The Pennsylvania State University

The Graduate School

Department of Engineering Science and Mechanics

DESIGN ANALYSIS OF THE STRESS DISTRIBUTION AT THE

INTERFACE OF A SKIN-STIFFENER

A Thesis in

Engineering Mechanics

by

Jamal Carver

Submitted in Partial Fulfillment

of the Requirements

for the Degree of

Master of Science

May 2010

The thesis of Jamal Carver was reviewed and approved* by the following:

Clifford J. Lissenden

Professor of Engineering Science and Mechanics

Thesis Advisor

Nicholas J Salamon

Professor Emeritus Engineering Science and Mechanics

Reginald F. Hamilton

Assistant Professor of Engineering Science and Mechanics

Judith A. Todd

Professor of Engineering Science and Mechanics

Department Head of Engineering Science and Mechanics

*Signatures are on file in the Graduate School

iii

ABSTRACT

Engineers service humanity by building things and thus students need to know about the

design process. The EMCH 213D gives the student an opportunity to better comprehend design

by its increased focus and the semester long project. The students are asked to use simplified

design principles to build a combination playground. Using the principles of simplified design

this thesis set out to analyze an actual structure where the design analysis presents challenges.

The structure that is chosen is an aircraft fuselage with many stiffeners attached to it. This is

difficult to analyze due to the complicated geometry and singularities that arise. The method our

study undertook was to analyze the stress distribution at the interface, by using a sample

representative substructure of an aircraft skin with a perfect adhesively bonded „T‟ stiffener.

Finite element analysis was used which allowed for a variety of representative loading

conditions that the aircraft fuselage may be subjected to. The study on the interface analysis

consisted studying the effect that the stiffener material and geometry has on the stress distribution

at the interface. The skin was selected as 2014-T6 aluminum and the stiffener was either the

2014-T6, A36 structural steel or a unidirectional graphite fiber reinforced polymer (GFRP). The

stiffener geometry was changed by tapering the stiffener flanges by either a 1:1 or 1:4 ratio of

thickness to length.

Results showed the effect material discontinuity has on the stress distribution within the

skin and stiffener. Singularities often occurred at the edges of the interface and the skin stress of

had higher magnitudes than the stiffener stress. The GFRP stiffener provided the lowest stresses

because of the increased stiffness in the fiber direction. The 1:1 taper did nothing to lower the

edge singularities as the change in geometry was still too sharp. The 1:4 taper did effectively

lower the singularities in both the skin and stiffener and allowed for the singularities to be

distributed along the contact interface which results in a more uniform stress distribution.

iv

TABLE OF CONTENTS LIST OF FIGURES ........................................................................................................ vi LIST OF TABLES ......................................................................................................... x ACKNOWLEDGEMENTS ........................................................................................... xi

Chapter 1 Introduction ........................................................................................................... 1

1-1. Engineering Design ................................................................................................. 1 1-2. Problem Statement .................................................................................................. 3 1-3. Literature Review .................................................................................................... 5

1.3-1 Hyer and Cohen ............................................................................................. 6 1.3-2 Kassapoglou .................................................................................................. 7 1.3-3 Interlaminar Stresses in Composite Layers, Salamon .................................... 9 1.3-4 Experimental Results Volpert ........................................................................ 10 1.3-5 Yap et al. ....................................................................................................... 10 1.3-6 Debonding Analysis ...................................................................................... 11 1.3-7 2D vs. 3D Finite Element Comparisons ........................................................ 12 1.3-8 Interface Analysis .......................................................................................... 13

1-4. Objectives ............................................................................................................... 14

Chapter 2 Modeling Methods ................................................................................................. 15

2-1. Finite Element Formulation .................................................................................... 15 2-2. Construction of the Model ...................................................................................... 18 2-3. Loading and Boundary Conditions .......................................................................... 20 2-4. Geometric and Material Considerations .................................................................. 22 2-5. Mesh of the Model .................................................................................................. 24 2-6. Plane of Data Collection ......................................................................................... 26 2-7. Convergence Study ................................................................................................. 28

Chapter 3 Results: Loading Case 3 ........................................................................................ 32

3-1. Skin-Stiffener Interface Analysis ............................................................................ 32 3-2. Non-Tapered Results............................................................................................... 33

3-2.1 System Displacement and Stresses ................................................................ 33 3-2.2 Alum/Alum Stress Distribution ..................................................................... 36 3-2.2. Alum/Steel Stress Distribution ..................................................................... 43 3-2.3. Alum/GFRP Stress Distribution ................................................................... 46

3-3 Tapered Stiffener Flange Results ............................................................................. 49 3-3.1 Alum/Alum 1:1 Taper Stress Distribution ..................................................... 50 3-3.2. Alum/Steel 1:1 Taper Stress Distribution ..................................................... 52 3-3.3. Alum/GFRP 1:1 Taper Stress Distribution ................................................... 55 3-3.4. Alum/Alum 1:4 Taper Stress Distribution .................................................... 57 3-3.5. Alum/Steel 1:4 Taper Stress Distribution ..................................................... 61 3-3.6. Alum/GFRP 1:4 Taper Stress Distribution ................................................... 63

Chapter 4 Loading Case 1 and Loading Case 2 ...................................................................... 66

v

4-1. Loading Case 1 ....................................................................................................... 66 4-1.1. System Displacements and Stresses ............................................................. 66 4-1.2. Alum/GFRP Stress Distribution ................................................................... 69 4-1.3. Alum/GFRP 1:1 and 1:4 Taper Stress Distributions ..................................... 72

4-2. Loading Case 2 ....................................................................................................... 74 4-2.1. System Displacements and Stresses ............................................................. 75 4-2.2. Alum/GFRP Base Model Stress Distributions .............................................. 77 4-2.3. Alum/GFRP 1:1 and 1:4 Tapered Model Stress Distribution ....................... 81

Chapter 5 Conclusions and Future Work ............................................................................... 84

5.1 Summary .................................................................................................................. 84 5.2 Future Work ............................................................................................................. 85 References: ..................................................................................................................... 87 Appendix A Solid 95 Shape Functions .......................................................................... 90 Appendix B Extra Skin-Stiffener Data .......................................................................... 91

Loading Case 1 Alum/GFRP Data .......................................................................... 91 Loading Case 2 Alum/GFRP Data .......................................................................... 93

vi

LIST OF FIGURES

Figure 1-1. Fuselage Skin with Stiffeners 3

Figure 1-2. Hyer and Cohen shear stress results………………………………………… 7

Figure 1-3. Kassapoglou Shear Stress results …………………………………………… 9

Figure 2-1. Solid 95 element ………………………………………………………….. 16

Figure 2-2, 3D View with Dimensions ………………………………………………… 19

Figure 2-3. Loading Case 1…………………………………………………………….. 20

Figure 2-4. Loading Case 2…………………………………………………………….. 21

Figure 2-5. Loading Case 3 …………………………………………………………..... 21

Figure 2-6. Boundary Conditions……………………………………………………… 22

Figure 2-7. Tapered Naming Convention ………………………………………………23

Figure 2-8. 3D View of Mesh ………………………………………………………… 25

Figure 2-9. Mesh Close-Up Views …………………………………………………… 26

Figure 2-10. Data Collection Location ……………………………………………… ...27

Figure 2-11. Convergence Study mesh Comparison …………………………………. 29

Figure 2-12. σz Skin: Mesh Convergence Study………………………………………. 30

Figure 2-13. σz Stiffener: Mesh Convergence Study …………………………………. 30

Figure 3-1. Loading Case 3 Deformed Shape ………………………………………… 33

Figure 3-2. σz Alum/Alum Loading Case 3 …………………………………………... 34

Figure 3-3. σz Section View ………………………………………………………… . 35

Figure 3-4. σyz Isometric View ……………………………………………………….. 35

Figure 3-5.σyz Section View ………………………………………………………..… 36

Figure 3-6. σx Alum/Alum ……………………………………………………………. 37

vii

Figure 3-7. σy Alum/Alum ……………………………………………………………. 38

Figure 3-8. σz Alum/Alum ……………………………………………………………. 39

Figure 3-9. σxy Alum/Alum …………………………………………………………… 39

Figure 3-10. σyz Alum/Alum ………………………………………………………….. 41

Figure 3-11. σxz Alum/Alum ………………………………………………………….. 41

Figure 3-12. σx, Alum/Steel …………………………………………………………... 43

Figure 3-13. σz Alum/Steel …………………………………………………………… 44

Figure 3-14. σxy Alum/Steel …………………………………………………………... 44

Figure 3-15. σx Alum/GFRP …………………………………………………………. 46

Figure 3-16. σz Alum/GFRP ………………………………………………………….. 47

Figure 3-17. σxy Alum/GFRP …………………………………………………………. 47

Figure 3-18. 1:1 Taper Close Up ………………………………………………………49

Figure 3-19. σx Alum/Alum 1:1 Taper ………………………………………………..50

Figure 3-20. σz Alum/Alum 1:1 Taper ……………………………………………….. 50

Figure 3-21. σxy Alum/Alum 1:1 Taper ………………………………………………. 51

Figure 3-22. σx Alum/Steel 1:1 Taper …………………………………………………53

Figure 3-23. σz Alum/Steel 1:1 Taper ………………………………………………... 53

Figure 3-24. σxy Alum/Steel 1:1 Taper ………………………………………………. 54

Figure 3-25. σx Alum/GFRP 1:1 Taper ………………………………………………… 55

Figure 3-26. σz Alum/GFRP 1:1 Taper ………………………………………………… 56

Figure 3-27. σxy Alum/GFRP 1:1 Taper ……………………………………………...... 56

Figure 3-28. 1:4 Taper Close Up ……………………………………………………..... 58

Figure 3-29. σx Alum/Alum 1:4 Taper …………………………………………………. 58

Figure 3-30. σz Alum/Alum 1:4 Taper …………………………………………………. 59

viii

Figure 3-31. σxy Alum/Alum 1:4 Taper ………………………………………………... 59

Figure 3-32. σx Alum/Steel 1:4 Taper …………………………………………………. 61

Figure 3-33. σz Alum/Steel 1:4 Taper …………………………………………………. 62

Figure 3-34. σxy Alum/Steel 1:4 Taper ………………………………………………… 62

Figure 3-35. σx Alum/GFRP 1:4 Taper ………………………………………………… 64

Figure 3-36. σz Alum/GFRP 1:4 Taper.. ………………………………………………. 64

Figure 3-37. σxy Alum/GFRP 1:4 Taper ……………………………………………..... 65

Figure 4-1. Deformed Shape Loading Case 1 …………………………………………. 66

Figure 4-2. σx Skin-Stiffener . …………………………………………………………. 67

Figure 4-3. σz Skin-Stiffener ...……………………………………………………..…. 68

Figure 4-4. σx Alum/GFRP Loading Case 1 …………………………………………... 70

Figure 4-5. σz Alum/GFRP Loading Case 1 ……………..……………………………. 70

Figure 4-6. σxy Alum/GFRP Loading Case 1 …………………………………….……. 71

Figure 4-7. σx Alum/GFRP 1:4 Taper Loading Case 1 ………………………………... 72

Figure 4-8. σz Alum/GFRP 1:4 Taper Loading Case 1 ………………………………... 73

Figure 4-9. σxy Alum/GFRP 1:4 Taper Loading Case 1 ………………………….……. 73

Figure 4-10. Deformed Shape Loading Case 2 ………………………………………… 75

Figure 4-11. σx Skin-Stiffener ………………………………………………………… 76

Figure 4-12. σz Skin-Stiffener ………………………………………………………… 76

Figure 4-13. σx Alum/GFRP Loading Case 2 ………………………………………….. 77

Figure 4-14. σz Alum/GFRP Loading Case 2 ……………………………………….…. 78

Figure 4-15. σxy Alum/GFRP Loading Case 2 ………………………………………… 78

Figure 4-16. σyz Alum/GFRP Loading Case 2 ………………………………………… 79

Figure 4-17. σy Alum/GFRP Loading Case 2 ………………………………………… 80

Figure 4-18. σxz Alum/GFRP Loading Case 2 ………………………………………… 81

ix

Figure 4-19. σx Alum/GFRP 1:4 Taper Loading Case2 ……………………………….. 82

Figure 4-21. σz Alum/GFRP 1:4 Taper Loading Case2 ……………………………….. 82

x

LIST OF TABLES

Table 2-1. Boundary Conditions ……………………………………………………… 22

Table 2-2. Material Properties ………………………………………………………... 24

Table 2-3. Skin-Stiffener Mesh Information ….……………………………………… 26

Table 2-4. Convergence study mesh comparison ……………………………………. 28

Table 3-1. Material and Taper Combinations of Skin-Stiffener ……………………... 32

xi

ACKNOWLEDGEMENTS

I would like to thank Dr. Lissenden for his help with the development of a design

criterion to analyze skin-stiffeners and the design of the EMCH 213D web site. His help and

support with has been invaluable. I would like to thank Dr. Salamon for his help with the website

and knowledge of skin-stiffeners. I appreciate Dr. Salamon and Dr. Hamilton for their

considerate review.

This material is based upon work supported by the National Science Foundation under

Grant No. 0633602 under the DUE‟s CCLI program.

Chapter 1

Introduction

1-1. Engineering Design

The function of an engineer is to create things that serve making, thus engineering

students need to learn how to design. Penn State University offers an alternative to the EMCH

213 Strength of Materials course that concentrates on design. This course, EMCH 213 D

Strength of Materials with Design, covers all of the analysis topics covered in the traditional

course, in addition to a semester long open-ended design project. The design project gives

students experience with the entire design methodology, not just the design analysis phase

(Salamon & Engel, 2001). Teams of students are formed and asked to design a combination

playground set using ASTM standards. In order to guide the students through the design process

a web site (http://www.esm.psu.edu/courses/emch213d/) was created. One section of the site

contains an interactive sample design problem to help the students learn the process. The web

site provides details that help the students with their design projects and provides resources and

links for them to use. The web site contains an interactive sample project

(http://www.esm.psu.edu/courses/emch213d/procedures/interactive/footbridge/) in which the

students are prompted through a simplified design example. The sample project asks the students

to design the beams of a simply supported pedestrian bridge. The bridge is subjected to a uniform

load from the foot traffic per AASHTO guidelines (1997). The user is asked to choose the beam

section and materials before they go through the design analysis. First, the user starts with

determining the free body diagram and then computes the shear and moment diagrams by

choosing the correct option from a list. Next, the required section modulus is to be computed

2

from the flexure formula. The size of the beam section is then selected and the student is asked to

compute actual the section modulus for the selected beam. The beam deflection is then computed

and compared with the allowable beam deflection from the AASHTO guidelines, which is L/500.

Finally, the dead load of the beam sections are taken into account and then a cost analysis is done

to identify the best design. The students can use this example along with the remaining web site

content to help them complete their project designs.

A design assessment quiz was created and completed by both EMCH 213 and 213D

students (Lissenden, Salamon, & Carver, 2010). Both design ability and knowledge of design

resources are assessed by the quiz. This assessment was used to determine whether or not the

design project and supplemental web site help students understand the design process.

Assessment results are positive. The focus of this thesis is on design analysis. The subject matter

parallels what is taught in E MCH 213 and 213D, but is more advanced.

The structure that is chosen for this design analysis is an aircraft fuselage that is

constructed as skin and stiffeners. Analyzing stiffened panels for design can be difficult because

it is difficult to quantify the stress distribution at the interface of the skin and stiffeners.

Stiffeners are bonded to the fuselage skin and give rise to singularities. Methods used by industry

are simplified 2D models and mechanical testing (Kreuger & Minguet, 2002). The 2D models

are used because they are fairly inexpensive, do not take much computation time and provide

upper and lower bounds for the mechanical testing results (Kreuger & Minguet, 2002). 3D

models provide greater accuracy but are not used because individuals within industry feel the

increase in computation time is too much (Kreuger & Minguet, 2002). Much of the current

analysis uses 2D models to study failure between the skin and the stiffener through a fracture

mechanics approach. However, herein the focus is on design analysis to study the stress

distribution at the interface of the skin and stiffener early in the design phase. The design

3

analysis uses finite element analysis of a representative substructure of a stiffened panel subjected

to a variety loading conditions and considers different material combinations

1-2. Problem Statement

The use of skin-stiffened panels is prevalent in many aerospace structures due to the

ability for stiffeners to provide extra stiffness while keeping the structure light. The DC6 and

DC7 aircrafts used stringers to carry compressive loads (McDonnell-Douglass, 1973). Stiffeners

provide extra load carrying capabilities, high bending stress resistance and can carry loads past

the point of initial buckling (Collier, Yarrington, & Van West, 2002). Stiffeners can be fastened,

riveted, or adhesively bonded to the fuselage skin to run longitudinally along the structure.

Figure 1-1. Fuselage Skin with Stiffeners (Boeing Co.)

Stiffeners must be lightweight and can be made of isotropic or composite materials.

However there are still some design considerations that need to be studied. A primary failure

4

mode with adhesively bonded stiffeners is separation of the skin or stiffener from the bond.

Separation between the skin and stiffener can be caused by buckling in the skin and due to peel

stress. Peel stress is associated with normal separation and acts at a right angle to the bond

(Savage, 2007). This results in the skin and stiffener being pulled in separate directions and

adhesives have weaker peel strengths (Savage, 2007). Adhesive bonds are used because they are

effective in transferring the load from the skin to the stiffener. Composite stiffeners need to be

designed for delamination between the ply layers. Delamination can lead to primary failure

within the structure and is something that must be accounted for. Another design consideration

with skin-stiffener systems is that stress singularities arise due to the geometry of the stiffener.

Stiffeners come in a variety of shapes, “ „T‟, „Z‟, „J‟, „I‟, blade, and hat” (Collier, Yarrington, &

Van West, 2002) and the singularities occur due the abrupt changes in geometry near the edges of

the flange. The singularities are where stresses are driven to infinity and are very hard to account

for in the design process.

In order to remedy this problem engineers have tried to quantify what occurs along the

interface between the skin and stiffener. Analytical models were created to characterize the stress

state of the interface by treating the stiffened panel as a two-dimensional structure where plane

stress and strain assumptions are used. Early versions of this work done by Hyer and Cohen

(1987) and Kassapoglou (1992, 1993) and finite element analysis were used to help verify the

models. Due to the difficult nature of determining a suitable stress function for these analytical

models energy minimization techniques have to be used. Various fracture mechanics criteria

have also been used to try to predict the initial loads upon which debonding occurs between the

skin and stiffener.

The current research attempts to quantify the stress state of a skin-stiffener panel under

in-plane, out-of-plane and pressure loading. This is done by modeling and loading a three

dimensional representative substructure of one skin attached to a stiffener. Finite element

5

modeling was used because it can be used in conjunction with the design phase to find a simple

way to predict the stress state at the interface. Knowing the stress distribution at the interface will

provide design engineers with valuable information early in the design phase. When using finite

element modeling the size of the model and computation time must be taken into account.

Industry prefers to use smaller models that do not require too much computation time. Two

dimensional simplifications are often made, but these require certain stresses or strains to be

neglected and therefore can cause a reduction in accuracy. Therefore the representative

substructure of the stiffened panel was modeled in three dimensions to provide a simple and easy

method to use during the design phase. These three dimensional models also can be used to

effectively model the stress singularities that occur near the flanges of the stiffener. The

singularities are problematic and they may be able to be reduced by altering dimensions or

materials in either the skin or stiffener. The study investigated effects of geometry, stiffener

material and tapering the stiffener flange.

1-3. Literature Review

The analysis of skin-stiffener panels has grown much in the past 30 years. The growth in

the fields of composite materials helped to spur this, and as a result the aviation industry has used

them to strengthen airframe components. Many different methods have been used to determine

numerical and analytical solutions of the stresses at the interface of the skin and stiffener. These

solutions have been used in conjunction with experimental data to ensure their viability. Growth

in the field of finite element analysis has also allowed for these solutions to be verified.

Now many of the methods for determining the stresses at the interface of the stiffened

panel will be presented. The literature review will start with the development of analytical

6

solutions for the stress distribution, experimental results, debonding analysis, fracture mechanics

and 3D finite element models.

1.3-1 Hyer and Cohen

Hyer and Cohen (1987) used the theory of elasticity to determine a solution for the

stresses along the interface of skin and a tapered flange, i.e. stiffener. In their model the bond line

was assumed to be perfect, and it was not included in their analytical and finite element methods.

The boundary conditions and equilibrium equations of the system were used in the development

of stress functions, F and ψ, to determine the displacement,

𝜎𝑥 = 𝜕2𝐹

𝜕𝑦2 𝜎𝑦 = 𝜕2𝐹

𝜕𝑥2 𝜏𝑥𝑦 = −𝜕2𝐹

𝜕𝑥𝜕𝑦 𝜏𝑦𝑧 = −

𝑑𝜓

𝑑𝑥 𝜏𝑥𝑧 =

𝜕𝜓

𝜕𝑦

The strain-displacement relations are then used to solve for the stress functions,

𝛽22𝜕4𝐹

𝜕𝑥4 + 2𝛽12 + 𝛽66 𝜕4𝐹

𝜕𝑥 2𝜕𝑦2 + 𝛽11𝜕4𝐹

𝜕𝑦4 = 0 𝛽44𝜕2𝜓

𝜕𝑥2 + 𝛽22𝜕2𝜓

𝜕𝑦2 = −2𝛽4 𝑤𝑖𝑡𝑕 𝛽𝑖𝑗 =

𝑎𝑖𝑗 −𝑎𝑖3𝑎𝑗3

𝑎33, 𝑖 = 1,2,4,5,6 𝑤𝑕𝑒𝑟𝑒 𝑎𝑖𝑗 𝑎𝑟𝑒 𝑡𝑕𝑒 𝑐𝑜𝑚𝑝𝑜𝑛𝑒𝑛𝑡𝑠 𝑜𝑓 𝑡𝑕𝑒 𝑐𝑜𝑚𝑝𝑙𝑖𝑎𝑛𝑐𝑒 𝑚𝑎𝑡𝑟𝑖𝑥

The equations are then solved by series approximations to determine the eigenvalues and

eigenvectors of the in-plane and out-of-plane stresses. Hyer‟s results focused on comparing the

elasticity and FEA solutions of the shear stress, τxy, in the skin and stiffener flange. Hyer‟s

solution is limited by the 2D assumption that was used to complete the analysis. Figure 1-2

shows Hyer and Cohen‟s finite element model and analytical results compared to the finite

element results. Their model used a composite skin with a “T-type” stiffener bonded perfectly.

The skin and stiffener both used the smeared orthotropic properties of a multiple laminate

composite (Hyer & Cohen, 1987).

7

Figure 1-2. Hyer and Cohen (1987) shear stress results

Figure 1-2 shows how the normalized shear stress varies along the interface of the skin-

stiffener. The stress is higher near the edge of the interface and decreases as the middle is

approached.

1.3-2 Kassapoglou

Kassapoglou and DiNicola (1992) also developed an analytical solution for the stresses at

the interface as a function of the applied load. Their approach varied from Hyer and Cohen‟s by

applying a different stress functions to solve the elasticity problem. The assumption that the

stress and strain did not vary in the axial direction of the stiffener was used to transform a three

8

dimensional problem into simplified two-dimensional one. The strain-displacement and

equilibrium equations of the problem were combined to determine an expression for the stresses

(Kassapoglou & DiNicola, 1992).

𝑆12

𝜕2𝜎𝑥𝜕𝑧2 + 𝑆22

𝜕2𝜎𝑦

𝜕𝑧2 + 𝑆23

𝜕2𝜎𝑧𝑥𝜕𝑧2 = 0, 𝑆12

𝜕2𝜎𝑥𝜕𝑥2 + 𝑆22

𝜕2𝜎𝑦

𝜕𝑥2 + 𝑆23

𝜕2𝜎𝑧𝑥𝜕𝑥2 = 0

A stress function dependent upon σx and σy was then used along with the equilibrium

equations to determine expressions for the peel and shear stress, σx and τxy (Kassapoglou &

DiNicola, 1992).

𝜕4𝜎𝑥𝜕𝑥4 +

𝑆55𝑆22 + 2𝑆13𝑆22 − 2𝑆12𝑆23

𝑆33𝑆22 − 𝑆232

𝜕4𝜎𝑥𝜕𝑥2𝜕𝑧2 +

𝑆11𝑆22 − 𝑆122

𝑆33𝑆22 − 𝑆232

𝜕4𝜎𝑥𝜕𝑧4 = 0

𝜕2𝜏𝑥𝑦

𝜕𝑥2 +𝑆66

𝑆44

𝜕2𝜏𝑥𝑦

𝜕𝑧2 = 0

The solutions to these two differential equations are found by determining the

eigenvalues and then applying the boundary conditions in the skin and stiffener. While doing this

Kassapoglou and DiNicola noticed a discontinuity between the stresses at the interface and had to

apply energy minimization to determine an effective solution (1992). FEA and the analytical

solution were then compared for the interlaminar stresses at the stiffener flange and skin

interface.

After developing this analytical solution Kassapoglou used it to determine the stress in

the skin-stiffener interface under generalized and shear loading. For the generalized loading

study Kassapoglou (1993a) studied how the thickness ratio would affect the stress at which

delamination between the skin and stiffener would occur. Kassapoglou (1993b) also noticed a

stress singularity that occurred in the interlaminar shear stresses that occurred near the flange

edges (1993b). Figure 1-3 shows the 3D finite element model that Kassapoglou used to compute

the shear stress at the interface of the skin and stiffener, and the comparison the analytical and

9

finite element solution (1993b). Kassapoglou concluded that the singularities at the flange ends

can be computed as long as the stress function has enough terms in the series.

Figure 1-3. Kassapoglou Shear Stress results (1993b)

1.3-3 Interlaminar Stresses in Composite Layers, Salamon

Salamon (1978) studied the interlaminar shear and normal stresses in a composite

analytically by finite difference approximation. The relations between stress, strain and

displacement from the theory of elasticity were solved by this finite difference method. Salamon

noticed that singularities occurred near the free edges of the composite laminate and studied how

the laminate‟s orientation affected the singularities.

10

1.3-4 Experimental Results Volpert

Volpert and Gottesman (1995) also used an analytical method to determine the interface

stresses and they attempted to verify this model with both finite element analysis and

experimentally. Their analytical model was also based on the theory of elasticity where the

equilibrium equations, boundary conditions and compatibility equations were formulated for the

“run-out zone” (Volpert & Gottesman, 1995). Their run-out zone was the region toward the end

of the skin and stiffener interface. A two dimensional bend test was then used to simulate the

stress field in the interface with the following stress function (1995).

𝜎11 =𝑀 𝑥1 𝑌 𝑥1, 𝑥2

𝐼 𝑥1 , 𝑥2

Where M(x1) is the internal moment from external forces, Y(x1,x2) is the difference from the

neutral surface, and I(x1,x2) is the moment of the cross-section with respect to the neutral axis

(Volpert & Gottesman, 1995). The stress function was then used with the equilibrium equations

and boundary conditions to determine the shear stress and other normal stresses. The stresses are

functions of the angle of a taper and thickness of the skin so that the geometry of the flange could

be studied. Volpert and Gottesman‟s experimental data and finite element analysis concluded

that the geometry of the “run-out zone” had a great effect on the stress distribution.

1.3-5 Yap et al.

Yap et al. (2002) developed an analytical tool with finite element analysis using fracture

mechanics principles to analyze debonding between a composite skin and stiffener. The initial

debonding between the skin and stiffener was treated as a crack and the strain energy release rate

was studied. This analysis was formulated for Mode I in-plane normal stress, Mode II in-plane

shear stress and Mode III combined loading (Yap et al., 2002). This criterion was used in

11

conjunction with loads and strains from finite element analysis to predict debonding between the

skin and stiffener. The buckling load that initiated crack growth was compared with experimental

tests to ensure the viability of the analytical method

1.3-6 Debonding Analysis

Determining how to best predict or prevent debonding within the stiffened panel has been

a huge area of research. A majority of this analysis has focused on using fracture criteria in

conjunction with the strain energy near the bond interface. The load at which debonding starts or

grows is also calculated by finite element modeling or experimental tests.

1.3-6a Benzeggagh and Kenane Fracture Criterion

Benzeggagh and Kenane (1996) set out to determine a fracture criterion to predict the

onset of delamination under mode I and mode II loading criterion. Experimental tests were

conducted so that the critical strain energy release rate, GTC, could be compared to the total

fracture resistance, GTR (Benzeggagh & Kenane, 1996)

𝐺𝑇𝐶 = 𝐺𝐼𝐶 + 𝐺𝐼𝐼𝐶 − 𝐺𝐼𝐶 𝐺𝐼𝐼

𝐺𝑇 𝑚

Where Gi are various fracture criterion that are dependent upon the loading mode of the

specimen and m is a parameter relating GTC and GTR. The authors have noted that m depends

upon the resin of the composite (Benzeggagh & Kenane, 1996). This fracture criterion has been

used by Bertolini et al. (2009), Krueger et al. (2002) and Minguet (1997) to study debonding in a

stiffened panel and has been tweaked to include mode III loading.

12

1.3-6b Bertolini et al.

Bertolini et al. (2009) set out to study the debonding by measuring the effect of the fiber

orientation, bonding method, temperature and geometry of the stiffener. Experimental results

concluded that adding a taper to the flange caused cracking to initiate at the first ply above the

interface and not the interface between the skin and stiffener (2009). Adding a taper to the

flanges also causes delamination to initiate later. When studying the fracture criterion

preliminary analysis noticed that mode 3, GIII, was the dominate mode and the authors believe

this mode should be studied further (Bertolini et al., 2009).

1.3-7 2D vs. 3D Finite Element Comparisons

Current design standards within the aviation industry allow for the modeling of a

stiffened panel in two dimensions due to assumptions about the behavior. Plane stress,

σ3=σ13=σ23=0, and plane strain, ε33= γ13=γ23=0, criterion are used to simplify the three

dimensional model down to two dimensions. People within the industry use these assumptions

because of the reduced modeling, computation and analysis time (Krueger, Paris, O'Brien, &

Minguet, 2002). Krueger et al. set out to determine how accurate the two dimensional models

were when compared to a three dimensional model. A generalized plane strain model, εzz= -νLεxx

where νL is poisson‟s ratio for a laminate and where γxz =0, γyz= 0, was also used to generate a

smaller simplified three dimensional model (Kreuger & Minguet, 2002). The virtual crack

closure technique, VCCT, (Krueger, Paris, O'Brien, & Minguet, 2002) was used to compute and

compare the strain energy against its critical value for delamination growth. The authors

determined the stress distribution along the bond line because of the influence that it has on

13

matrix cracking in the composite by determining the maximum principal transverse tensile stress

(Kreuger & Minguet, 2002),

𝜎𝑡𝑡 = 𝜎22 +𝜎33

2+

𝜎22−𝜎33

2

2+ 𝜏23

2.

Krueger and Minuet (2002) concluded that the plane strain and stress models should be

used as upper and lower bounds respectively for the analysis of a stiffened panel when compared

to experimental data. The generalized plane strain model allows for some of the three

dimensional strain affects to be included to a two dimensional model without significantly

increasing the computation time (Kreuger & Minguet, 2002). The results from the generalized

plane strain model follow the experimental results more closely and fit between the bounds of the

plane stress and strain cases. When applying the fracture criterion in the two dimensional models

the mode III strain energy GIII is not included and some studies have suggested that this is a

primary mode for delamination. The results from the full three dimensional model were similar

to that of the generalized plane strain model and allows for the inclusion of GIII. The effects of

the stress singularities at the free edges were seen in the three dimensional model but could not be

accurately quantified due to a lack of mesh refinement in the region.

1.3-8 Interface Analysis

Determining what happens at the interface of the skin and stiffener is paramount due to

the fact that one of the primary failure modes is debonding at that interface. The stress

distribution along the interface, fracture analysis and failure criterion was studied by Minguet

(1997). The stiffened panel was analyzed by the application of an applied pressure load and force

and moment resultants applied to the skin. However, Minguet noticed that Nyy and Myy did not

have to be analyzed due to their lack of effect on the interlaminar stresses (1997). The y-direction

14

runs vertically through the thickness of the stiffened panel. Sharp spikes near the flange edges

showed the effect of the stress singularities near the flange edges. Minguet concluded that

debonding in the frame (stiffener) occurs under combined axial and pressure loading from the

high strain energy release rate (1997).

1-4. Objectives

The objectives of the current study are to:

Determine the stress distribution at the interface of the skin and stiffener

Quantify the affect that the material of the stiffener has on the stress distribution

Characterize the effect that a tapered stiffener flange has on the stress distribution.

15

Chapter 2

Modeling Methods

This section outlines the methods employed to build the finite element model of a skin -stiffener

panel substructure.

2-1. Finite Element Formulation

Traditional structural mechanics problems typically involve solving governing differential

equations (equilibrium, strain-displacement, and material law),

𝜎𝑖𝑗 ,𝑗 + 𝑏𝑖 = 0 (2.1)

휀𝑖𝑗 = 1

2 𝜕𝑢𝑖

𝜕𝑥𝑗+

𝜕𝑢𝑗

𝜕𝑥𝑖 (2.2)

𝜎𝑖𝑗 = 𝐷𝑖𝑗𝑘𝑙 휀𝑘𝑙 (2.3)

subject to some known boundary conditions. The finite element method is a way to obtain

approximate solutions to problems having complex geometries and boundary conditions using the

principle of minimum potential energy (or other energy methods such as the principle of virtual

work). The principle of minimum potential energy states that the kinematically admissible

displacement field that corresponds to equilibrium minimizes the total potential energy. The

basis of the finite element method is that the domain can be discretized into generic finite

elements, with each element having a prescribed set of shape functions that represent the

displacement field within an element to the nodal displacements,

𝑢 = 𝑁 𝑢𝑖 (2.4)

where 𝑢 is the element displacement vector, 𝑁 is the 3 x n matrix of shape functions and 𝑢𝑖

are the nodal displacements. Shape functions are interpolation functions that depend upon the

type of element that is selected due to the type of problem that is being analyzed. The shape

16

functions depend upon the element that is used and for this analysis those of a 10-node tetrahedral

element are:

𝑢 = 𝑢𝐼 2𝐿1 − 1 𝐿1 + 𝑢𝐽 2𝐿2 − 1 𝐿2 + 𝑢𝐾 2𝐿3 − 1 𝐿3 + 𝑢𝐿 2𝐿4 − 1 𝐿4 + 4𝑢𝑀𝐿1𝐿2 +

𝑢𝑁𝐿2𝐿3 + 𝑢𝑂𝐿1𝐿3 + 𝑢𝑃𝐿1𝐿4 + 𝑢𝑄𝐿2𝐿4 + 𝑢𝑅𝐿3𝐿4

𝑣 = 𝑣𝐼 2𝐿1 − 1 𝐿1 + 𝑣𝐽 2𝐿2 − 1 𝐿2 + 𝑣𝐾 2𝐿3 − 1 𝐿3 + 𝑣𝐿 2𝐿4 − 1 𝐿4 + 4𝑣𝑀𝐿1𝐿2 +

𝑣𝑁𝐿2𝐿3 + 𝑣𝑂𝐿1𝐿3 + 𝑣𝑃𝐿1𝐿4 + 𝑣𝑄𝐿2𝐿4 + 𝑣𝑅𝐿3𝐿4

𝑤 = 𝑤𝐼 2𝐿1 − 1 𝐿1 + 𝑤𝐽 2𝐿2 − 1 𝐿2 + 𝑤𝐾 2𝐿3 − 1 𝐿3 + 𝑤𝐿 2𝐿4 − 1 𝐿4 + 4𝑤𝑀𝐿1𝐿2 +

𝑤𝑁𝐿2𝐿3 + 𝑤𝑂𝐿1𝐿3 + 𝑤𝑃𝐿1𝐿4 + 𝑤𝑄𝐿2𝐿4 + 𝑤𝑅𝐿3𝐿4

(ANSYS 11.0)

Figure 2-1. Solid 95 element (ANSYS 11.0)

where ui are the displacements at the respective nodes and Li are interpolating parameters related

to volume fractions within the tetrahedron (Hutton, 2004). The shape functions of the solid 95

element are shown in appendix A.

17

Substituting Equation 2.4 into the strain-displacement relations (Equation 2.2), :

휀11

휀22휀33

휀12휀13

휀23

=

𝜕

𝜕𝑥0 0

0𝜕

𝜕𝑦0

0𝜕

𝜕𝑦𝜕

𝜕𝑧0

0𝜕

𝜕𝑥0𝜕

𝜕𝑧

𝜕

𝜕𝑧0𝜕

𝜕𝑥𝜕

𝜕𝑦

𝑢𝑣𝑤 =

𝜕

𝜕𝑥0 0

0𝜕

𝜕𝑦0

0𝜕

𝜕𝑦𝜕

𝜕𝑧0

0𝜕

𝜕𝑥0𝜕

𝜕𝑧

𝜕

𝜕𝑧0𝜕

𝜕𝑥𝜕

𝜕𝑦

𝑁1 0 00 𝑁1 00 0 𝑁1

𝑁2 0 00 𝑁2 00 0 𝑁2

⋯⋯⋯

𝑢1

𝑣1𝑤1

𝑢2𝑣2𝑤2

⋮

which has a simplified form,

휀 = 𝐵 𝑢𝑖 . (2.5)

The total potential energy of a body, Π is given by

𝛱 = 𝑈 + 𝑊, (2.6)

where W is the external work potential and U is the strain energy.

The strain vector is, 휀 , the elastic constants for the material 𝐷 and the stress vector is, 𝜎 .

This is then integrated over the entire body to yield, 𝑈 = 1

2 휀 𝑇 𝐷 휀 and 𝑊 = 𝛿 𝑇 𝑓

and f are external the forces at the nodes. Now using the relation between elemental and nodal

stress, then factoring out terms the total potential energy for the system is: Π = 𝑈𝑒 − 𝑊 =

1

2 𝑢𝑖

𝑇 𝐵 𝑇 𝐷 𝐵 𝑑𝑉 𝑢𝑖 − 𝑢𝑖 𝑇 𝑓 . (2.7)

In order to obtain the minimum potential energy this is then minimized with respect to the nodal

displacements, 𝑢𝑖 , which provides the classical finite element system of element equations

𝐾 𝑢𝑖 = 𝐹 . (2.8)

where K = 𝐵 𝑇 𝐷 𝐵 𝑑𝑉 and is called the element stiffness matrix and 𝐹 = 𝑓 is

the nodal force vector. The respective element matrices throughout the entire solid are assembled

18

to obtain the global system of equations. The nodal displacements are then used to determine the

element strains and subsequently the element stresses

휀𝑖 = 𝐵 𝑢𝑖 (2.9)

𝜎 = 𝐷 𝐵 𝑢𝑖 (2.10)

2-2. Construction of the Model

The design of an aircraft fuselage is a very difficult task due to the many loads and design

requirements that must be accounted for. The fuselage is designed for mechanical loads, noise

requirements and thermal requirements (van Tooren & Krakers, 2007). The mechanical loads in-

service are often combined and may be a result of shear, torsion, bending and internal cabin

pressurization (van Tooren & Krakers, 2007). Stiffeners are placed longitudinally along the

length of the fuselage and frames run circumferentially around the fuselage. Our simplified

design analysis used a small representative substructure of one stiffener adhesively bonded to the

skin. Finite element analysis was used so that different loading conditions could be studied to

focus on the stress and the skin-stiffener interface. Elastic analysis was used in conjunction with

a 3D model of one mesh to study the affect of the different loading conditions. While industry

prefers doing a two dimensional analysis a three dimensional model was used because it allows

for all of the stress and strain data to be studied and provides the most accurate results. This

substructure contains skin and a „T‟ shaped stiffener. In order to allow some variability in the

analysis, the dimensions of the model were based on the thickness of the skin, which was taken to

be 2 mm. The thickness of the stiffener flange and web is the same as that of the skin. The rest

of the dimensions of the panel can be seen in Figure 2.2.

19

Figure 2-2, 3D View with dimensions in mm

To keep the model and analysis simple the skin and stiffener were assumed to be bonded

perfectly, which eliminated the need to model the adhesive. The adhesive bond was simulated by

gluing the skin and stiffener together via the glue command in ANSYS. The glue command tied

the nodes on the skin and stiffener together and prevented them from moving relative to each

other. The model was constructed in the x-y plane and extruded in the Z direction to create solid

volumes for both the skin and stiffener. The stress distribution will be reported along the x-axis

at the skin-stiffener interface. However, extruding the model as previously mentioned did not

create the elements and nodes along the x-axis for stress output. In order generate the nodes

along the x-axis half of the skin-stiffener was created, and then the remainder was created by

reflection about the x-y plane. This reflection matched the nodes and elements together to create

the solid model.

20

2-3. Loading and Boundary Conditions

In the analysis of the skin-stiffener three different loading conditions were studied, in-

plane loading in the x-direction (case 1), in-plane loading in the z-direction (case 2), and fuselage

internal pressure loading in the y-direction (case 3). These three were studied because an aircraft

fuselage can be subjected to each of these loads. The loading was such that a 1 MPa traction was

applied to the skin as shown in Figures 2-3, 2-4, and 2-5. Since the areas to which the traction

was applied are different for the three loading cases, the resultant forces are different. The

resultant forces for cases 1, 2, and 3 are 240 N, 128 N, and 7680 N, respectively.

Figure 2-3. Loading Case 1, traction in x-direction

21

Figure 2-4. Loading Case 2, traction in z-direction

Figure 2-5. Loading Case 3, traction in y-direction

22

The same boundary conditions, prescribed displacements, were applied for each loading

condition. Boundary conditions are applied only to the front and back faces of the model, which

are shown in Figure 2.6.

Figure 2.6. Boundary Conditions

Table 2-1. Boundary conditions

Front Edge v(x, y = 0, z = 60) = 0, w (x, y = 0, z = 60) = 0

Back Edge v(x, y = 0, z = -60) = 0

Middle Node along the Front Edge u (x = 0, y = 0,z = 60) = 0

2-4. Geometric and Material Considerations

The analysis of the skin-stiffener was conducted to gain a better understanding of the stress

distribution at the interface. This analysis was done by considering geometry, material mismatch

and material anisotropy. The effect that tapering the stiffener flange has on the stress distribution

is studied as well. The taper ratios were measured with respect its height and length and has the

23

following naming convention, height/ length, as shown in Figure 2-7.The analysis was of a

stiffener with a 1/1 and 1/4 taper. The combined effect of tapered flanges and material

discontinuity was also investigated.

Figure 2.7. Tapered naming convention

Representative materials were selected to give insight into the stress distribution of the

skin-stiffener. The skin was selected to be aluminum 2014-T6. The stiffener was selected to be

aluminum 2014-T6, structural A36 steel, or a unidirectional graphite fiber reinforced polymer,

(GFRP). The unidirectional GFRP is extruded in the z-direction with fiber orientations of 0° that

run parallel to the z-axis. This allowed for the input of orthotropic material properties for the

GFRP. The material properties are shown in Table 2-2.

24

Table 2-2. Elastic Material Properties

Isotropic Properties (Hibbeler, 2008)

Material E ν

2014-T6 Aluminum 73.1 GPa .35

A36 Structural Steel 200 GPa .32

GFRP Properties (Hyer, 1998)

Ez=155 GPa Ex=12.1 GPa Ey=12.1 GPa

νxz = .248 νxy = .458 νyz = .248

Gxz = 4.4 GPa Gxy = 3.2 GPa Gyz = 4.4 GPa

The models are named with respect to the material in the skin followed by the stiffener

material, i.e. Alum/Alum, Alum/Steel and Alum/GFRP.

2-5. Mesh of the Model

The skin-stiffener substructure was created using solid tetrahedral elements through

deconvolution of hexahedral (brick) elements. These elements have 10 nodes as shown in Figure

2.1, and each node has three degrees of freedom (ANSYS 11.0). The elements allow for the input

of orthotropic material properties and for large deflection. The shape functions for this element

are listed in the appendix.

The finite element mesh of the skin-stiffener substructure was generated using the mesh tool

command through the GUI, where the first step was to define the material properties of the skin

and stiffener. The elements were defined globally using the area size control prompt where each

element had a constant edge length. The maximum element edge length was 6.5 mm and the

25

mesh was refined using a refinement level of 3 to create a finest mesh possible. The refine

command was used to split all of the elements roughly in half to generate a more fine mesh. This

was limited because edge length values smaller than this failed during mesh refinement because

of the element limit criterion. The educational version of ANSYS that was used in the analysis

has an element limit of 256000 and any model with more elements than this causes an error that

shuts off ANSYS. Global mesh refinement was used so that the mesh would be constant

longitudinally through the skin-stiffener. Figure 2-8 shows the meshed skin-stiffener substructure

and Figure 2-9 shows two close-up views. Details for three of the finite element models are

provided in Table 2-3.

Figure 2-8. 3D View of Mesh

26

Figure 2-9. Mesh Close-Up Views

Table 2-3. Skin-Stiffener Mesh Information

Nodes DOF Elements

Non-Tapered 223097 669291 149696

1:1 Tapered 242728 726834 164607

1:4 Tapered 220362 661086 147953

2-6. Plane of Data Collection

The model results at the interface of the skin and stiffener was acquired by a number of

different methods available within ANSYS before the best method was selected. The preliminary

method that was used was the „path‟ command. The path command takes two specified nodes

along a line and then extrapolates the nodal data that is requested by the user. In this case, the

stress components were averaged at the nodes that exist along that line to create graphs of the

averaged skin and stiffener data. Paths were created along the front, middle, and back as shown

in Figure 2-10. Close examination of these results indicated that stresses at a given node were

calculated by averaging element stresses from elements in the skin and the stiffener. This was

particularly unsuitable for stress components that are not required to be continuous across the

27

interface. The use of nodal results was also investigated, but suffered from the same problem that

nodal stresses were computed by averaging element stresses at a common node.

Figure 2-10. Data Collection Location

In order to assess the stresses in the skin and the stiffener at the interface, the element stresses are

directly used with no averaging, only extrapolation from the integration points to the nodes. This

uses Equation 2.10 and results in multiple stresses being output at a given node, one from each

element connected to that node. The disadvantage associated with using this method for

examining the stress distribution at the middle location shown in Figure 2-9 is that it requires

more effort to automate output. The procedure is to identify nodes along the middle line and then

find all the elements connected to these nodes and save the element stresses at these nodes.

28

2-7. Convergence Study

To verify that the stress distribution obtained from the finite element results is accurate, a

convergence study was conducted. Three meshes were compared to characterize mesh size

dependence. The stress distributions along the interface between the skin and the stiffener for

loading case 3 were computed and compared for the Alum/ Alum skin-stiffener at the middle

location. As mentioned in the previous section the finest mesh was created by using the element

edge length command. The coarsest mesh, named Coarse, has an element edge length three times

finest mesh (named Fine) and the medium size mesh has an element edge length two times the

finest mesh. The mesh details are provided in Table 2-4, along with the maximum stresses in

both the skin and stiffener. The maximum stresses do not provide the basis for the convergence

study, but are given for completeness. Effectively, the medium mesh density is two times the

density of the coarsest mesh and the finest mesh is three times the size of the coarsest mesh. The

mesh densities from each version of the mesh can be seen in Figure 2-11.

Table 2-4. Convergence study mesh comparison

Coarse Medium Fine

Maximum Mesh edge

length (mm)

19.5 13 6.5

# of nodes 53554 68581 223097

Degrees of freedom 160662 205743 669291

Max Stresses Skin (MPa) -156.35 -90.88 -120.71

Max Stress Stiffener (MPa) -63.79 -101.24 -78

29

Figure 2-11. Convergence Study Mesh Comparison

The element stresses at the nodes of the skin and stiffener along the interface are plotted. The

stress, σij, was normalized by the applied traction, T, and this quantity is defined as 𝑆𝑖𝑗 =𝜎𝑖𝑗

𝑇. The

position was normalized by half the length of the interface, d = 16 mm.

30

Figure 2-12. σz Skin: Mesh Convergence, z = 0

Figure 2-13. σz Stiffener: Mesh Convergence Study, z = 0

31

When comparing the results for Sz in the skin from different meshes as shown in Figure

2-12, the main difference is that the fine mesh has more data recorded along the interface. This is

a result of the fine mesh having more nodes and therefore allows for collection of data at

locations not found in the coarser meshes. Each version of the mesh has a different number of

elements so this results in a range of varying stress values among the elements. There also is less

oscillation between the ranges of stresses at each element as the mesh size decreases. The finest

mesh is giving the best results as the elemental stress values are more centralized around a

singular value. This same trend can be seen in Figure 2-13 for the stiffener. As the mesh gets

finer the elemental stress values oscillate less and become more concentrated. When looking at

the other stress distributions the same thing occurs and it is known that the mesh size does have

an effect upon the accuracy of the data. As a result the fine mesh will be used in the rest of the

analysis.

32

Chapter 3

Results: Loading Case 3

3-1. Skin-Stiffener Interface Analysis

In light of the loading cases, material combinations, and stiffener flange taper there are

many possible combinations as shown in Table 3-1. The analyses completed and presented in this

thesis have boxes marked with „X‟.

Table 3-1. Material and Taper Combinations of Skin-Stiffener

33

3-2. Non-Tapered Results

This chapter shows the results of loading case 3 for the skin-stiffener, where loading case

3 is shown in Figure 2-5 and represents an external pressure load. The boundary conditions are

shown in Table 2-1 and with the applied traction result in beam bending. The displacement of the

system is discussed, along with the stress distribution along the interface. The effects of material

discontinuity and stiffener geometry are shown and discussed to provide analysis that can be used

in the design of skin-stiffener panels.

3-2.1 System Displacement and Stresses

Results from the Alum/GFRP model were used to get a feel of the displacement in the

skin-stiffener and to verify that the model showed beam bending. Figure 3-1 shows the deformed

shape plotted on top of the undeformed outline.

Figure 3-1. Loading Case 3 Deformed Shape

34

Upon the combination of the boundary conditions and the applied traction the deformed

shapes of the model appear as that of a bend test. As expected the maximum displacement occurs

at the middle of the model. The skin-stiffener has a maximum displacement of .001763 m.

Figure 3-2, σz Alum/GFRP Loading Case 3

The flexural stress, σz, shown in Figure 3-2 indicates that stress fluctuates in the skin-

stiffener and is symmetric with respect to the x-y plane. Peak stresses also occur in the top and

bottom of the skin-stiffener which are the furthest points from the neutral axis.

A Section view was taken at the origin of the skin-stiffener (see Figure 3-3), and provides

some insight into the stress state at the interface. The stress at the interface is in compression and

is higher in the middle of the interface. Stresses have similar magnitudes until the ends of the

interface where an increase in magnitude occurs due to the change in geometry.

35

Figure 3-3. σz Section View

Figure 3-4. σyz Isometric View

36

The shear stress, σyz, shown in Figure 3-4 indicates that the stress concentrations are

increasing near the front and back edges of the skin-stiffener. The stress throughout the middle of

the skin-stiffener was fairly uniform.

The shear stress is shown in Figure 3-5 is another section view taken at the origin. The

shear stress throughout the skin stiffener does not have much variation until the geometric

changes are reached. An abrupt change is stress is seen at the intersection of the stiffener web

and flange and at the ends of the interface of the flange and skin.

Figure 3-5. σyz Section View

3-2.2 Alum/Alum Stress Distribution

The distributions of all of the stress components for this case are analyzed and discussed

to provide insight into the analysis of the skin and stiffener at the interface. It is expected that the

skin and stiffener would have roughly the same stress distribution at the interface because there

37

was no material discontinuity, only a geometric discontinuity. This model provides a baseline for

the stress distribution in the skin-stiffener and will be used as a basis for comparison with the

other resulting data. The normalized stress components, Sij, are plotted as function of position

along the middle interface, which is normalized with respect to the half interface length, d= 16

mm, in Figures 3-6 to 3-11. Stresses in both the skin and the stiffener are shown in each figure.

Figure 3-6. σx Alum/Alum, z = 0

38

Figure 3-7. σy Alum/Alum

The σx stresses (see Figure 3.6) have the highest order singularities in any of the interface

stresses. The stress magnitudes remain near zero along the interface until the interface ends are

reached. This is where the geometry causes orders of singularities to arise that affect the stress

distribution and this study will attempt to reduce them. The stress in the skin and stiffener is the

same until the end of the interface where the singularities are located. The „T‟ shaped geometry of

the stiffener is causing singularities to occur where the interface ends. We must keep in mind that

true singularities are not reached as the stress does not reach infinity, but the order of singularity

that is seen is still high and can lead to separation in the skin-stiffener.

The σy stress distribution (see Figure 3.7) shows that the stress in the skin and stiffener is

the same. This is because traction continuity is required across the interface, where 𝑇𝑦 = 𝜎𝑦𝑛𝑦

and Ty is a traction vector and ny is the unit outward normal of the interface of the skin-stiffener.

Traction continuity is required as the bond is perfect and no matter the materials used the stress in

39

the skin and stiffener is the same for this stress component. Singularities occur at the ends of the

interface but there is not nearly the difference in skin and stiffener stresses as there was with σx.

The stresses are also symmetric about the y-axis which is what was expected.

Figure 3-8, σz Alum/Alum

Figure 3-9. σxy Alum/Alum

40

As expected for the stress distribution of σz (Figure 3.8) the skin and stiffener have the

same stresses. The difference for this case occurs with the order singularities where the skin has a

higher stress state than the stiffener, which is the same phenomenon as with the σx stresses.

Although the minimum stresses here are less than that for σx and the stress in the skin roughly 60

percent higher than that of the stiffener. The interface stresses σz are under compression, which

agrees with Figure 3-3.

The shear stress σxy (Figure 3.9) provided the most interesting stress distribution of any

of the shear stresses for the Alum/Alum model. The skin and stiffener once again had the same

state of stress along most of the interface. The stress goes from positive to negative over the

length of the interface and this switch occurs in the center. There is also a small stress peak that

occurs at x/d = ±.2 where the stresses increase in magnitude from the origin and then slightly

decline until the ends of the interface. The overall shape of this graph looks similar to the

analytical and finite element results obtained by Hyer (1987) and by Kassapoglou (1993b) seen in

Figures 1-2 and 1-3 respectively. Differences occur with Hyer‟s data with the stress in the skin

and Kassapoglou‟s because he did not differentiate between what was happening in the skin or

stiffener. However it was unexpected that the magnitudes in the order of stress magnitudes that

occur in the stiffener are higher than those of the skin. Traction continuity was also required for

this stress component as 𝑇𝑥 = 𝜎𝑥𝑦𝑛𝑦 , where Tx is a traction vector.

41

Figure 3-10. σyz Alum/Alum

Figure 3-11, σxz Alum/Alum

42

The σyz stress distribution (Figure 3.10) has the lowest stress magnitude at the interface

and the stress state is uniform between the skin and stiffener. As a result of the stresses being so

low the results that were discussed earlier with the sectioned contour plot make sense (Figure 3-

5). The effect of low order magnitude singularities can be seen, but since the overall stress state

is so low overall they are not too concerning. This is the final stress component for which stress

continuity was required as 𝑇𝑧 = 𝜎𝑦𝑧𝑛𝑦 , where Tz is a component of the traction vector.

The σxz stress distribution (see Figure 3.11) looks very similar to the σyz stresses and the

magnitudes are a little larger. The stresses in the skin and stiffener are most nearly uniform and it

is an interesting note that the magnitude of singularities that occur in the stiffener are a somewhat

higher than those in the skin.

Overall the stresses of most interest for the Alum/Alum model are the normal stresses σx,

σz and the shear stress σxy. The stresses, σx, σy, σz σxy, are all self-equilibrating and the stress

components in the y-direction have traction continuity at the interface. The stresses should be

self-equilibrating as the resultant forces are zero and there is no moment in the model. The

traction should be continuous because the bond between the skin and stiffener is perfect. The

stress distribution also shows that singularities of varying orders of magnitude occur at the ends

of the interface and are results of the change in geometry at the end of the stiffener flange. The

next section will focus on the affect that material discontinuity has on the stress distribution at the

interface. The remaining analysis will focus on the aforementioned stresses (σx, σy, σz σxy)

because of the difference in the skin and stiffener stresses. The stresses also have singularities

with the highest order. It is also interesting to note that the stress distribution for loading case 3 is

the highest and this is a result of it having the highest resultant force of our loading cases.

43

3-2.2. Alum/Steel Stress Distribution

While in actuality aluminum and steel would not be paired in a structure, it is done here

to characterize the effect of material property mismatch between two isotropic materials in this

type of structure. The steel stiffener shows the effect that material discontinuity has upon the

stress distribution.

Figure 3-12. σx, Alum/Steel

44

Figure 3-13. σz Alum/Steel

Figure 3-14. σxy Alum/Steel

45

The σx distribution is shown in Figure 3-12 and when compared with Figure 3-6 shows

the effect of the difference in isotropic material properties between aluminum and steel. The

stresses along the center have the same peaks and dips and this shows the effect of steel‟s elastic

properties. This stress distribution is also very similar to the Alum/Alum model, with the primary

difference being the increase in stress along middle of the interface. This may be a result of the

increased elastic modulus and increase in area as the web is located at the center of the interface.

This provides an increase in stiffness and the stress magnitudes will be higher as it carries more

of the stress. Singularities kick in at the edges of the stiffener flange, but the singularity

magnitudes in the steel stiffener are higher than those of the aluminum one shown in Figure 3-6.

The steel stiffener has caused the stress in the aluminum skin to be higher than that of the skin in

the Alum/Alum model.

The results for σz shown in Figure 3-13 from this case differed from the previous model

because the stress in the skin had lower magnitudes than the stiffener. Once again the stiffener

singularities are higher than the skin in the Alum/Alum model. The shapes of the stress curves

are also different and this can be seen at the position x = 0. The steel stiffener has lower

magnitude stresses at this point, and this did not occur in the previous model (Figure 3-8).

However the aluminum skin does have lower stresses over a majority of the interface of the skin

and stiffener, except where the singularities occur.

In the shear stress distribution σxy shown in Figure 3-14 the skin and stiffener have the

same stress magnitudes until the interface ends are reached. This stress component required

traction continuity across the interface, so the stress in the skin and stiffener is the same. The

magnitudes at the flange edges have magnitudes that are higher by 20 which may be a result of

the difference in the material parties of steel. The stress in the stiffener is higher than what is in

the skin and a little higher that of the Alum/Alum model (Figure 3-9).

46

Due to the perfect bond σy and σyz have traction continuity across the interface; therefore

the skin and stiffener have the same stresses. The steel stiffener is not effective in the reduction of

singularities and the effect of a tapered stiffener will have to be studied to see if steel would be an

effective isotropic material replacement.

3-2.3. Alum/GFRP Stress Distribution

The affect of a property mismatch between an isotropic skin and an anisotropic stiffener

is also investigated, with the Alum/GFRP. Perhaps the high longitudinal stiffness of GFRP will

provide an effective way to reduce the high stresses in the skin and stiffener. During the analysis

of the stress distributions differences were noticed with these results and those of the previous

models.

Figure 3-15. σx, Alum/GFRP

47

Figure 3-16. σz, Alum/GFRP

Figure 3-17. σxy, Alum/GFRP

48

The stress distribution for σx shown in Figure 3-15 has a variety of differences than what

occurred in the two previous models (Figures 3-6 and 3-12). Once again the σx stress components

were the highest and there were large differences between the stress in the skin and stiffener.

While both are still in compression the GFRP stiffener has normalized stresses in the -10 to -30

range. The aluminum stiffener‟s normalized stresses range from approximately -80 to -200. The

stresses also seem to converge to a value which is nearly the same as the stress at the ends of the

interface. The high stress magnitudes may be a result of the increased stiffness and extra area as

this is where the stiffener web is located. This same affect was seen with the Alum/Steel model

(Figure 3-12) as it also is stiffer in the region of the stiffener web.

The results from the σz stresses (see Figure 3-16) are similar in fashion to those of σx.

The major difference is that the values around which the stresses converge to in the middle of the

interface are higher than the singularities, which is a result of the increased stiffness in the region

of the stiffener web. In the stiffener the stresses in the middle are nearly doubled that at the

edges. Overall, the singularities for this skin-stiffener are lower than the singularities with the

two other models.

The shear stress for this case (shown in Figure 3-17) had the same expected results as

those of the aluminum and steel stiffeners, which is traction continuity. The skin stresses at the

ends have lower magnitudes than those seen in the Alum/Alum and Alum/Steel models, Figures

3-9 and 3-14 respectively. The singularities in the stiffener were of lower order than those of the

two previous stiffeners.

The increased stiffness of the GFRP causes high stress magnitudes to occur at the center

of the interface in all of the normal stresses because of the increased area due to the stiffener web.

As seen in the Alum/Alum and Alum/Steel models the other stress components have traction

continuity, σy and σyz, or have low magnitudes, σxz. Overall the stress magnitudes and order of

49

singularities for this model are lower than those of the two previous models. As a result it was

concluded that the GFRP is the most effective material for the stiffener.

In the next section the effect that tapering the ends of the stiffener flange has on the stress

distribution will be considered. The tapered flanges were made by removing some of the area

from the stiffener and while keeping the interface length constant.

3-3 Tapered Stiffener Flange Results

In order to reduce the high stresses that occur at the edges of the skin-stiffener interface a

taper was introduced. The intent was that the taper would reduce the abrupt change in geometry

and would result in a better stress distribution. The results from the 1:1 taper at the tip of the

stiffener flange are presented.

Figure 3-18. 1:1 Taper Close Up

50

3-3.1 Alum/Alum 1:1 Taper Stress Distribution

For the Alum/Alum skin-stiffener the shapes of all stress distributions were symmetric

about the y axis and looked similar in shape to the non tapered model.

Figure 3-19. σx Alum/Alum 1:1 Taper

Figure 3-20. σz Alum/Alum 1:1 Taper

51

Figure 3-21. σxy Alum/Alum 1:1 Taper

The σx stresses (Figure 3-19) were similar to the base model (no taper) with exception of

the stresses in the stiffener being much higher. Once again the stresses in the skin and stiffener

are the same along the interface. The taper allows the high stresses to be distributed over more of

the interface and this can be seen by the uniform stress distribution in this graph as compared to

its non tapered counterpart (Figure 3-6).

The stress singularities that occurred in the skin for σz (see Figure 3-20) had nearly the

same magnitudes as in the base model. The singularities are less concentrated at the ends of the

interface due to the taper causing the stress to be more uniform along the interface. The stress

values in the stiffener had higher order magnitude singularities than the skin, which was not the

case in the base model. The highest stress magnitude for this model was -141 which was almost

double what the highest magnitude of stress in the non-tapered model.

52

The shear stress for this tapered model (Figure 3-21) differs in the skin where the order of

singularities are higher than that of the non-tapered model. The stress magnitudes are nearly

doubled that of the other model and these elevated stresses occur along more of the interface.

Once again all of the other stresses in the skin and stiffener were the same and the

singularities in the stiffener are higher. As a result of the extremely high stresses in the skin it

was concluded that the geometry in this 1:1 model was too sharp and has no effect on reducing

the order of stress singularities for the Alum/Alum model.

3-3.2. Alum/Steel 1:1 Taper Stress Distribution

The 1:1 taper was applied to the steel stiffener to analyze it‟s affect of the stress

distribution. The taper has a large negative effect on the stresses in the steel stiffener and

virtually no effect on the stress skin for σz and σx cases. The σx stress distribution (See Figure 3-

22) has the same shape as the non-tapered model but the main difference is the increase in the

magnitude of the stiffener stresses. The stresses in the stiffener are nearly double what they are in

the non-tapered model along the entire length of the interface. However, the stress in the

aluminum skin is about the same as it was in the non-tapered model (Figure 3-6).

53

Figure 3-22. σx Alum/Steel 1:1 Taper

Figure 3-23. σz Alum/Steel 1:1 Taper

54

Figure 3-24. σxy Alum/Steel 1:1 Taper

The same pattern that was seen with the σx stress distribution is seen with the σz interface

stresses. Figure 3-23 really shows the effect of the taper allows the stress singularities to be

distributed across more of the interface. From x/d = ± .8 the increase in stresses due to the

singularities can be seen and in the non-tapered model singularities are only seen at the very ends

of the interface, x/d = ± 1.

Figure 3-24 shows the shear stress results for this tapered model and the results are

similar to the non-tapered Alum/Steel model (see Figure 3-14). The taper has allowed for the

stress to be a little more evenly distributed along the interface and the magnitudes at the flange

ends are more than doubled.

The tapered ends had the same effect on the stress in the stiffener in the other tapered

model as the stress at the interface ends has doubled from that of the non-tapered model. The

skin stress distribution is slightly smaller than the non-tapered model.

55

The 1:1 tapered Alum/Steel model is not particularly effective in reducing the order of

the singularities. The stresses in the cases not shown are roughly the same, while the stress

magnitudes in the stiffener for σx and σz double. Tapering the ends has not had the intended

effect for the Alum/Steel skin-stiffener.

3-3.3. Alum/GFRP 1:1 Taper Stress Distribution

The same 1:1 taper was applied to determine the effect that it has on the stress

distribution for the Alum/GFRP model.

Figure 3-25. σx Alum/GFRP 1:1 Taper

56

Figure 3-26. σz Alum/GFRP 1:1 Taper

Figure 3-27. σxy Alum/GFRP 1:1 Taper

57

The Alum/GFRP 1:1 taper σx (see Figure 3-24) provides better results than the aluminum

and steel stiffeners do. The stress distributions are very similar to the non-tapered model, and the

two main differences are the more uniform stress distribution and stress in the center of the

interface. The value around which the stresses converge at the center of the interface is slightly

lower than the peaks seen at the ends of the interface. This peak in stress is still a result of the

increased stiffness due to the stiffener web and increased modulus of the GFRP. The stress

magnitude in the aluminum skin is a little larger than what is seen in the non-tapered model.

Increases in stress due to the singularities occur at x/d = ± 0.7, whereas in the non-tapered model

this increase is only seen at the ends of the interface.

The taper did nothing to the overall stress distribution for the σz stresses (Figure 3-25).

The stress distributions are more uniform as what is expected as a result of tapering the ends and

the value where the stresses converge in the center of the interface have slightly higher

magnitudes.

The tapered ends also had no effect on the shear stress (see Figure 3-26), as the overall

stress distribution in the skin and stiffener appears to be the same.

Since the 1:1 taper ratio did not meet our expectations it was decided to increase the taper

so that the change in geometry is not as sudden. The primary effect of the taper is that it causes

the stress singularities to affect the stress distribution across more of the interface.

3-3.4. Alum/Alum 1:4 Taper Stress Distribution

The length of the taper was increased to a 1:4 height to length ratio with the hope that this

would provide the desired decrease in stress at the edges. This model has the same interface

length as the non-tapered and 1:1 tapered models. This more gradual taper can be seen in figure

3-28.

58

Figure 3-28. 1:4 Taper Close Up

When analyzing the stress distributions along the interface the results are better than that

of the previous taper. The first thing that is noticed is the shapes of the stress curves have

changed so that the effects of the singularities are seen along more of the interface. Increases in

stresses start to be seen at the x/d = ± 0.5 position, whereas in the non-tapered model this only

happens at the edges and with the 1:1 taper it starts at the x/d = ± 0.7 or 0.8 position.

Figure 3-29. σx, Alum/Alum 1:4 Taper

59

Figure 3-30. σz, Alum/Alum 1:4 Taper

Figure 3-31. σxy, Alum/Alum 1:4 Taper

60

The σx stresses (see Figure 3-29) show stress reductions in the skin and stiffener and a

more uniform distribution across the interface. However the stresses in the stiffener are much

higher and have nearly the same values as the skin. The maximum stress magnitudes near the

ends of the interface are almost double that of what was occurring in the non-tapered model

(Figure 3-6), but lower than what was seen in the 1:1 model.

The σz distribution (see Figure 3-30) has better stress results when compared to the non-

tapered and the 1:1 tapered models. The minimum stresses at the interface for σz are a little

higher in the stiffener than what is expected, -95 as compared to -76 in the non-tapered. The taper

provides a less abrupt geometry change which helps to reduce the stresses. The stresses in the

skin and the stiffener were the same over a majority of the length on the interface of the skin and

stiffener. The highest stress magnitude in the skin for this taper ratio was nearly 30 percent less

than what was seen in the base model (Figure 3-7).

The shear stress results (Figure 3-31) for this tapered model produced the desired

reduction in the order of the singularities. The stress is much smaller than what is seen in the

non-tapered and 1:1 tapered model. The affects of the stress singularities are seen across more of

the interface.

The high stresses in the stiffener may have been caused by the lack of elements at both

ends of the contact interface in this model. When looking at this version of the stiffener mesh it

was noticed the end nodes only have one element connected to it as compared to 3 or 4 in the

other models because of the mesh differences. An increase in the mesh near the stiffener flanges

will provide more data which gives more accurate results.

61

3-3.5. Alum/Steel 1:4 Taper Stress Distribution

Increasing the taper ratio produced results similar to the Alum/Steel 1:1 taper with the

difference being in the increased distance over which the effects of the stress singularities occur.

Figure 3-32. σx, Alum/Steel 1:4 Taper

62

Figure 3-33. σz, Alum/Steel 1:4 Taper

Figure 3-34. σxy, Alum/Steel 1:4 Taper

63

For the σx stress distribution (see Figure 3-32) the taper was effective in reducing the

stresses at the ends of the interface and a slight decrease in stress along the interface.

The σz stresses (Figure 3-33) show the response in the skin was better as the value of the

stress at the interface ends have lower magnitudes. The steel skin also has lower stresses than

that of the 1:1 model, and the effects of the singularities are spread across more of the interface.

The shear stress (Figure 3-34) in this tapered models in an improvement of that of the 1:1

Alum/Steel model as the stress magnitudes are much smaller. The stress distribution and stress

magnitudes are similar to the non-tapered Alum/Steel model. The taper causes the stresses to be

less abrupt and results in a more uniform stress distribution.

The stress in the skin and stiffener is the same in the resulting stress distributions, which

has been a theme throughout the analysis. Despite the increased taper ratio the stresses are still

too high and steel is not the optimum choice for a stiffener.

3-3.6. Alum/GFRP 1:4 Taper Stress Distribution

Increasing the taper ratio for this material configuration has provided the most interesting

results to date in the analysis. The distribution of stresses in the aluminum and graphite model

with a taper ratio of 1:4 had the expected symmetry and similar stresses as in the previous models

with this material configuration. The stresses in the x direction (see Figure 3-34) showed the

same convergence in the middle of the interface due to the increased stiffness because of the

stiffener web. What was unexpected though was how much lower the magnitudes of the

singularities were because of the geometry of the stiffener. The stresses in this case started to

increase in a more positive manner near the interface edges.

64

Figure 3-35. σx Alum/GFRP 1:4 Taper

Figure 3-36. σz Alum/GFRP 1:4 Taper

65

Figure 3-37. σxy Alum/GFRP 1:4 Taper

The σz stress distribution had a reduction of singularity order in both the skin and

stiffener. The resulting stress in the stiffener is odd and difficult to explain. Instead of the

stresses heading towards infinity, they reverse course and continue to increase positively.

The shear stress had excellent results as well and this was the first model in which the

stiffener had lower singularity magnitudes than the skin under this shear load. The stress

magnitude is approaching zero near the edges in both the skin and stiffener which is what should

have happened as the stress state at the end of the stiffener is traction free.

Increasing the taper for the Alum/GFRP was an effective way to decrease the order of

singularities that occurred near the edges and allow the singularities to distribute across more of

the interface. Since the GFRP stiffener provided the best overall results the remaining portions of

our analysis will use this material and then study the affect of the flange geometry.

66

Chapter 4

Loading Case 1 and Loading Case 2

4-1. Loading Case 1

When studying the in-plane loading of the skin-stiffener the study focused on the

Alum/GFRP material configuration as it gave the most interesting results. The stress magnitudes

that were caused due to the „T‟ shaped geometry of the stiffener were the lowest, which is what is

desired. The analysis will focus on the overall stress and displacement of the system and the

interface stress distribution at the plane of interest in the skin-stiffener.

4-1.1. System Displacements and Stresses

Figure 4-1 shows the displacement of the skin-stiffener under the applied traction in the

skin.

Figure 4-1. Deformed Shape Loading Case 1

67

As a result of the boundary conditions on the front and back edges, tensile tractions on

both faces of the skin in the x-direction (see Figure 2-3) should result in the deformed shape that

is seen. The maximum displacement is in the center of the interface of the skin, which can be

seen in the isometric and section views in Figure 4-2.

(a) Isometric View

(b) Cross Section

Figure 4-2. σx Skin-Stiffener, z = 0

68

Figure 4-3. σz Skin-Stiffener

The contour plot shown in Figures 4-2 and 4-3 show σx and σz respectively, and shows

the stress variation in the skin-stiffener under this loading condition. Stress concentrations

(Figure 4-2 a) can also be seen where there are sudden geometric changes and along the interface

of the skin-stiffener. The σz stresses show that the skin is loaded in tension, while the stiffener is

in compression. The stresses in the skin become more uniform as the distance from end of the

stiffener to the end of the skin increases. The magnitude of the stress in the stiffener is also lower

than that of the skin. It is more difficult to distinguish why there is such a variation of stresses for

σz. A majority of the skin is stress in tension and the stress switches to compression at the front

and back of the skin. Most of the stiffener with the exception of the top and small regions near

the front and back are in compression.

69

4-1.2. Alum/GFRP Stress Distribution

Once again the focus is on stress distribution analysis of the normal stresses σx, σz and the

shear stress σxy because of the difference in stress in the skin and stiffener. For loading case 1

traction continuity is seen in σy, σxy, and σyz for the same reasons discussed for LC3 as the

stresses are in the direction of the normal to the interface. The magnitudes of these stresses are

much lower than for loading case 3, due to the large difference in resultant forces. The maximum

magnitudes in the normalized stress distributions were around 1, whereas in the out-of-plane

loaded case (LC3) the magnitudes were up to two orders of magnitude larger.

Both the skin and stiffener are in tension for σx (see Figure 4-4) and the shapes of both

stress distributions are similar. As a result of the traction on both faces in the x-direction the

stress along the center of the interface is fairly uniform and the stress begins to increase due to the

change in geometry and the ends of the interface. The stress in the skin has a higher magnitude

than that of the stiffener. Both have stresses that have little variation along the center of the

interface and at x/d = ±0.5 the stresses start to increase as a result of the geometric affects. The

effect of the singularity magnitudes cannot fully be quantified due to a lack of elements at the

ends of the interface.

70

Figure 4-4. σx Alum/GFRP Loading Case1

Figure 4-5. σz Alum/GFRP Loading Case1

71

Figure 4-6. σxy Alum/GFRP Loading Case1, z = 0

When analyzing σz (see Figure 4-5) the skin is in tension while the stiffener is loaded in

compression. Stress singularities can be seen in the skin but they are much smaller in the

stiffener. The lack of nodes in the stiffener once again prevents more information about the high

stresses at the ends of the interface from being provided. The remaining stresses have lower

magnitude stress distributions and the same stresses within the skin and stiffener.

The shear stress distribution (see Figure 4-6) demonstrates traction continuity in the

central region while an increase occurs near x/d = 1.0. The discretization is not fine enough to

show that the stress goes back to zero at x/d = 1.0 as required by the boundary conditions. Closer

examination of the element stresses in the skin (not shown in Figure 4-6) indicates this.

72

4-1.3. Alum/GFRP 1:1 and 1:4 Taper Stress Distributions

Tapering the ends of the stiffener produced some of the same effects that were previously

discussed with loading case 3. Tapering the ends of the stiffener was also effective in slightly

reducing the stresses that occur in both the skin and stiffener in σx (see Figure 4-7). Overall there

is not much of a difference between the tapered and non-tapered model for this stress component.

Figure 4-7. σx Alum/GFRP 1:4 Taper Loading Case1

73

Figure 4-8. σz Alum/GFRP 1:4 Taper Loading Case1

Figure 4-9. σxy Alum/GFRP 1:4 Taper Loading Case1

74

The results for the σz stress distribution (Figure 4-8) shows the positive affect of adding a

slight taper to the model. The singularities that occurred in the skin were slightly reduced and the

stress distribution was spread across more of the interface. The slight high stresses at the ends of

the interface that were seen in the non-tapered model do not occur in this one and as a result there

is a more uniform stress distribution in the stiffener.

The shear stress results in this case are small and low order stress singularities occur in

the stiffener. Traction continuity requires that the shear stress should be zero at x/d = ± 1. This

appears to happen in the skin as two of the elements are located past the end of the interface.

However, this does not happen in the stiffener due to the coarseness of the mesh at x/d = ±1 as the

there is only one element at this point. Further analysis was conducted to determine what the

shear stress is in the skin, but results concluded that the results in the stiffener are non-zero due to

the mesh coarseness..

The 1:1 tapered model‟s results lie in between that of the 1:4 and the base model and do

not provide any additional insight for discussion. The results are shown in the appendix for

completeness.

4-2. Loading Case 2

The analysis for this loading case focuses on the same things as the two previous cases,

the displacement, stresses in the system and the stress distribution along the interface. As

mentioned in the procedures section, loading case 2 consists of a traction of 1 MPa applied to the

skin on both faces in the z-direction.

75

4-2.1. System Displacements and Stresses

Under the specified loading and boundary conditions the skin-stiffener has a positive

bending moment as a result of the load and the boundary conditions. Once again the maximum

displacement occurs at the center of the skin-stiffener.

Figure 4-10. Deformed Shape Loading Case 2

76

Figure 4-11. σx Skin-Stiffener

Figure 4-12. σz Skin-Stiffener

For the same reasons as specified earlier the stress distribution at the interface was shown

for the normal stresses σx and σz. The contour plot for σx (see Figure 4-11) shows that most of

the beam is in compression and the stress is fairly uniform. Stress concentrations are seen where

the changes in geometry are abrupt as would be expected. The σz normal stresses (Figure 4-12) a

77

majority of the skin is in tension and the stress in the stiffener is positive in the middle, then starts

to decrease as the you move away. The interface stresses appear to remain uniform and have

higher tension values near the middle of the skin-stiffener.

4-2.2. Alum/GFRP Base Model Stress Distributions

The stress distribution in the x direction, σx, (see Figure 4-13) had the expected results as

the skin has a higher stress distribution along the interface. Peak stress magnitudes occur near the

ends of the interface in the skin and the stresses there were nearly double that of a majority of the

interface. The stiffener stresses were uniform along the interface and the normalized magnitudes

were very close to zero. There were no singularities in either the skin or stiffener.

Figure 4-13. σx Alum/GFRP Base Model Loading Case 2

78

Figure 4-14. σz Alum/GFRP Base Model Loading Case 2

Figure 4-15. σxy Alum/GFRP Base Model Loading Case 2

79

The stress distribution in the direction of the applied load, σz, (see Figure 4-14) had

unexpected stress results as the stiffener had higher stresses than the skin. This is due to the

increased stiffness from the unidirectional fibers of the GFRP stiffener that is parallel to the

applied load. There were no singularities in the stiffener as the stress increased gradually as you

move away from the center of the interface. However singularities did occur in the skin and the

stress near the edges was almost double that of what occurred in the middle of the interface.

The shear results σxy (see Figure 4-15) for this loading case have extremely low

magnitude stiffener stresses and remain near zero. The resulting stress in the skin is very low, but

singularities do occur.

The σyz stress distributions (Figure 4-16) have small magnitudes and the traction

continuity requirement leads to the skin and stiffener having the roughly the same stresses.

Figure 4-16. σyz Alum/GFRP Base Model Loading Case 2

80

The traction continuity requirement is also seen with σy, shown in Figure 4-17. High

stresses are seen near the center of the interface which may be a result of the increased stiffness of

the GFRP and the stiffener web. Figure 4-18 shows σxz and the stress the skin has low

magnitudes; however, the stress in the skin is greater and singularities with low orders of

magnitude are seen at the ends of the interface. The next step in the analysis of loading case 2

was to introduce the tapered flange to see the effect it has on the stress distributions.

Figure 4-17. σy Alum/GFRP Base Model Loading Case 2

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Figure 4-18. σxz Alum/GFRP Base Model Loading Case 2

4-2.3. Alum/GFRP 1:1 and 1:4 Tapered Model Stress Distribution

Tapering the edges of the GFRP stiffener does not have much of an effect on the stress

distributions, as the increased stiffness of the GFRP kept the overall stress magnitudes low.

Figure 4-19 shows the x stress component and the primary difference from this and the non

tapered model (Figure 4-13) is the difference in stress magnitude. The stress in the center of the

interface is higher because of the added area of the stiffener web.

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Figure 4-19. σx Alum/GFRP 1:4 Taper Loading Case2

Figure 4-20. σz Alum/GFRP 1:4 Taper Loading Case2

83

The taper has little effect on σz (see Figure 4-20) because the stress distribution in the non

tapered model is nearly zero (Figure 4-14). The taper causes a slight decrease in the magnitude of

the stress distribution across the interface. There is a singular nature of stress that occurs near the

ends of the interface.

None of the shear stress results will be shown because of their very small magnitudes that

are effectively zero.

As mentioned with loading case 1 the 1:1 tapered results are very similar to the 1:4 taper

and the results are not shown. Overall, the GFRP is effective in reducing the singularities and

stress in the skin. This material configuration with the 1:4 tapered flanges presents the best

overall approach to provide the best overall stress distributions.

84

Chapter 5

Conclusions and Future Work

5.1 Summary

This thesis presents a design analysis that can be applied to a difficult stress analysis

problem that extends the elementary EMCH 213D design analysis concepts. Analysis of an

aircraft fuselage skin that has adhesively bonded stiffeners attached is difficult due to the variety

of applied loads, complicated geometry and stress singularities that arise. A design analysis was

applied to a representative fuselage substructure that consists of an aircraft skin and stiffener that

are perfectly bonded. Finite element analysis was used because it provides a way to approximate

the exact analytical solution due to the complex geometry of the model. FEA allowed for study

of the stress distribution along the interface of the skin-stiffener under a variety of loading

conditions.

The analysis consisted of studying the stress distributions associated with altering the

material of the stiffener, changing the geometry of the stiffener, and applying three different

loading conditions. The materials that were used to represent the stiffener were an aluminum

alloy, a steel alloy and a unidirectional GFRP with fiber orientation that runs longitudinally along

the stiffener. The geometry was analyzed by tapering the ends of the stiffener to determine the

effects on the stress distribution.

The affect of the „T‟ shaped stiffener geometry was computed for the non-tapered

Alum/Alum model (Figures 3-6 to 3-11). The geometry causes high edge stresses in and these

stresses were all self-equilibrating with traction continuity across the interface. Material

discontinuity was studied by using an isotropic steel stiffener and an orthotropic GFRP stiffener.

The Alum/Steel model had higher stress components than the Alum/Alum skin-stiffener due to

85

the material properties of the steel stiffener combined with the geometry. The Alum/GFRP was

effective in reducing the stiffener stresses in σx and reducing the stress in the skin and stiffener in

σxy. However the GFRP causes an increase in the skin stress in σx and both skin and stiffener

stresses in σy due the increased stiffness because of the stiffener web. The GFRP also has high

stresses in the middle of the interface in both the skin and stiffener that occur in σx and σy. Overall

the best material was the unidirectional GFRP as the overall stress in the skin and the stiffener

was lower for stresses in the loading cases that were analyzed. Tapering the ends of the stiffener

reduced the singularities and caused the high stress that was seen only at the edges to distribute

across more of the interface. However, the taper must be large enough so that the change in

geometry is not too sharp which will cause the stress at the ends of the interface to increase.

The design analysis provides valuable insight into the stress distribution at the skin-

stiffener interface. This simplified design criterion can be expanded upon to apply analytical

methods, fracture mechanics and other geometric considerations to further study a skin-stiffener.

5.2 Future Work

The analysis concluded that a unidirectional GFRP provided the best overall stress

distribution in the skin-stiffener. The simplified analysis required all of the composite fibers to be

unidirectional, and using an actual layered composite with varying ply orientations would provide

a more effective way to reduce the stress distribution and lower edge singularities. Finite element

analysis was used and an analytical approach from the theory of elasticity can be applied to find

the stress distribution by using a different stress function than Hyer and Cohen‟s (1987),

Kassapoglou and DiNicola‟s (1992) or Volpert and Gottesman‟s (1995). An increase in the mesh

density near the stiffener flanges will provide more elements that will help analyze the stress at

86

the interface ends. The analysis method can also be used to help with the design of an aircraft

fuselage skin with attached stiffeners.

87

References:

AASHTO. (1997). Guide Specifications for Design of Pedestrian Bridges. Washington D.C.: AASHTO.

ANSYS 11.0. (n.d.). Ansys Users Manual.

Benzeggagh, M. L., & Kenane, M. (1996). Measurement of mixed-mode delamination fracture

goughness of unidirectional glass/epoxy composites with mixed-mode bending appartus.

Composites Science and Technology Vol. 56 , 439-449.

Bertolini, J., Castanié, B., B. J.-J., & Navarro, J.-P. (2009). Multi-Level experimental and numerical

analysis of composite stiffener debonding. Part 1: non-specific specimen level. Composite

Structures Vol. 57 , 381-391.

Collier, C., Yarrington, P., & Van West, B. (2002). Composite, Grid-Stiffened Panel Design for Post

Buckling Using Hypersizer. Reston: American Institute of Aeronautics and Astronautics, Inc.

Corporation, M. S. (1998). Composite Materials Handbook 17. Fort Washington, PA, United States of

America.

Hibbeler, R. C. (2008). Mechanics of Materials 7th Edition. New Jersey: Pearson Prentice Hall.

Hutton, D. V. (2004). Fundamentals of Finite Element Analysis. New York: McGraw-Hill.

Hyer, M. W. (1998). Stress Analysis of Fiber-Reinforced Composite Materials. Boston: WCB McGraw

Hill.

88

Hyer, M. W., & Cohen, D. (1987). Calculation of Stresses in Stiffened Composite Panels. American

Institute of Aeronautics and Astronautics Vol. 26 No. 7 , 852-857.

Kassapoglou, C. (1993b). Calculation of Stresses at Skin-Stiffener Interfaces of Composite Stiffened

Panels Under Shear Loads. International Journal of Solids and Structures Vol. 30 No. 11 ,

1491-1501.

Kassapoglou, C. (1993a). Stress Determination at Skin-Stiffener Interfaces of Composite Stiffened

Panels Under Generalized Loading. AIAA Journal , 1753-1760.

Kassapoglou, C., & DiNicola, A. J. (1992). Efficient Stress Solutions at Skin-Stiffener Interfaces of

Composite Stiffened Panels. AIAA Journal Vol. 30, No. 7 , 1833-1839.

Kreuger, R., & Minguet, P. J. (2002, March). Influence of 2D Finite Element Modeling Assumptions on

Debonding Prediction for Composite Skin-stiffener Specimens Subjected To Tension and

Bending. ICASE Nasa Langley Research Center . Hampton, Virginia, United States of America.

Krueger, R., Paris, I. L., O'Brien, T. K., & Minguet, P. J. (2002). Comparison of 2D finite element

modeling assumptions with resuts from 3D analysis for composite skin-stiffener debonding.

Composite Structures Vol. 57 , 161-168.

Lissenden, C., Salamon, N., & Carver, J. (2010). Internet based tools for design in strength of

materials. 1-10.

McDonnell-Douglass. (1973). Isogrid Design Handbook. Huntington Beach: McDonnell Douglass

Astonautics Company.

89

Minguet, P. J. (1997). Analysis of the strength of the interface between frame and skin in a bonded

composite fuselage panel. 38th AIAA/ASME/ASCE/AHS/ASC structures, structural dynamics

and materials conference (pp. 2783-2790). Kissimmee: American Institure of Aeronautics

and Astronautics.

Salamon, N. J. (1978). Interlaminar Stresses in a Layered Composite Laminate in Bending. Fibre

Science and Technology Vol. 11 , 305-317.

Salamon, N., & Engel, R. (2001). Design projects for mechanics courses. ASEE Annual Conference

Proceedings, American Society for Engineering Education (pp. 3377-3390). Albuquerque:

ASEE.

Savage, G. (2007). Failure prevention in bonded joints on primary load bearing structures.

Engineering Failure Analysis Vol. 14 , 321-348.

Stinson, D. (1998). The Anatomy of the Airplane. Reston: American institure of Aeronautics and

Astronautics, Inc.

van Tooren, M., & Krakers, L. (2007). Multi-Disciplinary Design of Aircraft Fuselage Structures. 45th

AIAA Aerospace Sciences Meeting and Exhibition (pp. 1-13). Reno: American Institure of

Aeronautics and Astronautics.

Volpert, Y., & Gottesman, T. (1995). Skin-Stiffener Interface Stresses in Tapered Composite Panel.

Composite Structures Vol. 33 , 1-6.

Yap, J. W., Scott, M. L., Thomson, R. S., & Hachenberg, D. (2002). The Analysis of Skin-to-Stiffener

Debonding in Composite Aerospace Structures. Composite Structures Vol 57 , 425-435.

90

Appendix A

Solid 95 Shape Functions

𝑢 = 1

8 𝑢𝑖 1 − 𝑠 1 − 𝑡 1 − 𝑟 −𝑠 − 𝑡 − 𝑟 − 2 + 𝑢𝑗 1 + 𝑠 1 − 𝑡 1 − 𝑟 𝑠 − 𝑡 − 𝑟 − 2

+ 𝑢𝑘 1 + 𝑠 1 + 𝑡 1 − 𝑟 𝑠 + 𝑡 − 𝑟 − 2 + 𝑢𝐿 1 + 𝑠 1 − 𝑡 1 − 𝑟 −𝑠 + 𝑡 − 𝑟 − 2 + 𝑢𝑀 1 − 𝑠 1 − 𝑡 1 + 𝑟 −𝑠 − 𝑡 + 𝑟 − 2 + 𝑢𝑁 1 + 𝑠 1 − 𝑡 1 + 𝑟 −𝑠 − 𝑡 − 𝑟 − 2 + 𝑢𝑂 1 + 𝑠 1 + 𝑡 1 + 𝑟 𝑠 + 𝑡 + 𝑟 − 2 + 𝑢𝑃 1 − 𝑠 1 + 𝑡 1 + 𝑟 −𝑠 + 𝑡 + 𝑟 − 2

+1

4 𝑢𝑄 1 − 𝑠2 1 − 𝑡 1 − 𝑟 + 𝑢𝑅 1 + 𝑠 1 − 𝑡2 1 − 𝑟

+ 𝑢𝑆 1 − 𝑠2 1 + 𝑡 1 − 𝑟 + 𝑢𝑇 1 − 𝑠 1 − 𝑡2 1 − 𝑟 + 𝑢𝑈 1 − 𝑠2 1 − 𝑡 1 + 𝑟 + 𝑢𝑉 1 + 𝑠 1 − 𝑡2 1 − 𝑟 + 𝑢𝑊 1 − 𝑠 1 + 𝑡 1 − 𝑟2 + 𝑢𝑋 1 − 𝑠 1 − 𝑡2 1 + 𝑟 + 𝑢𝑌 1 − 𝑠 1 − 𝑡 1 − 𝑟2 + 𝑢𝑍 1 + 𝑠 1 − 𝑡 1 − 𝑟2

+ 𝑢𝐴 1 + 𝑠 1 + 𝑡 1 − 𝑟2 + 𝑢𝐵 1 − 𝑠 1 + 𝑡 1 − 𝑟2

𝑣 =1

8 𝑣𝑖 1 − 𝑠 … 𝑎𝑛𝑎𝑙𝑎𝑔𝑜𝑢𝑠 𝑡𝑜 𝑢

𝑤 =1

8 𝑤𝑖 1 − 𝑠 … 𝑎𝑛𝑎𝑙𝑎𝑔𝑜𝑢𝑠 𝑡𝑜 𝑢

𝑇 =1

8 𝑇𝑖 1 − 𝑠 … 𝑎𝑛𝑎𝑙𝑎𝑔𝑜𝑢𝑠 𝑡𝑜 𝑢

𝑉 =1

8 𝑉𝑖 1 − 𝑠 … 𝑎𝑛𝑎𝑙𝑎𝑔𝑜𝑢𝑠 𝑡𝑜 𝑢

𝛷 =1

8 𝛷𝑖 1 − 𝑠 … 𝑎𝑛𝑎𝑙𝑎𝑔𝑜𝑢𝑠 𝑡𝑜 𝑢

91

Appendix B

Extra Skin-Stiffener Data

Loading Case 1 Alum/GFRP Data

σx Alum/GFRP 1:1 Taper

92

σz Alum/GFRP 1:1 Taper

σxy Alum/GFRP 1:1 Taper

93

Loading Case 2 Alum/GFRP Data

σx Alum/GFRP 1:1 Taper

σy Alum/GFRP 1:1 Taper

94

σxy Alum/GFRP 1:1 Taper