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INELASTIC STABILITY OF STEEL MEMBERS AND FRAMES USING COMPUTERS WITH VACUUM TUBES IN ‘60s TO SUPERCOMPUTERS IN ‘90s .. 4

1. Introduction ................................................................................................................ 4

2. Computers Used in Various Inelastic Stability Analyses Described in the Report ... 4

3. Inelastic Stability of Planar Frames. ZEBRA Computer at EPUL 1964-66 .............. 6

3.1 Inelastic In-plane Stability of Unbraced Steel Frames [Vinnakota, 1967, 1971] .................................................................................................................... 6

3.2 Inelastic Stability of a Braced Hinged Base Portal Frame Studied by Ojalvo & Lu [1961] ........................................................................................... 12

3.3 Inelastic Buckling of Three Unbraced Hinged Base Portal Frames Tested by Yen, Lu and Driscoll [1962] ......................................................................... 13

3.4 Inelastic Stability of an Unbraced Hinged Base Portal Frame Studied by Yura & Galambos [1965] .................................................................................. 14

3.5 Inelastic Stability of a Fixed Base Gable Frame Tested by Baker and Eickhoff [1956] .................................................................................................. 14

3.6 Inelastic Stability of an Unbraced Fixed Base Hybrid Portal Frame Tested by Adams [1966] ............................................................................................... 15

4. Inelastic Planar Stability of Beam-Columns. ZEBRA Computer at EPUL 1964-65 and IBM 7040 Computer 1966-67. ..................................................................... 17

4.1 Inelastic Stability of Rotationally and/or Directionally Restrained Beam-Columns [Vinnakota, 1967; Vinnakota and Badoux, 1970a, 1971] .................. 17

4.2 Planar Strength of Pin-Ended Beam-Columns Studied by Galambos & Ketter [1959]; Ojalvo & Fukumoto [1962]........................................................ 23

4.3 Planar Strength of Rotationally Restrained Beam-Columns. Levi, Driscoll & Lu [1965]; Ketter and Beedle [1955]............................................................. 24

4.4 Full-Scale Beam-Columns Tested by Van Kuren & Galambos [1964] ............. 25

4.5 Inelastic Stability of Rotationally Restrained Beam-Columns Free to Sway. Driscoll et al. [1965] .......................................................................................... 30

4.6 Inelastic Stability of a Rotationally and Directionally Restrained Beam-Column. Driscoll et al. [1965] ........................................................................... 32

5 Inelastic Spatial Stability of Beam-Columns. CDC 7326 Computer at EPFL 1972-77 .................................................................................................................... 33

5.1 Inelastic Spatial Stability of Rotationally and Directionally Restrained Imperfect Beam-Columns. Vinnakota [1973, 1974, 1977b], Vinnakota & Aoshima [1974a, b], Vinnakota & Aysto [1974]............................................... 34

5.2 Biaxially-Bent Beam-Columns Tested by Birnstiel [1968] ............................... 42



5.3 Rotationally Restrained Biaxially-Bent Columns Tested by Gent and Milner [1968] and Milner and Gent [1971] ....................................................... 44

5.4 Inelastic Lateral-Torsional Buckling of Beam-Columns Tested at CTICM-Paris [Sfintesco, 1973] ....................................................................................... 46

5.5 Biaxially-Bent Beam-Columns Studied by Tebedge and Chen [1974] ............. 46

5.6 Rotationally Restrained Beam-Column Studied by Santathadaporn and Chen [1973] ....................................................................................................... 47

5.7 Inelastic Lateral-Torsional Buckling of Beams Tested by Kitipornchai and Trahair [1975b] .................................................................................................. 48

5.8 Influence of Level of Loading on Inelastic Lateral-Torsional Buckling of Beams, Studied by Yoshida and Imoto [1973] .................................................. 49

5.9 Beam-Columns Subjected to Thrust and Biaxial Bending. Full-Scale Tests Conducted at Liege, Belgium [Anslijn & Massonnet, 1977, Vinnakota, 1975] .................................................................................................................. 50

5.10 Influence of Imperfections on Strength of Biaxially Loaded Beam-Columns [Vinnakota, 1976a] ............................................................................. 54

5.11 Ultimate-Strength of Crooked Steel Beam-Columns Subjected to Axial Thrust and Major-Axis End Moments [Galambos (ed.), 1988] ......................... 56

6 Inelastic Planar Stability of Columns with Small End Restraints. VAX 11/780 Computer at MU 1982-85 ........................................................................................ 59

6.1 Planar Strength of Singly Symmetric Restrained Beam-Columns [Vinnakota, 1982, 1983a, b, c, 1985a] ............................................................... 59

6.2 Column with Small End-Restraint Tested by Bergquist [1977] ........................ 64

6.3 Beam-Columns with Lateral Loads Studied by Lu and Kamalvand [1968] ...... 64

6.4 Influence of Small End Restraint, Geometrical Imperfections, and Residual Stresses on Column Strength. Vinnakota [1982] ............................................... 65

6.5 Directionally and Rotationally End-Restrained Columns. Vinnakota [1983b] ............................................................................................................... 67

7 Inelastic Planar Stability of Large Unbraced Steel Frames. Cray C90 at Pittsburgh Supercomputing Center 1990s ................................................................................. 69

7.1 Inelastic Stability of Partially Restrained Unbraced Steel Frames Using Supercomputers. Foley [1996], Foley and Vinnakota [1997, 1999a, 1999b] .... 69

7.2 Inelastic Stability of Partially Restrained Single-Bay Three-Story Frame [Foley, 1996] ...................................................................................................... 73

7.3 Inelastic Stability of Partially Restrained Eight-Bay Sixteen-Story Frame. Foley [1996] ....................................................................................................... 75

7.4 Inelastic Stability of Partially Restrained Five-Bay Sixty-Story Frame. Foley [1996] ....................................................................................................... 77

8 Conclusions ............................................................................................................... 78



Acknowledgements ....................................................................................................... 78

Publications Vinnakota, on Inelastic Stability of Members and Frames ...................... 78

Some Ph. D Dissertations on Inelastic Stability ........................................................... 81

References ..................................................................................................................... 82

END OF PAPER ........................................................................................................... 85

Note: This report forms the basis for the SSRC Beedle Award Presentation by Vinnakota in Toronto on March 28, 2014.

In 1961 as a young engineer working in New Delhi, India, the author was asked to design a steel portal frame with crane loads for a power house structure. It was then that he came across and read the book Plastic Design of Steel Frames written by Prof. Lynn S. Beedle. He was fascinated and inspired by the book. That led him to go to Lausanne, Switzerland in Fall 1962 on a Swiss Government Scholarship for higher studies in structural engineering at the Institute of Technology at University of Lausanne (EPUL).

Prof. Maurice Cosandey, the then Professor of Steel Structures at EPUL and an eminent practicing engineer, guided and challenged Vinnakota’s doctoral research work. This, in spite of the fact that less than six months after Vinnakota joined him, he was named Director of EPUL and undertook the Herculean task of federalizing EPUL. Note: If you have any questions or comments on this report please send an email to: [email protected]



INELASTIC STABILITY OF STEEL MEMBERS AND FRAMES USING COMPUTERS WITH VACUUM TUBES IN ‘60s TO

SUPERCOMPUTERS IN ‘90s

Sriramulu Vinnakota1 Emeritus Professor, Marquette University, WI, USA.

1. Introduction

Equation Section (Next) This paper summarizes a number of research studies on inelastic stability of steel columns, beams, and beam-columns, and planar stability of steel frames. This research, with which the author is associated, stretched over a period of 40 years, and made use of several generations of computers.

In 1961 the author designed a steel portal frame (Fig. 1.1) for a small power house structure using, a slide rule, moment distribution method for analysis, standard tables from hand books and allowable stresses for beam and column design. It was then that he came across the book Plastic Design of Steel Frames written by Prof. Lynn S. Beedle. That led him to go to Lausanne, Switzerland in Fall 1962 and pursue research on stability of steel frames. This report is a summary of that research and its continuation.

2. Computers Used in Various Inelastic Stability Analyses Described in the Report

Equation Section (Next) ZEBRA computer, available at the Institute of Technology at University of Lausanne (EPUL), Switzerland, was used by Vinnakota from 1964 to 1966 to study the inelastic stability of steel beam-columns and (small) frames. (See Section 3.) It was one of the

Figure 1.1: Steel portal frame subject to roof, wind, and crane loads,



first computers installed in Switzerland. ZEBRA is an acronym for the Dutch words “Zeer Eenvoudage Binaire Reken-Automat”, which means “Very Easy Binary Calculating Machine”. The single-user computer, developed in the Netherlands, was a serial machine with a 33-bit word length. Main storage was provided by a magnetic drum memory holding 8K words and running at 6000 rpm. It had a fast-access storage consisting of 12 single-word registers. Input and output were via a perforated paper tape reader, a paper tape punch, and a teleprinter. Normal Code was the name of the assembler and occupied the top 1K of storage. Simple Code was the name of the programming system for casual users, in which most scientific and engineering programs were written and included many built-in functions. It occupied 2K of storage just below Normal Code. It was said that a system of 87 linear equations in 87 unknowns submitted on Friday evening was solved successfully by Saturday evening. The IBM 7040 computer installed in 1964 at EPUL had a magnetic drum memory of 32K words of 36 bits. Interface was by punched cards and processing was in batch mode. FORTRAN was the programming language. The IBM 7040 was used by Vinnakota from 1965 to 1968 to study the inelastic planar stability of restrained steel beam-columns. (See Section 4.) The CDC 7326 computer installed in 1972 at the Swiss Federal Institute of Technology, Lausanne (EPFL) had a capacity of 128K words of 60 bits magnetic drum. Interface was by punched cards and magnetic tapes, and batch processing. The CDC 7326 was used by Vinnakota from 1972-1977 to study the inelastic spatial stability of restrained steel beam-columns. (See Section 5.) The Digital Equipment Corporation (DEC) VAX 11/780 computer first introduced in 1977 was used by Vinnakota at Marquette University from 1982-85 to study the inelastic planar stability of steel beam-columns with small end-restraints. (See Section 6.). The VAX 11/780 was 32-bit single-processor computer that ran the DEC VAX/VMS operating system and implemented an Instruction Set Architecture (ISA). The Cray C90 (Y-MP) supercomputer, available at the Pittsburgh Supercomputing Center, Pittsburgh, PA, USA was used by Foley from 1994 to 1996, under a grant from the PSC to Vinnakota, to study the inelastic in-plane stability of large partially restrained steel frames using parallel processing and supercomputers as part of his dissertation work with Vinnakota (Section 7). The Cray C90 was a 16 vector processor supercomputer launched by Cray Research in 1991. The C90 ran on the UNICOS operating system.

In Sections 3 to 7 of this report, main points of the inelastic stability problem studied in that section are given to start with, followed by results for several control problems analyzed to check the procedures developed in the section.



3. Inelastic Stability of Planar Frames. ZEBRA Computer at EPUL 1964-66

Equation Section (Next) This section summarizes the results of a research program undertaken at Technical Institute of Lausanne University, EPUL (Director: Prof. M. Cosandey) on the inelastic behavior of steel frames. The displacement method, generalized to include plastic hinges in members, was used for the numerical studies to solve the problem on the ZEBRA computer. A generalized transfer matrix method was used to determine the stiffness matrix elements of partially plastified members. The spread of plasticity, both within the cross sections and along the length of the members, residual stresses, P and Pmoments, were all included in the analysis.

3.1 Inelastic In-plane Stability of Unbraced Steel Frames [Vinnakota, 1967, 1971]

Vinnakota [1967] developed an analytical method to study the inelastic planar stability of steel frames (Fig. 3.1.1) up to the point of instability (Fig. 3.1.2). The loading on the frame may consist of joint loads, and/or distributed transverse loads on the members (bars). The loads considered in this study were non-proportional. Thus, each applied load may consist of a constant part and/or a variable part. For example the external loads such as H and q in Fig. 3.1.1 may be written as: c v c v etc.H H H q q q (3.1)

where represents the load factor or load level; subscript “c” stands for the constant part of a force; and subscript “v” for the variable part of a force corresponding to the value of

1. In the numerical studies, was increased monotonically and the corresponding deformations, , are calculated (Fig. 3.1.2) . The maximum value of for which the

calculations converge is taken as the ultimate strength u of the structure.

The study was based on the following assumptions: (1) Cross sections of all

members retain their original shape, (2) deformations are small, (3) shearing strains are negligible and yielding is governed by normal stress only, (4) loads are conservative, (5) strain hardening is not considered, and (6) there is no stress reversal of material stressed beyond the elastic limit.

Figure 3.1.1: Planar steel frame subject to non-proportional loading

Figure 3.1.2: Load-deformation response of the frame.



It was further assumed that the material behavior is elastic-perfectly plastic (Fig. 3.1.3). So, if a unit-length of a beam-column is partially plastified under the action of an axial thrust P and a bending moment M (Fig. 3.1.4), the curvature can be expressed in the form:

( ) ( ) ( ), , , ,y g rF M P f f f (3.2)

Here, F represents a functional relationship, while ( ) ( ) ( ), ,y g rf f f are functions representing

the influence of the variation of the yield stress, the geometry of the cross section and the residual stress pattern (Fig. 3.1.5), respectively, on the curvature . Figure 3.1.6 shows the piecewise linear type of residual stress distribution found in rolled steel beam shapes. A particular case of this pattern is the Lehigh type, wherein:

0.32

f frc y rj rw rc rt

f f w f

b tF

b t t d t

(3.3)

Procedures for constructing P˗ M˗ curves for I-shapes based on Eq. (3.2), have been developed in Ketter, Kaminsky and Beedle [1955] and Vinnakota [1967]. Equation (3.2) can be put in the form:

2

2eq

d v M

dz EI (3.4)

where eqEI is the equivalent flexural rigidity of the inelastic beam-column at section z.

The value of eqEI corresponding to a set of M and P values can be obtained with the help

of the P˗ M˗ curves from the relation:

eq

MEI

(3.5)

P˗ M˗ eqEI curves constructed for an 8WF31 shape are shown in Fig. 3.1.7 in non-

dimensionalized form [Vinnakota, 1967]. Here, Py is the yield load and My is the yield moment of the cross section. Continuous curves correspond to an annealed section and the broken lines correspond to the Lehigh type residual stress pattern with 0.3rc yF

and 0.19rt yF .

The P˗ M˗ eqEI curves represent the influence of the residual stresses and spread of

plasticity in the cross section, on the flexural rigidity of a partially plastified section. The spread of plasticity along the length of a member of a frame is schematically shown in Fig. 3.1.8. To capture the influence of this spread of plasticity on member behavior, the member is subdivided into intervals so that the bending moment and hence the equivalent flexural rigidity EIeq can be considered constant for each interval.

A planar frame and its loading are shown in Fig. 3.1.9. The joints of the frame were considered rigid; but the individual members may be connected rigidly to the joints or through pin connections.



Figure 3.1.4: Plastification of a section. Figure 3.1.3: Material stress-strain diagram.

Figure 3.1.5: Yield stress and residual stress distribution in a rolled HEA section.

Figure 3.1.8: Distribution of plastic zones along member length.

Figure 3.1.6: Linear distribution of residual stresses.

Figure 3.1.7: Non-dimensionalized - - eqP M EI curves.



Two coordinate systems were used: a global coordinate system (Y Z ) is used for the structure in its entirety. In addition, a local coordinate system (YZ) is fixed to each bar. The origin of this system is arbitrarily located at one end of the bar (called End 1). The Z axis coincides with the longitudinal axis of that bar. Consider the bar Bb shown in Fig. 3.1.10 connecting joints Ai and Aj of a frame (say, bar B2 connecting joins A2 and A4 of Fig. 3.2.9). We assume that the loads cause certain parts of this bar to plastify, so that plastic hinges may eventually form at one or both ends. Four cases of connections at the member ends are to be considered at any level of loading, as shown in Fig. 3.1.10: (a) The bar is rigidly connected to joints at both ends, (b) the connection of End 2 to joint Aj is rigid; whereas the connection of End 1 to joint Ai is by a mechanical or plastic hinge, (c) the connection of End 1 to joint Ai is rigid; whereas the connection of End 2 to joint Aj is by a mechanical or plastic hinge, and (d) the connections of both ends of the bar are formed by mechanical or plastic hinges. Figure 3.1.11 shows the forces and deformations at the Ends 1 and 2 of the bar Bb in the deformed position of the frame. These forces and deformations are linked by the matrix relationships:

1 11 1 12 2 1 ( )

2 21 1 22 2 2 ( )

ˆ ˆ

ˆ ˆb b b b Q

b b b b Q

F R D R D F

F R D R D F

(3.6)

where 1 2,b bD D represent the deformation vectors at the ends of the bar Bb; 1 2,b bF F represent the force vectors; and 1 ( ) 2 ( ),b Q b QF F represent the influence of the transverse loads Q acting

on the bar. Further, 11 12 21 22ˆ ˆ ˆ ˆ, , , ,R R R R are the rigidity matrices of the bar in the local

coordinate system. We have:

1 1 1 1 2 2 2 2

1 1 1 1 2 2 2 2

b b b b b b b b

b b b b b b b b

F P V M F P V M

D w v D w v

(3.7)

Also, the first relation in Eq. (3.6) could be explicitly written as:

11 11

22 23 22 23 2

32 33 32 33 31 11 1 12 2 1 ( )

ˆ ˆ0 0 0 0 0

ˆ ˆ ˆ ˆ0 0

ˆ ˆ ˆ ˆ0 0b b b b Q

P r w r w

V r r v r r v f

M r r r r f

(3.8)

The elements of the matrices R may be identified with, and computed as the forces developing at the bar ends when one of the deformations 1 1 1, , ,b b bw v

2 2 2, , or b b bw v is in turn set equal to unity, while all the other deformations are set equal

to zero. External loads Q on the bar do not enter these calculations. The elements of the vectors 1 ( ) 2 ( ),b Q b QF F are the forces at the bar ends induced by the applied loads Q on the

bar, if all end deformations are prevented. However, axial loads P should be included in all these second-order calculations. The presence of any mechanical or plastic hinge must be taken into account in the above calculations. The generalized transfer matrix method described in Section 4 was used to determine the elements r of the partially plastified bars of the frame, at each cycle of calculations at each level of loading.



Figure 3.1.9: Planar frame and member sub-division into segments and intervals.

Figure 3.1.10: Member types based on end connections.

Figure 3.1.11: Forces at member ends in local coordinate system.

Figure 3.1.12: Forces at member ends in global coordinate system.

Figure 3.1.13: Equilibrium of forces acting at a joint.

Figure 3.1.14: Procedure to determine the ultimate strength of a steel frame.



The traditional matrix methods are then used to resolve the member end forces to the global coordinate system (Fig. 3.1.12) and then to write the joint equilibrium equations (Fig. 3.1.13) to obtain the matrix relation for the entire frame:

RD F (3.9)

where R is the stiffness matrix of the partially plastified frame at the level of loading considered, D is the deformation vector (unknown joint displacements and rotations) and F is the force vector (For details, see Vinnakota [1967, 1971]). The deformation vector is then obtained from the relation: 1D R F (3.10)

Knowing the joint deformations, defined by the vector D, the axial and shear forces, and the moments acting at the ends of each bar could be determined using Eq. (3.8). From these we determine the bending moments and normal forces at the ends of each interval, using the generalized transfer matrix method. These values yield, with the help of P˗ M˗

eqEI curves, the effective rigidities of the bar in each interval. Calculations for the joint

deformations of the frame are then repeated with the new rigidities. This process is repeated until the calculations converge. The iteration procedure used to determine the ultimate strength of a frame, u is graphically shown in Fig. 3.1.14. Let uc be the last converged value of , and ud (=

uc i ) be the next diverged value of . The calculations are repeated with decreasing values of load increment such that 100.i uc The ultimate strength was then taken as

2.uc ud

As mentioned earlier, the inelastic stability analysis method for planar steel frames described here includes the influence of residual stresses and spread of plasticity in the section (through the use of P˗ M˗ eqEI curves); spread of plasticity along the length

of the member, P moments, Pmoments, and transverse loads along the length of the member through the use of the generalized transfer matrix method; influence of the joint loads, influence of axial shortening of the members though the use of the displacement method (generalized slope-deflection method). Several portal and gable frames, for which earlier results were available, including some full scale test results, were analyzed using the computer program developed. Some of these examples are given in Sections 3.2 to 3.6. The size of the frames that could be analyzed was limited by the memory of the ZEBRA computer to that corresponding to a 12x12 matrix, for frame stiffness matrix R. The computer time for analysis of frames studied varied between 8 and 12 hours (single user). The computer program developed could not trace the unloading path of the load-deformation curve, and unloading of plastic hinges, if any, was not included.



3.2 Inelastic Stability of a Braced Hinged Base Portal Frame Studied by Ojalvo & Lu [1961]

A symmetric and symmetrically loaded hinged base portal frame ABCD having a span,

gL and a height, cL was considered by Ojalvo and Lu [1961]. The columns were of

30WF108 and the girder was of 30WF 132. The frame was subjected to gravity loads only,

consisting of concentrated loads, Q at the roof joints B and C, and a uniformly

distributed load, q on the girder BC, where is the load factor. A lateral restraint at

roof level was assumed to prevent sidesway motion of the frame. Failure occurs by excessive symmetrical bending deformations. The graphical approach used by Ojalvo and Lu was based on expressing the equilibrium of moments at a joint (B or C) using two moment-rotation curves developed for the beam-column and the beam. The stress-strain diagram for the rolled steel members was assumed to be ideally elastic perfectly plastic with a yield stress level of 33 ksi. A Lehigh type residual stress pattern with a maximum residual compressive stress of 0.3 yF was assumed. The study included the effect of the

spread of plasticity in the section and along the length of the member, P moments in the columns and residual stresses. Their study was limited to small, non-sway frames. The maximum value of O&Lu was estimated to be 4.11. The maximum value according to the

simple plastic theory pl was 4.94. The value obtained by Vinnakota [1967] SVu was

3.93. Table 3.2.1: Inelastic Stability of a Braced Hinged Base Portal Frame Studied by Ojalvo

& Lu [1961] Data: Hinged base portal frame

Columns: 30WF108 cL 59 ft

Girder: 30WF132 gL 59 ft

Yield stress: yF = 33 ksi

Lehigh type Residual stresses,

0.3rc yF

g2.5 2oQ qL q 1.0 kip/ft

Results: O&L

SV

First-order Plastic Hinge method: 4.94

Ojalvo and Lu [1961] Inelastic Stability Limit Load: 4.11

Vinnakota [1967] Inelastic Stability Limit Load: 3.93

pl

u

u



3.3 Inelastic Buckling of Three Unbraced Hinged Base Portal Frames Tested by Yen, Lu and Driscoll [1962]

Three sets of symmetric and symmetrically loaded hinged base welded rectangular portal frames having a span length ( gL ) of 88 in. and heights ( cL ) of 44, 66 and 88 in.,

respectively, were tested to failure [Yen, Lu and Driscoll 1962]. The columns and the girder were of M2362 (a small 2 ⅝ × 1½ in.) wide flange shape. The frame was subjected to gravity loads only, consisting of concentrated loads, Q at the roof joints B and C, and three concentrated loads, Q acting at the third points of the girder. Here is the load

factor, 1.00 kipQ and 3 2 .oQ n Q Thus, for 2,n the frame could be considered as

the first story of a three-story building. The ultimate loads u TEST obtained from the tests,

the inelastic buckling loads Luu calculated by Lu [Lu, 1960] and the inelastic stability

limit loads SVu given by Vinnakota [1967] are given in Table 3.3.1. Lu’s method was

based on the modified moment distribution method [Winter, Hsu, Koo and Loh, 1948] in which the member stiffnesses were modified for the effects of axial force present in the member at a given load level and the effect of yielding along the member length.

Table 3.3.1: Inelastic Buckling of Unbraced Hinged Base Portal Frames Tested by Yen, Lu & Driscoll [1962]

Data: Hinged base portal frames (3) g 80 87.6 in.xL r

Section: M-2362 Wide flange section. yF 42.7 ksi (average value)

h 2.625 in.; fb 1.813 in.

ft 0.207 in.; wt 0.156 in.

xr 1.095 in.; xI 1.251 in.4

Loading: 3 2Q n Q and 1 kipQ

c

x

L

r cL

in. n pl

TESTu

Test. Yen et al. [1962]

Luu

Inelastic buckling Lu [1965]

SVu

Inelastic stabilityVinnakota [1967]

W-1 40 43.8 2.0 2.76 2.475 2.36 2.30

W-2 60 65.7 2.0 2.76 2.253 2.26 2.25

W-3 80 87.6 1.8 2.72 2.182 2.05 2.02



3.4 Inelastic Stability of an Unbraced Hinged Base Portal Frame Studied by Yura & Galambos [1965]

Charts to determine the inelastic stability limit load of unbraced hinged base portal frames ABCD were presented by Yura and Galambos [1965]. The column height was cL

and the girder length was g.L The frames were subjected to gravity loads, Q which were

kept constant, acting at joints B and C. In addition, the frames were subjected to a horizontal load, H at joint C, where 1.0 kip,H and is the load factor. Their graphical method makes use of the nomographs for beam-columns developed earlier by Ojalvo and Fukumoto [1962], for 8WF31columns of A7 steel ( E 29,000 ksi, yF 33 ksi)

having Lehigh type residual stresses. Yura and Galambos assumed that the beam BC behaves elastically, that ,H Q and that the horizontal reactions were 2.H

Table 3.4.1: Inelastic Stability of an Unbraced Portal Frame Studied by Yura & Galambos [1965]

Data: Unbraced Pined Base Portal Frame:

Columns: c 40 xL r

Girder: g 120 xL r

Section: 8WF31

Material: A7 steel 33 ksiyF

Residual stresses: Lehigh type Loading:

0.3oyQ P H 1.0 kip

Results: Y&G

SV

7.00 by Yura and Galambos [1965]

6.88 by Vinnakota [1967]

u

u

Inelastic Stability Analysis


3.5 Inelastic Stability of a Fixed Base Gable Frame Tested by Baker and Eickhoff [1956]

Baker and Eickhoff [1956] conducted a full scale test on a symmetric, fixed base, gable frame ABDFG constructed throughout of 7 4 16 lb. per foot Rolled Steel Joist. The frame had a span ,L a column height c ,L and a roof slope of 1

2( 22 ) . It was

subjected to a horizontal load ( 1.70 tons)H at joint F that was kept constant, and a

vertical load Q at joint D. Here, 1.0 tonQ and is the load factor. The vertical load was increased gradually until collapse occurred at a vertical load of 14.20 tons. The plastic limit load using first-order plastic hinge method, assuming plastic hinges to form at B, D, F and G was found to be 13.36. Inelastic stability analysis by Vinnakota [1967]



resulted in a value of 12.80. The difference between u TEST and pl was attributed to the

strengthening of the frame due to the influence of strain hardening at the four plastic hinges that formed at B, D, F and G.

Table 3.5.1: Inelastic Stability of a Fixed Base Gable Frame Tested by Baker and Eickhoff [1956]

Data: Fixed Base Gable Frame:

Column height: c 8 0L

Span 16 0L

Rafter inclination: 1222

5

Section: 7 4 16 lb/ft RSJ

247 ton-in. 5.73 10 ton-in.pM EI

H 1.70 tons and 1.0 tonQ

Results:

TEST

SV

14.20 by Baker and Eickhoff [1956]

13.36 - Baker and Eickhoff

12.80 by Vinnakota [1967]

u

pl

u

Full Scale Test

First order Plastic Hinge Method


3.6 Inelastic Stability of an Unbraced Fixed Base Hybrid Portal Frame Tested by Adams [1966]

A full-scale test on an unbraced, fixed-base, hybrid portal frame ABCD was reported by Adams [1966] The columns were 5WF18.5 ( HEB200 sections of A441 high strength steel ( yF = 50 ksi) while the beam was a 10I25.4 ( IPER 200) of A36 low-carbon structural

steel ( yF = 36 ksi). The frame was subjected to concentrated vertical loads Q at the

quarter points of the girder, and to concentrated vertical loads Q (= 3Q ) over the column tops B and C. These vertical loads were held constant, during the test at a value which produced an axial load in the columns of 0.26 .yP The sway deformation was produced by

a horizontal load, ( ),H H applied at the mid-depth of the girder at joint C, with

1.00H (metric ton). The frame failed in a combined mechanism, forming two hinges in the leeward column, one at the base of the windward column and one in the beam at the windward load point.

The experimental results are shown in Fig. 3.6.1 by the solid curve. The data points marked “x” in this figure were those predicted by Vinnakota [1971, 1974c]. The



inelastic stability analysis included Pmoments, P moments, reduction in stiffness due to the spread of plasticity in the section and along the length of members, and the influence of Lehigh type residual stresses.

Table 3.6.1: Inelastic Stability of an Unbraced Fixed Base Hybrid Portal Frame Tested by Adams [1966]

Fixed Base Portal Frame:

Column height: c 265 cmL

Span 456 cmgL α = 0.25

Column: 5WF18.5

A441 Steel 50 ksiyF

Girder: 10I25.4

A36 Steel 36ksiyF

9.0 tonQ , H H

Figure 3.6.1: Load versus sway deflection of a hybrid portal frame tested by Adams [1966].



4. Inelastic Planar Stability of Beam-Columns. ZEBRA Computer at EPUL 1964-65 and IBM 7040 Computer 1966-67.

Equation Section (Next) This section summarizes the results of research at EPUL on the inelastic behavior of restrained steel beam-columns. A generalized transfer matrix method was used for the numerical studies to solve the problem on the ZEBRA computer. Spread of plasticity in the cross section and along the length of the member, residual stresses, P and Pmoments and accidental load eccentricity were all included in the analysis.

4.1 Inelastic Stability of Rotationally and/or Directionally Restrained Beam-Columns [Vinnakota, 1967; Vinnakota and Badoux, 1970a, 1971]

Columns in framed structures are connected at their ends to beams and other columns. These adjacent members provide rotational and translational restraints to the columns (Fig. 4.1.1). In this section, a generalized transfer matrix method developed by Vinnakota in 1965, to study the inelastic stability of restrained I-section steel beam-columns, is described and several numerical examples studied to validate the method are given. Consider the initially straight beam-column of length, L shown in Fig. 4.1.2. A rectangular coordinate system OYZ was chosen such that the origin O coincides with the End 1 of the member. The Z- axis coincides with the longitudinal axis of the member in its unloaded position, with its positive sense directed toward End 2. The positive sense of the Y-axis is directed upward. The plane OYZ lies in a principal plane of the section. The member is subjected to an axial thrust .P Let 1 1and x xV M represent the transverse load and moment acting at End 1; while 2 2 and x xV M represent the corresponding force/moment acting at End 2. In addition, the member is subjected to concentrated transverse loads ,Q concentrated moments ,C and distributed transverse loads, .q The restraints, loads, moments, and the deformations all lie in one and the same plane. As shown in Fig. 4.1.3, the load-deformation response of the rotational and directional restraints is considered to be linearly elastic-perfectly plastic. In this figure, rrk and rpM

represent the stiffness and the plastic moment, respectively, offered by the rotational restraint. Similarly, drk and rpV represent the stiffness and the maximum force,

respectively, offered by the directional restraint. In regions where the member enters the inelastic range, the member is subdivided into intervals so that the bending moment and hence the equivalent rigidity, ,eqEI can be

considered constant for each interval. Starting from End 1 of the bar, the intervals are numbered successively as 1, 2, … i, i+1, … n (Fig. 4.1.4). The junctions of the intervals are numbered 1, 2, 3, … j, …n+1, the junction 1 coinciding with End 1 of the bar, and the junction n+1 coinciding with End 2 of the bar (we have j = i +1). In addition, the left end of each interval is given the subscript l and the right end the subscript r. In the development of the theory, lateral loads and deflections in the positive Y-direction are considered positive. Also, external moments and slopes are considered positive when counter-clockwise. So, all the forces and all the deformations shown in Fig. 4.1.2, are positive.



.

Figure 4.1.1: Restrained beam-column as part of an unbraced frame.

Figure 4.1.2: Rotationally and directionally restrained beam-column

with transverse loads.

Figure 4.1.3: Load-deformation response of rotational and

directional restraints.

Figure 4.1.4: Designations of elements, i; junctions, j; and Ends 1

and 2 of a bar.

Figure 4.1.5: Designation, ns, for type of end-support present, at level

of loading considered.



Let iT be the transfer matrix connecting the state vector ,i rS at the right end of

interval i, with the state vector ,i lS at its left end. Note that the displacement ;v slope ;

shear force ;V and moment M are the four elements of the state vector at a point. We have: , ,i r i i lS T S (4.1)

or, explicitly, in the case of interval i with a uniformly distributed transverse load of intensity q over its entire length, we have [Vinnakota, 1967]:

2 3 4 2

2 3 4

, 2 3

, 2 3

,

2,

sin 1 cos sin 2 2cos1

sin 1 cos sin0 cos

sin sin 1 c0 cos

1

i i i i i i i i ii

i eqi i eqi i eqi i

i r

i i i i i i ii r i

eqi i eqi i eqi ii r

i ii r i i i i i

i i

L L qLL

EI EI EIv

L L qL

EI EI EIM

V PL L qL

,

,

,

,2

os

10 0 0 1

0 0 0 0 1

i l

i l

i l

ii l

i

i

v

M

V

qL

(4.2)

where 2 2 .i i i eqiPL EI Note that these relations are derived by writing the equilibrium of

the element in its deformed position. Let ,j jQ C be the transverse load and external

moment , if any, acting at the junction j of intervals i and i+1. Let ,i rS be the state vector

at the right end of interval i, and 1,i lS the state vector at the left end of interval i+1. We

have: 1, , ,i l j i r j i i lS G S G T S (4.3)

The matrix jG could be written explicitly as:

1 0 0 0 0

0 1 0 0 0

0 0 1 0

0 0 0 1

0 0 0 0 1

jj

j

CG

Q

(4.4)

Equation (4.3) relates the state vector at the left end of the interval i+1 to the state vector at the left end of interval i and the external forces, if any, acting over that portion of the bar. By extension, we can write:

, 1 2 1 1,

2 1

... ... or simply, n r n n i j i lS T G T G T G T S

S B S

(4.5)



The matrix B is a transfer matrix connecting the state vector 1S at the left end of interval

i = 1 (i.e., End 1 of the beam-column) to the state vector 2S at the right end of interval i =

n (i.e., End 2 of the beam-column). We have: 1 2 1 , 2 1, 1... ... ; ; n n i j i n r lB T G T G T G T S S S S (4.6)

Equation (4.5) could be rewritten as:

2 11 12 13 14 15 1

2 21 22 23 24 25 1

2 31 32 33 34 35 1

2 41 42 43 44 45 1

1 0 0 0 0 1 1

v b b b b b v

b b b b b

M b b b b b M

V b b b b b V

(4.7)

Column 2 of Table 4.1.1 shows the 9 types of supports encountered in practice, each of which is given an identification number designated by sn (Fig. 4.1.5 and Column

1 of Table 4.1.1). In any given problem the existing support conditions at each end give two elements of the state vector or two relations between the elements of the state vector at that end. Corresponding to the nine types of supports at End 1, the two unknown elements 1 2,X X and the two known elements (or known relations) in state vector 1S are

identified in Cols. 3 to 6 of this table. Similarly, corresponding to the nine types of supports at End 2, the two unknown elements 3 4,X X and the two known elements (or

known relations) in state vector 2S are identified in Cols. 8 to 11 of Table 4.1.1.

In any given problem, Eq. (4.7) can be rearranged by making use of the existing boundary conditions, so as to separate the four unknowns. All such equations, corresponding to any combination of boundary conditions in Cols. 2 and 7 of Table 4.1.1, can be represented by a single relation given below:

15 25 2 29 213 13 12 12 14 14 11 11 21 1

2513 23 12 22 14 24 11 21 22 2

3513 33 12 32 14 34 11 31 23 3

4513 43 12 42 14 44 11 41 24 4

0

0

0

0

x xb v Vb b b b X

bb b b b X

bb b b b X

bb b b b X

26 2 2,10 2

27 2

28 2

15 11 1 16 12 1 18 14 19 11 1 17 13 1,10 12 1

15 21 1 16 22 1 18 24 19 21 1 17 23 1,10 22 1

15 31 1 16 32 1 18 34 19 31 1 17 33

x x

x

x

x x x x

x x x x

x x x

M

M

V

b v b b b V b b M

b v b b b V b b M

b v b b b V b

1,10 32 1

15 41 1 16 42 1 18 34 19 41 1 17 43 1,10 42 1

x

x x x x

b M

b v b b b V b b M

(4.8)



There are in all twenty coefficients , ten corresponding to each end. (Note: the first subscript of a coefficient represents the end to which the coefficient is associated.) The value of a coefficient depends upon the type of support at the end to which it is associated and can be obtained with the help of Table 4.1.2. Equation (4.8) can be rewritten in the form: DX F (4.9) where 1 2 3 4, , ,X X X X X (4.10)

The solution of Eq. (4.9) is given by the relation:

1X D F (4.11) where 1D is the inverse of the matrix D . Knowing X and hence 1 2, ,X X the state vector

1S can be written with the help of Table 4.1.1. Once 1S is known, Eqs. (4.1) and (4.3) are

used to find the state vector at each end of each interval i of the beam-column (i = 1 to n). Using the axial force and the average bending moment in each interval, the equivalent flexural stiffness eqiEI is obtained from the appropriate - - eqP M EI curves of Fig. 3.1.7.

Table 4.1.1: Identification of known and unknown elements of state vectors at End 1 and

End 2 of a bar based on the type of end supports.

sn

End 1

v1

1

M1

V1

End 2

v2

2

M 2

V2

1 2 3 4 5 6 7 8 9 10 11

1

vx 1 X1 M x 1 X2 vx 2 X3 M x 2 X4

2

vx 1 x 1 X1 X2

vx 2 x 2 X3 X4

3

vx 1 ♣ X1 X2 vx 2 ♣ X3 X4

4

♠ X1 M x 1 X2 ♠ X3 M x 2 X4

5

♠ x 1 X1 X2

♠ x 2 X3 X4

6

♠ ♣ X1 X2 ♠ ♣ X3 X4

7 X2 X1 M x 1 Vx 1 X4 X3 M x 2 Vx 2

8

X2 x 1 X1 Vx 1

X4 x 2 X3 Vx 2

9

X2 ♣ X1 Vx 1 X4 ♣ X3 Vx 2

- ♠ Calculated from the relation ( ) /J xJ J drJv V V k after obtaining JV from Eq. (4.11) - ♣ Calculated from the relation ( ) /J xJ J rrJM M k after obtaining JM from Eq. (4.11)



Table 4.1.2: Values of the coefficients in matrix D , as a function of the type of support at each end of the member.

sn

Support

1J

2J

3J

4J

5J

6J

7J

8J

J9

10J

1 2 3 4 5 6 7 8 9 10 11 12

1

0 1 0 1 1 0 1 0 0 0

2

0 0 1 1 1 1 0 0 0 0

3

0 1

rr Jk 1 1 1 0 0 0 0

1

rr Jk

4

1

dr Jk 1 0 1 0 0 1 0

1

dr Jk 0

5

1

dr Jk 0 1 1 0 1 0 0

1

dr Jk 0

6

1

dr Jk

1

rr Jk 1 1 0 0 0 0

1

dr Jk

1

rr Jk

7

1 1 0 0 0 0 1 1 0 0

8

1 0 1 0 0 1 0 1 0 0

9 1 1

rr Jk 1 0 0 0 0 1 0

1

rr Jk

- The values of are obtained from this table corresponding to the type of support at each end. - For End 1 use J = 1. For End 2 use J = 2.

The unknown vector X, given by Eq.(4.11) , is determined again and the process

repeated till convergence occurs. The iteration procedure used to determine the ultimate strength u of a beam-column is similar to that shown in Fig. 3.1.14 for a frame [For details see Vinnakota, 1967].

The computer program developed, first on the ZEBRA computer using simple code and then updated to the IBM 7040 computer using FORTAN language, permitted the inelastic stability analyses of laterally loaded restrained beam-columns having any combination of the nine types of end support conditions usually encountered in practice. Consideration of these nine types of support conditions permitted one to take into consideration the changes in the end conditions that occur due to the formation of a plastic hinge in the member or the plastification of an end restraint. This program (as a subroutine) was used to determine the stiffness matrix elements, r of partially plastified members of frames studied in Section 3. (See Eq. (3.8).)



4.2 Planar Strength of Pin-Ended Beam-Columns Studied by Galambos & Ketter [1959]; Ojalvo & Fukumoto [1962]

Galambos and Ketter [1959] presented a procedure to determine the maximum planar strength of pin-ended rolled-steel beam-columns subject to end moment(s) using Newmark’s numerical integration method. In one example of a pin-ended column of length ( 40 )xL r subjected to an axial compressive force ( 0.80 )yP P and a moment at

one end only, they determined the maximum value of this end moment to be 0.233 .yM

The strength of this beam-column was determined by Vinnakota [1967] using the generalized transfer matrix method and the results obtained are reproduced in Table 4.2.1.

Table 4.2.1: Planar Strength of Pin-Ended Beam-Columns

Reference Data uγ

Reference Vinnakota [1967]

I Galambos

& Ketter [1959]

1 2

40.0 0.80

0.0 0.05x y

e e y

L r P P

M M M

γ

4.66 4.67

IIa

Ojalvo & Fukumoto

[1962]

1 2

136.5 1.0 kip

0.60 0.30x

e y e y

L r P

M M M M

90.3 88.6

IIb

1 2

100.0 1.0 kip

0.00 0.712x

e e y

L r P

M M M

90.3 87.2

Section: 8WF31 A7 Steel: 33 ksiyF E 30,000 ksi Bending about major axis

Residual Stresses: Lehigh type with 0.3rc yF

xr radius of gyration of the section for bending about x-axis

yM yield moment of the section for bending about x-axis

yP yield load of the section

Ojalvo and Fukumoto [1962] presented a graphical method to determine the load-

deformation behavior of beam-columns using nomographs developed by Ojalvo [1960]. In one example, Ojalvo and Fukumoto determined that the maximum length of a pin-ended rolled-steel wide-flange beam-column in stable equilibrium with an axial



compressive force 0.3 yP P and unequal end moments 0.60 and 0.3y yM M producing

single curvature bending, to be 136.5 .xr In another example, Ojalvo and Fukumoto

determined that a pin-ended rolled steel wide-flange beam-column of length ( 100 )xL rsubject to an axial compressive force 0.3 yP P and a moment at one end only can carry

a maximum end moment of 0.712 .yM These two examples were later studied by

Vinnakota [1967], wherein the axial compressive load was increased proportionally while the moments were kept constant, using the generalized transfer matrix method. The results obtained are summarized in Table 4.2.1. The three studies referred to above all include the influence of second-order effects, spread of plasticity in the section and along the length of the member and Lehigh-type residual stresses.

4.3 Planar Strength of Rotationally Restrained Beam-Columns. Levi, Driscoll & Lu [1965]; Ketter and Beedle [1955]

Levi, Driscoll and Lu [1965] presented a procedure to determine the maximum planar strength of rotationally restrained rolled-steel beam-columns subjected to axial compression and to any combination of end moments. It was essentially a graphical method and depended on the availability of charts developed by Levi [1962]. As an example, Levi et al. have studied a rotationally restrained beam-column having a slenderness ratio of 30 subjected to a constant axial force 0.80 ,yP P and two equal and

opposite external end moments producing symmetric single curvature bending. The linearly-elastic end rotational-restraints had a stiffness 12.5 ,rr yk M where yM is the

yield moment of the cross section. Using their graphical procedure Levi et al. have determined the maximum value of the external end moments to be 0.285 .yM This beam-

column was later studied by Vinnakota [1967] using the generalized transfer matrix method, keeping the external end moments constant at a value of 0.285 ,yM and

proportionally increasing the axial load to failure. The results are shown in Table 4.3.1. Bijlaard, Fisher and Winter [1955] presented test results on columns provided with equal elastic end rotational-restraints with various end eccentricities producing symmetric single curvature bending. The small 4I9.5 steel columns were bent about their minor axis. In a discussion to this paper, Ketter and Beedle [1955] presented two sets of design curves for maximum strength of elastically restrained wide flange columns bent about their major axis under symmetric single curvature applied moments. Separate curves were given for 4rrk EI L and 6 ,EI L for different slenderness ratios (L/rx) and axial strength ratios (P/Py). Vinnakota studied four 8WF31columns (one of L/rx = 50 and three of L/rx = 100) provided with elastic end rotational-restraints having stiffness,

4rrk EI L ) subjected to various levels of end moments. The end moments were applied first and held constant, while the axial thrust was increased proportionally. The results from Vinnakota [1967] are summarized in Table 4.3.1.



Table 4.3.1: Planar Strength of Restrained Beam-Columns

Reference Data

uγ

Reference Vinnakota [1967]

I Levi, Driscoll & Lu [1965]

12.5rr yk M

* *1 2

30.0 1.00 kip

0.285 0.285

x

e y e y

L r P

M M M M

240.8

( )= 0.8u y

P P

233.0

IIa

Ketter & Beedle [1955]

4rrk EI L

* *1 2

50.0 1.00 kip

0.645 0.645

x

e y e y

L r P

M M M M

240.8

( )= 0.8u y

P P

238.5

IIb * *1 2

100.0 1.00 kip

0.290 0.290

x

e y e y

L r P

M M M M

240.8

( )= 0.8u y

P P

237.4

IIc * *1 2

100.0 1.00 kip

0.740 0.740

x

e y e y

L r P

M M M M

180.6

( )= 0.6u y

P P

184.6

IId 1 2

100.0 1.00 kip

1.460 1.460x

e y e y

L r P

M M M M

120.4

( )= 0.4u y

P P

119.6

Section: 8WF31. Residual Stresses: Lehigh type with 0.3rc yF

rrk Stiffness of the end rotational restraint

4.4 Full-Scale Beam-Columns Tested by Van Kuren & Galambos [1964]

Van Kuren and Galambos [1964] reported tests on full-scale, as-rolled, wide-flange beam columns conducted at Lehigh University to study their strength and deformation behavior in the inelastic range. A total of 37 experiments were performed. Of those, ten tests were made on laterally supported members (A-series tests) and twenty-seven tests were made on laterally unsupported members (T-series tests). In each test the axial force and the end moment(s) were applied independently of each other and testing was continued until after the member failed. For the majority of the tests a predetermined axial force was first applied to the member. This axial force was kept constant while the end moment(s) were increased from zero to their maximum value(s). This process of loading was reversed for



Tests T-8 and T-22, in which the moment was held constant and the axial load was varied from zero to its maximum value. In most of the tests, bending was about the strong axis of the member. Exceptions to this were Tests T-25 and T-27 (not reported here), in which bending was about the weak axis of the wide flange shape.

The beam-columns were pin-ended in the plane of the applied moments and essentially fixed in the plane perpendicular to the plane of bending. Most of the experiments were on column shapes (8WF31, 4WF13) and a beam shape (8B13). Column lengths varied from 6 ft to 16 ft, giving slenderness ratios ranging from 20 to 112. Four different combinations of end moments designated as conditions “a,” “b,” “c,” and “d” were used in the tests. Loading condition “a” was that in which two equal end moments were applied in the same direction, creating double curvature deformation. In loading condition “b,” the two end moments were applied in such a way that the slope at one end was zero. This loading condition simulates a fixed-end column. For condition “c” two equal end moments were applied in opposite directions, resulting in symmetric single curvature deformation. Loading condition “d” involved moment applied at one end only. It was observed that failure of the test specimens was due to one or a combination of several of the following phenomena: (1) Excessive deflection in the plane of bending as a result of yielding (FB), (2) lateral-torsional buckling (LTB), and (3) local buckling (PLB). With the exception of two tests, all of the braced beam-columns failed by excessive deflection. The majority of the unbraced specimens tested failed by lateral-torsional buckling. The in-plane strength of all of the tested columns were determined by Vinnakota [1967] using the generalized transfer matrix method and these results along with test results are shown in Table 4.4.1 for case “d” tests, in Table 4.4.2 for case “a” tests, in Table 4.4.3 for case “b” tests, and in Table 4.4.4 for case “c” tests. In these tables, G&KuM

represents the ultimate strength of the column using the procedure given in Galambos and Ketter [1959]; Py represents the yield load of the column cross section and Mpx the plastic moment of the column cross section for bending about its major axis. The end moment versus end slope curves for Tests A-7 and T-31 are shown in Figure 4.4.1. Both specimens were 16-ft-long 4WF13 members and both were subject to case “d” loading. The axial force was approximately the same on each column. Test A-7 was provided with lateral bracing, whereas Test T-31 was not braced. The end moment vs. end slope curves for Tests T-13 (8WF31) and A-9 (8B13) are shown in Fig. 4.4.2. The condition of loading and the axial force ratio were equal for both columns with only a slight difference in slenderness ratio. The 8B13 specimen, having less resistance to lateral-torsional and local buckling, failed earlier. Also shown in these two figures were the results obtained by Vinnakota [1967].



Table 4.4.1: Pin-ended Columns under Moment at One End (Case d) Tested by Van Kuren and Galambos [1964]

Shape x

L

r

lbn y

P

P

u u pxM M M SV

TEST

u

u

M

M

Failure

Mode TESTuM G&KuM SVuM

1 A-1 4WF13 84.0 2 0.33 0.73 0.66 0.63 87 % FB

2 A-2 8WF31 55.0 1 0.65 0.37 0.35 0.34 93 % FB

3 A-3 8WF31 55.0 1 0.33 0.81 0.76 0.76 94 % FB

4 A-4 8WF31 55.0 1 0.49 0.60 0.55 0.55 92 % FB

5 A-5 8WF31 110.0 2 0.33 0.47 0.48 0.45 97 % FB

6 A-6 4WF13 112.0 2 0.50 0.14 0.13 0.13 93 % FB

7 A-7 4WF13 112.0 2 0.16 0.88 0.87 0.84 95 % FB

8 A-8 8B13 52.0 3 0.30 0.78 0.77 0.79 101 % LTB

9 A-9 8B13 52.0 3 0.12 0.96 0.96 0.94 97 % LTB+PLB

10 A-10 4WF13 52.0 3 0.60 0.46 0.40 0.40 87 % FB

11 T-1 8WF31 21.0 0 0.13 1.03 0.96 0.94 91 % FB

12 T-2 8WF40 20.0 0 0.15 1.13 0.94 0.94 83 % FB

13 T-13 8WF31 55.0 0 0.12 1.02 0.96 0.95 93 % FB+PLB

14 T-23 4WF13 83.0 0 0.11 0.93 0.96 0.94 101 % LTB

15 T-31 4WF13 112.0 0 0.12 0.84 0.88 0.89 107 % LTB

lbn number of lateral braces

TABLE 4.4.2: Pin-Ended Columns under Anti-symmetric End moments (Case a) Tested by Van Kuren and Galambos [1964].

Shape

x

L

r lbn

y

P

P

u u pxM M M SV

TEST

u

u

M

M

Failure


16 T-14 8WF31 55.0 0 0.23 0.90 0.87 0.89 99 % FB+PLB

17 T-29 4WF13 84.0 0 0.12 1.12 0.95 0.95 85 % LTB

18 T-30 4WF13 112.0 0 0.12 0.97 0.95 0.94 97 % LTB



TABLE 4.4.3: Fixed-Pinned Beam-Columns (Case b) Tested by Van Kuren and Galambos [1964].

Shape x

L

r lbn

y

P

P

u u pxM M M SV

TEST

u

u

M

M

Failure


19 T-3 8WF31 56.0 0 0.50 0.59 0.56 0.59 100 % LTB+PLB

20 T-4 8WF31 56.0 0 0.12 0.95 0.95 0.94 99 % FB+PLB

21 T-9 4WF13 111.0 0 0.10 0.87 0.95 0.95 109 % LTB

22 T-17 4WF13 56.0 0 0.12 0.86 0.96 0.95 110 % FB+PLB

23 T-22 4WF13 56.0 0 (0.61) 0.38 (0.69) (0.66) 109 % LTB

24 T-24 4WF13 83.0 0 0.12 1.05 0.96 0.96 92 % LTB+PLB

25 T-6 4WF13 112.0 0 0.27 0.59 0.70 0.76 127 % LTB

Table 4.4.4: Pin-Ended Columns under Equal End Moments Applied in Opposite Directions (Case c) Tested by Van Kuren & Galambos [1964]

Shape x

L

r

lbny

P

P

u u pxM M M SV

TEST

u

u

M

M

Failure

Mode TESTuM TGuM SVuM

26 T-8 8WF31 55.0 0 (0.59) 0.16 0.25 (0.69) 117 % LTB

27 T-12 8WF31 55.0 0 0.12 0.76 0.76 0.80 105 % LTB

28 T-16 8WF31 41.0 0 0.12 0.75 0.75 0.83 110 % LTB

29 T-19 8WF31 28.0 0 0.12 0.78 0.79 0.88 113 % LTB

30 T-32 4WF13 112.0 0 0.12 0.64 0.62 0.60 93 % LTB



Sectio

n x

L

r

y

P

P

Van Kuren & Galambos [1964]

Vinnakota [1967]

TESTuM G&KuM Failure Mode SVuM

A-7 4WF13 112 0.158 0.88 0.87 FB 0.84

T-31 4WF13 112 0.122 0.84 0.88 LTB 0.89

Figure 4.4.1: End moment vs. end rotation curves for Tests A-7 and T-31.

Section x

L

r

y

P

P

Van Kuren & Galambos [1964]

Vinnakota [1967]

TESTuM G&KuM Failure Mode SVuM

A-9 8B13 52 0.12 0.96 0.96 LTB+PLB 0.94

T-13 8WF31 55 0.12 1.02 0.96 FB+PLB 0.95

Figure 4.4.2: End moment vs. end rotation curves for Tests A-9 and T-13.



4.5 Inelastic Stability of Rotationally Restrained Beam-Columns Free to Sway. Driscoll et al. [1965]

The results for two examples of rotationally restrained beam-columns in unbraced frames, studied by Driscoll et al. [1965] and analyzed by Vinnakota [1967], are shown in Figs. 4.5.1 and 4.5.2. A restrained column in an unbraced frame differs from a restrained column in a braced frame, in that in the former the top of the column deflects laterally with respect to its bottom, and thus an additional moment, ,P is introduced in the column. Driscoll et al. made use of design charts developed by Parikh et al. [1965] to determine the load-deformation response of such restrained beam-columns. The two columns considered here were of length L ; free to sway at the top and provided with rotational restraints at both ends. Restraint stiffnesses are 1 2 and rr rrk k at Ends 1 and 2,

respectively. Also, Mpcx represents the plastic moment of the column cross section for bending about its major axis, in the presence of the axial load, P.

Section: 8WF31 36 ksiyF 0.3rc yF

Column length, L 30 xr

Type of supports at End 1 and End 2 (see Table 4.1.1): 1 3sn (Rotationally restrained. Displacement zero.)

2 9sn (Rotationally restrained. Free to sway )

Restraint stiffness: 1 50rr pcxk M 2 25rr pcxk M

Applied loads: Axial load, 0.6 yP P

End moments: *1 0.4e pcxM M *

2 0.1e pcxM M

Figure 4.5.1: Load-deformation response of a rotationally restrained, sway permitted

beam-column. Driscoll et al. [1965].



In the first example (shown in Fig. 4.5.1) the axial thrust, P (= 0.6Py) and external moment, *

1eM (= 0.4 Mpcx) at End 1 of the column were kept constant, while the external moment at End 2 was increased proportionally.

In the second example (shown in Fig. 4.5.2) the column was subjected to an axial thrust P (= 0.6Py), external moment *

1eM (= 0.4 Mpcx) at End 1, and to a transverse load Q (= 0.002P) at End 2. All these forces were kept constant, while the external moment at End 2 was increased proportionally.




2 9sn (Rotationally restrained. Free to sway. )

Restraint stiffness: 1 50rr pck M 2 25rr pck M


Transverse load at End 2: 0.002Q P

End moments: *1 0.4e pcM M *

2 0.1e pcM M

Figure 4.5.2: Load-deformation response of a rotationally restrained, sway permitted




4.6 Inelastic Stability of a Rotationally and Directionally Restrained Beam-Column.

Driscoll et al. [1965]

The results for a rotationally and directionally restrained beam-column studied by Driscoll et al. [1965] and analyzed by Vinnakota [1967] are shown in Fig. 4.6.1. The column considered was of length ,L free to sway at the top, and provided with rotational restraints at both ends. Restraint stiffnesses, 1 2 and rr rrk k correspond to Ends 1 and 2,

respectively. In addition, the column was provided with a directional restraint of stiffness, .drk The axial thrust P (= 0.6Py) and external moment *

1eM (= 0.4 Mpcx) at End 1 were kept

constant, while the external moment at End 2 was increased proportionally.




2 4sn (Rotationally free. Sway restrained.)

Restraint stiffness: 1 50rr pck M 2drk P L


End moments: *1 0.4e pcM M *

2 0.1e pcM M

Figure 4.6.1: Load-deformation response of a rotationally and directionally restrained,




5 Inelastic Spatial Stability of Beam-Columns. CDC 7326 Computer at EPFL 1972-

77

Equation Section (Next) In addition to an axial load, corner columns in building structures are subjected, to bending moments acting in two perpendicular directions (Fig. 5.1a). Also, most interior columns in modern high-rise structures are often left unsupported (unbraced) between floor levels for architectural reasons. Space action of the entire three-dimensional framing system, introduces biaxial bending moments into these columns in addition to axial thrust. The worst possible bending condition (single curvature) occurs when an unsymmetrical live loading condition (Fig. 5.1b) or a checkerboard live loading condition are applied in the vicinity of a column. The thin-walled open-section steel members (such as wide flange shapes) generally used as columns in such structures are susceptible to torsional loads and their ultimate strength and behavior under combined axial thrust, biaxial bending, and torsion is therefore of utmost practical importance.

This section summarizes the results of a research program undertaken at the Institute of Steel Structures (Director: Prof. J. C. Badoux) at EPFL during the years 1972-77 on the behavior (Fig. 5.1c) and inelastic spatial stability of rotationally and directionally restrained crooked steel beam-columns. The finite difference method, CDC 7326 computer and double precision calculations were used for the numerical studies. Spread of plasticity in the cross section and along the length of the member, residual stresses, initial geometrical imperfections in space, and end restraints were all included in the study.

Figure 5.1: Biaxially bent beam-columns.



5.1 Inelastic Spatial Stability of Rotationally and Directionally Restrained Imperfect Beam-Columns. Vinnakota [1973, 1974, 1977b], Vinnakota & Aoshima [1974a, b], Vinnakota & Aysto [1974]

A beam-column of an arbitrary open cross section of length L is shown in Figs. 5.1.1 and 5.1.2. The member is provided with two rotational restraints at each end. End displacements and end twist are prevented. Warping at the ends could be either free to occur or completely restrained. The external loads acting on the beam-column are shown in Fig. 5.1.1, while the deformed configuration is indicated in Fig. 5.1.3. In the development of the theory it was assumed that the cross section retains its original shape during the deformation. The stress-strain diagram of the material was considered ideally elastic-perfectly plastic. Strain hardening, strain reversal, and shearing strain were neglected. It was also assumed that yielding is governed by normal stress only and that loads are conservative. The reference axes X Y Z correspond to a stationary right-handed rectangular coordinate system. The Z axis is directed along the member length and passes through an arbitrary point C of the cross section at the left end. At each section, a local coordinate systems ,x yz fixed to that cross section and deforming with it, was considered. Its origin coincides with the reference point C of the cross section at z and, in the unloaded position, the xz and yz planes coincide with the X Z and Y Z planes, respectively. A second local coordinate system ,X Y Z whose origin also coincides with the reference point C of the cross section at z and whose axes are always parallel to the X Y Z axes (Fig. 5.1.3), is also considered in the analysis. Let S (x, y) be a general point on the middle line of the cross section. The position of the point S can also be defined by its distance s measured from an arbitrary point D of the cross section (Fig. 5.1.2). Let A A A( , )x y denote an

arbitrary point of the cross section taken as the pole of the sectorial coordinates. Also, let

AD be the warping function or double sectorial area of the point S with A as the pole and

AD as the initial vector.

Consider a section mn at a distance z from the origin (Fig. 5.1.1). The deformation of this section is defined by the displacement components A A,u v of the pole A; and by the

angle of rotation of the cross section (Fig. 5.1.4). Also, let ,u v be the displacements of the point S. The angle of rotation is taken positive about the z axis according to the right hand rule and the displacements are considered positive in the positive direction of the corresponding axis. In the deformed position of the cross section, let ( ,X Y ) and ( A A,X Y ) represent the coordinates of the point S and the pole A, respectively. The following relations could be written with the help of Fig. 5.1.4:

A A A A A A

cos sin sin cos

cos sin sin cos

X x y Y x y

X x y Y x y

(5.1)



Figure 5.1.1: Biaxially bent restrained beam-column (Note: Axial thrust, P = N).

Figure 5.1.2: Cross section and definitions.

Figure 5.1.3: Deformed configuration of the bar.



As the section is considered rigid, the displacements ,u v of an arbitrary point S could be expressed in terms of the displacements A A,u v of the pole A, by using the fact

that the vector AS does not vary in length. Thus, we obtain:

A A A

A A A

sin 1 cos

sin 1 cos

u u y y x x

v v x x y y

(5.2)

By replacing the cosines of small angles by unity, and the sines of small angles by the angles themselves, these relations reduce to their linearized form: A A A Au u y y v v x x (5.3)

In the research reported here, the finite deformation field defined by Eq. (5.2) was retained.

The normal strain at any point S ( , ,X Y ) of the cross section mn, is related to the deformations A A A, ,u v of the pole A of this cross section as follows:

A A A ru X v Y (5.4)

where is a constant and r is the residual strain, if any, existing at the point S. Note

that, for a bar with initial imperfections, Eq. (5.1) written for an initially straight bar changes to:

cos sin

sin cos

gi gi

gi gi

X x y

Y x y

(5.5)

where giu represents the initial rotation of the section at .z If the strain given by Eq.

(5.4) is less than the yield strain y of the material, the fiber at S is still elastic and

A A A rE u X v Y (5.6)

Here, E is the modulus of elasticity of the material and r the residual stress at the fiber

at S. However, if the strain given by Eq. (5.4) is equal to or greater than the yield strain, the fiber at S plastifies and: y (5.7)

Here, y is the yield stress of the material. In what follows, we use the subscripts e and p

to identify the elastic and plastic parts of a partially plastified section.

Figure 5.1.4: Displacements of the pole A and of a general point S of the cross section.



A segment of the beam-column to the left of the section mn is shown in Fig. 5.1.5; while Figure 5.1.6 gives the projections, on to the XZ and YZ planes of the forces acting on that segment. By considering the equilibrium of this beam-column segment in its deformed position, three equilibrium equations could be written. They are: equilibrium of forces acting in the XZ plane, equilibrium of forces acting in the YZ plane, and equilibrium of twisting moments taken about the pole A of the section mn. We obtain [for details see Vinnakota, 1977b]

Figure 5.1.5: Forces acting on the portion of the beam-column to the left of the section m-n.

Figure 5.1.6: Projection on the XZ and YZ planes of the forces acting on the segment of the beam-column to the left of the section mn.



* *A A A 1 A1 2 A2 1 2

A A1

* *A A A 1 A1 2 A2 1 2

1 1

1 1

Ye XYe Y e rry rry y y qX

gi Y

XYe Xe X e rrx rrx x x qY

z z z zEI u EI v EI Pu k u k u M M M

L L L LX X u P

z z z zEI u EI v EI Pv k v k v M M M

L L L L

A A1

* *A A 1 21

gi X

Xe Ye e

m gi

Y Y v P

z zEI u EI v EI GJ K M M

L LM K

(5.8)

These are the three nonlinear, nonhomogeneous, coupled differential equations of equilibrium for an open-section thin-walled steel beam-column subject to end moments and transverse loads, and loaded into the plastic range. The equilibrium is written in the deformed position of the bar, for the segment to the left of the section mn situated at a distance z from the origin. Note that the origin O of the co-ordinate system XOY need not be the centroid, it is an arbitrary point C of the cross section. The pole A, whose displacements A A and u v appear in the above equations, is also an arbitrary point of the

cross section, not the shear center. The coordinates ,X Y are measured in accordance with a system of axes X-Y moving parallel to the arbitrary coordinate system X Y Z fixed in space. The deformation field considered is not linearized as was done in the majority of the earlier studies. Parameters , and gi gi giu v define the initial geometrical imperfection

of the bar at section .z

In deriving the equilibrium equations (5.8), use was made of the following section properties of the elastic core at section mn:

2

2 2

Xe Xe Ye Ye eXe Xe XYe XYe Xe Xe

e e e

Ye Xe e eYe Ye Ye Ye e e

e e e

S S S S SI I I I I I

A A A

S S S SI I I I I I

A A A

(5.9)

where

2 2 2

e

e e e

e e e

e e e

e A

Xe Ye eA A A

Xe Ye eA A A

X e Y e XYeA A A

A dA

S Y dA S X dA S dA

I Y dA I X dA I dA

I Y dA I X dA I XY dA

(5.10)

Here, eA is the area of the elastic core; ,Xe YeS S the static moments of area about X and Y axes; ,Xe YeI I the moments of inertia about X and Y axes; XYeI the product of inertia with



respect to those same axes; eS the sectorial static moment; eI the sectorial moment of

inertia; ,Xe wYeI I the sectorial products of inertia with respect to the Y and X axes, respectively.

The functions , and X Y are given by:

p p

p

Ye eY e r e rA A

e e

XeX e rA

e

S SP X dA P dA

A A

SP Y dA

A

(5.11)

where eP represents the part of the axial thrust P resisted by the elastic core, and:

A A AYe Xe e e

re e e e

S S S PE X u Y v

A A A A

(5.12)

The integrals in Eq. (5.11) represent the part of the bending moments and bi-moment resisted by the plasticized part pA of the cross section.

In Eq. (5.8), G represents the shear modulus of the material and J is the the St.

Venant torsional constant of the section. Further, we have

2 2

A AAK x x y y dA (5.13)

Parameters , and qX qY mM M M represent the influence of the external transverse loads,

and ,x yq q on the bending and torsional moments at the section mn. We have:

(0) ( )

(0) ( )

A A A

A (0) ( )

0

0

( )

( ) ( ) 0

qX x qX qX L

qY y qY qY L

m x q y q gi Y

gi X gi y m m L

M q M M

M q M M

M q Y Y q X X v v M PX

u u M PY K q M M

(5.14)

Influence of the residual stresses appears in the functions , and ;X Y

indirectly in the determination of the elastic core; and in the calculation of the term .K In the derivation of the equilibrium equations, it was assumed that the residual stress distribution satisfies the conditions:

0r r r rA A A AdA X dA Y dA dA (5.15)

For a self-equilibrating system of residual stresses the first three conditions are necessary, while the fourth condition was added to simplify the final equations. For the rotationally restrained member considered, compatibility requires that the member ends and the attached rotational restraints rotate through the same angle. The



moment-rotation response of the restraints was assumed to be linear-elastic with a

stiffness, .rrk The end conditions could be written as:

*A1 A1 1 1 1 1 1

*A2 A2 2 2 A2 2 2

*A1 A1 1 1 A1 1 1

A2 A2 A2 2 2

Ye XYe Y e rry A y Y

Ye XYe Y e rry y Y

XYe Xe X e rrx x X

XYe Xe X e rrx

EI u EI v EI k u M

EI u EI v EI k u M

EI u EI v EI k v M

EI u EI v EI k v

*

2 2

A1 A1 1 1 1

A2 A2 2 2 2

A1 A1 1 A2 A2 2

or 0

or 0

0 and 0

x X

Y e X e we

Y e X e we

M

EI u EI v EI

EI u EI v EI

u v u v

(5.16)

where subscripts 1 and 2, correspond to Ends 1 and 2 of the beam-column.

The three coupled differential equations, given in Eq. (5.8), along with appropriate boundary conditions specified in Eq. (5.16), were solved numerically using a computer program based on the finite difference method.

Figure 5.1.7 shows the arbitrary shape of the initial (or measured) geometrical imperfections utilized in the computer program to study the inelastic behavior of rolled or welded I-shaped steel beam-columns. Also included were the sine and parabolic variations for geometrical imperfections. Three patterns of residual stress distributions, generally found in rolled sections (namely, Lehigh, linear and parabolic variations), were included in the program.

For carrying out the numerical calculations the I-section was divided into small elements (Fig. 5.1.8a). As shown in Fig. 5.1.8b, the actual surface of the strain distribution across an element, under the combined influence of axial, bending, and warping strains may not be a plane. Figure 5.1.8b also shows the trapezoidal variation of the strain distribution across each element, assumed in the determination of the zones of plasticity, calculation of the geometrical properties of the elastic core, etc. Note that the influence of the flange web fillet was included in the calculations.

All of the external forces considered in the program were non-proportional. In effect, each applied load may consist of a constant part and/or a variable part. Thus

c v y c v c v

* * * * * *1 c 1 v 1 1 c 1 v 1 etc.

y y x x x

x x x y y y

P P P q q q q q q

M M M M M M

(5.17)

where represents the load factor or load level; subscript c stands for a constant part of a force; and subscript v for the variable part of a force corresponding to the value of 1.

Typically the member length was divided into 8 to 10 segments. The cross-sectional division typically consisted of, 8 to 20 flange divisions, and 8 to 20 web divisions. For the examples presented in the publications the calculation time for the CDC 7326 computer available at EPF-Lausanne in the ‘70s was approximately 20 seconds.



Figure 5.1.7: Geometric imperfections of the member.

Figure 5.1.9: Possible shift in the centroid and shear center, and rotation of the principal axes of the elastic core, with plastification of the section.

(schematic)

Figure 5.1.8: Partitioning of a cross section for numerical calculations.



5.2 Biaxially-Bent Beam-Columns Tested by Birnstiel [1968]

Sixteen isolated steel H columns subject to biaxially eccentric thrust were tested to failure by Birnstiel [1968]. The columns were held in position against horizontal displacement at the ends, which were also prevented from twisting about the longitudinal axis. The end sections were permitted to rotate about any axis lying in the x-y plane. Warping at the ends was prevented by welding very thick plates. Ultimate strengths of two of these columns (Specimen numbers 4 and 15) were determined by Vinnakota and Aoshima [1974a]. For Specimen number 4 the eccentricities of loading were substantially alike at both ends, resulting in single symmetric curvature with respect to both principal axes. On the other hand, for Specimen number 15 the eccentricities of loading were unlike at both ends resulting in double curvature. The relevant data is given in Table 5.2.1. Load-deflection curves obtained by Vinnakota and Aoshima [1974a] are compared with the data obtained by Birnstiel [1968] in Fig. 5.2.1 for Specimen number 4, and in Fig. 5.2.2 for Specimen number 15. In these figures, the small circles connected by thin lines represent the experimental results while the theoretical curves obtained by Vinnakota and Aoshima are shown by thick solid lines.

Table 5.2.1: Biaxially Bent Beam-Columns Tested by Birnstiel [1968]

Data Specimen 4 Specimen 15

Section 6 6H 5 5H

Length, L 96 in. 120 in.

yF 36 ksi 65 ksi

1xe 1.67 in. 1.75 in.

2xe 1.66 in. 1.47 in.

1ye 2.95 in. 2.96 in.

2ye 2.95 in. 2.96 in.

Result

TESTu yP P 0.327 0.353 Test Birnstiel [1968]

V&Au yP P 0.315 0.319 Vinnakota and Aoshima [1974a]



Figure 5.2.1: Symmetrically bent isolated column; Specimen No. 4 tested by Birnstiel [1968].

Figure 5.2.2: Antisymmetrically bent isolated column; Specimen No. 15 tested by Birnstiel [1968].



5.3 Rotationally Restrained Biaxially-Bent Columns Tested by Gent and Milner [1968] and Milner and Gent [1971]

The test setup used by Gent and Milner [1968] to study the behavior of biaxially loaded rotationally restrained beam-columns is shown in Table 5.3.1. The column was first bent by turnbuckle loads 1W and 2.W Then, while clamping these turn buckles rigidly, the axial load P was increased to failure. With the increase of the axial load the column deformed, thereby increasing its joint rotations and relaxing its end moments which were controlled by the beam stiffness. Table 5.3.1 gives a summary of the data for the H-section columns and the major- and minor-axis rectangular beams for Specimen A4 and B3 considered here. Milner and Gent [1971] also developed a computer program to study the inelastic behavior of the restrained columns, based on the finite difference method. Variations of the column end moments and of the mid-height displacements, as a function of the axial load, measured in the experiments and those calculated by Vinnakota and Aoshima [1974a, 1974b] are compared in Figs. 5.3.1 and 5.3.2, for Specimens A4 and B3. Table 5.3.1: Restrained Beam-Columns Tested by Gent and Milner [1968]

A4 B3

Column (H section) Length, L 16.5 18.0 fd b 1.000 0.750

ft (in.) 0.101 0.055

wt (in.) 0.060 0.035

Major axis beams (Rectangular) Length (in.) 11.7 12.1 b (in.) 0.375 0.375 h (in.) 0.500 0.850 Minor axis beams (Rectangular) Length (in.) 11.7 12.1 b (in.) 0.375 0.375 h (in.) 0.850 0.850

Initially applied beam moments ↑ Test set-up of Gent and Milner [1968] Major axis (in-kip) 0.500 0.210 Minor axis (in-kip) 0.510 0.170 Result:

TESTu yP P 0.82 0.90 Measured. Gent and Milner [1968]

V&Au yP P 0.80 0.84 Calculated. Vinnakota and Aoshima [1974a]



Figure 5.3.1: Rotationally restrained biaxially bent beam-column (Specimen A4) tested by Gent and Milner [1968] and analyzed by

Vinnakota and Aoshima [1974b].

Figure 5.3.2: Rotationally restrained biaxially bent beam-column (Specimen B3) tested by Gent and Milner [1968] and analyzed by

Vinnakota and Aoshima [1974a].



5.4 Inelastic Lateral-Torsional Buckling of Beam-Columns Tested at CTICM-Paris [Sfintesco, 1973]

Four lateral-torsional buckling tests were carried out at CTICM, Paris on isolated beam-columns of HEA 160. Warping was restrained at both ends. The loading consisted of a longitudinal compressive force, P acting in the plane of the web with eccentricities

1 2and y ye e at the Ends 1 and 2, respectively (Table 5.4.1). This load was increased

proportionally until failure occurred TEST( ).uP Also given in the Table are the calculated

values of the ultimate strengths, SVuP presented by [Vinnakota, 1974b]. For these

calculations a fictitious eccentricity of 0.1cm, perpendicular to the plane of the web, was used (i.e., 1 2 0.1 cmx xe e ). Two sets of calculations were done: one for annealed

shapes and one for shapes having Lehigh-type residual stresses with ( 0.3 ).rc yF

Table 5.4.1: Inelastic LTB of Beam-Columns Tested at CTICM, Paris [Sfintesco, 1973]

L cm

xL r 1ye

cm

2ye

cm

TESTuP

tons

SVuP

rc

0.0 rc

0.3

1 240 36.5 16.90 16.90 28.5 28.19 27.17

2 240 36.5 8.45 16.90 31.3 31.58 30.24

3 321 48.9 5.60 5.60 47.0 49.58 45.51

4 321 48.9 2.80 5.60 52.5 55.02 52.30

HEA 160: d 15.2 cm, fb 16 cm, ft 0.6 cm, wt 0.9 cm;

yF 2.95 t/cm2

5.5 Biaxially-Bent Beam-Columns Studied by Tebedge and Chen [1974]

Tebedge and Chen [1974] determined the maximum strength of W8x31 rolled mild steel members under axial force and symmetric single curvature end moments acting about the major and minor axes. The ends of the columns were free to warp. In each case the axial force P and the major axis bending moments xM were applied first and maintained constant. The column was then bent about the weak axis until it failed. The non-dimensionalized values for the minor-axis end moments u y pyM M were determined for

several slenderness ratios, ,xL r and for constant values of the axial force yP P and major-

axis moment .x pxM M For the purpose of comparison, Vinnakota [1974b] analyzed five

columns of slenderness ratio xL r 40. Those results are reproduced in Table 5.5.1.



Table 5.5.1: Strength of Biaxially Bent Beam-Columns [Tebedge and Chen, 1974]

y

P

P x

px

M

M u y pyM M Data:

xL r 40

Section: W8x31 36 ksiyF

Residual stresses: Lehigh Type; 0.3rc yF

Tebedge & Chen [1974]

Vinnakota [1974b]

1 0.10 0.20 0.782 0.759 2 0.10 0.60 0.381 0.219 3 0.40 0.20 0.361 0.339 4 0.40 0.50 0.093 0.00

5 0.60 0.20 0.100 0.016

5.6 Rotationally Restrained Beam-Column Studied by Santathadaporn and Chen [1973]

The symmetric and symmetrically loaded restrained beam-column analyzed by Satathadaporn and Chen [1973] is shown in Fig. 5.6.1. In the analysis a 220-in.-long W14x43 column was subjected to a doubly eccentric, longitudinal load, ( 5.0 in., yP e

0.5 in.).xe The stiffness rrk of the end rotational-restraints about the x and y axes was

2,700 kip-in/radian. The yield stress of the material was 36 ksi. The sway and twist at the ends as well as warping of column ends were prevented. Residual stresses and geometrical imperfections were not considered. Figure 5.6.1 shows the variation of the mid-height deformations calculated by Vinnakota and Aoshima [1974b] as a function of the axial load. The ultimate load calculated (= 0.34 of the yield load yP ) compares favorably with the value (0.33) given

by Santathadaporn and Chen [1973].

Figure 5.6.1: Load-deformation response of biaxially eccentrically loaded, rotationally restrained column studied by Santathadaporn and Chen [1973].



5.7 Inelastic Lateral-Torsional Buckling of Beams Tested by Kitipornchai and Trahair [1975b]

Kitipornchai and Trahair [1975b] presented results of six lateral buckling tests on full- scale simply supported 10UB29 beams subjected to central concentrated vertical loads. The point of application of the load was in the web plane, 3.5 in. above the top flange. Four beams were tested in the as-rolled condition with the residual stresses resulting from the cooling process. After stress relieving two of these four beams, two more tests were carried out. The maximum loads observed from the tests are compared in Table 5.7.1 and Fig. 5.7.1 with those calculated by Vinnakota [1977a]. For the calculations the yield stress was taken as 43.8 ksi. The measured residual stress distribution was replaced by a cosine pattern with compressive stresses of 0.3 yF at the flange tips and 0.5 yF at the web

center. In addition, the beams were considered to have in their x-x plane, an initial crookedness of half sine shape with a maximum ordinate 1000gmu L at mid-span.

Numerical results obtained by Kitipornchai and Trahair [1975a] for simply

supported 10UB29 beams with central concentrated loads Q acting at the geometrical

axis ( 2 0b h ) and above the top flange ( 2 1.76,b h which corresponds to the value used in the tests) are also shown in Fig. 5.7.1. In this figure, ( )yx x yM S F is the initial

yield moment of the section about its x-axis.

Table 5.7.1: Lateral Buckling of Beams Tested by Kitipornchai and Trahair [1975b]

Beam Beam

condition Span (in.)

TESTuQ

(kips)

SVuQ

(kips) SV

TEST

u

u

Q

Q

S1-20 As-rolled 240 12.8 12.02 0.94 S2-10 As-rolled 120 41.6 42.50 1.02 S3-12 As-rolled 144 32.6 31.87 0.98 S4-8 As-rolled 96 52.8 58.28 1.10 S2-10 Annealed 120 43.6 45.70 1.05 S2-12 Annealed 144 31.5 34.23 1.09

For the longer beams S1-20 and S3-12, the initial deformations considered in Vinnakota [1977a] are likely to be more severe than the combined effect of initial deformations and load eccentricities of test pieces, while the reverse is likely to be true for the short beam S2-10 and more so for S4-8. For the shortest beam, S4-8, the large difference between the calculated and experimental result is considered to result from the assumption that contribution of shear stress on yielding was neglected. As is to be expected, the annealed beams have a higher calculated strength compared to the corresponding rolled beams.



5.8 Influence of Level of Loading on Inelastic Lateral-Torsional Buckling of Beams, Studied by Yoshida and Imoto [1973]

Yoshida and Imoto [1973] presented a transfer matrix method to determine the inelastic lateral-torsional buckling strength of beams. They studied, among other factors, the influence of the level of loading on the LTB strength of simply supported 8WF31 beams subjected to a concentrated load, ,Q at mid-span. The load positions considered were: load at the lower flange, load at the center of the web, and load at the upper flange. Figure 5.8.1 shows a comparison of their results for buckling strength, with the ultimate strength values determined by Vinnakota [1977a] assuming initial geometric imperfection in the beam XOZ plane, of symmetric half-sine shape with an amplitude

1000.gmu L A linear pattern of residual stress distribution with 0.3rc rw yF and

0.3rj yF were also assumed in both calculations. In Fig. 5.8.1, plQ is the transverse load

Q corresponding to the formation of a plastic hinge at the mid-span of the beam. Yoshida and Imoto neglected the effect of the shift in the shear center due to plasticity (Fig. 5.1.9) and the influence of pre-buckling deformations on plasticity. This, in addition to the effect of initial geometrical imperfections considered by Vinnakota, explains the difference between the two results in the plastic range. In the elastic range, the values given by Vinnakota are higher than those of Yoshida and Imoto due to the influence of the nonlinear deformation field included by Vinnakota.

Figure 5.7.1: Inelastic LTB of beams; comparison with tests by Kitipornchai and Trahair [1975b].



5.9 Beam-Columns Subjected to Thrust and Biaxial Bending. Full-Scale Tests Conducted at Liege, Belgium [Anslijn & Massonnet, 1977, Vinnakota, 1975]

In 1970s Prof. Massonnet at Liege University, Belgium, developed a program to conduct about 90 full-scale tests on Fe360 steel HEA 200 (which is almost identical to the American profile 8WF31) beam-columns subjected to compression and biaxial bending. The different parameters retained for the tests were essentially selected from the following combinations [Vinnakota and Anslijn, 1977]:

- Slenderness ratios, yL r : 40, 60, 80, 100, 130

- Load eccentricities: 1xe : 0.5, 1.0, 3.0 times yr

- and 1ye : 0.5, 1.0, 3.0 times xr

- End moment ratio, 2 1M M : 1.0, 0.0, 0.5, 1.0 in XOZ and YOZ planes of the member.

The column ends could be considered pinned-pinned for flexural buckling about the x- and y-axes. Rotation at the supports was prevented around a vertical axis. Thick plates were welded at the ends of the specimens and the plates bolted to the plateau of the testing machine. Thus, the column ends could be considered restrained against warping. More than a year before the tests were begun, Vinnakota [1975] determined the ultimate strength of these biaxially bent beam-columns using the computer program at EPFL, Switzerland. These calculations, used nominal lengths for the columns, nominal dimensions for the cross section, nominal yield stress, and a Lehigh-type residual stress distribution with 0.3 .rc yF These calculations also included initial geometrical

imperfections, sinusoidal in shape with an amplitude 1000,gm gmu v L in the X°OZ° and

Y°OZ° planes. The results for L/ry = 60 for five different sets of end-eccentricity combinations are given in Tables 5.9.1 to 5.9.5 [Vinnakota, 1975].

Figure 5.8.1: Influence of level of loading on inelastic LTB of beams [Yoshida and Imoto, 1973]



Table 5.9.6 gives a comparison of the ultimate strength values predicted by Vinnakota [1975] with the test results, for 11 specimen having a nominal slenderness ratio of 60 [Vinnakota and Anslijn, 1977; Anslijn and Massonnet, 1978]. The predicted values were lower than the experimental values, as they should be, as the calculations were based on nominal values for yield stress, section dimensions, residual stresses and initial geometrical imperfections. In practice, the yield strength of the material in the flanges and, more so, in the web is generally much higher than the nominal yield stress stipulated by the specifications.

Table 5.9.1: Ultimate Strength of Biaxially Bent Beam-Columns ( 60,xL r Series A)

Tested at Liege, Belgium [Vinnakota, 1975]

No. 1x

y

er

er

1y

x

2x

y

er

er

2y

x

SV1975u yP P maxucm

maxvcm

max Rad.

A-1 1.0 1.0 1.0 1.0 0.347 2.08 (5) 0.77 (5) 0.0243 (5) A-2 1.0 1.0 1.0 1.0 0.384 2.70 (5) 0.40 (7) 0.0031 (3) A-3 1.0 1.0 1.0 1.0 0.422 0.98 (7) 0.83 (5) 0.0282 (6) A-4 1.0 1.0 1.0 1.0 0.463 0.66 (7) 0.41 (7) 0.0535 (5) A-5 1.0 1.0 0.0 1.0 0.403 1.34 (4) 0.86 (5) 0.0236 (4) A-6 1.0 1.0 0.0 1.0 0.459 1.38 (4) 0.37 (7) 0.0172 (5) A-7 1.0 1.0 1.0 0.0 0.381 3.06 (5) 0.47 (4) 0.0166 (5) A-8 1.0 1.0 1.0 0.0 0.463 0.51 (7) 0.33 (3) 0.0203 (5) A-9 1.0 1.0 0.0 0.0 0.447 1.41 (4) 0.47 (3) 0.0180 (3) A-10 1.0 1.0 1.0 0.5 0.359 2.22 (5) 0.55 (5) 0.0164 (5) A-11 1.0 1.0 0.5 1.0 0.378 1.96 (5) 0.93 (5) 0.0261 (5) A-12 1.0 1.0 0.5 0.5 0.397 1.90 (4) 0.64 (4) 0.0187 (4) A-13 1.0 1.0 0.5 2.0 0.313 1.19 (5) 0.90 (5) 0.0186 (5) A-14 1.0 1.0 2.0 0.5 0.291 3.07 (6) 0.50 (5) 0.0175 (5) A-15 1.0 1.0 2.0 2.0 0.247 3.06 (6) 1.37 (6) 0.0498 (6) A-16 1.0 0.0 0.0 1.0 0.472 1.96 (4) 0.57 (6) 0.0365 (5)

60yL r ; Section: HEA 200; Steel: Fe 360; 0.3rc yF GI: Sinusoidal, 1000gm gmu v L .

Figure 5.9.1: Loading for biaxially bent beam-columns tested at Liege, Belgium.



Table 5.9.2: Ultimate Strength of Biaxially Bent Beam-Columns ( 60,xL r Series B)


No 1x

y

er

er

1y

x

2x

y

er

er

2y

x

SV1975u yP P maxucm

maxvcm

max Rad.

B-1 3.0 0.5 3.0 0.50 0.184 2.64 (5) 0.02 (5) 0.0046 (5) B-2 3.0 0.5 3.0 0.50 0.191 2.71 (5) 0.31 (6) 0.0019 (5) B-3 3.0 0.5 3.0 0.50 0.244 0.68 (8) 0.03 (5) 0.0056 (8) B-4 3.0 0.5 3.0 0.50 0.244 0.66 (8) 0.32 (5) 0.0029 (8) B-5 3.0 0.5 0.0 0.50 0.244 1.91 (3) 0.11 (2) 0.0169 (3) B-6 3.0 0.5 0.0 0.50 0.244 1.73 (4) 0.27 (6) 0.0076 (2) B-7 3.0 0.5 3.0 0.00 0.194 2.39 (5) 0.16 (5) 0.0084 (5) B-8 3.0 0.5 3.0 0.00 0.244 0.61 (8) 0.17 (5) 0.0029 (2) B-9 3.0 0.5 0.0 0.00 0.244 1.78 (3) 0.12 (6) 0.0119 (3) B-10 3.0 0.5 3.0 0.25 0.191 3.33 (5) 0.05 (6) 0.0048 (5) B-11 3.0 0.5 1.5 0.50 0.222 2.40 (4) 0.07 (3) 0.0114 (4) B-12 3.0 0.5 1.5 0.25 0.222 2.71 (4) 0.04 (2) 0.0086 (4) B-13 3.0 0.5 1.5 1.00 0.222 2.79 (4) 0.28 (4) 0.0139 (5) B-14 3.0 0.5 6.0 0.25 0.122 2.38 (6) 0.17 (5) 0.0622 (6) B-15 3.0 0.5 6.0 1.00 0.122 1.10 (4) 0.08 (4) 0.0146 (7) B-16 3.0 0.0 0.0 0.50 0.247 2.12 (3) 0.15 (5) 0.0069 (4)


Table 5.9.3: Ultimate Strength of Biaxially Bent Beam-Columns ( 60,xL r Series C)


No 1x

y

er

er

1y

x

2x

y

er

er

2y

x

SV1975u yP P maxucm

maxvcm

max Rad.

C-1 0.5 3.0 0.50 3.0 0.228 1.02 (5) 1.61 (5) 0.0429 (5) C-2 0.5 3.0 0.50 3.0 0.263 0.16 (5) 0.44 (7) 0.0028 (7) C-3 0.5 3.0 0.50 3.0 0.228 1.10 (5) 1.56 (5) 0.0509 (5) C-4 0.5 3.0 0.50 3.0 0.259 0.36 (6) 0.42 (7) 0.0236 (5) C-5 0.5 3.0 0.00 3.0 0.216 0.08 (7) 0.98 (5) 0.0045 (7) C-6 0.5 3.0 0.00 3.0 0.263 0.10 (6) 0.44 (7) 0.0127 (4) C-7 0.5 3.0 0.50 0.0 0.263 0.33 (3) 0.62 (3) 0.0081 (2) C-8 0.5 3.0 0.50 0.0 0.263 0.46 (5) 0.58 (3) 0.0214 (3) C-9 0.5 3.0 0.00 0.0 0.263 0.09 (2) 0.55 (3) 0.0040 (5) C-10 0.5 3.0 0.50 1.5 0.256 0.77 (4) 1.20 (4) 0.0291 (3) C-11 0.5 3.0 0.25 3.0 0.228 0.72 (5) 1.43 (5) 0.0338 (5) C-12 0.5 3.0 0.25 1.5 0.259 0.76 (3) 1.35 (4) 0.0386 (3) C-13 0.5 3.0 0.25 6.0 0.147 0.13 (7) 1.05 (6) 0.0121 (7) C-14 0.5 3.0 1.00 1.5 0.253 1.21 (4) 1.34 (4) 0.0379 (4) C-15 0.5 3.0 1.00 6.0 0.141 0.29 (7) 0.97 (6) 0.0069 (7) C-16 0.5 0.0 0.00 3.0 0.266 0.08 (8) 0.78 (7) 0.0127 (8)




Table 5.9.4: Ultimate Strength of Biaxially Bent Beam-Columns ( 60,xL r Series D)

Tested at Liege, Belgium [Vinnakota, 1975].

No 1x

y

er

er

1y

x

2x

y

er

er

2y

x

SV1975u yP P maxucm

maxvcm

max Rad.

D-1 2.0 0.5 2.0 0.50 0.250 2.73 (5) 0.10 (5) 0.0059 (5) D-2 2.0 0.5 2.0 0.50 0.253 2.72 (5) 0.32 (6) 0.0022 (5) D-3 2.0 0.5 2.0 0.50 0.347 0.75 (8) 0.14 (7) 0.0100 (8) D-4 2.0 0.5 2.0 0.50 0.347 0.69 (8) 0.34 (6) 0.0033 (5) D-5 2.0 0.5 0.0 0.50 0.331 1.88 (4) 0.22 (4) 0.0174 (3) D-6 2.0 0.5 0.0 0.50 0.338 1.88 (4) 0.29 (6) 0.0041 (2) D-7 2.0 0.5 2.0 0.0 0.253 2.77 (5) 0.11 (6) 0.0040 (5) D-8 2.0 0.5 2.0 0.0 0.347 0.65 (7) 0.11 (6) 0.0037 (7) D-9 2.0 0.5 0.0 0.0 0.338 2.06 (4) 0.08 (2) 0.0117 (3) D-10 2.0 0.5 2.0 0.25 0.253 2.95 (5) 0.03 (5) 0.0050 (5) D-11 2.0 0.5 1.0 0.50 0.294 2.56 (4) 0.19 (4) 0.0112 (5) D-12 2.0 0.5 1.0 0.25 0.297 2.71 (4) 0.11 (3) 0.0092 (5) D-13 2.0 0.5 1.0 1.00 0.288 2.48 (4) 0.39 (5) 0.0150 (5) D-14 2.0 0.5 4.0 0.25 0.175 2.48 (6) 0.10 (5) 0.0068 (6) D-15 2.0 0.5 4.0 1.00 0.172 2.45 (6) 0.15 (7) 0.0131 (6) D-16 2.0 0.0 0.0 0.50 0.341 1.97 (4) 0.07 (5) 0.0090 (4)


Table 5.9.5: Ultimate Strength of Biaxially Bent Beam-Columns ( 60,xL r Series E)

Tested at Liege, Belgium [Vinnakota, 1975].

No 1x

y

er

er

1y

x

2x

y

er

er

2y

x

SV1975u yP P maxucm

maxvcm

max Rad.

E-1 0.5 2.0 0.50 2.0 0.297 1.22 (5) 1.45 (5) 0.0410 (5) E-2 0.5 2.0 0.50 2.0 0.350 0.39 (5) 0.46 (7) 0.0020 (7) E-3 0.5 2.0 0.50 2.0 0.291 0.67 (6) 0.97 (5) 0.0189 (5) E-4 0.5 2.0 0.50 2.0 0.350 0.38 (6) 0.44 (7) 0.0375 (5) E-5 0.5 2.0 0.00 2.0 0.303 0.69 (5) 1.35 (5) 0.0355 (5) E-6 0.5 2.0 0.00 2.0 0.360 0.08 (1) 0.46 (7) 0.0179 (4) E-7 0.5 2.0 0.50 0.0 0.347 0.69 (4) 0.65 (3) 0.0154 (3) E-8 0.5 2.0 0.50 0.0 0.341 0.43 (6) 0.43 (3) 0.0176 (3) E-9 0.5 2.0 0.00 0.0 0.350 0.23 (2) 0.57 (3) 0.0045 (1) E-10 0.5 2.0 0.50 1.0 0.334 1.15 (4) 1.21 (4) 0.0342 (4) E-11 0.5 2.0 0.25 2.0 0.297 0.75 (5) 1.18 (5) 0.0250 (5) E-12 0.5 2.0 0.25 1.0 0.338 0.84 (4) 1.10 (4) 0.0258 (4) E-13 0.5 2.0 0.25 4.0 0.206 0.32 (6) 1.08 (6) 0.0186 (7) E-14 0.5 2.0 1.00 1.0 0.322 1.48 (5) 1.11 (4) 0.0304 (4) E-15 0.5 2.0 1.00 4.0 0.197 0.61 (6) 1.00 (6) 0.0125 (7) E-16 0.5 0.0 0.0 2.0 0.350 0.18 (7) 0.59 (7) 0.0157 (7)




Table 5.9.6: Biaxially Bent Beam-Columns Tested at Liege, Belgium

[Vinnakota & Anslijn, 1977; Anslijn & Massonnet, 1978]

Test No.

x-axis y-axis SV1975u

y

P

P

TEST1977u

y

P

P

SV1975

TEST1977

u

u

P

P

% 1x

y

e

r 1y

y

e

r 2

1

x

x

e

e 2

1

y

y

e

e

1 A-2 1.0 1.0 1.0 1.0 0.384 0.44 87%

2 A-3 1.0 1.0 1.0 1.0 0.422 0.49 86%

3 A-4 1.0 1.0 1.0 1.0 0.463 0.57 81%

4 A-5 1.0 1.0 0.0 1.0 0.403 0.43 94%

5 A-7 1.0 1.0 1.0 0.0 0.381 0.42 93%

6 A-8 1.0 1.0 1.0 0.0 0.463 0.49 95%

7 A-9 1.0 1.0 0.0 0.0 0.447 0.50 89%

8 A-10 1.0 1.0 1.0 0.5 0.359 0.42 86%

9 A-12 1.0 1.0 0.5 0.5 0.397 0.53 75%

10 A-13 1.0 1.0 0.5 2.0 0.313 045 70%

11 A-16 1.0 0.0 0.0 1.0/0.0 0.472 0.52 91%

60yL r ; Section: HEA 200; Steel: Fe 360; 0.3rc yF

GI: Sinusoidal, 1000gm gmu v L .

5.10 Influence of Imperfections on Strength of Biaxially Loaded Beam-Columns [Vinnakota, 1976a]

The ultimate strength of biaxially bent beam-columns is influenced not only by the magnitude of the geometrical imperfections, but also by the sign of these imperfections as seen in Fig. 5.10.1 [Vinnakota, 1976a]. The results are for pin-ended, biaxially bent, W8x31 rolled-steel beam-columns of length L = 30rx. They were all subjected to a constant axial force P (= 0.3Py) and to symmetric, single-curvature moments in the two principal planes. Residual stresses were considered to be of Lehigh-type with

0.3 .rc yF The initial rotation was taken as 0.001 radian for all columns. The shape of

the initial crookedness was assumed to be a half sine wave. For the purpose of comparison, columns with the four possible combinations of signs of initial deflections in the x and y planes were considered, namely, ( , )gm gmu v , ( , )gm gmu v , ( , )gm gmu v and

( , )gm gmu v . The results are shown in Fig. 5.10.1 for an amplitude of 500.L They show

that for a given value of the minor-axis bending moment, the maximum value of Mx is obtained with ( , )gm gmu v and the minimum value of Mx with ( , )gm gmu v .



The influence of imperfections, as a function of the column slenderness, on the ultimate strength of biaxially loaded beam-columns is shown in Fig. 5.10.2 [Vinnakota, 1976b]. Imperfections considered are the initial crookedness and the residual stresses. The results shown are for pin-ended W8x31 steel columns with xL r 30 and 50. The columns were subjected to an axial compressive force P that was kept constant at 0.3Py. In addition, they were subjected to symmetric, single-curvature moments in both principal planes. Maximum values of My were calculated for selected values of major axis moments Mx, which were kept constant during the loading. Four different combinations of geometrical imperfections and residual stresses were considered. From the figure, we observe that the influence of initial geometrical imperfections and residual stresses is more important for xL r 50 than for xL r 30.

Figure 5.10.2: Variation of the influence of imperfections, with slenderness, on the strength of biaxially bent columns [Vinnakota, 1976a].

Figure 5.10.1: Influence of the sign of geometrical imperfection ( 500gm gmu v L ) on the strength of biaxially bent columns [Vinnakota, 1976a].



5.11 Ultimate-Strength of Crooked Steel Beam-Columns Subjected to Axial Thrust and Major-Axis End Moments [Galambos (ed.), 1988]

Using the study described in Section 5.2, Vinnakota [1977a] determined the ultimate strength, ucxM of laterally unsupported imperfect rolled steel I-section beam-columns,

subjected to major-axis end moments xM producing symmetric, single-curvature bending, and axial compressive loads, P. The ends were simply supported for the lateral displacements and rotations, and warping was permitted. The numerical calculations were done using a W8x31 section having the following characteristics:

All of the members were assumed to have small initial geometrical imperfections of sine shape, with a maximum mid-span out-of-straightness given by 1000gm gmu v L

and 0.001gm radian. Note that the signs of and gm gmu v were chosen to obtain the worst

effect of the initial imperfections. In the calculations, the torsional rigidity GJ was assumed to remain constant during plastification of sections. The results obtained for

ucxM are reproduced in Table 5.11.1 in non-dimensional form. Values in the first line of

this table (p = 0) are the lateral-torsional strengths of imperfect beams ( uxM ). Also, given in this table, in brackets, are the ultimate strengths, Pu of axially loaded, unbraced, pin-ended columns with residual stresses and geometrical imperfections (spatial).

Inelastic lateral-torsional strengths uxM of imperfect beams from Table 5.11.1 were compared with the results for the lateral-torsional buckling strengths for beams given by the then AISC design equation

CRC 1.073160

y y

ux px px

L r FM M M

(5.18)

The results are shown in Fig. 5.11.1 [AISC, 1969; Vinnakota and Nethercot, 1982; Galambos (ed.) 1988].

Inelastic lateral-torsional strengths uxM of imperfect beams from Table 5.11.1 were also compared with the results for the lateral-torsional buckling strengths for beams given by the then LRFD and ECCS design methods. These results are shown in Fig. 5.11.2 [Vinnakota and Nethercot, 1982; Galambos (ed.) 1988].

Data: Section: 8WF31 bf = 8 in. d = 8 in. tf = 0.433 in. tw = 0.288 in. A = 8.983 in.2 J = 0.49 in.4 Iw = 528.9 in.6 rx = 3.464 in. ry = 2.028 in. ro = 4.015 in. Material: A36 steel Fy = 36 ksi E = 30,000 ksi G = 12,000 ksi Yield load, Py = 323.4 kips Residual stresses: Lehigh-type with 0.3rc yF



Table 5.11.1: Non-dimensional Lateral-Torsional Strengths, ,xm of Imperfect I-Section Members under Axial Thrust and Uniform Major Axis Moment [Vinnakota, 1976a]

Figure 5.11.2: Lateral torsional beam strengths versus lateral-torsional buckling strengths from ECCS and LRFD design

methods [Galambos (ed.), 1988].

Figure 5.11.1: Lateral torsional beam strengths versus lateral-torsional buckling strengths from Eq. 5.18

[Galambos (ed.), 1988].

Note: x ucx pxm M M yp P P

ucxM Lateral-torsional strength of members under axial thrust and major axis moment

uxM Lateral-torsional strength of beams

yP Yield load of the cross section

pxM Major axis plastic moment of the cross section

- Values in the first line (p = 0) are the lateral-torsional strengths of imperfect beams ( uxM )

- Values in the brackets are the ultimate spatial strengths of nominally axially loaded imperfect columns ( uP )



Inelastic lateral torsional strengths ucxM of imperfect beam-columns were used to check the validity of the then AISC interaction equation for the design of beam-columns:

11uy u ex

P M

P M P P

(5.19)

- By substituting in the LHS of Eq. (5.19): Puy as Puy obtained from SSRC Curve 2 and CRCu uxM M given by Eq. (5.18) we can construct Fig. 5.11.3 [Vinnakota and

Nethercot, 1982; Galambos (ed.), 1988]. - By substituting in the LHS of Eq. (5.19): , and uy u u ux ucxP P M M M M from

Table 5.11.1, we can construct Fig. 5.11.4 [Vinnakota and Nethercot, 1982; Galambos (ed.), 1988].

Note that, in Figs. 5.11.3 and 5.11.4, the straight line joining the end points (1, 1) is

the graphical representation of the interaction equation (5.19).

After the publication in 1976 of the results for lateral-torsional stability of unsupported beam-columns given in Table 5.11.1, Lindner, Heil, and other researchers in Europe studied these beams and beam-columns using completely different numerical procedures and confirmed the results given here [see Heil, 1979, for example].

Figure 5.11.3: Verification of LRFD interaction equation for beam-columns with data from Table 5.11.1, using uM

from Eq. 5.18 [Galambos (ed.), 1988].

Figure 5.11.4: Verification of LRFD interaction equation for beam-columns with data from Table 5.11.1, using uM values also taken from

that table [Galambos (ed.), 1988].



6 Inelastic Planar Stability of Columns with Small End Restraints. VAX 11/780

Computer at MU 1982-85

Equation Section (Next)

Technical Memorandum 5 of the Structural Stability Research Council [Ziemian, 2010] states that “Maximum strength, determined by evaluation of those effects that influence significantly the maximum load-resisting capacity of a frame, member or element, is the proper basis for the establishment of strength design criteria.” In the case of columns, this memorandum requires explicitly the inclusion of the following three main factors in the determination of the load-carrying capacity, using a load-deflection approach: (1) residual stresses, (2) out-of-straightness, and (3) end restraints [Chen, W. F., 1980]. To this end, the SSRC Task Group 23 entitled “Effect of End Restraint on Initially Crooked Columns,” was established in January 1979.

The studies presented in this section were undertaken primarily under the guidance of SSRC Task Group 23, to evaluate the combined influence of residual stresses, geometrical imperfections and small end restraints on column strength. Note that the theory presented in this section is a particular case of the studies described in Section 5 on spatial stability of restrained crooked beam-columns undertaken during the period 1972-77.

6.1 Planar Strength of Singly Symmetric Restrained Beam-Columns [Vinnakota, 1982, 1983a, b, c, 1985a]

Figure 6.1.1 shows a prismatic beam-column of length .L The cross section is a singly-symmetric I-shape, whose only section axis of symmetry is that through the middle of the web (Fig. 6.1.2). G is the centroid of the section, while C is an arbitrary point (corresponding to the support point) on the middle line of the web at a distance CGd from G. It is assumed that the origin O of the co-ordinate system coincides with the point C of the section at End 1 of the beam-column, and that the ZY plane lies in the web plane of the member. The beam-column is provided with a rotational restraint at each end and, in addition, a directional restraint at End 2. The beam-column has an initial geometrical imperfection or crookedness, ,gv in the plane of symmetry. Its value at End 2, namely

,o represents the initial out-of-plumbness of the member. Variable gcv in Fig. 6.1.1

indicates the initial chord deflection at abscissa .z

In addition to longitudinal thrust, ( ),P N the member may be subjected to

external applied end moments * *1 2, ;M M an arbitrarily distributed lateral load, ( );zq and/or

a concentrated transverse load *2Q applied at End 2. Under the action of these forces, an

additional deflection ,v will develop, as shown in Fig. 6.1.3. Here, ( )Lv represents

the additional deflection or sway of End 2 of the beam-column. It is assumed that the supports, the restraints, the applied forces and bending moments, and the deflections, all



lie in the plane containing the axis of symmetry of the section and the Z-axis. The deflections are considered small.

The rotational restraints at Ends 1 and 2 react to end rotations 1 2 and with

moments 1 2 and ,rr rrM M respectively. Compatibility at the support points requires that the member ends and the attached rotational restraints rotate through the same angle, v (Fig. 6.1.4).

Figure 6.1.1: Rotationally and directionally restrained initially crooked member.

Figure 6.1.3: External loads and additional deformations. (Note: P = N)



Figure 6.1.5: Moment-rotation response of end rotational-restraint.

Figure 6.1.6: Force-displacement response of end directional-restraint.

Figure 6.1.4: Moment equilibrium of support at End 1.

Figure 6.1.2: Section and coordinate system considered.

Figure 6.1.8: Connection moment-rotation data from Sugimoto and Chen [1981].

Figure 6.1.7: Characteristics of a partially plastified section.



In actual structures, the restraints offered by the connections and the adjacent beams are complex nonlinear functions of deformations. A general piecewise linear moment-rotation response shown in Fig. 6.1.5 could be considered to be composed of several straight lines (a-b, b-c, c-d, etc.). In the development of the theory, the moment-rotation response of the connection was considered to be elastic with prestress as shown by the line b-c in Fig. 6.1.5. Here, 0

rrM represents the connection moment at zero rotation,

while rrk represents the connection stiffness, valid for b c( ). The particular

straight line (such as a-b, b-c, or c-d, etc.) and thus the values of 0and rr rrk M to be used in

the equilibrium equations will depend on the connection rotation at the load level considered. (Note that, a linearly elastic rotational restraint, corresponds to the particular case with 0 0.)rrM

The directional restraint at End 2 reacts with a force drF to the lateral

displacement 2( )v at End 2. The force-displacement relation of the directional

restraint is considered to be elastic with prestress as shown in Fig. 6.1.6. In this figure, 0

drF represents the restraint force at zero displacement, while drk is the restraint stiffness.

Under the loading considered, if the section at z plastifies partially (Fig. 6.1.7), the

centroid of its elastic core shifts from G to Ge . Moment equilibrium of forces acting on the segment to the left of the section at z in the deformed position of the member, results in [Vinnakota, 1982; 1983b; 1985a]:

* *1 1 2 2 1 2 ( )

0 0rr1 rr2

1 1

1e

rr rr q z

g e G C Ap

z z z zEI v Pv k v k v M M M

L L L L

z z zPv M M P d M P

L L L

xe

(6.1)

Equation (6.1) is the nonlinear, nonhomogeneous, second-order differential equation of equilibrium for a rotationally and directionally restrained, initially crooked, steel beam-column subject to end moments and transverse loads and loaded into the plastic range. In this equation, the terms on the right have the following significance:

- The first term represents the influence of the external moments at the Ends 1 and 2.

- The second term represents the influence of transverse loads on the bending moment at section z.

- The third term represents the influence of the initial geometrical imperfection. - The fourth term represents the moment intercepts of the end connections, if any

(see Fig. 6.1.5). - The fifth term represents the influence of the shift in the centroid of the elastic

core of the section at ,z with loading. - The sixth term is the part of the moment at section z resisted by the plastified

zones, if any, of the section at ,z at the load level considered. - The last term represents the influence of the drift of the member (and the presence

of the directional restraint).



The following notation is used in Eq. (6.1) :

2

2

eCG Ap aAp

xexe xe exe

e

SI I I A d M y dA

A (6.2)

Therefore, I xe represents the moment of inertia of the elastic core about its centroid, and

ApM is the moment at section z resisted by plastic zones. Note that eCGd varies along the

length of the member.

Moment equilibrium of supports at Ends1and 2 gives:

* 01 1 1 1 rr11 1

* 02 2 2 2 rr22 2

e

e

rr Ap e G C

rr Ap e G C

EI v k v M M M P d

EI v k v M M M P d

x

x

(6.3)

while shear equilibrium at End 2 gives:

* **1 2 1 2

1 2 2 2

0 001 2

2

rr rr odr q

rr rrdr

P k k M M Pk v v R Q

L L L L L

M MF

L

(6.4)

Finally, the end condition for displacement at End 1 is 1 0v (6.5)

In studies referred to above (in this section), nonlinear moment rotation

characteristics, identified in Fig. 6.1.8 by labels T-1, T-2 and T-3, were used for end connections. Type 1 data is a tri-linear fit used by Sugimoto and Chen [1981] to the test data reported originally by Bergquist [1977] for an assembly consisting of a W10x21 beam connected to weak axis of a W10x29 column using wide cleats fastened with A325 bolts. Type 2 and Type 3 data are bilinear relations used by Sugimoto and Chen [1981] for studying members consisting of W12x30 beams connected to the weak and strong axes, respectively, of W12x65 columns. All of the connection moments are nondimensionalized by dividing them by the plastic moment of the column section about the axis under consideration. The theory presented above took into account the influence of the three parameters stipulated by a Technical Memorandum No. 5 of the SSRC, namely, residual stresses, geometrical imperfections, and nonlinear end restraints. Based on this theory, computer programs using the finite difference technique were developed to determine the ultimate strength of doubly symmetric restrained I-section beam-columns bent about their major or minor axis. [For details see Vinnakota, 1982.]



6.2 Column with Small End-Restraint Tested by Bergquist [1977]

The column with small end-restraint tested by Bergquist [1977] consisted of a W10x29 column of slenderness 189.7yL r with W10x21 beams attached to the minor axis using

web cleats fastened with A325 bolts. The moment-rotation response of the connection used was the tri-linear type labelled T-1. As seen in Fig. 6.2.1, agreement between the Bergquist’s experimental results and the analytical results from Vinnakota [1982] were generally quite good, with the calculated curve slightly underestimating the stiffness. In the analysis, the column was assumed to possess geometrical imperfections of half sine wave with a central ordinate of 1000.L A linear pattern of residual stresses (Lehigh type) with 0.3rc yF was included in the calculations. Also shown in the figure are the

analytical results of Jones, Kirby and Nethercot [1982] using a finite element technique and a moment-rotation response of the connection that is slightly modified from the measured response of the connection.

6.3 Beam-Columns with Lateral Loads Studied by Lu and Kamalvand [1968]

The relations between the lateral load and the end slope for a pin-ended steel beam-column subjected, in addition to an axial thrust ,P to a concentrated transverse load Q at mid-span is shown in Fig. 6.3.1. Figure 6.3.2 shows the corresponding results for a beam-column subjected to a uniformly distributed transverse load, .q In both cases bending is about the major axis. The numerical computations [Vinnakota, 1982] were for W8x31 A36 steel columns of length 60 xL r and an axial compressive load ( 0.4 ).yP P The

four curves shown in these figures correspond to: (1) An initially crooked, pin-ended column; (2) an initially straight, pin-ended column; (3) an initially crooked, rotationally restrained column; and (4) an initially straight, restrained column. The geometrical imperfections, when included, were sinusoidal in shape, with an amplitude of 1000.L The restraint considered was linear, with a stiffness .rr xk EI L The loading path adopted

Figure 6.2.1: Column with small end restraints, tested by Bergquist [1977].



consisted of first applying the selected axial load on the column and then increasing the lateral load from zero to failure. The lateral loads and Q q in Figs. 6.3.1 and 6.3.2 are shown as fractions of the simple plastic limit load, or ,pl plQ q as the case may be.

Assuming that no axial force is applied to the member, the plastic limit loads for a pure beam are:

2

4 8;px px

pl pl

M MQ q

L L (6.6)

in which pxM is the plastic moment of the section about its major axis.

Also shown in these figures are the results obtained by Lu and Kamalvand [1968] for initially straight, pin-ended columns using a numerical integration technique.

6.4 Influence of Small End Restraint, Geometrical Imperfections, and Residual Stresses on Column Strength. Vinnakota [1982]

Influence of the three parameters, namely, geometrical imperfections, residual stresses and small end rotational restraints, on the strength of A36 steel W8x35 columns bent about their minor axis was studied by Vinnakota [1982]. The geometrical imperfections considered were sinusoidal in shape, with maximum central ordinates of 5,000,L

1000,L and 500.L The first value corresponds to an almost straight column, the second value represents the maximum allowable out-of-straightness generally specified by standards, and the last value may be considered as an upper limit for out-of-straightness. Except for annealed columns, all strength calculations included the influence of an assumed, Lehigh type residual stress pattern with 0.30 .rc yF The two end rotational

restraints have the same moment-rotation characteristics, taken to be of the tri-linear Type T-1 shown in Fig. 6.1.8. The nonlinear characteristic considered correspond to a

Figure 6.3.1: Load-deformation response of a column subject to a

concentrated transverse load.

Figure 6.3.2: Load-deformation response of a column subject to a

uniformly distributed transverse load.



small end restraint provided by simple cleat angle connections about the minor axis of the column. Note that such columns are traditionally designed as pin-ended columns (i.e., K is taken as 1.0).

The results are shown in Figs. 6.4.1 and 6.4.2 in the form of a non-dimensional

slenderness factor E versus the strength ratio .uP Here, EE

1,y yFL

r E

P

P ,u

uy

PP

P and

2 2EP EI L is the Euler buckling load of the column (K = 1.0). Figure 6.4.1 shows the

influence of two parameters considered in that study, namely, geometrical imperfections and small end restraints; while Figure 6.4.2 shows the influence of the third parameter, namely, residual stresses, on the ultimate strength of columns. In all of these cases, small end restraints raise the ultimate strength of pin-ended columns. The spread of results due to imperfections (initial crookedness or residual stresses) appeared to be about the same for pinned columns and columns with simple end connections.

The results for the ultimate strength of W8x35, W10x29, and W12x65 columns with geometrical imperfections of 1000L , having Lehigh type residual stresses with

0.3rc yF and provided with small end restraints (T-1 and T-2 type connections) bent

about their minor axis are shown in Fig. 6.4.3 in a slightly modified form. The abscissa is

now the effective slenderness factor, .y y

ee

F PKL

r E P

Here, K is the effective length

factor and 2 2( )eP EI KL the elastic buckling load of the restrained column. For

restrained columns K could be obtained from an elastic eigenvalue analysis. For columns with equal end rotational restraints having linear moment-rotation response, the effective length factor could be calculated, say, from the following approximate formula [Kavanagh, 1962]:

Figure 6.4.1: Influence of geometrical imperfections and small end restraints on column strength [Vinnakota, 1982].

Figure 6.4.2: Influence of residual stresses and small end restraints on column strength [Vinnakota, 1982].



2

2

2

4rr rr

rrrr

k kK where k

k EI L

(6.7)

Figure 6.4.3 also includes the SSRC Curve 2 and the corresponding 97½ percentile curve for flexural strength of pin-ended steel columns [Bjorhovde, 1972]. It can be seen that the

spread of results is much less with the use of the effective slenderness factor ( e ) as the

abscissa instead of the slenderness factor ( E ).

6.5 Directionally and Rotationally End-Restrained Columns. Vinnakota [1983b]

Results for the ultimate strength of restrained A36 steel W12x65 columns, hinged at the left end and provided with rotational and directional restraints at the right end, obtained by Vinnakota [1983b] are shown in Fig. 6.5.1. The columns were bent about their major axis. The rotational restraints correspond to the Type-3 cleat angle connection shown in Fig. 6.1.8. A linear pattern of residual stresses (Lehigh type) with 0.3rc yF was

included in the calculations. The geometrical imperfections consisted of an out-of-plumbness of 500L and an initial chord deflection sinusoidal in shape with a maximum

(central) value of 1000.L The stiffness drk of the directional restraint was expressed as:

ydr

Pk C

L (6.8)

Results for two sets of columns with C 0.5 and 1.0, and xL r 30, 40, 60, and 80 are shown in Fig. 6.5.1. Values for the elastic effective length factors K for these rotationally and directionally restrained columns are obtained from charts in Bridge and Trahair [1976]. Figure 6.5.1 shows that the use of the elastic effective length factor and SSRC Curve 2 result in a conservative estimate for the strength of directionally and rotationally restrained imperfect columns.

Figure 6.4.3: Ultimate strength versus effective slenderness factor data for imperfect columns with small end rotational restraints [Vinnakota, 1982].



An alternative method of presentation of these results is shown in Fig. 6.5.2. The inelastic effective length factors iK used to determine the slenderness factors

y y

ie

ii

F PK L

r E P

are now calculated using the iterative concept proposed by Yura

[1971]. The improvement of the results using the inelastic effective length factor iK (Fig.

6.5.2) over the results using the elastic effective length factor eK (Fig. 6.5.1) is evident.

The general conclusion of these studies is that small end rotational restraint from simple beam-to-column connections (such as cleat angle connections) has a significant effect on the load carrying capacity of geometrically imperfect, pin-ended columns.

Figure 6.5.1: Ultimate strength vs. elastic slenderness factor for rotationally and directionally restrained steel columns [Vinnakota, 1983b].

Figure 6.5.2: Ultimate strength vs. inelastic slenderness factor for rotationally and directionally restrained steel columns [Vinnakota, 1983b].



7 Inelastic Planar Stability of Large Unbraced Steel Frames. Cray C90 at Pittsburgh Supercomputing Center 1990s

Equation Section (Next)

7.1 Inelastic Stability of Partially Restrained Unbraced Steel Frames Using Supercomputers. Foley [1996], Foley and Vinnakota [1997, 1999a, 1999b]

A second-order analysis method, using supercomputers, to study the inelastic in-plane behavior including post-critical response of large unbraced steel frames with fully restrained (FR) and/or partially restrained (PR) moment connections was developed by Foley [1996]. The spread of plasticity method (fiber element method) was used in the analysis. The method of substructuring was used in conjunction with parallel processing and vectorization to reduce the computation times required for the analysis of large multistory steel frames. The program was run on the Cray C90 Y-MP supercomputer at the Pittsburgh Supercomputing Center, USA. The following assumptions were made in the development of the research: (1) plane sections remain planar after bending, (2) out-of-plane deformations were neglected, (3) local buckling of the plate elements was not considered, (4) shearing strains are negligible, (5) yielding of the fibers is governed by normal stress alone, (6) member deformations are small, but overall structure deformations may be large, and (7) strain hardening is neglected.

The planar beam-column finite element used by Foley is shown in Fig. 7.1.1 along with assumed positive directions of the nodal displacements, nodal forces, and transverse member forces. The finite element considered both concentrated and uniformly distributed transverse loads applied between the element nodes and also loads applied directly to the element nodes.

The material model was assumed to be linearly elastic-perfectly plastic in which unloading occurs with a stiffness equal to the initial modulus of elasticity. Sixty-six fiber elements (27 in each flange and twelve in the web), as shown in Fig. 7.1.2, were used within the cross section of each finite element. In addition to the assignment of initial stress states corresponding to a variety of residual stress patterns, this discretized “fiber configuration” analytically allows for accurate modeling of cross sectional yielding (plastic zones). Two of the linear residual stress patterns used in the frame analyses are shown in Fig. 7.1.3. As shown in Fig. 7.1.4, plastification along the length of a member was tracked via utilization of a number of finite elements to model each beam and column in the frame (20 elements per column member and 30 elements per beam member).

Partially restrained (semi-rigid) moment connections were modeled using a linearized moment-rotation curve. A quadrilinear representation of the moment rotation curve was used (Fig. 7.1.5). The final stiffness was assumed to be zero and corresponds



to the plastic moment capacity of the connection, .rpM Unloading of the connection was

assumed to follow the initial stiffness. The Rayleigh-Ritz method and principle of minimum potential energy were used

in the development of the finite element stiffness equations. Assuming linearly elastic-perfectly plastic material behavior (unloading occurs with the initial modulus), the total potential energy for the partially plastified planar beam-column element is given by [Foley, 1996]:

22 2 2 42 2 2

2 2 20 0

2 2

2

22 2 4

2p

p p

L Le

e ze ze e ze

A

A A

AE du d v du d v E du dv d v dv dvA S I dx A S dx

dx dx dx dx dx dx dx dx dx

Pdu dv d vP M

dx dx dx

0 0

( )2

pL LA T

y

Pdx w v dx Q v Q r d

(7.1) Note that, the first moment of area, second moment of area, and cross-sectional

area terms are based on the remaining elastic core with reference to the fixed coordinate system shown in Fig. 7.1.1. These terms are similar to those used by Vinnakota [1982] for planar stability of beam-columns and by Vinnakota [1977] for spatial stability of beam-columns. Again, the nonmoving coordinate system allows for unsymmetric yielding within the cross section to be tracked and for the eccentricity of the applied loads (whose point of application remains in the fixed system), with respect to the remaining elastic core to be included. The cross-sectional properties may be defined with the help of Fig. 7.1.6 and can be expressed as the following integrals:

2, , ,e e e p

e e ze e e CGe ze e p pA A A AA dA S y dA A d I y dA A dA (7.2)

In addition, the generalized stress relations for the plastified areas [Vinnakota, 1982; Foley, 1996] may be written as: ,

p pp p

A y p A y pA AP dA M y dA (7.3)

Partially restrained ends (for beam members only) were simulated through two spring elements assembled in line with the beam elements to give an eight degree of freedom member (Fig. 7.1.4). The two unneeded degrees of freedom were then condensed out to obtain a six degree of freedom planar element for each main beam member in a frame.

Two types of loading were used in the study. The first was proportional, in which all loads applied to the structure were incrementally varied from an input level to the ultimate load factor condition through the post critical response. The second type of loading was termed non-proportional, where the lateral (gravity) loads were varied from input level to their ultimate load factor (and then decreased as the limit point is surpassed), while the gravity (lateral) loads were applied first then held constant throughout the analysis. Three load cases were considered in the analysis:



Load Case I: (1.2D + 0.5L + 0.5S + 1.3W) Load Case II: (1.2D + 0.5L + 0.5S) + (1.3W) Load Case III: (1.2D + 0.5L + 0.5S) + (1.3W) Load Case I increased all loads on the framework until collapse. Load Case II applied lateral loads first and then applied and increased gravity loads until collapse. Load Case III applied gravity loads first and then applied and increased lateral loads until collapse



Figure 7.1.1: Beam-column element with nodal loads and displacements

[Foley, 1996].

Figure 7.1.5: Linearized moment-rotation curve used for PR connections [Foley, 1996].

Figure 7.1.3: Residual stress patterns assumed in frame analyses

[Foley, 1996].

Figure 7.1.2: Fiber elements used to track yielding within a cross

section [Foley, 1996].

Figure 7.1.4: Finite element mesh used to track yielding along the member length [Foley, 1996].

Figure 7.1.6: Nomenclature for partially plastified cross section

[Foley, 1996]



7.2 Inelastic Stability of Partially Restrained Single-Bay Three-Story Frame [Foley, 1996]

The first frame presented here (designated as Frame 1) is a single-bay three-story frame, having the geometry and member sizes given in Fig. 7.2.1. Three connection types were used in its analysis. The first was the fully restrained (idealized) connection. The second and third connections (both partially restrained) were the same as those used by Barakat [1988] and Barakat and Chen [1991] and designated there as III-17 and III-11, respectively. The linearized representation of the partially restrained moment-rotation curves used in the analysis of Frame 1 is shown in Fig. 7.2.2.

Figure 7.2.3 shows the loading and unloading parts of the load-deformation response of the frame for the three connection types considered. Also given in this figure is the state of connection stiffness at the ultimate load level. The symbols at the beam ends indicate the branch of linearized stiffness corresponding to the connection rotation demand being imposed. Note that the unloading paths for the three frame configurations do not follow a common second-order rigid-plastic unloading curve. Ultimate load factors ( u ) and lateral deflections of the top story at ultimate load ( u ) for frames with the three connection types are summarized in Table 7.2.1. Table 7.2.1: Ultimate load factors and lateral deflections at ultimate load for single-bay

three-story frame [Foley, 1996]

Connection u u (in.)

FR 2.188 7.57 PR-Type III-17 1.872 12.22 PR-Type III-11 1.622 14.58

The spread of plasticity (indicated as a percentage of yielded cross section) at the

ultimate load condition for Frame 1 with each connection configuration is shown in Figs.7.2.4 to 7.2.6. The plastification for Frame 1 changes significantly as the connection type is changed. Note especially the change in yielding within the columns as the connection changes from FR to PR connections. The yielding tends to concentrate at the ends of the column when the beam-to-column connections are stiff, while the yielding spreads out over the length of the lower story columns when the PR connections are used. In all cases the frame’s design appears to be weak-column/strong-beam type as yielding at the end of the beam was minimal.

The presence of PR connections significantly alters the failure mechanism at the strength limit state. The typical sway mechanism is expected and seen for the frame with FR connections (Figure 7.2.4). When PR connections are introduced into the framework, a failure mode that involves significantly reduced plastic hinging within the beam spans occurs with a reduction in column plastic hinging in the region of the beam-to-column connections. The PR III-17 connection frame has a yielding distribution that is similar in nature to a sway mechanism and the PR III-11 connection frame has a distribution of yielding that deviates significantly from the typical sway mechanism.



Figure 7.2.1: Single-bay three-story frame studied by Foley

[1996].

Figure 7.2.2: Linearized moment-rotation curves used in analysis of Frame 1 [Foley, 1996; Barakat & Chen 1991].

Figure 7.2.3: Load-drift behavior and state of connection stiffness

for Frame 1 [Foley, 1996].

Figure 7.2.4: Spread of plastification for Frame 1 with fully restrained

connections [Foley, 1996].



7.3 Inelastic Stability of Partially Restrained Eight-Bay Sixteen-Story Frame. Foley [1996]

The second frame presented here (designated as Frame 2) is an eight-bay sixteen-story frame with an aspect ratio of 1.0, having the geometry and member sizes given in Fig. 7.3.1. The frame has fully restrained base plate connections. It was designed for office type gravity loading and ANSI standard wind loading and the LRFD Specification [AISC, 1994] assuming FR moment connections and elastic analysis. A single beam size was utilized throughout the frame (with the exception of the roof beams) and the columns within the frame were continuous for two stories. The method of implementing live load reduction into frame analysis proposed by Ziemian [1992] and Ziemian and McGuire [1993] was also implemented. The frame was studied with three distinct types of beam-to column connections: FR moment connections, extended end plate moment connections and flange plate moment connections [Fig. 7.3.2]. Three loading combinations were utilized as described earlier. Load Case I was considered proportionally applied loading, while Load Cases II and III where lateral-constant and gravity-constant non-proportional loading conditions, respectively. The effect of connection type on frame behavior can be seen in Fig. 7.3.3; while the variation in load-deformation response for the three load sequences is shown in Fig. 7.3.4. It is interesting to note that all connections behave essentially as FR connections demonstrated by the very small differences in ultimate load levels and lateral displacements corresponding to the strength limit state. However, the load sequence for the fully restrained connections significantly affects the load deflection and strength limit state load factors as seen in Figure 7.3.3. Note that, for the inelastic stability analysis of this frame 6690 sub-elements; 441,540 fiber elements; and 19,716 degrees of freedom were used to solve the problem.

Figure 7.2.6: Spread of plastification for Frame 1 with partially restrained connections of Type III-17 [Foley,

1996].

Figure 7.2.5: Spread of plastification for Frame 1 with partially restrained connections of Type III-11 [Foley,

1996].



Figure 7.3.3: Load-displacement relationships for eight-bay, sixteen-

story frame for various beam-to-column connection types [Foley, 1996].

Figure 7.3.4: Load-displacement relationships for eight-bay, sixteen-story frame for various proportional and non-proportional load cases [Foley, 1996].

Figure 7.3.1: Eight-bay sixteen-story frame (6690 elements, 19716 degrees of freedom and 441,540 fiber elements within the structure).

Figure 7.3.2: Typical connections used in the

analysis of frames.



7.4 Inelastic Stability of Partially Restrained Five-Bay Sixty-Story Frame. Foley [1996]

The third frame presented here (designated as Frame 3) is a five-bay sixty-story frame with an aspect ratio of 6.0, having the geometry and member sizes given in Fig. 7.4.1. For partially restrained connection details refer to Foley [1996].

The effect of connection type on behavior of Frame 3 with proportional loads applied can be seen in Fig. 7.4.2. The significant effect of connection type on behavior with proportionally applied loads is seen in Fig. 7.4.2. The behavior of the frames appears to be consistent across all connection types and is attributed to the slender nature of the framework. The failure mechanism tended to include along-length column yielding in the leeward columns and column end yielding and beam end yielding in the upper stories.

Figure 7.4.2: Load-displacement behavior of Frame3

with proportional loads applied [Foley, 1996]. Figure 7.4.1: Five-bay sixty-

story frame [Foley, 1996].



8 Conclusions

From the description of the research given in Sections 3 to 7, it could be seen that as the size and speed of the computers increased, the complexity of the inelastic stability problems that could be solved using them also progressed. It is also interesting to note that, so did the sophistication of the specification rules that govern the design of steel members and frames [AISC, 1962; AISC, 2010].

Acknowledgements

I am indebted to Professors M. Cosandey, J. C. Badoux, and T. V. Galambos for their help in the pursuit of this research over the years. My thanks also go to P. Aysto, Y. Aoshima and especially to Chris Foley for their contribution to different parts of the research presented here.

Publications Vinnakota, on Inelastic Stability of Members and Frames

1. Vinnakota, S. [1967]: Inelastic Stability Analysis of Rigid Jointed Frames (Flambage des Cadres dans le domaine elasto-plastique), Thesis submitted on: December 13, 1967; successfully defended on May 22, 1968, published on February 10, 1970, Federal Institute of Technology, Lausanne, Switzerland.

2. Vinnakota, S. and Badoux, J. C. [1970a]: “Strength of Restrained Beam-Columns,” Construction Metallique, Paris, No. 2, pp. 5-16, June.

3. Vinnakota, S. and Badoux, J. C. [1970b]: “Elastic Buckling of Rectangular Frames,” (in French), Bulletin Technique de la Suisse Romande, Lausanne, No. 23, pp. 335-348, November.

4. Vinnakota, S. [1971]: “Inelastic Stability of Frames,” (in French), Bulletin Technique de la Suisse Romande, Lausnne, No. 21, pp. 491-499, October.

5. Vinnakota, S. and Badoux, J. C. [1971]: “Strength of Laterally Loaded (Inelastically) Restrained Beam-Columns,” The Civil Engineering Transactions, The Institution of Engineers, Australia, pp. 107-114, October.

6. Vinnakota, S. [1972]: “Elastic-Plastic Instability of Multi-Story Frames,” Invited Discussion, Technical Committee 16, ASCE-IABSE International Conference on Tall Buildings, Lehigh University, PA, August 1972. Preprints: Discussion and Summary, pp. 568-574.

7. Vinnakota, S. [1973]: “Calcul et comportement des poteaux encastres sollicitees en flexion biaxiale,” Colloquium on Column Strength, Paris 23-24 November, 1972. Published in Construction Metallique, Paris, No. 2, 1973, pp. 7-16.

8. Vinnakota, S. [1974a]: “Design and Analysis of Restrained Columns Under Biaxial Bending,” Paper presented at the International Conference, “Tall Buildings – Planning, Design and Construction,” Bratislava, April 9th to 12th, 1973. Published in Stavebincky Casopis SAV, Bratislava, Czechoslovakia, No. 4, Vol. 22, 1974, pp. 182-206.



9. Vinnakota, S. [1974b]: “Some Remarks on Proposed Revisions to 1969 AISC Specification, Section 2.4 Columns,” Report submitted to the Task Group 3 of the American Column Research Council, March, 1974.

10. Vinnakota, S. [1974c]: “Elasto-Plastic Stability of Frames,” International Civil Engineering, Israel, Vol. III, pp.37-48.

11. Vinnakota, S. and Aoshima, Y. [1974a]: “Inelastic Behavior of Rotationally Restrained Columns Under Biaxial Bending,” Journal of the Structural Engineers, London, Vol. 52, July, pp. 245-255.

12. Vinnakota, S. and Aoshima, Y. [1974b]: “Spatial Stability of Rotationally and Directionally Restrained Beam-Columns,” Publications IABSE, Zurich, Vol. 34-II, 1974, pp. 169-194.

13. Vinnakota, S. and Aysto, P. [1974]: “Spatial Stability of Restrained Beam-Columns,” Journal of the Structural Division, Proc. ASCE, New York, Vol. 100, November 1974, pp. 2235-2254.

14. Aoshima, Y., Vinnakota, S. and Badoux, J. C. [1974]: “Comportement elasto plastique des poutres-colonnes soumisesa la flexion biaxiale et presentant des coditions limites quelconques,” Proceedings, Japan Society of Civil Engineers, Tokyo, pp. 23-31, April 1974.

15. Vinnakota, S. [1975]: Report sent to Prof. Ch. Massonnet, Chairman, Commission 5, Plasticity, ECCS, May.

16. Vinnakota, S., Badoux, J.C. and Aoshima, Y. [1975]: “Fundamental Equations Governing the Behavior of Thin-Walled Open-Section Beam-Columns,” Bulletin Technique de la Suisse Romande, Lausanne, No. 26, Vol. 101, December, 1975, pp. 437-445.

17. Vinnakota, S. [1976a]: “Influence of Imperfections on the Maximum Strength of Biaxially Bent Columns,” Report presented at the Annual Technical Session of the American Column Research Council, Toronto, May 6, 1975 [Also published in the Canadian Journal of Civil Engineering, June 1976, pp. 186-197].

18. Vinnakota, S. [1976b]: Discussion of “Buckling of Inelastic I-Beams Under Moment Gradient,” by Kitipornchai, S. and Trahair, S., Proceedings ASCE, No. ST4, April, 1976.

19. Vinnakota, S. [1976c]: Discussion of “Inelastic Buckling of Simply Supported Steel I-Beams,” by Kitipornchai, S. and Trahair, S., Proceedings ASCE, No. ST6, June 1976, pp. 1263-1265.

20. Vinnakota, S. [1977a]: “Inelastic Stability of Laterally Unsupported I-Beams,” Presented at the Second National Symposium on Computerized Structural Analysis and Design at the School of Engineering and Applied Science, George Washington University, Washington, DC., March 1976. Published in: Computers and Structures, Vol. 7, No. 3, June 1977, pp. 377-389.

21. Vinnakota, S. [1977b]: Finite Difference Method for Plastic Beam-Columns, Chapter 10 of the Book, Theory of Beam-Columns, Vol. 2, by W. F. Chen and T. Atsuta, McGraw-Hill Book Co., Ltd., London, 1977, pp. 451-503.

22. Vinnakota, S. [1977c]: “Theoretical Research, Practical Recommendations and Some Remarks on Stability of Structures,” (in French), Bulletin Technique de la Suisse Romande, Lausanne, Vol. 103, No. 17, August 1977.



23. Vinnakota, S. and Anslijn, R. [1977]: “Tests on Columns under Compression and Biaxial Bending and Theoretical Evaluation of Their Limit Load,” Final Report, Second International Colloquium on Stability of Steel Structures, Liege, Belgium, April 1977.

24. Vinnakota, S. and Beer, J. [1977]: “STELCA, A Computer Program to Study Elastic Buckling of Rectangular Multistory Frames,” (in French), Bulletin Technique de la Suisse Romande, Lausanne, Vol. 103, No. 11, 1977.

25. Vinnakota, S. [1981]: “Influence of Imperfections on the Maximum Strength of Restrained Beam-Columns,” Proceedings SSRC, Bethlehem, PA, 1981.

26. Vinnakota, S. [1982]: “Planar Strength of Restrained Beam-Columns,” Journal of the Structural Division, ASCE, Vol. 108, No. ST11, November, 1982, pp. 2496-2516.

27. Vinnakota, S. and Nethercot, D. [1982]: “Verification of the SSRC Interaction Formula for Lateral Buckling of Beam-Columns,” presented to Task Group 3, SSRC, at the Annual Technical Session held at New Orleans, April 1982, 42 pages.

28. Vinnakota, S. [1983a]: “Design of columns as Part of Frames – Some Remarks,” Presented to Task Group 3, SSRC, at the Annual Technical Session held in Toronto, May 1983, pp. 1-26.

29. Vinnakota, S. [1983b]: “Planar Strength of Directionally and Rotationally Restrained Steel Beam-Columns,” Proceedings of the 3rd International Colloquium on Stability of Metal Structures, Toronto, Canada, May 9-11, 1983.

30. Vinnakota, S. [1983c]: “Inelastic Stability of Directionally and Rotationally Restrained Imperfect Steel Columns,” Proceedings of the ASCE-EMD Specialty Conference, Purdue University, Lafayette, IN, May, 1983.

31. Vinnakota, S. [1985a]: “Planar Strength of Restrained Mono-Symmetric Beam-Columns,” Proceedings SSRC Annual Technical Session, April, 1985, pp. 57-70.

32. Vinnakota, S. [1985b]: “Beam Design Under AISC Allowable Stress Design, Plastic Design and Load and Resistance Factor Design,” Presented at ASCE Structural Engineering Congress, Chicago, September, 1985.

33. Foley, C. M. and Vinnakota, S. [1994a]: “Parallel Processing in the Elastic Nonlinear Analysis of High Rise Frameworks,” Journal of Computers and Structures, UK, Vol. 52, No. 6, 1994, pp. 1169-1179.

34. Foley, C. and Vinnakota, S. [1994b]: “Nonlinear Analysis of Structural Frameworks Using Supercomputing Techniques,” Proceedings 1st Congress on Computing in Civil Engineering, ASCE, Washington, DC, June 1994, pp. 2058-2065.

35. Foley, C. and Vinnakota, S. [1995]: “Toward Design Office Moment-Rotation Curves for End-plate Beam-to-Column Connections,” Journal of Constructional Steel Research, Vol. 8, pp. 217-253.

36. Foley, C. M. and Vinnakota, S. [1997]: “Inelastic Analysis of Partially Restrained Steel Frames,” Engineering Structures, Vol. 19, No. 11, 1997, pp. 891-902.

37. Vinnakota, S. and Foley, C. M. [1998]: “Second-Order Elasto-Plastic Analysis of Steel Frames – Review and Comments,” Proceedings SSRC Annual Technical Session, 1998.

38. Foley, C. M. and Vinnakota, S. [1999a]: “Inelastic Behavior of Multi-Story Partially Restrained Steel Frames: Part 1,” Journal of Structural Engineering, ASCE, Vol. 125, ST8, 1999, pp. 854-861.



39. Foley, C. M. and Vinnakota, S. [1999b]: “Inelastic Behavior of Multi-Story Partially Restrained Steel Frames: Part II,” Journal of Structural Engineering, ASCE, Vol. 125, ST8, 1999, pp. 862-869.

Some Ph. D Dissertations on Inelastic Stability

1. Vinnakota, S. [1967]: Inelastic Stability Analysis of Rigid Jointed Frames (Flambage des Cadres dans le domaine elasto-plastique), Thesis submitted on: December 13, 1967; successfully defended on May 22, 1968, published on February 10, 1970, Federal Institute of Technology, Lausanne, Switzerland.

2. Galambos, T. V. [1959]: Inelastic Lateral-Torsional Buckling of Eccentrically Loaded Wide-Flange Columns, Ph. D. Dissertation, Lehigh University, PA.

3. Lu L.W. [1960]: Stability of Elastic and Partially Plastic Frames, Ph. D. Dissertation, Lehigh University, PA.

4. Ojalvo, M. [1960]: Restrained Columns, Ph. D. Dissertation, Lehigh University, PA. 5. Levi, V. [1962]: Plastic Design of Braced Multi-Story Frames, Ph. D. Dissertation,

Lehigh University, PA. 6. Fukumoto, Y. [1963]: Inelastic Lateral-Torsional Buckling Under Moment Gradient, Ph.

D. Dissertation, Lehigh University, PA. 7. Nishino, F. [1964]: Buckling Strength of Columns and Their Component Plates, Ph. D.

Dissertation, Lehigh University, PA. 8. Lay, M. G. [1964]: The Static Load-Deformation Behavior of Planar Steel Structures, Ph.

D. Dissertation, Lehigh University, PA. 9. Milner, H. R. [1965]: The Elastic-Plastic Stability of Stanchions Bent about two-Axes,

Dissertation, University of London, December. 10. Parikh, B. P. [1966]: Elastic-Plastic Analysis and Design of Unbraced Multi-Story Steel

Frames, Ph. D. Dissertation, Lehigh University, PA. 11. Adams, P. F. [1966]: Plastic Design in High-Strength Steel, Ph. D. Dissertation, Lehigh

University, PA. 12. Wright, E. W. [1966]: Analysis of Multi-Story Steel Rigid Frames Subject to Sidesway,

Ph. D. Dissertation, Univ. of Illinois, at Urbana, IL. 13. Yarimci, E. [1966]: Incremental Inelastic Analysis of Framed Structures and Some

Experimental Verifications, Ph. D. Dissertation, Lehigh University, PA. 14. Daniels, J. H. [1967]: Combined Load Analysis of Unbraced frames, Ph. D. Dissertation,

Lehigh University, PA. 15. McName, B. M. [1967]: The General Behavior and strength of Unbraced Multi-Story,

Unbraced Planar Frames, Ph. D. Dissertation, Lehigh University, PA. 16. Korn, A. [1967]: The Elastic-Plastic Behavior of Multi-Story, Unbraced Planar Frames

under gravity loading , Ph. D. Dissertation, Washington University, MO. 17. Trahair, N. S. [1967]: Elastic Stability of Frame Structures, Ph. D. Dissertation,

University of Sydney, Australia, November.

18. McDonald, J. R. [1969]: Inelastic Analysis of Multi-Story Frames, Ph. D. Dissertation, Purdue University, IN.

19. Kim, S. W. [1971]: Elastic-Plastic Analysis of Unbraced Frames, Ph. D. Dissertation, Lehigh University, PA.

20. Bjorhovde, R. [1972]: Deterministic and Probabilistic Approaches to the Strength of Steel Columns, Ph. D. Dissertation, Lehigh University, Bethlehem, PA.



21. Ackroyd, M. H. [1979]: Nonlinear Inelastic Stability of Flexibly Connected Plane Steel Frames, Ph. D. Dissertation, University of Colorado, Boulder, Colo.

22. Orbison, J. G. [1982]: Nonlinear static Analysis of Three Dimensional Steel Frames, Ph. D. Dissertation, Cornell University, Ithaca, NY.

23. Cook, N. E. [1983]: Strength and Stiffness of AISC Type 2 Steel Frames, Ph. D. Dissertation, University of Colorado, Boulder, Colo.

24. Yang, Y. B. [1984]: Linear and Nonlinear analysis of Space Frames with Non-Uniform Torsion Using Interactive Computer Graphics, Ph. D. Dissertation, Cornell University, Ithaca, NY.

25. Barakat, M. [1988]: Simplified Design Analysis of Frames with Semi-Rigid Connections, Ph. D. Dissertation, Purdue University, West Lafayette, IN.

26. Al-Mashary, F. A. [1989]: Simplified nonlinear analysis for steel frames, Ph. D. Dissertation, Purdue University, West Lafayette, IN.

27. King, W. S. [1990]: Simplified Second-order Inelastic Analysis for Frame Design, Ph. D. Dissertation, Purdue University, West Lafayette, IN.

28. Lee, S. L. [1990]: Limit Strength of Semi-rigid Frames, Ph. D. Dissertation, Vanderbilt University, Nashville, TN, USA.

29. Ziemian, R. D. [1990]: Advanced Methods of Inelastic Analysis in the Limit States of Steel Structures, Ph. D. Dissertation, Cornell University, Ithaca, NY.

30. Abdel-Ghaffar, M. M. [1992]: Post-Failure Analysis of Steel Structures, Ph. D. Dissertation, Purdue University, West Lafayette, IN.

31. Liew, J. Y. R. [1992]: Advanced Analysis for Frame Design, Ph. D. Dissertation, Purdue University, West Lafayette, IN.

32. Attalla, M. R. [1995]: Inelastic Torsional-Flexural Behavior and Three-Dimensional Analysis of Steel Frames, Ph. D. Dissertation, Cornell University, Ithaca, NY.

33. Foley, C. M. [1996]: Inelastic Behavior of Partially Restrained Steel Frames using Parallel Processing and Supercomputers, Ph. D. Dissertation, Marquette University, Milwaukee, WI.

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20. Euler, L. [1744]: “Methodus Inveniendi Lineas Curvas…,” Bousquet, Lausanne. 21. Galambos, T. V. [1968]: Structural Members and Frames, Prentice Hall, Inc, Englewood

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Thrust,” Journal Proceedings ASCE, Vol. 85, EM2, p. 1900, April. 24. Galambos, T.V. and Lay, M.G. [196x]: End Moment-End Rotation Characteristics for

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26. Heil, W.: Tranglastermittlung von raumlich belasteten Durchlauy-tragern mit of fenem, dumnwandigem Duerschnitt bei beliebigem Werk-stoffgesetz, Ph. D. Dissertation, University of Karlsruhe, West Germany, 1979.



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