UNIVERSITY OF CINCINNATI

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1 UNIVERSITY OF CINCINNATI Date: I,, hereby submit this work as part of the requirements for the degree of: in: It is entitled: This work and its defense approved by: Chair:

2 Structural Behavior and Design of Two Custom Aluminum Extruded Shapes in Custom Unitized Curtain Wall Systems Thesis for Master s degree in Civil Engineering by Yongbing Wang May 15, 2006

3 ABSTRACT The focus of the research in this thesis is on the structural behavior and design of two aluminum E type structural sections that are used in unitized curtain wall systems today. Moment capacities of the two E-type sections as one are analyzed by three methods - hand calculations using Minimum Moment Capacity Approach and Total Moment Capacity Approach, the finite element method using ABAQUS program, and via an experimental investigation. A comparison of moment capacities determined using the three methods is presented. Upon the evaluation and comparison of the results obtained by the three methods in this study, the Total Moment Capacity Approach is determined to be an accurate method to predict moment capacities of the two E-type structural sections functioning as one structural member and is recommended for the evaluation of moment capacities of complex thin-walled sections and assembled sections to supplement the Specification for Aluminum Structures (2000).

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5 Acknowledgements First of all, I would like to express my gratitude to my research advisor, Dr. James A. Swanson, for his support, patience and guidance. Next, I must thank my committee member, Dr. Gian Andrea Rassati, for his great help on my research. Also, I would like to thank Dr. T. Michael Baseheart, for serving on my committee and for his continued interest. I would like to thank Dr. Cynthia Tsao for her service in my thesis committee as well. Second, I would like to thank Mr. Keith Smith from Waltek & Company for his donation of aluminum mullion specimens. Also, I would like to thank Mr. Wayne Toenjes from Major Industries for providing further assistance with machined parts used in my research experiments. Finally, I would like give my sincere thankfulness to my wife, Cathy Chen, and to my son, Jianuo Wang, for their love and sacrifices.

6 TABLE OF CONTENTS List of Tables... v List of Figures....vii Chapter 1 Introduction History and Application of Aluminum Design and Application of Custom Extruded Aluminum Shapes 2 Chapter 2 Moment Capacity Calculation Material Properties of Aluminum Column Equations Plate Equations Under Uniform Compression Plate Equations Under Bending in Their Own Plane AA Specification s Formulas Total Moment Capacity Approach...11 Chapter 3 Preliminary Finite Element Analysis Using ABAQUS Introduction Definitions and Analysis of Finite Element Model Material Properties of Aluminum Alloy 6063-T Finite Element Analysis Geometrical Imperfection Nonlinear Finite Element Analysis Evaluation of Failure Load Proportion Factor (LPF) ABAQUS Analysis Comparison of MMCA, TMCA and FEA Results Chapter 4 Experimental Program Objective Preparation of Experiments Definition of Failure Model...30 i

7 4.4 Testing Comparison of Results of Hand Calculation, FEA and Experiments Chapter 5 Modified ABAQUS Analysis...40 Chapter 6 Conclusions Summary Conclusions References Appendix A Hand Calculations of Moment Capacities.. 52 A.1 Cross-section Geometry of Two E-type Structural Profiles...52 A.1.1 Two E-type Sections...52 A.1.2 Assumption of Stress Distribution...53 A.2 Section Properties A.2.1 Basic Geometric Properties of E-type Structural Section TD A.2.2 Basic Geometric Properties of E-type Structural Section TD A.2.3 Ultimate Shape Factor of Web...54 A.3 Mechanical Properties of Aluminum Alloy 6063-T6 from AA Specification for Aluminum Structures A.4 Moment Capacity Calculation with AA Specification for Aluminum Structures A.4.1 Moment Capacity of E-type Structural Section TD A Moment Capacity of E-type Structural Section TD-01 with Top Flange under Compression.. 55 A Moment Capacity of E-type Structural Section TD-01 with Bottom Flange under Compression A Ultimate Moment Capacity of E-type Structural Section TD A.4.2 Moment Capacity of E-type Structural Section TD ii

8 A Moment Capacity of E-type Structural Section TD-02 with Top Flange under Compression.. 57 A Moment Capacity of E-type Structural Section TD-02 with Bottom Flange under Compression A Ultimate Moment Capacity of E-type Structural Section TD A.4.3 Moment Capacity of TD-01 and TD-02 as One.59 A.5 Moment Capacity by Total Moment Capacity Approach (TMCA).60 A.5.1 Moment Capacity of E-type Structural Section TD A Moment Capacity of E-type Structural Section TD-01 with Top Flange under Compression.. 63 A Moment Capacity of E-type Structural Section TD-01 with Bottom Flange under Compression A Ultimate Moment Capacity of E-type Structural Section TD A.5.2 Moment Capacity of E-type Structural Section TD A Moment Capacity of E-type Structural Section TD-02 with Top Flange under Compression.. 66 A Moment Capacity of E-type Structural Section TD-02 with Bottom Flange under Compression A Ultimate Moment Capacity of E-type Structural Section TD A.5.3 Moment Capacity of TD-01 and TD-02 as One.68 Appendix B ABAQUS Input Files B.1 ABAQUS Input File of S1b100 (Beam 1 Computer Model) B.2 ABAQUS Input File of S4b100 (Beam 1 Computer Model) B.3 ABAQUS Input File of S1t100 (Beam 2 Computer Model) 73 iii

9 B.4 ABAQUS Input File of S4t100 (Beam 2 Computer Model) 75 Appendix C Diagrams of ABAQUS Analysis Results C.1 ABAQUS Analysis Result of Beam 1 under Pure Bending.77 C.2 ABAQUS Analysis Result of Beam 1 under Four-point Bending...78 C.3 ABAQUS Analysis Result of Beam 2 under Pure Bending.79 C.4 ABAQUS Analysis Result of Beam 2 under Four-point Bending C.5 Modified ABAQUS Analysis Result of Beam 1 under Four-point Bending...81 C.6 Modified ABAQUS Analysis Result of Beam 2 under Four-point Bending...82 iv

10 LIST OF TABLES Table 2.1 B, C, and D Factors of the Current AA Specification for Aluminum Structures...8 Table 2.2 B, C, and D Factors of the Current AA Specification for Aluminum Structures for Plates under Uniform Compression Table 2.3 B, C, and D Factors of the Current AA Specification for Aluminum Structures for Plates under Flexural Compression Table 2.4 Formulas for Determining Factored Limit State of Compressive Stresses for Flat Plates Supported on One Edge (AA Specification for Aluminum Structures, Section ) Table 2.5 Formulas for Determining Factored Limit State of Compressive Stresses for Flat Plates Supported on Both Edges (AA Specification for Aluminum Structures, Section ) Table 2.6 Formulas for Determining Factored Limit State of Compressive Stresses for Flat Plates (Web) with Both Edges Supported (AA Specification for Aluminum Structures, Section ) Table 2.7 Formulas for Determining Factored Limit State of Compressive Stresses for Flat Plates (Web) with Horizontal Stiffener, Both Edges Supported (AA Specification for Aluminum Structures, Section ) Table 3.1 Table 3.2 Table 4.1 Table 4.2 Parametric Study Results of AA-MMCA, TMCA and FEA Comparison of Parametric Study Results (Normalization by FEA). 27 Load and Deflection Estimation for Testing Summary of Results of Hand Calculation, FEA and First-series Experiments..33 Table 4.3 Comparison of Results of Hand Calculation, FEA and First-series Experiments (Normalization by Experimental Results)...33 Table 4.4 Summary of Results of Hand Calculation, FEA and Experiments v

11 Table 4.5 Comparison of Results of Hand Calculation, FEA and Experiments (Normalization by Experimental Results) Table 5.1 Summary of Results of Hand Calculation, Modified FEA and Experiments...45 Table 5.2 Comparison of Results of Hand Calculation, Modified FEA and Experiments (Normalization by Experimental Results)..46 vi

12 LIST OF FIGURES Figure 1.1 Figure 1.2 Figure 1.3 Figure 1.4 Figure 1.5 Figure 1.6 Figure 2.1 Figure 2.2 Figure 3.1 Figure 3.2 Figure 3.3 Figure 3.4 Figure 3.5 Figure 4.1 Figure 4.2 Figure 4.3 Figure 4.4 Figure 4.5 Figure 4.6 Figure 4.7 Figure 4.8 Figure 4.9 Mullion Design in Unitized Curtain Wall Systems...3 Mullion Design in Unitized Curtain Wall Systems.3 Two E-type Structural Shapes as One in Unitized Curtain Wall Systems...3 Simplified E-type Section for Mullion TD Simplified E-type Section for Mullion TD Two Simplified E-type Sections as One Stress-Strain Relationship for Aluminum Alloy 6063-T6..6 Column Curves for Aluminum Alloys 7 Tri-linear Law of Aluminum Alloy 6063-T6 19 ABAQUS Analysis Result of Beam 1 Model under Pure Bending..22 ABAQUS Analysis Result of Beam 1 Model under Four-point Bending...23 ABAQUS Analysis Result of Beam 2 Model under Pure Bending..24 ABAQUS Analysis Result of Beam 2 Model under Four-point Bending...25 Model of Aluminum Beam with Two E-type Sections.28 Test Set-up of Specimen...30 Local Buckling of Beam 1 after Testing...32 Local Buckling of Beam 2 after Testing...32 Moment Displacement Diagram of Beam Moment Displacement Diagram of Beam Modified Test Set-up of Specimen...36 Test Results of Beam 3, 4 and 6 36 Moment Displacement Diagram of Beam 1 Group...37 Figure 4.10 Moment Displacement Diagram of Beam 2 Group...38 Figure 5.1 Figure 5.2 Coupon Specimen.40 Coupon Test of Aluminum Alloy 6063-T vii

13 Figure 5.3 Modified ABAQUS Analysis Result of Beam 1 Model under Four-point Bending.42 Figure 5.4 Modified ABAQUS Analysis Result of Beam 2 Model under Four-point Bending.43 Figure 5.5 Figure 5.6 Moment Displacement Diagram of Beam 1 Group...44 Moment Displacement Diagram of Beam 2 Group...45 Figure A.1.1 Mullion Design in Unitized Curtain Wall Systems Figure A.1.2 Mullion Design in Unitized Curtain Wall Systems Figure A.1.3 Two E-type Structural Shapes as One in Unitized Curtain Wall Systems Figure A.1.4 Simplified E-type Section for Mullion TD Figure A.1.5 Simplified E-type Section for Mullion TD Figure A.1.6 Two Simplified E-type Sections as One.. 52 Figure A.1.7 Linear Approximation for A Non-linear Stress Distribution of TD Figure A.5.1 Analysis Result of Finite Strip Method CUFSM for E-type Section TD Figure A.5.2 Analysis Result of Finite Strip Method CUFSM for E-type Section TD Figure A.5.3 ABAQUS Analysis Result of E-type Section TD Figure A.5.4 ABAQUS Analysis Result of E-type Section TD Figure C.1 ABAQUS Analysis Result of Beam 1 under Pure Bending..77 Figure C.2 ABAQUS Analysis Result of Beam 1 under Four-point Bending Figure C.3 ABAQUS Analysis Result of Beam 2 under Pure Bending..79 Figure C.4 ABAQUS Analysis Result of Beam 2 under Four-point Bending Figure C.5 Modified ABAQUS Analysis Result of Beam 1 under Four-point Bending...81 viii

14 Figure C.6 Modified ABAQUS Analysis Result of Beam 2 under Four-point Bending...82 ix

15 1.1 History and Application of Aluminum Chapter 1 Introduction Aluminum markets and applications for aluminum grew quickly following the discovery of the Hall-Héroult process. In 1886, two young scientists, Paul Louis Toussaint Héroult (France) and Charles Martin Hall (USA), working separately and unaware of each other's work, simultaneously invented a new electrolytic process, the Hall-Héroult process, which is the basis for all aluminum production today. The physical and mechanical properties of aluminum alloys are generally responsible for the success of aluminum as an alternative construction material. Owing to the advantages of its lightness, corrosion resistance, ductility and good strength, aluminum has become one of the most common metals used for structural applications. Early use of aluminum structural components in construction began around Aluminum was first used to envelope buildings as curtain walls after World War II (Kissell and Ferry 2002). Since then, more and more building envelope systems have employed aluminum structures for support framing. These exterior building envelope systems typically consist of aluminum materials along with glass or granite, and are usually called aluminum and glass or aluminum and granite curtain wall systems. Today, aluminum applications can be categorized into two groups: structural and nonstructural. For the applications of aluminum in the construction industry, one of the largest segments of its use is in aluminum curtain wall systems. Other uses include sign structures, windows, etc. With the broad use of aluminum in envelope systems of modern commercial buildings, factories and some residential buildings, aluminum is becoming one of the most widely used construction materials. 1

16 1.2 Design and Application of Custom Extruded Aluminum Shapes In many cases, custom extruded shapes are more practical and structurally effective for specific applications than standardized shapes, particularly in unitized curtain wall systems. Curtain wall designers can integrate multiple features into the shape of a support mullion that can meet several functions. The following two E-type mullions with edge stiffeners on the flanges (hereafter called E-type structural sections or shapes, Figure 1.1 through Figure 1.3) are connected so as to function as one section while sustaining wind loads in practical applications of unitized curtain wall systems. There has been a lot of study on aluminum members or structural sections that are symmetrical along one axis (Clift and Austin 1989, Xiao and Craig 2003, Kim 2003). However, there is very little documentation of the structural behavior of unsymmetrical aluminum sections, especially in the application of unitized curtain wall systems. Similar E-type structural sections are one of the most common structural shapes in use for unitized curtain wall systems today. The other is a type of structural sections similar to S-type shapes. The focus of this study is on E-type structural sections with the goal of providing accurate predictions of the bending performance of members when two halves are combined to act as a single section, which is important for reliable and efficient design of aluminum supporting frames of unitized curtain wall systems. This is of particular importance considering that no detailed provisions or formulas for the evaluation of flexural moments of E-type sections appear in Aluminum Association (AA) Specification for Aluminum Structures (2000). The following aluminum profiles are presented as a conceptual design for the E-type structural sections used in unitized curtain wall systems (Figure 1.1 through Figure 1.3). The two E-type sections called TD-01 and TD-02 are a revised design. The original design is a concept of Builders Federal (Hong Kong) Ltd. for the bid of a 2

17 curtain wall project in China. Given the complexity of these shapes, the sections shown in Figure 1.4 through Figure 1.6 are assumed to be equivalent in this study for the purposes of structural analysis and simulation via finite element modeling. Figure 1.1 Mullion Design in Unitized Curtain Wall Systems Figure 1.2 Mullion Design in Unitized Curtain Wall Systems Figure 1.3 Two "E-Type" Structural Shapes as One in Unitized Curtain Wall Systems Figure 1.4 Simplified "E-type" Section for Mullion TD-01 Figure 1.5 Simplified "E-type" Section for Mullion TD-02 Figure 1.6 Two Simplified "E-type" Sections as One Glass curtain wall systems generally consist of large glass panes supported by vertical and horizontal mullions. Wind forces acting on the exterior surfaces of the curtain wall system are transferred to the main structural system by anchors. Generally speaking, the vertical mullions are subjected only to bending moments and a low level of torsion. The small torsion is considered to be negligible and is typically ignored. The connections 3

18 between vertical mullions and horizontal mullions, as well as the glazing constraints, can be considered as an adequate lateral support to resist the lateral buckling and the low level of torsion (Clift and Austin 1989), although the glazing constraint is often ignored for structural analysis of vertical and horizontal mullions. 4

19 Chapter 2 Moment Capacity Calculation 2.1 Material Properties of Aluminum Aluminum as a material is much different than other metals such as steel because of the heating effects and fabrication processes employed. Additionally, different aluminum alloys have considerably different mechanical properties. The stress-strain relationship of an aluminum alloy cannot be simplified to an elastic or elastic-perfectly-plastic behavior like that of mild steel. The Ramberg-Osgood (Ramberg and Osgood, 1943) continuous model that presents stress-strain relationships as a continuous nonlinear relationship is typically used for aluminum alloys. The classical form of the Ramberg-Osgood law is shown in Equation 2.1, where n is the Ramberg-Osgood parameter and strength of the aluminum alloy. f 0.2 is yielding n σ σ E f ε = (2.1) 0.2 The aluminum alloy and temper used in this study is 6063-T6. The nominal yield strength is 25 ksi, the nominal ultimate strength is 30 ksi, and Young s modulus is E= 10,100 ksi (see Figure 2.1). These values are obtained from the AA Specification for Aluminum Structures (2000). The specific form of the Ramberg-Osgood model for the 6063-T6 alloy is shown as Equation 2.2. Kim derived the value of n for most aluminum alloys in 2000 (Kim 2003). A multi-linear stress-strain relationship for the 6063-T6 alloy consisting of three segments is employed for the current investigation σ σ ε = (2.2) E B 5

20 Stress (ksi) Strain (in/in) Figure 2.1 Stress Strain Relationship for Aluminum Alloy 6063-T6 2.2 Column Equations Columns usually fail in one of four ways: (1) flexural buckling, (2) torsional buckling, (3) combined flexural-torsional buckling, or (4) local element buckling. The equations for column buckling are as follows (Sharp 1993, Clark and Rolf 1966): 2 π E σ = 2 λ (2.3) 2 π Et σ = Bc Dcλ, (or σ = ) 2 λ (2.4) where B, D =buckling formula constants for compression in columns. c c The modulus of elasticity is the material property that must be known for computing the Euler stress. When inelastic buckling occurs the elastic modulus E in Equation 2.3 6

21 may be replaced by the tangent modulus E t. The tangent modulus can be obtained from the Ramberg-Osgood equation. The inelastic curve, based on the tangent modulus, is approximated by a straight line conservatively; see Figure 2.2 (Sharp 1993, Clark and Rolf 1966). The material constants for Equation 2.4 are given in Table 2.1. C c defines the intersection of the elastic regions. σ c B c C λ c ( T 6, T 7, T 8, T 9 ) C c 2 π E 2 λ C c B c D λ c ( O, H, T 1, T 2, T 3, T 4, T 5) λ Figure 2.2 Column Curves for Aluminum Alloys For the category of aluminum alloys, the AA Specification for Aluminum Structures categorizes aluminum alloys into two groups: (1) temper designations -T5, -T6, -T7, -T8 and -T9; (2) temper designation -O, -H, -T1, -T2, -T3 and -T4. The aluminum profiles examined in this study are 6063-T6. The related formulas are shown in Table

22 Table 2.1 B, C and D Factors of the Current AA Specification for Aluminum Structures Temper Designation T5, -T6. T7, -T8, or T9 Intercept, ksi Slope, ksi Intersection B c = F cy Fcy / 2 D c Bc 6B = 20 E c 1/ 2 C c 2B = 3D c c where C c is a buckling formula constant for compression in columns; F cy is the compression yield strength. 2.3 Plate Equations under Uniform Compression Clark and Rolf (1966) used the straight-line column formula to develop the plate equation under uniform compression. The development of the model is shown as Equations 2.5 through 2.8. F cr 2 kπ E = 2 12(1 ν ) t b 2 (2.5) F cr = B D λ (2.6) p p p 1/ 3 1 f f2 B p = f f1 E (2.7) Dp = 0.5Bp Bp f 2 1 E f1 (2.8) Table 2.2 B, C and D Factors of the Current AA Specification for Aluminum Structures for Plates under Uniform Compression Temper Designation Intercept, ksi Slope, ksi Intersection T5, -T6. T7, -T8, or T9 B p ( F ) 1/ cy = Fcy D p Bp Bp = 10 E B C p = D p p 8

23 where B, C, D =buckling formula constants for compression in flat plates. The p p p slenderness ration of a plate, λ, is 1.63b/t for a plate with both edges simply supported. p 2.4 Plate Equations under Bending in Their Own Plane Clark and Rolf (Clark and Rolf 1966) used Equation 2.6 to develop equations for a plate under bending in its own plane. Table 2.3 B, C and D Factors of the Current AA Specification for Aluminum Structures for Plates under Flexural Compression Temper Designation Intercept, ksi Slope, ksi Intersection T5, -T6. T7, -T8, or T9 B b ( F ) cy = Bbr = 1.3Fcy / 3 D b = D br Bb = 20 6B E b C br 2B = 3D br br In Table 2.3, BBb, D b are buckling formula constants for bending; br B, D br and C br are buckling formula constants for bending in rectangular bars. The formulas in the Table 2.1, Table 2.2 and Table 2.3 are from Table and Table of AA Specifications for Aluminum Structures (pages I-B-21 and I-B-22). A detailed derivation of these formulas is presented by Kim (2000). 2.5 AA Specification s Formulas Other related sections of AA Specification for Aluminum Structures are shown in Tables 2.4 through 2.7. The formula in Tables 2.4 through 2.6 are general formula used to calculate the factored limit state stresses of flat plates for three cases: (a) compression stresses for flat plates supported by one edge with the other edge free; (b) compression stresses for flat plates supported by both edges; (c) bending compression stresses for flat plates supported by both edges. The formula in Table 2.7 are general formula to compute 9

24 the factored limit state stresses of flat webs with one horizontal stiffener. Table 2.4 Formulas for Determining Factored Limit State of Compressive Stresses for Flat Plates Supported on One Edge (AA Specification for Aluminum Structures, Section ) φ F L for Slenderness φ F L for Slenderness φ F L for Slenderness Limit S1 Slenderness Limit S2 Slenderness b / t < S 1 S 1 < b / t < S 2 b / t > S 2 φ yf cy b t B φ F /φ p y cy b b = φb 5.1D Bp 5. 1D p p t b t k1b p = 5.1D p φbk2 BpE 5.1b / t In Table 2.4, F F L cy is the limit state stresses; is the compression yielding strength; φ φ y is the resistance factor ( is 0.95; and φc are 0.85); BBp and D p are the buckling formula constants for compression in flat plates; k 1 is the coefficient for determining slenderness limit S 2 for sections for which the limit state compressive stress is based on ultimate strength; k 2 is the coefficient for determining limit state compressive stress in sections with slenderness ratio above S 2 for which the limit state compressive stress is based on ultimate strength. Table 2.5 Formulas for Determining Factored Limit State of Compressive Stresses for Flat Plates Supported on Both Edges (AA Specification for Aluminum Structures, Section ) φ F L for Slenderness b / t < S 1 φ yf cy b t Slenderness Limit S1 B φ F /φ φ F L for Slenderness S 1 < b / t < S 2 p y cy b b = φb 1.6D Bp 1. 6Dp p t Slenderness Limit S2 b t k1b p = 1.6D p φ F L for Slenderness b / t > S 2 φ B k b 2 p 1.6b / t E 10

25 Table 2.6 Formulas for Determining Factored Limit State of Compressive Stresses for Flat Plates (Web) with Both Edges Supported (AA Specification for Aluminum Structures, Section ) φ F L for Slenderness h / t < S 1 1.3φ y F cy h t Slenderness B Limit S1 1.3φ F / φ φ F L for Slenderness S 1 < h / t < S 2 br y cy b h = φb 0.67D Bbr 0. 67Dbr br t Slenderness Limit S2 h t k1bbr = 0.67D br φ F L for Slenderness h / t > S 2 φ B bk2 br 0.67h / t E Table 2.7 Formulas for Determining Factored Limit State of Compressive Stresses for Flat Plates (Web) with Horizontal Stiffener, Both Edges Supported (AA Specification for Aluminum Structures, Section ) φ F L for Slenderness h / t < S 1 1.3φ y F cy h t Slenderness B Limit S1 1.3φ F / φ φ F L for Slenderness S 1 < h / t < S 2 br y cy b h = φb 0.29D Bbr 0. 29Dbr br t Slenderness Limit S2 h t k1bbr = 0.29D br φ F L for Slenderness h / t > S 2 φ B bk2 br 0.29h / t E The formulas for determining the factored limit state of tensile stresses in rectangular tubes, structural shapes bent about their strong axis, extreme fiber, (AA Specification for Aluminum Structures, Section 3.4.2) are φ or φ uf tu / kt. is the tensile yielding yfty strength, F is the tensile ultimate strength and is the coefficient for tension members. tu k t F ty 2.6 Total Moment Capacity Approach (TMCA) Kim proposed several approaches including a Total Moment Capacity Approach (TMCA) for evaluating complex sections in flexure (Kim 2003). He extended this approach to the application of most possible cross-sectional geometries. He carried out a full analysis of a complex I-shape mullion section that is singly-symmetric with vertical edge lips. The results of analyses by TMCA, finite element analysis with ABAQUS, and 11

26 physical tests matched very well according to his research. The TMCA approach is used in this study to estimate the moment capacity of the two E-type structural sections with multiple horizontal stiffeners of different sizes. The equations (Kim 2003) used to calculate the moment capacities of these sections are shown as Equations 2.9 and The minimum ultimate moment capacity of Equations 2.9 and 2.10 will be the ultimate moment capacity of the member. F I F I F M + or tf tf bf bf tw tw u = M tf + M bf + M tw = + (2.9) ctf cbf ctw F I F I F M + tf tf bf bf bw bw u = M tf + M bf + M bw = + (2.10) ctf cbf cbw I I where =ultimate moment of a member, M u M tf =moment capacity of top flange, M bf =moment capacity of bottom flange, M tw =moment capacity of top part of web, M bw =moment capacity of bottom part of web, F tf = limit state stress of top flange, F bf = limit state stress of bottom flange, F tw = limit state stress of top part of web, F tw = limit state stress of bottom part of web, I tf = moment of inertia of top flange about neutral axis, I bf = moment of inertia of bottom flange about neutral axis, I tw = moment of inertia of top part of web about neutral axis, I bw = moment of inertia of bottom part of web about neutral axis, c tf = distance from center or extreme edge of top flange to neutral axis, c bf = distance from center or extreme edge of bottom flange to neutral axis, c tw = distance from top of web to neutral axis, 12

27 c bw = distance from bottom of web to neutral axis The most obvious difference between the TMCA approach and the current AA specification is that the TMCA employs a numerical slenderness approach to calculate the minimum elastic buckling stress and then uses the minimum elastic buckling stress to calculate the equivalent slenderness ratio. ABAQUS is used in this study along with CUFSM, a numerical buckling analysis tool, to calculate the minimum elastic buckling stress. Dr. Schafer developed the CUFSM program (Schafer 2003). In the AA Specification for Aluminum Structures (2000), boundaries between component elements are assumed as ideally simply supported. For complex extruded shapes, this boundary idealization is not totally appropriate. The minimum critical elastic buckling stress is obtained by ABAQUS or CUFSM. The equivalent slenderness ratio is then calculated with the minimum critical elastic buckling stress. The relevant item of those equations in AA specification for Aluminum Structures (2000) is replaced with the equivalent slenderness ratio. This method is called a numerical slenderness approach. A shape factor may need to be calculated first before preceding to the evaluation of moment capacity of the E-type structural sections by hand calculation. The ratio of the plastic moment to the yield moment is a function of the cross section and is called the shape factor f (Gere 2000). f M Z = p = (2.11) M S y Equation 2.11 is valid for a fully elastic-plastic material. For an aluminum material, since it is an elastic hardening material, the shape factor is not only dependent on the shape of the cross section but also on the material behavior of aluminum. Also, there are no simplified plastic design techniques similar to those used for steel structures. For aluminum members, it is difficult to achieve strengths above those corresponding to initial yielding. Sharp suggested Equations 2.12 and 2.13 for the estimation of shape factors (Sharp 1993). 13

28 For yielding: Z = Z p (2.12) For ultimate strength: Z = Z (2.13) p Where Z = shape factor of aluminum alloys Z p = Shape factor for rigid-plastic case In 2003, Kim proposed a simplified ultimate shape factor for rectangular web elements with a neutral axis not at mid-depth (Kim 2003). The equation is shown Equation Ftu yna yna + Ftu α + w = a (2.14) Fty h h Fty y where, 0 NA h The minimum value of the shape factors from the Equations 2.11, 2.13 and 2.14 are used to evaluate the moment capacity of webs of the two E-type sections in this study. Kim also proposed a shape factor for rectangular flange elements under compression instead of a unity (Kim 2003). F tu α f = (2.15) Fty Since the extrusion of aluminum profiles is now considered to be relatively routine, longitudinal stiffeners are widely employed for various kinds of functions. Longitudinal stiffeners will definitely affect the local buckling of webs and flanges. In this study, the effects of intermediate longitudinal stiffeners attached to the webs will not be examined in detail because (1) the numerical slenderness method is used with ABAQUS and the finite strip software, CUFSM, to predict the minimum buckling stress for computing the equivalent slenderness ratio, (2) the webs usually provide less moment strength than the flanges, and (3) it is complicated to estimate the exact increase in the strength due to the longitudinal stiffeners. The relevant equations for calculating equivalent slenderness ratio are shown as Equations 2.16 through

29 b λ p = k, t k 2 12(1 v ) = (2.16) k p F cr 2 2 π E π E = k = (2.17) p (1 v )( b / t) λ p E λ p = π (2.18) F F cr = M S (2.19) cr cr / M = k M (2.20) cr LPF y M y = FyS (2.21) S = I / c (2.22) For edge stiffeners and intermediate stiffeners of thin-walled members, Desmond fully described how to compute the contribution of moment capacity of those stiffeners to members (Desmond et al. 1981A, Desmond et al. 1981B). For flanges with edge stiffeners, he suggested that in the fully effective range, adequate stiffener rigidity is determined by Equation 2.23 and in the post-buckling range, stiffener rigidity is determined by Equation AISI employed an effective width approach and use Equation 2.25 to calculate the ultimate strengths (AISI 1986). 3 ( / ) I s w t = A B (2.23) t w adequate t a w I s t eff adequate ( w / t) 115 = w t a + 5 (2.24) k w t k we 0.95t (2.25) σ y w σ = y where w is the width of edge stiffeners or intermediate stiffeners, 15

30 t is the thickness of edge stiffeners or intermediate stiffeners, k w is the coefficient of edge stiffeners or intermediate stiffeners, w eff is the effective width of edge stiffeners or intermediate stiffeners. The AISI Specification states that the minimum I s of edge stiffeners or intermediate stiffeners is I 2 4 ( w / t) ( 0.136E) F 4 = 3 y t or 18.4t. If the min.66 / I min 4 moment inertia of a stiffener is less than or 18.4t, the effect of the stiffener will be disregarded. According to this requirement, the small intermediate stiffeners for the E-type structural sections in this study are ignored. The AA Specification for Aluminum Structures considers the effect of edge stiffeners. For intermediate longitudinal stiffeners, AA Specification for Aluminum Structures (Section ) only considers the effect of one intermediate longitudinal stiffener. In this study, the influence of the intermediate stiffeners is accounted for in the hand calculation with TMCA as well as the FEA analysis. 16

31 Chapter 3 Preliminary Finite Element Analysis Using ABAQUS 3.1 Introduction The objective of this analytical study is to evaluate the ultimate moment capacity of the two E-type structural sections in a custom unitized curtain wall system. A finite element model of a beam consisting of the two E-type sections is developed with the ABAQUS program (ABAQUS 6.5-1). The finite element analysis results of the models are presented. 3.2 Definitions and Analysis of Finite Element Model Four computer models of an aluminum beam consisting of two E-type mullions under pure bending or four-point bending will be simulated using the finite element program ABAQUS. The width of the beam is 4 in. and the height is in. The length of the beam in the model is 100 in. The S4R shell element is employed to obtain reliable results with reasonable computation time. The total numbers of elements is 36,792 for the model and the total number of nodes is 37,146 in the model. The total number of degree of freedom is approximately 222,000. The model required approximately 30 hours of run-time each on a typical desktop PC. The aspect ratios of all shell elements are kept as close to unity as is practical. The true stress-log-strain relationship is incorporated instead of the true stress-strain relationship for input files. Finally, when the failure load factor is determined after the analysis, the Von-Mises stresses are carefully monitored together with the ultimate load factor. Nonlinear finite element analysis is employed for the best approximate solutions. The arc length method called the modified Riks method is employed. This method is used for cases where the loading is proportional; that is, where the load magnitudes are governed by a single scalar parameter. It can provide solutions even in cases of complex, 17

32 unstable response. Isotropic hardening is considered in the finite element analysis to consider the material properties of aluminum. The model consists of two E-type of structural sections TD-01 and TD-02. Multi-point constraint (MPC) ties are used to connect the two halves of the section at the top flanges and intermediate stiffeners. At the ends of the beam, for pure bending, three layers of elements are set up as rigid bodies to simulate the end support and lateral support and to make the analysis stable (i.e. avoid the development of plastic strain at the areas of load application). For four-point bending, fourteen layers of elements at each end are set up as a rigid body to simulate the end support and lateral support. Also, fifteen stiffeners along the webs of each section of TD-01 and TD-02 are employed at each end support and each loading area. Spring element Spring2 is employed between web stiffeners and flanges and longitudinal intermediate stiffeners. The Spring2 element is defined as compressive spring and it can only transfer the compressive forces, which are produced by shear forces from loading. Therefore, there is no additional moment capacity increased by adding web stiffeners. 3.3 Material Properties of Aluminum Alloy 6063-T6 A tri-linear approximation was used for the true strain-stress behavior of aluminum alloy 6063-T6. For details on material behavior of aluminum alloy, please refer to section 2.1. Figure 3.1 shows the trilinear law of the aluminum material. The yield strength is 25 ksi, the ultimate strength 30 ksi and Young s modulus is E= 10,100 ksi. The ultimate strain is 4% - 8% for aluminum alloy 6063-T6. 18

33 Stress (ksi) Strain (in/in) 3.4 Finite Element Analysis Figure 3.1 Tri-linear Law of Aluminum Alloy 6063-T6 Usually for finite element analysis on structural behavior of aluminum material, non-linear analysis is required. For input of geometric imperfections, an eigenvalue buckling analysis is used Geometric Imperfection A geometric imperfection pattern is generally introduced in a model for a post-buckling load-displacement analysis. Initial geometric imperfections were generated by an elastic eigenvalue analysis. This was the best choice since currently no imperfection measurement data is available. Imperfections are usually introduced by perturbations in the geometry. Usually there are three ways to define an imperfection: as a linear superposition of buckling eigenmodes, from the displacements of a static analysis, or by specifying the node number and imperfection values directly. A general eigenvalue buckling analysis can provide useful estimates of collapse mode shapes. An eigenvalue buckling analysis was 19

34 used in this study to introduce the imperfections for the model of the aluminum beams. Also, the imperfections consisting of multiple critical buckling modes were introduced because the precise shape of an imperfection is unknown Nonlinear Finite Element Analysis It is often necessary to obtain nonlinear static equilibrium solutions for unstable problems, particularly where the load and/or the displacement may decrease as the solution evolves during periods of the response. A nonlinear finite element solution technique called the modified Riks method is used. It is assumed that the loading is proportional that is, that all load magnitudes vary with a single scalar parameter. The essence of the method is that the solution is viewed as the discovery of a single equilibrium path in a space defined by the nodal variables and the loading parameter. 3.5 Evaluation of Failure Load Proportion Factor (LPF) The failure of a compact plate might be due to one of two major sources. One is a failure due to the instability resulting from inelastic buckling and the other is due to reaching the failure criteria without instability. There are three possibilities for the finite element analysis results. First, when the analysis reaches the maximum LPF, the maximum Von-Mises stress is less than the ultimate stress. The maximum LPF is employed directly as the capacity. Second, before the analysis reaches the maximum LPF, the maximum Von-Mises stress reaches the ultimate stress. In this situation, LPF when the stress reaches the ultimate stress is taken as the failure load factor. Third, the analysis terminates due to convergence problems. The final Maximum LPF may be acceptable as the failure load factor upon a careful check on the analysis result. 3.6 ABAQUS Analysis Two types of analysis have been carried out by ABAQUS. One is a pure bending 20

35 loading with concentrated moments applied at both ends of the beam model. The second is a four-point bending loading with two concentrated forces applied at the two load points at a distance of 30 in. from each end of the beam model. Two loading directions are employed. For the pure bending, both a counter-clockwise moment and a clockwise moment are applied to the beam model separately to get the moment capacities of the beam under two loading directions: positive moment for Beam 1 and negative moment for Beam 2. The same method is employed for the four-point bending. The following are the screen shots of ABAQUS analysis results. 21

36 22 Figure 3.2 ABAQUS Analysis Result of Beam 1 Model under Pure Bending

37 23 Figure 3.3 ABAQUS Analysis Result of Beam 1 Model under Four-point Bending

38 24 Figure 3.4 ABAQUS Analysis Result of Beam 2 Model under Pure Bending

39 25 Fig ure 3.5 ABAQUS Analysis Result of Beam 2 Model under Four-point Bendin g

40 3.7 Comparison of MMCA, TMCA and FEA Results Local buckling was observed from the finite element analysis of the model of Beam 1 under both pure bending and four-point bending. Local buckling was also observed from the finite element analysis of the model of Beam 2 under four-point bending. No apparent local buckling was observed from the finite element analysis of the model of Beam 2 under pure bending. There are two main reasons why no apparent local buckling happened for the model Beam 2 under pure bending. First, a rectangular cavity was formed on the compression side of Beam 2 after the assembly of TD-01 and TD-02 and it made local buckling hard to occur under pure bending. Another reason is that the input imperfections were too small to induce local buckling. Since local buckling occurred in the four-point bending model of Beam 2, it is not necessary to pursue local buckling in the pure bending model of Beam 2 such as increasing imperfections to the computer model of Beam 2 and running ABAQUS analysis again. For the FE models of Beam 1 and Beam 2, there is an assumption that the beams have full longitudinally lateral support considering the horizontal mullions support along with glazing panes. At each end support, three-layer elements for pure bending and fourteen-layer elements were set up as rigid bodies to resist the local deformation caused by loadings or reactions. The longitudinal screw races were transformed into equivalent rectangular bars for convenient finite element modeling. These assumptions and simplifications have little effect on the results of the analyses. The moment capacities of the beams by TMCA and FEA are very close. The moment capacities of the beams by AA-MMCA are very conservative (see Table 3.1). The hand calculations are done in Appendix A. One reason is that the method of AA-MMCA uses the minimum buckling stress of all elements of a beam without considering load redistribution after the local buckling occurs at one element. Another reason is that current AA-MMCA only considers one longitudinal stiffener for the calculation of web 26

41 moment capacities. Table 3.1 Parametric Study Results of AA-MMCA, TMCA and FEA AA-MMCA (in-k) TMCA (in-k) FEA (in-k) M u of Beam M u of Beam Table 3.2 Comparison of Parametric Study Results (Normalization by FEA) AA-MMCA TMCA FEA M u of Beam M u of Beam

42 Chapter 4 Experimental Program 4.1 Objective The objective of the experimental portion of this stu dy is to investigate the failure of two E-type mullions under four-point bending and determine their actual moment capacity. Although the numerical simulations documented in Chapter 3 are very useful and are considered to be the representative of actual conditions, physical tests are required to evaluate the simulations in this study since the sections under investigation are quite complex. 4.2 Preparation of Ex periments Experimental Model: the specimens tested in this study were loaded using a four-point-bending scheme. The model of test specimens is shown as Figure 4.1. This type of bending test is widely used for evaluating flexural capacity since it has a pure bending zone between points C and D without any shear force. The failure location will most likely occur within the inner span where only a pure bending moment exists. P P C D 6" A B c a b a c x L 4" Figure 4.1 Model of Aluminum Beam with Two E-type Sections Six pieces of aluminum alloy 6063-T6 aluminum profile TD-01 and seven pieces of 28

43 aluminum alloy 6063-T 6 profile TD-02 were donated by Waltek & Company for testing. First, two aluminum beams called Beam 1 and Beam 2 were prepared for the first series of tests. Dimensions a, b and c are 30 in., 36 in. and 2 in. respectively. Four-inch-long wooden stiffening blocks were inserted into the aluminum beams to guard against web crippling at the load and support points. Eight sheet metal screws were used to connect the two halves of the section and the wooden stiffening blocks together. Sheet metal screws were also used to connect the top flanges of the two halves after the assembly to make the whole beam more stable for loading as well as to simulate the support of anchors, horizontal mullions, and panels typical in curtain wall systems. The spacing between screws for the connection was 9 inches. Test Procedure: A 400 kip capacity Tinius-Olsen Super-L Universal Test Machine was used for the experiments. The machine was set up with one steel beam below the specimens acting as a supporting platform and a second steel beam above the specimens to act as a load transfer beam from the compression platen. A W10 x 26 shape (10 3/8 x 5 3/4 x 120, flange thickness: 7/16 ; web thickness: 5/16 ) steel beam was used as the support platform to sustain an aluminum beam for loading. A 10 1/2 x 5 7/8 x 56 1/8 (flange thickness: 7/16 ; web thickness: 5/16 ) section was used as the load transfer beam to apply loads to the aluminum specimen. A photo is shown in Figure 4.2 to illustrate the test set-up. 29

44 4.3 Definition of Failure Model Figure 4.2 Test Set-up of Specimen Since aluminum alloys are ductile materials, it is reasonable to employ the maximum energy theory to compute the stress and to predict the failure of the mullions under loading. The types and locations of failure are critical to the performance of a successful bending test. Failure in this research study is defined as a beam that is no longer able to support additional load. At this stage, massive local buckling was generally observed. A large deformation should take place at the inner span under pure bending only for successful bending tests. 4.4 Testing Two E-type sections TD-01 and TD-02 were assembled as one with bulb gaskets. The center point and points with a distance of 2 in. and 30 in. from each end were marked and right-section lines through these points were drawn. Beam supports were set on the designated points of the supporting steel beam and the specimen was placed in the testing 30

45 position. The load transfer beam was then set on the top of the specimen with two beam supports. Linear variable differential transformers (LVDTs) were placed at the supports, loading points and the center of span of the specimen. Strain gages were not used during the experiments since only the moment capacities of beams are investigated and compared with other hand calculation methods and ABAQUS analysis and measuring the strains are not necessary. A pre-load test was run and the data was recorded with MEGADAC Test Control Software (TCS) data system. The estimated ultimate load and deflection are listed in Table 4.1. Table 4.1 Load and Deflection Estimation for Testing Ultimate Load Magnitude (kips) Deflection (in.) The testing was formally run at the sampling rate of 4 samples per second, load and displacement readings were recorded simultaneously with TCS data system. Photographs were taken after each experiment to record failure modes for the test specimens. 31

46 Figure 4.3 Local Buckling of Beam 1 after Testing Figure 4.4 Local Buckling of Beam 2 after Testing 32

47 4.5 Comparison of Results of Hand Calculation, FEA and Experiments The predicted strength from finite element analysis results by ABAQUS is about 12% - 24% less than the experimental results of the first series test. The strength by TMCA is about 18% larger than the strength by AA-MMCA. The strength results and comparison are listed in Table 4.2 and Table 4.3. Table 4.2 Summary of Results of Hand Calculation, FEA and First-series Experiments AA-MMCA (in-k) TMCA (in-k) FEA (in-k) Experimental Result (in-k) M u of Beam M u of Beam Table 4.3 Comparison of Results of Hand Calculation, FEA and First-series Experiments (Normalization by Experimental Results) Average AA-MMCA TMCA FEA Experimental Result M u of Beam M u of Beam

48 Displacement - Moment Diagram Test _Beam ABAQUS_BeamPB ABAQUS_Beam1 ip-in) Moment (k Displacement at Center (in) Figure 4.5 M oment Displacement Diagram of Beam 1 34

49 Test_Beam2 Moment (in-k) ABAQUS_PB ABAQUS_Beam Displacement at Center (in) Figure 4.6 Moment Displacement Diagram of Beam 2 After the completion of experiments on the first two beams, Beam 1 and Beam 2, some modifications were made based on the test results as well as the design concept of the typical curtain wall system. First, 1/8 in. thick aluminum end plates were used instead of wooden stiffening blocks to constrain the ends of aluminum beams for the best simulation of curtain wall design and to achieve maximum loading during the test. Second, wooden stiffening blocks were not used at loading points as they were during the first two tests since the beams were stable at the two loading points during the first two tests. Third, coupon specimens were cut from a spare aluminum profile TD-02 for testing to determine the actual stress-strain relationship of the alloy. A photo is shown in Figure 4.7 for the new beam test setup for the second series testing of the remaining four 35

50 aluminum beams called Beam 3, Beam 4, Beam 5 and Beam 6. For the remaining four beams, Beam 3 and Beam 4 are tested with loading designed to cause a negative moment like Beam 2 (hereafter called Beam 2 Group ) and while Beam 5 and Beam 6 tested with loading designed to cause a positive moment like Beam 1 (hereafter called Beam 1 Group). Figure 4.7 Modified Test Set-up of Specimen 36

51 Figure 4.8 Test Results of Beam 3, 4 and 6 Displacement - Moment Diagram Test _Beam1 Test_Beam5 Test_Beam6 ABAQUS_BeamPB ABAQUS_Beam1 Moment (kip-in) Displacement at Center (in) Figure 4.9 Moment Displacement Diagram of Beam 1 Group 37

52 Test_Beam3 Test_Beam2 Test_Beam ABAQUS_PB Moment (in-k) ABAQUS_Beam Displacement at Center (in) Figure 4.10 Moment Displacement Diagram of Beam 2 Group Table 4.4 Summary of Results of Hand Calculation, FEA and Experiments Average TMCA Experimental AA-MMCA (in-k) (in-k) FEA (in-k) Result (in-k) M u of Beam M u of Beam

53 Table 4.5 Comparison of Results of Hand Calculation, FEA and Experiments (Normalization by Experimental Results) Average AA-MMCA TMCA FEA Experimental Result M u of Beam M u of Beam The final experimental result documents greater strength than that predicted from the finite element analysis results and the strength by TMCA. The main reason is that the actual yield and ultimate strengths of aluminum alloy 6063-T6 for these aluminum beams is higher than the theoretical yield and ultimate strengths. Additional coupon tests were carried out to measure the actual yield and ultimate strengths for modified ABAQUS analyses in Chapter 5. 39

54 Chapter 5 Modified ABAQUS Analysis Since the ABAQUS results presented in Chapter 3 are more than 20% lower than the experimental results presented in Chap ter 4, the FE model was modified to include the actual aluminum material properties. Coupon specimens are fabricated from spare material from specimen TD-02. The dimens ions of the coupon specimens are shown in Figure 5.1. Three coupon test results are shown in Figure " 4" 1 2 " " " 1 8 " 3 4 " R1" " " Figure 5.1 Coupon Specimen 40

55 Stress (k si) 40 Test B 35 Test C Test A Strain (in/in) Figure 5.2 Coupon Test of Aluminum Alloy 6063-T6 After the coupon tests, the material properties of the aluminum alloy was modified in the computer models. The experimentally measured yield strength was ksi instead of 25 ksi and the new ultimate strength was ksi instead of 30 ksi. The ultimate strain was 8% instead of the 4% that was used before. After the modification of the material properties, the moment capacity found by the ABAQUS analysis is in-kips for beam type 1 and in-kips for beam type 2. Stress contours obtained from the ABAQUS analyses are shown in Figures 5.3 and 5.4. Table 5.1 is a summary of results for hand calculation, the ABAQUS analyses, and experimental testing. Various comparisons of the results for the three methods are presented in Figure 5.5, Figure 5.6 and Table

56 42 Figure 5.3 Modified ABAQUS Analysis Result of Beam 1 Model unde r Four-point Bending

57 43 Figure 5.4 Modified ABAQUS Analysis Result of Beam 2 Model under Four-point Bending

58 Displacement - Moment Diagram Test_Beam6 ABAQUS_Beam1new Test _Beam1 Test_Beam5 ABAQUS_BeamPB ABAQUS_Beam1 Moment (kip-in) Displacement at Center (in) Figure 5.5 Moment Displacement Diagram of Beam 1 Group 44

59 Test_Beam3 Test_Beam2 Test_Beam4 ABAQUS_Beam2new ABAQUS_PB Moment (in-k) ABAQUS_Beam Displacement at Center (in) Figure 5.6 Moment Displacement Diagram of Beam 2 Group Table 5.1 Summary of Results of Hand Calculation, Modified FEA and Experiments Average AA-MMCA (in-k) TMCA (in-k) FEA (in-k) Experimental Result (in-k) M u of Beam M u of Beam In the Table 5.1, the results found by the modified ABAQUS analyses using the actual material properties are about 15% higher than that of the preliminary ABAQUS analyses discussed in Chapter 3. The modified ABAQUS results are much closer to the experimental results. A comparison of the normalized data is provided in Table

60 Table 5.2 Comparison of Results of Hand Calculation, Modified FEA and Experiments (Normalization by Experimental Results) Average AA-MMCA TMCA FEA Experimental Result M u of Beam M u of Beam The modified finite element analysis results by ABAQUS match well with the experimental results in the above comparison. The finite element analysis can be a reliable method to predict the actual moment capacity of complex thin-wall sections. The analysis input files and calculations are attached in the Appendix B. 46

61 Chapter 6 Conclusions 6.1 Summary The structural behavior and de sign of the two E-type sections are investigated in this study. Two methods for calculating the s trength, the Minimum Moment Capacity Approach and the Total Moment Capacity Approach, are fully explored in Chapter 2. The ultimate moment capacities of the two E-type sections are calculated step by step in Appendix A with the Minimum Moment Capacity Approach and the Total Moment Capacity Approach. The finite element method approach for this study is discussed in Chapter 3. The finite element modeling using ABAQUS is thoroughly described and the first of a series of four ABAQUS analyses is implemented based on the theoretical mechanical properties of aluminum alloy 6063-T6. The results of the finite element analyses are presented in Chapter 5. The results of the Minimum Moment Capacity Approach, the Total Moment Capacity Approach and the finite element method approach are closely compared and clearly tabulated. The experimental effort is fully documented in Chapter 4. The first series of tests for Beam 1 and Beam 2 are carried out after the first series of ABAQUS analysis. The results of the three methods of hand calculation, the finite element method and the first series of experiments are tabulated and compared. The differences among the three methods are investigated and the possible explanations for the differences are discussed at length. Some modifications are made to the second series of tests after the observation of the results of the first series of tests. The second series of tests for Beam 3, 4, 5 and 6 are implemented successfully, including the coupon tests which were carried out to characterize the actual aluminum material properties. Upon completion of all the experiments, some modifications such as the actual material properties of aluminum alloy 6063-T6 from the coupon tests are integrated with the ABAQUS models for the final 47

62 analysis. ABAQUS analysis results predict greater strength in the final analysis and more accurately correlate with the observed experimental results. Finally, the results of the three methods of hand calculation, the finite element method and the experiments are thoroughly discussed and evaluated. 6.2 Conclusions It is observed that the bending capacities calculated using the fundamental component formulas from the Specification for Aluminum Structures are very conservative based on the results from the experimental tests and the finite element analyses in this study. In the thorough evaluation of results of hand calculations, the finite element method approach and the experiments, the Total Moment Capacity Approach much better predicts the actual moment capacity of complex thin-walled sections than the Minimum Moment Capacity Approach. ABAQUS modeling can be a reliable method of evaluating the moment capacities of complex structural sections instead of experiments to save the cost of experiments as well as time. Also, it would be very beneficial to curtain wall companies to use the Total Moment Capacity Approach to compute a more accurate bending capacity of custom aluminum shapes which will allow designers to use less aluminum material thereby saving money. Eventually, it would be beneficial for building owners to save unnecessary cost for using curtain wall systems. The Total Moment Capacity Approach is an accurate and very valuable structural design approach to evaluate the bending capacities of the two complex E-type sections as one from this study. The equivalent slenderness ratios of the two complex E-type sections are computed with their minimum buckling stresses via finite element analysis. It is safe to conclude that the Total Moment Capacity Approach is a reasonable conservative alternative for any arbitrary thin-walled structural sections. Actually, there is no specific formula to compute complex thin-walled sections and any arbitrarily assembled sections in the Specification for Aluminum Structures (2000). It is recommended that the Total 48

63 Moment Capacity Approach is used to accurately compute the bending capacity of complex thin-walled sections to supplement the Specification for Aluminum Structures (2000). 49

64 REFERENCES AA (2000) The Aluminum Design Manual 2000, Aluminum Association, NW, Washington, DC AA (1993) Aluminum Standards and Data 1993, Aluminum Association, Arlington, VA ABAQUS 2004 Analysis User s Manual, version 6.5-1, ABAQUS, Inc., Providence, RI AISI (1986) Cold-Formed Steel Design Manual, American Iron and Steel Institute, Washington, D.C. Clark, J. W., Rolf, R. L., (1966). Buckling of Aluminum Columns, Plates and Beams, Proceedings of the American Society of Civil Engineers, Journal of the Structural Division, Vol. 92, No. 3, June, 1966, pp Clift, C. D., Austin, W. J., (1989). Lateral Buckling in Curtain Wall Systems, Journal of Structural Engineering, Vol. 115, No. 10, pp Desmond, T. P., Pekoz, T., and Winter, G., (1981A). Edge Stiffeners for Thin-walled Members, Journal of the Structural Division, Vol. 107, No. 2, February 1981, pp Desmond, T. P., Pekoz, T., and Winter, G., (1981B). Intermediate Stiffeners for Thin-walled Members, Journal of the Structural Division, Vol. 107, No. 4, April 1981, pp Gere, J. M., (2000). Mechanics of Materials, Fifth Edition, Brooks/Cole Thomson Learning, pp 457 Kim, Y., (2003). "Behavior and Design of Aluminum Members in Bending", Ph.D. Thesis, Cornell University, Ithaca, NY Kim, Y., (2000). "Behavior and Design of Laterally Supported Doubly Symmetric 50

65 I-Shaped Extruded Aluminum Sections", M.S. Thesis, Cornell University, Ithaca, NY Kissell, J. R., Ferry, R. L., (2002). What Is Aluminum?, Aluminum Structures, 2 nd Ed., John Wiley & Sons, Inc., pp 5 Ramberg, W and Osgood, W (1943). Description of Stress Strain Curves by Three Parameters. Technical Note No. 902, National Advisory Committee for Aeronautics, Washington, DC. Schafer, B. W., (2003). CUFSM v2.6 (finite strip software), Johns Hopkins University, Baltimore, Maryland Sharp, M. L., (1993). Behavior and Design of Aluminum Structures, McGraw-Hill, Inc., pp 72-73, pp 96 Xiao, Y., Craig, C., (2003). Ultimate Compressive Strength of Aluminum Plate Elements, Journal of Structural Engineering, pp

66 Appendix A Hand Calculations of Moment Capacities A.1 Cross-section Geometry of Two E-type Structural Profiles A.1.1 Two E-type Sections Figure A.1.1 Mullion Design in Unitized Curtain Wall Systems Figure A.1.2 Mullion Design in Unitized Curtain Wall Systems Figure A.1.3 Two "E-Type" Structural Shapes as One in Unitized Curtain Wall Systems Figure A.1.4 Simplified "E-type" Section for Mullion TD-01 Figure A.1.5 Simplified "E-type" Section for Mullion TD-02 Figure A.1.6 Two Simplified "E-type" Sections as One 52

67 A.1.2 Assumption of Stress Distribution F tf compression flange grop F tf web group F tw C tf C tw = + C bf C bw F bf tension flange group F bf Fbw Figure A.1.7 Linear Approximations for A Non-linear Stress Distribution of TD-01 A.2 Section Properties A.2.1 Basic Geometric Properties of E-type Structural Section TD-01 Width: in. Height: in. I y : in I : x 4 in r x : in. Area: in ybottom y NA : in. y : in. : in. top ctf : in. I : tf 4 in cbf ctw : in. I : : in. I : bf tw 4 in 4 in 53

68 c bw : in. I bw : A.2.2 Basic Geometric Properties of E-type Structural Section TD-02 Width: in. Height: in. 4 in I y : I x in : in r : in. Area: x 2 in ybottom y NA : in. y : in. : in. top ctf : in. I : tf 4 in cbf ctw : in. I : : in. I : bf tw 4 in 4 in c bw : in. I bw : A.2.3 Ultimate Shape Factor of Web 4 in Z x : in S _ : x bottom 3 in S : x _ top 3 in bottom : f f : : top Z p _ bottom Z p _ top : A.3 Mechanical Properties of Aluminum Alloy 6063-T6 from AA Specification for Aluminum Structures (2000) Fty : 25 ksi F : 30 ksi : 25 ksi E : ksi F cy tu 54

69 A.4 Moment Capacity Calculation with AA Specification for Aluminum Structures A.4.1 Moment Capacity of E-type Structural Section TD-01 A Moment Capacity of E-type Structural Section TD-01 with Top Flange under Compression 1) Top Flange under Compression Width: in. Thickness: in. Slenderness: 10.8 Intercept BBp: Slope: ksi ksi 0.18 ksi Intersection: Slenderness limit S 1 : Factored limited state stress φ F L : ksi 2) Bottom Flange under Tension Factored limited state stress φ FL : ksi controls Factored limited state stress 3) Compression Side of Web φ FL : 25.5 ksi Hei ght: in. Thickness: in. Slenderness: ksi Intercept B p : Slope: 0.38 Intersection: ksi Slenderness limit S 1 : ksi Factored Limited State Stress 4) Tension Side of Web φf L : ksi Factored limited state stress φ : controls FL ksi Factored limited state stress φ FL : ksi 5) Moment Capacity M M d d 4 = F I / c = 23.75ksi *11.167in / 2.784in = 95. 3in k tf tf 4 = F I / c = 23.75ksi *11.167in / 3.403in = 77. 9in k bf bf 55

70 6.08 M M d d 4 = zf I / c = * 29.25ksi *11.167in / 2.659in = in k tw tw 4 = zf I / c = * 23.75ksi *11.167in / 2.903in = in k bw bw Ultimate moment capacity of TD-01 with top flange under compression: M u = 77.9in k / 0.95 = 82. 0in k A Moment Capacity of E-type Structural Section TD-01 with 1) Top Flange under Tension Factored limited state stress Bottom Flange under Compression φ FL : ksi controls Factored limited state stress φf L : 25.5 ksi 2) Bottom Flange under Compression Width: i n. Th ickness: in. Slenderness: 10.0 Intercept BBp: ksi ksi Slope: 0.18 ksi Intersection: Slenderness limit S 1 : Factored limited state stress 3) Tension Side of Web Factored limited state stress φ FL : ksi φ FL : ksi controls Factored limited state stress φ : ksi 4) Compression Side of Web F L Height: in. Th ickness: in. Slenderness: Intercept BB p: 4 ksi Slope: 0.38 ksi Intersection: ksi Slenderness limit S 1 : Factored limited state stress : ksi φf L 56

71 1 5) Moment Capacity M M M d d d 4 = F I / c = 23.75ksi *11.167in / 2.909in = 91. 2in k tf 4 = zf I / c = * 23.75ksi * in / 2.659in = in k tw tf 4 = F I / c = 23.75ksi *11.167in / 3.403in = 77. 9in k bf bf tw M d = zf bw I / c bw 4 = * 29.25ksi *11.167in / 2.903in = in k Ultimate momen t capacity of TD-01 with bottom flange under compression: M u = 77.9in k / 0.95 = 82. 0in k A Ultimate Moment Capacity of E-type Structural Section TD-01 Ultimate moment capacity of TD-01: M = 82. 0in k A.4.2 Moment Capacity of E-type Structural Section TD-02 u A Moment Capacity of E-type Structural Section TD-02 with Top Flange under Compression 1) Top Flange under Compression Width: in. Thickness: in. Slenderness: 7.50 Intercept BB p: 31.4 ksi Slope: 0.18 ksi i Intersection: ks Slenderness Limit S 1 : 3.89 Factored limited state stress φf L : 21.0 ksi 2) Bottom Flange under Tension Factored limited state stress Factored limited state stress 3) Compression Side of Web φ FL : ksi controls φ FL : 25.5 ksi Height: in. Thickness: in. Slenderness:

72 6.08 Slope: Intercept BB p: 4 ksi Slope: 0.38 ksi Intersection: ksi Slenderness limit S 1 : Factored limited state stress 4) Tension Side of Web Factored limited state stress Factored limited state stress 5) Moment Capacity φ F L : ksi φ FL : ksi controls φ FL : ksi M M d d 4 = F I / c = 21.0ksi *9.980in / 2.941in = 71. 3in k tf tf 4 = F I / c = 23.75ksi *9.980in / 3.247in = 73. 0in k bf bf M d 4 = zf I / c = * ksi *9.980in / 2.816in = in k tw tw M d = zf bw I / c bw 4 = * 23.75ksi *9.980in / 2.747in = in k Ultimate moment capacity of TD-02 with top flange under compression: M u = 71.3in k / 0.95 = 75. 1in k A Moment Capacity of E-type Structural Section TD-02 with Bottom Flange under Compression 1) Top Flange under Tension Factored limited state stress φf : ksi L controls Factored limited state stress 2) Bottom Flange under Compression φ FL : 25.5 ksi Width: in. Thickness: in. Slenderness: 10.0 Intercept BB p: ksi 0.18 ksi Intersection: ksi Slenderness limit S : Factored limited state stress : ksi φf L 58

73 3) Tension Side of Web Factored limited state stress φ FL : ksi Factored limited state stress φf L : ksi controls 4) Compression Side of Web Height: in. Th ickness: in. Slenderness: Intercept B p : ksi Slope: 0.38 ksi Intersection: ksi Slenderness limit S 1 : ) Moment Capacity Factored limited state stress φ F L : ksi M M M M d d d d 4 = F I / c = 23.75ksi *9.980in / 3.066in = 77. 3in k tf 4 = zf I / c = * 29.25ksi *9.980in / 2.816in = in k 4 = zf I / c = * ksi *9.980in / 2.747in = in k bw tf 4 = F I / c = 23.75ksi *9.980in / 3.247in = 73. 0in k bf tw bf tw bw Ultimate moment capacity of TD-02 with bottom flange under compression: M u = 73. 0in k / 0.95 = 76. 8in k A Ultimate Moment Capacity of E-type Structural Section TD-02 Ultimate moment capacity of TD-02: = 75. 1in k M u A.4.3 Moment Capacity of TD-01 and TD-02 as One Ultimate moment capacity of TD-01 and TD-02 as one with top flanges under compression: M u = in k Ultimate moment capacity of TD-01 and TD-02 as one with bottom flanges under compression: M u = in k Ultimate moment capacity of TD-01 and TD-02 as one: M u = in k 59

74 A.5 Moment Capacity by Total Moment Capacity Approach (TMCA) Both the finite element method and the finite strip method were used to predict the minimum buckling stresses and use the smaller one. The minimum buckling stress for TD-01 is ksi from ABAQUS analysis. The equivalent slenderness ratio is λ p = π E = π = 63.1 F cr The minimum buckling stress for TD-02 is 25.4 ksi from ABAQUS analysis. The equivalent slenderness ratio is λ = π E = π p = 62.6 F 25.4 cr Figure A.5.1 Analysis Result of Finite Strip Method CUFSM for E-type Section TD-01 60