anti-nodes. The helical segments that in-

tersect the longitudinal member within the buckling span contribute to the flexural rigidity. Six-

node IsoTruss structures are inherently stable from shell-like buckling due to triangular geometry.

10

IsoTruss structures with more than six nodes that have inner longitudinal members, however, be-

come susceptible to shell-like buckling as the outer diameter increasingly resembles a cylindrical

shell. Figures 1.10 and 1.11 demonstrate shell-like buckling as exhibited in an FE analysis per-

formed in ANSYS WorkBench with a mesh density of 5.08 in.−1 (200 m−1). Fig. 1.11 is the same

buckling mode configuration as Fig. 1.10, rotated 90° about the longitudinal axis to demonstrate

the 3-dimensional deformation. These structures are shorter than those analyzed in subsequent

chapters in order to show the buckling mode more clearly. The longer structures exhibit the same

buckling shape, with the end constraints preventing shell-like buckling in the outer-most bays.

Figure 1.10: Shell-like buckling as exhibited in FE analysis (side).

Figure 1.11: Shell-like buckling as exhibited in FE analysis (rotated side).

11

Sui et al. derived an analytical expression to predict shell-like buckling. The expression

calculates the critical buckling load of the longitudinal members, which are constrained by the

planar X-style helical trusses. The expression omits the rigidity that is contributed by the pyramidal

helical trusses. Chapter 2 documents the derivation of the shell-like buckling load as outlined by

Sui et al., with slight adjustments to the final expression that are explained therein.

Longitudinal Strut Buckling

Longitudinal strut buckling occurs on the unbraced lengths of the longitudinal struts. It

governs failure when the helical members are robust enough to constrain against shell-like buck-

ling, and the bay length is too long for global buckling to govern. The lengths of the longitudinal

struts are defined by the spans between transition nodes. These spans are referred to as l1 and

l2 (see Kesler [4]). The term l1 refers to the short span of longitudinal members between transi-

tion nodes, whereas l2 refers to the longer span between transition nodes. The unbraced length of

longitudinal strut buckling is defined by l2.

Longitudinal strut buckling is predicted with the Euler-buckling equation provided in Eq.

1.3. The boundary constraint conditions are defined by the strut connectivity at the transition

nodes. A boundary condition factor, µl , is presented by Rackliffe [5] and provided below. Sui et

al. refer to longitudinal strut buckling as mono-cell buckling and approximate the flexural rigidity

of the strut connectivity with the term, α , shown in Eq. 1.3.

Pl =N · π2 ·Ez · IL

(µl · l2)2

µl =

(1+

α · l22 ·Ez · IL

)·(

1+α · l2Ez · IL

)−1

α =4 ·Ez · IL

l1

(1.3)

12

CHAPTER 2. DERIVATION OF THE SHELL-LIKE BUCKLING LOAD OF 8-NODEISOTRUSS® STRUCTURES

2.1 Synopsis

The purpose of the current chapter is to document the derivation of the critical buckling load

of the shell-like buckling mode of 8-node IsoTruss® structures. The derivation is based on energy

methods to determine the least energy state of a longitudinal member, constrained by intersecting

helical struts, subject to uniaxial compression. The wavelength of the shell-like buckling mode

is approximately two bays, therefore, the derivation is limited to a segment of the longitudinal

member that is two bay lengths. The derivation is based on structural mechanics presented by

Megson [8] and the methods outlined by Sui et al. [6], with slight deviations that are documented

herein. The derivation of the shell-like buckling load is implemented in subsequent studies as an

analytical prediction of the critical buckling load. The analytical predictions are compared with

finite element predictions and employed in optimization routines to minimize the mass of IsoTruss

structures in uniaxial compression.


D Flexural rigidity (typical)

DX Equivalent plate flexural rigidity of planar X-style helical trusses


Estrain Strain energy of the two-bay buckled region

EX Equivalent plate stiffness of planar X-style helical trusses in the z-direction

I Moment of inertia of longitudinal (L) or helical (H) member

M Bending moment

N Number of nodes


Psb Buckling load of shell-like buckling mode

13

Si Area of a representative unit cell occupied by the ith strut

U Strain energy (typical)

b Bay length (one repeating unit consisting of a planar X-section truss and a pyramid)



r Radius of longitudinal (L) or helical (H) member

t Plate thickness (typical)

w Transverse deflection

z Coordinate reference axis that is parallel with the global length of IsoTruss

αi Directional angle of the ith strut

δ2 One-half the base length of the planar X-style helical struts

φ Inclination angle of helical members from horizontal

µsb Boundary constraint coefficient of shell-like buckling (converts length to effective length)

ν Poisson’s ratio

θ Configuration reference angle(360°

N

)

2.2 Introduction and Background

Shell-like buckling is a typical failure mode of IsoTruss structures, with greater than six

nodes, subject to compression loading. Rackliffe et al. [2] observed shell-like buckling in 8-node

IsoTruss structures with bay lengths greater than 2.5 in. (64 mm) that were tested experimentally

and supplemented with finite element (FE) predictions. Shell-like buckling is referred to therein as

local buckling. Sui et al. [6] performed a numerical analysis of the shell-like buckling of IsoTruss

structures using FE methods. They derived an analytical expression for shell-like buckling via

energy methods. The purpose of the current study was to derive an analytical expression to predict

the shell-like buckling load of IsoTruss structures. Traditional mechanics provided in Megson [8]

were used to derive an energy equation that describes the net energy of the buckled system. The

derivation is based on the process and methods outlined by Sui et al., with minor adjustments.

14

The shell-like buckling load presented by Sui et al. is derived by first, writing an expression

to approximate the net energy of the buckled system; second, taking the derivative of the energy

equation with respect to the initial deflection; and third, setting the derivative equal to zero (thereby

minimizing the total potential energy of the system) and solving for the critical buckling load. The

buckling mode has a wavelength of two bays, therefore, the buckled system consists of one longi-

tudinal member of length 2 · b, including the helical struts that intersect the longitudinal member.

Within the two-bay span, both planar X-style helical trusses and pyramidal style helical trusses

intersect and constrain the longitudinal member. Nevertheless, the derivations presented in Sui et

al. and in the current study neglect the energy contributed by the pyramidal style helical trusses.

The resulting expression for the shell-like buckling load represents the load required to buckle N

longitudinal members of length 2 · b, constrained at the transition nodes by planar X-style helical

trusses.

2.3 Methods

Fig. 2.1 shows a free-body diagram of the structural components that are included in the

buckled system. The longitudinal segment of interest is parallel with the z-axis. The helical struts

that make-up the pyramidal truss are shown with dashed lines, and are not included in the deriva-

tion. Conversely, the helical struts that make-up the planar X-style trusses are shown with solid

lines, diverging from the longitudinal member. The buckled energy of these struts is included in

the derivation by approximating the inclined struts as plates in bending.

Figure 2.1: Wavelength of shell-like buckling mode.

15

2.3.1 Strain Energy Derivation

Sui et al. start with Eq. 2.1 to approximate the deflection of the longitudinal member

along the z-axis, imposed by the bending moment, Mz, about the θ -axis. Equation 2.1 is plotted in

Fig. 2.2, using the reference values 1.0 in. (25 mm) and 2.5 in. (64 mm) for the initial transverse

deflection, w0, and bay length, b, respectively. The plot shows the shape of a longitudinal member

that is deflected under shell-like buckling. The same shape can be observed in the FE depiction of

shell-like buckling provided in Fig. 1.10.

w(z) = w0 · cos(

π · zb

)(2.1)

Figure 2.2: Transverse deflection of a longitudinal member during shell-like buckling as calculatedby Eq. 2.1.

The strain energy of the buckled system (the system that takes the shape of the curve shown

in Fig. 2.1) is approximated in Eq. 2.2 as the summation of three terms (see also Eq. 16 [6]). Term

1 represents the energy of the longitudinal segment in bending; Term 2 approximates the energy

of the planar X-style helical trusses; and Term 3 estimates the residual energy of the post-buckled

16

longitudinal segment. The following sections outline the derivation of each term of the energy

equation.

Estrain =12

∫ 2·b

0Ez · IL ·

(d2 w(z)

d z2

)2

dz Term 1

+4∫ l1

2

0

12· (2 ·δ2) ·DX ·

(d2 w(z)

d z2

)2

dz Term 2

− 12· P

N

∫ 2·b

0

(d w(z)

d z

)2

dz Term 3

(2.2)

Strain Energy Term 1: Longitudinal Segment

The first term in Eq. 2.2 can be derived from the general equation for the bending energy

of a beam of wavelength a, shown in Eq. 2.3.

U =12

∫ a

0

M(z)2

E · Idz (2.3)

Substitute 2 · b for the wavelength of the beam and M(z) = Ez · IL · d2 w(z)d z2 for the bending

moment to derive the bending energy of the longitudinal segment. The resulting expression is

provided below in Eq. 2.4.

U =12

∫ 2·b

0Ez · IL ·

(d2 w(z)

d z2

)2

dz (2.4)

Strain Energy Term 2: Intersecting Helical Struts

The second term in Eq. 2.2 can be derived by approximating the planar X-style helical

struts as plates in bending. Equation 2.5 is the general equation for the bending strain energy of a

plate with dimensions δx by δy imposed by Mz (see Section 7.6.1 of Megson [8]). For clarity, the

expression has been transposed from the x-y-z coordinate system to the r-θ -z coordinate system

used herein.

17

U =12·Mz ·δθ ·

(−d2 w(z)

d z2 ·δ z)

(2.5)

The general equation for the bending moment of a plate, Mz, in terms of curvature and

flexural rigidity is shown in Eq. 2.6.

Mz =−D ·(

d2 w(z)d z2 +ν · d2 w(z)

d θ 2

)(2.6)

Substituting Eq. 2.6 into Eq. 2.5, the equation for the energy of the intersecting helical

segments is approximated in Eq. 2.7.

U =12·(−D ·

(d2 w(z)

d z2 +ν · d2 w(z)d θ 2

))·δθ ·

(−d2 w(z)

d z2 ·δ z)

(2.7)

Equation 2.7 can be simplified by assuming that the bending moment, Mz, does not incur

curvature with respect to the θ -axis. In addition, the dimensions of the plate segment of interest,

δθ and δ z, can be replaced with the dimensions 2 ·δ2 and the integral of w(z) from zero to l12 (see

Fig. 2.1). Next, the equation is multiplied by 4 to account for the helical segments in the two-bay

shell-buckling region. The simplified version of Eq. 2.7 (i.e., the energy of the helical struts) is

shown in Eq. 2.8.

U = 4∫ l1

2

0

12· (2 ·δz) ·DX ·

(d2 w(z)

d z2

)2

·dz (2.8)

Next, the general plate flexural rigidity, D, is replaced with the effective plate flexural

rigidity of the planar X-style truss, DX . Two methods are explored to approximate the effective

rigidity of the plate. The first method assumes that shell-like buckling occurs across the plane of

the X-style truss (see Fig. 2.3 (a)). In which case, the equivalent plate thickness is the diameter

of the helical segments, 2 · rH . The second method assumes that shell-like buckling occurs on the

plane tangent to the inner diameter of the IsoTruss structure (see Fig. 2.3 (b)). In this scenario, the

helical segments are inclined at an angle, θ/2, from the plane of buckling. Therefore, the thickness

of the plate in bending is the height of the triangle formed by the inclined helical segments (i.e.,

δ2 · sin(θ/2)). The equivalent plate flexural rigidity of both methods are shown with respect to the

18

general equation for flexural rigidity in Eq. 2.9. In this study, it is assumed that shell-like buckling

occurs about the plane of the X-style truss, thus, the flexural rigidity is calculated via Eq. 2.9 (a).

Figure 2.3: Potential planes of shell-like buckling: (a) one of the two planes of the planar X-styletruss; and, (b) the plane tangent to the inner diameter.

D =E · t3

12 · (1−ν2)(general)

DX =EX · (2 · rH)

3

12 · (1−ν2)(a)

DX =1

cos(

θ

2

) · EX ·(δ2 · sin

(θ

2

))3

12 · (1−ν2)(b)

(2.9)

The effective stiffness of the planar X-style truss, EX , is based on the derivation of the

effective stiffness of a lattice composite regarded as a continuum material, derived by Kollar et al.

[9]. This derivation is for a single-layer grid with equidistant bar rows, and provided for reference

in Eq. 2.10.

19

E =

∑ri cos4 αi ∑ri cos2 αi sin2

αi ∑ri cos3 αi sinαi

∑ri sin4αi ∑ri cosαi sin3

αi

∑ri cos2 αi sin2αi

(2.10)

Equation 2.10 is adjusted by Fan et al. [10] for composite structures composed of periodic

lattice cells. Therein, ri is defined as Eq. 2.11 such that Si is the area of a representative unit cell

occupied by the ith strut.

ri =Ez ·AH

Si(2.11)

Assuming that the helical struts buckle about the plane that is parallel to the planar X-style

truss, the area of the representative unit cell, Si, is calculated by Eq. 2.12.

Si = (2 ·δ2) · (2 · rH) (2.12)

The equivalent stiffness of the planar X-style truss, EX , is approximated from the first term

of Eq. 2.10. The cosαi represents the directional cosine of the helical strut with respect to the

z-axis. Thus, the cosine of the angle α is equivalent to the sine of the helical inclination angle,

φ , such that the equivalent stiffness is calculated via Eq. 2.13. The stiffness is multiplied by 2 to

account for both of the helical struts that make-up the planar X-style truss.

EX = 2 · Ez ·AH

(2 ·δ2) · (2 · rH)· sin4(φ) (2.13)

Strain Energy Term 3: Post-buckled Longitudinal Segment

The third term of Eq. 2.2 represents the residual energy of the longitudinal segment in its

post-buckled configuration. The post-buckling energy of the longitudinal member comes from the

in-plane axial force, PN . The energy can be calculated by multiplying the in-plane axial force by the

corresponding axial deformation. The axial deformation caused by the force is derived in Section

20

7.6.3 of Megson from the binomial expansion of the element-wise deformation, λ , of a beam of

length a in bending, truncated after the first term (see Eq. 2.14).

λ =12

∫ a

0

(dw(x)

dx

)2

dx (2.14)

The general form of the axial deformation shown in Eq. 2.14 is converted to the r-θ -z

coordinate system and multiplied by the axial force to get the energy of the post-buckled system in

Eq. 2.15.

Vz =−12· P

N

∫ 2·b

0

(dw(z)

dz

)2

dz (2.15)

2.3.2 Critical Shell-like Buckling Load

The critical shell-like buckling load is derived from Eq. 2.2 by taking the derivative of the

function with respect to the initial deflection, w0, and setting the derivative equal to zero, thereby

minimizing the total potential energy of the system. The resulting expression is shown in Eq. 2.16.

Psb =1b3 ·2 ·π

2 ·DX · l1 ·δ2 ·N +π2 ·Ez · IL ·N ·b+2 ·π ·DX ·δ2 ·N ·b · sin

(π · l1

b

)(2.16)

Equation 2.16 can be reformatted to resemble the traditional Euler-buckling equation by

introducing the boundary constraint coefficient, µsb. The final expression for shell-like buckling

is shown in Eq. 2.17. The expression is equivalent to that of Sui et al. with the exception of the

equivalent planar X-style truss stiffness, EX . The stiffness derived by Sui et al. has a sin3(φ) rather

than the sin4(φ) shown in Eq. 2.17.

21

Psb =N ·π2 ·Ez · IL

(µsb ·b)2

1µ2

sb=1+

2 ·DX · l1 ·δ2

b ·Ez · IL+

2 ·DX ·δ2 · sin(

π·l1b

)π ·Ez · IL

DX =EX · (2 · rH)

3

12 · (1−ν2)

EX =2 · Ez ·AH

(2 ·δ2) · (2 · rH)· sin4(φ)

(2.17)

2.4 Summary

2.4.1 Conclusion

The purpose of this chapter was to provide an overview of the derivation of the critical

buckling load of the shell-like buckling mode of 8-node IsoTruss structures. The derivation is

based on energy methods to determine the least energy state of a longitudinal member, constrained

by intersecting helical struts, in uniaxial compression. The wavelength of the shell-like buckling

mode is approximately two bays, therefore the derivation is limited to a segment of the longitudinal

member that is two bay lengths. The derivation is based on the methods provided by Megson [8]

and Sui, et al. [6]. The final expression is equivalent to that derived by Sui et al. with the exception

of the equivalent stiffness of the planar X-style truss that is approximated as a plate in bending.

2.4.2 Implications for Further Investigation

The derivation of the shell-like buckling load is implemented in subsequent studies as an

analytical prediction of the critical buckling load. Chapter 3 implements the analytical expression

to predict the critical buckling load of IsoTruss structures tested in experimental studies. The devi-

ations of the prediction from experimental results and FE methods are presented therein. Chapter 4

performs a dimensional study of shell-like buckling and includes a broad analysis of the deviation

of the analytical predictions from FE predictions. The analytical expression is used in Chapter 6 to

minimize the mass of IsoTruss structures in uniaxial compression.

22

CHAPTER 3. VALIDATION OF A FINITE ELEMENT MODEL IN ANSYS WORK-BENCH FOR ISOTRUSS® STRUCTURES IN UNIAXIAL COMPRESSION

3.1 Synopsis

The purpose of this study was to develop, verify, and validate a finite element (FE) model

of IsoTruss® structures subject to uniaxial compression. The FE model was developed in ANSYS

WorkBench and verified using axial stiffness and critical buckling loads that were calculated from

traditional mechanics. The buckling failure modes include global buckling and local/shell-like

buckling. The predictions from the FE model and mechanics calculations are validated by experi-

mental data generated in preceding studies. Results indicate that the effective material modulus is

highly affected by variations in manufacturing. The FE models accurately predict the axial stiff-

ness and critical buckling loads demonstrated in experimental testing when the nominal material

properties are adjusted to account for: imperfections in manufacturing; imperfections in material

properties; and/or, the limitations of micromechanics equations. The FE model developed in this

study is used in subsequent chapters to perform dimensional analyses and analyze novel configu-

rations that have not been manufactured or tested experimentally.

3.2 Nomenclature


Atow Cross-sectional area of carbon fiber tow

D Outer diameter of IsoTruss

DX Equivalent plate flexural rigidity of planar X-style helical trusses

Ez Young’s modulus of composite in z-direction

E Effective buckling (b) or stiffness (s) modulus

Et Young’s modulus of constituent in tension


23

G Shear modulus

Ig Moment of inertia of global IsoTruss structure

Kz Axial stiffness of structure in the z-direction


N Number of nodes

Nb Number of bays

Nt Number of fiber tows per longitudinal (L) or helical (H) member

P Buckling load of global (g) or shell-like (sb) buckling mode




k Boundary condition factor (converts length to effective length)



v f Fiber volume fraction

δ2 One-half the base length of the planar X-style helical members

δmax Maximum axial deflection of IsoTruss structure


γ Localized rotation of longitudinal strut within a bay

µsb Boundary constraint coefficient for shell-like buckling (converts length to effective length)


ρ Material density


N

)

3.3 Introduction

Finite element (FE) analysis is a highly robust, numerical method to assess the structural

performance and capacity of untested configurations of IsoTruss® structures. Before implementing

FE analysis in the design process, however, the FE model must be validated with experimental data

24

and verified with traditional mechanics to ensure accurate results. The purpose of the current study

is to develop an FE model, validate it with experimental data gathered in preceding studies, and

verify the results with traditional mechanics calculations.

The current study is limited to IsoTruss structures subject to uniaxial compression. Typical

failure modes include material failure, global buckling, and local/shell-like buckling. The FE

model is validated with experimental data including the axial deflection and critical buckling loads

of the specimens that were manufactured and tested by McCune [1] and Rackliffe et al. [2].

3.4 Background

Rackliffe et al. and McCune constructed carbon/epoxy IsoTruss structures to test under

uniaxial compression. Both studies manufactured the specimens by consolidating the members at

the nodes with spiral-wrapping. Rackliffe et al. tested four long IsoTruss structures with 8-nodes to

determine the critical buckling loads, whereas McCune analyzed both 6- and 8-node configurations

that were shorter and failed by crushing. McCune and Rackliffe et al. present the total deformation

and ultimate load of each specimen. The specimens tested by Rackliffe et al. failed in local/shell-

like buckling and global buckling, thereby providing critical buckling loads to validate eigenvalue

buckling models. The data of McCune and Rackliffe et al. are used in the current study to validate

the FE model with experimental results.

Sui et al. [6] developed a FE model of IsoTruss structures with Patran Command Language

that is verified with theoretical equations derived from traditional mechanics. The IsoTruss struc-

tures were modeled with 8-nodes to correspond with the experiments performed by Rackliffe et

al. The model incorporates four potential failure modes previously observed in experiments and

FE models: fracture of the struts, global buckling, shell-like buckling, and mono-cell buckling.

Mono-cell buckling is the buckling of longitudinal members at the bay-level and is referred to

herein as longitudinal strut buckling. In Chapter 2, the shell-like buckling equation of Sui et al. is

re-derived from energy methods. The resulting expression is implemented in the current chapter

to compare the analytical predictions with the FE predictions. Global buckling and axial deflec-

tion are predicted analytically with the traditional equations for column buckling and stiffness,

respectively.

25

3.5 Methods

This section outlines the procedure for developing, verifying, and validating the FE model.

The nominal material properties of the composite were calculated from the fiber and matrix prop-

erties documented by Rackliffe et al. and McCune. The geometry of each structure was drafted in

ANSYS SpaceClaim. Static structural analyses were performed on each unique configuration to

predict the axial stiffness. The results are verified with mechanics calculations and validated with

the axial stiffness reported from the physical testing. The nominal modulus of the composite is

reduced to an effective stiffness modulus to correlate the model with the experimental results.

Once the axial stiffness predictions from ANSYS have been verified with mechanics and

correlated with the experimental data, eigenvalue buckling (EB) analyses were performed for each

of the specimens tested by Rackliffe et al. The critical buckling loads predicted by the FE model are

verified with the analytical predictions calculated using the shell-like buckling equation presented

in Chapter 2 and the Euler-buckling equation for column buckling. The nominal material modulus

is adjusted to an effective buckling material modulus and re-analyzed to correlate the model with

the experimental results.

The specimens that were tested by Rackliffe et al. are referred to throughout this chapter

as Rackliffe1, Rackliffe2, Rackliffe3, and Rackliffe4. They correspond to the specimens referred

to by Rackliffe et al. as IsoTruss1, IsoTruss2, IsoTruss3, and IsoTruss4, respectively. Likewise,

the specimens that were manufactured by McCune are referred to in this paper as McCune8, Mc-

Cune12, McCune16, and McCune20. They correspond to the specimens manufactured by McCune

with 8, 12, 16, and 20 carbon fiber tows in the longitudinal members.

3.5.1 Composite Material Properties

Tables 3.1 and 3.2 present the material properties of the carbon fiber and resin used by

McCune and Rackliffe et al., respectively, to manufacture the physical specimens for experimental

testing. Poisson’s ratio was not documented in the publications, therefore, the values shown below

were selected based on the material properties of similar materials.

26

Table 3.1: Material Properties of Fiber and Resin (McCune [1])

Carbon Fiber ResinMaterial Name Thornel 12K T300C EPON 826

Et [ksi (GPa)] 33000 (228) 400 (2.76)

ν 0.32 0.33

ρ

[lb.in.3

(kgm3

)]0.063 (1750) 0.042 (1162)

Table 3.2: Material Properties of Fiber and Resin (Rackliffe et al. [2])

Carbon Fiber ResinMaterial Name IM7 6K UF3325-95

Et [ksi (GPa)] 40000 (276) 410 (2.83)

ν 0.32 0.33

ρ

[lb.in.3

(kgm3

)]0.064 (1780) 0.044 (1207)

Atow[in.2

(mm2)] 1.94e-04 (0.125) -

The material properties of the fiber and resin provided in Tables 3.1 and 3.2 were used to

calculate the orthotropic material properties of the composite using the micromechanics equations

presented by Kollar and Springer [7] and provided for reference in Appendix A. Since McCune and

Rackliffe et al. only provide the tensile modulus of the carbon fiber, the compression modulus was

approximated by applying the compressive-to-tensile stiffness ratio of a HexTow IM7 composite

to the fiber tensile modulus. The resulting material properties of the composites are provided in

Table 3.3.

3.5.2 Geometry

Table 3.4 presents the geometric properties of the experimental specimens prepared by

McCune and Rackliffe et al. that are used to draft the model geometry in ANSYS SpaceClaim.

The radii of the McCune specimens are based on the axial area and the fiber volume fractions

reported by McCune.

27

Table 3.3: Nominal Properties of Composite Materials

Rackliffe McCune8 McCune12 McCune16 McCune20v f 0.65 0.44 0.42 0.44 0.43

ρ

[lb.in.3

(kgm3

)]0.049 (1360) 0.0517 (1431) 0.0521 (1443) 0.0517 (1431) 0.0519 (1437)

Ez [ksi (GPa)] 23300 (161) 13200 (90.7) 12600 (86.7) 13200 (90.7) 12900 (88.7)

Eθ [ksi (GPa)] 966 (6.66) 645 (4.45) 628 (4.33) 645 (4.45) 636 (4.39)

Er [ksi (GPa)] 966 (6.66) 645 (4.45) 628 (4.33) 645 (4.45) 636 (4.39)

νzθ 0.32 0.33 0.33 0.33 0.33

νzr 0.32 0.33 0.33 0.33 0.33

νθr 0.33 0.33 0.33 0.33 0.33

Gzθ [ksi (GPa)] 432 (2.98) 266 (1.83) 257 (1.77) 266 (1.83) 261 (1.80)

Gzr [ksi (GPa)] 432 (2.98) 266 (1.83) 257 (1.77) 266 (1.83) 261 (1.80)

Gθr [ksi (GPa)] 364 (2.51) 243 (1.67) 236 (1.63) 243 (1.67) 239 (1.65)

Table 3.4: Geometric Parameters of Experimental Specimens

L NNNtL NNNtH rrrLLL rrrHHH NNNbbb b DSpecimen [in. (m)] [in. (mm)] [in. (mm)] [in. (cm)] [in. (cm)]Rackliffe1 104 (2.64) 9 4 0.029 (0.743) 0.019 (0.495) 24 4.33 (11.0) 5.9 (15.0)

Rackliffe2 116 (2.94) 11 3 0.032 (0.821) 0.017 (0.429) 27 4.33 (11.0) 5.9 (15.0)

Rackliffe3 110 (2.80) 11 3 0.032 (0.821) 0.017 (0.429) 44 2.50 (6.35) 5.9 (15.0)

Rackliffe4 114 (2.89) 11 3 0.032 (0.821) 0.017 (0.429) 53 2.17 (5.51) 5.9 (15.0)

McCune8 14.1 (0.358) 8 4 0.064 (1.62) 0.045 (1.14) 5 2.84 (7.21) 5.0 (12.7)

McCune12 14.1 (0.358) 12 4 0.080 (2.03) 0.046 (1.17) 5 2.84 (7.21) 5.0 (12.7)

McCune16 14.1 (0.358) 16 4 0.090 (2.29) 0.045 (1.14) 5 2.84 (7.21) 5.0 (12.7)

McCune20 14.1 (0.358) 20 4 0.102 (2.58) 0.046 (1.17) 5 2.84 (7.21) 5.0 (12.7)

3.5.3 Finite Element Modeling

FE analyses were developed in the programs IsoTruss.exe and ANSYS WorkBench (ver-

sions 19.1 and R2). The WorkBench platform employs ANSYS SpaceClaim to define the struc-

tures’ geometry and ANSYS Mechanical to generate the mesh, define constraints and load criteria,

and solve the mathematical model. Static structural analyses were performed to predict the axial

stiffness of the specimens tested by Rackliffe et al. and McCune. Eigenvalue buckling (EB) anal-

28

yses were performed on the specimens tested by Rackliffe et al. to predict the critical buckling

loads and modes.

Pre-processing

The pre-processing of the FE model includes four main components, summarized herein:

geometry, mesh generation, boundary conditions, load criteria, and analysis type. The geometry

of the IsoTruss structure is generated in the application Isotruss.exe. This program uses the inputs

of bay length, number of bays, overall length, and number of nodes to create a wireframe structure

based on the geometric patterns of IsoTruss structures. The wireframe is exported as a *.dxf

CAD file and uploaded into ANSYS SpaceClaim (ASC). Within ASC, the wireframe geometry is

converted into beam elements with the corresponding cross-sectional dimensions (see Table 3.4).

The connectivity of the individual strut components is merged by defining the component topology

as ‘shared’.

Using the appropriately-developed geometry, the model is opened in ANSYS Mechanical

where the FE mesh is generated and boundary conditions are defined. Figure 3.1 provides data

from a mesh refinement study, demonstrating how the critical buckling load predicted by the model

is affected by increasing the coarseness of the FE mesh. The percent deviation of the critical

buckling load was calculated relative to a standard mesh density of 5.08 in.−1 (200 m−1). A coarse

mesh of 0.25 in.−1 (10 m−1) was used in the current study to reduce computation time and expense.

The refinement study suggests that the coarse mesh can demonstrate approximately 2.7% deviation

from the standard, fine mesh.

The final step of pre-processing is to define boundary and loading conditions. The boundary

conditions of the global truss are defined as fixed-free by fixing the nodes at one end (both the

longitudinal and helical members). The uniaxial load is defined as a remote force, applied to the

nodes of the longitudinal members on the opposite end of the structure. The remote force acts as a

rigid load, distributed uniformly to the members. The remote force allows the end of the structure

to deflect laterally during global buckling, but constrains the nodes as if the load were applied on

a top plate.

29

Figure 3.1: Percent deviation vs. FE mesh density.

Static Structural Analysis

After pre-processing, a static structural analysis is performed in ANSYS Mechanical to

evaluate the axial stiffness. The loading is applied in ten steps, and the displacement is plotted

against the force reaction to produce a load-displacement curve. The slope of this curve is ex-

tracted as the FE axial stiffness to be verified with mechanics calculations and validated with the

experimental results.

Eigenvalue Buckling Analysis

The settings of the eigenvalue buckling model are programmed to derive the lowest five

buckling modes and corresponding load multipliers. The critical buckling load is calculated from

the load multiplier and the ultimate load applied in the analysis. The magnitudes of the applied

loads are defined in the model to match the ultimate loads documented in the experimental results

of Rackliffe et al. (see Table 3.6). The pre-stress environment of the eigenvalue buckling model is

defined by the preceding static structural analysis.

30

3.5.4 Verification of FE Model using Mechanics

The axial stiffness, Kz, and critical buckling loads that are predicted by the FE models are

verified using analytical expressions. Equation 3.1 predicts the axial stiffness from the modulus in

the principal direction, Ez, the overall length of the IsoTruss structure, and the cumulative cross-

sectional area of the longitudinal members.

Kz =Ez · (N ·AL)

L(3.1)

Global buckling is predicted analytically with the Euler-buckling equation for column

buckling provided in Eq. 3.2. The boundary conditions of the column are defined as fixed-free,

therefore, the boundary constraint coefficient, kg, is 2.0.

Pg =π2 ·Ez · Ig

(kg ·L)2

kg =2.0

(3.2)

The global moment of inertia can be calculated using two methods. The first method is

presented by Winkel [3], and uses Eq. 3.3 with moment of inertia coefficients derived for IsoTruss

structures (see Table 3.5). The second method is presented by Sui et al., and uses Eq. 3.4. The

methods are compared in Appendix B for 8-node IsoTruss structures and produce the same result.

The method presented by Winkel is implemented in the calculations presented herein.

Ig = c ·AL ·R2 (3.3)

Table 3.5: Moment of Inertia Coefficients for IsoTruss Structures [3]

Nodes 6 7 8 9 10 11 12Inner Longis 1 1.676 2.343 2.990 3.618 4.228 4.823

Outer Longis 3 3.5 4 4.5 5 5.5 6

31

Ig =4 ·π · r2L · (d2

1 +d22)

d1 =R · sin(θ)

d2 =R · cos(θ) · tan(0.5 ·θ)(3.4)

The analytical expression used to predict local/shell-like buckling is based on the derivation

presented in Chapter 2. Shell-like buckling has a wavelength of two bays, and is localized to a

single longitudinal segment that is intersected by helical struts within the buckling wavelength.

The critical shell-like buckling load is calculated from the analytical expression provided in Eq.

3.5 for reference.

Psb =N ·π2 ·Ez · IL

(µsb ·b)2

1µ2

sb=1+

2 ·DX · l1 ·δ2

b ·Ez · IL+

2 ·DX ·δ2 · sin(

π·l1b

)π ·Ez · IL

DX =EX · (2 · rH)

3

12 · (1−ν2)

EX =2 · Ez ·AH

(2 ·δ2) · (2 · rH)· sin4(φ)

(3.5)

3.5.5 Validation of FE Model using Experimental Data

Table 3.6 summarizes the experimental data that is used to validate the FE model. The

effective stiffness moduli, Es, of the Rackliffe specimens were taken from the slopes of the stress-

strain curves generated from the experimental test data. The effective stiffness moduli of the Mc-

Cune specimens were calculated from the axial area, length, and average stiffness, K, reported by

McCune in Table 4.5 [1]. The average modulus of elasticity reported by McCune in Table 5.4 [1]

are less than 1% different than those shown in Table 3.6, except for the McCune16 specimen. The

effective stiffness modulus reported in Table 3.6 for McCune16 is -5.6% different relative to the

32

average modulus reported by McCune. The ‘Axial Area’ is the cumulative cross-sectional area

of the eight longitudinal members (i.e., N ·AL). The cross-sectional area of a single longitudinal

member is calculated from the number of tows per longitudinal member (see Table 3.4) and the

nominal fiber volume fraction (see Table 3.3).

Table 3.6: Previous Experimental Results [1] [2]

EEEsss Axial Area KKKzzz PPPult FailureSpecimen [ksi (GPa)] [in.2 (mm2)] [kip/in. (kN/mm)] [lb. (kN)] ModeRackliffe1 4610 (31.8) 0.0215 (13.9) 0.955 (0.167) 126 (0.560) Local Buckling

Rackliffe2 8960 (61.8) 0.0263 (16.9) 2.03 (0.356) 135 (0.602) Local Buckling

Rackliffe3 6420 (44.2) 0.0263 (16.9) 1.53 (0.268) 266 (1.18) Global Buckling

Rackliffe4 5810 (40.1) 0.0263 (16.9) 1.34 (0.235) 251 (1.12) Global Buckling

McCune8 8820 (60.8) 0.102 (65.8) 63.8 (11.2) 3600 (16.0) Crushing

McCune12 7790 (53.7) 0.160 (103) 88.4 (15.5) 5380 (23.9) Crushing

McCune16 8230 (56.7) 0.204 (132) 119 (20.8) 7920 (35.2) Crushing

McCune20 8710 (60.1) 0.259 (167) 160 (28.0) 10400 (46.1) Crushing

3.6 Results and Discussion

This section presents the axial stiffness and buckling results of the FE models, and discusses

their correlations with mechanics calculations and experimental data. These correlations are used

to determine the effectiveness of the FE models in predicting the total deflection, buckling loads,

and buckling modes of IsoTruss structures under uniaxial compression.

Five FE models were tested with various geometric and material properties to determine

the parameters that would accurately predict the results of the experimental testing. The difference

of the FE results with respect to the analytical (i.e., Mechanics or Mech.) predictions and the

experimental (i.e., Exp.) results were calculated using Eq. 3.6.

FE Dif. Relative to Mech./Exp. =FE−Mech./Exp.

Mech./Exp.·100 (3.6)

Table 3.7 provides a summary of the FE models and the corresponding input parameters

(i.e., the radii of the longitudinal and helical members and the material properties that include

33

Young’s modulus, Poisson’s ratio, and the shear modulus). The input parameters of each model are

either nominal or effective values. The ‘nominal radii’ were calculated from the reported number

of tows in each member, the fiber volume fraction, and the area of a single tow. The ‘nominal

material properties’ refer to the values that were calculated from the micromechanics equations

provided in Appendix A, and are shown in Table 3.3. The ‘effective material properties’ of FE

Models C, D, and E are explained in greater detail in the subsequent sections. The ‘measured

radii’ of Model B were measured physically from an experimental specimen. The results from FE

Model B did not contribute to the conclusions of this study, and are therefore omitted from this

chapter.

Table 3.7: FE Models and Corresponding Parameters

Model ID Radii Material PropertiesA Nominal Nominal (Table 3.3)

B Measured Nominal (Table 3.3)

C Nominal Effective per Axial Stiffness (Table 3.6)

D Nominal Effective per Global Buckling Load (Ez = 142 GPa)

E Nominal Effective per Shell-like Buckling Load (Ez = 185 GPa)

3.6.1 Axial Stiffness

FE Model A

FE Model A implements the nominal length, nominal radii, and nominal material proper-

ties. The nominal length and nominal radii are based on the measurements, number of tows, and

fiber volume fractions reported by Rackliffe et al. and McCune (see Table 3.4). The nominal mate-

rial properties of the composite were calculated from the material properties of the fiber and resin

using micromechanics equations presented by Kollar and Springer (see Appendix A). Table 3.8

presents the theoretical stiffness calculated from the mechanics equation, Eq. 3.1, and the stiffness

predicted by FE Model A.

34

Table 3.8: Axial Stiffness from Mechanics and FE Model A

Axial Stiffness[kip/in. (kN/mm)] FE Dif. Relative to

Specimen Mech. FE Mech. Exp.Rackliffe1 4.82 (0.845) 4.81 (0.843) -0.24% 404%

Rackliffe2 5.24 (0.918) 5.23 (0.916) -0.16% 157%

Rackliffe3 5.57 (0.975) 5.56 (0.973) -0.20% 262%

Rackliffe4 5.33 (0.933) 5.32 (0.931) -0.20% 296%

Average 5.24 (0.918) 5.23 (0.916) -0.20% 280%

Std. Dev. 0.27 (0.047) 0.27 (0.047) 0.03% 87.9%

McCune8 95.2 (16.7) 95.7 (16.8) 0.58% 50.1%

McCune12 143 (25.0) 143 (25.0) 0.11% 61.6%

McCune16 190 (33.3) 189 (33.1) -0.69% 58.9%

McCune20 236 (41.4) 237 (41.6) 0.45% 48.4%

Average 166 (29.1) 166 (29.1) 0.11% 54.7%

Std. Dev. 52.8 (9.24) 52.7 (9.23) 0.49% 5.63%

The average total deformation predicted by the FE models for the Rackliffe specimens was

less than 1% different than that calculated from mechanics. When compared to the total defor-

mation reported in the experiment, the average difference was 280%, with a standard deviation of

87.9%. Rackliffe et al. also acknowledged the large discrepancy between FE predictions and ex-

perimental results, attributing these differences to manufacturing and developmental factors. The

discrepancy can also be attributed to the limitations of the micromechanics equations used to cal-

culate the nominal properties.

Another possible source of decreased stiffness exhibited in the experimental testing could

be the rotation of longitudinal struts within each bay, represented in Fig. 3.2. If the maximum

deflection, δmax, was distributed among all the bays, Nb, the average deflection per bay would be

4.12 ·10−3 in. (0.105 mm). This local deflection corresponds to an average of 2.91° of rotation, γ ,

within each bay:

γ = arccos

(b− δmax

Nb

b

)(3.7)

35

Figure 3.2: Potential rotation of the longitudinal strut within each bay (NTS).

Incorporating the local strut rotation of the bays in the FE model could enhance the accu-

racy of the FE model in predicting the experimental behavior of the specimens.

On the other hand, the axial stiffness predicted by FE Model A for McCune’s specimens

were, on average, 54.7% different than the axial stiffness exhibited in the experiment. While this

is not an accurate prediction of the total displacement, the average standard deviation was 5.63%,

showing greater precision than the Rackliffe models. The McCune structures were smaller and

manufactured with more consistent methods and geometries than those constructed by Rackliffe

et al. This implies that as specimens are manufactured with greater consistency, a more accurate

prediction of the total deformation can be made using FE models. The effective stiffness modulus,

based on experimental data, can be implemented in FE models to produce an accurate and precise

prediction of total deflection. Figure 3.3 is a plot of the effective stiffness modulus of McCune’s

samples (demonstrated during experimental testing) versus the nominal modulus calculated from

micromechanics. It suggests that the correlation factor that should be applied to the nominal mod-

ulus is approximately 0.65, to match the 8-node compression specimens tested by McCune.

FE Model C

FE Model C was implemented to explore how an FE model could be defined to accurately

predict the axial stiffness demonstrated by the Rackliffe experiments. Each Rackliffe specimen was

36

Figure 3.3: Effective stiffness vs. nominal stiffness of McCune specimens.

modeled using the nominal length, nominal radii, and effective material properties. The effective

material properties were derived from the effective moduli of elasticity, Es, that were extracted

from stress-strain curves generated from experimental data and are documented in Table 3.6. The

axial stiffness predicted by FE Model C and by the mechanics equation, Eq. 3.1, are presented in

Table 3.9.

The average difference of the FE predictions of axial stiffness relative to the predictions

from mechanics is once again less than 1%. The difference relative to the experimental data was

decreased from 280% to -0.83%, with a standard deviation of 0.50%. This analysis demonstrates

that an accurate prediction of the total deformation can be made by adjusting the nominal material

properties by a correlation factor to account for manufacturing inconsistencies, imperfections in

the apparatus, and/or limitations of the micromechanics equations. It also implies that Eq. 3.1

is an accurate representation of the stiffness of IsoTruss structures under uniaxial compression.

Additional studies could be performed to find correlations between the correlation factors and the

structural design parameters such as the bay length, total length, and fiber volume fraction.

37

Table 3.9: Axial Stiffness from Mechanics and FE Model C

Axial Stiffness[kip/in. (kN/mm)] FE Dif. Relative to

Specimen Mech. FE Mech. Exp.Rackliffe1 0.954 (0.167) 0.951 (0.167) -0.29% -0.42%

Rackliffe2 2.01 (0.352) 2.01 (0.352) -0.17% -1.23%

Rackliffe3 1.53 (0.268) 1.53 (0.268) -0.20% -0.26%

Rackliffe4 1.32 (0.232) 1.32 (0.232) -0.19% -1.42%

Average 1.45 (0.254) 1.46 (0.256) -0.21% -0.83%Std. Dev. 0.38 (0.067) 0.38 (0.067) 0.05% 0.50%

3.6.2 Eigenvalue Buckling

FE Model C

An eigenvalue buckling (EB) analysis was performed on each Rackliffe specimen to find

a correlation between the critical buckling load predicted by FE methods and that exhibited in

experimental testing. The first EB analysis was performed using the effective material properties

from the FE Model C static structural analysis. This model severely under-estimated the buck-

ling capacity of the specimens, with an average difference of -72.5% relative to the experimental

data. Table 3.10 presents the buckling loads predicted from FE Model C and from the mechanics

expressions, Eq. 3.2 and Eq. 3.5. Table 3.10 also provides the difference of the FE predictions

relative to the mechanics predictions and experimental results. The FE predictions were less than

3% different than the mechanics predictions when global buckling governed, however, there was

greater deviation when local buckling governed.

Despite the disparity between the predicted and experimental buckling loads, FE Model C

accurately predicted the experimental buckling modes of each Rackliffe specimen, except Rack-

liffe3. One possible reason for the discrepancy is that Rackliffe3 was constructed with a bay length

of 2.5 in. (64 mm). According to Rackliffe [5], the IsoTruss structures designed within the spec-

ified design space that have a bay length greater than 2.5 in. will fail in local buckling, whereas

structures with a bay length less than 2.5 in. will fail in global buckling. Since the bay length of

38

Rackliffe3 is the transitional length between local and global buckling, it is not unreasonable for

either failure mode to be predicted numerically by FE methods or mechanics calculations.

Table 3.10: Critical Buckling Loads from Mechanics and FE Model C

Buckling Load Predicted[lb. (N)] FE Dif. Relative to Buckling Mode

Specimen Mech. FE Mech. Exp. Mech. FE Exp.Rackliffe1 12.4 (55.3) 21.7 (96.5) 74.6% -82.8% Local Local Local

Rackliffe2 33.7 (150) 45.6 (203) 35.1% -66.3% Local Local Local

Rackliffe3 70.3 (313) 82.0 (365) 16.6% -69.1% Local Local Global

Rackliffe4 72.5 (323) 70.9 (315) -2.16% -71.7% Global Global Global

Average 47.2 (210) 55.1 (245) 31.0% -72.5% - - -

Std. Dev. 25.3 (113) 23.4 (104) 28.3% 6.26% - - -

FE Models D and E

Two additional models (i.e., FE Models D and E) were developed to find a correlation be-

tween the critical buckling loads predicted by FE methods and those demonstrated in experimental

testing. The effective material properties used in FE Models D and E were derived from the load

multipliers and material properties of FE Model C by the following process. Eq. 3.8 was used to

calculate a new Ez for each specimen by dividing the Ez from FE Model C by the corresponding

load multiplier, λ , predicted by the eigenvalue buckling analysis. The new Ez is referred to as the

effective buckling modulus, Eb, whereas the Ez from FE Model C is referred to as the effective

stiffness modulus, Es. The remaining material properties were calculated from the new Ez using

micromechanics.

Eb =Es

λ(3.8)

The average effective buckling modulus of Rackliffe3 and Rackliffe4 was 20600 ksi (142

GPa) with a standard deviation of 120 ksi (0.83 GPa). This average modulus was used to derive

the remaining material properties for FE Model D. The average effective buckling modulus of

39

Rackliffe1 and Rackliffe2 was 26800 ksi (185 GPa) with a standard deviation of 119 ksi (0.82

GPa). This average modulus was used to derive the remaining material properties for FE Model

E. The results from FE Models D and E are presented in Table 3.11 and Table 3.12, respectively.

These tables also include the mechanics predictions calculated with Eq. 3.2 and Eq. 3.5.

Table 3.11: Critical Buckling Loads from Mechanics and FE Model D

Buckling Load Predicted[lb. (kN)] FE Dif. Relative to Buckling Mode

Specimen Mech. FE Mech. Exp. Mech. FE Exp.Rackliffe3 226 (1.00) 258 (1.15) 14.5% -2.71% Local Local Global

Rackliffe4 257 (1.14) 251 (1.12) -2.52% 0.02% Global Global Global

Average 241 (1.07) 255 (1.13) 5.99% -1.34% - - -

Std. Dev. 15.8 (0.070) 3.86 (0.017) 8.52% 1.37% - - -

Table 3.12: Critical Buckling Loads from Mechanics and FE Model E

Buckling Load Predicted[lb. (N)] FE Dif. Relative to Buckling Mode

Specimen Mech. FE Mech. Exp. Mech. FE Exp.Rackliffe1 72.3 (322) 123 (549) 70.7% -2.02% Local Local Local

Rackliffe2 101 (450) 135 (599) 33.1% -0.52% Local Local Local

Average 86.7 (386) 129 (574) 51.9% -1.27% - - -

Std. Dev. 14.4 (64.1) 5.63 (25.0) 18.8% 0.75% - - -

The relative differences between the FE predictions and the experimental data are, on av-

erage, -1.34% and -1.27% for FE Model D and FE Model E, respectively. The differences relative

to mechanics predictions are less than 3% when global buckling governs, however, the deviation

varies significantly between local buckling (i.e., shell-like buckling) predictions. FE Models D and

E predict the critical buckling load with greater accuracy than Model C with respect to experimen-

tal data. Additional exploration is needed to increase the accuracy of the prediction of shell-like

40

buckling from mechanics. The standard deviation between the predictions of FE Model E and

mechanics calculations is 18.8%.

Predictions from FE Models D and E demonstrate significant improvement from FE Model

C, reducing the relative difference from -72.5% to -1.34% and -1.27%. This demonstrates that the

FE model is viable in predicting the critical buckling load when the modulus is adjusted to account

for manufacturing imperfections and micromechanics limitations. The correlation factors of 0.88

and 1.15 were applied to the nominal modulus (i.e., 23300 ksi or 161 GPa), to calculate the effective

buckling moduli of FE Models D and E, respectively. The adjustment factors 0.88 and 1.15 may

depend on the failure mode (i.e., global versus shell-like buckling) or the geometric parameters of

the IsoTruss structure (i.e., 2.17- and 2.50-inch bay length versus 4.33-inch bay length). Additional

research is needed to determine the interrelations between geometric parameters and the correlation

factors.

3.7 Summary

A finite element (FE) model is presented to predict the total deflection and critical buckling

load of IsoTruss structures subject to uniaxial compression. The model is verified with traditional

mechanics calculations and validated with experimental data from preceding studies.

3.7.1 Conclusion

The FE predictions of total deflection are within 1% to those predicted with traditional me-

chanics, but underestimate the deflection exhibited by experimental testing. The total deflection

from experimentation can be accurately predicted with mechanics and the FE model if the nominal

modulus of elasticity (calculated with micromechanics) is changed to an effective modulus using

a correlation factor. The correlation factor accounts for imperfections in the physical specimens,

imperfections in the testing apparatus, an overestimated fiber volume fraction, and/or limitations of

the micromechanics equations. The factor is approximately 0.65 for the specimens tested by Mc-

Cune. The correlation factors for the specimens tested by Rackliffe et al. had significant variation

between samples due to manufacturing inconsistencies.

41

The critical buckling loads predicted in the FE model are within 3% of the predictions from

Euler-buckling calculations when global buckling is the governing buckling mode. When shell-like

buckling governs, there is greater variation in the predictions from the FE model and those from

the mechanics equation derived in Chapter 2. The discrepancies between FE predictions and the

analytical equation are explored further in Chapter 4 to identify potential sources of deviation. The

buckling models also use a correlation factor to accurately predict the buckling loads exhibited

in experiment. The factors fluctuate depending on the geometry and failure mode of the distinct

configuration. More research is needed to determine how the correlation factors vary with respect

to the design parameters of the IsoTruss structure.

The finite element models predicted the same critical buckling modes as the analytical

expressions for each tested configuration. The FE and analytical methods predicted the same

critical buckling modes exhibited in experiment except for Rackliffe3. The Rackliffe3 specimen

has a bay length of 2.5 in. (64 mm). This particular bay length has been identified previously by

Rackliffe et. al as the transition between local and global buckling for configurations within the

design space tested.


In addition to enhancing the accuracy of the finite element model, improvements are needed

to more accurately predict shell-like buckling via engineering mechanics. The method presented in

Chapter 2 is further explored in Chapter 4 to determine why there is variation in the results and how

to improve the accuracy and precision. With a greater understanding of the mechanics of shell-like

buckling, a model can be implemented that will consistently predict the governing buckling mode.

While shell-like buckling and global buckling were the only buckling modes demonstrated by the

FE models of this study, longitudinal strut buckling is a potential buckling mode if the helical

members are robust enough to constrain the longitudinal members.

The model developed in this study is an effective tool for the numerical analysis of IsoTruss

structures in subsequent studies. It provides a framework for dimensional and optimization analy-

ses of IsoTruss structures, including a method to gauge the relative performance of novel configu-

rations of IsoTruss structures such as those with outer longitudinal members.

42

CHAPTER 4. DIMENSIONAL ANALYSIS OF SHELL-LIKE BUCKLING IN ISOTRUSS®

STRUCTURES USING BUCKINGHAM’S PI THEOREM

4.1 Synopsis

The purpose of the current study was to perform a dimensional analysis of IsoTruss® struc-

tures to: first, interrelate design parameters with the shell-like buckling mode; and second, to

correlate analytical predictions for shell-like buckling with finite element (FE) predictions. Buck-

ingham’s Pi Theorem (BPT) is implemented to reduce the critical geometric and material design

parameters to a smaller set of independent, non-dimensional Pi (Π) variables. The shell-like buck-

ling load is also made non-dimensional and is referred to as the dependent Π variable. FE analyses

are performed to predict the shell-like buckling loads of various configurations of IsoTruss struc-

tures. Data from the FE analyses are plotted in the form of Π variables to characterize trends

between the design parameters and the shell-like buckling load. The FE data are plotted with the

data predicted by analytical expressions to correlate FE predictions with analytical predictions.

Results indicate that the analytical expression accurately represents the influence of the longitudi-

nal radius and bay length on the shell-like buckling load, however, the contribution of the helical

radius is underestimated by the analytical prediction.

4.2 Nomenclature

DX Equivalent plate flexural rigidity of planar X-style helical struts

Ez Young’s modulus of the composite in the z-direction



L Base dimension of length or global length of IsoTruss

M Base dimension of mass

N Number of nodes

43

Nb Number of bays

Nt Number of carbon tows per longitudinal (L) or helical (H) member

Psb Shell-like buckling load


R2 Statistical coefficient of determination (i.e., R-squared value)


l1 Short span of longitudinal members between transition nodes


t Base dimension of time

α Coefficient of quadratic term in best-fit curves

β Coefficient of linear term in best-fit curves

δ2 One-half the base length of the planar X-style helical struts


γ Coefficient of constant term in best-fit curves

Γ Dimensional matrix derived from Buckingham’s Pi Theorem

µsb Boundary constraint coefficient for shell-like buckling (converts length to effective length)


Π Non-dimensional Pi variable derived from Buckingham’s Pi Theorem

ξ Coefficient of power term in best-fit curves

4.3 Introduction

In preceding studies, including Chapter 3, finite element (FE) methods are used in the

structural design process and preliminary assessment of IsoTruss® structures. Numerical methods

such as FE methods facilitate efficient and prolific analyses without the costs of manufacturing

and physical testing. To ensure the accuracy of the FE predictions, models are typically validated

with experimental data and verified with analytical expressions derived from traditional mechanics.

Deviations between FE predictions and experimental data can be attributed to imperfections in the

testing apparatus, material microstructure, manufacturing process, etc. Deviation from analytical

44

expressions are often attributed to the limited scope or over-simplified assumptions of the analytical

expression.

In Chapter 3, FE models are used to predict the axial deflection and critical buckling loads

of 8-node IsoTruss structures subject to uniaxial compression. The models are validated with ex-

perimental data produced by Rackliffe et al. [2] and McCune [1], and verified with traditional

mechanics. The FE predictions of axial deflection and global buckling correlate well with both

the analytical predictions and experimental results when the nominal modulus of elasticity is re-

placed with an effective modulus of elasticity that accounts for diverse sources of imperfections

in the physical specimens. The FE predictions of shell-like buckling likewise correspond with ex-

perimental data after implementing effective material properties. The FE predictions of shell-like

buckling, however, do not consistently correlate with the analytical expression that is derived in

Chapter 2.

The purpose of the current study is to perform a dimensional analysis that explores the

shell-like buckling behavior of 8-node IsoTruss structures. The results are used for two main

objectives: first, to characterize interrelations between the design parameters and the critical shell-

like buckling load; and second, to determine potential causes for deviation between the FE pre-

dictions and analytical predictions. The study implements Buckingham’s Pi Theorem (BPT) to

perform the dimensional analysis. BPT uses the rank-nullity theorem of linear algebra to deter-

mine non-dimensional Pi (Π) variables from a complete set of critical geometric and material, i.e.,

dimensional, parameters.

4.4 Background

Parametric analyses have been performed on open-lattice composite structures in preced-

ing studies to identify critical design parameters and explore the extent to which they influence

structural behavior. Winkel [3] performed a parametric analysis to explore the following design

parameters of IsoTruss structures: bay length, outer diameter, number of nodes, and the number of

tows in helical and longitudinal members. These parameters are plotted with respect to the axial

stability and critical buckling load of the IsoTruss structure subject to uniaxial compression. The

parameters were analyzed with respect to global and strut buckling modes, however, no interrela-

tions were analyzed with respect to shell-like buckling.

45

Lai et al. [11] performed a similar study that relates the critical buckling load of com-

posite grid cylindrical shells to the helical angle. Both Lai et al. and Winkel implemented FE

methods to perform parametric analyses to generate plots that characterize trends and derive in-

terrelations between the design parameters and the buckling load. The current study utilizes the

same design parameters as Winkel, however, the scope is expanded by employing BPT to derive

non-dimensional Π variables from the design parameters. The Π variables are plotted with respect

to the critical shell-like buckling load.

Additional parametric methods are implemented by Rackliffe et al. and Belardi et al. [12].

Rackliffe et al. use FE analysis to iterate various structural configurations and determine the best

compromise between bay length and total length to resist buckling and maximize the compressive

strength-to-weight ratio. These iterations are used to select a configuration to be manufactured

and tested experimentally. Belardi et al. present a parametric analysis with FE methods that uti-

lizes both discrete and continuous parameters. The definition of these parameters allows anisogrid

structures to be consistently defined based on geometric constraints, and facilitates optimization.

Table 4.1 summarizes relevant parametric analyses performed previously, the design parameters

identified therein, and the performance metric (i.e., dependent parameter of interest).

Table 4.1: Parametric Studies on Open-lattice Composite Structures

Author Independent Design Parameters Dependent Parameter(s)Belardi et al. [12] Continuous: radial thickness; Mass

width of helical ribs; and, width of hoop ribs.

Discrete: number of helical ribs with

the same slope and number of cells arranged

alongside the axis of the lattice shell.

Lai et al. [11] Helical angle and grid type. Critical buckling load and

weight efficiency.

Winkel [3] Bay length; global diameter of truss; Compression stability;

number of nodes; critical buckling load; and,

inner-helical to inner-longitudinal ratio; and, weight.

sizes of helical and longitudinal members.

46

IsoTruss structures with 8-nodes that are subject to uniaxial compression are susceptible

to buckling globally, locally in the struts, and locally in shell-like buckling. Both global and local

strut buckling are addressed in the parametric analysis performed by Winkel, however, shell-like

buckling is not considered. Shell-like buckling is a buckling mode with a wavelength of approx-

imately two bays that can occur in extremely light-weight IsoTruss structures with greater than

6-nodes. In addition to FE methods, Sui et al. implement an analytical expression to predict shell-

like buckling. The expression is derived with slight adjustments in Chapter 2, and implemented in

the current study via Eq. 4.1.

Psb =N · π2 ·Ez · IL

(µsb ·b)2

µsb =1+2 ·δ2 ·DX

Ez · IL·(

l1b+

1π· sin

π · l1b

)

DX =EX · (2 · rH)

3

12 · (1−ν2zθ)

EX =2 ·Ez ·π · r2

H2 ·δ2 · (2 · rH)

· sin4φ

(4.1)

Equation 4.1 is derived by finding the minimum energy solution of a longitudinal strut that

spans two bays and is constrained mid-span with helical struts that form planar X-style trusses.

The derivation of the minimum energy solution and resultant expression are included in Chapter 2.

The critical buckling load predicted analytically from Eq. 4.1 varies from the critical buckling load

predicted by FE models (see Chapter 3). The current study explores the deviation between the

analytical and FE predictions with respect to the design parameters.

4.5 Methods

The general process used to perform the dimensional analysis is summarized in the steps

below and expounded in the subsequent sections.

1. Select a dependent parameter of interest;

47

2. Identify the independent design parameters that influence the dependent parameter;

3. Apply Buckingham’s Pi Theorem to derive the non-dimensional Π variables; and,

4. Perform a set of finite element analyses to gather data and analyze trends.

4.5.1 Selecting Parameters

In this analysis, the dependent parameter of interest is the critical shell-like buckling load.

Referring to the shell-like buckling expression derived in Chapter 2, the independent design param-

eters that influence the dependent parameter (i.e., the shell-like buckling load) include the radius

of the longitudinal members, the radius of the helical members, the bay length, the outer radius,

and the modulus of elasticity in the longitudinal direction.

4.5.2 Dimensional Analysis

Buckingham’s Pi Theorem

Buckingham’s Pi Theorem (BPT) reduces the independent and dependent parameters to a

smaller subset of non-dimensional Π variables. The theorem is based on the rank-nullity theorem

of linear algebra, and derives the Π variables from the nullspace of a dimensional matrix that is

composed of columns that represent the design parameters and rows that represent the correspond-

ing base dimensions of the design parameters. The base dimensions of mass (M), length (L), and

time (t) are implemented in this analysis. Table 4.2 summarizes the design parameters and the

corresponding base dimensions.

The dimensional matrix, Γ, is defined in Eq. 4.2 from the independent parameters, depen-

dent parameter, and the corresponding base dimensions. Each column of Γ represents a parameter,

whereas the rows of Γ represent the corresponding base dimensions. The parameters are assigned

to columns in the same order as they are listed in Table 4.2. The three base dimensions are repre-

sented as rows in the order of M, L, and t.

48

Table 4.2: Parameters and Corresponding Dimensions

Parameter Typical Units Base DimensionsR in. (mm) L1

rL in. (mm) L1

rH in. (mm) L1

b in. (mm) L1

Ez 106 psi (GPa) M1 ·L−1 · t−2

Psb lb. (N) M1 ·L1 · t−2

Γ =

R rL rH b E Psb −

0 0 0 0 1 1 M

1 1 1 1 −1 1 L

0 0 0 0 −2 −2 t

(4.2)

Equation 4.3 is the reduced-row echelon form (rref) of Γ, and is used to find the nullspace

of the matrix. According to BPT, the vectors that span the nullspace of the dimensional matrix

represent the non-dimensional variables of the system.

Γ =

R rL rH b E Psb −

1 1 1 1 0 2 M

0 0 0 0 1 1 L

0 0 0 0 0 0 t

(4.3)

The vectors that span the nullspace (null) of Γ are shown in Eq. 4.4 and are used to derive

the non-dimensional Π variables of the system. Each vector represents the Π variable shown in

the top row of the vector, and each row of the vectors corresponds to the design parameter shown

in the right-most vector. The value of each index indicates the power of the corresponding design

parameter in the associated Π variable. For example, the first index in the first vector indicates that

Π1 has a value R−1

49

null (Γ) = span

Π1

−1

1

0

0

0

0

Π2

−1

0

1

0

0

0

Π3

−1

0

0

1

0

0

Π0

−2

0

0

0

−1

1

−

R

rL

rH

b

E

Psb

(4.4)

The nullspace vectors shown in Eq. 4.4 are reformatted as Π variables in Eq. 4.5. The

first three Π variables (i.e., Π1, Π2, and Π3) are composed of the independent design parameters,

whereas Π0 is the non-dimensional form of the dependent design parameter of interest (i.e., the

critical shell-like buckling load, or Psb).

Π1 =rL

R

Π2 =rH

R

Π3 =bR

Π0 =Psb

E ·R2

(4.5)

According to Eq. 4.5, the outer radius, R, and modulus of elasticity, E, are a subset of

independent parameters that non-dimensionalize the parameters rL, rH , b, and Psb. Since rL, rH ,

and b have the same base dimensions as R, any of these could have been used with E to non-

dimensionalize the remaining parameters (e.g., b and E could have been implemented to derive Π

variables rLb , rH

b , Rb , and Psb

E·b2 ). In the current study, R and E are used to non-dimensionalize the

remaining parameters, as shown in Eq. 4.5.

Trend Analysis

A trend analysis is performed on each independent Π variable (i.e., Π1, Π2, and Π3) to

identify the interrelations between the independent Π variable and the dependent Π variable of

50

interest (i.e., Π0). Each trend analysis consists of three sets of FE analyses. Each set of FE analyses

has different design parameters to demonstrate how the interrelations may vary with respect to

different geometric dimensions.

For example, the trend analysis of Π1 consists of three sets of eighteen FE analyses. Within

one set, all eighteen configurations have the same design parameters except the longitudinal radius.

The longitudinal radius varies between 0.0169 and 0.0436 in. (0.429 and 1.11 mm, respectively),

which correspond to 3 and 20 carbon fiber tows. The predicted shell-like buckling loads are divided

by Ez ·R2 to get the Π0 value, and plotted with respect to Π1. Two additional sets of eighteen FE

analyses are performed with different design parameters.

The three curves of the Π1 trend analysis are plotted together to demonstrate how the in-

terrelation between Π0 and Π1 may vary for different design parameters. Each set of design pa-

rameters used in the Π1 trend analysis is distinguished by the ratio Π3-to-Π2 (or Π3 / Π2 as shown

in the tables and plots). In like manner, the ratios Π3-to-Π1 and Π1-to-Π2 are used to distinguish

each set of design parameters used in the Π2 and Π3 trend analyses, respectively. The Π variables

that are fixed in each set of FE analyses are provided in Table 4.3. The ratios of the Π variables

that are used to distinguish between FE sets are also provided in Table 4.3.

Table 4.3: Fixed Π Variables of Dimensional Analyses

Π Variable Set (1) Set (2) Set (3)Π1 9.91 ·10−3 1.20 ·10−2 1.10 ·10−2

Π2 6.61 ·10−3 6.25 ·10−3 6.81 ·10−3

Π3 1.47 0.93 1.95

Π3/Π2 222 148 287

Π3/Π1 148 77 178

Π1/Π2 1.50 1.92 1.61

51

4.5.3 Finite Element Models

Both static structural and eigenvalue buckling analyses are performed for each unique con-

figuration, and the shell-like buckling load is recorded. The design parameters that are fixed in

each set of FE analyses are summarized in Table 4.4.

Table 4.4: Fixed Design Parameters of FE Analyses

Parameter Units Set (1) Set (2) Set (3)R [in. (mm)] 2.95 (74.9) 2.70 (68.6) 3.20 (81.3)

NtL N/A 9 11 13

rL [in. (mm)] 0.0292 (0.743) 0.0323 (0.821) 0.0351 (0.893)

NtH N/A 4 3 5

rH [in. (mm)] 0.0195 (0.495) 0.0169 (0.429) 0.0218 (0.554)

b [in. (mm)] 4.33 (110) 2.50 (63.5) 6.25 (159)

Nb N/A 23 40 16

L [in. (m)] 99.6 (2.53) 100 (2.54) 100 (2.54)

Ez [106 psi (GPa)] 23.3 (161) 23.3 (161) 23.3 (161)

In addition to the FE predictions, the shell-like buckling load of each configuration is also

predicted using the analytical expression, Eq. 4.1. As demonstrated in Chapter 3, deviation exists

between the analytical predictions and the FE predictions. The deviation is plotted versus the

independent Π variables to explore trends between the design parameters and deviation.

4.6 Results

The results from the trend analyses are analyzed for two main objectives: first, to explore

interrelations between the design parameters and the shell-like buckling capacity predicted by FE

methods; and second, to identify sources of deviation between FE predictions and the analytical

predictions. The unfilled markers indicate design configurations that failed in global buckling. In

these cases, the unfilled plotted values correspond with the first shell-like buckling loads.

52

4.6.1 Shell-like Buckling Interrelations

Data from the FE analyses are used to characterize trends between the non-dimensional Π

variables. The independent variables Π1, Π2, and Π3 are plotted against the dependent variable, Π0

in Figures 4.1, 4.2, and 4.3, respectively. The figures are followed by expressions that are derived

for the best-fit curves. The best-fit curves are represented in the figures by the dotted lines.

Figure 4.1: Π0 vs. Π1.

Π0 =α ·Π21 +β ·Π1 + γ (4.6)

Table 4.5: Coefficients relating Π0 to Π1

ΠΠΠ333///ΠΠΠ222 ααα βββ γγγ R2

148 5.16 ·10−2 −7.16 ·10−4 2.95 ·10−6 0.99

222 1.88 ·10−2 −2.41 ·10−4 1.09 ·10−6 0.99

287 9.15 ·10−3 −1.07 ·10−4 5.62 ·10−7 0.99

53


Π0 =α ·Π22 +β ·Π2 + γ (4.7)


ΠΠΠ333///ΠΠΠ111 ααα βββ γγγ R2

77 1.04 ·10−1 −1.30 ·10−3 5.81 ·10−6 0.99

148 7.14 ·10−2 −8.36 ·10−4 2.87 ·10−6 0.99

178 4.82 ·10−2 −5.22 ·10−4 1.73 ·10−6 0.99

54


Π0 =α ·Πξ

3 (4.8)


ΠΠΠ111///ΠΠΠ222 ααα ξξξ R2

1.50 1.55 ·10−6 −1.66 1.00

1.61 1.30 ·10−6 −1.48 1.00

1.92 9.52 ·10−7 −1.41 1.00

55

4.6.2 Global and Local Buckling Curves

The trend analyses of the non-dimensional Π variables, Π1 and Π2 demonstrated both local

and global failure modes. Figures 4.4 and 4.5 plot the global and local loads, that are represented by

Π0, with respect to Π1 and Π2, respectively. The shell-like buckling loads (i.e., the local buckling

loads) are represented by filled markers whereas the global buckling loads are represented by

unfilled markers.

As shown in Fig. 4.4, both the global and local buckling capacity increase as the longitudi-

nal radius, represented by Π1, increases. Figure 4.5, however, indicates that increasing the helical

radius, represented by Π2, does not increase the buckling capacity of the configuration when global

buckling governs.

Figure 4.4: Π0 vs. Π1 Global and local buckling curves.

56

Figure 4.5: Π0 vs. Π2 Global and local buckling curves.

4.6.3 Analytical Predictions vs. FE Predictions

To identify potential sources of deviation between analytical and FE methods, the predic-

tions from both methods are plotted together in Figures 4.6, 4.8, and 4.10 for Π1, Π2, and Π3,

respectively. The percent deviation of the analytical predictions (Psbmech) from the FE predictions

(PsbFE ) is calculated for each configuration per Eq. 4.9. The deviation varies with changes in design

parameters, therefore, the percent deviation is plotted with respect to each Π variable in Figures

4.7, 4.9, and 4.11. Solid lines represent the FE predictions whereas the dashed lines represent the

analytical predictions.

Percent Deviation =Psbmech −PsbFE

PsbFE

·100 (4.9)

57

Figure 4.6: Π0 vs. Π1 Analytical and FE predictions.

Figure 4.7: Percent deviation of analytical predictions from FE predictions vs. Π1.

58



59



60

4.7 Significance of Results

4.7.1 Shell-like Buckling Interrelations

As depicted in Figures 4.1 and 4.2, increasing the non-dimensional variables, Π1 and Π2

induces a quadratic increase in Π0. It follows that increasing design parameters rL and rH induces

a quadratic increase in the shell-like buckling load. The coefficients of the best-fit curves indicate

that changes in rH induce a steeper change in the shell-like buckling load than equivalent changes

in rL. As rL and rH increase, there exists a transition from shell-like buckling to global buckling.

It appears that increasing the rH induces the transition sooner than equivalent changes in rL, in

the current design space. Additional research is needed to prescribe the transition from global to

shell-like buckling, as affected by the helical radius.

Fig. 4.3 indicates that increasing the bay length has an inverse affect on the shell-like buck-

ling load, as anticipated. Within the design space of the current study, the rates of change of the

curves in Fig. 4.3 are not as rapid as those demonstrated in Figures 4.1 and 4.2, by inspection.

Figures 4.1, 4.2, and 4.3 indicate that increasing the Π variable ratio does not change the interre-

lation trend between the independent Π variable and the dependent Π variable. On the other hand,

each figure does indicate that changing the Π variable does change the coefficients of the best-fit

curves, and therefore changes the rate at which the independent Π variables affect the dependent

Π variable. It follows that a dimensional analysis is relevant within the design space of interest.

The dimensional analysis could be improved by selecting Π variable ratios that are relevant to a

specific design space of interest. The ratios used herein are selected to correspond with the design

space of the experimental specimens tested by Rackliffe et al.

4.7.2 Global and Local Buckling Curves

Plotting the global and local buckling capacities together in Figures 4.4 and 4.5 reveals

further interrelations between the radii of the members and the overall buckling capacity. While

increasing the helical radius does induce a steep increase in the shell-like buckling capacity, it does

not increase the global buckling capacity. Conversely, the longitudinal radius continues to increase

the buckling capacity even after the configuration has transitioned to global buckling. Therefore,

the overall buckling capacity of the configuration is improved in greater measure by increasing

61

the longitudinal radius rather than increasing the helical radius. In addition, increasing the helical

radius increases the mass two-fold because there are twice as many helical members as there are


4.7.3 Analytical Predictions vs. FE Predictions

According to Figures 4.6 and 4.10, changes in rL and b demonstrate similar effects in both

the FE and analytical predictions. Changes in the rH demonstrated in Fig. 4.8, however, do not

induce similar effects in the FE and analytical predictions. While both methods demonstrate a

quadratic interrelation between rH and Psb, the FE prediction increases significantly more than the

analytical prediction for equivalent changes in rH . Therefore, it can be deduced that the influence

of rH is not adequately represented in the analytical expression, Eq. 4.1.

Additional research is required to more accurately represent the influence of rH in Eq. 4.1.

Potential sources of deviation include the following areas. First, Eq. 4.1 does not account for

the rigidity of the pyramidal style helical trusses that intersect the longitudinal members within the

shell-like buckling wavelength. The constraining influence of these members could be incorporated

in the energy equation of the system during the derivation of the shell-like buckling load. The

expression might also be improved by re-deriving the influence of the planar X-style helical trusses.

The influence of the planar trusses is currently derived by approximating the inclined struts as

plates in bending. It may be that the energy of the struts in bending is not accurately represented

as a plate in bending, despite the reduction factors to decrease the equivalent plate stiffness and

equivalent plate flexural rigidity.

4.8 Summary

A dimensional analysis was performed to analyze the shell-like buckling behavior of IsoTruss

structures with 8-nodes under uniaxial compression. The shell-like buckling load of various ge-

ometric configurations was predicted using finite element (FE) models and the analytical expres-

sion derived in Chapter 2. Each set of FE models has a different combination of geometric de-

sign parameters. The design parameters and the critical buckling load are reconfigured into non-

dimensional Π variables via Buckingham’s Pi Theorem (BPT). The Π variables are plotted with

62

respect to each other to characterize trends between design parameters and the shell-like buckling

load. In addition, the deviation between the FE predictions and analytical predictions is plotted

versus the Π variables to find correlations between the FE and analytical predictions.

4.8.1 Conclusion

Results indicate that increasing the radii of the longitudinal and helical members induces

quadratic increases in the shell-like buckling load. Increasing the bay length, however, decreases

the shell-like buckling load at the rate of a shallow power curve. Increasing the longitudinal radius

is more effective than increasing the helical radius with respect to increasing the strength-to-weight

ratio.

The analytical expression and the finite element model demonstrate the same trends be-

tween design parameters rL and b and the shell-like buckling load. Conversely, the shell-like buck-

ling load predicted by the analytical expression does not increase as rapidly as the FE predictions

with respect to rH . Therefore, it can be concluded that the analytical expression underestimates the

constraining influence of the helical members against shell-like buckling. A substantial portion of

the deviation between analytical and FE predictions can be attributed to this limitation.


The expressions and interrelations produced in the current study are relevant within the

design space of long IsoTruss structures with thin members subject to uniaxial compression. Ad-

ditional dimensional analyses should be performed for different design spaces that are relevant to

specific applications to derive the corresponding best-fit expressions.

Since increasing the helical radius is not as effective for increasing strength-to-weight, it

is not implemented in the optimization framework of Chapter 6 as a design variable, but rather as

a function of the longitudinal radius and a ratio of the longitudinal-to-helical radii (prescribed by

heuristics). The longitudinal radius is defined as a design variable that is adjusted by the optimizer

to minimize the weight of the structure.

As stated previously, the analytical expression, Eq. 4.1, accurately demonstrates how

changing the longitudinal radius or changing the bay length influences the shell-like buckling load.

63

On the other hand, the results indicate that the expression does not accurately account for the in-

fluence of the helical radius. The expression could be improved by deriving the contribution of the

pyramidal style helical trusses and re-deriving the contribution of the planar X-style helical trusses

as inclined struts (rather than approximated as a plate).

64

CHAPTER 5. GRADIENT-BASED VERSUS GRADIENT-FREE OPTIMIZATION OFISOTRUSS® STRUCTURES IN UNIAXIAL COMPRESSION

5.1 Synopsis

The objective of the current study was to explore and compare gradient-free and gradient-

based optimization techniques in the design of IsoTruss® structures under uniaxial compression.

The efficiency and robustness of each method were compared to determine which method is prefer-

able for further development. The optimization problem was designed to minimize the mass of the

structure with respect to the number of bays, number of carbon tows in both longitudinal and helical

members, and the outer diameter. Additional design parameters were fixed according to prescribed

design criteria. The analysis was subject to the constraints induced by structural failure modes,

including material strength and buckling. The optimization objective and constraints were defined

from mechanics equations that have been verified by finite element analysis and validated with

experimental data. The results indicate that gradient-based methods are both consistent and robust

in producing optimized configurations that correlate with typical design standards. Gradient-free

methods also produce viable results, however, they are not consistent between iterations and not

robust in producing designs that correlate with general manufacturing practices.

5.2 Nomenclature




Ig Moment of inertia of global IsoTruss



M Mass of IsoTruss

65

N Number of nodes

Nb Number of bays



P Buckling load of global (g), shell-like (sb), or longitudinal strut (l) buckling mode



k Boundary condition factor (converts length to effective length)



ll Length of longitudinal strut buckling region

lsb Length of shell-like buckling region



βsb Empirically derived correlation coefficient for shell-like buckling (simplified method)

λ Load multiplier

ρ Material density

σall Allowable stress of member (material property)

σact Actual stress in member

5.3 Introduction

Many of the preceding studies on IsoTruss® structures have relied on physical testing and

hand calculations supplemented with finite element (FE) predictions. While physical testing pro-

duces reliable data on the physical behavior of the structures, they are time and cost intensive. By

correlating FE models and analytical mechanics equations with the experimental test data, novel

configurations can be explored with less time and at lower cost. The design process can be fur-

ther enhanced by implementing the FE models and analytical methods in optimization routines to

identify the structural configuration with the least mass that can support a prescribed load criteria.

66

The purpose of the current study is to explore and compare gradient-free and gradient-

based optimization techniques in the design of IsoTruss structures under uniaxial compression.

The optimization framework is defined to minimize the mass of 8-node IsoTruss structures with

respect to the number of bays, number of tows in longitudinal members, number of tows in helical

members, and outer diameter. The objective is constrained by the structural failure modes that

typically govern IsoTruss structures in compression. These include global column buckling, shell-

like buckling, strut buckling, and material failure. Another failure mode is radial crushing of the

cross-section due to excessive bending of the struts induced by locally applied transverse loads.

Radial crushing is based on an empirical formula for short spans and is not implemented in the

current study. The design is also constrained by lower and upper bounds defined according to

typical manufacturing limitations. The results of this study provide a framework for more robust

optimization routines implemented in subsequent research.

5.4 Background

Preceding studies have implemented diverse methods to predict structural failure and op-

timize open-lattice composite structures similar to IsoTruss structures. Sayad [13] implements

analytical expressions to define the structural failure of a composite grid stiffened cylinder. These

expressions predict the ultimate loads of the three prevailing failure modes. Buragohain et al. [14]

also uses analytic expressions to predict the failure of a composite hexagonal lattice cylindrical

shell, and validates the expressions with FE analysis. While the results from analytical expressions

and FE solutions are not exact, they are sufficiently validated to compare configurations in the

preliminary design process.

Converse to the analytical methods of the studies discussed previously, Belardi et al. [12]

uses a discrete approach and implements full parametric FE modeling to predict the failure of

anisogrid composite lattice cylindrical shells. This method accurately predicts the failure load

regardless of the failure mode, facilitating a comprehensive structural analysis. Similar to the

approaches taken by Sayad and Buragohain et al., the current study implements analytical expres-

sions to define the optimization objective and constraints from design variables that define the

geometric configuration of the structure. These expressions are verified with FE analysis. Like

Buragohain et al., the results indicate that the analytic expressions are similar to FE predictions

67

and the error varies with geometric parameters. The FE models correlate with experimental data

when correlation factors are applied to the nominal material modulus to account for imperfections

in manufacturing (see Chapter 3).

In addition to the diverse methods of defining structural failure, many different methods

are used to perform the optimization analysis. Sayad implements the graphical method to optimize

the structure. The optimization objective is to minimize mass subject to the constraints defined

by analytical expressions. The graphical method is a gradient-based method that determines the

allowable range of each design variable with respect to each constraint, plots the corresponding

mass, and identifies the minimum of the plot. Conversely, Belardi et al. implements the non-

dominated sorting genetic algorithm II (NSGA-II) to find the optimum configuration with respect

to the full parametric FE modelling. Belardi et al. selected the NSGA-II method because it is

widely used and can explicitly optimize discrete and continuous variables.

Kim et al. [15] also implements a genetic algorithm to optimize the mass of an open-lattice

composite structure. The mass is minimized with respect to geometric design variables and subject

to strength and stiffness constraints. The optimization is performed in Matlab. The current study

implements a gradient-free optimization method (i.e., the NSGA-II method) and a gradient-based

optimization method in Matlab to compare the efficiency and robustness of the methods. The

optimization is defined to minimize the mass subject to constraints that are defined analytically

with respect to discrete and continuous geometric parameters.

5.5 Methods

5.5.1 Resources

The optimization code is implemented in Matlab for both the gradient-based and gradient-

free analyses. The fixed parameters are input in imperial units. The built-in constrained optimizer

‘fmincon’ is used to perform the gradient-based optimization. For the purposes of the current

study, the gradients are calculated using finite difference methods, which is the default setting of

‘fmincon’. A third-party toolbox called NGPM, an NSGA-II program written for Matlab [16],

was selected as the gradient-free optimizer. The NGPM toolbox employs intermediate crossover

68

and Gaussian mutation to generate sample populations. NGPM also provides the option to define

design variables as continuous or discrete.

The FE program ANSYS WorkBench (versions 19.1 and R2) was used to verify the an-

alytical equations used in the optimization. The FE model is validated with data collected by a

preceding experimental study in Chapter 3. The validated model was used to derive the simplified

shell-like buckling expression presented hereafter as Eq. 5.2. The simplified expression is replaced

in Chapter 6 by the mechanics-based method derived in Chapter 2.

5.5.2 Design Variables

The design criteria of IsoTruss structures include geometric and material parameters, in-

cluding the following:

1. Fiber volume fraction, v f

2. Fiber and matrix properties

3. Number of nodes, N

4. Applied load, P

5. Global length, L

6. Number of bays, Nb

7. Number of carbon tows, Nt

8. Outer diameter, D

Of these parameters, the number of bays, number of carbon tows, and outer diameter are

implemented as design variables in the optimization routine. The number of bays and carbon

tows are discrete, integer-value variables, whereas the outer diameter is a continuous variable. The

remaining parameters are typically prescribed in the concept design criteria, and are passed through

the code as fixed values. The design variables are scaled to the order of one in the gradient-based

analysis to enhance the performance of the optimizer.

69

5.5.3 Design Properties

The geometric properties of the IsoTruss specimen used in the analysis are based on the

Rackliffe4 specimen (see Chapter 3). The material properties are based on the properties used in

FE Model D that are correlated with the results from the physical test data. The geometric and

physical properties are summarized in Table 5.1.

Table 5.1: Material and Geometric Properties Implemented in Optimization Analysis

Property Implemented Valuev f 0.65

Fiber HexTow IM7 6K

Matrix UF3325-95

Ez [ksi (GPa)] 20740 (143)

σall [ksi (MPa)] 326 (2250)

ρ

[lb.in.3

(kgm3

)]0.0491 (1360)

N 8

P [lb. (N)] 251 (1120)

L [in. (m)] 115 (2.92)

5.5.4 Constraints

The failure modes implemented in the current study include global buckling (Eq. 5.1),

shell-like buckling (Eq. 5.2), longitudinal strut buckling (Eq. 5.3), and material strength (Eq.

5.4). These equations are simplified versions of the expanded expressions presented in Chapter 1.

In Chapters 6 and 7, the expanded expressions are implemented to increase the accuracy of the

optimization analysis.

Global Buckling

Global buckling is predicted using Euler-buckling. The effective length is calculated from

the global length of the IsoTruss and the boundary condition factor, kg = 2.0, for a fixed-free

70

column. The moment of inertia of the structure is calculated with the moment of inertia coefficient

for 8-node IsoTruss structures with inner longitudinal members, c, presented by Winkel [3].

Pg =π2 ·Ez · Ig

(kg ·L)2

kg =2.0

Ig = c ·AL ·R2

c =2.343

(5.1)

Shell-like Buckling (Simplified Method)

The expression that is used in this chapter to predict shell-like buckling is a simplified

method that was developed before the mechanics-based equation presented in Chapter 2 was de-

rived. It is sufficient for the purposes of the current chapter, however, the mechanics-based ex-

pression is implemented in subsequent analyses. The simplified method uses a correlation coeffi-

cient, βsb, that was derived empirically by correlating the Euler-buckling equation to the FE model

validated in Chapter 3. The boundary condition coefficient, ksb = 1.0, is approximated as pinned-

pinned at the outer transition nodes. The effective shell-like buckling length used in the current

chapter, lsb, is highlighted in Fig. 5.1 in green.

Psb =βsb ·N · π2 ·Ez · IL

(ksb · lsb)2

lsb =2 · l2 + l1

ksb =1.0

IL =π · r4

L4

βsb =3.726

(5.2)

71

Figure 5.1: Effective length of shell-like buckling (simplified method) [4].

Longitudinal Strut Buckling

Longitudinal strut buckling refers to the tendency of a longitudinal member to buckle in

the largest span between transition nodes. The effective length is taken as the maximum unbraced

length, i.e., l2. The boundary condition coefficient, kl = 1.0, is approximated as pinned-pinned at

the outer transition nodes.


(kl · ll)2

ll = l2

kl =1.0

IL =π · r4

L4

(5.3)

72

Material Failure

Material failure is implemented in the optimization routine as shown in Eq. 5.4.

σact =P

N ·AL(5.4)

Lower and Upper Bounds

In addition to the constraints derived from structural failure modes, the code constrains the

lower and upper bounds of each design variable. The lower bounds for the number of tows in the

helical and longitudinal members are based on typical manufacturing practice.

5.5.5 Problem Definition

The problem definition of the optimization analysis is summarized mathematically in the

following statement:

minimize M

with respect to Nb,NtL ,NtH ,D

subject to λg > 1.0

λsb > 1.0

λl > 1.0

σact < σult

[Nb NtL NtH D]> [20 8 2 4]

[Nb NtL NtH D]< [100 13 5 8]

(5.5)

73

5.6 Procedure

Prescribed design criteria are provided as input parameters to be used in both the gradient-

based and gradient-free optimization methods. The optimizer uses the micromechanics equations

presented by Kollar and Springer [7] that are provided in Appendix A to calculate the composite

properties from fiber properties, matrix properties, and the fiber volume ratio. The composite

properties are stored in an object class that can be referenced throughout the analysis.

The gradient-free optimizer, NGPM, computes the corresponding mass and constraints for

100 generations, each with 50 design points. The results of each sample is compiled in a text file.

The text file is processed, the optimum point of each generation is plotted with its correspond-

ing generation number, and the viable sample with the lowest mass is output in the results. The

gradient-based optimizer, ‘fmincon’, initially optimizes the mass with continuous design variables.

Once optimized, the discrete design variables (i.e., Nb, NtL , and NtH ) are rounded up to the nearest

integer, and the outer diameter is re-optimized as a continuous variable. Fig. 5.2 summarizes the

optimization process.

Figure 5.2: Process of optimization analysis.

74


The gradient-based optimizer consistently produces the same optimum variables, provided

that the initial design variables are not changed. The solution changes slightly if the initial design

variables are changed. This could indicate multiple local minima in the design space. Conversely,

the gradient-free optimizer produces a different optimum with each iteration. The gradient-free

results of one iteration are shown in Table 5.2 together with the results of the gradient-based opti-

mizer, and the results of experimental specimen, Rackliffe4. Rackliffe4 was used as the baseline

design for the fixed parameters provided to the optimizer.

Table 5.2: Optimized Results Compared to Rackliffe Specimen [2]

Gradient-based Gradient-free Rackliffe4 SpecimenMass [lb. (kg)] 0.258 (0.117) 0.217 (0.098) 0.353 (0.160)

[Nb NtL NtH D [in.]] [52 9 2 6.05] [37 8 2 6.42] [53 11 3 5.90]

Constraints [4.7e-6 -0.28 -1.5 -3.1e5] [2.8e-6 0.49 -5.7e-3 -3.1e5] [-0.16 -0.98 -2.9 -3.2e5]

Function Calls 199 5000 N/A

For the iterations shown in Table 5.2, the gradient-based and gradient-free methods pro-

duced a similar optimum mass and corresponding design variables. Both optimizers produced

specimens with a lower mass than that tested experimentally. In a subsequent iteration, the lower

and upper bounds were adjusted to observe how the results would be affected. The results from

the gradient-free optimizer varied significantly with slight changes in the bounds, and the lower

bound of the number of longitudinal tows governed the result. Varying the bounds did not affect

the optimum produced by the gradient-based optimizer, unless the lower bound for the number of

longitudinal tows was increased more than the preceding optimum.

With each analysis, the number of helical tows resorted to the prescribed lower bound.

In the analytical approach taken in this study, the helical members do not explicitly affect the

constraint violations because the boundaries are approximated as pinned-pinned. In a discrete

approach with comprehensive finite element analysis, the number of helical tows may demonstrate

a greater influence.

75

The number of function calls required by the gradient-free optimizer is consistently 5000.

The number of function calls is the product of the number of generations and the population size of

each generation. Conversely, the gradient-based optimizer required only 199 function calls. Even

though the gradient-based optimizer required less function calls, neither method required more

than thirty seconds to run. Fig. 5.3 shows a typical convergence rate of the gradient-based method

and the gradient-free method with respect to the iteration/generation. The gradient-based (1x) and

gradient-based (2x) data demonstrate the convergence before and after the discrete variables are

fixed, respectively. The figure suggests that the minimum produced by the gradient-free optimizer

is more optimal than the gradient-based optimum, however, the settings of the NSGA-II toolbox

were such that the constraints could be violated according to an internally prescribed factor. The

gradient-free analysis could be improved by redefining the internal settings of the NGPM program

to prohibit constraint violations.

Figure 5.3: Convergence of gradient-based and gradient-free optimization algorithms.

76

5.8 Summary

5.8.1 Conclusion

Both the gradient-based and gradient-free methods produced IsoTruss structures with less

mass than the baseline design produced for experimental testing. Both methods are capable of

producing discrete or continuous design variables as appropriate. The gradient-based and gradient-

free optimizers performed the optimization under 30 seconds with 199 and 5000 function calls,

respectively. While the gradient-free optimizer produced a different optimum with each analysis,

the gradient-based method produced consistent results, provided that the initial design variables

were consistent. The gradient-based optimizer is preferred over the gradient-free optimizer for its

consistency and robustness. It did not vary with changes in bounds and produced a design that

followed general manufacturing practices.


The gradient-based optimizer can be developed further to improve the optimality of the

solution and increase the accuracy of the analytical expressions. The optimality can be improved

by using more accurate methods to calculate the gradients. Algorithmic differentiation is recom-

mended since it can be implemented with relatively few changes to the source code and calculates

accurate gradients. The optimality could also be improved by optimizing with diverse initial design

variables to determine the global optimum from potential local minima.

To increase the accuracy of the analytical predictions, the most critical improvement to be

made is in the prediction of shell-like buckling. According to Rackliffe [5], shell-like buckling is

the governing failure mode for extremely light-weight IsoTruss structures with bay lengths greater

than 2.5 in. (64 mm) that possess the distinct design criteria of the specimen used as a base design

in this study. In the optimized configurations of this study, the bay length is about 2.45 in. (62 mm),

therefore shell-like buckling does not theoretically govern. An equation that accurately predicts

shell-like buckling will be critical in testing novel configurations in order to safeguard premature

failure. The mechanics-based method derived in Chapter 2 is implemented in subsequent studies.

77

CHAPTER 6. GRADIENT-BASED OPTIMIZATION OF ISOTRUSS® STRUCTURESIN UNIAXIAL COMPRESSION

6.1 Synopsis

The purpose of this study was to implement gradient-based optimization techniques in the

preliminary design process of IsoTruss® structures to minimize mass with respect to prescribed

loading criteria. The optimization framework developed in Chapter 5 was enhanced to improve

the analytical expressions used to define the constraints and enhance the optimality of the solution.

Improvements include: verifying and implementing the analytical expression derived in Chapter 2;

applying algorithmic differentiation to calculate the gradients of the objective and constraint func-

tions; analyzing the sensitivity derivatives and Lagrange multipliers; and, exploring the modality

of the design space to identify local minima. The optimization analysis identified two local minima

in the defined design space. The enhanced optimization routine provided herein is implemented in

subsequent chapters to compare the structural performance and efficiency of novel configurations

of IsoTruss structures.

6.2 Nomenclature


Atow Cross-sectional area of fiber tow


DX Equivalent plate flexural rigidity of the planar X-style helical trusses


E f 1 Young’s modulus of the fiber in compression

Em Young’s modulus of the matrix in compression


H Overall length of helical member

78


L Overall length of IsoTruss

M Mass

N Number of nodes

Nb Number of bays


P Total applied axial load

Pg Buckling load of global buckling mode

Psb Buckling load of shell-like buckling mode

Pl Buckling load of longitudinal strut buckling mode






α Rotational stiffness

δ2 One-half the base length of the planar X-style helical members

φ Inclination angle of helical member relative to the zθ -plane

λg Eigenvalue of global buckling

λsb Eigenvalue of shell-like buckling

λl Eigenvalue of longitudinal strut buckling

λm Lagrange multiplier

µsb Local boundary constraint coefficient for shell-like buckling

µl Local boundary constraint coefficient for longitudinal strut buckling

νzθ Poisson’s ratio in the zθ -plane (for a planar X-section)

ρ Material density

σult Ultimate strength of member (material property)

σact Actual design stress in member


N

)

79

6.3 Introduction

The optimization study presented in the preceding chapter, Chapter 5, is a preliminary anal-

ysis performed to select an optimization technique to minimize the mass of IsoTruss structures.

The preliminary analysis indicates that both gradient-based and gradient-free optimization are ef-

ficient approaches, however, gradient-based techniques are selected over gradient-free techniques

for further development in the current study. The purpose of the current study is to enhance the

gradient-based optimization analysis so that it can be applied to the preliminary design process of

IsoTruss structures. Improvements are made to increase the accuracy of the analytical expressions

used to predict structural failure and enhance the optimality of the optimized solution.

Improvements to the gradient-based optimization include: verifying and implementing the

analytical expressions derived in Chapter 2; applying algorithmic differentiation to calculate the

gradients of the objective and constraint functions; analyzing the sensitivity derivatives and La-

grange multipliers; and, exploring the modality of the design space for local minima.

6.4 Background

The optimization analysis performed in Chapter 5 is a preliminary analysis to select either

gradient-based or gradient-free methods for further development. The optimization framework

therein implements analytical expressions to predict structural failure and to constrain the optimal

solution. The analytical expressions include equations for global buckling, shell-like buckling, lon-

gitudinal strut buckling, and material failure. All equations are derived explicitly from traditional

mechanics except shell-like buckling.

The expression for shell-like buckling uses an empirically-derived correlation factor, βsb,

to correlate the Euler-buckling equation to finite element (FE) predictions of the critical shell-

like buckling load of Rackliffe et al. [2] specimens IsoTruss1 and IsoTruss2. While the empirical

expression is sufficient for the purposes of the preliminary analysis, the current study employs

the expression derived in Chapter 2. The shell-like buckling expression presented in Chapter 2

is verified with FE models and validated with experimental data provided by Rackliffe et al. in

Chapter 3. Only two experimental specimens failed in local (i.e., shell-like) buckling, providing

80

only two data points for validation. The FE difference relative to the mechanics equation varies

from about 75% (Rackliffe1 specimen) to about 35% (Rackliffe2 specimen).

Buragohain et al. [14] also use FE models to validate analytical expressions that predict

local buckling in composite hexagonal lattice cylindrical shells. In one analysis, the deviations

between FE predictions and analytical expressions vary from 4% to 25%. The variation between

specimens is influenced by the geometric configuration. For example, the specimen with 4% devi-

ation has a rib cross-sectional depth-to-width ratio of 0.5, whereas the specimen with 25% devia-

tion has a rib cross-sectional depth-to-width ratio of 1.5. Chapter 4 presents a dimensional analysis

that demonstrates how the deviation between analytical and FE predictions varies with respect to

diverse geometric configurations of IsoTruss structures. The deviation varies with respect to dif-

ferent ratios of geometric parameters, as manifest by Buragohain et al. Despite the deviations from

FE predictions, the analytical expressions are sufficiently validated to be used in the preliminary

design process.

6.5 Methods

The optimization framework developed in Chapter 5 is enhanced in the current study by

implementing the following improvements:

1. Verify and implement analytical expressions presented by Sui et al. [6] and Chapter 2;

2. Implement algorithmic differentiation to calculate gradients;

3. Redefine design variables to accommodate design rules of thumb; and,

4. Explore the multi-modal design space to identify global minimum.

6.5.1 Analytical Expressions

The analytical expressions used to predict failure are enhanced by verifying and incorpo-

rating the methods presented by Sui et al. and derived in Chapter 2. The global buckling, ultimate

stress, and mass expressions provided in Sui et al. yield the same results as the expressions imple-

mented in Chapter 5, however, the shell-like buckling and longitudinal strut buckling expressions

81

differ. The longitudinal strut buckling expression presented by Sui et al. and the shell-like buckling

expression derived in Chapter 2 are implemented in the current study. This section summarizes the

analytical expressions implemented in the optimization framework.

The expressions for global buckling and ultimate stress remain as previously defined in

Chapter 5 as Eq. 5.1 and Eq. 5.4, respectively. The expression that calculated the total mass in the

previous framework of Chapter 5 is verified with that of Sui et al. in Appendix B. Both expressions

produce the same result, therefore, the original equation is left unaltered and provided in Eq. 6.1

for reference.

M =ρ · (N ·AL ·L+2 ·N ·AH ·H)

H =Nb ·√

b2 +(D · sinθ)2 (6.1)

Equation 6.2 is the critical shell-like buckling load of IsoTruss structures derived in Chap-

ter 2. DX represents the equivalent plate flexural rigidity of the planar X-style helical members.

This method is implemented in the optimization framework of the current study to replace the

empirically-derived equation, Eq. 5.2 used in Chapter 5.

Psb =N · π2 ·Ez · IL

(µsb ·b)2

µsb =1+2 ·δ2 ·DX

Ez · IL·(

l1b+

1π· sin

π · l1b

)

DX =EX · (2 · rH)

3

12 · (1−ν2zθ)

EX =2 ·Ez ·π · r2

H2 ·δ2 · (2 · rH)

· sin4φ

(6.2)

Sui et al. present an expression for longitudinal strut buckling (referred to therein as mono-

cell buckling) that incorporates the stiffness coefficient, µl , derived by Rackliffe et al. [5]. Sui et

al. use an expression for the rotation stiffness due to bending, α , that represents the continuity of

82

the longitudinal members through the joint but neglects the helical members. This method (shown

in Eq. 6.3) is implemented in the code framework of the current study.


(µl · l2)2

µl =

(1+

α · l22 ·Ez · IL

)·(

1+α · l2Ez · IL

)−1

α =4 ·Ez · IL

l1

(6.3)

6.5.2 Algorithmic Differentiation

In the previous framework of the optimization analysis (implemented in Chapter 5), the

gradients were calculated using finite difference methods. Algorithmic differentiation (AD) is

implemented in the current study to increase the accuracy of the gradients supplied to the objective

and constraint functions. The AD method requires minimal changes to the original source code and

calculates gradients with greater accuracy than finite difference methods. The AutoDiff toolbox

was selected and implemented in the optimization routine [17]. The toolbox provides the gradients

for sensitivity analyses by producing the Jacobian matrix.

6.5.3 Design Variables

After implementing the AD toolbox, a sensitivity analysis was performed to calculate the

sensitivity derivatives of the objective and constraint functions with respect to the design vari-

ables. It shows that the number of helical tows contributed the least to the buckling capacity of the

structure. While increasing the number of helical tows does increase the resistance to shell-like

bucking, it does not increase the resistance to global buckling. In each optimization iteration, the

number of tows in each helical member resorted to the lower bound that was prescribed in the

problem definition. Consequently, the number of helical tows was removed as a design variable.

In the new optimization framework, the number of helical tows is calculated based on the number

of longitudinal tows and a prescribed, minimum ratio of helical-to-longitudinal tows. The ratio

83

is prescribed according to manufacturing rules of thumb, and is typically 0.25, 0.33, or 0.50, for

structures loaded primarily in compression.

6.5.4 Multi-modal Analysis

The preliminary gradient-based analysis presented in Chapter 5 demonstrates varying re-

sults depending on the initial dimensions provided to the optimizer. Varying solutions indicate the

possibility of a multi-modal solution, i.e., local minima in the design space. To identify the global

minimum, a multi-start approach is used in the current study to optimize the structure with diverse

initial dimensions.

A broad population of initial dimensions is generated using Latin-hypercube sampling. A

vector of six evenly spaced values is generated for each design variable (i.e., Nb, NtL , D) from the

lower and upper bounds. The mean of each vector and the covariance matrix of the combined

vectors are provided to the built-in Matlab function, ‘lhsnorm’. The lhsnorm function generates

a normally distributed, random sample of initial dimensions. Each set of initial dimensions is

optimized individually and the results of each optimization analysis are compared to identify a

global minimum.

6.5.5 Problem Definition

The problem definition of the optimization analysis is summarized mathematically in Eq.

6.4. The optimization framework is built to minimize the mass of the structure with respect to the

number of bays, the number of tows in each longitudinal member, and the outer diameter. The

minimization is subject to constraints that are defined by structural failure and upper and lower

bounds.

84

minimize M

with respect to Nb,NtL ,D


λsb > 1.0

λl > 1.0

σact < σult

[Nb NtL D]> [20 8 4]

[Nb NtL D]< [100 13 8]

(6.4)

The constraints that prevent global, shell-like, and longitudinal strut buckling are phrased

such that the load multipliers (i.e., the ratios of critical load to actual design load), λ , must be

greater than 1.0 (see Eq. 6.5).

λg =Pg

P

λsb =Psb

P

λl =Pl

P

(6.5)

The lower and upper bounds used in this optimization analysis were determined by refer-

encing typical designs of IsoTruss structures. By restricting the number of bays between 20 and

100, the bay length of a 115-inch (2.92-meter) IsoTruss structure is bounded between 1.15 and

5.75 in. (29.2 and 146 mm). The values of the bounds can be easily changed in the optimization

analysis to accommodate specific design criteria.

The fixed design parameters implemented in the optimization study are summarized in

Table 6.1. The properties are based on the experimental specimen Rackliffe4. The ‘crown’ refers

to the geometric configuration at the end of the IsoTruss structure. The specimen of interest does

85

not have a crown at the ends, therefore, the structure begins and ends with nodes rather than

transition nodes. The minimum ratio of helical-to-longitudinal tows (NtH -to-NtL) is 0.25.

Table 6.1: Material and Geometric Properties Implemented in Optimization Analysis

Property Implemented ValueFiber HexTow IM7 6K

Matrix UF3325-95

v f 0.65

E f 1 [106 psi (GPa)] 35.7 (246)

Em [106 psi (GPa)] 0.411 (2.83)

Ez [106 psi (GPa)] 23.3 (161)

νzθ 0.324

σult [ksi (MPa)] 326 (2250)

ρ

[lb.in.3

(kgm3

)]0.0491 (1360)

N 8

P [lb. (N)] 251 (1120)

L [in. (m)] 115 (2.92)

Crown None

Atow[in.2

(mm2)] 1.94e-04 (0.125)

NtH -to-NtL 0.25


The results of the optimization analysis and the corresponding properties of the experi-

mental specimen, Rackliffe4, are shown in Table 6.2 and Table 6.3, respectively. Of the thirty-six

analyses performed with varying initial dimensions, only two local minima were found. Local

Minimum 1 and Local Minimum 2 are 0.325 lb. (0.147 kg) and 0.372 lb. (0.169 kg), respectively,

therefore Local Minimum 1 is the global minimum of the design space. The Lagrange multipliers,

λm, are shown with respect to the lower bound, upper bound, and structural failure constraints (i.e.,

86

[λg λsb λl σult]). The Lagrange multiplier for the global buckling constraint is the largest, indicat-

ing that global buckling governs both minima. Conversely, the ultimate strength is the constraint

with the least effect.

Table 6.2: Mass and Dimensions of Optimized IsoTruss Structures

Property Local Minimum 1 Local Minimum 2Mass [lb. (kg)] 0.325 (0.147) 0.372 (0.169)

[Nb NtL NtH D [in. (mm)]] [45 12 3 5.24 (133)] [41 13 4 5.03 (128)]

λm (lower) 1.62e-8 1.95e-8

λm (upper) 7.22e-9 6.73e-9

λm (const.) [0.061 1.2e-7 2.2e-9 6.3e-14] [0.069 1.3e-7 2.3e-9 6.3e-14]

Table 6.3: Mass and Dimensions of Rackliffe4 Specimen

Property Rackliffe4 SpecimenMass [lb. (kg)] 0.353 (0.160)

[Nb NtL NtH D [in. (mm)]] [53 11 3 5.90 (150)]

The corresponding sensitivity derivatives of Local Minimum 1 and Local Minimum 2 are

shown in Eqs. 6.6 and 6.7, respectively.

JLM1 =

∂M∂Nb

∂M∂NtL

∂M∂D

∂λg∂Nb

∂λg∂NtL

∂λg∂D

∂λsb∂Nb

∂λsb∂NtL

∂λsb∂D

∂λl∂Nb

∂λl∂NtL

∂λl∂D

∂σact∂Nb

∂σact∂NtL

∂σact∂D

=

0.0260 0.0135 0.224

0 −0.0833 −0.382

−5.10 −19.1 0.0003

−45.0 −169 0

0 −729 0

(6.6)

87

JLM2 =

∂M∂Nb

∂M∂NtL

∂M∂D

∂λg∂Nb

∂λg∂NtL

∂λg∂D

∂λsb∂Nb

∂λsb∂NtL

∂λsb∂D

∂λl∂Nb

∂λl∂NtL

∂λl∂D

∂σact∂Nb

∂σact∂NtL

∂σact∂D

=

0.0333 0.0135 0.271

0 −0.0769 −0.397

−5.46 −17.2 0.0007

−48.1 −152 0

0 −621 0

(6.7)

Row 1 of Eq. 6.6 indicates that all design variables are directly proportional to the mass of

the IsoTruss structure. Row 2 of Eq. 6.6 implies that the outer diameter has the greatest relative

effect (i.e., inversely) on the global buckling load of the three design variables, while the number

of bays is negligible. Row 3 indicates that the shell-like buckling load is inversely proportional to

the number of bays and number of longitudinal tows and directly proportional to the outer diameter

(i.e., directly proportional to the bay length). Row 4 implies that buckling of the struts is inversely

related to the number of bays and the number of tows in each longitudinal member, and has no

relation with the outer diameter. The last row, Row 5, indicates that the ultimate stress is only

affected by the number of longitudinal tows, and is inversely proportional.

6.7 Summary

6.7.1 Conclusion

The gradient-based optimization techniques presented in Chapter 5 are improved by ver-

ifying and implementing the analytical expressions presented by Sui et al. [6], using algorithmic

differentiation to calculate the gradients of the objective and constraint functions, analyzing the

sensitivity derivatives and Lagrange multipliers, and exploring the modality of the design space

for local minima. Thirty-six initial dimensions were generated using Latin-hypercube sampling

to implement a globalization strategy and identify local minima in the design space. An opti-

mization analysis (with fixed design parameters based on the Rackliffe4 experimental specimen)

was performed on each set of initial dimensions and two local minima were identified. For the

configuration investigated (based on the design parameters summarized in Table 6.1), the global

minimum has a mass of 0.325 lb. (0.147 kg) and dimensions of 45 bays, 12 tows in each longitudi-

88

nal member, and an outer diameter of 5.24 in. (133 mm). The sensitivity derivatives and Lagrange

multipliers were calculated at the global minimum and indicate that global buckling governs the

optimal design.


The optimization framework developed in the current study provides an infrastructure for

the optimization of novel configurations of IsoTruss structures. The analytical expressions can

be applied to structures with different geometric configurations such as number of nodes, length,

compressive load, and/or outer longitudinal members. Chapter 7 includes an optimization analysis

of IsoTruss structures with outer longitudinal members. The gradient-based methods presented

herein are expanded to include outer longitudinal failure modes. Two recommendations are made

to increase the accuracy and robustness of the optimization analysis.

First, the analytical expression for shell-like buckling must be enhanced to reduce the de-

viation between the analytical predictions and the FE predictions. The inaccuracy of the shell-like

buckling load reduces the accuracy of the optimized results, which could lead to over- or under-

designed configurations.

Second, FE methods could be implemented in place of the analytical expressions to perform

the structural analysis within the optimization routine. While FE methods would significantly

increase the accuracy of the optimized results, the computational expense of each iteration would

also increase. Another challenge posed by intermediate FE analyses is generating the geometry

of the IsoTruss structures. The geometry of the FE models used in the preceding studies were

generated individually in IsoTruss.exe and ANSYS SpaceClaim (see Chapter 3). Autonomous

methods that generate the IsoTruss geometry from design parameters could be implemented in

ANSYS or other FE software.

89

CHAPTER 7. DIMENSIONAL ANALYSIS OF BUCKLING IN ISOTRUSS® STRUC-TURES WITH OUTER LONGITUDINAL MEMBERS

7.1 Synopsis

The purpose of this study was to analyze the buckling behavior of 8-node IsoTruss® struc-

tures with outer longitudinal members, loaded in uniaxial compression. IsoTruss structures with

outer longitudinal members have the same mass and helical configuration as IsoTruss structures

with inner longitudinal members (with equivalent dimensions), however, the longitudinal members

are placed at the outer diameter of the structure at the nodes. Finite element (FE) analyses were

performed in ANSYS WorkBench to predict the critical buckling loads of IsoTruss structures with

various dimensions. Buckingham’s Pi Theorem (BPT) was implemented in a dimensional analysis

to derive non-dimensional Π variables from the governing design parameters. The Π variables

are plotted to characterize trends between the governing design parameters and the corresponding

critical buckling loads. The relative performance of the outer longitudinal configuration (OLC) is

analyzed with respect to the internal longitudinal configuration (ILC) of an IsoTruss structure with

the same bay length, outer diameter, longitudinal radius, helical radius, and mass. The dimensional

analysis demonstrates that the buckling capacity of the ILC exceeds that of the OLC equivalent for

the dimensions that are fixed and tested herein. An optimization analysis was performed by opti-

mizing both the ILC and OLC configurations with respect to mass for the same load criteria. The

optimized OLC has 10% less mass than the ILC by reducing the outer diameter whilst maintaining

the same global moment of inertia. Additional research is needed to delineate the design spaces in

which the OLC and ILC govern strength-to-weight.

7.2 Nomenclature

AL Cross-sectional area of longitudinal member


90

Ig Moment of inertia of IsoTruss structure

IL Moment of inertia of longitudinal member

J Jacobian matrix of sensitivity derivatives

L Length of the global IsoTruss

M Mass

Nb Number of bays

Nt Number of carbon fiber tows in longitudinal (L) or helical (H) members

Pcr Critical buckling load

Pg Buckling load of the global buckling mode

Pb Buckling load of the bay-level buckling mode


R2 Statistical coefficient of determination (i.e., R-squared value)



kg Boundary constraint coefficient of global buckling


α Coefficient of quadratic term in best-fit curves

β Coefficient of linear term in best-fit curves

λ Load multiplier or Lagrange multiplier

µb Boundary constraint coefficient of bay-level buckling

Π Non-dimensional Π variables derived from Buckingham’s Pi Theorem

σact Actual applied stress

σult Ultimate stress (material property)

ξ Coefficient of power term in best-fit curves

7.3 Introduction

The purpose of the current study is to expand the scope of the analysis methods, presented

in preceding chapters, to 8-node IsoTruss structures with outer longitudinal members. The outer

91

longitudinal configuration (OLC) possesses the same geometric characteristics of the inner longi-

tudinal configuration (ILC) except that the longitudinal members are placed at the outer diameter

of the structure, spanning between the nodes. Figure 7.1 is the end view of an IsoTruss structure,

indicating the location of the longitudinal members in the inner and outer longitudinal configura-

tions. A side view of the OLC is shown in Fig. 7.2.

Figure 7.1: Location of longitudinal members in the: a) inner longitudinal configuration (green);and, b) outer longitudinal configuration (blue).

OLC and ILC configurations of equal bay length, outer diameter, and member radii are

equivalent in mass. By pushing the longitudinal members to the outer diameter, the global moment

of inertia of the structure is increased without increasing the mass. Hence, the OLC is inherently

more resistant to global buckling than the ILC of equal dimensions. On the other hand, the place-

ment of the longitudinal members in the OLC increases the span of the longitudinal struts, thereby

increasing the susceptibility to local buckling.

Due to inherent manufacturing complexity, experimental testing has not been widely per-

formed on the OLC, therefore, there is limited physical data to demonstrate the structural perfor-

mance and buckling behavior. The current study produces data from dimensional analysis (akin to

that performed in Chapter 4), finite element modeling, and optimization techniques (based on the

framework presented in Chapter 6 and provided in Appendix C) to explore four subtopics. First,

the data are used to characterize trends between the OLC design parameters and the buckling ca-

pacity. Second, the FE predictions are plotted with analytical predictions to verify the accuracy

92

Figure 7.2: IsoTruss structure with outer longitudinal members (i.e., OLC).

of an analytical expression presented herein. Third, the relative performance of the OLC with re-

spect to the ILC is analyzed via dimensional analysis. Finally, the OLC and ILC are optimized

with respect to mass to indicate potential advantages of each configuration under the same loading

criteria.

7.4 Methods

Three methods of analysis are implemented in the current study to analyze the buckling be-

havior of the OLC and compare its relative performance to the ILC. First, a dimensional analysis

is performed to characterize the interrelations between the governing design parameters and the

critical buckling load. The parameters are reduced to three non-dimensional independent Π vari-

ables via Buckingham’s Pi Theorem (BPT). Likewise, the critical buckling load is also reduced to

a non-dimensional term via BPT. Next, FE methods are used to predict critical buckling loads for

diverse structural configurations. FE analyses are performed in ANSYS WorkBench based on the

validated methods discussed in Chapter 3. The predictions are used to assess the relative accuracy

of analytical expressions for local buckling in the OLC. Finally, the optimization techniques pre-

93

sented in Chapter 6 are implemented to optimize the OLC and ILC with respect to mass. These

methods are expounded in the subsequent sections.

7.4.1 Dimensional Analysis

Buckingham’s Pi Theorem

The governing design parameters of the OLC are the same as those identified in Chapter 4 to

govern the buckling behavior of the ILC (i.e., rL, rH , b, R, and Ez). Therefore, the three independent

Π variables derived in Chapter 4 are used in the current study, and are provided in Eq. 7.1 for

reference. In the current study, both global and local buckling modes are considered, hence, the

critical buckling load, Pcr is selected as the dependent variable of interest in place of the shell-like

buckling load used in Chapter 4. While the global length, L, is not explicitly defined as a design

parameter in BPT, it is implicitly incorporated in the FE predictions of the critical buckling load of

the global buckling mode.

Π1 =rL

R

Π2 =rH

R

Π3 =bR

Π0 =Pcr

Ez ·R2

(7.1)

Trend Analysis

Trend analyses are performed for the OLC in the same manner as those performed in Chap-

ter 4 for the ILC. That is, a trend analysis is performed for each independent Π variable with respect

to the dependent Π variable. Each trend analysis consists of three sets of FE analyses, and each

set of FE analyses has different design parameters to demonstrate how the interrelations may vary

with respect to different geometric dimensions. Each set of geometric dimensions is distinguished

by the ratios Π3-to-Π2, Π3-to-Π1, or Π1-to-Π2 for the trend analyses of variables Π1, Π2, and

94

Π3, respectively. The independent Π variables and the Π ratios of each FE set are provided in

Table 7.1. The values of the variables were selected to provide Π ratios that are round numbers

within the design space of the long, light-weight IsoTruss structures typical of the Rackliffe et

al. [2] specimens.

Table 7.1: Fixed Π Variables of Dimensional Analyses

Π Variable Set (1) Set (2) Set (3)Π1 1.29 ·10−2 1.13 ·10−2 1.00 ·10−2

Π2 8.72 ·10−3 6.50 ·10−3 4.82 ·10−3

Π3 1.31 1.30 1.21

Π3/Π2 150 200 250

Π3/Π1 100 115 120

Π1/Π2 1.5 1.75 2

Trend analyses are also used in the current study to compare the relative performance of

the OLC and the ILC. A trend analysis is performed for each independent Π variable of the ILC

configuration using the design parameters of Set 2. The results of each ILC analysis are plotted

with the corresponding results of the Set 2 OLC analysis.

7.4.2 Finite Element Models

The FE analyses consist of static structural analyses and eigenvalue buckling analyses to

predict the critical buckling load and mode of each distinct configuration. The boundary conditions

were defined as fixed-free at the ends of the IsoTruss structure, and the compression load was

defined as 112 lb. (500 N). The density of the FE mesh was 0.25 in.−1 (10 m−1). The fixed design

parameters that correspond with each set of trend analyses described in the preceding section are

summarized in Table 7.2.

95

Table 7.2: Fixed Design Parameters of FE Analyses

Parameter Units Set (1) Set (2) Set (3)NtL N/A 11 12 13

rL [in. (mm)] 0.0323 (0.821) 0.0338 (0.859) 0.0351 (0.893)

NtH N/A 5 4 3

rH [in. (mm)] 0.0218 (0.554) 0.0195 (0.495) 0.0169 (0.429)

b [in. (mm)] 3.27 (83.1) 3.90 (99.1) 4.22 (107)

R [in. (mm)] 2.50 (63.5) 3.00 (76.2) 3.50 (88.9)

Nb N/A 30 25 23

L [in. (m)] 98.1 (2.49) 97.5 (2.48) 97.1 (2.47)

Ez [106 psi (GPa)] 23.3 (161) 23.3 (161) 23.3 (161)

The FE models demonstrate two general buckling modes: global buckling and local buck-

ling. The global buckling mode follows the typical model and expression of Euler-buckling of a

cantilever column. The local buckling mode is exhibited in the longitudinal members, buckling

either inward or outward symmetrically with a wavelength of two bays. Figures 7.3 and 7.4 are

images produced by the FE models in ANSYS Mechanical, demonstrating local buckling from the

side and top of the structure, respectively. The models shown in the images have a mesh density of

5.08 in.−1 (200 m−1).

Figure 7.3: Local buckling of OLC (side view).

96

Figure 7.4: Local buckling of OLC (top view).

7.4.3 Optimization Techniques

The gradient-based techniques presented in Chapter 6 are implemented in the current study

to optimize the OLC with respect to the same bounds and constraints as those imposed on the ILC

that was optimized in Chapter 6. The problem definition summarized as Eq. 6.4 in Chapter 6

includes a constraint for the longitudinal strut buckling mode, λl , and shell-like buckling mode,

λsb, that are typical for the ILC. Contrary to the shell-like buckling mode exhibited by the ILC, the

local bay-level buckling of the OLC demonstrates complete radial symmetry, with the longitudinal

struts all buckling either outward or inward at a given point in the cross-section. The optimization

code was updated to include a new calculation that predicts the local buckling mode in the OLC.

This local buckling mode, referred to as bay buckling, λb, is implemented to replace λl and λsb

of the OLC. The analytical expression that is implemented to predict local buckling in the OLC

is shown in Eq. 7.2. The boundary constraints imposed by the helical struts are approximated as

pinned joints.

97

Pb =N · π2 ·Ez · IL

(µb ·b)2

µb =1.0 (7.2)

The global buckling load is predicted using the Euler-buckling equation for a cantilever

column. The moment of inertia coefficient, c, is selected based on the derivation by Winkel [3] for

outer longitudinal members (see Eq. 7.3).

Pg =π2 ·Ez · Ig

(kg ·L)2

kg =2.0

Ig = c ·AL ·R2

c =4.0

(7.3)

The adjusted problem definition of the optimization analysis performed in the current study

is summarized mathematically in Eq. 7.4.

minimize M

with respect to Nb, NtL , D


λb > 1.0

σact < σult

[Nb NtL D]> [20 8 4]

[Nb NtL D]< [100 13 8]

(7.4)

98

7.5 Results

The results from the FE analyses and analytical predictions are presented as four subtopics

in the subsequent sections. The first two subtopics focus on characterizing the buckling behavior

of the OLC. First, the FE analyses of the OLC are used in trend analyses to assess the interrelations

between each independent Π variable and the dependent Π variable. Second, the analytical pre-

dictions of the OLC critical buckling loads are plotted with the FE predictions. The plots indicate

the extent to which the analytical expression adequately predicts critical buckling with respect to

the FE results. The next two subtopics compare the performance of the OLC with that of the ILC.

First, data collected for the OLC and ILC trend analyses are plotted together to indicate the rela-

tive performance of the configurations within the design space of the trend analyses. Second, the

analytical expression for bay-level buckling in the OLC is implemented in the gradient-based opti-

mization routine presented in Chapter 6 to compare the OLC and ILC structures that are optimized

for mass.

7.5.1 Trend Analyses of OLC

Data from the OLC trend analyses are first used to characterize trends between the non-

dimensional, independent Π variables. The independent Π variables Π1, Π2, and Π3 are plotted

against Π0 in Figures 7.5, 7.6, and 7.7, respectively. Local buckling loads are represented in the

plots by solid markers, whereas global buckling loads are represented by markers that are unfilled.

The dotted lines represent the best-fit curves.

Fig. 7.5 indicates that increasing Π1 induces a quadratic increase in Π0. It follows that

increasing the radius of the longitudinal members induces a quadratic increase in the critical buck-

ling load. Fig. 7.5 also indicates that the Π0 vs. Π1 curve shifts downward as the ratio of b-to-rH

increases. The general quadratic expression that relates Π1 to Π0 is provided in Eq. 7.5. The

coefficients of the quadratic expressions (i.e., α and β ) vary with the ratio of Π3-to-Π2. The co-

efficients and R-squared values that correspond to the curves shown in Fig. 7.5 are provided in

Table 7.3. The expressions are derived such that the ordinate intercept is set to zero.

99

Figure 7.5: OLC Π0 vs. Π1.

Π0 =α ·Π21 +β ·Π1 (7.5)


ΠΠΠ333///ΠΠΠ222 ααα βββ R2

150 1.81 ·10−2 −1.02 ·10−4 0.99200 1.28 ·10−2 −6.70 ·10−5 0.99250 1.11 ·10−2 −5.59 ·10−5 0.98

Fig. 7.6 indicates that Π0 increases with respect to increases in Π2. It follows that the

critical buckling load, Pcr, increases with respect to increases in the radius of the helical members,

until global buckling becomes the governing buckling mode. Once global buckling occurs, the

curve flattens with respect to Π2, as shown in the curve b/rH = 100 where Π2 is approximately

0.015. As the b-to-rL ratio increases, the Π0 vs. Π2 curve shifts downward. The generalized

quadratic expression that relates Π2 to Π0 is provided in Eq. 7.6. The coefficients of the expression

(i.e., α and β ) vary with the ratio of Π3-to-Π1. The coefficients that correspond to the curves shown

100

in Fig. 7.6 are provided in Table 7.4. The expressions are derived such that the ordinate intercept

is zero. The corresponding R-squared values are also provided in Table 7.4.


Π0 =α ·Π22 +β ·Π2 (7.6)


ΠΠΠ333///ΠΠΠ111 ααα βββ R2

100 5.60 ·10−3 1.44 ·10−4 0.99115 3.71 ·10−3 1.03 ·10−4 0.99200 3.68 ·10−3 8.89 ·10−5 0.99

Fig. 7.7 presents the interrelations of Π0 and Π3 for three values of the ratio rL-to-rH .

As the rL-to-rH ratio increases, the Π0 vs. Π3 curve shifts downward. The curve of best-fit that

characterizes the trends between Π0 and Π3 is a power curve, provided in general terms in Eq.

7.7. The coefficients, α and ξ , of the power curves vary with respect to the rL-to-rH ratio. The

coefficients are provided in Table 7.5 with the corresponding R-squared values.

101


Π0 =α ·Πξ

3 (7.7)


ΠΠΠ111///ΠΠΠ222 ααα ξξξ R2

1.5 2.55 ·10−6 −1.73 1.001.75 1.26 ·10−6 −1.78 1.00

2 7.19 ·10−7 −1.85 1.00

7.5.2 Analytical vs. FE Predictions of OLC

In this section, the analytical predictions of the critical buckling loads of the OLC are

compared with the FE predictions. Figures 7.8, 7.10, and 7.12 plot the analytical predictions and

FE predictions of Π0 vs. Π1, Π2, and Π3, respectively. The percent deviation (calculated via Eq.

7.8) of the analytical predictions with respect to the FE predictions is plotted against Π1, Π2, and

Π3 in Figures 7.9, 7.11, and 7.13, respectively. Solid lines represent the FE predictions whereas

the dashed lines indicate the analytical predictions.

102

Percent Deviation =Pcranal −PcrFE

PcrFE

·100 (7.8)

Figure 7.8: OLC Π0 vs. Π1 Analytical and FE predictions.

Figure 7.9: OLC Π0 vs. Π1 Percent deviation of analytical predictions from FE predictions.

103



104



105

7.5.3 Trend Analyses of OLC vs. ILC

In this section, the buckling capacity of the OLC and ILC are compared to assess the relative

performance of the two configurations. The independent Π variables of the OLC and ILC Set 2

configurations are plotted with respect to Π0 in Figures 7.14, 7.15, and 7.16.

Fig. 7.14 demonstrates the interrelation of Π0 and Π1 of both the ILC and OLC, where

the b-to-rH ratio is 200. Both configurations demonstrate a quadratic relation between Π1 and Π0.

While the ILC curve indicates a greater buckling capacity than the corresponding OLC curve, the

difference of the OLC curve relative to the ILC curve decreases dramatically from -60% to -4% as

Π1 increases from approximately 0.006 to 0.015.

Figure 7.14: Π0 vs. Π1 of OLC and ILC.

Fig. 7.15 demonstrates the interrelation of Π0 and Π2 of the ILC and OLC structures,

where the b-to-rL ratio is 115. The plot once again demonstrates that the ILC possesses greater

buckling capacity than the OLC for the Set 2 design space. As rH increases, the critical buckling

load of the ILC increases quadratically, whereas the critical buckling load of the OLC increases

more proportionally. At approximately Π2 of 0.010, the ILC buckling mode transitions from local

106

to global buckling, and the critical buckling load plateaus. Conversely, the OLC continues to be

controlled by local buckling. This can be attributed to the placement of the longitudinal members

at the outer diameter. With the longitudinal members at the outer diameter, the unbraced length of

the longitudinal struts is increased compared to the ILC equivalent. In addition, the global moment

of inertia of the OLC is greater than that of the ILC when both IsoTruss structures have the same

number of nodes and outer radius (see Winkel [3]). Thus, the OLC is more susceptible to bay-level

buckling and less susceptible to global buckling than the ILC equivalent. The difference of the

OLC curve relative to the ILC curve decreases to less than -1% when Π2 approaches 0.014.


Fig. 7.16 demonstrates the interrelation of Π0 and Π3 of the ILC and OLC, where the rL-

to-rH ratio is approximately 1.75. The critical buckling load of the ILC once again exceeds that

of the OLC in each case. The difference between the OLC design point and the ILC design point

increases from -2% to -17% as Π3 increases from 0.88 to 1.83.

107


7.5.4 Optimization of OLC vs. ILC

This final section incorporates the analytical expressions for the OLC configuration in the

gradient-based optimization routine. The OLC structure is optimized for mass with the same

bounds as the ILC structure presented in Chapter 6. The constraints are also maintained the same,

with the exception of local bay-level buckling. With the longitudinal members placed at the outer

diameter of the structure, the OLC is not susceptible to shell-like buckling. The local bay buckling

shown in Fig. 7.3 is the same failure mode as longitudinal strut buckling in the OLC.

Table 7.6 presents the dimensions and mass of the optimized OLC and ILC structures. The

ILC configuration is the Local Minimum 1 configuration from Chapter 6. The optimized OLC

has 10% less mass than the optimized ILC for the prescribed constraints and bounds. The OLC

optimum has more bays than the ILC optimum, however, the outer diameter of the OLC optimum

is 16% smaller than that of the ILC optimum. Even though the outer diameter has been reduced,

the global moment of inertia is the same between structures. The Lagrange multipliers, λm, are

shown with respect to the lower bound, upper bound, and structural failure constraints (λg, λsb, λl ,

and σult , respectively).

108

Table 7.6: Results of OLC and ILC Multi-modal Optimization Analysis

Property OLC Minimum ILC Local Minimum 1 OLC-to-ILC RatioMass [lb. (kg)] 0.292 (0.132) 0.325 (0.147) 0.898

Nb 51 45 1.13

NtL 10 12 0.833

NtH 3 3 1.00

D [in. (mm)] 4.39 (112) 5.24 (133) 0.838

b [in. (mm)] 2.26 (57.4) 2.56 (65.0) 0.882

Ig [in.4 (cm4)] 5.75e-2 (2.39) 5.76e-2 (2.40) 0.999

λm (lower) 5.19e-8 1.62e-8 -

λm (upper) 5.53e-9 7.22e-9 -

λm (gb) 5.7e-2 6.1e-2 -

λm (sb) N/A 1.2e-7 -

λm (b and l) 8.2e-7 2.2e-9 -

λm (σ ) 6.3e-14 6.3e-14 -

A local sensitivity analysis of each optimized configuration was performed by calculating

the sensitivity derivatives of the mass and the constraints with respect to each design variable. The

Jacobian matrices of the optimized OLC and ILC structures are presented in Eqs. 7.9 and 7.10,

respectively.

JOLC =

∂M∂Nb

∂M∂NtL

∂M∂D

∂λg∂Nb

∂λg∂NtL

∂λg∂D

∂λb∂Nb

∂λb∂NtL

∂λb∂D

∂σact∂Nb

∂σact∂NtL

∂σact∂D

=

0.0218 0.0135 0.253

0 −0.0100 −0.455

−4.02 −20.5 0

0 −1050 0

(7.9)

JILC =

∂M∂Nb

∂M∂NtL

∂M∂D

∂λg∂Nb

∂λg∂NtL

∂λg∂D

∂λsb∂Nb

∂λsb∂NtL

∂λsb∂D

∂λl∂Nb

∂λl∂NtL

∂λl∂D

∂σact∂Nb

∂σact∂NtL

∂σact∂D

=

0.0260 0.0135 0.224

0 −0.0833 −0.382

−5.10 −19.1 0.0029

−45.0 −169 0

0 −729 0

(7.10)

109

Row 1 of Eqs. 7.9 and 7.10 indicate that the design variables of both configurations are

directly proportional to the mass of the overall structures. Row 2 of Eqs. 7.9 and 7.10 imply that the

outer diameter has the greatest relative effect (i.e., inversely) on the global buckling load, while the

number of bays is negligible. Row 3 of Eq. 7.9 and Row 4 of Eq. 7.10 indicate that the longitudinal

buckling load of each configuration (manifest in the OLC as local bay-level buckling) is inversely

proportional to the number of bays and number of longitudinal tows (i.e., directly proportional

to the bay length). The sensitivity of the ILC with respect to the number of longitudinal tows is

steeper than that of the OLC. Row 4 of Eq. 7.9 and Row 5 of Eq. 7.10 imply that the ultimate

material stress is only affected by the number of longitudinal tows, and is inversely proportional.

The sensitivity of the material stress of the OLC optimum is much steeper than that of the ILC

optimum with respect to the number of longitudinal tows (for these particular optima).

7.6 Significance of Results

7.6.1 Influence of Helical Members

One of the most prominent themes from the analyses of the current study is the unprece-

dented contribution of the helical members to the critical buckling load of the OLC structures.

The influence of the helical members can be assessed by the OLC interrelation curves, the plots

comparing the analytical and FE predictions, and the percent deviation curves.

Figures 7.8, 7.10, and 7.12 demonstrate the extent to which the independent Π variables

influence the dependent Π variable predicted from both analytical and FE predictions. The shapes

of the curves shown in Fig. 7.8 and Fig. 7.12 indicate that the independent Π variables, Π1 and

Π3, induce similar effects in Π0 whether predicted using analytical or FE methods. Fig. 7.8, con-

versely, indicates that changes in Π2 affect the analytical and FE predictions differently. While the

analytical predictions are not affected by changes in Π2, the FE predictions indicate that increasing

Π2 increases the FE prediction of Π0.

Likewise, the percent deviation curves indicate that the radius of the helical members has

significant effect on the deviation between analytical and FE predictions. Fig. 7.9 demonstrates

that as the ratio b/rH decreases, the percent deviation curve is shifted downward, indicating an

increase in the percent deviation. Thus, if all design parameters are fixed, and the helical radius

110

is increased, the percent deviation will also increase. Fig. 7.11 demonstrates that as Π2 increases

along each curve, the percent deviation also increases until global buckling is induced, at which

point the critical buckling load is not changed with respect to the helical radius.

These results indicate that the analytical expression Eq. 7.2 can be improved by incorpo-

rating the helical radius. One method would be to include the helical radius in the calculation of

the boundary constraint coefficient, µb. The analytical expressions for ILC strut buckling and ILC

shell-like buckling both include derivations for boundary constraint coefficients as shown in Chap-

ter 6. The strut buckling derivation calculates the flexural rigidity of the helical struts at the nodes,

whereas the shell-like buckling derivation incorporates the bending energy from the intersecting

helical members.

Figures 7.17 and 7.18 are images produced from FE Models of OLC Set 2 structures. The

figures have the same design parameters except the helical radius. Figure 7.17 has two carbon

tows in the helical members, whereas Fig. 7.18 has eleven carbon tows in each helical member. By

increasing the number of carbon tows, the rotation at the IsoTruss nodes is noticeably decreased,

thereby increasing the flexural rigidity and localizing the deflection to the buckled longitudinal

strut. The colors of the figures represent deflection.

While the boundary constraint of Fig. 7.17 acts similar to a pinned connection, the bound-

ary constraint of Fig. 7.18 approaches the behavior of a fixed connection. The rotation of the

helical constraints at the nodes of the OLC are magnified for clarity in Fig. 7.19. Note that the

helical members with two carbon tows (in the left image) show enough rotation at the nodes to

resemble a smooth inflection point, whereas the nodes of the eleven tow helical members (shown

on the right) do not rotate as much, and flatten the longitudinal member at the nodes. It is recom-

mended that a boundary constraint coefficient be derived for bay buckling of OLC structures that

incorporates the flexural rigidity demonstrated in the images.

111

Figure 7.17: Local buckling of Set 2 OLC with two carbon tows in helical members.

Figure 7.18: Local buckling of Set 2 OLC with eleven carbon tows in helical members.

Figure 7.19: Rotation at the nodes of helical members with: a) two carbon tows; and, b) elevencarbon tows.

112

7.6.2 OLC vs. ILC Performance

The relative performance of the OLC and ILC with respect to buckling is assessed from the

comparative trend analyses and the optimization study. The trend analyses presented in Figures

7.8, 7.10, and 7.12 each demonstrate that the buckling capacity of the ILC structures exceeds that

of the corresponding OLC structures that possess the same outer radius, and are not independently

optimized. Furthermore, Fig. 7.10 demonstrates that the Π0 of the ILC structure increases quadrat-

ically with respect to Π2 until it transitions to global buckling. Conversely, the Π0 of the OLC

structure increases at a more shallow rate. The curves meet at approximately Π2 = 0.014 where

the OLC local buckling load corresponds with the ILC global buckling load.

While the design space of the trend analyses favored the ILC, the optimization analysis

favored the OLC where both configurations were optimized with respect to mass. The optimized

OLC has a shorter bay length, which increases the total mass due to the longer helical member

length, but the OLC also has a smaller outer diameter and fewer longitudinal tows. The bottom

line is that the OLC strength-to-weight exceeds that of the ILC, in part, by reducing the outer

diameter. The outer diameter is 16% smaller than the ILC configuration, and the overall weight

is reduced by 10%. The influence of the outer diameter in the ILC and OLC buckling behavior

could have been manifest in the dimensional analysis if a trend analysis had been performed with

respect to the outer diameter. One such analysis could be performed by plotting the Π variablesPcr

E·r2L

versus RrL

where R varies for a fixed value of rL.

7.7 Summary

7.7.1 Conclusion

The purpose of the current study is to characterize the buckling behavior of 8-node IsoTruss

structures with outer longitudinal members. A dimensional analysis is performed to analyze the

interrelations between the governing design parameters and the critical buckling load. The critical

buckling loads of diverse geometric dimensions are predicted using finite element (FE) modeling

in ANSYS WorkBench. The best-fit curves that indirectly relate the longitudinal radius, the helical

radius, and the bay length to the critical buckling load are characterized as quadratic and power

113

expressions. The FE predictions are also plotted with analytical predictions to assess the accuracy

of the analytical expression for bay-level buckling with respect to FE methods. Changes in the

longitudinal radius and the bay length induce similar trends in the FE and analytical predictions.

Increasing the helical radius, however, does not induce the same trends in the analytical and FE

predictions. While increasing the helical radius increases the FE prediction, there is no change in

the prediction from the analytical expression.

Trend analyses are also performed on corresponding 8-node IsoTruss structures with inner

longitudinal members. The buckling data of the inner longitudinal configurations (ILC) are plotted

with the data of the outer configurations (OLC) to analyze the relative performance of the config-

urations with respect to buckling resistance. Each plot indicates that the ILC has greater buckling

resistance than the outer longitudinal counter-part within the design space of the trend analysis

where the dimensions of the ILC and OLC are equivalent. The relative performance of the OLC

and ILC is also analyzed by optimizing both configurations with respect to mass. The optimized

structures are subject to the same bounds, and the constraints are defined by analytical expressions

that predict the relevant buckling modes of each configuration. The optimized OLC has 10% less

mass than that of the optimized ILC.


First, a boundary constraint coefficient should be derived for the analytical expression that

predicts local buckling in the OLC. The coefficient should incorporate the flexural rigidity of the

helical members, thereby capturing the effect of the helical radius on the buckling stability. Once

derived, another trend analysis of Π2 can be performed to determine if the analytical expression

and FE model predict similar trends in the local buckling load by varying rH . The improved

analytical expression could be re-implemented in the gradient-based optimization code to improve

the accuracy of the bay-level buckling constraint.

Second, additional research should be performed to delineate the design spaces where the

ILC and OLC are preferred. While the results of the trend analyses indicate that the ILC has

greater resistance to buckling than the OLC counter-part, the optimization analysis indicates that

the optimized OLC has less mass than the optimized ILC. The advantage can be attributed to the

fact that the OLC has a greater global moment of inertia than the ILC of equivalent outer radius.

114

The design space could be delineated by performing a trend analysis with respect to the outer

radius and the bay length.

115

CHAPTER 8. CONCLUSIONS AND RECOMMENDATIONS

8.1 Conclusions

The following conclusions are valid within the design space investigated. The analyses

were restricted to 8-node IsoTruss structures loaded in uniaxial compression, with fixed-free end

conditions. Geometric and material properties were defined based on graphite/epoxy specimens

that were manufactured for light-weight space applications. These structures are on the scale of 10

ft. (3 m) long, weighing approximately 0.3 lb. (0.14 kg). The bay length is analyzed from 2.17

to 6.25 in. (55 to 160 mm), with the outer diameter ranging between 4 and 8 in. (100 and 200

mm). The compression loads were defined in the finite element models based on the ultimate loads

demonstrated in experimental testing, and ranged between 125 and 265 lb. (560 and 1180 N). In

the finite element models used for dimensional analysis, the load was defined as 112 lb. (500 N).

8.1.1 Validity of Finite Element Modeling

The axial stiffness and global buckling loads of IsoTruss structures were predicted with

finite element models to determine the extent to which the finite element predictions correlate with

results from experimental testing and predictions from traditional mechanics. The finite element

predictions of axial stiffness and global buckling are within 1% and 3% of the predictions from

traditional mechanics, respectively. Therefore, it can be concluded that the finite element predic-

tions of axial stiffness and global buckling loads are adequately verified with respect to traditional

mechanics.

The finite element and mechanics predictions severely overestimate the axial stiffness and

global buckling capacity demonstrated by the specimens in experimental testing. The overesti-

mation can be attributed to a number of sources including: imperfections in the manufacturing,

material, or testing apparatus; limitations of the micromechanics equations used to calculate the

116

composite material properties; and/or, an inaccurate estimation of the fiber volume fraction. The

finite element models accurately and consistently predicted the experimental results when the ma-

terial properties were adjusted by a correlation factor. The correlation factor of one set of data

had a standard deviation of 5.6% for a set of specimens manufactured with consistent methods

and materials. The standard deviation increased if the specimens were manufactured with different

methods or different bay lengths. As manufacturing methods and configurations are standardized,

the correlation factors become more consistent, facilitating more precise finite element predictions

of experimental results.

The buckling modes exhibited in experimental testing were accurately predicted by the

finite element models and mechanics for all specimens except the specimen with a bay length of

2.5 in. (64 mm). This particular bay length has been previously identified as a critical transition

region between global buckling and local bay-level buckling [5].

8.1.2 Analytical Predictions of Bay-level Buckling

Bay-level buckling is manifest in inner longitudinal IsoTruss structures as shell-like buck-

ling, and outer longitudinal structures as bay buckling. An analytical equation was derived from

fundamental mechanics to predict shell-like buckling of 8-node IsoTruss structures with inner lon-

gitudinal members. A similar equation was written for bay buckling of 8-node IsoTruss structures

with outer longitudinal members. Both expressions underestimate the constraining effect imposed

by the flexural rigidity of the helical members on the longitudinal members. As a result, the analyt-

ical predictions are consistently lower (i.e., more conservative) than the finite element predictions.

The variation between the analytical and finite element predictions increases as the robustness of

the helical members increases, effectively changing the boundary constraint from almost pinned to

nearly fixed (within the defined limits).

8.1.3 Design Parameter Interrelations with Buckling Capacity

The critical design parameters that are optimized with respect to failure include the num-

ber of longitudinal tows in each member, the number of helical tows in each member, the outer

diameter, and the bay length. Of these parameters, increasing the number of helical tows is the

117

least effective for optimizing the strength-to-weight ratio of the structure. When performing an op-

timization analysis, it is more effective to optimize the number of helical tows based on heuristics

rather than via optimization algorithms. For example, the number of helical tows is determined

in the optimization code of Appendix C as a function of the number of longitudinal tows and a

helical-to-longitudinal ratio that is determined by a manufacturing rule-of-thumb. The remaining

design parameters are optimized by the gradient-based optimization algorithm.

8.1.4 Preferred Optimization Techniques

The gradient-based optimizer, ‘fmincon’, is preferred over the gradient-free optimizer,

NSGA-II for the analyses performed herein. The gradient-based analysis produced the same result

with each iteration, and identified local minima using the multi-start approach. Each iteration of

the gradient-free analysis, however, produced a different optimum, depending on the randomly

generated population and mutation of the analysis.

8.1.5 Relative Buckling Capacity of Outer vs. Inner Longitudinal Members

The optimization analysis, based on analytical expressions, indicates that the inner longi-

tudinal structure manufactured and tested by Rackliffe (Rackliffe4) could have been 8% lighter by

adding one longitudinal tow, increasing the bay length by 18%, and decreasing the outer radius

by 11%. The outer longitudinal structure, optimized for the same length and loading criteria, is

10% lighter than the optimized inner longitudinal structure. The governing failure mode of both

configurations is global buckling, and the optimized structures have equivalent moments of iner-

tia. The configuration with outer longitudinal members has superior strength-to-weight, within the

relevant design space, because the global moment of inertia is the same as the structure with inner

longitudinal members, but the outer diameter is 16% smaller. The optimized OLC is 10% lighter

than the optimized ILC structure.

8.2 Recommendations

First, the analytical predictions of shell-like buckling in the inner longitudinal configura-

tion and local buckling in the outer longitudinal configuration must be improved to reflect the

118

contribution of the helical members. The trend analyses of the inner and outer longitudinal con-

figurations both indicate that the analytical predictions of shell-like buckling and local buckling

underestimate the contribution of the helical members. The shell-like buckling expression could

be improved by incorporating the constraints imposed by the pyramidal style helical trusses. It

could also be improved by using finite element modeling to analyze the buckling behavior of the

planar X-style helical trusses. While the current derivation approximates the trusses as a plate in

bending, there may be another representation that is more accurate. The equation for bay-level

buckling of the outer longitudinal configuration could be improved by incorporating a boundary

constraint coefficient that includes the flexural rigidity at the ends of the longitudinal strut provided

by the connection to the helical members.

Second, the accuracy of the optimization analysis could be significantly increased by im-

plementing finite element methods in the structural analysis. In the current framework, analytical

expressions are implemented to predict structural failure. While these expressions are sufficiently

accurate for preliminary design, finite element methods could facilitate an exhaustive analysis of

the optimized design space, including additional loading scenarios. One disadvantage to imple-

menting finite element methods is the computational expense of the analysis. The current frame-

work can complete a multi-modal analysis with automatic differentiation within a matter of min-

utes, whereas the time and computational expense of a finite element analysis would increase the

computational expense of each iteration. Another challenge posed by finite element methods is

to generate a unique configuration of the IsoTruss with each iteration. The geometry of the finite

element models analyzed in this research were generated individually for distinct geometric con-

figurations, except for the models analyzed in the trend analyses where the bay length and outer

radius were held constant. The geometry could potentially be autonomously generated by using

the IsoTruss.exe software or Mechanical APDL that use batch file analysis.

Finally, the dimensional and optimization analyses indicate different scenarios in which the

inner longitudinal and outer longitudinal configurations are preferred. Additional research should

be performed to delineate the design space in which the structure with outer longitudinal members

governs structural efficiency. The design space could be delineated by performing another dimen-

sional analysis that analyzes the buckling capacity of the outer and inner longitudinal configura-

tions with respect to the outer radius. These analyses should include additional loading scenarios

119

to indicate the relative advantages of the configurations with diverse types of loading. Identifying

the design space where the outer longitudinal members are preferred will indicate novel config-

urations that could be manufactured, tested, and implemented in diverse applications of IsoTruss

structures.

120

REFERENCES

[1] McCune, A. M., 2001. “Tension and compression of carbon/epoxy IsoTrussTM grid struc-tures.” Thesis, Brigham Young University, August. viii, 25, 27, 32, 33, 45

[2] Rackliffe, M. E., Jensen, D. W., and Lucas, W. K., 2006. “Local and global buckling of ultra-lightweight IsoTruss® structures.” Composites Science and Technology, 66(2), pp. 283–288.viii, ix, 3, 6, 14, 25, 27, 33, 45, 75, 80, 95, 126

[3] Winkel, L. D., 2001. “Parametric investigation of IsoTrussTM geometry using linear finiteelement analysis.” Thesis, Brigham Young University, December. viii, 1, 10, 24, 31, 45, 46,66, 71, 91, 98, 107, 126, 129, 130

[4] Kesler, S. L., 2006. “Consolidation and interweaving of composite members by a continuousmanufacturing process.” Thesis, Brigham Young University, December. ix, 7, 12, 72, 126,128, 129

[5] Rackliffe, M. E., 2002. “Development of ultra-lightweight IsoTrussTM grid structures.” The-sis, Brigham Young University, December. 6, 12, 38, 77, 82, 117

[6] Sui, Q., Fan, H., and Lai, C., 2015. “Failure analysis of 1D lattice truss composite structurein uniaxial compression.” Composites Science and Technology, 118, pp. 207–216. 7, 13, 14,16, 22, 25, 81, 88, 126, 128, 129, 130, 131

[7] Kollar, L. P., and Springer, G. S., 2003. Mechanics of Composite Structures. CambridgeUniversity Press. 9, 27, 74, 123

[8] Megson, T. H. G., 2013. Introduction to Aircraft Structural Analysis. Butterworth-Heinemann. 13, 14, 17, 22

[9] Kollar, L., and Hegedus, I., 1985. “Analysis and design of space frames by the continuummethod.” Developments in Civil Engineering, 10. 19

[10] Fan, H., Meng, F., and Yang, W., 2006. “Mechanical behaviors and bending effects of carbonfiber reinforced lattice materials.” Archive of Applied Mechanics, 75(10-12), pp. 635–647.20

[11] Lai, C., Wang, J., and Liu, C., 2014. “Parameterized finite element modeling and bucklinganalysis of six typical composite grid cylindrical shells.” Applied Composite Materials, 21(5),pp. 739–758. 46

[12] Belardi, V., Fanelli, P., and Vivio, F., 2018. “Structural analysis and optimization of anisogridcomposite lattice cylindrical shells.” Composites Part B: Engineering, 139, pp. 203–215. 46,67

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[13] Sayad, K., 2010. “Optimization for minimum mass of composite grid stiffened cylindri-cal shells under axial compressive load based on analyitical and graphical methods.” The2nd International Conference on Composites: Characterization, Fabrication and Application(CCFA-2), 12. 67

[14] Buragohain, M., and Velmurugan, R., 2009. “Buckling analysis of composite hexagonallattice cylindrical shell using smeared stiffener model.” Defence Science Journal, 59(3). 67,81

[15] Kim, Y., Kim, P., Kim, H., and Park, J., 2019. “An optimization of composite lattice cylinderusing the approximate method.” Advanced Composite Materials, 28(3), pp. 287–320. 68

[16] Song, L., 2011. “NGPM: A NSGA-II Program in Matlab.” User Manual. 68

[17] Reif, U., 2020. “AutoDiff R2016b.” MATLAB Central File Exchange. 83

122

APPENDIX A. COMPOSITE MICROMECHANICS EQUATIONS

The following equations (except Eqs. A.1 and A.2) are presented by Kollar and Springer

[7] to calculate the orthotropic material properties of a composite material from the constitutive

properties of the fiber and resin. These equations are implemented in Chapter 3 to calculate the

composite material properties to be used in the finite element models. The equations are also used

in the optimization code of Chapter 6 and Chapter 7 that is provided in Appendix C.

Longitudinal compression modulus of fiber (approximation)

E f 1 = E f 1c = E f 1t ·Ecompositec

Ecompositet

(A.1)

Transverse compression modulus of fiber (approximation)

E f 2 = 0.10 ·E f 1 (A.2)

Longitudinal compression modulus

E1 = v f ·E f 1 + vm ·Em (A.3)

Transverse compression modulus

E2 =

(v f

E f 2+

1− v f

Em

)−1

(A.4)

123

Longitudinal shear modulus

G12 =

(v f

G f 12+

1− v f

Gm

)−1

(A.5)

Transverse shear modulus

G23 =

(v f

G f 23+

1− v f

Gm

)−1

(A.6)

Longitudinal Poisson’s Ratio

ν12 = v f ·ν f + vm ·νm (A.7)

Transverse Poisson’s Ratio

ν23 =E2

2 ·G23−1 (A.8)

Composite Material Properties (rθz)

Ez

Eθ

Er

=

E1

E2

E2

(A.9)

Gzθ

Gzr

Gθr

=

G12

G12

G23

(A.10)

124

νzθ

νzr

νθr

=

ν12

ν12

ν23

(A.11)

125

APPENDIX B. GEOMETRIC CORRELATIONS OF ISOTRUSS® STRUCTURES

The geometric properties of IsoTruss structures have been defined and calculated using

diverse methods in various studies. The purpose of this section is to document the methods that

are relevant to this research and prove correlations between studies. The methods are cited from

studies performed by Winkel [3], Sui et al. [6], Kesler [4], and the chapters of this thesis. The terms

are correlated by comparing results that are calculated using the sample values from Rackliffe

Specimen 4 [2] (i.e., Racliffe4).

B.1 Nomenclature

c1 Fractional portion of b between upper and lower transition nodes (see [4])

h Length of helical segment

h1 Hypotenuse of a planar X-style helical truss

rL Radius of longitudinal member

rH Radius of helical member


Atow Area of one tow

LH Length of helical member from start to end of IsoTruss

N Number of nodes

NtL Number of tows in each longitudinal member


φ Helical angle


N

)

126

Additional nomenclature employed by the research studies affiliated with BYU (partic-

ularly by Winkel and Kesler) and the nomenclature employed by Sui et al. are documented in

Table B.1.

Table B.1: Varying Nomenclature of IsoTruss Structures

BYU Sui et al. Descriptionb LB Bay length

l1 L2 Short span of longitudinal member between transition nodes

l2 L1 Long span of longitudinal member between transition nodes

Ig Ic Global moment of inertia of IsoTruss structure

L H Global length of IsoTruss structure

N n Number of nodes

δ2 d2 One-half the base length of the planar X-style helical truss

B.2 Sample Values of Rackliffe4

N = 8

θ = 45°

NtL = 11

NtH = 3

Atow = 1.94 ·10−4 in.2 (0.125 mm2)

v f = 0.65

rL = 0.03233 in. (0.821 mm)

rH = 0.01688 in. (0.429 mm)

R = 2.95 in. (74.9 mm)

b = 2.17 in. (55.1 mm)

Nb = 53

127

B.3 Geometric Correlations

δ [in. (mm)] (Eqs. 1.8, 1.9, & 1.11 [4])

δ1 =R ·

(cos(θ)cos(

θ

2

))= 2.26 (57.3)

δ2 =R · cos(θ) · tan(

θ

2

)= 0.864 (21.9)

δ3 =R ·(

1− cos(θ)sin(θ)

)= 1.22 (31.0)

(B.1)

d [in. (mm)] (Eq. 10 [6])

d1 =R · sin(θ) = 2.09 (53.0)

d2 =R · cos(θ) · tan(

θ

2

)= 0.864 (21.9)

d1 −d2 =1.22 (31.0)

(B.2)

φ (Eq. 1 [6])

φ =arctan(

LB

2 ·R · sin(θ)

)= 0.480 (B.3)

l2 [in. (mm)] (Eq. 1.18 [4])

l2 =b ·(

11+ cos(θ)

)= 1.27 (32.3) (B.4)

128

φ (Fig. 1.6 [4])

φ =arctan

(l22

R · sin(θ)−R · cos(θ) · tan θ

2

)= 0.480 (B.5)

φ (Eq. 3.4 [3])

φ =arctan(

bR ·2 · sin(θ)

)= 0.480 (B.6)

LB (Eq. 1 [6])

LB =2 ·R · sin(θ) · tan(φ) = 2.17 (55.1) (B.7)

L1 (Eq. 1 [6])

L1 =LB

1+ cos(θ)= 1.27 (32.3) (B.8)

L2 (Eq. 2 [6])

L2 =L1 · cos(θ) = 0.899 (22.8) (B.9)

l1 (Eq. 1.23 [4])

l1 =b ·(

cos(θ)1+ cos(θ)

)= 0.899 (22.8) (B.10)

129

Ic [in.4 (mm4)] (Eq. 9 [6])

Ic =4 ·π · rL · (d21 +d2

2) = 0.067 (2.79 ·104) (B.11)

Ig [in.4 (mm4)] (Eq. 3.1 [3])

Ig =c ·AL ·R2 = 0.067 (2.79 ·104)

c =2.343 (8-node IsoTruss structures)(B.12)

M [lb. (kg)] (Eq. 6.1 Chapter 6)

D =2 ·R = 5.9 in. (150mm)

h =

√b2 +D2 · sin2(θ) = 4.70 in. (119mm)

Nb =53

LH =h ·Nb = 249 in. (6330mm)

M =ρ · (N ·AL ·L+2 ·N ·AH ·LH) = 0.324 (0.147)

(B.13)

M [lb. (kg)] (Eq. 4 [6])

M =2 ·n ·π ·H ·ρ · r2

Hsin(φ)

+N ·π ·H ·ρ · r2L = 0.324 (0.147) (B.14)

130

Directional Cosines (Chapter 2)

c1 =cos(θ)

1+ cos(θ)= 0.414

h =

√b2 +D2 · sin2(θ) = 4.70 in. (119mm)

h1 =c1 ·h2= 0.974 in. (24.7mm)

a =arcsin(

δ2

h1

)= 1.09

α =cos(a) = 0.461

(B.15)

Directional Cosines (Eq. 15 [6])

α = sin(φ) = 0.461 (B.16)

131

APPENDIX C. GRADIENT-BASED OPTIMIZATION CODE

The following pages document the gradient-based optimization code written in Matlab.

The optimization analysis is set up and run from the live script files, isotruss master sweep.mlx or

isotruss master.mlx. The live script files provide a framework to perform a multi-modal analysis

with thirty-six multi-start iterations (i.e., isotruss master sweep.mlx) or a single iteration analysis

with manually prescribed starting dimensions (i.e., isotruss master.mlx).

132

C.1 Multi-start Optimization Framework

IsoTruss Optimization - Multistart approach for multi-modal analysis

clc; clear; close all;

% Generate fiber material propertiesv_f = 0.65fiber = fiber_IM7_6K_Imperial;matrix = matrix_UF3325_95;compositeProps = compositeProps(fiber,matrix,v_f)

% Define fixed variables to be carried through optimizationN_nodes = 8; E_x = compositeProps.E_x;L = 115.01; %InchP = 250.6; %poundstype_longi = 0; %Internal = 1; External = 0;crown = 0; %Crown = 1; No Crown = 0;rho = compositeProps.rho; %pcisigma_ult = compositeProps.sigma_ult; %psiA_tow = fiber.A_tow; %inchnu_xy = compositeProps.nu_xy; HL_ratio = 0.25; %ratio of helical to longitudinal tows

% Variables = [N_bays, N_tow_L, N_tow_H, D]% Units = lbs, in

[p] = [N_nodes E_x L P type_longi crown rho sigma_ult A_tow v_f nu_xy HL_ratio];

lb = [20/10, 8, 4.0];ub = [100/10, 13, 8.0];

N_bay_vect = transpose(linspace(lb(1), ub(1), 6));N_tow_L_vect = transpose(linspace(lb(2), ub(2), 6));D_vect = transpose(linspace(lb(3), ub(3), 6));mean = [mean(N_bay_vect), mean(N_tow_L_vect), mean(D_vect)];dim_comb = horzcat(N_bay_vect,N_tow_L_vect,D_vect);cov = cov(dim_comb);

dim0_mat = lhsnorm(mean,cov,36);

options = optimoptions('fmincon', 'Display', 'iter-detailed', ... 'StepTolerance', 1e-10, ... 'MaxFunctionEvaluations', 3e10, ... 'MaxIterations', 1e06, ... 'SpecifyObjectiveGradient', true, ... 'SpecifyConstraintGradient', true, ... 'PlotFcn', @optimplotfval);

localMin = zeros(1,5);row = 1;while row < length(dim0_mat) dim0 = dim0_mat(row,:);

1

133

if fmincon_const(dim0, p) < 1 [dim_opt] = fmincon(@(dim)fmincon_calcMass_grad(dim, p),dim0,[],[],[],[], ... lb,ub,@(dim)fmincon_const_grad(dim, p),options); % Round up discrete variables N_bays = ceil(dim_opt(1) * 10); N_tow_L = ceil(dim_opt(2)); N_tow_H = ceil(N_tow_L * HL_ratio); D0 = dim_opt(3); % Reoptimize with discrete variables fixed (no scaling) fmincon_calcMass_fixed = @(D)%%2(N_bays, N_tow_L, N_tow_H, D, p); fmincon_const_fixed = @(D)fmincon_const2(N_bays, N_tow_L, N_tow_H, D, p); lb_D = lb(3); ub_D = ub(3); options = optimoptions('fmincon', 'Display', 'iter-detailed', ... 'StepTolerance', 1e-10, ... 'MaxFunctionEvaluations', 3e10, ... 'MaxIterations', 1e06, ... 'SpecifyObjectiveGradient', false, ... 'SpecifyConstraintGradient', false, ... 'PlotFcn', @optimplotfval); [D_opt, mass_opt, ~, ~, lambda] = fmincon(fmincon_calcMass_fixed,D0,[],[],[],[], ... lb_D,ub_D,fmincon_const_fixed,options); % Calculate the final constraint parameters dim_final = [N_bays, N_tow_L, D_opt] N_tow_H HL_ratio_final = N_tow_H / N_tow_L mass_final = mass_opt dv_final = [N_bays, N_tow_L, N_tow_H, D_opt] lagr_mult_lb = lambda.lower lagr_mult_ub = lambda.upper lagr_mult_const = lambda.ineqnonlin const_final = fmincon_const(dim_final, p); % Calculate the Jacobian [dobj/dx1 ... dobj/xn; dconst1/dx1 ... ] Jobj = transpose(autoDiff(@fmincon_calcMass_audi,dim_final,p,3,1)); Jconst = transpose(autoDiff(@fmincon_const_audi,dim_final,p,3,4)); J = cat(1,Jobj,Jconst) localMin = cat(1,localMin,[mass_final, dv_final]); row = row + 1; else row = row + 1; endend

2

134

C.2 Single Iteration Optimization Framework

IsoTruss Optimization - Single iteration

clc; clear; close all;

% Generate fiber material propertiesv_f = 0.65fiber = fiber_IM7_6K_Imperial;matrix = matrix_UF3325_95;compositeProps = compositeProps(fiber,matrix,v_f)

% Define fixed variables to be carried through optimizationN_nodes = 8; E_x = compositeProps.E_x;L = 115.01; %InchP = 250.6; %poundstype_longi = 1; %Internal = 1; External = 0;crown = 0; %Crown = 1; No Crown = 0;rho = compositeProps.rho; %pcisigma_ult = compositeProps.sigma_ult; %psiA_tow = fiber.A_tow; %inchnu_xy = compositeProps.nu_xy; HL_ratio = 0.25; %ratio of helical to longitudinal tows

% Variables = [N_bays, N_tow_L, N_tow_H, D]% Units = lbs, in

[p] = [N_nodes E_x L P type_longi crown rho sigma_ult A_tow v_f nu_xy HL_ratio];

dim0 = [39/10, 11, 5.9]; %[N_bay(scaled), N_tow_L, D]

% Boundslb = [20/10, 8, 4.0];ub = [100/10, 13, 8.0];

options = optimoptions('fmincon', 'Display', 'iter-detailed', ... 'StepTolerance', 1e-10, ... 'MaxFunctionEvaluations', 3e10, ... 'MaxIterations', 1e06, ... 'SpecifyObjectiveGradient', true, ... 'SpecifyConstraintGradient', true, ... 'PlotFcn', @optimplotfval);

if fmincon_const(dim0, p) < 1 [dim_opt] = fmincon(@(dim)fmincon_calcMass_grad(dim, p),dim0,[],[],[],[], ... lb,ub,@(dim)fmincon_const_grad(dim, p),options); % Round up discrete variables N_bays = ceil(dim_opt(1) * 10); N_tow_L = ceil(dim_opt(2)); N_tow_H = ceil(N_tow_L * HL_ratio); D0 = dim_opt(3); % Reoptimize with discrete variables fixed (no scaling)

1

135

fmincon_calcMass_fixed = @(D)fmincon_calcMass2(N_bays, N_tow_L, N_tow_H, D, p); fmincon_const_fixed = @(D)fmincon_const2(N_bays, N_tow_L, N_tow_H, D, p); lb_D = lb(3); ub_D = ub(3); options = optimoptions('fmincon', 'Display', 'iter-detailed', ... 'StepTolerance', 1e-10, ... 'MaxFunctionEvaluations', 3e10, ... 'MaxIterations', 1e06, ... 'SpecifyObjectiveGradient', false, ... 'SpecifyConstraintGradient', false, ... 'PlotFcn', @optimplotfval); [D_opt, mass_opt, ~, ~, lambda] = fmincon(fmincon_calcMass_fixed,D0,[],[],[],[], ... lb_D,ub_D,fmincon_const_fixed,options); % Calculate the final constraint parameters dim_final = [N_bays, N_tow_L, D_opt] N_tow_H HL_ratio_final = N_tow_H / N_tow_L mass_final = mass_opt dv_final = [N_bays, N_tow_L, N_tow_H, D_opt] lagr_mult_lb = lambda.lower lagr_mult_ub = lambda.upper lagr_mult_const = lambda.ineqnonlin const_final = fmincon_const(dim_final, p);

% Calculate the Jacobian [dobj/dx1 ... dobj/xn; dconst1/dx1 ... ] Jobj = transpose(autoDiff(@fmincon_calcMass_audi,dim_final,p,3,1)); Jconst = transpose(autoDiff(@fmincon_const_audi,dim_final,p,3,4)); J = cat(1,Jobj,Jconst) else msgbox('Initial guess does not satisfy constraints') end

2

136

C.3 Composite Material Properties

%% Calculate orthotropic material properties

classdef compositeProps < handle

properties

E_x

E_y

E_z

G_xy

G_xz

G_yz

nu_xy

nu_xz

nu_yz

rho

sigma_ult

end

methods

function obj = compositeProps(fiberProps,matrixProps,v_f)

obj.E_x = v_f * fiberProps.E_f1 + (1-v_f) * matrixProps.E_m;

obj.E_y = (v_f/fiberProps.E_f2 + (1-v_f)/matrixProps.E_m)^(-1);

obj.E_z = obj.E_y;

obj.G_xy = (v_f/fiberProps.G_f12 + (1-v_f)/matrixProps.G_m)^(-1);

obj.G_xz = obj.G_xy;

obj.G_yz = (v_f/fiberProps.G_f23 + (1-v_f)/matrixProps.G_m)^(-1);

obj.nu_xy = v_f * fiberProps.nu_f12 + (1-v_f) * matrixProps.nu_m;

obj.nu_xz = obj.nu_xy;

obj.nu_yz = obj.E_y / (2*obj.G_yz) - 1;

obj.rho = (fiberProps.rho_f * matrixProps.rho_m) / ...

(v_f*fiberProps.rho_f + (1-v_f)*matrixProps.rho_m);

obj.sigma_ult = (fiberProps.sigma_f_comp * v_f) + ...

(matrixProps.sigma_m * (1-v_f));

end

end

end

% Material properties of IM7 6K Fiber (Imperial Units)

classdef fiber_IM7_6K_Imperial < handle

properties

unit_length = 'inch'

unit_modulus = 'psi'

E_f1_tension = 40.03 * 10^6

ratio_compression_E = 0.891

137

ratio_compression_sigma = 0.62

nu_f12 = 0.32

rho_f = 0.0643

A_tow = 1.94 * 10^(-4)

sigma_f_ten = 799883

end

properties

E_f1

E_f2

G_f12

G_f23

sigma_f_comp

end

methods

function e_f1 = get.E_f1(obj)

e_f1 = obj.E_f1_tension * obj.ratio_compression_E;

end

function e_f2 = get.E_f2(obj)

e_f2 = obj.E_f1 * 0.1;

end

function g_f12 = get.G_f12(obj)

g_f12 = obj.E_f1 / (2*(1+obj.nu_f12));

end

function g_f23 = get.G_f23(obj)

g_f23 = obj.E_f2 / (2*(1+obj.nu_f12));

end

function Sigma_f_comp = get.sigma_f_comp(obj)

Sigma_f_comp = obj.sigma_f_ten * obj.ratio_compression_sigma;

end

end

end

% Material properties of UF3325 95 Matrix (Imperial Units)

classdef matrix_UF3325_95 < handle

properties

unit_length = 'inch'

unit_modulus = 'psi'

E_m = 0.41 * 10^6

nu_m = 0.33

rho_m = 0.0436

G_m

sigma_m = 10000

end

methods

function g_m = get.G_m(obj)

138

g_m = obj.E_m / (2*(1+obj.nu_m));

end

end

end

139

C.4 Objective Function

%% Calculate mass of IsoTruss structure

% Calculate mass of IsoTruss structure

% Called in the fmincon_calcMass_grad function

function [mass] = fmincon_calcMass(dim, p)

[N_bays] = dim(1)*10;

[N_tow_L] = dim(2);

[D] = dim(3);

%[N_nodes E_x L P type_longi crown rho sigma_ult A_tow v_f N_tow_H] = p;

N_nodes = p(1);

E_x = p(2);

L = p(3);

P = p(4);

type_longi = p(5);

crown = p(6);

rho = p(7);

sigma_ult = p(8);

A_tow = p(9);

v_f = p(10);

nu_xy = p(11);

HL_ratio = p(12);

% Calculate radii

r_L = calcRadius(v_f,N_tow_L,A_tow);

N_tow_H = ceil(N_tow_L * HL_ratio);

r_H = calcRadius(v_f,N_tow_H,A_tow);

theta = 360 / N_nodes;

b = L / N_bays;

A_L = pi()*r_L^2;

A_H = pi()*r_H^2;

h = sqrt(b^2 + D^2*sind(theta)^2); % Length of helical from node to node

h_1 = h/2*(cosd(theta)/(1+cosd(theta)));

h_2 = h/2*(1/(1+cosd(theta)));

if crown == 1

H = h * N_bays + 2*(h_2 + 2*h_1); % Length of one helical

else

H = h * N_bays;

end

N_H = 2 * N_nodes; % Number of helicals

mass = rho * (N_nodes * A_L * L + N_H * A_H * H);

end


% Objective function that supplies mass and gradient of obj function

% Called in isotruss_master_sweep

function [mass, mass_grad] = fmincon_calcMass_grad(dim, p)

140

[mass] = fmincon_calcMass(dim, p);

n_x = size(dim);

n_f = size(mass);

[mass_grad] = autoDiff(@fmincon_calcMass_audi, dim, p, n_x, n_f);

end


% Used in the intermediate analysis to handle audi variables for automatic

% differentiation

function [mass] = fmincon_calcMass_audi(dim, p)

[N_bays] = dim(1)*10;

[N_tow_L] = dim(2);

[D] = dim(3);


N_nodes = p(1);

E_x = p(2);

L = p(3)

P = p(4);

type_longi = p(5);

crown = p(6);

rho = p(7);

sigma_ult = p(8);

A_tow = p(9);

v_f = p(10);

nu_xy = p(11);

HL_ratio = p(12);

% Calculate radii


N_tow_H = ceil(N_tow_L.c(1) * HL_ratio);



b = L / N_bays;

A_L = pi()*r_L^2;

A_H = pi()*r_H^2;

h = sqrt(b^2 + D^2*sind(theta)^2); % Length of helical from node to node

h_1 = h/2*(cosd(theta)/(1+cosd(theta)));

h_2 = h/2*(1/(1+cosd(theta)));

if crown == 1


else

H = h * N_bays;

end



141

end


% Used in the second stage optimization to optimize the outer diameter

% after the discrete variables have been optimized and fixed

function [mass] = fmincon_calcMass2(N_bays, N_tow_L, N_tow_H, D, p)

%[N_nodes E_x L P type_longi crown rho sigma_ult A_tow v_f HL_ratio] = p;

N_nodes = p(1);

E_x = p(2);

L = p(3);

P = p(4);

type_longi = p(5);

crown = p(6);

rho = p(7);

sigma_ult = p(8);

A_tow = p(9);

v_f = p(10);

nu_xy = p(11);

% Calculate radii




b = L / N_bays;

A_L = pi()*r_L^2;

A_H = pi()*r_H^2;

h = sqrt(b^2 + D^2*sin(theta)^2); % Length of helical from node to node

h_1 = h/2*(cos(theta)/(1+cos(theta)));

h_2 = h/2*(1/(1+cos(theta)));

if crown == 1


else

H = h * N_bays;

end



end

142

C.5 Constraint Functions

%% Calculate structural failure constraints

% Calculate structural failure constraints

% Called in the fmincon_const_grad function

function [c, ceq] = fmincon_const(dim, p)

% Scaling

dim(1) = dim(1)*10;

dim(2) = dim(2);

dim(3) = dim(3);

%[N_nodes E_x L P type_longi crown rho sigma_ult A_tow] = p;

N_nodes = p(1);

E_x = p(2);

L = p(3);

P = p(4);

type_longi = p(5);

crown = p(6);

rho = p(7);

sigma_ult = p(8);

A_tow = p(9);

v_f = p(10);

nu_xy = p(11);

% c <= 0

% Buckling constraint

if type_longi == 1

[eigValue] = eigenBuckl_int(dim, p);

const_buck = 1 - eigValue;

elseif type_longi == 0

[eigValue] = eigenBuckl_ext(dim, p);


end

% Stress constraint

[sigma] = ultStress(dim, p);

const_stress = sigma - sigma_ult;

% Compile constraints

c = [const_buck, const_stress];

% Scale constraints to an order of one

c = [c(1), c(2), c(3), c(4)];

c = transpose(c);

ceq = [];

end


% Constraint function that supplies constraints and gradient of const function

% Called in isotruss_master_sweep

function [c, ceq, c_grad, ceq_grad] = fmincon_const_grad(dim, p)

143

[c, ceq] = fmincon_const(dim, p);

n_x = size(dim);

n_f = size(c);

[c_grad] = autoDiff(@fmincon_const_audi, dim, p, n_x, n_f);

[ceq_grad] = [];

end


% Used in the intermediate analysis to handle audi variables for automatic

% differentiation

function [c, ceq] = fmincon_const_audi(dim, p)

% Scaling

dim(1) = dim(1)*10;

dim(2) = dim(2);

dim(3) = dim(3);

%[N_nodes E_x L P type_longi crown rho sigma_ult A_tow] = p;

N_nodes = p(1);

E_x = p(2);

L = p(3);

P = p(4);

type_longi = p(5);

crown = p(6);

rho = p(7);

sigma_ult = p(8);

A_tow = p(9);

v_f = p(10);

nu_xy = p(11);

% c <= 0


if type_longi == 1

[eigValue] = eigenBuckl_int_audi(dim, p);



[eigValue] = eigenBuckl_ext_audi(dim, p);


end

% Stress constraint





% Scale constraints to an order of one

c = [c(1), c(2), c(3), c(4)];

144

c = transpose(c);

ceq = [];

end


% Used in the second stage optimization to optimize the outer diameter

% after the discrete variables have been optimized and fixed

function [c, ceq] = fmincon_const2(N_bays, N_tow_L, N_tow_H, D, p)

dim = [N_bays, N_tow_L, D];

%[N_nodes E_x L P type_longi crown rho sigma_ult A_tow v_f HL_ratio] = p;

N_nodes = p(1);

E_x = p(2);

L = p(3);

P = p(4);

type_longi = p(5);

crown = p(6);

rho = p(7);

sigma_ult = p(8);

A_tow = p(9);

v_f = p(10);

nu_xy = p(11);

HL_ratio = p(12);

% c <= 0


if type_longi == 1

[eigValue] = eigenBuckl_int(dim, p);



[eigValue] = eigenBuckl_ext(dim, p);


end

% Stress constraint





c = transpose(c);

ceq = [];

end

145

C.6 Failure Modes

%% Analytical expressions used to calculate failure modes

% Eigenvalue buckling of IsoTruss structures with outer longis

function [eigValue] = eigenBuckl_ext(dim, p)


N_nodes = p(1);

E_x = p(2);

L = p(3);

P = p(4);

type_longi = p(5);

crown = p(6);

rho = p(7);

sigma_ult = p(8);

A_tow = p(9);

v_f = p(10);

nu_xy = p(11);

HL_ratio = p(12);

[N_bays] = dim(1);

[N_tow_L] = dim(2);

[D] = dim(3);

% Calculate radii





b = L / N_bays;

% Global truss buckling

c = inertiaCoeff(p);

k = 2; % For fixed-free cantilever

A_L = pi()*r_L^2;

I_global = c * A_L * (D/2)^2;

P_cr_global = pi()^2 * E_x * I_global / (k * L)^2;

eig_global = P_cr_global / P;

% Local bay buckling (Outward buckling of outer longis; wavelength = 2*b)

I_L = pi()*(r_L*2)^4 / 64;

mu_l = 1;

P_cr_strut = N_nodes * pi()^2 * E_x * I_L / (b * mu_l)^2;

eig_local = P_cr_strut / P;

% Local strut buckling (same as Local Bay Buckling for OLC)

eig_bay = eig_local;

% Eigenvalues of each buckling mode

eigValue = [eig_global eig_bay eig_local];

end

% Eigenvalue buckling of IsoTruss structures with outer longis

% Formatted for audi variables for automatic differentiation

146

function [eigValue] = eigenBuckl_ext_audi(dim, p)


N_nodes = p(1);

E_x = p(2);

L = p(3);

P = p(4);

type_longi = p(5);

crown = p(6);

rho = p(7);

sigma_ult = p(8);

A_tow = p(9);

v_f = p(10);

nu_xy = p(11);

HL_ratio = p(12);

[N_bays] = dim(1);

[N_tow_L] = dim(2);

[D] = dim(3);

% Calculate radii





b = L / N_bays;




A_L = pi()*r_L^2;

I_global = c * A_L * (D/2)^2;



% Local bay buckling (Outward buckling of outer longis; wavelength = 2*b)

I_L = pi()*(r_L*2)^4 / 64;

mu_l = 1;

P_cr_strut = N_nodes * pi()^2 * E_x * I_L / (b * mu_l)^2;

eig_local = P_cr_strut / P;

% Local strut buckling (same as Local Bay Buckling for OLC)

eig_bay = eig_local;


eigValue = [eig_global eig_bay eig_local];

end

% Eigenvalue buckling of IsoTruss structures with inner longis

function [eigValue] = eigenBuckl_int(dim, p)


N_nodes = p(1);

E_x = p(2);

L = p(3);

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P = p(4);

type_longi = p(5);

crown = p(6);

rho = p(7);

sigma_ult = p(8);

A_tow = p(9);

v_f = p(10);

nu_xy = p(11);

HL_ratio = p(12);

[N_bays] = dim(1);

[N_tow_L] = dim(2);

[D] = dim(3);

% Calculate radii





b = L / N_bays;




A_L = pi()*r_L^2;

I_global = c * A_L * (D/2)^2;



% Local strut buckling (SUI; mono-cell buckling)

I_L = pi()*(r_L*2)^4 / 64;

l_1 = b * (cosd(theta)/(1+cosd(theta)));

l_2 = b / (1+cosd(theta));

l_strut = l_2;

alpha = 4 * E_x * I_L / l_1;

mu_m = (1 + alpha*l_2/(2*E_x*I_L)) * (1+alpha*l_2/(E_x*I_L))^(-1);

P_cr_strut = N_nodes * pi()^2 * E_x * I_L / (l_strut * mu_m)^2;

eig_longi = P_cr_strut / P;

% Local bay buckling (SUI; shell-like buckling; wavelength = 2*b)

theta_rad = theta * 2*pi() / 360;

d_2 = (D / 2) * cos(theta_rad) * tan(theta_rad/2); % Half the base of the X

phi_rad = atan(b / (D*sin(theta_rad))); % Helical angle

E_sb = 2*E_x*(pi()*r_H^2*sin(phi_rad)^4) / (2*d_2*2*r_H);

D_sb = E_sb * (2*r_H)^3 / (12*(1-nu_xy^2));

L_2 = l_1;

L_B = b;

mu_sb = sqrt( 1 / (1+2*d_2*D_sb/(E_x*I_L) * ...

(L_2/L_B + 1/pi() * sin(pi()*L_2/L_B))));

P_cr_bay = (N_nodes * pi()^2 * E_x * I_L) / (mu_sb * b)^2;

eig_bay = P_cr_bay / P;


eigValue = [eig_global eig_bay eig_longi];

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end

% Eigenvalue buckling of IsoTruss structures with inner longis

% Formatted for audi variables for automatic differentiation

function [eigValue] = eigenBuckl_int_audi(dim, p)


N_nodes = p(1);

E_x = p(2);

L = p(3);

P = p(4);

type_longi = p(5);

crown = p(6);

rho = p(7);

sigma_ult = p(8);

A_tow = p(9);

v_f = p(10);

nu_xy = p(11);

HL_ratio = p(12);

[N_bays] = dim(1);

[N_tow_L] = dim(2);

[D] = dim(3);

% Calculate radii





b = L / N_bays;




A_L = pi()*r_L^2;

I_global = c * A_L * (D/2)^2;



% Local strut buckling (SUI; mono-cell buckling)

I_L = pi()*(r_L*2)^4 / 64;

l_1 = b * (cosd(theta)/(1+cosd(theta)));

l_2 = b / (1+cosd(theta));

l_strut = l_2;

alpha = 4 * E_x * I_L / l_1;

mu_m = (1 + alpha*l_2/(2*E_x*I_L)) * (1+alpha*l_2/(E_x*I_L))^(-1);

P_cr_strut = N_nodes * pi()^2 * E_x * I_L / (l_strut * mu_m)^2;

eig_longi = P_cr_strut / P;

% Local bay buckling (SUI; shell-like buckling; wavelength = 2*b)

theta_rad = theta * 2*pi() / 360;

d_2 = (D / 2) * cos(theta_rad) * tan(theta_rad/2); % Half the base of the X

phi_rad = atan(b / (D*sin(theta_rad))); % Helical angle

E_sb = 2*E_x*(pi()*r_H^2*sin(phi_rad)^4) / (2*d_2*2*r_H);

D_sb = E_sb * (2*r_H)^3 / (12*(1-nu_xy^2));

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L_2 = l_1;

L_B = b;

mu_sb = sqrt( 1 / (1+2*d_2*D_sb/(E_x*I_L) * ...

(L_2/L_B + 1/pi() * sin(pi()*L_2/L_B))));

P_cr_bay = (N_nodes * pi()^2 * E_x * I_L) / (mu_sb * b)^2;

eig_bay = P_cr_bay / P;


eigValue = [eig_global eig_bay eig_longi];

end

% Material failure from ultimate compression stress

function [sigma] = ultStress(dim, p)

[N_bays] = dim(1);

[N_tow_L] = dim(2);

[D] = dim(3);

%[N_nodes E_x L P type_longi crown rho sigma_ult D] = p;

N_nodes = p(1);

E_x = p(2);

L = p(3);

P = p(4);

type_longi = p(5);

crown = p(6);

rho = p(7);

sigma_ult = p(8);

A_tow = p(9);

v_f = p(10);

nu_xy = p(11);

% Calculate radius


A_L = pi()*r_L^2;

sigma = P / (N_nodes * A_L);

end

150

C.7 Supplementary Functions

%% Supplementary functions

% Algorithmic Differentiation

% Forward mode

% Using Auto_Diff2016b Toolbox

function [J] = autoDiff(func, dim, p, n_x, n_f)

% Convert design variables A into dual numbers

dim = amatinit(dim, 1);

y = func(dim, p);

% Calculate the first partial derivative **NOT FORMATTED CORRECTLY**

solution = adiff(y);

% Extract jacobian from the output

J = transpose(ajac(solution, 0));

end

% Calculate the radius of the longitudinal and helical members from the

% fiber volume fraction, number of tows, and cross-sectional area of fiber

% tow

function r_strut = calcRadius(v_f, N_tow, A_tow)

A_strut = N_tow * A_tow / v_f;

d_strut = sqrt(4*A_strut/pi());

r_strut = d_strut / 2;

end

% Select the appropriate moment of inertia coefficient based on the

% outer/inner longitudinal configuration and the number of nodes

function [c] = inertiaCoeff(p)

%[N_nodes E_x L P type_longi crown rho sigma_ult D] = p;

N_nodes = p(1);

E_x = p(2); %Pascals

L = p(3); %Meters

P = p(4); %Newtons

type_longi = p(5);

crown = p(6);

rho = p(7); %kg / m^3

sigma_ult = p(8); %Pascals

A_tow = p(9); %inch

v_f = p(10);

nu_xy = p(11);

nodes = {6, 7, 8, 9, 10, 11, 12};

if type_longi == 1

coeff = [1 1.676 2.343 2.990 3.618 4.228 4.823];

else

coeff = [3 3.5 4 4.5 5 5.5 6];

end

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coeff_table = containers.Map(nodes,coeff);

c = coeff_table(N_nodes);

end

152

investigation of isotruss® structures in compression using

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