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A contribution to the computer aided design of optimized structures for the steel industry Lo, David Siu-Kau 1988

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A CONTRIBUTION TO THE COMPUTER AIDED DESIGN OF OPTIMIZED STRUCTURES FOR THE STEEL INDUSTRY BY DAVID SIU-KAU LO B.A.Sc, The University of British Columbia, 198 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE in THE FACULTY OF GRADUATE STUDIES Department of C i v i l Engineering We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA March 1988 (5) David Siu-Kau Lo, 1988 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department of G i v i l E n g i n e e r i n g The University of British Columbia 1956 Main Mall Vancouver, Canada V6T 1Y3 D ' Fe b r u a r y 24, 1988 DE-6(3/81) Abstract A p r a c t i c a l method of incorporating r e a l i s t i c f l e x i b l e connections including the effect of connection sizes and shear deflection i n plane frame analysis i s presented. The general algorithm can be e a s i l y implemented i n a standard plane frame analysis program and once implemented i t can be an ideal tool for production work i n the steel industry. In t h i s approach connection s t i f f n e s s i s programmed d i r e c t l y into the analysis by u t i l i z i n g the connection moment-rotation equations developed by Frye and Morris but i t may also be entered separately as data. Nonlinear connection analysis i s carried out by the procedure outlined by Frye and Morris. P r a c t i c a l application of this method of analysis i s demonstrated by modifying a standard plane frame analysis program to include the effect of f l e x i b l e connections. The v a l i d i t y of the modified program, CPlane, was v e r i f i e d against the findings of Moncarz and Gerstle. Using CPlane, a simple plane frame structure was analyzed under various l a t e r a l load i n t e n s i t i e s for different connection assumptions. I t was found that the inclusion of connection behavior s i g n i f i c a n t l y altered the internal force d i s t r i b u t i o n and design of the structure. i i Table of Contents Abstract i i 1 Introduction 1 2 The Refined Member-Connection Model 4 3 General Procedure for Assembling the Refined Member Stiffness Matrix ... 10 3.1 Assembling the Refined Fix-Fix Member Stiffness Matrix 19 3.1.1 The Local Bending Stiffness Matrix of F l e x i b l y Connected Fix-Fix Members 19 3.1.2 Introducing the Effect of Connection Sizes 35 3.1.3 Transforming to Global Coordinates 38 3.1.4 Verify i n g the Refined Fix-Fix Member Stiffness Matrix 42 3.1.4.1 Morforton and Wu's Derivation 42 3.1.4.2 The Stiffness Matrix of Members with Rigid Ends 45 3.1.4.3 The Conventional Fix-Fix Member Stiffness Matrix .... 48 3.2 Assembling the Refined Fix-Pin Member Stiffness Matrix 51 3.2.1 The Local Bending Stiffness Matrix of F l e x i b l y Connectected Fix-Pin Members 51 3.2.2 Introducing the Effect of Connection Sizes 60 3.2.3 Transforming to Global Coordinates 63 3.2.4 Verify i n g the Fix-Pin Member Stiffness Matrix 66 3.3 Assembling the Refined Pin-Fix Member Stiffness Matrix 69 3.4 Assembling the Refined Pin-Pin Member Stiffness Matrix 72 i i i 4 Modified Fixed End Forces 73 4.1 Uniformly Distributed Load on F l e x i b l y Connected Fix - F i x Members ... 73 4.2 Uniformly Distributed Load on Fl e x i b l y Connected Fix-Pin Members ... 78 4.3 Uniformly Distributed Load on Fl e x i b l y Connected Pin-Fix Members ... 81 4.4 Point Load on F l e x i b l y Connected Fix-Fix Members 83 4.5 Point Load on Fl e x i b l y Connected Fix-Pin Members 88 4.6 Point Load on Fl e x i b l y Connected Pin-Fix Members 92 5 Member Forces Calculation '. . . . . 94 5.1 Calculating Shears and Moments of Fix-Fix Members 98 5.2 Calculating Shears and Moments of Fix-Pin Members 101 5.3 Calculating Shears and Moments of Pin-Fix Members 104 6 Connection Stiffness 107 7 Programming Details 110 7.1 Modifying Input Format ' I l l 7.2 Modifying Stiffness Matrix I l l 7.3 Modifying Fixed End Forces 113 7.4 Calculating Member Forces 113 7.5 Modeling Nonlinear Connection Response 113 i v 8 Analysis of a Simple Plane Frame Structure with Flexible Connections ... 116 8.1 V e r i f i c a t i o n of CPlane 119 8.2 Girder and Connection Moments 123 8.3 Column Moments 128 8.4 Maximum Top Story Sway 129 8.5 Linear versus Nonlinear Connection Behavior 131 9 Conclusion .. 132 Acknowledgment 134 References 135 Bibliography 136 Appendices 138 Appendix A: L i s t i n g of the Connection Stiffness Subroutine 139 Appendix B: L i s t i n g of the Modified Local Stiffness Matrix 141 Appendix C: L i s t i n g of the Modified Fixed End Forces 145 Appexdix D: L i s t i n g of the Modified Member Forces 152 Appendix E: CPlane User Manual 159 v L i s t of Figures Figure 1 An Idealized Structure 1 Figure 2 A Real Connection 2 Figure 3 A Member with Rigid Ends 3 Figure 4 Joint F l e x i b i l i t y 4 Figure 5 Connection F l e x i b i l i t y 4 Figure 6 Joint Stiffener 5 Figure 7 Nonlinear versus Linear Connection Response 6 Figure 8 Test Frames of Moncarz, Marley and Gerstle 7 Figure 9 A Four-Element Connection Model 8 Figure 10 A Special Connection Element Model 8 Figure 11 A Member-Connection Model by Static Condensation 9 Figure 12 The Refined Member-Connection Model 9 Figure 13 The Adopted Sign Convention 10 Figure 14a The Relationship between Joint and Member End Rotations for Rigi d Connections 11 Figure 14b The Relationship between Joint and Member End Rotations for Fle x i b l e Connections 12 Figure 15a A Simply Supported Beam 13 Figure 15b A Simply Supported Beam under a Uniformly Distributed Load .. 13 Figure 16 The Effect of Shear Strain on Beam Deflection 14 v i Figure 17a A Typical Refined Member i n Its Deformed Position 15 Figure 17b Coordinate Transformation 16 Figure 18a Transfer of Forces 16 Figure 18b Transfer of Forces 17 Figure 19 The Adopted Sign Convention of Conjugate Beam 19 Figure 20 Boundary Condition of Real Beam versus Support Condition of Conjugate Beam 20 Figure 21 Deriving the Second Column of (khll) 20 Figure 22 Conjugate Beam Load : Fix-Fix Member, d 2=l 21 Figure 23 Deriving the F i f t h Column of {kbll} ' 26 Figure 24 Deriving the Third Column of (kbll) 27 Figure 25 Conjugate Beam Load : Fix-Fix member, d 3=l 27 Figure 26 The (ku) matrix 35 Figure 27 The Local Refined Member Stiffness Matrix of Fi x - F i x Members 36 Figure 28 The (K1X) matrix 38 Figure 29 Morforton and Wu's f k u } Matrix 43 Figure 30 The (K^) Matrix of Members with Rigid Ends 46 Figure 31 The Conventional (Klx) Matrix 49 Figure 32a Deriving the Second Column of (kbl0 ) 51 Figure 32b Conjugate Beam Load : Fix-Pin Member, d 2=l 51 Figure 33 Deriving the F i f t h Column of {kbl0} 54 v i i Figure 34a Deriving the Third Column of (khl0) 55 Figure 34b Conjugate Beam Load : Fix-Pin Member, d 3=l 55 Figure 35 The (k10) matrix 60 Figure 36 The Local Refined Member Stiffness Matrix of Fix-Pin Members 61 Figure 37 The (Kw) Matrix 63 Figure 38 The Conventional (K10) Matrix 66 Figure 39 The (K0l) Matrix 69 Figure 40 The (KQ0) Matrix 72 Figure 41a A Simply Supported Beam under a Uniformly Distributed Load .. 73 Figure 41b A Fix- F i x Beam with d 3=l 74 Figure 41c A Fix- F i x Beam with d 6=-l 75 Figure 41d A Fix- F i x Beam under a Uniformly Distributed Load 76 Figure 42a A Simply Supported Beam under a Uniformly Distributed Load .. 78 Figure 42b A Fix-Pin Beam with d 3=l 78 Figure 42c A Fix-Pin Beam under a Uniformly Distributed Load 79 Figure 43a A Simply Supported Beam under a Uniformly Distributed Load .. 81 Figure 43b A Pin-Fix Beam with d 6—1 81 Figure 43c A Pin-Fix Beam under a Uniformly Distributed Load 81 Figure 44a A Simply Supported Beam under a Point Load 83 Figure 44b A Fix- F i x Beam with d 3=l 84 Figure 44c A Fix- F i x Beam with d6=-1 85 v i i i Figure 44d A Fix- F i x Beam under a Point Load 86 Figure 45a A Simply Supported Beam under a Point Load 88 Figure 45b A Fix-Pin Beam with d 3=l 89 Figure 45c A Fix-Pin Beam under a Point Load 90 Figure 46a A Simply Supported Beam under a Point Load 92 Figure 46b A Pin-Fix Beam with d 6=-l 92 Figure 46c A Pin-Fix Beam under a Point Load 93 Figure 47 Member Forces Sign Convention ..• 94 Figure 48 Calculating Shears and Moments 96 Figure 49 Calculating Shears and Moments at Connection Ends by Superposition 97 Figure 50 Shears and Moments of Fix-Fix Members by Superposition 98 Figure 51 Shears and Moments of Fix-Pin Members by Superposition 101 Figure 52 Shears and Moments of Pin-Fix Members by Superposition 104 Figure 53 Common Types of Connections and Their Standardization Parameters 108 Figure 54 Incorporating Flexible Connections i n Plane Frame Analysis ... 110 Figure 55 Input Format for Connection Data I l l Figure 56 Typical Moment-Rotation Relationship of a Nonlinear Connection 114 Figure 57 The Convergence of k for Nonlinear Connections 115 Figure 58a A Simple Plane Frame 116 i x Figure 58b Connection Properties 117 Figure 59a Leeward Column Top Moment versus Lateral Load Intensity 120 Figure 59b C r i t i c a l Moment of Lower Girder versus Lateral Load Intensity 121 Figure 60 Normalized Moment of Lower Girder versus Lateral Load Intensity 122 Figure 61 C r i t i c a l Moment of Lower Girder versus Lateral Load Intensity for Various Connection Assumptions 124 Figure 62 Girder End Moment of Lower Girder versus Lateral Load Intensity for Various Connection Assumptions 125 Figure 63 Difference i n Lower Girder Design Moments for Various Connection Assumptions, w=0.02 • 126 Figure 64 Difference i n Lower Connection Design Moments for Various Connection Assumptions, w=0.02 127 Figure 65 Leeward Lower Column Top Moment versus Lateral Load Intensity for Various Connection Assumptions 128 Figure 66 Maximum Top Story Sway versus Lateral Load Intensity for Various Connection Assumptions 130 Figure 67 Difference i n Maximum Top Story Sways for Various Connection Assumptions, w=0.04 131 L i s t of Tables Table 1 Standardized Connection Moment-Rotation Functions 109 Table 2 Common Connection Types 112 Table 3 L i s t of Girder-Column Connection Design Assumptions 118 x 1 Introduction I n t h e c o n v e n t i o n a l a n a l y s i s o f f ramed s t e e l s t r u c t u r e s , member end-c o n n e c t i o n s a r e i d e a l i z e d as f i n i t e p o i n t s and a r e assumed to behave as p e r f e c t l y h i n g e d o r p e r f e c t l y r i g i d ( F i g u r e 1). I n g e n e r a l t h e s e i d e a l i z a -t i o n s a r e c o n t r a r y to a c t u a l c o n n e c t i o n c o n f i g u r a t i o n and c o n n e c t i o n b e h a v i o u r b u t a r e a d o p t e d b e c a u s e o f the s i m p l i c i t y i n a n a l y s i s and d e s i g n . T y p i c a l l y c o n n e c t i o n s , a r e a b o u t f i v e p e r c e n t the l e n g t h o f a member and t h e y a l l p o s s e s s c e r t a i n amount o f f l e x i b i l i t y i n r o t a t i o n ( F i g u r e 2 ) . A l t h o u g h c o n n e c t i o n s may a p p e a r to be s m a l l i n s i z e and c o n t r i b u t e l i t t l e t o the o v e r a l l w e i g h t o f a s t r u c t u r e , t h e i r b e h a v i o u r c a n s i g n i f i c a n t l y a l t e r the i n t e r n a l f o r c e d i s t r i b u t i o n and hence a f f e c t i n g the o v e r a l l d e s i g n o f a s t r u c t u r e . The common a s s u m p t i o n o f p e r f e c t l y r i g i d c o n n e c t i o n s n e g l e c t i n g c o n n e c t i o n f l e x i b i l i t y and c o n n e c t i o n s i z e s may l e a d to u n d e r e s t i m a t i o n o f the sway o f b a r e f rames and o v e r e s t i m a t i o n o f the f o r c e s a t t h e c o n n e c t i o n s r e s u l t i n g i n o v e r l y h e a v y co lumns and c o n n e c t i o n s . F u r t h e r , c o n n e c t i o n s have a r e l a t i v e l y h i g h l a b o u r c o n t e n t and t h e y c a n r e p r e s e n t a s u b s t a n t i a l p o r t i o n o f the o v e r a l l c o s t o f a s t r u c t u r e . Idealized Connect ion Real Connect ion / F l e x i b l e C o n n e c t i o n M <t> = Slip Angle Figure 2 A Real Connection The topic of connection behaviour has been researched as early as 1917 [1] . More recent research include studies performed by Douty i n 1964 [2], Popov and Pinkney i n 1969 [3], Frye and Morris i n 1975 [4], and Stelmack, Marley and Gerstle i n 1986 [5].. Test result [5] indicates that connection response i s nonlinear i n nature, but can be approximated as l i n e a r l y e l a s t i c within the working range of the frames. Many a n a l y t i c a l methods of modeling connections have been proposed over the years. One commonly accepted method i s the use of a member with r i g i d ends (Figure 3). This method correctly accounts for connection sizes, but i t neglects the rotati o n a l f l e x i b i l i t y of connections. Gere and Weaver [6], Morforton and Wu [7], and Livesley [8] have each presented methods of modeling line a r e l a s t i c connections. However thei r methods a l l neglect the effect of connection sizes and shear deflection. More elaborate methods of modeling nonlinear connection response have been presented by Romstad and Subramania [9], and Moncarz and Gerstle [10]. Unfortunately the i r methods also neglect the effect of connection sizes and shear deflection. In .addition p r a c t i c a l implementation of th e i r methods involve extensive and expensive programming work. A p r a c t i c a l method of incorporating connection behaviour i n plane frame analysis for o f f i c e use i s much i n need. 2 R i g i d E , I, A R i g i d L 1 L Figure 3 A Member with Rigid Ends The object of th i s work i s twofold. The f i r s t i s to present a p r a c t i c a l method of modifying an existing l i n e a r e l a s t i c plane frame analysis program to perform refined f l e x i b l e frame analysis for o f f i c e use. The second objective i s to demonstrate the usefulness of such a method of analysis by applying i t to a t y p i c a l unbraced ste e l frame and comparing i t s response with the response from commonly accepted methods of analysis. I t i s hoped that t h i s w i l l serve to encourage the use of this type of analysis i n practice i n the future. 3 2 Refined Member-Connection Model Conventional s t r u c t u r a l analysis assumes that connections have negligible dimensions and behave as perfectly r i g i d or perfectly hinged. But real connections have f i n i t e dimensions and possess some degree of f l e x i b i l i t y whether they are r i g i d or hinged, and their behaviour i s rather complex. To be perfectly precise one should r e a l l y distinguish between j o i n t f l e x i b i l i t y and connection f l e x i b i l i t y . Joint f l e x i b i l i t y refers to the a b i l i t y of the j o i n t to deform i n shear or i n bending (Figure 4) while connection f l e x i b i l i t y refers to the s l i p i n rotation between a j o i n t and the member end (Figure 5 ) . — T S H E A R D E F O R M A T I O N (fJr) ^ V X B E N D I N G D E F O R M A T I O N B E N D I N G D E F O R M A T I O N Figure 4 Joint F l e x i b i l i t y Flexible Connection M J o i n t R o t a t i o n <t> - S l i p A n g l e Figure 5 Connection F l e x i b i l i t y 4 A properly detailed j o i n t should not undergo s i g n i f i c a n t deformation i n shear or i n bending; i n practice j o i n t s which are weak i n shear are strengthened with the addition s t i f f e n e r s (Figure 6). Therefore the effect of j o i n t f l e x i b i l i t y i s usually small and i t w i l l not be considered herein. S t i f f ener \ 1^ \ y " * — Figure 6 Joint Stiffener However, connections may possess s i g n i f i c a n t f l e x i b i l i t y i n rotation [1,2,3,4,5] and i t s effect should be considered. The degree of rotat i o n a l f l e x i b i l i t y exhibited by a connection depends very much on the type of connection i n question; a bolted or l i g h t l y welded connection i s l i k e l y to be more f l e x i b l e than a f u l l y welded connection [4]. I t i s desirable to have a computer model capable of accounting for the effect of connection sizes as well as connection f l e x i b i l i t y . Many investigators [1,2,3,4,5] have studied the behaviour of f l e x i b l e connections. The most comprehensive formulation of f l e x i b l e connection behaviour i s perhaps the one presented by Frye and Morris [4]. They tested a wide variety of connections under different monotonic loading conditions. Their r e s u l t indicates that the response of f l e x i b l e connection i s nonlinear i n nature. However, Moncarz and Gerstle [10] observed that f l e x i b l e 5 connections may be connection response response with l i n e a r s a t i s f a c t o r i l y modeled by assuming l i n e a r e l a s t i c They compared a n a l y t i c a l l y nonlinear connection connection response (Figure 7) and the i r conclusion : "The assumption of linear response of flexible connections seems reasonable and appears to give a good prediction of the bare frame response " " 1 They further remarked that the sequence of load application only played a minor influence i n the sways of the multistory frames which they investigated. Wi W 2 g + p t w t t t m 1 2 ' - 0 " 12'-0" 2 4 ' - 0 " Column Design Moment vs. Lateral Load o Rigid Connections - Nonlinear Connections • Linear Elastic Connections 0.01 0.02 0.03 w (k ip / f t"2) 0.04 Figure 7 Nonlinear versus Linear Connection Response 1 Moncarz, P.D. and Gerstle K.H., [Ref 10] p.1440. The findings of Moncarz and Gerstle [10] are la t e r confirmed by the experimental r e s u l t of Stelmack, Marley and Gerstle [5]. Stelmack, Marley and Gerstle performed a t o t a l of 10 tests of 2 frame configurations (Figure 8) and here are some of their observations : "1. The connection response remained essentially linear elastic within the working range of the frames. Accordingly linear elastic frame analysis is adequate for predicting frame response to service loads, 2. No evidence of incremental deflections or other instabilities was observed under a significant number of cycles at high loads".2 In l i g h t of the a n a l y t i c a l and experimental evidence presented, i t appears reasonable to assume linea r e l a s t i c connection response for l i n e a r e l a s t i c s t r u c t u r a l analysis. w2 E 3 p p E 3 3'-0- 3'-0 - 3'-0-(a) One-Bay, Two-Story Frame 3'-0" 3'-0 - 3*-0- 3'-0 - 3'-0' 3'-0" (b) Two-Bay, Single-Story Frame Figure 8 Test Frames of Moncarz, Marley and Gerstle Several d i f f e r e n t approaches may be adopted to model li n e a r e l a s t i c connections. The simplest and most obvious one i s to treat each connection as four elements joined together r i g i d l y at one point l i k e a cross (Figure 9). Connection behaviour may be modeled by specifying appropriate value of length, E I X 3 and EA4 for each element. This method can be applied to any standard 2 Stelmack, T.W., Marley, M.J., and Gerstle, K.H., [Ref.5] p.1586. 3 EI X represents the bending s t i f f n e s s of the connection. 4 EA represents the a x i a l s t i f f n e s s of the connection. 7 s t r u c t u r a l analysis program. Unfortunately i t also introduces 12 extra degrees of freedom and 4 extra elements at each connection. This greatly reduces the size of structure which can be analyzed on a computer and renders i t impractical. Figure 9 A Four-Element Connection Model An alternative approach i s to introduce a special connection element [11] at each connection (Figure 10). Connection behaviour may be modeled by assigning appropriate value of length, height and E I X 5 to each special element. This method introduces only 1 extra degree of freedom and 1 extra element at each connection, but i t i s s t i l l inconvenient for p r a c t i c a l use because one has to specify e x p l i c i t l y how the member elements are connected with the connection elements. I f one wishes to a l t e r the structural configuration the connection sequence would have to be specified again. Figure 10 A Special Connection Element Model 5 E I X represents the bending s t i f f n e s s of the connection. A better approach i s to incorporate the connection elements d i r e c t l y into the member element (Figure 11). This can be achieved by combining two connection elements together with a member element and then removing the s i x i n t e r i o r degrees of freedom by means of s t a t i c condensation. This method i s fine except for the fact that i t s implementation involves substantial modification to a standard structural analysis program which can be both expensive and time consuming. I t seems best to take a direct approach i n modeling connection response. Figure 12 depicts the proposed refined member-connection model; i t consists of a member element and two r i g i d end pieces connected together by two rotati o n a l springs. The r i g i d end pieces model the effect of connection,sizes and the two ro t a t i o n a l springs model the s l i p between the connections and the member ends (Figure 6). This approach i s deemed direct because each element of the model represents a s p e c i f i c aspect of connection behaviour. There are several advantages to th i s model. F i r s t of a l l i t does not introduce any additional degrees of freedom to the system. Secondly, connection elements are d i r e c t l y incorporated into the member elements and the tedious task of jo i n i n g member elements to connection elements i s avoided. Lastly this model can be ea s i l y implemented i n a standard plane frame analysis program as w i l l be shown i n l a t e r sections. Figure 11 A Member-Connection Model by Static Condensation Figure 12 The Refined Member-Connection Model g 3 General Procedure for Assembling Refined Member Stiffness Matrix The f i r s t step i s to derive a l o c a l member s t i f f n e s s matrix, (k) which includes the effect of connection f l e x i b i l i t y and shear deflection. The effect of connection sizes w i l l be added on l a t e r . One should note that the a x i a l and bending component of a member s t i f f n e s s matrix are uncoupled and they could be separated into a l o c a l a x i a l s t i f f n e s s matrix {ka} and a l o c a l bending s t i f f n e s s matrix {kh}. Connection f l e x i b i l i t y and shear deflection only affect the l o c a l bending s t i f f n e s s matrix (kh), thus only {kb} needs to be derived. The adopted sign convention i s i l l u s t r a t e d i n Figure 13; The d's and f's denote the l o c a l degrees of freedom and member end-forces associated with the member. [3-1] d 2 d5. d 3 d 6 + ' v e r o t a t i o n + ' v e V & M Figure 13 The Adopted Sign Convention 1 0 As explained e a r l i e r i n Section 2 connection f l e x i b i l i t y may be modeled. by means of two rot a t i o n a l springs, one at each end of the member. Connection behaviour i s governed by the rotati o n a l s t i f f n e s s of the springs, Kx and K2 such that: 0, = - ^ [3- 2 a] * 2 " where ' 1 > v 2 = member-end rotations, /a./6 = member-end forces, K\,K2 = spring s t i f f n e s s constants. I f the connections are perfectly r i g i d (Figure 14a), the slope of member cross section at either end of the member y>m2 must equal to'- the corresponding j o i n t rotation, p n , v/2. i i I f the connections are f l e x i b l e (Figure 14b) and the s l i p i n rotati o n between connections and member ends are denoted as <t>2, the relationship between p/s and p m ' s now become: P m 2 = Vj2 + <t>2 [3-3a] [3-3b] Figure 14b The Relationship between Joint and Member End Rotations for Flexible Connections The method of modeling shear deflection w i l l now be explained. Consider a beam (Figure 15a) with a series of lines a-a painted on i t s surface at right angle to the neutral axis. These pained l i n e s represent the cross sections along the beam. The same beam i s shown i n i t s deformed po s i t i o n after a uniformly d i s t r i b u t e d load i s applied to i t (Figure 15b). Under the engineering beam theory the lines a-a must remain at right angle to the neutral axis such that the i r rotation denoted as a i s dy/dx. In other words, the slope of member cross sections with respect to the v e r t i c a l axis, >i>m must equal to the slope of the neutral axis of the member, a. 12 Figure 15a A Simply Supported Beam Y ym=a 'a — ~ \ 1 ••••A'—\ a X ^ E d ^ d x Figure 15b A Simply Supported Beam under a Uniformly Distributed Load However shear stresses i n the beam w i l l cause the lin e s to rotate to the dotted positions a'-a' (Figure 16) and the cross sections w i l l no longer be at right angle to the neutral axis. The rotation of the cross sections from a-a to a'-a' i s defined as the shear s t r a i n , y. y -v [3-4] where V G = shear force at member cross section, = shear area, = shear modulus. 13 Figure 16 The Effect of Shear Strain on Beam Deflection This would mean that: pm = a-y [3-5] where yjm = slope of member cross section, a = rotation of neutral axis, dy/dx, y = shear s t r a i n . Combining the re s u l t of [3,-3] and [3-5]: a-i=Vix+4>i+y\ [ 3 " 6 a ] «2 = f > 2 + 02 + y 2 [3-6b] The combined effect of connection f l e x i b i l i t y and shear deflection w i l l included i n the derivation i f the the boundary conditions as prescribed [3-6a,b] are enforced at both ends of the member. Having derived (kb), the next step i s to introduce the r i g i d end pieces to model connection sizes. This i s done by means of transformation. Figure 17a depicts a t y p i c a l refined member i n i t s deformed position. The relationship between the displacements of the f l e x i b l y connected member denoted as d's and that of the refined member denoted as D's i s summarized i n the same figure. D 5 D 4 d 5 dNSf d 2 d, 1 — D 6 Di ' *~ X dl =D i d 4 = D 4 d 2 = D 2 + D 3 ^ i d 5 = D5-D6-ll2 d 3 = Da d 6 = D 6 Figure 17a A Typical Refined Member i n I t s Deformed Position Arranging these relationships i n matrix notation: {d} = {T c}-{£> [3-7] where {d> = displacement vector of f l e x i b l y connected member (TcJ = displacement transformation matrix {D} = displacement vector of refined f l e x i b l y connected member 15 The matrices of [3-7] are summarized i n Figure 17b. <D} = W ( 1 0 0 0 0 0 0 1 ' l 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 -I 0 0 0 0 0 1 J d0 {d} d. d. Figure 17b Coordinate Transformation S i m i l a r l y , Figure 18a depicts the relationships between the forces of the f l e x i b l y connected member denoted as f's and that of the refined member denoted as F's. Figure 18a Transfer of Forces 16 Arranging these relationships i n matrix notation: <F} = {Tf}-{f} [3-8] where {/} (rf) = force vector of f l e x i b l y connected member = force transformation matrix = force vector of refined f l e x i b l y connected member The matrices of [3-8] are summarized i n Figure 18b. ''A r 2 F, F, \ F ' I VJ ( 0 I 0 0 0 0 0 1 0 0 0 0 ' l 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 -h 1 {/> = J / 2 / 3 Noting that: one could define: Figure 18b Transfer of Forces <7} = {T C} = { T / } T From theory of e l a s t i c i t y : {/>= <fc>-{d> Substituting [3-6] and [3-8] into [3-9] {f}={k}-{T}-{D} [3-9] [3-10] [3-lla] Pre-multiplying both sides of [3-lla] by (T)T and substituting [3-8] 17 {F}={T}T-{k}-{T}-{D} Define: [3-llb] {K} = {T}T-{*:}• {7} [3-12] where = the refined s t i f f n e s s matrix i n l o c a l coordinates. F i n a l l y , the refined s t i f f n e s s matrix i n global coordinates may be obtained by applying the standard transformation procedure: {K} = {Tr}T-{X}-{Tr} o r {Tr}T-{T}T-{k}-{T}-{Tr} where {Tr} I c s 0 0 0 0 -s c 0 0 0 0 0 0 1 0 0 0 0 0 0 c s 0 [ ° 0 0 - s c 0 V o 0 0 0 0 1 Note 1 L Ax Ay c s {7> = [3- 13] / I 0 0 0 0 0 \ 0 1 ' i 0 0 0 \ 0 0 1 0 0 0 0 0 0 1 0 • 0 1 0 0 0 0 1 \ V o 0 0 0 0 1 J = length of connection ( r i g i d end pieces) = length of member, = t o t a l span = 1^ + 1 + lz, — change i n x form j o i n t 1 to j o i n t 2, = change i n y from j o i n t 1 to j o i n t 2, = cos(Ax/L), = sin (Ay / L) . 18 3.1 Assembling the Refined Fix-Fix Member Stiffness Matrix 3.1.1 The Local Bending Stiffness Matrix of Flexibly Connected Fix-Fix Members The derivation of (khll) as outlined i n Section 3 may be carried out by any c l a s s i c a l methods. The method of conjugate beam i s used here out of personal preference. The adopted sign convention i s i l l u s t r a t e d i n Figure 19. The relationship between the boundary conditions of a re a l beam and the corresponding support conditions of the conjugate beam i s summarized i n Figure 20. M o m e n t D i a g r a m of R e a l B e a m M x C o n j u g a t e B e a m L o a d i n g M Figure 19 The Adopted Sign Convention of Conjugate Beam 19 Real Beam Conjugate B e a m t t Ri=©i R 2 = S 2 Figure 20 Boundary Condition of Real Beam versus Support Condition of Conjugate Beam The f i r s t and fourth column of ( kbli) are zero vectors because a x i a l displacements do not affect ( k b l l ) . The second column of (k b l l ) i s obtained by setting d 2 equal to 1 (Figure 2 1 ) . This i s a 2 ° indeterminate system. I f the two fixed-end moments, f 3 and f 6 are chosen to be the redundant, the conjugate beam w i l l be loaded as shown i n Figure 22 . Figure 21 Deriving the Second Column of (k b l l ) 20 EI Figure 22 Conjugate Beam Load : Fix - F i x Member, d 2=l Because the shears at the ends of the conjugate beam are equal to the net angle changes at the ends of the actual beam, one could write from [3-6]: . ^ 2 = a.2 " #>,2 + y 2 + 0 2 [3-14a] [3-14b] Also, the moment m1 of the conjugate beam equals to the deflection d 2 of the re a l beam. Therefore, m , = l [3-14c] Substituting [3-2] and [3-4] into [3-14], V Au-C l l [3-15a] Note : The -'ve sign i s due to the conjugate beam's d e f i n i t i o n of counter clockwise rotation as positive rotation (See Figure 13). v _ n Av - C K2 [3-15b] But, d 3 = ^ ; 1 = 0, d 6 - p j 2 = 0 21 Therefore [3-15] becomes, V /3 /?,= + — [3-16a] R7 = — [3-16b] 2 A„-G K2 From equilibrium of conjugate beam: fz' I f 6 ' I nti R.= -— + — + — [3-17a] 1 3-EI 6 • EI I f 6 ' I / T " I fTl, R-= — - — - — [3-17b] 2 3-EI 6- EI I where m, = l from [3-14c] Combining [3-16a] and [3-17a] and c o l l e c t i n g terms, 3 \ Kt 3- EI J 6- EI Au-G I Multiplying both sides by: 3 - EI I and defining dimensionless constants: 12 • EI A„- G- / 2 [3-18] v,=[ 1+ | [3-19] K \ ' I Resulting i n : / 3 ft V - g - l 3-EI I-[3-20] Si m i l a r l y , combining [3-16b] and [3-17b] and c o l l e c t i n g terms, 1 I \ f3.-l V 1 ^ 6 1 K2 3-EI J 6-EI Av-G I 22 Multiplying both sides by: 3 - EI I and defining dimensionless constants: \2 - EI v ' - l 1 + ^ r f r ' - ,3-211 Resulting i n : U.tl. V - B - ' + ^ I L [3-22] v2 2 4 I2 ' From equilibrium of r e a l beam: V = / 3 + /< I [3-23] Substituting [3-23] into [3-20],[3-22] and c o l l e c t i n g terms, f: ( 4- v V 4 + vy • g 4 [3-24] / 6 = 4 • v -4 + v 2 - g 3 - £ 7 / 3 [3-25] Substituting [3-25] into [3-24], / 3 = 4 • v 4 + v , • g 3 - £ 7 Z 2 4- v 2 4 + v 2 • g 3 - £ 7 / V + T - ( 2 - g ) ( 2 - g ) 23 Collecting terms, (4 + v , • g)-(4 + v 2 - g ) - v , • v 2 - ( 2 - g ) : / 4 - v , - ( 4 + v 2 - g ) y 3 - £ 7 f 4 - ( 4 + v 2 ) + 8 • v 2 - 4 • v2 • g I' 4 - ( 4 + v 2 - g ) or, v'i+v2 + v i - v 2 \ 6-EI f v 1 - ( 2 + v 2 ) ^ 4 - v,•v2 4 - v , • v i ' " 2 y Defining dimensionless constants: 4 - v ! • v 2 [3-26] C 2 = v , 2 + v -4 - v , • v 2 [3-27] Therefore, 6- EI f / 3 - — — - c V 1+sf-c, [3-28] S i m i l a r l y , substituting [3-24] into [3-25] and c o l l e c t i n g terms: / 6 = 4 • v • 4 + V ! • g 3- £ 7 4 • v2 4 + v 2 • g 3 • £ 7 / + — - ( 2 - g ) ( 2 - g ) or, ' • ' l 1 * 9 ' j = — ' h ' T ^ ; 24 D e f i n i n g d i m e n s i o n l e s s c o n s t a n t s : _ v , + v2 + v , •v2 L 1 ~ 4 - v , • v 1 " 2 C 4 = v ' - l 2 » " ' J I-3-29] 4 - v , • v 2 T h e r e f o r e , / 6 i 2 4 V l + g - c , ; [3-30] S u b s t i t u t i n g [ 3 - 2 8 ] , [3-30] i n t o [ 3 - 2 3 ] , 1 + g - C , 7 [3-31] I n t r o d u c i n g a n o t h e r d i m e n s i o n l e s s c o n s t a n t to [ 3 - 2 8 ] , [3-30] and [ 3 - 3 1 ] , 1 _ [3-32] 1 + g • C , f 3 , f 6 a n d V now become: 6 • EI f3 = -j—-C2-S1 [3-33] r / 6 = 7 T — - C 4 - 5 , [3-34] K = 1 2 ^ 3 £ / - C . - 5 , [3-35] 25 In summary the second column of (khll) i s : fan '2 0 12: EI ? 6- EI \ 2 ' C 2 ' $ 1 0 12 • EI I3 6- EI 2" where 4 - v , • v 1 " 2 1 + 3- EI K, • I - I C 0 = 4 - v , • v 2 v , = 1 + 3- £7 A", • / -1 4 - v , • v 2 12 • £ 7 y4„- G - / : 1 + g - c , The f i f t h column of f/c b l l) i s obtained by setting d 5 equal to 1 (Figure 23), and i t can be derived i n a simi l a r fashion as before. Figure 23 Deriving the F i f t h Column of (khll) 26 But noting that Figure 23 i s a mirror r e f l e c t i o n of Figure 21, the column vector can be written d i r e c t l y as: 0 12 • El 6- EI \ I-0 V 12 • EI I3 6-EI I2 C l - S l c0 - s J The t h i r d column of (k h l l ) i s obtained by setting d 3 equal to 1 (Figure 24). Once again this i s a 2° indeterminate system. I f the two fixed-end moments, f 3 and f 6 are chosen to be the redundant, the conjugate beam w i l l be loaded as shown i n Figure 25. Figure 24 Deriving the Third Column of {kbll} Figure 25 Conjugate Beam Load : Fix-Fix member, d 3=l 27 Because the shears at the ends of the conjugate beam are equal to the net angle changes at the ends of the actual beam, one could write: [3-36a] Substituting [3-2] and [3-4] into [3-36], [3-36b] R V f: A„-G K, [3-37a] Note : The -'ve sign i s due to the conjugate, beam's d e f i n i t i o n of counter clockwise rotation as positive rotation (See Figure 13). [3-37b] But, d3 = P J l = 1. Therefore [3-37] becomes, R , = - \ + V Av-G K, ft V • * 2 ~ ~ Av-G K From equilibrium of conjugate beam: [3-38a] [3-38b] 3-EI 6-EI f 6 - l f 3 - l 3-EI 6-EI [3-39a] [3-39b] Combining [2-38a] and [2-39a] and c o l l e c t i n g terms, /3- IT 1 I f 6 ' l V K{ 3-EI ) 6-EI Av-G •+ 1 28 Multiplying both sides by: 3 - EI I and defining dimensionless constants as i n [3-18], [3-19] 12 • EI g Av-G- V 1 + 3 - EI K i • I -1 Resulting i n : l l = l l v, 2 V-g - I 3 - E I 2 + [3-40] Si m i l a r l y , combining [3-38b] and [3-39b] and c o l l e c t i n g terms, I fz-L V K2 3-EI J 6-EI Av-G Multiplying both sides by: 3-EI I and defining dimensionless constants as i n [3-18], [3-21]: \2-EI 9 = Av - G - I ( , 3 - £ 7 V 1 v 2 = 1 + — V K 2 - l J Resulting i n : / 6 _ / 3 v - g - i v 2 2 4 [3-41] 29 From equilibrium of r e a l beam: V = f3 + f< I [3-42] Substituting [2-42] into [2-40],[2-41] and c o l l e c t i n g terms, 4 • V , A [3-43] ft V2 • /: ( 2 - f i O [3-44] Substituting [3-44] into [3-43], f: 4 • v 4 + v , • g 3 - £ 7 , v 2 - / 3 ( 2 - g ) ( 2 - g ) — I 4 + v2-g Collecting terms, f (4 + v , - g ) - ( 4 + v 2 - g ) - y 1 - v 2 - ( 2 - g ) : V or, 4- v , • (4 + v 2 • g) 4- EI y , + v2 + v{ • v2 4 - v , • v 2 Z 3 - £ 7 3 • v 4 - v , • v 2 ( i Defining dimensionless constants: C, = 4 - v ! • v 2 [3-45] C, 4- v ! • v 2 [3-46] Therefore, ' . - ^ • M - T ^ I T -30 S i m i l a r l y , substituting [2-43] into [2-44] and c o l l e c t i n g terms: v , + v 2 + v , - v 2 \ 3-EI vl - v2-2-(1 - g / 2 ) 4- - v l • v 2 J I 4 - v l - v 2 Defining dimensionless constants: C, = 4 - v , • v 2  3 - v x - v 2 5 4 - v , - v 2 [3-48] Therefore, 2- EI f 1 - g / 2 A r , Substituting [2-28], [2-30] into [2-23], V= 6 ' f 7 - C 2 - l | [3-50] Introducing dimensionless constants to [3-47], [3-49] and [3-50], s i + v 2 - g / 4 2 [3-51] 5 i - g / 2 l + g - C i ' [3-52] f 3, f 6 and V now become: 4-EI f 3 , 3 ' 2 J [3-53] 2- £7 J 6 , ^ 5 ' 3 I [3-54] 6 • £7 / 2 [3-55] 31 In summary the t h i r d column of (kbll) i s : V 0 6- EI I2 4- EI \ C 2 • S j C 3 1 S 2 0 6- EI I2 2- EI I C 2 • 5 [ where c 4 4 - v , • v 2 v,-{2+v2) 4 - V ! • v 2 3 • v , 4 - v , • v 2 4 - v , • v 1 " 2 C 5 = 4 - v , • v 2 C6 = 3 • v. 4 - v , • v 2 v, = 1 + v 2 = 1 + 3 - £ 7 K x - I 3 - £ 7 - I - I g 5 , = 5 4 = # 2 ' / , 12- EI Au-G- I2 1 1 + g - C , 1+ v 2 • g / 4 i + g - c , 1 - g / 2 l + g - c , 1 + v t • g / 4 l + g - c , 32 The s i x t h column of {kbn} i s obtained by setting d 6 equal to 1 , and i t can be derived i n a si m i l a r fashion as before. The resu l t i s shown below: Where V 0 6-EI • C 4 • S ! \ Z' 2- EI I C q • S 5 ^3 0 6- EI V 4- EI } c 6 - s 4 ^ 4 = 3 -4 - v , • v 2 1+ vj • g / 4 1 + g - C , In summary: /0 0 o c , - s , ZJ 6- £ 7 0 2 C 2 ' S j 0 12-EI 3 £ 1 ' 1 0 6-EI 2 • C 2 • S j z-4- £ 7 Z C 3 • v? 2 \ 0 6- £ 7 2 ^ 2 ' $ 1 6- £ 7 2 ^ 4 • 5 , 2- £ 7 Z £ 5 ' $ 3 \ 33 0 0 1 2 - EI —r* c . ' 5 > I' 6- EI Z 2 0 6-EI ,2 * C 4 ' S 1 4 ^ 1 Z' Z c 6 • s 4 \ J where C-4 - v i • v 2 f r ( 2 + r 2 ) 4 - V ! • v 2 3 • V j 4 - V i - V j 4 - v , • v 2 3 ' • ^ 2 4 - v , • v 2 4 - v , • v 2 v , = 1 + v 2 = 1 + 3 - EI K, • Z 3 - £ 7 ^2 = 5-^ 4 K2 • I J 12 • EI Av - G • I2 1 1 + g r - C , 1+ v 2 • g / 4 l + g - C , i - g / 2 i + g - c , 1+ v , • g / 4 l + g - c , 34 3 . 1 . 2 I n t r o d u c i n g t h e E f f e c t o f C o n n e c t i o n S i z e s The e f f e c t o f c o n n e c t i o n s i z e s i s i n t r o d u c e d b y means o f t r a n s f o r m a t i o n as o u t l i n e d i n [ 3 - 1 2 ] : {Zn} = {T}T-{ku}-{Ty Where (T) = t h e l o c a l member s t i f f n e s s m a t r i x i n c l u d i n g t h e e f f e c t o f f l e x i b l e c o n n e c t i o n s and s h e a r d e f l e c t i o n s , = t h e t r a n s f o r m a t i o n m a t r i x d e f i n e d b y [ 3 - 9 ] , = t h e r e f i n e d s t i f f n e s s m a t r i x i n l o c a l c o o r d i n a t e s . The l o c a l member s t i f f n e s s m a t r i x f k n } c o m p r i s e s o f (ka) and (khll). {ka} i s the s t a n d a r d p i n - p i n member m a t r i x and (kbll) has j u s t b e e n p r e s e n t e d i n the p r e v i o u s s e c t i o n . The f u l l {kn} m a t r i x i s shown i n F i g u r e 26. 11 f o V 0 AE 0 V 0 / AE I 0 V 0 o 0 \ 12- El I3 6- EI I2 C 2 ' S ! 6- EI I2 4- EI I C 2 • S , C 3 • S 2 J 0 12 • EI ~T3 6-EI I2 0 6- EI I7 2- EI C 2 ' S , ^ 5 ' -^3 J 0 0 \ 12 • EI ~ T 3 6-EI I2 6- EI I2 4- EI • I c 6 - s 4 J F i g u r e 26 The (kix) m a t r i x 35 The l o c a l refined member s t i f f n e s s matrix [~K xx) i s i l l u s t r a t e d i n Figure 27 AE I 0 0 0 12 • £ 7 ,3 ' C 1 ' S 1 1 2 - £ 7 - Z, Z 3 6 - £ 7 Z 2 c 2 • s, 0 1 2 - El - Z, Z 3 6- EI C . - 5 , Z' 4- £7 Z C 2 • 5 i C 3 • >? 2 1 2 - £ 7 - Z, Z 1 12 • £ 7 Z 3 r • 9 2 J 1 AZ£ Z 0 0 0 12 • EI Z' 12 • EI • I Z 3 6- EI I2 C l - S l c 2 • s } 0 12 • EI - I. Z 1 6- EI C x - S x 2- EI C 4 • S j C 5 * ^ 3 z 6- EI - li Z 2 6- £ 7 • Z: Z 2 12 • EI • Z, • Z 2 C 4 • 5 , C 2 • S , c , - s , Figure 27 The Local Refined Member Stiffness Matrix of Fix- F i x Members 36 AE I 0 0 0 1 2 • EI I3 c r s l 1 2 • EI • l-l3 6-EI I2 • C { - S , C 4 • S j 0 1 2 • EI • l: T3 6-EI c r s l V 4- EI I 1.2 • EI • I. ? 1 2 - EI I3 C 4 • S j c 6 - s 4 c\4 • s, Figure 27(cont'd) The Local Refined Member Stiffness Matrix of Fix-Fix Members where C , = v i + v 2 + v i • v 2 4 - y j • y 2 v, = 1 + 3 - EI -1 C 4 - y , • y 2 v , - 1 + 3 - £ 7 3 • y, C. Cc C , 4 - v-1 • y 1 " 2 " 2 / ( 2 + v i , 4 - • v 2 3 - ^ 1 ' ^ 2 4 - v , • y 2 3 • y-4 - y , • v 2 llt lz = length of connection 2 = length of member L = t o t a l span = 11 + 1 + 2 2 5. 5 4 = 1 2 • £ 7 / ! „ • C • I2 1 l + .g-c , . 1+ y 2 • g / 4 1 + g - c x 1 - g / 2 1 + g - c , 1+ y ! • g / 4 37 3.1.3 Transforming to Glocal Coordinates F i n a l l y , the global refined member s t i f f n e s s matrix (K^) i s obtained by means of r o t a t i o n a l transformation as outlined by [3-13]: {Ku} = {Tr}T-{Xn}-{Tr} {K^} i s i l l u s t r a t e d i n Figure 28. AE 2 AE 12; EI I3 12-EI 75 C i - 5 , - c - s 12- EI • t, I3 6- EI C 2 • S , • s AE AE K ) I 12 • £ 7 • • • C t • S x • c • s + \1-El I3 12' f / • Z, Z3 6- EI H • z2 — 7 5 C,-5,-s C , - 5 , - c 1 C 3 - S -6 - £ 7 _ 0 6 - £ 7 ^ o 1 2 - f Z - Z , • C 2 - S , - s + T — C 2 - S , - c + - • C 2 - S 1 Z2 - i - Z2 - 2 - i ~ [ 2 12- £7 2 ,T ' ' 1 ' C 1 ' ^ 1 Figure 28 The (K^} matrix 38 AE 12 • EI Z3 Ct • 5,•s' AE 12-EI „ „ + - r C j • 51 • c • s 12- £ 7 • Z2 Z3 6 - £ 7 C , • S, • s 2 ' "^4 ' >5*1 " S / I f + r i C , - 5 , - f s / I f 12-r/ z3 C , • S, • c 2 12- £"/• Z3 6-EI C, • 6", • c Z' C 4 • S , • c 12-///• Z, Z3 6 - £ 7 Z2 C , • S , • s \1-EI • Z, Z3 6-EI • C , • S , • c Z' 2 • £ 7 Z C V S , 6-EI- L C 4 - S , 6-EI- l2 I2 \2 - EI • li - I I3 C 9 * 5* i • C . - S , Figure 28(cont'd) The f£ n} matrix 39 AE 2 12-F/ „ „ 2 AE c • s 1 2 - F / „ „ 75 C , - S , - c - s 12-F/- Z; Z3 6- EI Z' C 4 - 5-, • s y4F 1 2 - F / . _ 75 C , - 5 , - c - s /4F 12- F/ Z3 C , • Si • c2 12 • EI • L2 I3 6 • EI C.- S,•c 12-EI- Z2 ? 6- EI Cv- Si - s V C4-SL • s \2-El • l2 I3 6- EI V C4- S r c 4- El I 12-EI - l2 * j ^ - c . - S i 12-EI I3 l2 - C i • S [ Figure 28(cont'd) The f£ n} matrix where 4 - v , • v 1 " 2 C 2 = 1 2 + v-r - V , • V 2 3 • V, 1 4 - V [ • V 2 c 4 = C 5 = 2+ v C , 4 - v , • v 2 3 • v i • v 2 4 - v i • v 2 3 • v 2 4 - v , • v 2 1 + 1 + 3 - F / A , • Z 3 - £ 7 -1 g 5-S-K2 • I 12-EI Av-C- I2 1 1 + g - C , 1+ v 2 ' g / 4 1 + g - C , 1 - g/2 1 + g - C , 1+ v,•g/4 1 + g - C , 40 length of connection ( r i g i d end pieces length of member, t o t a l span = lx + 1 + 12, change i n x form j o i n t 1 to j o i n t 2, change i n y from j o i n t 1 to j o i n t 2, cos(<dx/L), sin (Ay / L) . 3.1.4 Verifying the Refined Fix-Fix Member Stiffness Matrix 3.1.4.1 Morforton and Wu's Derivation Morforton and Wu [7] also used the conjugate beam method to derive the l o c a l s t i f f n e s s matrix for f l e x i b l y connected members but neglecting the effect of connection sizes and shear deflection (Figure 29). The st i f f n e s s matrix {KZ1} presented i n the previous section should reduce to that of Morforton and Wu i f the effect of connection sizes and shear deflection are eliminated. This i s done by setting: h,l2 - 0, g - o . and considering l o c a l coordinates: c = 1, s = 0. The dimensionless constants now become: C , = V , + V 2 + V , • V 2 4 - y , • v2 v , = 1 + 3 - EI 2+ v. 4 - y i • y 1 " 2 y 2 = l 1 + 3-EI -1 3 • y -4 - y , • v 1 " 2 5 ,= l +g-c = l y-i-(2 + vXl 4 - y , • y 2 l + y 2 - ( g / 4 ) 1+g-C, C. 3 • v i • y 2 4 - v , • y 2 5 3 = 1 -g/2 l+g-c, = 1 42 :C, 3 • v. 4 - v , • v 2 I+v . , - (g /4 ) , •J 4 - 1 l+g-c One sees that the s t i f f n e s s matrix presented i n the previous section reduces to that of Morforton and Wu. k, k 11, / AE_ I 0 AE I 0 0 f * . . L - { * n ) « / AZJ Z 0 V 0 0 12-EI ~T3 6- EI C 2 C2 0 I3 6- EI 0 6 - EI Z 2 4- EI I C C \ J C 0 6- EI 2 C2 2 C 4 2- £ 7 Z C, \ 0 12-EI ~ T 3 6- EI C 0 6- EI 2 C 4 2 C 4 4- EI I C, \ J Figure 29 Morforton and Wu's {klx} Matrix 43 Where _ -v, + v2 + v[ • v2 f 3-EI L , v = 1 + 3 • v, C -3 A 4 - v , • v 2 L 4 4 - v , • v 1 " 2 4 - v , • V 2 3 • v 2 r = 4 - v , • v 2 4 - v , • v 2 1 I K,- I C * = A ^ 2 = 1 1 + 4 - v , - v , V * w 44 3.1.4.2 The Stiffness Matrix of Members with Rigid Ends Another useful v e r i f i c a t i o n of {K1X) i s to compare i t with the st i f f n e s s matrix of a member with r i g i d ends (Figure 30) . This can be done simply by setting: v , = 1 + 3-EI V Kx • I 3-EI I j - h i \ 1 V 0 = 1 + K2 • I J The dimensionless constants now become: v \ + v2 + v \ ' v 2 \2-EI C , = 1 * - * - = l g 4 - v , • v2 Av - G - I' 2 4 - v , - v 2 l + Q C 3 = * -=1 5 2 = — 4 - v , - v 2 i + g v 2 - [ 2 + v , j - l - g / 2 C 4 = ; =1 J 3 4 - v , • v 2 1 + 9 3 • v , • v , ' „ 1 + g / 4 4 - v i ' v 2 i + g 3 • v2 c 6 = - 2 =1 • 4 - v , • v 2 One sees that (Kxl) correctly reduces to the s t i f f n e s s matrix of members with r i g i d ends. 45 AE 12-EI I3 S, • s' AE c • s 12-EI Z 3 S, • c • s 12-EI- Zt I3 6-EI 51! • s 2 s r s AE c • s 12- EI Z 3 S x - c • s AE 12-EI I3 S, • c' 12- EI- Z, Z3 • S, • c 6-EI + ~^2 Src 12-EI- lt I3 6-EI • St - s Z2 S{ - s 12-EI • Z, Z3 6-EI 51! • c Z2 C 2 - c 4-EI 12- £"/• Z, ? s< AE 2 -T'c 12-EI Z3 AE 12- EI • S, • c • s 12- £ / • Z; Z3 6-EI S, • s Z' 5, • s 12- £ 7 Zc 5", • c • s 12- EI 12-EI- l7 ? 6 • EI H Z2 S [ • C 5, • c 12 • £ 7 • Zi Z3 6 - £ 7 5 [ • s Z: 5, • s 12- EI- Z, I3 6-EI Z2 S. • c 5. • c 2- £7 6-EI • I, + — • 9 z 2 6 ' 6 - EI - l2 V 12-EI • Z, Z2 + ! £_ . c F i g u r e 30 The Z £ n } M a t r i x o f Members w i t h R i g i d Ends 46 AE 12-EI „ 2 AE • c - s 12-EI Z' 12 • EI • I, I3 6- EI S, • c • s Si • s a 5, - S y4F 12- F/ Z3 S,•c•s +-AE 12-EI I3 S, • c2 12-EI- Z, Z3  6 • El I2 S, • c S, • c 12-F/- L2 Z3 6-EI • S, • s Z2 S, • s 12-F/- Z; Z3 6- £ • / S, • c Z' 4- EI S.-c I 12-EI- Z; Z2 12- F7 Z3 I 9 ' ^ 1 Figure 30(cont'd) The fK^) Matrix of Members with Rigid Ends where S, = 12-EI Au-G- Z2 1 1 + g / 4 i + g l - g / 2 i+g 1 +g/4 i + g 47 3.1.4.3 The Conventional Fix-Fix Member Stiffness Matrix One f i n a l check i s to compare the derived {K^} with the conventional (K1X] (Figure 31). This can be done simply by setting: /, , i2 = o, v , = 1 + = 1 1 I K r l J f 3-EI V 1 The dimensionless constants now become: v \ + V2 + V 1 ' v2 12 - EI c , = — l - — - = i g 1 4 - v , - v 2 Av-Q-L v 2 4 - v , - v 2 1 + 9 3 - v , 1 + g / 4 c > = 1 s 7 = —-z 4 - v , • v , i+g V ; 4 4 - v , - v 2 1 + 9 3 • v , • v 9 „ 1 + g / 4 4 - v , • v 2 i + g 3 • c 6 = - - = 1 4 - v , • v 2 One sees that (K u} correctly reduces to the conventional member st i f f n e s s matrix. 48 AE 12-EI „ 2 + 51! • s 2 13 1 AE I c • s 12- EI I3 5 j • c • s 6-EI - f — S , - s 71E I c • s 12-EI I3 Sx- c • s AE Z 12- EI I3 Sx- c 2 6-EI I2 5 , • c 6-EI I2 5, • s 6-EI I2 S , • c 4- EI —•s> AE ~ C 12-EI I3 S,- sJ AE I c • s 12- EI I3 S, • c • s 6 • £ 7 Z2 5 , • s A" i I AZf c • s 12- EI I3 S j • c • s ^ £ 2 \2-El I3 S , • c 2 6 7 / ,2 - S l ' C 6 - £ / „ ,2 ' ^ l - S Z' 6 7 / Z2 5 , • c 2- EI I Figure 31 The Conventional (K1X) Matrix 49 AE 12-EI 2  + 7-3 S t - s a AE c - s 12-EI —r3 S^-c-s 6- EI —T2 - V S AE c • s 6- EI I2 S [ • s 12-EI Ti Sx-c-s AE 6-EI I2 5, • c 12-EI 2 —T* <VC 6- El S! • c 4- EI I Figure 31(cont'd) The Conventional {Kn} Matrix where 5\ = S4 1 2 - EI Av-C - I2 1 1 + g/4 i + g 1 - g / 2 i + g i + g / 4 i + g 50 3.2 Assembling the Refined Fix-Pin Member Stiffness Matrix 3.2.1 The Local Bending Stiffness Matrix of F l e x i b l y Connected Fix-Pin Members The derivation of (kbl0) i s carried out i n the same manner as i n the previous section. The f i r s t and fourth column of (kblQ) are zero vectors because a x i a l displacements do not affect (kbl0). The second column of {kbl0} i s obtained by setting d 2 equal to 1 (Figure 32a). This i s a 1° indeterminate system. I f the fixed-end moment, f 3 i s chosen to be the redundant, the conjugate beam w i l l be loaded as shown i n Figure 32b. Because the shears at the ends of the conjugate beam are equal to the net angle changes at the ends of the actual beam, one could write from [3-6]: Figure 32a Deriving the Second Column of {khl0) EI Figure 32b Conjugate Beam Load : Fix-Pin Member, d 2=l [3-56] 51 Also, the moment m2 of the conjugate beam equals to the deflection d 2 of the real beam. Therefore, m, = l [3-57] Substituting [3-2] and [3-4] into [3-56], ' V fz \ A„-Q Ki J [3-58] Note : The -'ve sign i s due to the conjugate beam's d e f i n i t i o n of counter clockwise rotation as positive rotation (See Figure 13). But, d3 = pJl = 0, Therefore [3-58] becomes, R ~ V i l l 1 Av-Q Kx [3-59] From equilibrium of conjugate beam: f3 - I m l 3-EI I [3-60] where m, = l f r o m [3-57] Combining [3-59] and [3-60] and c o l l e c t i n g terms, f,-\ - - — L _ ] — ^ 1 y 3 1 ^ , 3-EI J Au-G I Multiplying both sides by: 3-EI I 52 and d e f i n i n g d i m e n s i o n l e s s c o n s t a n t s : 12-EI Av-Q-r [3-61] 3 - EI x v. = 1 + 1 1 A , - / 7 [3-62] R e s u l t i n g i n : /3_ V•g-I f 3 - EI [3-63] From e q u i l i b r i u m o f r e a l beam: 1/ = ^ I [3-64] S u b s t i t u t i n g [3-64] i n t o [3-63] and c o l l e c t i n g terms _ 3-EI ( 1 A / 3 I2 V ' \ 1 + v , • g / 4 J [3-65] S u b s t i t u t i n g [3-65] i n t o [ 3 - 6 4 ] , 3-EI f • 1 Z 3 V 1 • g / 4 y [3-66] I n t r o d u c i n g a d i m e n s i o n l e s s c o n s t a n t to [3-65] and [ 3 - 6 6 ] , 5 s l-l + v , - g / 4 [3-67] 53 f 3 , and V now become: / 3 = 3 - EI I2 v , • S 1 ^ 5 V 3-EI I3 1 ^ s [3-68] [3-69] In summary the second column of (khl0) i s •2 0 3 - EI - i i — y . - s , 3-EI -T2 v , ' S s \ I' 0 V 3-EI — r * — v r s s Z' 0 where 1 + v ! • g / 4 v , = 1 + 3 - £ 7 A", • Z 12 • EI A v - Q - I-The f i f t h column of (k b l 0) i s obtained by setting d 5 equal to 1 (Figure 33) and i t can be derived i n a sim i l a r fashion as before. Figure 33 Deriving the F i f t h Column of {kj •blO i 54 But noting that Figure 33 i s a mirror r e f l e c t i o n of Figure 32, the column vector can be written d i r e c t l y as: The t h i r d column' of (kbl0] i s obtained by setting d 3 equal to 1 (Figure 34a). Once again this i s a 1° indeterminate system. I f the fixed-end moment, f 6 i s chosen to be the redundant, the conjugate beam w i l l be loaded as shown i n Figure 34b. f 3 h Figure 34a Deriving the Third Column of {khl0} R i i , R 2 i EI Figure 34b Conjugate Beam Load : Fix-Pin Member, d 3=l 55 Because the shears at the ends of the conjugate beam are equal to the net angle changes at the ends of the actual beam, one could write: R , - a 1 = i O > 1 + y 1 + 0 , [3-70] Substituting [3-2] and [3-4] into [3-70], ' V f 6 \ Av-G KiJ [3-71] Note : The -'ve sign i s due to the conjugate beam's d e f i n i t i o n of counter clockwise rotation as positive rotation (See Figure 13). But, d3 = ipjl = \ , Therefore [2-71] becomes, V / a R, = - 1 + + — Av-G R) [3-72] From equilibrium of conjugate beam: R f 3 ' 1 1 3-EI [3-73] Combining [3-72] and [3-73] and c o l l e c t i n g terms, 1 I \ V ^3 ' ' K , + 3 - EI ) A„ • G + 1 Multiplying both sides by: 3 - EI I 56 and d e f i n i n g d i m e n s i o n l e s s c o n s t a n t s as i n [ 3 - 6 1 ] , [3-62] 1 2 - EI A v - C • I - I i n 3 ' E I v . = 1 + 1 1 Ki • I R e s u l t i n g i n : / 3 V-g-l 3-EI vx 4 I [3-74] From e q u i l i b r i u m o f r e a l beam: V = !-± ' [ 3 -75 ) S u b s t i t u t i n g [3-75] i n t o [3-74] and c o l l e c t i n g t e r m s , 3-EI ( 1 "\ / s I ' V i ' { l + v , - g / 4 J [3-76] S u b s t i t u t i n g [3-74] i n t o [ 3 - 7 5 ] , 3 - EI V = ^ 2 v , l + V i - g / 4 7 [3-77] I n t r o d u c i n g a d i m e n s i o n l e s s c o n s t a n t as i n [3-65] and [3-66] 5*5 = 1+ vi • g/4 57 f3, and V now become: 3-EI / 3 = 7 — - v r S 5 „ 3-EI V = —-i Vi'Ss [3-78] [3-79] In summary the t h i r d column of {khl0} i s 0 3-EI ~T2 v v 5 5 I 3-EI I 0 V 3-EI —T5 * V 5 s I-0 where ^ 5 = 1 1 + v i • g / 4 3 - £ 7 - I The s i x t h column of {khl0} i s a zero vector because of the pin connection. 58 In summary: ! * * i o } „ -r 0 o c 3-EI 3-EI 3- EI V 3-EI I 2 -v,-.S 5 Z 0 0 3-EI -^3— - v . - S j 0 z; 3 - £ • / Z 2 1 ^ 5 { ^ . o } , = ( ^ n } ^ . { * « n o } u f° 0 V o 0 3 • EI 0 o j where 1 v , = 1 + 1 + v! • g / 4 3 - 2 ? / A", • Z 1 2 7 / >4„- G - Z 2 59 3.2.2 Introducing the Effect of Connection Sizes The effect of connection sizes i s introduced by means of transformation as outlined i n [3-12]: {7T10} = { T } T - { A : I 0 } - { T } Where (k 10 ' (T) = the l o c a l member s t i f f n e s s matrix including the effect of f l e x i b l e connections and shear deflections, = the transformation matrix defined by [3-9], = the refined s t i f f n e s s matrix i n l o c a l coordinates. The l o c a l member s t i f f n e s s matrix (kbl0) comprises of (ka) and (khl0). {ka} i s the standard pin-pin member matrix and (khl0) has j u s t been presented i n the previous section. The f u l l (k10) matrix i s shown i n Figure 35. { * i o } , f V o 0 V AE I 0 0 { * . l ) i y - ( * l i > i o V o 0 3-EI — 7 — v . - s 5 3- EI —T2 v > ' 5 s 0 3-EI — 7 2 — v r $ s v 3- EI I J 0 3 - EI —73 0 3 - EI I7 v r S s 0 0 3 - EI v,-ss 0 0 oJ Figure 35 The (klQ) matrix 60 The l o c a l ref ined, member st i f f n e s s matrix j i s i l l u s t r a t e d i n Figure 36. ( * . o } „ -{ * . o } „ = AE Z 0 0 AE Z 0 0 0 3-EI —T3 v , ' 5 5 3- El - Z, ? 3-EI i Z 2 v r s s o 3 - EI Z 3 v r S . 3- El - Z, Z 3 3- El v r s s I-v r s 5 o 3-El-l Z 1 3 - £ 7 i Z 2 3 7 / Z 6-EI- It Z 2 3 - £ 7 Z 3 0 3- EI - l: Z 3 3-EI - I. v , 7 5 Z 2 3 • £ 7 • Z, • Z 2 _ { ^ i o } w - ( ^ i o } ; Figure 36 The Local Refined Member Stiffness Matrix of Fix-Pin Members 61 AE I 0 0 0 3-EI o 3-EI - l. 3 3-EI - l2 I3 3 - £ 7 2 Z* Figure 36(cont'd) The Local Refined Member Stiffness Matrix of Fix-Pin Members where f 3- EI N v , = 1 + ' V AT, - Z 12 • £"/ S 5 = Av-Q- Z 2 1 1+ v i • g / 4 1 1 ( 1 2 = l e n g t h o f c o n n e c t i o n 2 = l e n g t h o f member L = t o t a l s p a n = lx + 1 + 2 2 62 3.2.3 Transforming to Global Coordinates F i n a l l y ; the refined member s t i f f n e s s matrix (K10) i n global coordinates i s obtained by means of rotatio n a l transformation as outlined by [3-13]: [Ki0} = { T r V - { X ] 0 } . { T r } {K10} i s i l l u s t r a t e d i n Figure 37. AE 2 .3-EI ^ + v , • Ss- s AE AE • c • s 3- EI 7s AE 2 3-EI 3-EI 2 — v 1 - S 5 - c - s + - 5 — - v . - S g - c 3 - £ / - Z , 3 - £ 7 - Z , v , ' 5 s ' S ^ V i - 5 s - c Z3 3- £ 7 Z2 v , • s y s z-3- EI I" 3- EI- 7, ? 3-EI v, • s y s I' 3 • EI - 7, Z3 3-EI v , • s y s v , • S 5 • c 3- £7 Z 2 v , • S s - c v, - S s 6- £•/• Z, Z2 3-£"/ , 2 — 7 1 — 2 - v , - s s Figure 37 The (K10) Matrix 63 AE I ' C 3 • EI I3 • v , - S s - s a AE 3- EI /= vx- S5-c- s 3- EI - Z2 Z3 • v , • 5 5 - s { * > o } „ -/IF 3- £ 7 Z3 3 - £ 7 z3 v , - S s - c 2 3- EI - Zs Z3 • y , - 5 s - c 3-EI • lj I3 3 • £ 7 ». . Z2 • v , - 5 s - s v r S 5 - s 3- EI- Z, Z3 3- EI v, • S y c Z2 v , • S y c 3- EI - I, I2 3 - EI • Lj l2 I3 Figure 37(cont'd) The (K10) Matrix 64 AE 3 • EI AE c • s 3 - EI - I. v , • Ss-s 3- EI Z3 V , • Sg • C • S ( * > o } „ = AE AE 3- EI- Z2 Z1 v, • 5 S - c 3 - £ 7 3 - £ 7 „ 2 — v , - 5 s - c - s + ~J v , • 5 g • c 3 - F 7 - Zs Z3 V , • Sg - S 3- £ 7 - Z2 Z1 V , • Sg- C 3 7 / , 2 75 '2 ' V . - S g Figure 37(cont'd) The fJC10} Matrix where 1 + • 3-EI K, • I - I 12- EI A„-G- i' 1 + v ! • g / 4 Note : llt 12 =* length of connection ( r i g i d end pieces), 1 = length of member, L = t o t a l span = 1^^ + 1 + 1 2, Ax = change i n x form j o i n t 1 to j o i n t 2, z)y = change i n y from j o i n t 1 to j o i n t 2, c = cos(z!x/Z.), s = sin ( z l y / L ) . 65 3.2.4 Verifying the Fix-Pin Member Stiffness Matrix As a check of the above derivation i t i s useful to compare the derived (K10) with the conventional {K10} (Figure 38). This can be done simply by setting: Z, , Z 2 = 0 , EI \ _ 1 1 + 3 Z = 1 . * i J The dimensionless constants now become: g S5 1 2 - 7 7 / Av-Q- Z 2 1 1+ v, • g / 4 One sees that (K10) correctly reduces to the conventional f i x - p i n member s t i f f n e s s matrix. AE 2 c Z 3 - EI H Z 3 AE I c • s 3-EI I3. Sr- c- s 3-EI Z 2 S5-s AE I c • s 3 • EI I3 Ss-c-s + AE 2 s Z 3-EI z-3 • EI Z 2 S s • c 3 • EI Z 2 Ss- s 3-EI Z 2 S.R • c 3 • £ 7 Z 5, Figure 38 The Conventional {K10} Matrix 66 AE - T ' c 3-EI I3 S~ • s' AE I 3-EI I3 c • s S5- c-s 0 K 10) AE I 3-EI I3 c • s Ss-c- s AE 2 s I 3-EI I3 s 5 - c o 3- EI I2 S5-s 3-EI I2 S5-c 0 \ K i o } l 7 - { ^ I O } ^ -AE AE I c • s 3; EI I3 S5-s-3-EI I3 5 S • c • s AE AE c • s I 3 • EI I3 S5- c- s 3-EI I3 S5-c-0 0 0 0 0 where Figure 38(cont'd) The Conventional (K10) Matrix 9 ^ 5 = 12 • EI Av-G - I2 1 1 +g/4 67 length of connection ( r i g i d end pieces), length of member, t o t a l span = 11 + 1 + 12, change i n x form j o i n t 1 to j o i n t 2, change i n y from j o i n t 1 to j o i n t 2, cos(Ax/L), sin {Ay / L) . 68 3.3 Assembling the Refined Pin-Fix Member Stiffness Matrix The (Koi) matrix i s derived i n the exact same manner as presented i n the previous section therefore only the result w i l l be quoted below (Figure 39): AE 3-EI I3 v 2 - S 6 - s ' AE • c - s AE c • s 3-EI- Z, Z3 v2- S&- s 3- EI I3 v2 • S6• c• s AE 2 3- EI • Zt Z3 3-EI o 3 - E I —- v 2 - S 6 - c - s + — v2-Sb-c' 3-EI- Z, F v 2 • S6•s 3-El • Zt Z3 y2-S6-c 3- EI V2- S6 AE AE 3- EI v 2 - S 6 - s 3- EI I3 v 2 - S 6 - s ' 3- EI Z3 '2 ' ^ 6 * c • s 3- EI • l2 i~3 >K 01} ij-AE 3 • EI Z3 c • s AE 2 3- EI I3 3- EI • v I2 3-EI•I 2 ' ^ 6 ' C 3 - £ 7 - Z , n 3-EI-l, 3-EI-l, 75 V 2 - - V c — j 2 v 2 - S 6 3 • EI • Z, Z v 2 ' s & Figure 39 The {K01} Matrix 69 { * o i L - { * o . } y , AE AE • • c • s 3-EI • v , • S , - s 2 " 2 ^ 6 3- EI 3-EI I3 • • v 2 • 5 6 • c • s 3- El - I, 3 V 2 ' $ 6 ' S ( * o . ) „ -AE • c - s 3 • EI I3 3-EI v2•S6•c•s I2 3-El • l2 I3 AE 2 ~ 8 3- EI 3 "2 ^6 3-EI ,2 - * V - V c 3- £ /• Z2 Z3 • v 2 - 5 y c 3- £ / . Z2 v 2 - 5 y c 3- EI - l7 Z3 v 2 - S 6 - c 3- £ 7 Z 6- El- I-Z2 3 • El I3 v2- S6-s L2 - v 2 - S 6 where Figure 39(cont'd) The (K01) Matrix v , = 1 + 3-EI K7- I g = ^ 6 = 12-EI Av-C - Z 2 1 1 + v 7 - g / 4 Note lx, 12 = length of connection ( r i g i d end pieces), 70 length of member, t o t a l span = lx + 1 + I 2 , change i n x form j o i n t 1 to j o i n t 2, change i n y from j o i n t 1 to j o i n t 2, cos(Ax/L), sin (Ay /L). 3.4 Assembling the Refined Pin-Pin Member Stiffness Matrix Since connection sizes and connection f l e x i b i l i t y have no effect on the a x i a l forces of a member, (K00) matrix i s the same as the conventional pin-pin member s t i f f n e s s matrix (Figure 40). ( c • c c • s 0 - c • c - c • s 0 c • s s • s 0 - c • s - s • s 0 0 0 0 0 0 0 - c • c -c • s 0 c • c c • s 0 - c • s -s • s 0 c • s s • s 0 0 0 0 0 0 0 Figure 40 The {K00} Matrix Note 2 Ax Ay c s length of member, change i n x form j o i n t 1 to j o i n t 2, change i n y from j o i n t 1 to j o i n t 2, cos(Ax/L), sin (Ay /L) . 72 4 Modified Fixed End Forces Connection sizes and connection f l e x i b i l i t y also affect the fix e d end forces. The method of modifying fixed end forces for various loading conditions are presented i n the following sections. 4.1 Uniformly Distrinuted Load on Fl e x i b l y Connected Fi x - F i x Members The f i r s t step i s to derive the fixed end forces for a f l e x i b l y connected f i x - f i x member under uniformly distributed load. The effect of connection sizes can be introduced by means of transfer of forces at a l a t e r stage. The f i x e d end forces of a f l e x i b l y connected f i x - f i x member i s obtained by combining the following three load cases (Figure 41a,b,c). C a s e a + / ? f a - C a s e b + / ? C - C a s e c Case a Figure 41a A Simply Supported Beam under a Uniformly Distributed Load / 2 = / s = w • I 2 w 2 4 - EI 73 Case b Figure 41b A Fix-Fix Beam with d 3=l From the t h i r d column of (khll): d 3=l, see [3-53],[3-54], and [3-55] / 2 = / 3 = / 5 = V 4- EI I C 6- El I-2 - EI I C. 1 + v 2 • g / 4 l+g-c, 1 i-g/2 l+g-c, where C 2 = c,= 4 - v , • v 2 ^ r ( 2 + v 2 ) 4 - v , • v 2 4 - v , • v 2 3 - v , • v 2 4 - v , • V 2 74 for d3=pb 6 - EI I2 4- EI I C. C 1 6-EI fs--P*-—a C2 / « = /?« V 2 - EI I C. 1+g-C, 1 + v2•g/4 l+g-c, i - g / 2 ^ Case c Figure 41c A Fix-Fix Beam with d 6=-l From the s i x t h column of (khll): d 6=l / 2 = / 3 = / 5 = / 6 = 6 - £" / 1 2- EI I •6 • EI V A- EI Z C, i + g - d ; 1+ v2•g/4 l+g-c, 1 l+g-c, i - g / 2 N 75 where C , = c. c 6 = 4 - v i • v 2 4 - v , • v 2 3 • v i • v 2 4 - v , • v 2 3 • v 2 4 - v , • v 2 for dR=-6 - / 3 ( 6- EI ( 1 , - _ / ? 2 ' g / ^ f l + v 2 - g / 4 6 - F / ( I z 2 v i +g• C i 4 - F / f i - g / 2 / V i +g• c, Therefore, Figure 41d A Fix-Fix Beam under a Uniformly Distributed Load 76 / 2 = / 3 = / 5 = / 6 = w I f ( v, - V 1 + V 1 V 1 2 <  4 - v , • v 2 I + g - C Z 2 3 - v 1 - ( 2 - v 2 . ( l - g r ) ) 12 ( 4 - v , - v 2 ) - ( l + g - C , ) i f • / 1 + 4 - v , • v 2 1 + g • C , i f / 2 3 - v 2 - ( 2 - v , - ( l - g r ) ) 12 ( 4 - v , - v 2 ) . ( l + g . C , The effect of connection sizes i s introduced by means of transfer of forces as outlined i n [3-8] (Figure 18b). In summary, F • F, Where w • I ~~2 w I2 12 w • I 2 w I2 1 + v 4 - V j • v2 l + g- C, 3 - v 1 - ( 2 - v 2 - ( l - g ) ) ( 4 - v , - v 2 ) . ( l + g - C , ) v 2 - v , 1 + F 2 - ' i 1 +. 4 - v , • v 2 1 + g - C , 3 - v 2 - ( 2 - v , - ( l - g ) ) 12 ( 4 - y , - y 2 ) - ( l + g - C , ) - f s - Z 5 1 2 1 + V,=- 1 + 3 - EI 3- EI lx, lz = length of connection 1 = length of member L = t o t a l span = 11 + 1 + lz 9 12 • EI Av- G- I'-ll 4.2 Uniformly Distributed Load on F l e x i b l y Connected Fin-Pin Members The fi x e d end forces for a f i x - p i n member under uniformly dis t r i b u t e d load i s derived by the method of superposition as presented i n the previous section (Figure 42). Case a + Bb• Case b Case a Figure 42a A Simply Supported Beam under a Uniformly Distributed Load / 2 = / s w • I 2 w 2 4 - EI Case b Figure 42b A Fix-Pin Beam with d 3=l 78 From the t h i r d column of (khl0): d 3=l, see [3-78] and [3-79] / 3 = / 5 = For d3=£?( / 2 = / V / 3 = /? b Therefore, 3 • EI I2 3 • EI I 3 • EI 1 2 Vl 1 + v j • g / 4 1 1+ v j • g / 4 1 1+ v , • g / 4 3 • EI I2 3 • EI I 3 • EI 1 2 V l 1+ v ! • g / 4 1 1+ v , • g / 4 1 1+ vx • g / 4 Figure 42c A Fix-Pin Beam under a Uniformly Distributed Load 79 w I f 4 + v , • (1 + g ) ^ / 3 = J i T V l ' ( l + v 1 1 . g / 4 ) / s 8 \ l + v , - g / 4 J The effect of connection sizes i s introduced by means of transfer of forces as outlined i n [3-8] (Figure 18b). In summary, 2 8 '{ l + v , . g / 4 J w I ( 1 ^ F3 = - — v.-l \ + F 2 - L 8 1 { l + v r g / 4 J 2 1 5 8 A l + v , - g f / 4 J Where - ( ' ^ ) " ' g _ Av-C- I2 l l t 12 = length of connection ( r i g i d end pieces) 1 = length of member L = t o t a l span = 11 + 1 + lz 80 4.3 Uniformly Distributed Load on F l e x i b l y Connected Pin-Fix Members The fix e d end forces for a p i n - f i x member under uniformly d i s t r i b u t e d load are analogous to that of a f i x - p i n member (Figure 43). Therefore only the result w i l l be quoted here. Figure 43b A Pin-Fix Beam with d 6—1 W U U U U U I I Figure 43c A Pin-Fix Beam under a Uniformly Distributed Load 81 w I ( 4 - v 2 - ( l + g ) 8 1 + v 2 - g / 4 F3 = F0-l 2 1 1 w I ( 4 + v 2 •( 1 -g) 8 w • I 1 + v2-g/4 8 1 V 1 + v , • g / 4 1 - ^ 5 ^ 2 Where 1 + 3 - EI -1 K2 - I J = 12 • EI  9 Av-Q-l2 llt 12 = length of connection ( r i g i d end pieces) 2 = length of member L = t o t a l span = 11 + 1 + 2 2 82 4 . 4 P o i n t L o a d on F l e x i b l y C o n n e c t e d F i x - F i x Members Once a g a i n the f i x e d end f o r c e s f o r a f l e x i b l y c o n n e c t e d f i x - f i x member u n d e r a p o i n t l o a d i s d e r i v e d f i r s t . The e f f e c t o f c o n n e c t i o n s i z e s w i l l be i n t r o d u c e d b y means o f t r a n s f e r o f f o r c e s a t a l a t e r s t a g e . The f i x e d end f o r c e s o f a f l e x i b l y c o n n e c t e d f i x - f i x member i s o b t a i n e d b y c o m b i n i n g t h e f o l l o w i n g t h r e e l o a d c a s e s ( F i g u r e 4 4 ) . C a s e a + / ? b - C a s e b + / 3 C - C a s e c Case a F i g u r e 44a A S i m p l y S u p p o r t e d Beam u n d e r a P o i n t L o a d P • b / 2 I P - a Z P - a - b •(b+Z) 6 • EI • I P - a - b • (a+Z) Po 6-EI-I 83 Case b F i g u r e 44b A F i x - F i x Beam w i t h d 3 = l From t h e t h i r d co lumn o f {khll}: d 3 = l , see [ 3 - 5 3 ] , [ 3 - 5 4 ] , and [3-55] IT. I3 Is Is 6-EI I2 4- EI I 6-EI I2 2 • EI } ( C C 1 C f 1+ v 2 - g / 4  1 l ' + S f - C , 1 -g/2 1+ g - C ] where C, V , + V 2 + V x • V 2 4 - v , • v2 4 - v , • v 2 3 • v j ' 4 - v , • v 2 3 • v , • v 2 4 - V , • V 2 84 For d3=Bb f2 = Bb f5 6- EI I2 4- EI I C. 6- EI B b - — ^ — - C 2 f6 = Bt 2- EI I 1 l+g-c , J 1 + v 2 •g/4 . i+g-c, l i+g-c , i - g / 2 ' i+g-c , , Case c Figure 44c A Fix-Fix Beam with d 6=-l From the s i x t h column of ( k b l l ) : dfi=l / 2 = / 3 = / 5 = fe = 6 - EI I2 2 - EI I C. 1 6 - EI 4 - El C 4 i+g-c, 1+ v2•g/4 i+g-c, 1 i+g-c, l - g / 2 l+g-C; 85 where 4 - y , • y 2 ^ • ( 2 + 1 ^ ) 4 - v , • v 2 3 • v , • y 2 4 - v , • v 2 3 - v 2 4 - V , • V 2 F o r d 6 = - / j c F i g u r e 44d A F i x - F i x Beam u n d e r a P o i n t L o a d c 6 = 86 P-b I a 4 - z ( v 1 - v 2 ) - v , - a - ( 2 + v 2 ) + v 2 - b - ( 2 + v,) I I' ; 4 - v 1 - v 2 ) - ( l + g - C , ; / 3 = / 6 = P-a-b2 f v , - Z-(4 + v 2 - ( l . 5 - g - l ) ) - v 1 - a - ( 2 + v 2 ) A f 1 Z 2 T3 • a 1 + b - ( 4 - v , - v 2 J ; v 1 + <7'C, b 4 - z ( v 2 - v 1 ) + v , - a - ( 2 + v 2 ) - v 2 - b - ( 2 + y 1 Z V" 1 2 ( 4 - v 1 - v 2 ) - ( l + g . C I ) /> -a 2 -b f v 2 - Z-(4 + v , - ( l . 5 - f l r - l ) ) - v 2 - b - ( 2 + v,; Z" a • ( 4 - v , • v ; The effect of connection sizes i s introduced by means of transfer of forces as outlined i n [3-8] (Figure 18b). In summary, F , = P • b f l + a 4 - Z ( v , - v 2 ) - v 1 - a-(2H - v 2 ) + v 2 - b'(2 + v 1 ) A *V Z2 ( 4 - v , - v 2 ) - ( l + g - C , ) J P-a-b2 ( y, • I-(4+y2-(1.5-g-l))-y, • a-(2 + y ; Z2 />• a b - ( 4 - y , - y 2 ) + F 2 - Z, 4- Z ( y 2 - y , ) + y , - a-(2 + y 2 ) - y 2 - b - ( 2 + y 1 ) 2 Z V" <2 ( 4 - v , - v 2 ) - ( l + S f C I ) P-a2-b f v 2 - Z-(4 + v , - ( 1 . 5 - f f - l ) ) - v 2 - b - ( 2 + v 1 ) Z^  a-(4 - v , - v 2 ) - ^ s - Z2 Where v, = 1 + V , = 1 + 3 - £ 7 K, • I 3- EI • I 1 1, 1 2 = length of connection 2 = length of member L = t o t a l span = lx + 1 + 12 g 12 • EI 87 4.5 Point Load on Flexibly Connected Fix-Pin Members The fixed end forces for a fix-pin member under a point load is derived by the method of superposition as presented in the previous section (Figure 4 5 ) . Case a + B b•Case b Case a Figure 45a A Simply Supported Beam under a Point Load For simply supported beams under point load / 2 = P • b P • a P-a-b 6-EI -I P-a-b 6-EI - I ( b + Z ) ( a + Z ) 88 Case b f 3 f 2 Figure 45b A Fix-Pin Beam with d 3=l From the t h i r d column of (khl0): d 3=l, see [3-78] and [3-79] f 3 ' E I ( 1 "\ / 2 = l2 'V,'{ l + V l . g / 4 J 3-EI ( 1 A / 3 _ I ' V , ' { l + v , - g / 4 J , _ 3 - £ 7 f 1 > / s = Z 2 * V ) A l + v , . g / 4 J For d3=y5b 3 - £ 7 f 1 ^ / , - ! > > — ? — * , • [ 1 + v , . f f / 4 J / , - / » . — j — v , - ^ 1 + V i . g / 4 J 3 - £ 7 f 1 A T h e r e f o r e , F i g u r e 45c A F i x - P i n Beam u n d e r a P o i n t L o a d / 2 = f3 = P-b P - a - b - ( a + 2 - b ) ( I 2 - Z 3 P • a • b • ( a + 2 - b ) / 2 ^ ? v » i P • a P - a - b • ( a + 2 - b ) 1 \, 1 + v, • g / 4 1 A 1+ v,•g/4 ( I 2- I' V 1 + v , - g / 4 The e f f e c t o f c o n n e c t i o n s i z e s i s i n t r o d u c e d b y means o f t r a n s f e r o f f o r c e s as o u t l i n e d i n [3-8] ( F i g u r e 1 8 b ) . I n summary, F-F* = Where P-b P • a-b-(a + 2 - b ) / I 2 - I3 P-a-b - ' (a + 2 - b ) o v 1 2 - Z 2 1 P • a P • a- b - ( a + 2 - b ) • v 1 ^ 1 + v , • g / 4 1 1 + V i • g / 4 + F 2 - Z, 2 - Z1 1+ v , • g / 4 F * = - F s - Z 2 v, = 1 + 3 - £ 7 K, - I - I 90 " = 1 2 - EI  9 ~ A v - C - I 2 12 = length of connection length of member t o t a l span = 11 + 1 + 12 4.6 Point Load on F l e x i b l y Connected Pin-Fix Members The f i x e d end forces for a p i n - f i x member under a point load are analogous to that of a f i x - p i n member (Figure 46) . Therefore only the resul t w i l l be quoted here. Figure 46a A Simply Supported Beam under a Point Load Figure 46b A Pin-Fix Beam with d 6—1 92 Figure 46c A Pin-Fix Beam under a Point Load P-b P-a-b-(b + 2- a ) ^2 = — , ~ - V 2 I F3 = F 2 - l x 2- t V l + v 2 - g / 4 F t ^ 6 = P-a P - a-b-(6 + 2 - a ) / Z 2- r 1 P-a-b - (b + 2-a) 2 ^ V l + v 2 - g / 4 1 V 1 + v2-g/4 5 L2 iere v 2 = | 1 + 3 - EI K2- I 12 • EI A v - C - l 2 llt lz = length of connection ( r i g i d end pieces) 1 = length of member L = t o t a l span = lx + 1 + 12 93 5 Member Forces Calculation With the addition of connections at either end of a member, nodal displacements now become the end displacements of the connections and not the member i t s e l f . Therefore member forces must be calculated d i f f e r e n t l y . By adopting the sign convention i l l u s t r a t e d i n Figure 47 and r e c a l l i n g the member forces relationships established i n Section 3 (See Figure 18a), one can relate forces at connection ends and member ends as follows: BM2 A X I A L bm2 a x i a l a x i a l Figure 47 Member Forces Sign Convention F,=-AXIAL f! = -axial f2 = shear 1 /3 = -bml /4 = axial F2= SHI F3 = -BMl F 4 = AXIAL F5=-SH2 F 6 = BM2 f 5 = -shear2 f6= bm2 94 Recalling the transfer of forces relationships of [3-8] and reorganizing: axial = AXIAL . [5-1] shearl=SHl [5-2] shear2 = SH2 [5-3] bml = BM 1 +shear 1 • I, [5-4] bm2 = BM2-shear2-l2 [5-5] Noting that connections are perfectly r i g i d , the a x i a l force can be written d i r e c t l y the same as before: axial= —j- * [{d 4 - d ,) • c + ( d 5 - d2) • s) where llt 12 = length of connection ( r i g i d end pieces), I = length of member, L «= t o t a l span = I x + 2 + 22, /jx = change i n x form j o i n t 1 to j o i n t 2, Ay = change i n y from j o i n t 1 to j o i n t 2, c = cos(z)x/Z.), s = sin (Ay/L). 95 Shears and moments at connection ends are obtained by reorganizing the s i x deflections into the three cases shown below: BM2 SHEAR 1 V5 ~ - " SHEAR2 d 7 case 1 BM2 > \ S ^ '•• Sf SHEAR2 -BM 1 ^ SHEAR 1 case 2 -BM2 SHEAR2 BMI -SHEAR 1 case 3 Figure 48 Calculating Shears and Moments 96 The above three cases have already been solved for unit deflections i n Section 3. Therefore the shears and moments at connection ends can be obtained simply by means of superposition: F F 2 d 7 =1 F 6 V5 - ' " F 5 case a d 3 = l —. F 3 F 2 case b - d 6 = l V7 1 3 " F2 / — ^ \ / case c Figure 49 Calculating Shears and Moments at Connection Ends by Superposition Once the forces at connection ends are calculated, the member forces can be obtained by [5-1] to [5-5]. 97 5.1 Calculating Shears and Moments of Fix- F i x Members The three cases used for calculating shears and moments of f i x - f i x members have already been solved i n Section 3.1 and are summarized below i n Figure 50. case c Figure 50 Shears and Moments of Fix- F i x Members by Superposition 98 By s u p e r p o s i t i o n : SH 1 = 1 2 £ 7 Z 3 C x - S x - d 7 a n d , 6EI C 2 • S , + ^ I, • C,•S j 6 £ 7 2 c 4 - s 1 + Z 3 1 2 £ 7 Z 3 d -d, [5-6] / 627/ \2EI \ BMl =-[ 0 - C , - 5 | + - Z t • C , • 5 , • d 7 Z 2 4EI 2EI I c 3 - s 2 + I3 \2EI I \2EI 2 Z , - C 2 - S 1 + — ,6 £ 7 ^ „ 6 £ 7 1 2 £ 7 Z 2 Z, • Z 2 - C , • 5 , ) - d 6 r e a r r a n g i n g and n o t i n g [5-6] BM\=-\ 6EI I2 4EI I 2EI I C2-Slj-dy 6EI C 3 - 5 2 + c 5 - s 3 + I2 6EI 11 • C" 2 * *5 , Z 2 ' C 2 ' S 1 ri-de - SH 1 [5-7] 99 Member force shearl i s calculated from [5-2]: ( \2EI \ shear 1 = I — — C , • S , I • d7 6EI \2EI \ C0 • S , + I, • C , • S , • d Z 2 2 1 z 3 1 1 1 3 6 £ 7 _ _ \2EI t2 C 4 * $ i + — T i Z 2 • C , • S , }• d 6 [5-8] From equilibrium, shear2 = shear \ [5-9] From [5-4] and noting that shear1=SH1, bml reduces to: i 6 E I bm 1 = - | —— C2 • S | j • d 7 4 £ 7 6 £ 7 C 3 * S 2 + * ' i ' C 2 ' - 5 ' ] J • d 3 z 2 £ 7 6 £ 7 j C 5 • S 3 + — Z 2 , C 2 , S 1 j , d 6 [5-10] From equilibrium,-bm2 = 6 m 1 + shear 1 • Z, [5-11] 100 5.2 Calculating Shears and Moments OF Fix-Pin Members The three cases used for calculating shears and moments of f i x - p i n members have already been solved i n Section 3.2 and are summarized below i n Figure 51. case c Figure 51 Shears and Moments of Fix-Pin Members by Superposition 101 By s u p e r p o s i t i o n : SHI = EI \ I3 1 5J d7 3EI I2 3EI I3 3EI l , - v r S 5 - d 3 [5-12] a n d , Z 3EI 3EI \ BMl ¥ _ . V l - 5 5 + —73—- Z , - v 1 - 5 5 J - d 7 3 £ 7 . 6 £ 7 . . -3 £ 7 Z V , - 5 5 + —-jy- • l X - V y - S z + — — • Z, 3 £ 7 3 £ 7 *\ —72—- l2- V \ • 5 5 + — • *1 • l2- V \ • S 5 J - d6 r e a r r a n g i n g and n o t i n g [5-6] 3 £ 7 BM\=-\ - j r - - v r S 5 ) - d 7 3 £ / „ 3 £ 7 , „ , , —j v i ' S s + t2 ' Z i • v i ' S s J' ^ 3 3 EI • l 2 - v r S 5 )-d6-SHl- Z, I' [5-13] 102 Member force shearl i s calculated from [5-2]: , 3 £ 7 A shear 1 = | — — v r • S5 J • d 7 3EI 3EI \ v, • SR + — • I, - v,- • d 3EI I [5-14] From equilibrium, shear2 = shear 1 [5-15] From [5-4] and noting that shear1=SH1, bml reduces to: 3_£7 i bml =- — T 5 v i " 5 s ' D 7 3 £ 7 3 £ 7 v , „ • v , - 5 5 - d 3 3 £ " / / 2 - v , - S 5 | - d , [5-16] By d e f i n i t i o n , bm2 = 0 [5-17] 103 5.3 Calculating Shears and Moments OF Pin-Fix Members The three cases used for calculating shears and moments of p i n - f i x members have already been solved i n Section 3.3 and are summarized below i n Figure 52. F 2 case b d 6 =-1 " F v " F 3 -F 2 \ F 5 case c Figure 52 Shears and Moments of'Pin-Fix Members by Superposition 104 By superposition: SHl=(3^-v2-S6)-d7 3EI , „ \ , 3EI I-3EI \ y 2 - S 6 + , 3 • V V2- $ 6 ' d 6 Z' [5-18] and, 3EI 3EI \ BM2-\ —pr--v2-S6 + —^--l2'V2-S6J-d7 ( 3EI 3EI „ , Z, • v 0 - 5 , + — • Z , - Z , - v , - 5 . - d 2^ <• 1 " 2 ^ 6 ' j 3 M " - 2 " 2 ^ 6 ) ^ 3 3EI 6EI , 3EI v2-S6+ -l2-v2-S6+ • I. I * 0 i rearranging and noting [5-6] I 3 £ 7 BM2,= \ ——• v 2 - S 6 | - d 7 Z , • v 0 • S , • d 3 £ 7 7i 1 1 "2 J 6 J " 3 3 / 7 / 3 £ 7 \ —7— • v 2 - ^ 6 + — T T " • ' 2 - v 2 - 5 6 J - d 6 + 5 / / 2 [5-19] 105 Member.force shearl is calculated from [5-2] shear 1 3 £ 7 ^ — [ r ~ - y 2 ' S 6 \-d7 3EI \ ,3 'I " 2 ^.6 ' J d3 3EI „ 3EI [5-20] From equilibrium, shear2 = shear 1 [5-21] From [5-5] and noting that shear2=SH2, bm2 reduces to: 3EI b m 2 = \ [ 2 , y 2 , 5 6 I'd? Z, • v 0 • 5 , • d 3 £ 7 75 1 1 " 2 ^ 6 j " 3 3EI 3EI —— -V2-S6+ i 2 • l 2 - v 2 - S6 \-d6 [5-22] By definition, bml=0 [5-23] 106 6 Connection Stiffness While the derivation of the s t i f f n e s s matrix, fi x e d end forces and member forces i s complete, there s t i l l remains the question of what s t i f f n e s s values to use for the various types of connections i n analysis. In 1969 Somner[16] devised a standardized procedure for expressing the moment-rotation characteristics for a l l connections of a given type i n a non-dimensional form. A few years l a t e r , Frye and Morris[4] u t i l i z e d the same procedure to develop a set of dimensionless equations which express the moment-rotation relationships of the seven most commonly used connections (Figure 53, Table 1). The general moment-rotation relationship i s of the form: 0 = c , • (kM) + c2- (kM)3 + c3- (kM)5 and, k=pa2-pb3-P4-pi where Cj^ , c 2, c 3, p x, p 2, p 3 and p A are constants which depend on the connection type. These moment-rotation relationships are very compact and well suited for programming. I t w i l l be shown i n the next section how these relationships can be incorporated into a plane frame analysis program. 107 V P3 = t P 2 = d ^ P l = Lj P4= g (a) SINGLE WEB A N G L E P3 = t P 2= d ^ P l = L l p4 = g (b) DOUBLE WEB A N G L E P,= t P q= w P l = L i p2= d V p4 = g (c) HEADER P L A T E R , - t PK = (fastene r I p2= d ^ In. P l = Lj (d) TOP & S E A T A N G L E P l = Li I P 4= f (only with column stiff eners) I P 9= d A column stiffener I P 3= t (e), (f) END P L A T E (WITH & WITHOUT COLUMN STIFFENERS) ^| (fastener ol) 1 P3 = t P s = 1 P 4= f P 9= d A J P l = L, (g) T - S T U B Figure 53 Common Types of Connections and Their Standardization Parameters 108 Table 1 Standardized Connection Moment-Rotation Functions Connection Type Standardized Function, 0 Standardization Constant, Ci C 2 c 3 a b s d single web angle 4 28E- 03 1. 45E-09 1 51E- 16 -2 .40 -1.81 +0 .15 0.00 double web angle 3 66E- 04 1. 15E-06 4 57E- 08 -2 .40 -1.81 +0 .15 0.00 header plate 5 10E- 05 6. 20E-10 2 40E- 13 -2 .30 -1.60 +1 .60 +0.50 top & seat angle 8 46E- 04 1. 01E-04 1 24E- 08 -1 .50 -0.50 -1 .10 -0.70 end plate (without column s t i f f e n e r ) 1 83E- 03 -1. 04E-04 6 38E- 06 -2 .40 -0.40 +1 .10 0.00 end plate (with column s t i f f e n e r ) 1 79E- 03 1. 76E-04 2 04E- 04 -2 .40 -0.60 0 .00 0.00 t-stub 2 10E- 04 6. 20E-06 7 60E- 09 -1 .50 -0.50 -1 .10 -0.70 109 7 Programming Details T h i s s e c t i o n d i s c u s s e s i n d e t a i l s some p r a c t i c a l a s p e c t s o f i m p l e m e n t i n g the a l g o r i t h m d e s c r i b e d above i n an e x i s t i n g p l a n e frame a n a l y s i s p r o g r a m . The g e n e r a l s t r a t e g y i s l a i d o u t ' i n F i g u r e 54. F o r l i n e a r c o n n e c t i o n b e h a v i o u r o n l y f o u r o f the s i x s e c t i o n s o f the p r o g r a m r e q u i r e s m o d i f i c a t i o n s . F o r n o n l i n e a r c o n n e c t i o n b e h a v i o u r , a d d i t i o n a l m o d i f i c a t i o n has to be made to the o u t p u t s e c t i o n . E x i s t i n g Program R E A D D A T A • geometry • material properti • loads C A L C U L A T E M E M B E R F O R C E S I W R I T E O U T P U T • deflections • member forces • support reaction Modif ica t ions A D D * connection data M O D I F Y local <k> M O D I F Y fixed end forces M O D I F Y • member forces U P D A T E connection data 1 1 Linear Connection Behavior Nonlinear Connection Behavior F i g u r e 54 I n c o r p o r a t i n g F l e x i b l e C o n n e c t i o n s i n P l a n e Frame A n a l y s i s no . 7.1 Modifying Input Format The input section of the program needs be modified to accommodate the additional connection data. Each connection i s i d e n t i f i e d by the j o i n t and member which i t i s associated with and the connection type. A maximum of si x parameters should be a l l o t t e d for describing a p a r t i c u l a r connection (Figure 55) with parameter Mi representing the moment at the pa r t i c u l a r connection, parameter p i describing the length of the connection and parameter p2, p3, p4, and p5 containing d e t a i l information about the connection (Figure 53). Connection Data n_num m_num typ Mi P i P2 P3 p4 P5 Figure 55 Input Format for Connection Data 7.2 Modifying Stiffness Matrix The global s t i f f n e s s matrix i s assembled d i r e c t l y from the l o c a l s t i f f n e s s matrices of indiv i d u a l members. However, before a l o c a l s t i f f n e s s matrix can be formulated, connection s t i f f n e s s must f i r s t be obtained. This can be accomplished simply by adding to the program a subroutine which reads i n connection properties d i r e c t l y as data or calculates connection s t i f f n e s s according to the connection parameters entered. A sample subroutine which incorporates the equations as outlined by Frye and Morris [4] for calculating connection s t i f f n e s s i s l i s t e d i n Appendix A. i n Table 2 Common Connection Types Type Description k value 1 perfectly hinged 0 2 single web angle calculated 3 double web angle calculated 4 header plate calculated 5 top & seat angle calculated 6 end plate (without column sti f f e n e r s ) calculated 7 end plate (with column sti f f e n e r s ) calculated 8 t-stub calculated 9 other user specified 10 perfectly r i g i d -1E+038 Type 9 connection i s to provide user with the option of using his own st i f f n e s s values and also to accommodate connections which are not covered i n the l i s t . The special value of -1E+38 i s adopted to represent the s t i f f n e s s value of a r i g i d connection. The l o c a l s t i f f n e s s matrix can be modified as discussed i n Section 3. Sample l i s t i n g of the modified l o c a l s t i f f n e s s assembly routine can be found i n Appendix B. The only point to note i s that when defining the dimensionless constants V j and v 2 , one should be careful to check for the value of k. I f k equals -1E+38 for a r i g i d connection, the program should automatically assign the value of 1 to the corresponding v1 and v 2. 112 7.3 Modifying Fixed End Forces The fi x e d end forces can be modified as discussed i n Section 4. Sample l i s t i n g of the modified fi x e d end forces can be found i n Appendix C. 7 .4 Calculating Member Forces The member forces can be modified as discussed i n Section 5. Sample l i s t i n g of the modified fi x e d end forces can be found i n Appendix D. 7.5 Modeling Nonlinear Connection Response The procedure outlined here i s the same one presented by Frye and Morris [4] and i t i s based on the premise that correct structural response and member forces can be obtained from a s i n g l e , l i n e a r analysis provided the correct connection s t i f f n e s s are used. The procedure i s therefore a simple i t e r a t i v e process which guesses at the connection s t i f f n e s s c h a r a c t e r i s t i c s i n a structure at every cycle. When the connection s t i f f n e s s characteristics converge to s u f f i c i e n t accuracy then these values are used to perform a linear analysis to calculate the correct deflections and member forces for a structure with nonlinear connections. Figure 56 depicts the t y p i c a l moment-rotation relationship of a nonlinear connection: 0 = / ( M ) where f(M) i s a nonlinear function of the moment acting on the connection. 113 T y p i c a l Momen t -Ro ta t i on Rela t ionship 2500 i 1 0 0.005 0.01 0.015 0.02 Rotat ion (rads) Figure 56 Typical Moment-Rotation Relationship of a Nonlinear Connection One begins the i t e r a t i v e procedure by assuming M=0, and perform a li n e a r analysis on the structure. This i s equivalent to setting the s t i f f n e s s of the connection equal to the i n i t i a l tangent of the moment-rotation relationship. *. 1  From the analysis one gets a new value of M: M = M , 114 W i t h t h i s one c a n c a l c u l a t e : ^ 1 = / ( M = M 1 ) , I f ka and kx a r e s u f f i c i e n t l y c l o s e t h e n the i t e r a t i v e p r o c e d u r e c a n be t e r m i n a t e d . O t h e r w i s e one c a n r e p e a t t h e a n a l y s i s u s i n g k1 t o o b t a i n a new v a l u e o f k2 and t h e n compare the v a l u e o f kx t o k2. T h i s p r o c e s s i s r e p e a t e d u n t i l t h e s t i f f n e s s v a l u e , k, a t each c o n n e c t i o n s t a b i l i z e s a t a c e r t a i n v a l u e w i t h s u f f i c i e n t a c c u r a c y . F i g u r e 57 d e p i c t s t h e c o n v e r g e n c e o f k f o r a t y p i c a l n o n l i n e a r c o n n e c t i o n . Iterative Procedure for Nonlinear Connections 2500 | 1 R o t a t i o n ( r a d s ) F i g u r e 57 The C o n v e r g e n c e o f k f o r N o n l i n e a r C o n n e c t i o n s The a l g o r i t h m f o r the i t e r a t i v e p r o c e d u r e d e s c r i b e d above i s e x t r e m e l y easy to i m p l e m e n t . One needs o n l y to u p d a t e the c o n n e c t i o n s t i f f n e s s o f the s t r u c t u r e a t the end o f e a c h c y c l e o f a n a l y s i s and to add a s t a t e m e n t f o r c o m p a r i n g t h e v a l u e s o f k . 115 8 Analysis of a Simple Plane Frame Structure with Flexible Connections Using the method outlined in previous sections, a standard plane frame analysis program was modified to include the effect of flexible connections. As means of verifying the val i d i t y of the resulting program, CPlane, the simple plane frame structure (Figure 58a) previously investigated by Moncarz and Gerstle [10] was analyzed for comparison purposes. The experimentally determined moment-rotation curves for the upper and lower connections are shown in Figure 58b along with the approximations used by Moncarz and Gerstle and the ones used by CPlane. The service load conditions are as follow: (1) Dead load, g = 1.86 kips/ft or 0.155 kips/in, (2) Live load, 1 = 1.20 kips/ft or 0.100 kips/in, (3) Lateral load intensity, w- 0.00, 0.01, 0.02, 0.03, 0.04 kips/sq f t Resulting in: W1 = 0.00, 2.88, 5.76, 8.46, 11.52 kips/sq f t W2 = 0.00, 1.44, 2.88, 4.32, 5.74 kips/sq f t Details of the results are discussed in Section 8.1. Wi j t t l i i i l i t ! Leeward ^ Column Upper Girder W 2 • g + p j i t t i i i i i i . J , - c Windward f Column d Lower / Girder 12'-0" 24'-0" Figure 58a A Simple Plane Frame 116 I a a B o 2 Moncarz & Gerstle (linearized) 2500 2000 -1S00 -1000 500 Moncarz & Gerstle (linearized) 0 0.005 0.01 0.015 0.02 Rotation (rads) CPlane Johnston & Hechtman (from experiments) Moncarz & Gerstle (trilinearized) CPlane Johnston & Hechtman (from experiments) Moncarz & Gerstle (trilinearized) Figure 58b Connection Properties In order to investigate the effect of f l e x i b l e connections on internal force d i s t r i b u t i o n of a structure, the same frame as i l l u s t r a t e d i n Figure 58a was analyzed using various member-connection models with di f f e r e n t assumptions on connection behavior. ' Details of the study are discussed i n Sections 8.2 to 8.5. The l i s t of connection models and assumptions used i n the study i s summarized i n Table 3. 117 Table 3 L i s t of Girder-Column Connection Design Assumptions C O N N E C T I O N T Y P E M O D E L C O N N E C T I O N L E N G T H S C O N N E C T I O N B E H A V I O R P e r f e c t l y P i n n e d (PIN) S o O K ® N / A P i n L i n e a r F l e x i b l e C o n n e c t i o n ( L F ) S « » S N / A L i n e a r E l a s t i c N o n l i n e a r F l e x i b l e C o n n e c t i o n ( N L F ) s« N / A N o n l i n e a r L i n e a r R e f i n e d C o n n e c t i o n ( L R ) S • •—S3 ^i - 2^ L i n e a r E l a s t i c N o n l i n e a r R e f i n e d C o n n e c t i o n ( N L R ) S e • — ^ l j , i2 N o n l i n e a r P e r f e c t l y R i g i d (RIGID) S S N / A R i g i d R i g i d E n d s ( R G E N D ) S S R i g i d 118 8.1 Verfication of CPlane Moncarz and Gerstle [10] investigated the effect of flexible connections on structural response under the assumption of linear and nonlinear connection behavior. However, their formulation neglects the effect of connection sizes and treats .connections as point connections. For verification purposes the same set of connection assumptions was used by CPlane and the member forces from the two studies are shown in Figure 59a and Figure 59b. The column design moments from the two studies are almost identical. It is of interest to note that the assumption of linear connection behavior produced nearly the same result as nonlinear connection behavior. The c r i t i c a l girder design moments from Moncarz and Gerstle and CPlane follow the same pattern but the values from CPlane are consistently higher by about 5% (Figure 59b) . What appears surprising is that the response for r i g i d connections from the two studies also differ by about the same amount. Surely i f identical parameters were used, the r i g i d connection response from both studies should be the same. Therefore by making the r i g i d connection response the benchmark for comparison, the results from Moncarz and Gerstle were normalized accordingly and Figure 60 shows the plot of normalized c r i t i c a l girder moments against lateral load intensity. It is clear from the plot that the results from the two studies are practically identical after normalization. The apparent discrepancy probably stems from the fact that the particular girder under investigation is not available from any Canadian steel mills and the section properties selected are slightly different from the ones used by Moncarz and Gerstle. As an independent check, the same structure was analyzed under the assumption of r i g i d connections by another structural analysis program, LPS, from the Institute fur Baustatik der Universitat Stuttgart, West Germany. The results from LPS for r i g i d connections are in exact, agreement with that of CPlane (Figure 59b). 119 Wi Upper ^ Girder W i n d w a r d ^ Column Leeward g ^ Column ff, + ,P Lower ^ Girder C o n n e c t i o n A s s u m p t i o n X R I G I D • L F ( C P l a n e ) N L F ( C P l a n e ) K « A L F ( M o n c a r z & G e r s t l e ) S « » S • N L F ( M o n c a r z & G e r s t l e ) 0 O.Ol 0 .02 0.03 0.04 L a t e r a l L o a d I n t e n s i t y , w ( k i p s / s q f t ) F i g u r e 59a L e e w a r d Column Top Moment v e r s u s L a t e r a l L o a d I n t e n s i t y 120 \ I CO 4-> c E o CP T 3 5 e 3 X Wi Upper Girder W2 W i n d w a r d ^ C o l u m n 160 150 140 130 120 1 10 Leeward g Column g + p Lower * Girder C o n n e c t i o n A s s u m p t i o n X RIGID ( C P l a n e ) • LF ( C P l a n e ) gj» ®S • N L F ( C P l a n e ) S«- ®S O RIGID ( M o n c a r z <& G e r s t l e ) A LF ( M o n c a r z & G e r s t l e ) • N L F ( M o n c a r z & G e r s t l e ) RIGID ( L P S ) 0 0.01 0.02 0.03 0.04 L a t e r a l L o a d I n t e n s i t y , w ( k i p s / s q f t ) Figure 59b C r i t i c a l Moment of Lower Girder versus Lateral Load Intensity 121 Wi Upper Girder W2 W i n d w a r d ^ Lower Co lumn 180 Leeward g At Column Girder Connec t ion A s s u m p t i o n X RIGID (CPlane ) • LF (CPlane ) N L F (CP lane ) ra.e » R O RIGID ( N o r m a l i z e d : M o n c a r z & G e r s t l e ) A LF ( N o r m a l i z e d : M o n c a r z & G e r s t l e ) A. N L F ( N o r m a l i z e d : M o n c a r z & G e r s t l e ) 0 0.01 0.02 0 .03 0.04 L a t e r a l L o a d I n t e n s i t y , w ( k i p s / s q f t ) Figure 60 Normalized Moment of Lower Girder versus Lateral Load Intensity From the close agreement between the normalized moments of Moncarz and Gerstle and that of CPlane and the excellent correlation between LPS and CPlane, i t appears reasonable to conclude that the method of incorporating connections i n plane frame analysis outlined i n this paper i s good and gives result comparable to that of Moncarz and Gerstle. 122 8.2 Girder and Connection Moments The structure as i l l u s t r a t e d i n Figure 58a was analyzed again using the various connection models and assumptions described i n Table 3 to study the effect of f l e x i b l e connections on maximum c r i t i c a l girder moments and girder end moments. Figure 61 and Figure 62- plot the c r i t i c a l girder moments and girder end moments of the lower girder against l a t e r a l load in t e n s i t y for the various connection assumptions. As expected, the c r i t i c a l girder moment varies nonlinearly with l a t e r a l load int e n s i t y for a l l connection assumptions with the exception of pin connections. The apparent nonlinearity of the c r i t i c a l girder moment i s caused by the variable location of the c r i t i c a l section for differenet l a t e r a l load intensity. From Figure 61 i t i s evident that the common assumption of pin and r i g i d connections leads to substantial overestimation and underestimation of c r i t i c a l girder moments respectively. The assumption of f l e x i b l e connections neglecting the effect of connection sizes leads to underestimation of girder s t i f f n e s s r e s u l t i n g i n somewhat lower c r i t i c a l girder moments. Since c r i t i c a l girder moments govern the design of girders, one expects the assumption of pin and r i g i d connections w i l l lead to conservative and unconservative design of girders respectively and the assumption of f l e x i b l e connections neglecting connection sizes w i l l result i n s l i g h t l y unconservative design. By making the response from nonlinear refined connections the basis for comparison, Figure 63 shows the difference i n girder design moments for the different connection assumptions with l a t e r a l load in t e n s i t y , w, equals 0.02 kips/sq f t . 123 Wi Upper ^ Girder W i n d w a r d ^ Column 240 220 I CO •2- 200 c CD 6 o CD T3 3 180 160 •2 140 -o 120>^ 100 Leeward q M- Co lumn r Lower ~ Girder C o n n e c t i o n A s s u m p t i o n O P I N oagj • L F N L F A L R -© &m • N L R -© &m X R I G I D R G E N D 0 0.01 0.02 0 .03 0.04 L a t e r a l Load I n t e n s t i y , w ( k i p s / s q f t ) Figure 61 C r i t i c a l Moment of Lower Girder versus Lateral Load Intensity for Various Connection Assumptions 124 L e e w a r d 4Q L_ i I i I i I i I 0 • 0.01 0.02 0.03 0.04 Lateral Load Intensity, w (kips/sq ft) Figure 62 Girder End Moment of Lower Girder versus Lateral Load Intensity for Various Connection Assumptions 125 C o n n e c t i o n A s s u m p t i o n RIGID R G E N D LF NLF LR _Q 0_ -30 - 2 0 - 1 0 0 10 D i f f e r e n c e f r o m N o n l i n e a r R e f i n e d • C o n n e c t i o n ( N L R ) R e s p o n s e l o w e r g i r d e r Figure 63 Difference i n Lower Girder Design Moments for Various Connection Assumptions, w=0.02 126 From Figure 62 i t i s clear that the assumption of r i g i d connections leads to substantial overestimation of girder end moments while the assumption of f l e x i b l e connections neglecting connection sizes leads to s l i g h t overestima-t i o n of girder end moments. Since girder end moments are used i n the design of connections, one expects the assumption of r i g i d connections w i l l lead to conservative design while the assumption of f l e x i b l e connections neglecting connection sizes w i l l lead to s l i g h t l y conservative design. Once again by making the response from nonlinear refined connections the basis for comparison, Figure 64 shows the difference i n connectionn design moments for the di f f e r e n t connection assumptions with l a t e r a l load i n t e n s i t y , w, equals 0.02 kips/sq f t . C o n n e c t i o n A s s u m p t i o n - 1 0 0 10 2 0 3 0 4 0 5 0 [%] D i f f e r e n c e f r o m N o n l i n e a r R e f i n e d C o n n e c t i o n ( N L R ) R e s p o n s e Figure 64 Difference i n Lower Connection Design Moments for Various Connection Assumptions, w=0.02 127 8.3 Column Moments Figure 65 plots the moments of the leeward lower column top against l a t e r a l load in t e n s i t y . The results show that the assumption of r i g i d connections leads to substantial overestimation of column moments. However, i n practice t h i s i s not as s i g n i f i c a n t as i t appears because columns are usually designed according to the magnitude of a x i a l forces and the magnitude of column end moments are not quite as important. Nevertheless i t i s interesting to observe how the different connection assumptions affect these moments. Leeward 0 0.01 0.02 0.03 0.04 L a t e r a l L o a d I n t e n s i t y , w ( k i p s / s q f t ) Figure 65 Leeward Lower Column Top Moment versus Lateral Load Intensity for Various Connection Assumptions 128 8.4 Maximum Top Story Sway The maximum top story sway varies l i n e a r l y with l a t e r a l load intensity as i l l u s t r a t e d i n Figure 66. As expected, the assumption of r i g i d connections results i n s t i f f e r structures and leads to underestimation of bare frame sways while the assumption of f l e x i b l e connections neglecting connection sizes results i n more f l e x i b l e structures and leads to overestimation of bare frame sways. By making the nonlinear refined connections the basis for comparison, Figure 67 shows the difference i n maximum top story sway for the different connection assumptions with the l a t e r a l load int e n s i t y , w, equals 0.04 kips/sq f t . 129 L e e w a r d 0 0.01 0.02 0.03 0.04 Lateral Load Intensity, w (kips/sq ft) Figure 66 Maximum Top Story Sway versus Lateral Load Intensity for Various Connection Assumptions 130 C o n n e c t i o n A s s u m p t i o n -50 -40 -30 -20 -10 0 10 [%] Difference f rom Nonl inear Ref ined Connection (NLR) Response Figure 67 Difference i n Maximum Top Story Sways for Various Connection Assumptions, w=0.04 8.5 Linear versus Nonlinear Connection Behavior From Figure 63, Figure 64, Figure 65 and Figure 67, i t appears that the assumption of lin e a r connection response yields satisfactory r e s u l t provided the connection moment-rotation relationships are correc t l y linearized. However l i n e a r i z a t i o n of these moment-rotation relatioships i s a subjective process and good results depend on judgement and experience. Poor l i n e a r i z a t i o n of these relationships could only lead to incorrect structural response and therefore caution should be exercised. 131 9 Conclusion A p r a c t i c a l method of incorporating r e a l i s t i c f l e x i b l e connections i n plane frame analysis has been presented. This method requires modification to the input format, l o c a l s t i f f n e s s matrix, fixed end forces and member forces of a standard plane frame analysis program. Connection s t i f f n e s s i s programmed d i r e c t l y into the analysis by u t i l i z i n g the connection moment-rotation equations developed by Frye and Morris [4] but may also be entered as data separately. The algorithm presented i s very general and i t can be used to model l i n e a r as well as nonlinear connection response. A standard plane frame analysis program was modified accordingly and the re s u l t i n g program, CPlane, was used to analyze the simple plane frame structure previously investigated by Moncarz and Gerstle [10]. The results from CPlane were found to be comparable to that of Moncarz and Gerstle. The same structure was analyzed again using different connection models and di f f e r e n t assumptions on connection behavior for various l a t e r a l load i n t e n s i t i e s . I t was found that the inclusion of f l e x i b l e connections i n analysis s i g n i f i c a n t l y a l t e r the internal force d i s t r i b u t i o n of a structure. By making the response from nonlinear refined connections the basis for comparison, here are the findings: Girder Design 1. Pin connections lead to overdesign of girders. 2. F l e x i b l e connections neglecting connection sizes lead to s l i g h t l y unconservative design. 3. Rigid or Rigid-end connections lead to unconservative design of girders. 132 Connection Design 4. Linear f l e x i b l e connections neglecting connection sizes lead to s l i g h t l y conservative design. 5. Rigid or Rigid-end connections lead to conservative design of connections. Column Design 6. While the column design moments vary substantially for different connection assumptions, this i s not s i g n i f i c a n t because i n design i t i s usually the magnitude of a x i a l forces and not the magnitude of column end moments which governs. Maximum Top Story Sway 7. Fle x i b l e connections neglecting connection sizes lead to overestimation of building sway. 8. Rigid or Rigid-end connections lead to underestimation of buil d i n g sway. Linear versus Nonlinear Connection Behavior 9. Proper l i n e a r i z a t i o n of connection moment-rotation relationships yields satisfactory structural response. However the l i n e a r i z a t i o n process i s subjective and the quality of the result depends strongly on judgement and experience. 133 Acknowledgment The research work was conducted at the University of B r i t i s h Columbia and CANRON Inc. (Western Bridge Division) under the sponsorship of the B r i t i s h Columbia Science Research Council and CANRON Inc. (Western Bridge Division). The LPS str u c t u r a l analysis program was supplied by the I n s t i t u t fur Baustatik der Universitat Stuttgart, West Germany. I would l i k e to take this opportunity to thank Dr.-Ing. S.F. Stiemer, P. Eng., Mr. M. Frank, P. Eng., Dr. D.L. Anderson, P. Eng. and Professor E. Ramm for t h e i r invaluable advice and assistance on this project. I would also l i k e to thank Dr. N. Stander and Ms. K. 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Douty, R.T., "Strength Characteristics of High Strength Bolted Connections with P a r t i c u l a r Application to the P l a s t i c Design of Steel Structures," d i s s e r t a t i o n presented to Cornell University, i n 1964, .in p a r t i a l f u l f i l l m e n t of the requirements for the degree of Doctor of Philosophy. 20. Wilson, W.M., and Moore, H.F., "Tests to Determine the R i g i d i t y of Riveted j o i n t s i n Steel Structure," B u l l e t i n 104, Engineering Experiment Station, University of I l l i n o i s , 1917. 21. Fielding, D.J. and Chen, W.F., "Steel Frame Analysis and Connection Shear Deformation," Journal of Structural Division, ASCE, Vol. 99, No. ST1, January 1973. 22. Krawinkler, H., Bertero,-V.V., and Popov, E.P., "Shear Behavior of Steel Frame Jo i n t s , " Journal of the Structural Division, ASCE, Vol. 101, No. ST11, November 1975. 23. Clough, R.W., Rectangular Plane Frame Joint Element, CE 220B lecture notes, 1980, University of C a l i f o r n i a Berkeley, D i v i s i o n of Structural Engineering and Structural Mechanics. 137 Appendices 138 Appendix A: L i s t i n g of the Connection Stiffness Subroutine #include <math.h> double ConnectKi(typ,mi,p2,p3,p4,p5) i n t typ; f l o a t mi,p2,p3,p4,p5; { /* Local Variables */ f l o a t c4,c5,c6,yl,y2,y3,y4; double k,phi,kmi,ki; k=ki=kmi=0.0; mi=fabs(mi); switch(typ) { case 1: /* perfectly hinged connection */ ki=0.0; break; case 2: /* single web connection */ y l — 2 . 4 ; •y2—1.81; y3=0.15; k=pow(p2,yl)*pow(p3,y2)*pow(p4,y3); c4=4.28e-3; c5=1.45e-9; c6=1.51e-16; kmi=k*mi; break; case 3: /* double web connection */ y l — 2 . 4 ; y2—1.81; y3=0.15; k=pow(p2,yl)*pow(p3 fy2)*pow(p4,y3); c4=3.66e-4; c5=1.15e-6; c6=4.57e-8; kmi=k*mi; break; case 4: /* header plate connection */ yl=-2.3; y2—1.60; y3=1.6; y4=0.5; k=pow(p2,yl)*pow(p3,y2)*pow(p4,y3)*pow(p5,y4); c4=5.10e-5; c5=6.20e-10; c6=2.40e-13; kmi=k*mi; break; case 5: /* top & seat angle connection */ y l — 1 . 5 ; y2=-0.5; y3—1.1; y4=-0.7; k=pow(p2,yl)*pow(p3,y2)*pow(p4,y3)*pow(p5,y4); c4=8.46e-4; c5=1.01e-4; c6=1.24e-8; kmi=k*mi; break; case 6: /* end plate connection (without column s t i f f e n e r s ) */ y l — 2 . 4 ; y2=-0.4; 139 V 3-1.1; k=pow(p2,yl)*pow(p3,y2)*pow(p4,y3); c4=1.83e-3; c5=-1.04e-4; c6=6.38e-6; kmi=k*mi; break; case 7: /* end plate connection (with column st i f f e n e r s ) */ y l — 2 . 4 ; y2=-0.6; k=pow(p2,yl)*pow(p3,y2); c4=1.79e-3; c5=1.76e-4; c6=2.04e-4; kmi=k*mi; • break; case 8: /* t-stub connection */ y 1—1.5; y2=-0.5; y3—1.1; y4=-0.7; k=pow(p2,yl)*pow(p3,y2)*pow(p4,y3)*pow(p5,y4); c4=2.1e-4; c5=6.2e-6; c6=7.60e-9; kmi=k*mi; break; case 9: /* other */ ki= P2; break; case 10: /* perfectly r i g i d */ ki—1.00e+38; break; } i f (typ>l && typ <9) { kmi=k*mi; phi=c4*kmi+c5*pow(kmi,3.)+c6*pow(kmi,5.); • i f (phi<=l.e-30) k i - l . / ( k * c 4 ) ; else ki=mi/phi; } r e t u r n ( k i ) ; 140 Appendix B: L i s t i n g of the Modified Local Stiffness Matrix #include <stdio.h> #include <math.h> #include "Globals.h" #include "inout.h" extern f l o a t E,G; extern struct BEAMSTIFFNESS theStorage; struct BEAMSTIFFNESS *BeamStiffnessFor(TPBeam) struct BEAM *TPBeam; { (TPBeam->nodel->X); (TPBeam->nodel->Y); /* Local Variables */ struct BEAMSTIFFNESS *Local = &theStorage; i n t D0F[6],kl2; f l o a t 11,12; double K[22] ; double c,cc,s,ss,cs,xm,ym,1,dm,xm2,ym2,dm2,g.nul,nu2; double CI,C2,C3,C4,C5,C6,SI,S2,S3,S4,S5,S6,f1,f2,f3,f4,kl,k2; i n t i ; c=cc=l.; s=ss=cs=0.; DOF[0] = TPBeam->nodel->DOF[0] D0F[1] = TPBeam->nodel->DOF[l] D0F[2] = TPBeam->nodel->D0F[2] D0F[3] = TPBeam->node2->D0F[0] D0F[4] = TPBeam->node2->D0F[l] D0F[5] = TPBeam->node2->DOF[2] kl2 = TPBeam->kl2; xm = (TPBeam->node2->X) ym = (TPBeam->node2->Y) xm2 = xm*xm; ym2 = ym*ym; dm2 = xm2+ym2; dm = sqrt(dm2); c=xm/dm; cc=c*c; s=ym/dm; ss=s*s; /* Connection Sizes */ i f (TPBeam->connectl—NULL) 11=0.; else ll=(TPBeam->connectl->pl); i f (TPBeam->connect2—NULL) 12=0.; else 12=(TPBeam->connect2->pl); l-dm-11-12; /* Flexible Connection */ kl-k2—1.00e+38; i f (TPBeam->connectl==NULL) nul=l; else { kl=(TPBeam->connectl->Ki); i f (kl>0. && kl<=l.e-30) nul=0.; i f (kl—1.00e+38) nul=l.; else nul=l./(I.+((3*E*(TPBeam->inertia))/(kl*l))); 141 } i f (TPBeam->connect2==NULL) nu2=l.; else { k2=(TPBeam->cormect2->Ki); i f (k2>0. && k2<=l.e-30) nu2=0.; i f (k2=-1.00e+38) nu2=l.; nu2=l./(I.+((3*E*(TPBeam->inertia))/(k2*l))); } /* Shear Deflection */ g - 0.0; i f ((TPBeam->ashear) != 0.0 && G != 0.0) g=12.*E*(TPBeam->inertia)/((TPBeam->ashear)*G*l*l); /* Storage A l l o c a t i o n for K 1 1 2 3 4 5 6 2 7 8 9 10 11 3 12 13 14 15 4 16 17 18 5 19 20 6 . 21 1 2 3 4 5 6 Member Degrees of freedom DOF 1 4 I I 0 ---> %= ==—% ---> 3 ' ->2 '->5 (+)ve: r i g h t , up, ccw */ /*** Defining constants for p i n j p i n member ***/ fl=(TPBeam->area)*E/l; /* f i l l i n pin_pin section of member s t i f f n e s s matrix K [1] = f l * c c ; K 2] = f l * c s ; K 3] 0.0; K 4] = - f l * c c ; K 5] = - f l * c s ; K L6] = 0.0; K .7] f l * s s ; K 8] = 0.0; K r9] = - f l * c s ; K L10] = - f l * s s ; K 11 = 0.0; K 12 = 0.0; K 13 = 0.0; K 141 0.0; K 15, 0.0; K 16: f l * c c ; K 17: f l * c s ; K '18' = 0.0; K 19 f l * s s ; K 20 0.0; K 21 = 0.0; 142 i f (kl2 != pin_pin) /*** Defining constants for f i x _ f i x member i f (kl2 — f i x _ f i x ) { Cl = (nul+nu2+nul*nu2)/(4-nul*nu2); C2 = nul*(2+nu2)/(4-nul*nu2); C3 = 3*nul/(4-nul*nu2); C4 = nu2*(2+nul)/(4-nul*nu2); C5 = 3*nul*nu2/(4-nul*nu2); C6 = 3*nu2/(4-nul*nu2); 51 = l/(l+g*Cl); 52 = (l+g*nu2/4)/(l+g*Cl); 53 = (l-g/2)/(l+g*Cl); 54 = (l+g*nul/4)/(l+g*Cl); f l = 12.*E*(TPBeam->inertia)/(l*l*l)*Cl*Sl; f2 = 12.*E*(TPBeam->inertia)/(l*l)*C2*Sl; f3 = 12.*E*(TPBeam->inertia)/(l*l)*C4*Sl; f4 => 4.*E*(TPBeam->inertia)/l; } else { ' i f (kl2 = fix_pin) { /* */ S5 f l f2 f3 f4 } else i f l/(l+g*nul/4); 3.*E*(TPBeam->inertia)/(l*l*l)*nul*S5; 6*E*(TPBeam->inertia)/(1*1)*nul*S5; 0; 3*E*(TPBeam->inertia)/l; (kl2 == p i n _ f i x ) S6 = l/(l+g*nu2/4); f l = 3.*E*(TPBeam->inertia)/(l*l*l)*nu2*S6; f2 = 0; f3 = 6*E*(TPBeam->inertia)/(l*l)*nu2*S6; f4 = 3*E*(TPBeam->inertia)/l; } f i l l i n terms which are common to p i n _ f i x , f i x _ p i n and f i x _ f i x members K [ l K[2 K[3 K" K K K K K 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 += f l * s s ; -= f l * c s ; -= f l * l l * s ; -= f l * s s ; += f l * c s ; -= fl*12*s; += f l * c c ; += f l * l l * c ; += f l * c s ; -= f l * c c ; += fl*12*c; += f1*11*11; += f l * l l * s ; -= f l * l l * c ; += f1*11*12; += f l * s s ; -= f l * c s ; += fl*12*s; += f l * c c ; -= fl*12*c; f1*12*12; /* f i l l i n remaining f i x _ f i x terms */ 143 i f (k!2 — f i x f i x ) K [3] -= K 6] .= K 8] += K 11] += K 12] += K 13] += K 14] -= K 15] += K 18] += K 20] -= K 21] += 0.5*f2*s 0.5*f3*s 0.5*f2*s; 0.5*f2*c; 0. 5*f 4*C5" 0.5*f3*s; 0.5*f3*c; + f 2 * l l ; + 0 . 5 * f 3 * l l + 0.5*f2*12; /* f i l l i n remaining f i x _ p i n terms */ else i f (kl2 = fi x _ p i n ) { K[3] -= 0.5*f2*s; K[8] += 0.5*f2*c; K[12] += f4*nul*S5; K[13] += 0.5*f2*s; K[14] -= 0.5*f2*c; K[15] += 0.5*f2*12; } /* f i l l i n remaining p i n _ f i x terms */ i f (kl2 == p i n _ f i x ) { K[6] K [ l l K[15 K[18 K[20 K[21 -= 0.5*f3*s; += 0.5*f3*c; += 0 . 5 * f 3 * l l ; += 0.5*f3*s; -= 0.5*f3*c; += f4*nu2*S6 + f3*12; for ( i - 1 ; i<-21; { Local->K[i] = K [ i ] ; } for (i=0; i<=5; ++i) Local->D0F[i] = D0F[i]; return(Local); 144 Appendix C: L i s t i n g of the Modified Fixed End Forces void PerpBeamLoad (shearl, shear2, bml, bm2, dm, dm2, TPBLoad, TPBeam) f l o a t *shearl; f l o a t *shear2; f l o a t *bml; f l o a t *bm2; f l o a t dm; double dm2; struct BLOAD *TPBLoad; struct BEAM *TPBeam; { f l o a t wl2, wl8, b, 11, 12, 1, k l , k2; f l o a t w, P, a; • double nul, nu2, CO, CI, g, SI, A l , A2, A3, A4, Rxl, Rx2, Ryl, Ry2; /* Connection Sizes */ l=sqrt(dm2); i f (TPBeam->connectl—NULL) 11=0.; else ll=(TPBeam->connectl->pl); i f (TPBeam->connect2==NULL) 12=0.; else 12=(TPBeam->connect2->pl); l=dm-ll-12; /* Flexible Connection */ i f (TPBeam->connectl==NULL) nul=l.; else { kl=(TPBeam->connectl->Ki); i f (kl==0.0) nul=0.; i f (kl—1.00e+38) nul=l.; else nul=l/(l+((3*E*(TPBeam->inertia))/(kl*l))); } i f (TPBeam->connect2—NULL) nu2=l.; else { k2=(TPBeam->connect2->Ki); i f (k2==0.0) nu2=0.; sXs6 i f (k2 1.00e+38) nu2=l.; nu2=l/(l+((3.*E*(TPBeam->inertia))/(k2*l))); } /* Shear Deflection */ g=0.0; i f ((TPBeam->ashear) !- 0.0 && G != 0.0) g=12.*E*(TPBeam->inertia)/((TPBeam->ashear)*G*l*l); 145 w=(TPBLoad->w)*dm/l; P=TPBLoad->P; a=TPBLoad->a-ll; b - 1-a; switch (TPBeam->kl2) { case f i x _ f i x : CO = l/(4-nul*nu2); Cl = (nul+nu2+nul*nu2)/(4-nul*nu2); SI = l/(l+g*Cl); i f (P != 0.0) { /* calculate the Beam Forces for f i n a l output */ A l = l+a/(l*l)*(4*l*(nul-nu2)-nul*a*(2+nu2)+nu2*b*(2+nul))*C0*S1; A2 = l+b/(l*l)*(4*l*(nu2-nul)+nul*a*(2+nu2)-nu2*b*(2+nul))*C0*Sl; A3 = (nul*l*(4+nu2*(1.5*g-l))-nul*a*(2+nu2))/b*C0*Sl; A4 = (nu2*l*(4+nul*(1.5*g-l))-nu2*b*(2+nul))/a*C0*Sl; Ryl = P*b/1*A1; Ry2 = P*a/1*A2; *shearl += Ryl; *shear2 -= Ry2; *bml -= P*a*b*b/(1*1)*A3+Ryl*ll; *bm2 -= P*a*a*b/(1*1)*A4+Ry2*12; } i f (w != 0.0) { /* calculate the Beam Forces for f i n a l output */ wl2 = w*l/2.0; Cl = (nul+nu2+nul*nu2)/(4-nul*nu2); A l = l+(nul-nu2)/(4-nul*nu2)/(l+g*Cl); A2 = l+(nu2-nul)/(4-nul*nu2)/(l+g*Cl); A3 = 3*nul*(2-nu2*(l-g))*C0*Sl; A4 = 3*nu2*(2-nul*(l-g))*C0*Sl; Ryl = wl2*Al; Ry2 = wl2*A2; *shearl += Ryl; *shear2 —. Ry2; *bml -= wl2*l/6.0*A3+Ryl*ll; *bm2 -= wl2*l/6.0*A4+Ry2*12; } break; case pin f i x : i f (P != 0.0) { /* calculate the Beam Forces for f i n a l output */ Al = nu2/(l+g*nu2/4); A2 = a*b*(b+2.0*a)/(2.0*1*1*1); Ryl = P*a/1 + P*A2*A1; Ry2 = P*b/1 - P*A2*A1; *shearl += Ryl; *shear2 -= Ry2; *bml -= R y l * l l ; *bm2 -= P*A2*1+Ry2*12; } i f (w != 0.0) { /* calculate the Beam Forces for f i n a l output */ wl8 = w*l/8.0; A l = (4-nu2*(l-g))/(l+g*nu2/4); A2 = (4+nu2*(l+g))/(l+g*nu2/4); A3 = nu2/(l+g*nu2/4)I 146 Ryl = wl8*Al; Ry2 - wl8*A2; *shearl += Ryl; *shear2 -= Ry2; *bml — R y l * l l ; *bm2 -= wl8*l*A3+Ry2*12; } break; case f i x pin: i f (P != 0.0) { /* calculate the Beam Forces for f i n a l output */ A l = mil/(l+g*nul/4); A2 = a*b*(a+2.0*b)/(2.0*1*1*1); Ryl = P*b/1 + P*A2*A1; Ry2 = P*a/1 - P*A2*A1; *shearl += Ryl; *shear2 -= Ry2; *bml -= P*A2*1+Ryl*ll; *bm2 -= Ry2*12; ) i f (w != 0.0) { /* calculate the Beam Forces for f i n a l output */ wl8 = w*l/8.0; A l = (4+nul*(l+g))/(l+g*nul/4); A2 = (4-nul*(l-g))/(l+g*nul/4); A3 = nul/(l+g*nul/4); Ryl = wl8*Al; Ry2 = wl8*A2; *shearl += Ryl; *shear2 -= Ry2; ' *bml -= wl8*l*A3+Ryl*ll; *bm2 -= Ry2*12; } break; case pin pin: i f TP != 0.0) { /* calculate the Beam Forces for f i n a l output */ *shearl = P*b/1; *shear2 =-P*a/l; ) i f (w != 0.0) { /* calculate the Beam Forces for f i n a l output */ wl2 = w*l/2.0; *shearl += wl2; *shear2 -= wl2; ) break; } ) 147 void XYdirBeamLoad (pxl, px2, pml, pm2, xym, dm2, TPBLoad, TPBeam) f l o a t * p x l ; f l o a t *px2; f l o a t *pml; f l o a t *pm2; double xym, dm2; struct BLOAD *TPBLoad; struct BEAM *TPBeam; { f l o a t wl2, wxy2, wl8, wxy8, b, 11, 12, 1, k l , k2; f l o a t w, P, a; double dm, nul, nu2, CO, CI, g, SI, A l , A2, A3, A4, Rxyl, Rxy2; dm=sqrt(dm2); /* Connection Sizes */ , i f (TPBeam->connectl==NULL) 11=0.; else ll=(TPBeam->connectl->pl)*xym/dm; i f (TPBeam->connect2==NULL) 12=0. ; else 12=(TPBeam->connect2->pl)*xym/dm; l-dm-11-12; /* Fle x i b l e Connection */ i f (TPBeam->connectl=NULL) nul=l; else { kl=(TPBeam->connectl->Ki); i f (kl==0.0) nul=0. ; B l SG i f ( k l — 1 . 0 0 e + 3 8 ) nul=l.; else nul=l/(l+((3*E*(TPBeam->inertia))/(kl*l))); } i f (TPBeam->connect2==NULL) nu2=l. ; else { k2=(TPBeam->connect2->Ki); i f (k2—0.0). nu2=0.; 6 X S 6 i f (k2 1.00e+38) nu2=l.; € l s G nu2=l/(l+((3*E*(TPBeam->inertia))/(k2*l))); } /* Shear Deflection */ g=0.0; i f ((TPBeam->ashear) != 0.0 && G != 0.0) g=12.*E*(TPBeam->inertia)/((TPBeam->ashear)*G*l*l); w=(TPBLoad->w)*dm/l; P=TPBLoad->P; a=TPBLoad->a-ll; b = 1-a; switch (TPBeam->kl2) ( 148 case f i x _ f i x : CO = l/(4-nul*nu2); Cl = (nul+nu2+nul*nu2)/(4-nul*nu2); SI = l/(l+g*Cl); i f (P ! = 0.0) { /* calculate the structure applied loads */ i f (1>0.0) { A l = l+a/(l*l)*(4*l*(nul-nu2) -nul*a*(2+nu2)+nu2*b*(2+nul))*C0*S1; A2 = l+b/(l*l)*(4*l*(nu2-nul) +nul*a*(2+nu2)-nu2*b*(2+nul))*C0*S1; A3 - (nul*l*(4+nu2*(1.5*g-l))-nul*a*(2+nu2))/b*C0*Sl; A4 = (nu2*l*(4+nul*(1.5*g-l))-nu2*b*(2+nul))/a*C0*Sl; *pxl += P*b/1*A1; *px2 += P*a/1*A2; Rxyl = * p x l ; Rxy2 = *px2; *pml -= sign(xym)*P*a*b*b/(l*l)*A3+Rxyl*ll; *pm2 += sign(xym)*P*a*a*b/(l*l)*A4+Rxy2*12; } else { *pxl += P*b/1; *px2 += P - (*pxl); } , ) i f (w!-0.0) { i f (1>0.0) { /* calculate the structure applied loads */ wl2 = w*l/2.0'; wxy2 = w*l/2.0*sign(xym); Cl = (nul+nu2+nul*nu2)/(4-nul*nu2); A l = l+(nul-nu2)/(4-nul*nu2)/(l+g*Cl); A2 = l+(nu2-nul)/(4-nul*nu2)/(l+g*Cl); A3 = 3*nul*(2-nu2*(l-g))*C0*Sl; A4 - 3*nu2*(2-nul*(l-g))*C0*Sl; *pxl += wl2*Al; *px2 += wl2*A2; Rxyl = * p x l ; Rxy2 = *px2; *pml -= wxy2*l/6.0*A3+Rxyl*ll; *pm2 += wxy2*l/6.0*A4+Rxy2*12; } else { *pxl += w*l/2.; *px2 += *p x l ; } } break; case pin f i x : i f ( P -!- 0.0) { i f (1>0.0) { /* calculate the structure applied loads */ A l = nu2/(l+g*nu2/4); A2 = a*b*(b+2.0*a)/(2.0*1*1*1); Rxyl = P*a/1 + P*A2*A1; Rxy2 = P*b/1 - P*A2*A1; 149 *pxl += Rxyl; *px2 += Rxy2; *pml -= R x y l * l l ; *pm2 += sign(xym)*P*A2*l+Rxy2*12; } else { *pxl = P*b/1; *px2 = P - (*pxl); } } i f (w != 0.0) { i f (1>0.0) { /* calculate the structure applied loads */ wl8 = w*l/8.0; wxy8 = w*l/8.0*sign(xym); A l = (4-nu2*(l-g))/(l+g*nu2/4); A2 = (4+nu2*(l+g))/(l+g*nu2/4); A3 = nu2/(l+g*nu2/4); Rxyl = wl8*Al; Rxy2 = wl8*A2; *pxl += Rxyl; *px2 += Rxy2; *pml -= R x y l * l l ; *pm2 += wxy8*l*A3+Rxy2*12; } else { *pxl += w*l/2.; *px2 += *p x l ; } } break; case f i x _ p i n : i f (P != 0.0) { i f (IX).0) { /* calculate the structure applied loads */ A l = nul/(l+g*nul/4); A2 = a*b*(a+2.0*b)/(2.0*1*1*1); Rxyl = P*b/1 + P*A2*A1; Rxy2 = P*a/1 - P*A2*Al; *pxl += Rxyl; *px2 += Rxy2; *pml -= sign(xym)*P*A2*l+Rxyl*ll; *pm2 += Rxy2*12; } else { *pxl = P*b/1; *px2 = P - (*pxl); } } i f (w != 0.0) { i f (1>0.0) I /* calculate the structure applied loads */ wl8 = w*l/8.0; wxy8 = w*sign(xym)*l/8.0; A l = (4+nul*(l+g))/(l+g*nul/4); 150 A2 = (4-nul*(l-g))/(l+g*nul/4); A3 = nul/(l+g*nul/4); Rxyl = wl8*Al; Rxy2 = wl8*A2; *pxl += Rxyl; *px2 += Rxy2; *pml -= wxy8*l*A3+Rxyl*ll; *pm2 += Rxy2*12; } else { *pxl += w*l/2.; *px2 += * p x l ; } } break; ase pin pin: i f (P != 0.0) { i f (1>0.0) { /* calculate the structure applied loads *pxl = P*b/1; *px2 = P*a/1; } else { *pxl = P*b/1; *px2 = P - (*pxl); } } i f (w != 0.0) { i f (1>0.0) { /* calculate the structure applied loads wl2 = w*l/2.0; *pxl += wl2; *px2 += wl2; ) else { *pxl += w*l/2.; *px2 += * p x l ; • } } break; 151 Appexdix D: L i s t i n g of the Modified Member Forces void Write(NodeVec, MaxNodeNum, struct NODE *NodeVec[' struct BEAM *BeamVec[; i n t MaxNodeNum, MaxBeamNum; BeamVec, MaxBeamNum, MaxSpringNum) /* Local Variables */ in t D0F1,D0F2,D0F3,bof,boflen,pcode,eof,eoflen,col,row,i,j; i n t cnheader,kl2,fmt,colwl,colwlen,width,Typ; double dl,d2,d3,d4,d5,d6,d7,c,s,Cl,C2,C3,C4,C5,SI,S2,S3,S5,S6; f l o a t kl,k2,df 1,df2,df 3,axiall,axial2,shear!,shear2,bml,bm2; f l o a t area,ashear,inertia,axial,shear,bm,kc,p2,p3,p4,p5; double nul,nu2,xm,ym,dm,xm2,ym2,dm2,1,11,12,g.ksl, ks2,ks3; struct BEAM *TPBeam; struct CONNECT *connectn; struct forces *reactns, *end_rctns; FILE * ) ; f l o a t , f l o a t , f l o a t , f l o a t ) ; /* /* /* /* /* / * / * /* /* Functions used */ void f l t _ r e c ( i n t , i n t , f l o a t , i n t , FILE * ) ; void header(int, i n t , char *, char *, char * char *, FILE * ) ; void i n t _ r e c ( i n t , i n t , i n t , FILE * ) ; void l b l _ r e c ( i n t , i n t , char *, char, void w r t i n t ( i n t , FILE * ) ; double ConnectKi(int, f l o a t , /* Defining constants */ row=col=0; bof=0; boflen=2; pcode=1030; colwl=8; colwlen=3; eof-1; eoflen=0; /* ### Write output ### */ wrtint(bof,fptarget); wrtint(boflen,fptarget); wrtint(pcode,fptarget); for (col=0; col<=6; col++) { wrtint(colwl,fptarget); wrtint(colwlen,fptarget); wrtint(col,fptarget); i f (col<=0) width=6; else { i f ( c o l — 3 | | c o l — 6 ) width=13; else width=10; } fputc(width,fptarget); char *, char *, char *, define column A & row 1 */ opcode for BOF marker */ length of BOF marker */ product code */ opcode for setting column width-*/ length of column width record */ opcode for EOF marker */ length of EOF marder */ col=0; 152 /* ### Write TITLE ### */ lbl_rec(col,row,ver,left,fptarget); row++; l b l _ r e c ( c o l , r o w , t i t l e , l e f t , f p t a r g e t ) ; row+=2; reactns = beg_rctns; /* i n i t i a l i z e linked l i s t of support reactions /* ### Write NODE displacements ### */ lbl_rec(col,row.nodedisp,left,fptarget); row++; header(2,row,"node","x_disp","y_disp","rotation","blank", "blank","blank",fptarget); row++; while (i<=MaxNodeNum) { i f (NodeVecfi] != NULL) { int_rec(0,row.NodeVec[i]->num,fptarget); fmt=0x92; dl=NodeVec[i]->d[0] d2=NodeVec[i]->d[l] d3=NodeVec[i]->d[2] d f l = ( f l o a t ) d l ; df2=(float)d2; df3=(float)d3; flt_rec(l,row,df1,fmt,fptarget); flt_rec(2,row,df2,fmt,fptarget); flt_rec(3,row,df3,fmt,fptarget); row++; /* id e n t i f y support nodes */ D0Fl=NodeVec[i]->D0F[0] D0F2=NodeVec[i]->D0F[1] D0F3=NodeVec[i]->D0F[2] i f (DOF1 ==0 || D0F2 — 0 || DOF3 — 0) { reactns->num=NodeVec[i]->num; reactns++; } . ) i++; }. end_rctns=reactns; reactns=beg_rctns; row++; /* ### Write MEMBER forces ### */ lbl_rec(col,row,beamforce,left,fptarget); row++; header(2,row,"member","axiall","shear1","bml","axial2", "shear2","bm2",fptarget); row++; while (i<=MaxBeamNum) ( i f (BeamVec[i] != NULL) { TPBeam = BeamVec[i]; d l = BeamVec[i d2 = BeamVec[i d3 = BeamVec[i d4 = BeamVec[i d5 = BeamVec[i d6 = BeamVec[i ->nodel->d[0] ->nodel->d[l] ->nodel->d[2] ->node2->d[0] ->node2->d[l] ->node2->d[2] 153 kl2= BeamVec[i]->kl2; area = BeamVec[i]->area; ashear = BeamVec[i]->ashear; i n e r t i a = BeamVec[i]->inertia; xm = (BeamVec[i]->node2->X) - (BeamVec[i]->nodel->X); ym = (BeamVec[i]->node2->Y) - (BeamVec[i]->nodel->Y); xm2 = xm*xm; ym2 = ym*ym; dm2 = xm2+ym2; dm = sqrt(dm2); c = xm/dm; s = ym/dm; /* Connection Sizes */ i f (TPBeam->connectl=NULL) 11=0.; else ll=(TPBeam->connectl->pl); i f (TPBeam->connect2==NULL) 12=0.; else 12=(TPBeam->connect2->pl); 1 = dm-11-12; /* Flexible Connection */ kl-k2—1.00e+38; i f (TPBeam->connectl==NULL) nul=l; else { kl=(TPBeam->connectl->Ki); i f (kl>0. && kl<=l.e-30) nul=0.; sXs© i f (kl—-1.00e+38) nul=l.; else nul=l./(l.+((3*E*(TPBeam->inertia))/(kl*l))); } i f (TPBeam->connect2—NULL) nu2=l.; else { k2=(TPBeam->connect2->Ki); i f (k2>0. && k2<=l.e-30) nu2=0.; sX ss i f (k2—1.00e+38) nu2=l.; sX S S nu2=l./(1.+((3*E*(TPBeam->inertia))/(k2*l))); } axiall=axial2 = ((d4-dl)*c+(d5-d2)*s)*area*E/l; i f (kl2 == pin_pin) shearl = shear2 = bml = bm2 = 0.0; else { d7 = (d2-d5)*c+(d4-dl)*s; /* Shear Deflection */ i f (G — 0.0 || ashear — 0.0) g=0.0; else g=12.*E*inertia/(ashear*G*l*l); 154 i f (kl2 == f i x _ f i x ) { CI = (nul+nu2+nul*nu2)/(4-nul*nu2); C2 = nul*(2+nu2)/(4-nul*nu2); C3 = 3*nul/(4-nul*nu2); C4 = nu2*(2+nul)/(4-nul*nu2); C5 = 3*nul*nu2/(4-nul*nu2); 51 = l/(l+g*Cl); 52 - (l+g*nu2/4)/(l+g*Cl); 53 = (l-g/2)/(l+g*Cl); shearl =((2.0*Cl/l)*d7+(C2+2*ll*Cl/l)*d3 +(C4+2*12*Cl/l)*d6)*6*E*inertia*Sl/(l*l); stlG3.1T2 — sll.G3.irl * bml = -((6.0*C2*Sl/l)*d7+(4.0*C3*S2+6.0*ll*C2*Sl/l)*d3 +(2.0*C5*S3+6.0*12*C2*Sl/l)*d6)*E*inertia/l; bm2 = bml+shearl*l; } i f (kl2 = p i n _ f i x ) { S6 = l/(l+g*nu2/4); shearl = (d7/l+ll/l*d3+(l+12/l)*d6)*3*E*iner-tia*n u 2 * S 6 / ( l * l ) ; bml =0.0; sliG3.ir2 = s l i 6 3 i r l " bm2 = (d7/l+il/l*d3+(l+12/l)*d6)*3*E*inertia*nu2*S6/l; } i f (kl2 == fix_pin) { S5 = l/(l+g*nul/4); shearl = (d7/l+(l+ll/l)*d3+12/l*d6)*3*E*iner-t i a * n u l * S 5 / ( l * l ) ; bml = -(d7/l+(l+ll/l)*d3+12/l*d6)*3*E*inertia*nul*S5/l; shear2 = shearl; bm2 =0.0; } } /* i f there were loads on the beam the end forces w i l l be atached to the beam i n the BFORCE structure. Add these to structural forces to get resultant */ i f (BeamVec[i]->bforce!=NULL) { a x i a l l +=BeamVec[i]->bforce->axiall; axial2 +=BeamVec[i]->bforce->axial2; bml +=(BeamVec[i]->bforce->bml) +((BeamVec[i]->bforce->shearl)*ll); bm2 +=(BeamVec[i]->bforce->bm2) -((BeamVec[i]->bforce->shear2)*12); shearl +=BeamVec[i]->bforce->shearl; shear2 +=BeamVec[i]->bforce->shear2; } /* write out resultant beam forces */ int_rec(0,row,BeamVec[i]->num,fptarget); fmt=0x81; fl t _ r e c ( l , r o w , a x i a l l , f m t , f p t a r g e t ) ; flt_rec(2,row,shearl,fmt,fptarget); flt_rec(3,row,bml,fmt,fptarget); flt_rec(4,row,axial2,fmt,fptarget); flt_rec(5,row,shear2,fmt,fptarget); flt_rec(6,row,bm2,fmt,fptarget); row++; 155 /* Assign appropriate forces to Connections i f (TPBeam->connectl != NULL) { connectn = TPBeam->connectl; connectn->Vc = shearl; connectn->Mc = bml; Typ = connectn->Typ; p2 = connectn->p2; p3 = connectn->p3; p4 = connectn->p4; p5 = connectn->p5; kc = (float) ConnectKi(Typ,bml,p2,p3,p4,p5); connectn->Kc = kc; } i f (TPBeam->connect2 != NULL) { connectn = TPBeam->connect2; connectn->Vc = shear2; connectn->Mc = bm2; Typ = connectn->Typ; p2 = connectn->p2; p3 = connectn->p3; p4 = connectn->p4; p5 = connectn->p5; kc = (float) ConnectKi(Typ,bm2,p2,p3,p4,p5); connectn->Kc = kc; } /* sum a l l beam forces at node supports reactns=beg_rctns; while (reactns != end_rctns) { i f (BeamVec[i] ->nodel->num=reactns->num) { a x i a l = a x i a l l * ( - l . ) ; shear=shearl; reactns->M+=bml*(-1.)+shear*ll; reactns->Rx+=axial*c-shear*s; reactns->Ry+=axial*s+shear*c; } i f (BeamVec [i] ->node2->num=reactns->num) { axial=axial2; shear=shear2*(-1.); reactns->M+=bm2-shear*12; reactns->Rx+=axial*c-shear*s; reactns->Ry+=axial*s+shear*c; } reactns++; } } i++; ) /* ### Write CONNECTION forces ### */ cnheader=0; i - i ; while (i<=MaxNodeNum) { i f (NodeVec[i] != NULL) { j - i ; while (j <= MaxBeamNum) I i f (BeamVec[j] != NULL) { 156 TPBeam = BeamVec[j]; i f (TPBeam->connectl->Jn==NodeVec[i]->num || TPBeam->connect2->Jn==NodeVec [ i ] ->num) { i f (cnheader=0) { row++; lbl_rec(col,row,connforce,left,fptarget) ; row++; header(2,row,"n_num","m_num","Vc","bmc","Kc" , "bmi","Ki".fptarget); row++; cnheader=l; } i f (TPBeam->connectl->Jn==NodeVec[i]->num) connectn=TPBeam->connectl; else connectn=TPBeam->connect2; int_rec(0,row,connectn->Jn,fptarget); int_rec(l,row,connectn->Mn,fptarget); fmt=0x81; flt_rec(2,row,connectn->Vc,fmt,fptarget); flt_rec(3,row,connectn->Mc,fmt,fptarget); fmt=0x92; i f ((connectn->Kc) < 0.) lbl_rec(4,row,"rigid",centre,fptarget); else flt_rec(4,row,connectn->Kc,fmt,fptarget); fmt=0x81; flt_rec(5,row,connectn->Mi,fmt,fptarget); fmt=0x92; i f ((connectn->Ki) < 0.) lbl_rec(6,row,"rigid",centre,fptarget); else flt_rec(6,row,connectn->Ki,fmt,fptarget); row++; ) } } } i++; } ' row++; /* ### Write Support Reactions ### */ lbl_rec(col,row,reactions,left,fptarget); row++; header(2,row,"node","Rx","Ry","M","blank","blank","blank",fptarget); row++; reactns=beg_rctns; while (reactns != end_rctns) { int_rec(0,row,reactns->num,fptarget); fmt=0x81; flt_rec(l,row,reactns->Rx,fmt,fptarget); flt_rec(2,row,reactns->Ry,fmt,fptarget); flt_rec(3,row,reactns->M,fmt,fptarget); row++; reactns++; } row++; 157 /* ### Write Spring Forces ### */ i f (MaxSpringNum>0) { lbl_rec(col,row,springforce,left,fptarget); row++; header(2,row,"node","Rsx","Rsy","Ms","blank","blank","blank",fptarget) row++; i - i ; while (i<=MaxSpringNum) { i f (NodeVecfi] != NULL) { i f (NodeVec[i]->spring!=NULL) { } i++; ksl=NodeVec[i]->spring->Ks[0] ks2=NodeVec[i]->spring->Ks[l] ks3=NodeVec[i]->spring->Ks[2] axiall=(float)(ksl*NodeVec[ij->d[0]*(-1.OL)) axial2=(float)(ks2*NodeVec[i]->d[1]*(-1.OL)) bml=(float)(ks3*NodeVec[i]->d[2]*(-1.OL)); int_rec(0,row,NodeVec[i]->num,fptarget); 1fmt=0x81; fl t _ r e c ( l , r o w , a x i a l l , f m t , f p t a r g e t ) ; flt_rec(2,row,axial2,fmt,fptarget); flt_rec(3,row,bml,fmt,fptarget); row++; } } /* ### EOF marker ### */ wrtint(eof,fptarget); wrtint(eof len, fptarget).; fclose(fptarget) ; 158 Appendix E: CPlane User Manual CPlane Highlight: a s t r u c t u r a l analysis program as an add-on for spreadsheets — includes the effect of f l e x i b l e connections l i n e a r and nonlinear simple geometry generation and post processing with spreadsheets ease of use — written i n C compatible with spreadsheets: LOTUS 1-2-3 (TM), SYMPHONY (TM), QUATTRO (TM). 159 Introduction The e l a s t i c structural analysis program "CPlane" was developed with the intention of devising a p r a c t i c a l method of incorporating behavior of f l e x i b l e connections i n analysis of st e e l structures. The program i s designed for use with personal computers equiped with a math coproccessor and requires DOS or OS/2 operating systems. Any personal computer capable of running LOTUS 1-2-3 (TM) or SYMPHONY (TM) should have no problem i n running "CPlane". "CPlane" works independently from Lotus 1-2-3, Release 2.0 and l a t e r . However, i t requires input (source) data and produces output (target) data i n LOTUS worksheet f i l e format ( f i l e s with extension ".WK1"). "CPlane" operates with a l l IBM PC's or compatibles and no s p e c i f i c requirements are necessary. The problem solution module i s centered around a unique s t i f f n e s s matrix program which i s written i n the C language. I t interfaces with the spreadsheet program d i r e c t l y without going through cumbersome data f i l e conversions. The spreadsheet allows for greatest s i m p l i c i t y and f l e x i b i l i t y i n creating input data and evaluating numerical r e s u l t s . A-set of pre- and post-processor templates i s provided to guide the user i n the use of "CPlane". User can ea s i l y implement this analysis module i n his custom designed spreadsheet templates as w i l l be shown i n an example l a t e r . "CPlane" can solve s t a t i c l i n e a r e l a s t i c s t r u c t u r a l systems with a multitude of load types. The user should be f a m i l i a r with basic engineering and computer terminology to use this program most e f f e c t i v e l y . We suggest that new users read parts of this manual to become f a m i l i a r with "CPlane"'s c a p a b i l i t i e s . In most cases, a thumbing through the manual i s s u f f i c e . "CPlane" accepts any consistent set of units. Geometry plots can be viewed on the screen when the preset templates are used for creating the program input. Hardcopy plots can be made using the spreadsheet routines (PRINTGRAPH, etc.). Once the structure's geometry i s described, i t i s 160 advisable to perform a geometry check i n order to help to locate any errors that were made during the input process. Loads and springs can be specified i n various ways and they can be automatically generated, which i s very h e l p f u l for sloping distributed loads or e l a s t i c foundations. The b r i e f manual consists mainly of a basic introduction to the program. The data input i s unparalleled i n terms of f l e x i b i l i t y . This manual assumes that you are a current 1-2-3 user, f a m i l i a r with i t s basic functions and operations. I t also assumes that you are f a m i l i a r with your computer, i t s keyboard and operating system. 161 Hard Disk I n s t a l l a t i o n of CPLANE ACCESS SYSTEM 1. Place floppy disk "CPLANE" into drive A and s t a r t the i n s t a l l a t i o n routine: A:\>installh[enter] 2. Change the LOTUS 1-2-3 default directory to "C:\PLANE" and update the directory. (command sequence i n LOTUS: "/wgddC:\PLANE[Enter]uq") 3. Include locations of the LOTUS system and "CPLANE" i n the path of your AUTOEXEC.BAT f i l e . I f you don't have an AUTOEXEC.BAT, rename the one supplied: C:\>ren planeaut.bat autoexec.bat[enter] 4. Reboot your system [ C r t l ] - [ A l t ] - [ D e l ] . Start the "CPLANE ACCESS SYSTEM" from the root directory C: C:\>caccess[enter] From now on a menu system w i l l guide you. 162 Floppy Disk I n s t a l l a t i o n of CPLANE ACCESS SYSTEM 1. Create your own "work disk". Place an empty floppy disk, which has been previously formatted, into drive B and the program disk "CPLANE" i n drive A. Start the i n s t a l l a t i o n routine: A:\>installf[enter] 2. Replace the "CPLANE" disk by your Lotus program disk i n drive A, st a r t 1-2-3, change the default directory to "B:\", and update the directory, (command sequence i n LOTUS: "/wgddB:\[Enter]uq") 3. Quit LOTUS. Start the "CPLANE ACCESS SYSTEM" from B: B:\>caccess[enter] From now on a menu system w i l l guide you. 163 Using "CPLANE" There are two ways to conduct a structural analysis with LOTUS 1-2-3 (TM): "CPLANE" and A. CPLANE ACCESS SYSTEM: ( i n s t a l l a t i o n required) Menu driven pre- and postprocessor templates have been prepared and can be used as i s or e a s i l y customized. (hard disk) C:\>caccess[enter] or (floppy disk) B:\>caccess[enter] From then on a menu system w i l l guide you through a l l modules including data generation, storing, p l o t t i n g , viewing, p r i n t i n g , analyzing, reducing, etc... The more advanced user might want to develop his own pre-postprocessor or prefers to work without any menu system, following procedure i s proposed: or Then the B. GENERIC PROCEDURE: (no i n s t a l l a t i o n required, j u s t copy CPLANE.EXE to your work disk or directory) 1. Load LOTUS 1-2-3 and create your input values for the structure and loads, safe the worksheet under a descriptive name ( i . e . FRAME1.WK1). Quit LOTUS. 2. Start "CPLANE" by CPLANE[enter] and respond to "CPLANE"'s prompts ( i t w i l l ask you for the name of the source data f i l e and target data f i l e (including f i l e extensions)) or CPLANE framel.wkl resultl.wkl[enter] The l a t t e r procedure enables you either to create a batch f i l e containing several input and output f i l e s or to use "CPLANE" as the analysis engine for your custom pre- or postprocessors. 3. Load LOTUS 1-2-3 and retrieve the target f i l e ( s ) . Now you can use the output data to plo t , to p r i n t etc. 164 Input F i l e Structure for "Cplane": CPlane t i t l e nodes n num i x i y im xcoor ycoor ngen members m_num fm j 1 J2 Area Ashea I jstep material E G wt springs s num Sx sy Sm sgen nodal loads 1 num Px Py Pm lgen member loads i num d i r w P a igen connection data n num m num typ Mi p i P2 p3 p4 p5 echo ite r a t e endata 165 L i s t of Symbols: CPlane t i t l e (one l i n e of text) nodes n num, i x , i y , im, xcoor, ycoor, ngen (one row for each node) n_num i x im xcoor ycoor ngen nodal number freedom i n x-direction freedom i n y-direction freedom to rotate i f = 0: fix e d i f = 1: free i f = J: linked to same degree of freedom as node J nodal coordinate x nodal coordinate y generation of n_num i n ngen-steps, same degrees of freedom, li n e a r interpolation of geometry, generation ends at node from next row n o d e : 1 1 1 n o d e : 0 1 1 n o d e : 1 0 1 n o d e : 1 1 0 n o d e : 0 0 1 n o d e : 0 0 0 x Fig. 1: Nodes and Coding of F i x i t i e s 166 members m_num,fm,j1,j2.Area,Ashea,I,jstep (one row for each member) m num fm j l j 2 Area Ashea I jstep member number end condition of member i f — p: member i s pinned - pinned i f = fp: member i s fixed - pinned i f = pf: member i s pinned - fixe d i f = f: member i s fix e d - fixed lesser nodal number greater nodal number t o t a l cross sectional area cross sectional area to carry shear, i f t h i s c e l l i s empty or equal to zero analysis i s done without consideration of shear deformation ( s i m p l i f i e d s t i f f n e s s analysis) moment of i n e r t i a generation of j l i n jsteps, keeps difference between j i and j2 constant, l i n e a r interpolation of Area, Ashea and I, ends at member form next row Local +-direction i s lower nodal to higher nodal number. +-member force i s tension, +-member bending exerts tension i n upper f i b r e s . member: p m e m b e r : pf m e m b e r : fp m e m b e r : f Fig. 2: Members and Coding of End F i x i t i e s 167 material E,G,wt E : modulus of e l a s t i c i t y G : shear modulus, i f t h i s c e l l i s empty or equal to zero, shear deformation i s not considered wt : weight of material (note: weight i s i n the current version of "plane2.0" not considered) springs s_num, Sx, Sy, Sm, sgen (one row for each spring) s_num : node that spring i s attached to Sx : spring s t i f f n e s s i n x-direction, positive i f +x-displacement causes t e n s i l e (+) reaction i n spring Sy : spring s t i f f n e s s i n y-direction, positive i f +y-displacement causes t e n s i l e (+) reaction i n spring Sm : rot a t i o n a l s t i f f n e s s , positive i f counter-clockwise rotation causes counter-clockwise moment i n spring sgen : generation of n num i n sgen-steps, l i n e a r interpolation ol s t i f f n e s s s p r i n g i n x - d i r e c t i o n s p r i n g i n y - d i r e c t i o n s p r i n g a b o u t z - a x i s Fig. 3: Elementary Springs, Combination with each other and with p a r t i a l l y fixed nodes possible 168 nodal_loads l_num, Px, Py, Pm, lgen (one row for each loaded node) l_num Px Py Pm lgen number of loaded node load i n x-direction load i n y-direction moment about node generation of n_num i n lgen-steps, lin e a r Interpol, of loads member_loads i num, d i r , w, P, a, igen (one row for each loaded member) i_num d i r w lgen member number the load i s applied to dire c t i o n the load i s applied to i f = i f -i f = x: y: p: :: i n x-direction (positive) r: i n y-direction (positive) >: perpendicular to member (positive when clockwise about lower node) uniformly distributed load i f direc = p: load = w i f direc = x: load = v e r t i c a l projection of w i f direc = y: load = horizontal projection of w concentrated load i f direc = p: load perpendicular i f direc = x: load i n x-direction i f direc = y: load i n y-direction distance from lower node of member to location of cone, load d i s t . along member from lower node to concentrated load d i s t . along y-axis from lower node to concentrated load d i s t . along x-axis from lower node to concentrated load generation of i_num i n igen-steps, l i n e a r interpolation of loads and load locations, the parameter direc of th i s l i n e i s used for a l l generated nodes; w, P, and a are incremented between the loads given i n this l i n e and the next l i n e i f direc = p: i f direc = x: i f direc = a = a = 169 1 w V n o d e 1 n o d e 2 Fig. 4: Uniformly Distributed Load Perpendicular to Member, direc=p p Fig. 5: Concentrated Load Perpendicular to Member, direc=p connections n_num, m_num, typ, Mi, p i , p2, p3, p4, p5 n_num m_num typ entered Mi pl-p5 j o i n t at which connection i s located member which the connection i s connected to connection type ( 1. perfectly hinged, 2. single web angle, 3. double web angle, 4. header plate, 5. top & seat angle, 6. end plate (without column s t i f f e n e r s ) , 7. end plate (with column s t i f f e n e r s ) , 8. t-stub, 9. other, (note: connection s t i f f n e s s i s as parameter p2) 10. perfectly r i g i d . ) connection moment (optional) default uses i n i t i a l tangent s t i f f n e s s value connection parameters. Please refer to Figure 6 for d e t a i l s . 170 P3 = t P 2 - d P i = Li p 4 = g (a) SINGLE WEB A N G L E P 3 " t P 2 = d P I - L l P4= 9 (b) DOUBLE WEB A N G L E P 3 - t P 5 = w P2= d V P l " Lj p4 - g (c) HEADER P L A T E P 3 ~ t P 5 -P 4 " k ( fastener Of) r p 2 = d ^ P l " (d) TOP & S E A T A N G L E P l = L, I P 4 - f (only w i t h column s t i f f ene rs ) P 2 = d ^ co lumn s t i f f ener I P 3 - t 1 P 3 " t P 5 = ' ? 4 ' k ( fas tener 0[) p 2 - d A J P l = L, (g) T - S T U B (e), ( f) END P L A T E (WITH & WITHOUT COLUMN STIFFENERS) Fig. 6: Connection Types and Their Standardization Parameters 171 echo indicator for creation of echofile i f t h i s keyword exists i n the source f i l e then ECH0.WK1 f i l e w i l l be created i t e r a t e indicator for nonlinear connections i f t h is keyword exists program automatically creates and updates the intermediate data f i l e ITERATE.WK1 for i t e r a t i o n purposes. endata indicator for end of data i f t h i s keyword exists i n the source f i l e then spreadsheet may contain other comments, macros or data (for i.e. plotting) 172 Example ^ o u r c e ^ i l e T CPlane n l f l e x 2 : s t e e l frame investigated by Moncarz & Gerstle, w=0.02 kips/sq f t . nodes 1 0 0 0 0 0 2 1 1 1 0 144 3 1 1 1 0 288 4 1 1 1 288 288 5 1 1 1 288 144 6 0 0 0 288 0 members 1 f 1 2 9.71 0.00 170.00 2 f 5 6 9.71 0.00 170.00 3 f 2 5 13.00 0.00 843.00 4 f 2 3 9.71 0.00 170.00 5 f 4 5 9.71 0.00 170.00 6 f 3 4 9.12 0.00 375.00 material 30000 member loads 3 y -0.255 6 y -0.155 nodal loads 2 5.76 3 2.88 connections 2 3 5 0 4.865 20.66 1.222 1 125 6 50 3 6 5 0 4.865 15.88 1.065 1 250 5 52 4 6 5 0 4.865 15.88 1.065 1 250 5 52 5 3 5 0 4.865 20.66 1.222 1 125 6 50 iter a t e echo 173 

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