II E> R.AR.Y 
 
 OF THE 
 UN IVLR.SITY 
 OF ILLINOIS 
 
 620- 1 123 
 
 1 1 59/ 
 v. 3-5 
 cop. 2 
 
The person charging this material is re- 
 sponsible for its return to the library from 
 which it was withdrawn on or before the 
 Latest Date stamped below. 
 
 Theft, mutilation, and underlining of books are reasons 
 for disciplinary action and may result in dismissal from 
 the University. 
 To renew call Telephone Center, 333-8400 
 
 UNIVERSITY OF ILLINOIS LIBRARY AT URBANACHAMPAIGN 
 
 CCQ 
 
 02 HI81 
 
 L161— O-1096 
 
Digitized by the Internet Archive 
 
 in 2011 with funding from 
 
 University of Illinois Urbana-Champaign 
 
 http://www.archive.org/details/inelasticdesigno35univ 
 
/ ? 
 
 WADD TECHNICAL REPORT 60-580 
 Part III 
 
 INELASTIC DESIGN OF LOAD CARRYING MEMBERS 
 
 Part III The Significance of an Inelastic Analysis of 
 Eccentrically-Loaded Members 
 
 O.M, Sidebottom 
 
 Theoretical and Applied Mechanics 
 University of Illinois 
 
 WRIGHT AIR DEVELOPMENT DIVISION 
 
WADD TECHNICAL REPORT 60-580 
 Part III 
 
 INELASTIC DESIGN OF LOAD CARRYING MEMBERS 
 
 Part III The Significance of an Inelastic Analysis of 
 Eccentrically-Loaded Members 
 
 O.M. Sidebottom 
 
 Theoretical and Applied Mechanics 
 University of Illinois 
 
 September 26, 1960 
 
 Materials Central 
 Contract No. AF 33(616)-5658 
 Project No. 7351 
 
 Wright Air Development Division 
 
 Air Research and Development Command 
 
 United States Air Force 
 
 Wright Patterson Air Force Base, Ohio 
 
CZ0.IIZ3 
 ,■3 ~ 6 
 
 ENGINEERING LIBRARY, 
 
 • 
 
 FOREWORD 
 
 7-yoo ^3 
 
 This report was prepared by the University of Illinois under USAF 
 Contract No. AF 33(616) -5658 . This Contract was initiated under Project No. 
 7351 "Metallic Materials", Task No. 73521, "Behavior of Metals". It was 
 administered under the direction of the Materials Central, Directorate of 
 Advanced Systems Technology, Wright Air Development Division with Mr. R.F. 
 Klinger acting as the Project Engineer. 
 
 This report covers work conducted from November 1, 1959 to October 
 
 31, 1960. 
 
 The work was conducted in the Department of Theoretical and Applied 
 Mechanics in the Engineering Experiment Station, University of Illinois, 
 Urbana, Illinois. Professor O.M„ Sidebottom was the Project Supervisor. 
 
 WADD TR 60-580 Pt III 
 
ABSTRACT 
 
 The author has worked with others on ten investigations, sponsored by 
 Wright Air Development Division, which have considered the theoretical and 
 experimental inelastic analyses of eccentrically-loaded tension and compression 
 members. In all cases good agreement was found between theory and experiment 
 for numbers tested at room temperature and at elevated temperatures . This in- 
 vestigation was undertaken to consider the significance of an inelastic analy- 
 sis of eccentrically-loaded members. If the inelastic deformation can be con- 
 sidered time independent, a choice has to be made between an elastic and an 
 inelastic solution. A study was made of the effect of several variables on the 
 ratio of the load necessary to produce a specified inelastic deformation to 
 the maximum elastic load. If the inelastic deformation is time dependent 
 (creep), the only choice is an inelastic solution. 
 
 PUBLICATION REVIEW 
 This report has been reviewed and is approved, 
 
 FOR THE COMMANDER 
 
 W .J . Trapp 
 
 Chief, Strength and Dynamics Branch 
 Metals and Ceramics Laboratory 
 Materials Central 
 
 WADD TR 60-580 Pt III iii 
 
TABLE OF CONTENTS 
 
 Page 
 No. 
 
 I . INTRODUCTION 1 
 
 II . THEORETICAL APPROACH 2 
 
 III. ELASTIC VERSUS INELASTIC ANALYSES 5 
 
 WADD TR 60-580 Pt III iv 
 
LIST OF FIGURES 
 
 Figure Page 
 
 No . No . 
 
 1. Idealized Stress-Strain Diagram for Time Independent In- 12 
 elastic Deformation 
 
 2. Constant Depth of Yielding Interaction Curves for Rectangular 13 
 Section Member ( oL = 0) . 
 
 3. Family of Curves for Rectangular-Section Member Giving Rela- 14 
 tion Between A, q, and K. 
 
 4. Family of Curves for Rectangular-Section Member Giving Rela- 15 
 tion Between P , q, and K. 
 
 5. Configurations of Eccentrically-Loaded Tension Members and 16 
 Columns . 
 
 6. Cross-Sections for Eccentrically-Loaded Members. 17 
 
 7. Ratio of Collapse Load to Maximum Elastic Load for Rectan- 18 
 gular-Section Eccentrically-Loaded Columns. 
 
 8. Ratio of Collapse Load tc Maximum Elastic Load for Circular- 19 
 Section Eccentrically-Loaded Columns. 
 
 9. Ratio of Collapse Load to Maximum Elastic Load for Angle- 20 
 Section Eccentrically-Loaded Columns. 
 
 10. Ratio of Collapse Load to Maximum Elastic Load for T-Section 21 
 Eccentrically-Loaded Columns. 
 
 11. Ratio of Collapse Load to Maximum Elastic Load for I-Section 22 
 Eccentrically-Loaded Columns. 
 
 12. Ratio of Collapse Load to Tangent Modulus Load for Eccentri- 23 
 cally-Loaded Columns Made of a Material that Creeps. 
 
 WADD TR 60-580 Pt III 
 
I. INTRODUCTION 
 
 Modern design, particularly in the aircraft and missile fields, requires 
 that load carrying members be used to the limit of their capacity. In many 
 applications the peak loads are applied only a small number 01 times so tnat 
 fatigue is not a problem. For the design of these members, the engineer has 
 to choose between an elastic ana an inelastic analysis. Because of its ease 
 of application and the confidence which the engineer has in its application, 
 the engineer wouia prefer to use the elastic solution. However, the elastic 
 solution is too safe in many applications, and the engineer must use an in- 
 elastic design to utilize the increase in load carrying capacity which is pos- 
 sible if small, inelastic strains are allowed. The percentage increase in 
 load above the maximum elastic load depends upon such variables as type of 
 member, shape of cross-section, shape of stress-strain diagram of the material, 
 and the amount of inelastic deformation. In the case of design lor an ele- 
 vated temperature of sufficient magnitude to produce creep, an inelastic anal- 
 ysis is always required. 
 
 The difficulty of an analysis depends upon the type of relation used to 
 represent the stress-strain diagram for the material. Linear elasticity uses 
 one of the simplest relations, namely, a straight line. If the material has 
 a yield point and the member is designed for strength rather than deformation, 
 a fully plastic analysis may be used; the resulting solution may be easier to 
 apply than an elastic solution. *'A fully plastic analysis cannot be used in a 
 large number of instances if any one of the following conditions are present: 
 
 1. The member is made of a material which does not have a yield point. 
 
 2. It is necessary to know the load-deformation relation for inelastic 
 conditions . 
 
 3. The member may become unstable before reaching the fully plastic con- 
 dition as is the case of an eccentrically-loaded column. 
 
 In most cases, the loads on the members are assumed to maintain their same re- 
 lative magnitude and to increase in magnitude so that a nonlinear elastic 
 analysis can be used. In deriving the theoretical relations, the stress-strain 
 diagram of the material is represented by one non-linear function or two func- 
 tions. At room temperature, two functions are usually used to represent the 
 stress-strain diagram for metals, and the resulting solution is called an elas- 
 tic-plastic solution. 
 
 Because of the ever increasing interest is the inelastic analysis of load 
 carrying members, it is necessary that theoretical inelastic analyses be de- 
 veloped for all types of load carrying members. Furthermore, it is necessary 
 that these theories be checked experimentally for the following reasons: 
 
 1. Frequently simplifying assumptions are used in order to reduce the 
 complexity of the theory so that the theory can be readily applied by 
 the design engineer. It is necessary to determine the influence of these 
 simplifying assumptions on the agreement between theory and experiment. 
 
 Manuscript released by author September 26, 1960, for publication as a 
 WADD Technical Report . 
 
 WADD TR 60-580 Pt III 1 
 
2. It is logical that the engineer does not have the same confidence in 
 an inelastic analysis which he has in an elastic analysis . Experimental 
 verification of inelastic theories helps promote confidence in those 
 theories . 
 
 Some of the investigations (1 through 10) sponsored by the Wright Air 
 Development Division have considered the theoretical and experimental inelas- 
 tic analyses of eccentrically-loaded tension and compression members. These 
 investigations have considered a large number of variables, namely: 
 
 1. Several types of ductile metals such as S.A.E. 4340 steel, 7075-T 6 
 aluminum alloy, type 304 stainless steel, 17-7FH stainless steel, and 
 Ti 155A titanium alloy. 
 
 2. Several types of cross-sections such as rectangular .angle , and T- 
 sections . 
 
 3. Several end conditions for the columns such as fixed ends and pivot 
 ends with equal and unequal end eccentricities of various magnitudes. 
 
 4. Several slenderness ratios for the columns, 
 
 5. Both room temperature and elevated temperatures. 
 
 In all cases good agreement was found between theory and experiment indicating 
 that the theory was sufficiently exact to be used in the design of eccentri- 
 cally-loaded members. Since the theory was found to be reliable, was the in- 
 crease in load carrying capacity of a member (which resulted from the inelastic 
 deformation) of sufficient magnitude to justify the increase in labor necessary 
 to complete an inelastic analysis? Answers to this question vary from yes for 
 some cases to no for other cases. In order to obtain definite answers to the 
 question, the theory will be reviewed so that the effect of various variables 
 on the increase in load carrying capacity of eccentrically-loaded members can 
 be determined . 
 
 II. THEORETICAL APPROACH 
 
 In deriving theoretical, load-deflection relations for eccentrically- 
 loaded members, the procedure is to equate the internal load and moment at 
 every section of the member to the applied load and moment. For elastic con- 
 ditions, Timoshenko (11) has presented closed solutions for rather general 
 loading conditions. Closed solutions have not been found for inelastic condi- 
 tions. Several analyses requiring a trial and error solution have been pre- 
 sented (2,3,5,6,7,8,9,10,12,13,14). In each case, a family of interaction 
 curves was constructed to give a relation between the internal load, moment, 
 and curvature at any section of the member. A relation between applied load, 
 moment, and curvature was obtained by approximating the configuration of the 
 deformed member , 
 
 * Numbers in parenthesis refer to correspondingly numbered entries in the 
 Bibliography . 
 
 WADD TR 60-580 Pt III 2 
 
At least three different types of interaction curves have been used in 
 the analyses of eccentrically-loaded columns in which the inelastic deforma- 
 tion was primarily time independent. Chwalla (13) and Bijlaard (12) used a 
 family of curves giving the relations between the average stresses of origi- 
 nally straight, centrally-loaded columns and the post-buckling deflection. 
 Galambos and Ketter (14) used dimensionless , moment versus curvature curves 
 for specified loads. Sidebottom and Clark (2) approximated the stress-strain 
 diagram of the material by two straight lines (see Fig. 1), and constructed 
 dimensionless, moment versus load curves for constant depths of yielding. 
 These curves are shown in Fig. 2 for rectangular-section members made of a 
 material with o^ = (see Fig. 1). In Fig. 2, P e = %& and M e = 0~ e I/c , 
 It is the author's opinion that constant depth of yielding interaction curves 
 are the easiest to construct and use. 
 
 In case the inelastic deformation is time dependent (creep), Sidebottom, 
 Clark, and Dharmarajan (6) approximated the isochronous stress-strain diagram 
 of the material by an arc hyperbolic sine curve of the form 
 
 0~~ 0~ arc sinh -gr- (1) 
 
 in which CJ and 6 are experimental constants. Since the stress distribu- 
 tion in the eccentrically loaded member changes with time and <T and 6 were 
 obtained from constant stress creep curves, Eq . 1 does not accurately repre- 
 sent the stress-strain relation in the member at a specified time. Good 
 agreement was found between theory and experiment (9,10) by decreasing <J by 
 10 per cent for eccentrically-loaded tension members and by increasing 0~ by 
 10 per cent for eccentrically-loaded columns. Based on Eq . 1, two families 
 of interaction curves were constructed as indicated in Figs. 3 and 4 for a 
 rectangular-section member. In these figures A is the distance from the ac- 
 tion line of the load to the centroidal axis of the member at the point of 
 maximum deflection, h is the depth of the member, qh is the distance from the 
 most strained fiber to the neutral axis, and K is Che ratio of the maximum 
 strain at a given section to £ . 
 
 Using the interaction curves shown in either Fig. 2 or Figs. 3 and 4, the 
 deflection, load, and curvature can be computed for any assumed point on any 
 of the curves. Another relation for the curvature of an eccentrically-loaded 
 member is obtained in terms of the configuration of the member. A trial and 
 error solution is required in order to determine the point on a given inter- 
 action curve which will make the two curvatures equal. One point on the the- 
 oretical load-deflection curve for the eccentrically-loaded member may be 
 obtained from each interaction curve. 
 
 Consider the eccentrically-loaded tension member shown in Fig. 5a. Since 
 the exact configuration of the member was difficult to obtain for inelastic 
 conditions, the axis of the member was assumed (2,5,6,7,8,9) to deform into a 
 
 Diagram obtained by plotting values of stress and strain obtained at a 
 given time from constant stress creep curves. 
 
 WADD TR 60-580 Pt III 3 
 
segment of a circle. This configuration assumes that every section has the 
 same curvature as that for section C. Since every other section has a curva- 
 ture greater than that at C, the resulting theoretical deflection will be less 
 than actual, i.e. nonconservative . The error is small if e is large compared 
 to /. 
 
 In the case of an eccentrically-loaded column (see Fig. 5b), Chwalla (13) 
 and Galambos and Ketter (14) have presented procedures to approximate the con- 
 figuration of the inelastically deformed column as accurately as desired. 
 These procedures could also be used in the analysis of eccentrically-loaded 
 tension members and columns having end conditions other than that shown in 
 Fig. 5b. However, these exact solutions are extremely time consuming. Since 
 a column having any known end condition deforms into a sine curve elastically, 
 Bijlaard (12) suggested that the inelastic column also be assumed to deform 
 into a sine curve. For the column shown in Fig. 5b, the curve DCF can be rep- 
 resented by the following relation: 
 
 TT x 
 y = A sin - TT~ (2) 
 
 The curvature of the column at section C can be obtained from Eq . 2 to be 
 equated with the curvature obtained from the assumed point on a given inter- 
 action curve. The correct point on the interaction curve will satisfy the 
 condition that y = e when x = u . Solutions for general end conditions are 
 presented in references 10 and 12. Theoretical (12) and experimental (10) 
 investigations have shown that relatively small errors are introduced by the 
 assumption that the inelastic column deforms into a sine curve. 
 
 In several investigations (3,5,9), the curve ACB was represented both by 
 a segment of a circle and a sine curve of the form 
 
 c . 1? x 
 y = ^sin-_ (3) 
 
 Since neither of these approximations introduced another unknown (such as L 
 when using Eq . 2), they are easier to work with than Eq. 2. For a given load, 
 the segment of a circle approximation gives a deflection greater than actual 
 (conservative), while Eq. 3 gives a deflection less than actual (nonconserva- 
 tive). Good agreement was found (3,5,9) between theoretical collapse loads 
 based on Eq . 3 and the experimental collapse loads. Furthermore, the ratios 
 of collapse loads to maximum elastic loads are nearly identical for theories 
 based on Eq. 2 and Eq. 3. Computations have been made for eccentricities as 
 large as 25 per cent of the column depth. 
 
 WADD TR 60-580 Pt III 
 
III. ELASTIC VERSUS INELASTIC ANALYSES 
 
 The discussion which follows will consider the ratio of the load neces- 
 sary to produce a specified amount of inelastic deformation to the maximum 
 elastic load for eccentrically-loaded members. For a given member this ratio 
 depends upon the amount of inelastic deformation, the shape of cross-section, 
 and the magnitude of c^ (see Fig. 1). The ratio is also greatly influenced 
 by the type of loading. In the case of an eccentrically-loaded tension mem- 
 ber, the ratio continues to increase with inelastic deformation (limited by 
 the fully plastic condition when ai = )• The ratio reaches a maximum at the 
 collapse load in the case of eccentrically-loaded columns. Separate discus- 
 sions will be presented for each type of member. In each case data will be 
 presented for the five cross-sections shown in Fig. 6. The I-section member 
 is assumed to deform without local buckling. 
 
 Eccentrically-Loaded Tension Members . As indicated in Fig. 5a, the sec- 
 tion of an eccentrically-loaded tension member having maximum moment is located 
 at either end of the member. Since small deformation theory is used, the mo- 
 ment remains constant and equal to Pe . If the inelastic deformation is 
 primarily time independent, the stress-strain diagram of the material is ap- 
 proximated by two straight lines as indicated in Fig. 1, and interaction curves 
 similar to those shown in Fig. 2 are used in the inelastic analysis. Since 
 the ratio of moment to load remains constant, the load necessary to produce 
 any depth of yielding in an eccentrically-loaded tension member may be obtained 
 from a radial line drawn from point in Fig. 2. The slope of line OB in Fig. 
 2 is equal to the ratio of M/M e to P/P e and can be shown to be equal to ec/r 2 
 where c is the distance from the centroidal axis to the outermost fiber of the 
 member, and r is the radius of gyration of the cross-section with respect to 
 the same centroidal axis . 
 
 The maximum elastic load and the load necessary to produce any depth of 
 yielding are easily obtained from the radial line in Fig. 2. The ratio of the 
 load necessary to produce a specified depth of yielding to the maximum elastic 
 load increases from 1.00 for G = (centrally-loaded tension member) to a 
 maximum for a value of between and Tr/<=. When & is equal to n/g , the 
 member is subjected to pure bending. 
 
 The line OB shown in Fig. 2 has a slope of 1.5. The ratio of the load 
 necessary to produce one half depth of yielding to the maximum elastic load 
 was computed for a slope of 1.5 for all of the cross-sections shown in Fig. 6. 
 These data are shown in Table 1 for values of oi =0 and oc =0.20. It will be 
 noted that an appreciable increase in load carrying capacity resulted from 
 allowing the inelastic deformation. The increase is greatly influenced by the 
 shape of cross-section and by the strain hardening factor, o^ . The increase 
 in deflection for this increase in load may be obtained using the theories pre- 
 sented in references 2,5,6,7,8, or 9. The deflection necessary to produce one 
 half depth of yielding at each end of an eccentrically-loaded tension member 
 is in the neighborhood of twice the maximum elastic deflection. 
 
 WADD TR 60-580 Pt III 
 
In case the inelastic deformation is time dependent (creep), there is no 
 choice between an elastic and an inelastic design since any design for creep 
 is based on an inelastic analysis. The load necessary to produce a specified 
 strain in the most strained fibers can be obtained using the procedure outlined 
 in references 6,7,8,9, or 10. This procedure uses interaction curves such as 
 those shown in Figs . 3 and 4 . 
 
 Eccentrically-Loaded Columns . As indicated in Fig. 5b, the maximum mo- 
 ment in the eccentrically-loaded column occurs at the section which has the 
 maximum deflection A „ Thus, an inelastic analysis requires that the load- 
 deflection curve for the column be obtained. If the inelastic deformation is 
 primarily time independent, constant depth of yielding interaction curves 
 similar to those shown in Fig. 2 are used in the inelastic analysis. By the 
 procedure outlined in references 3,5,9, or 10, a moment-load curve similar to 
 curve OD in Fig. 2 can be constructed for a given column from which the load- 
 deflection curve is obtained. It will be noted that the load carried by the 
 eccentrically-loaded column does not continue to increase with an increase in 
 depth of yielding but that the load is limited to the collapse load. 
 
 The effect of inelastic deformation on the load carrying capacity of ec- 
 centrically-loaded columns will be indicated by the ratio of the collapse load 
 to the maximum elastic load. This ratio is influenced by such variables as 
 cross-section, slenderness ratio, yield strain, strain hardening factor oc , 
 and initial eccentricity. The two variables, slenderness ratio and yield 
 
 strain, can be replaced by one variable £ (iVr) . 
 
 6 
 
 Ratios of the collapse load to the maximum elastic load are shown in 
 Figs. 7 through 11 for the 5 cross-sections shown in Fig. 6. Data is given 
 in each case for two initial eccentricities of 5 and 25 per cent of the column 
 depth and for two strain hardening factors of zero and 0.20. It will be noted 
 that the ratio of the collapse load to maximum elastic load approaches 1.0 
 for large values of £ p {Q/r} ; furthermore, the effect ofoZ.cn the ratio is 
 small for large values of £ e (iVr) . The magnitude of £ £ (P/r} at which 
 the ratio becomes small depends upon the shape of cross-section and upon the 
 initial eccentricity. The effect of shaoe of cross-section can be represented, 
 at least partially, by the variable ch/r where c is the distance from the 
 neutral axis to the outermost fibers of the member. Consider the dimensionless 
 product , 
 
 S = Ch 1 
 
 " r 2 (1- /h)* ViQ/rV€« (3) 
 
 If S is less than or equal to 17, the ratio of collapse load to maximum elas- 
 tic load will be less than 1.05 for all of the variables considered in Figs. 7 
 through 11. In the case of the T-section member, the ratio will be less than 
 1.05 for values of S as large as 28. When these conditions are satisfied, an 
 inelastic analysis is not justified. The minimum slenderness ratio, for sev- 
 eral materials and for three eccentricities, necessary to have the ratio less 
 than 1.05 were calculated and are shown in Table 2 for the cross-sections 
 
 WADD TR 60-580 Pt III 
 
shown in Fig. 6. As indicated in Table 2, there are many applications for 
 which an inelastic analysis is not justified, particularly for eccentrically- 
 loaded columns made of high strength, low modulus materials for which the 
 initial eccentricity is small. It should be noted that the forementioned 
 values of S are valid for unequal eccentricities if the larger value of e/h 
 is substituted in Eq. 3. 
 
 If S is greater than 28 for the T-section member and greater than 17 
 for the other cross sections shown in Fig. 6, an inelastic analysis may re- 
 sult in an appreciable saving in weight and cost. As indicated in Figs. 7 
 through 11, the ratio of collapse load to maximum elastic load may be as 
 large as 2.00 in some instances. This ratio will be larger for larger ini- 
 tial eccentricities. 
 
 In case the inelastic deformation is time dependent (creep), the design 
 of an eccentrically-loaded column must be based on an inelastic analysis. 
 For centrally-loaded columns with pivot ends, the collapse load is the tangent 
 modulus load which is easily computed using Eq. 1. Since Eq . 1 does not 
 accurately represent the stress-strain relation for the material in the column 
 at the specified time (see Section II), the actual collapse load will be 
 greater than the tangent modulus load by 10 per cent . The ratio of the col- 
 lapse load of an eccentrically-loaded column for a specified time to the tan- 
 gent modulus load for the same time was computed for the rectangular-section, 
 the T-section, and the I-section and are shown in Fig. 12. For zero initial 
 eccentricity the column is centrally-loaded and the collapse load is 10 per 
 cent greater than the tangent modulus load as explained in Section II. The 
 experimental data presented in references 9 and 10 indicate that , for a given 
 cross-section and a given initial eccentricity, the ratio of the collapse 
 load to the tangent modulus load is not influenced by either the slenderness 
 ratio or the time to collapse if the tangent modulus load is calculated for 
 the specified time. 
 
 WADD TR 60-580 Pt III 
 
BIBLIOGRAPHY 
 
 1. Brush, D.O., Sidebottora, O.M. and Smith, J. 0., "Axial Tension and Bending 
 Interaction Curves for Members Loaded Inelastically ," WADC Technical Re- 
 port, February, 1952. 
 
 2. Sidebottom, O.M. and Clark, M.E., "The Effects of Inelastic Action of the 
 Resistance to Various Types of Loads of Ductile Members Made from Various 
 Classes of Metals — Part I Eccentrically-Loaded Tension Members Having 
 Angle-and T-Sections," TAM Report No. 100, March, 1956, WADC Technical 
 Report, April, 1957. 
 
 3. Clark, M.E. and Sidebottom, O.M., "The Effects of Inelastic Action on the 
 Resistance to Various Types of Loads of Ductile Members Made from Various 
 Classes of Metals — Part IV Eccentrically-Loaded Columns Having Angle- 
 and T-Sections," TAM Report No. 112, November, 1956, WADC Technical Report. 
 
 4. Maeda, Y., Van Lierde, P., Sidebottom, O.M., and Clark, M.E., "The Effects 
 of Inelastic Action on the Resistance to Various Types of Ductile Members 
 Made from Various Classes of Metals — Part VI A Digital Computer Analysis 
 of Bending Moment — Axial Load Interaction Curves," TAM Report No. 115, 
 January, 1957, WADC Technical Report. 
 
 5. Sidebottom, O.M. and Clark, M.E., "A Theoretical and Experimental Analysis 
 of Members Loaded Eccentrically and Inelastically," Bulletin Series No. 
 447, Engineering Experiment Station, University of Illinois, 1958. 
 
 6. Sidebottom, .M . , Clark, M.E., and Dharmarajan, S., "The Effects of Inelas- 
 tic Action on the Resistance to Various Types of Loads of Ductile Members 
 Made from Various Classes of Metals — Part VIII Eccentrically Loaded Ten- 
 sion Members Made of Two Stainless Steels tested at Elevated Temperatures, 
 TAM Report No. 132, April, 1958, WADC Technical Report, October, 1958. 
 
 7. Sidebottom, O.M. and Dharmarajan, S., "The Effects of Inelastic Action on 
 the Resistance to Various Types of Loads of Ductile Members Made from 
 Various Classes of Metals — Part IX T-Section Eccentrically-Loaded Ten- 
 sion Members Made of Type 304 Stainless Steel and Tested at 1000 F," TAM 
 Report No. 134, May, 1958, WADC Technical Report, October, 1958. 
 
 8. Sidebottom, O.M. and Dharmarajan, S., "The Effects of Inelastic Action on 
 the Resistance to Various Types of Loads of Ductile Members Made from 
 Various Classes of Metals — Part X T-Section Eccentrically-Loaded Tension 
 Members of 17-7 PH Stainless Steel and Tested at 1000°F," TAM Report No. 
 144, November, 1958, WADC Technical Report, May 1959. 
 
 ii 
 
 WADD TR 60-580 Pt III 
 
9. Sidebottom, O.M., Dharmarajan, S., Gubser, I.L. and Leasure, J.D., "The 
 Effects of Inelastic Action on the Resistance to Various Types of Ductile 
 Members Made of Various Classes of Metals — Part XII Eccentrically-Loaded 
 Tension Members and Columns Made of 17-7 PH Stainless Steel and Ti 155A 
 Titanium Alloy and Tested at Various Temperatures," TAM Report No. 154, 
 October, 1959, WADC Technical Report. 
 
 10. Costello, G.A., Sidebottom, O.M., and Pocs, E.," Inelastic Design of Load 
 Carrying Members — Part II The Effects of End Conditions on the Collapse 
 Loads of Columns," TAM Report No. 178, September, 1960, WADD Technical Re- 
 port . 
 
 11. Timoshenko, S., "Theory of Elastic Stability," McGraw-Hill Book Co., Inc., 
 New York, 1936. 
 
 12. Bijlaard, P.P., "Buckling of Columns with Equal and Unequal End Eccen- 
 tricities and Equal and Unequal Rotational Restraints," Proceeding of the 
 Second U S. National Congress of Applied Mechanics, 1954, pp. 555-562. 
 
 13. Chwalla, E., " Aussermittig gedrcukte Baustahlstabe mit elastisch 
 eingespannten Enden und verscheiden grossen Augrif f shebeln," Der Stahlban, 
 1937, Nos. 7-8. 
 
 14. Galambos, T.V. and Ketter, R.L., "Columns Under Combined Bending and 
 Thrust," Proceedings ASCE, Vol. 85, No. EM 2, April 1959. 
 
 WADD TR 60-580 Pt III 
 
TABLE 1 
 
 Ratio of Load Necessary to Produce One Half Depth of Yielding in 
 an Eccentrically-Loaded Tension Member and Maximum 
 Elastic Load 
 
 Strain 
 Hardening 
 Factor 
 
 
 0.2 
 
 Ratio when tan = 1.5 
 
 Rectangular- Circular- Angle- 
 Section Section Section T-Section I-Section 
 
 1.43 
 1.66 
 
 1.53 
 1.74 
 
 1.51 
 1.78 
 
 1.27 
 1.62 
 
 1.22 
 1.46 
 
 WADD TR 60-580 Pt III 
 
 10 
 
IjA HI ,Vf 2 
 
 Minimum Slenderness Ratios for Eccentrically-Loaded Columns 
 for which an Inelastic Analysis is Not Justified 
 
 Material 
 
 % 
 
 (psi) 
 
 e e 
 
 Slenderness Ratio for 
 Angle- 
 
 Rectangular- Circular- 
 Section Section Section T-Section I-Section 
 
 Steel 
 Steel 
 Steel 
 17-7PH 
 
 30,000 .0010 
 
 90,000 .0030 
 
 180,000 .0060 
 
 183,000 .065 
 
 7075-T6 80,000 .0078 
 Ti 155A 164,000 .0092 
 
 Steel 
 Steel 
 Steel 
 17-7PH 
 
 30,000 .0010 
 
 90,000 .0030 
 
 180,000 .0060 
 
 183,000 .0065 
 
 7075-T6 80,000 .0078 
 Ti 155A 164,000 .0092 
 
 Steel 
 
 Steel 
 
 Steel 
 
 17-7PH 
 
 7075-T6 
 
 Ti 155A 
 
 30,000 
 
 90,000 
 
 180,000 
 
 183,000 
 
 80,000 
 
 164,000 
 
 .0010 
 .0010 
 .0060 
 .0065 
 .0078 
 .0092 
 
 e/h = 0.05 
 124 
 71 
 50 
 49 
 44 
 41 
 
 e/h = 0.25 
 198 
 115 
 
 81 
 
 78 
 
 71 
 
 65 
 
 e/h = 0.50 
 446 
 258 
 182 
 175 
 160 
 147 
 
 165 
 95 
 67 
 65 
 59 
 54 
 
 265 
 153 
 108 
 104 
 95 
 87 
 
 595 
 344 
 243 
 234 
 213 
 196 
 
 202 
 
 117 
 
 82 
 
 79 
 
 72 
 67 
 
 324 
 187 
 132 
 127 
 116 
 107 
 
 729 
 421 
 298 
 286 
 261 
 240 
 
 98 
 57 
 40 
 39 
 35 
 32 
 
 157 
 91 
 64 
 62 
 56 
 52 
 
 354 
 205 
 145 
 139 
 127 
 117 
 
 70 
 40 
 29 
 28 
 25 
 23 
 
 113 
 65 
 46 
 
 44 
 40 
 37 
 
 253 
 
 146 
 
 103 
 
 99 
 
 91 
 
 83 
 
 WADD TR 60-580 Pt III 
 
 11 
 
 UNIVERSITY Of 
 ILLINOIS LIBRAE 
 
(0 
 
 w 
 
 (0 
 
 0) 
 
 u 
 •p 
 
 CO 
 
 Slope = u E 
 
 o; 
 
 Strain, 
 
 Fig. 1 Idealized Stress-Strain Diagram for Time Independent 
 
 Inelastic Deformation 
 
 WADD TR 60-580 Pt III 
 
 12 
 
(U 
 
 s 
 
 1.0 
 
 P/Pe 
 
 Fig. 2 Constant Depth of Yielding Interaction Curves for 
 Rectangular-Section Member ( oi =0) 
 
 WADD TR 60-580 Pt III 
 
 13 
 
Fig. 3. Family of Curves for Rectangular-Section Member Giving Relation Between 
 A, q, and K 
 
 WADD TR 60-580 Pt III 
 
 14 
 
WADD TR 60-580 Pt III 
 
 15 
 
^ -£ 
 
 -J 
 
 ■S- 
 
 : C 
 
 (a) Eccentrically-Loaded Tension Member (b) Eccentrically-Loaded 
 
 Column 
 Fig. 5 Configurations of Eccentrically-Loaded Tension Members and Columns. 
 
 WADD TR 60-580 Pt III 16 
 
h b 
 
 (a) Rectangular-Section 
 
 (b) Circular-Section 
 
 ■4-t, 
 
 I 
 
 1 
 
 ii 
 
 -c 
 
 (d) T-Section 
 
 (e) I-Section 
 
 Fig. 6 Cross-Sections for Eccentrically-Loaded Members 
 
 
 WADD TR 60-580 Pt III 
 
 17 
 
Fig. 7 Ratio of Collapse Load to .vliximum Elastic Load for Rectangulai 
 Section Eccentrically-Loaded Columns 
 
 WADD TR 60-580 Pt III 
 
 18 
 
2.0 
 
 ^< <?/r)< 
 
 Fig. 8 Ratio of Collapse Load to Maximum Elastic Load for Circular-Section 
 
 Eccentrically-Loaded Columns 
 
 WADD TR 60-580 Pt III 
 
 19 
 
2.Q 
 
 1.4 
 
 20 
 
 e e (Mo 2 
 
 30 
 
 40 
 
 Fig. 9 Ratio of Collapse Load to Maximum Elastic Load for Angle-Section 
 
 Eccentrically-Loaded Columns 
 
 WADD TR 60-580 Pt III 
 
 20 
 
2.0 
 
 € e a/-r 
 
 Fig. 10 Ratio of Collapse Load to Maximum Elastic Load for T-Section 
 
 Eccentrically-Loaded Columns 
 
 WADD TR 60-580 Pt III 
 
 21 
 
2.Q 
 
 l.J 
 
 •a 
 o 
 
 (0 
 
 « 
 i-i 
 
 w 
 
 a 
 a 
 s 
 
 •H 
 CD 
 
 2 
 O 
 
 •o 
 oj 
 o 
 
 1.6 
 
 ft 1.4 
 at 
 
 O 
 
 h 
 
 8* 
 
 di 
 
 t — 
 
 n 
 
 *-> 
 
 OS 
 
 1.2 
 
 Curve 
 
 e/'h 
 
 0.05 
 0.05 
 0.25 
 0.24 
 
 06 
 
 0.00 
 0.20 
 0.00 
 0.20 
 
 1.0 
 
 10 
 
 20 
 £ e WrY 
 
 30 
 
 40 
 
 Fig. 11 
 
 Ratio of Collapse Load to Maximum Elastic Load for I-Section 
 Eccentrically-Loaded Columns 
 
 WADD TR 60-580 Pt III 
 
 22 
 
1.2 
 
 ■a 
 
 cs 
 o 
 
 (0 
 
 3 
 
 rH 
 
 3 
 
 •o 
 o 
 
 s 
 
 c 
 0) 
 
 c 
 
 a) 
 H 
 
 O 
 
 ■p 
 
 XJ 
 0) 
 
 o 
 
 V 
 (0 
 
 a 
 
 
 O 
 
 o 
 
 -p 
 at 
 
 1.0 
 
 0.8 
 
 0.2 
 
 -Ul- 
 
 k— 8t — -| 
 
 .05 0.10 0.15 
 
 INITIAL ECCENTRICITY, e/h 
 
 0.20 
 
 0.25 
 
 Fig. 12 Ratio of Collapse Load to Tangent Modulus Load for Eccentrically- 
 Loaded Columns Made of a Material that Creeps 
 
 WADD TR 60-580 Pt III 
 
 23 
 
WADD TECHNICAL REPORT 60-580 
 Part IV 
 
 INELASTIC DESIGN OF LOAD CARRYING MEMBERS 
 
 Part IV. The Behavior of Beam -Columns in the 
 Inelastic Range 
 
 B. B. Muvdi 
 O. M. Sidebottom 
 
 Theoretical and Applied Mechanics Department 
 University of Illinois 
 
 April , 1961 
 
 Materials Central 
 Contract No. AF 33 (6 16) -7600 
 Project No: 7351 
 
 Wright Air Development Division 
 Air Research and Development Command 
 United States Air Force 
 Wright Patterson Air Force Base, Ohio 
 
WADD TECHNICAL REPORT 60-580 
 Part IV 
 
 INELASTIC DESIGN OF LOAD CARRYING MEMBERS 
 
 Part IV. The Behavior of Beam -Columns in the 
 Inelastic Range 
 
 B. B. Muvdi 
 O. M. Sidebottom 
 
 Theoretical and Applied Mechanics 
 University of Illinois 
 
 April 1961 
 
 WRIGHT AIR DEVELOPMENT DIVISION 
 
FOREWORD 
 
 This report was prepared by the University of Illinois under USAF Contract 
 No. AF 33(6l6)-7600. This Contract was initiated under Project No. 7351, 
 "Metallic Materials", Task No. 73521, "Behavior of Metals". It was administered 
 under the direction of the Materials Central, Directorate of Advanced Systems 
 Technology, Wright Air Development Division with Mr. R. F. Klinger acting as the 
 Project Engineer. 
 
 This report covers work conducted from October 1 1960 to Setpember 31, 
 1961. 
 
 The work was conducted in the Department of Theoretical and Applied 
 Mechanics in the Engineering Experiment Station, University of Illinois, Urbana, 
 Illinois. Professor O. M. Sidebottom was the Project Supervisor. 
 
ABSTRACT 
 
 Two theories were presented for constructing either moment -load or load- 
 deflection relations as well as the collapse loads for beam-columns. In each case, 
 trial and error solutions were required which used constant depth of yielding interaction 
 curves. A "so called" exact theory was presented which gave results as accurate as 
 desired; however, the theory was not -practical because of the excessive time required. 
 An approximate theory was presented which gave results in close agreement with the 
 exact theory and with experimental data. This theory required the elastic solution 
 for maximum elastic conditions. 
 
 The experimental part of the investigation included tests of rectangular - and 
 T-section columns made of 2024-T4 aluminum alloy, SAE 1020 steel, and 17-7PH 
 stainless steel. Several slenderness ratios were considered. In addition to the 
 variable axial load, the columns wer subjected to a constant transverse load either at 
 midspan of at quarter span which produced a bending stress of 0. 25, 0. 50, or 0. 75 C . 
 
 PUBLICATION REVIEW 
 This report has been reviewed and is approved. 
 FOR THE COMMANDER: 
 
 W. J. Trapp 
 
 Chief, Strength and Dynamics Branch 
 Metals and Ceramics Laboratory 
 Materials Central 
 
 in 
 
TABLE OF CONTENTS 
 
 Page 
 
 I. INTRODUCTION 1 
 
 1. Preliminary Statement 1 
 
 2. Purpose and Scope 2 
 
 II. ANALYSIS 4 
 
 1. Interaction Curves 4 
 
 2. Elastic Solution for the Beam -Column 7 
 
 3. Method of Solution 11 
 
 4. Sample Problem 12 
 
 III. MATERIALS AND TEST PROCEDURE 15 
 
 1. Materials and Specimens 15 
 
 2. Apparatus and Test Procedure 16 
 
 IV. RESULTS AND DISCUSSION 18 
 
 1. Symmetrically Loaded Beam -Columns 18 
 
 2. Unsymmetrically Loaded Beam -Columns 20 
 
 V. SUMMARY AND CONCLUSIONS 23 
 
 VI. BIBLIOGRAPHY 25 
 VII. APPENDIX A 27 
 
 IV 
 
LIST OF ILLUSTRATIONS 
 Figure Page 
 
 1 T Section Member Subjected to Combined Loading. The 34 
 Sketch Shows the Stress Distribution When the Member is 
 
 Partly Elastic and Partly Inelastic. 
 
 2 A Schematic Representation of Stress -Strain Relations in 35 
 Tension and Compression. 
 
 3 A Schematic Representation of Interaction Curves for 36 
 Constant Depth of Yielding . 
 
 4 A Sketch Showing the Beam -Column Studied in this Program. 37 
 
 5 Dimensionless Interaction Curves for Rectangular Sections 38 
 Made of 2024-T4 Aluminum. 
 
 6 Dimensionless Interaction Curves for T-Sections Made of 39 
 2024-T4 Aluminum . 
 
 7 Dimensionless Interaction Curves for Rectangular Sections 40 
 Made of SAE 1020 Steel. 
 
 8 Dimensionless Interaction Curves for T-Sections Made of 41 
 SAE 1020 Steel. 
 
 9 . Dimensionless Interaction Curves for Rectangular Sections 42 
 
 Made of 17-7PH Stainless Steel. 
 
 10 Test Apparatus Used to Apply Transverse and Axial Loads 43 
 to Beam Columns. 
 
 11 Knife -Edge Fixtures Used to Apply Axial Load to Beam 44 
 Columns. 
 
 12 Comparison of Moment Load Curves for Symmetrically 45 
 Loaded 2024-T4 and SAE 1020 Steel Beam-Columns Determined 
 
 by Two Theoretical Methods and by Test. 
 
 13 Moment -Load Curves for Symmetrically Loaded, Rectangular 46 
 2024 -T4 Aluminum Beam-Columns. (M„/M = 0.25, 
 
 e x = e 2 = 0) <* e 
 
 14 Moment -Load Curves for Symmetrically Loaded, Rectangular 47 
 2024-T4 Aluminum Beam-Columns. (M n /M = 0. 50; 
 
 e = e 2 = 0) ^ e 
 
LIST OF ILLUSTRATIONS 
 Figure Page 
 
 15 Moment-Load Curves for Symmetraically Loaded, 48 
 Rectangular 2024-T4 Aluminum Beam -Columns. 
 
 (M Q /M e = 0. 25; e x = e £ = 0. 15h) 
 
 16 Moment -Load Curves for Symmetrically Loaded T-Section 49 
 Beam Columns of 2024-T4 Aluminum. 
 
 17 Moment-Load Curves for Symmetrically Loaded, Rectan- 50 
 gular SAE 1020 Steel Beam-Columns. (M n /M = 0. 50; 
 
 e x = e 2 = 0) Q e 
 
 18 Moment-Load Curves for Symmetrically Loaded Rectangular 51 
 SAE 1020 Steel Beam Columns, (M Q /M = 0. 75; e, = e 2 = 0) 
 
 19 Moment- Load Curves for Symmetrically Loaded T-Section 52 
 Beam -Columns of SAE 1020 Steel. 
 
 20 Moment -Load Curves for Symmetrically Loaded Rectangular 53 
 17-7PH Stainless Steel Beam-Columns, (e, = e~ =0) 
 
 21 Moment-Load Curves for Symmetrically Loaded; Rectangular 54 
 17-7PH Stainless Steel Beam-Columns, (e. = e = 0. 15h) 
 
 22 Moment- Load Curves for Unsymmetraicaily Loaded, 55 
 Rectangular 2024-T4 Aluminum Beam-Columns. (M n /M 
 
 = 0. 50; e x = e 2 = 0) g e 
 
 23 Moment-Load Curve for an Unsymmetric -Loaded T-Section 56 
 Beam -Column of 2024- T4 Aluminum. (M n /M =0.50; 
 
 e x = e 2 = 0) ^ e 
 
 24 Moment-Load Curves for Unsymmetrkaily Loaded, Rectangular 57 
 SAE 1020 Steel Beam -Columns. <M Q /M = 0. 50; e l - e 2 = 0) 
 
 25 Moment -Load Curves for Unsymmetrically Loaded, Rectangular 58 
 SAE 1020 Steel Beam-Columns, with Equal and Unequal End 
 Eccentricities 
 
 26 Moment- Load Curve for an Unsymmetrically Loaded T-Section 59 
 Beam-Columns of SAE 1020 Steel. (M Q /M e - 75; e 1 = e 2 = 0) 
 
 27 Moment- Load Curve for Unsymmetrically Loaded Rectangular 60 
 17-7PH Stainless Steel Beam Columns. (M n /M = 0. 25; 
 e 1 =e 2 =0) Q e 
 
 VI 
 
I. INTRODUCTION 
 
 1. Preliminary Statement 
 
 In modern design, particularly in the aircraft and missile fields, load carrying 
 members are, in general, used to the limit of their capacity. This stipulation requires 
 that the design of many load carrying members be based on an inelastic analysis . If 
 inelastic action is allowed to penetrate only a small depth into the member, the resulting 
 deformations will not, in general, become excessive and an inelastic analysis can be used 
 in design for either strength or deformation. Small amounts of inelastic action have 
 been found to impart to a member an appreciable increase in the load-carrying capacity 
 above that associated with initiation of yielding (1-3) . 
 
 This investigation was undertaken to examine the behavior of a certain class of 
 members known as beam-columns, when subjected to loads higher than those necessary 
 to initiate yielding in the member. The term beam -column is applied to long, slender 
 members which are subjected to transverse as well as axial loads, i. e. , members that 
 simultaneously perform the functions of beams and columns. 
 
 While the problem of column behavior beyond the elastic range has been examined 
 extensively in recent years (4-15), that of beam-columns has received very little 
 attention. Aside from the well-known solutions for the behavior of beam columns in the 
 elastic range (16), only two-published papers were found that treat the subject of beam- 
 columns in the inelastic range. The first and the oldest of the two papers was published 
 by Osgood (17) in 1947. In his paper, Osgood proposed an approximate method which 
 essentially consisted of replacing the given beam -column by an equivalent eccentrically 
 loaded column. Available methods were then applied to the resulting eccentrically 
 loaded column. Theoretical solutions were presented for several cases, but no test 
 data were given. The second paper was published in 1955 by Ketter (18) who presented 
 a solution based on the principal of virtual displacements. Ketter 's method is 
 essentially based on the premise that instability in the beam -column will set in when 
 the rate of increase of the internal resisting moment (M. ) with respect to the curvature 
 
 Numbers in parentheses refer to corresponding entries in the bibliography. 
 
 Manuscript released by authors April, 1961, for publication as a WADD Technical 
 Report. 
 
( $) becomes equal to the rate of increase of externally applied moment (M ) with 
 respect to curvature. Symbolically, this instability condition may be expressed as 
 follows: 
 
 AM., AM, 
 
 mt ext 
 
 A| A ip 
 
 By assuming ideal plasticity (i. e. non-strain hardening material) and a given configura- 
 tion for the deflection curve, Ketter determined the combination of axial load and 
 bending moment that would cause instability in members subjected to constant axial 
 load with increasing bending moment. However, all cases considered involved 
 members that were symmetrically loaded. 
 
 In the present paper, a new method is proposed that would make it possible to 
 analyse beam -columns made of materials with any strain hardening characteristics. 
 The approach is sufficiently general to consider symmetrically and unsymmetrically 
 loaded members, as well as concentrically and eccentrically applied axial loads. 
 Furthermore, the shape of cross-section of the member does not influence the case with 
 which the solution can be performed. 
 2. Purpose and Scope 
 
 The purpose of this investigation was to develop a procedure that would predict 
 the moment-load curve, or the load -deflection curve as well as the collapse load for 
 beam-columns which are subjected to constant transverse loads. While the analysis 
 was applied only to the case of a single concentrated transverse load, it it believed that 
 the theory is equally valid for cases in which the transverse loading is made up of 
 several concentrated loads, distributed loads, or any combination of the two. 
 
 Two theories are presented for predicting the load -deflection curves for beam- 
 columns. An exact numerical procedure is outlined in Appendix A. Although this 
 procedure will yield results as accurate as desired, several hours of computations are 
 required for a beam-column that is symmetrically-loaded. The amount of time is 
 increased many folds for an unsymmetrically loaded beam-column Since the exact 
 theory was found to be impractical, an approximate theory was developed. In this 
 theory the configuration of the inelastic beam -column was based on the elastic configura- 
 tion of the member when yielding was impending. 
 
 The experimental part of the investigation included tests on 46 beam-columns. 
 These tests covered several variables as follows: 
 
 2 
 
(a) Material: Three different materials were investigated, 
 namely, 2024-T4 aluminum alloy : SAE 1020 mild steel and 17-7PH stainless 
 steel. 
 
 (b) Cross -section: The two shapes of cross sections that were 
 examined were the rectangular and the T- sections. 
 
 (c) Slenderness ratio: In the case of the rectangular section, 
 slenderness ratios of 50 3 75, and 100 were used. A slenderness ratio of 60 
 was used in the case of the T- section. 
 
 (d) Position of transverse toad: The transverse load was either 
 applied at midspan or at the quarter point of the member. 
 
 (e) Magnitude of transverse load: The transverse load was 
 maintained constant throughout a given test. The magnitude of this load was 
 selected so as to produce maximum bending stresses in the member equal to 
 25, 50, and 75 per cent of the yield strength of the material. 
 
 (f) Eccentricity: The axial load was applied either concentrically 
 or with initial eccentricity of 15 per cent of the section depth. Furthermore, 
 equal and unequal eccentricities at the two ends of the beam-column were 
 examined. 
 
II. ANALYSIS 
 
 Two methods will be presented for constructing either moment-load or load- 
 deflection curves for beam -columns. The collapse load is defined as the maximum 
 load obtained from either the moment -load or load-deflection curve. In order to 
 construct these curves, two independent relations between the variables axial load, 
 moment and curvature are developed. 
 
 The first axial load-moment-curvature relation is obtained by considering the 
 equilibrium of internal forces and moments, and is represented graphically by 
 dimensionless interaction curves together with a moment-curvature equation. In this 
 investigation, interaction curves for constant depth of yielding are used. The second 
 axial load-moment-curvature relation is established by equating the interval and external 
 moments, and by considering the deformation characteristics of the beam-columns. 
 
 Using the interaction curves and a numerical procedure described by Chwalla (6), 
 the configuration of the inelastically deformed member can be derived as accurately as 
 desired. This so-called exact method is described in Appendix A; however, the method 
 is considered impractical because of the time required to obtain a solution for even the 
 simplest problem. 
 
 A theory based on certain approximations regarding the configuration of the 
 inelastically deformed beam -column was found to compare favorably with both the 
 exact theory and test results. This approximate theory requires that the elastic 
 solution to the beam -column be known, so that the magnitude of the axial load, maximum 
 moment, deflection and curvature at the critical section can be determined for 
 maximum elastic conditions. 
 1. Interaction Curves 
 
 Consider the T-section member shown in Fig. 1 which is subjected to an axial 
 load, P, at the centroid and to a moment, M. In order to construct interaction curves 
 for the given T-section, it is necessary to obtain expressions for P and M for any given 
 strain distribution. These expressions were previously developed by Sidebottom and 
 Clark (2) using the following assumptions: 
 
 The term "critical section" is used in its ordinary sense to signify the section of the 
 member at which the bending moment has its maximum value. 
 
(a) Plane sections remain plane in both the elastic and inelastic 
 ranges. This assumption has been shown to be reasonably accurate (19). 
 
 (b) The stresses and strains in the beam-column are related in 
 the same fashion as they are in simple tension and compression. Experimental 
 evidence exists to justify this assumption (19, 20). 
 
 (c) The stress-strain diagram for the material can be approxi- 
 mated by two straight lines intersecting at a value of stress considered to be the 
 yield stress of the material as shown schematically in Fig. 2. 
 
 (d) The modulus of elasticity in tension is the same as that in 
 compression. 
 
 Let the member shown in Fig. 1 be subjected to a load, P, and moment, M, of 
 such magnitude that yielding has penetrated to a depth, a., on the compression side and 
 to a depth, a 2> on the tension side. The remaining depth, a, is elastic. The 
 expressions for P and M as developed by Sidebottom and Clark (2), are given below 
 without derivation. 
 
 For a ^ t : a a b a 
 
 -5- = 1 " -(c, - a.) -£ - -i-s- (1 - a,) — 
 
 P a v 1 r<r aA 1 r a 
 
 e l e l 
 
 2 , 
 a 2 £ 1 % 
 
 e l 
 2c 2 a e a lb c 2 a % 
 
 Nf-ir^ rr~ (1 - a i )(c i- t } ~ 
 
 e e 2 e 2 
 
 2 
 a 9 t.c a 9 ct 
 
 e 2 
 
 Fora l^ t 2 : P 2, ,% a l\ n , % 
 
 T = l- T (c - a)_ + __(i_a 2 ) — 
 
 e e 
 
 (1 - a x ) 
 a A 
 
 baj - (b - t 1 )(a 1 - t 2 ) 
 
 C7 
 
 e 
 
 e l 
 
 (3) 
 
M - 2 ° 2 '• - ** V2 (1 _ a „ V ^e 
 
 M a <r a I 2 M 2 3 a 
 
 6 6 2 e 2 
 
 -(1-Oj) 
 
 2C 2 Vl, , .2, a+a 2, 
 
 a 
 
 -^ • (4) 
 
 a 
 
 e 2 
 
 In the above expressions P is the axial load required to initiate yielding when acting 
 
 alone; M is the moment required to initiate yielding when acting above; A is the cross - 
 
 sectional area of the member; I is the moment of inertia of the section; and 2<j = a 
 
 e e x 
 
 + a . The remaining symbols are as defined in Figs. 1 and 2. 
 
 e 2' 
 
 By making the thickness of the web t. equal to the width of the flange b, the 
 
 above equations become applicable to rectangular sections. In this case, however, 
 
 only Eqs. 1 and 2 are needed in which c. becomes equal to c„. Furthermore, since 
 
 for the materials considered a was equal to or greater than a , it is replaced by 
 
 e 2 e 1 
 
 a in Eq. 2. 
 
 e i 
 
 While expressions for the dimensionless quantities P/P and M/M can be 
 
 obtained separately in terms of the properties of the section and those of the material, 
 
 there is no simple way of directly relating P/P to M/M analytically. Resort is, 
 
 therefore, made to a graphical representation of the above relations in the form of 
 
 dimensionless interaction curves for constant depth of yielding as shown schematically 
 
 in Fig. 3. The procedure consists of assuming a given depth of inelastic penetration 
 
 (or a given value for a). Then values of a and a are assumed such that their sum 
 
 is always equal to the depth of plastic penetration. Equations 1 through 4 are then 
 
 evaluated to compute the values of P/P and M/M . This process is repeated as 
 
 many times as is necessary to obtain a smooth curve for the specified depth of yielding. 
 
 The same procedure is used for other depths of inelastic penetration. 
 
 A study of Fig. 3 indicates that, with. the exception of the straight line defining 
 
 initiation of yielding (i. e. , a = h), all curves consist of two parts, a straight line and a 
 
 curve. The combinations of P/P and M/M represented by the straight line portion 
 
 are those necessary to cause yielding on one side of the member only, while the other 
 
 side is still elastically strained. The point of intersection between the two portions 
 
 of the interaction curve defines the combination of P/P and M/M that produce the 
 
 specified depth of yielding on one side of the member and incipient yielding on the other 
 
 side. The curved portion, however, defines the various combinations of P/P and 
 
M/M that would cause inelastic action on both sides such that the total depth of 
 inelastic penetration is equal to the specified depth of yielding. 
 
 Once the interaction curves for constant depth of yielding are known (see Fig. 3), 
 they may be used to obtain a relation between axial load, moment, and curvature. It 
 has been shown by Sidebottom and Clark (2) that the curvature of the member for any 
 point on any one of the interaction curves shown in Fig. 3 is given by the relation 
 
 e + e M/M e + e M/M 
 
 e e 2 e e 1 e 2 e 
 
 * = ~~i M /M = kh WJM~ ^ 
 
 u e u e 
 
 where e and e define the strains corresponding to the compressive and tensile 
 
 6 1 e 2 
 
 yield stresses, respectively, as shown in Fig. 2. The quantity k is a proportionality 
 
 constant which may vary from zero to unity. M/M is the dimensionless moment 
 
 that exists at the given section of the member and M /M is the dimensionless moment 
 
 at the upper end of the straight line portion of the interaction curve. The ratio M/M 
 
 assumes a constant value of unity in the curved portion of the interaction curve. 
 
 A solution for the beam-column cannot be obtained using only the interaction 
 curves and Eq. 5 since the point on the interaction curve which is valid for a given 
 section of the beam -column is not known. Another relation between axial load, moment, 
 and curvature for the inelastically deformed beam -column will be obtained by con- 
 sidering the equality between internal and external moments as well as the deflection 
 characteristics of the member. The deflection characteristics of the inelastically 
 deformed member are developed from the elastic solution of the given beam-column. 
 2. Elastic Solution for the Beam-Column 
 
 For the beam -column shown in Fig. 4, the bending moment at any section a 
 distance x from the origin is given by the equation 
 
 M = P 
 
 y + e i " J < e i " e 2> 
 
 + M Q (a) 
 
 which applies to sections on either side of the transverse load Q. The symbols e 1 
 and e„ are the eccentricities of the load P at the left and right ends, respectively, y 
 is the deflection of the centroidal axis of the member at the point considered, 4 is the 
 length of the member and M n is the moment at the section in question due to the 
 transverse load Q acting alone. Equation a may be written in dimensionless form 
 by dividing both sides by the quantity M = P /y , in which y is a constant that 
 
depends on the properties of the cross -section. Thus 
 
 M . P . 
 
 M T V P ' 
 
 e e 
 
 y + e x -^(e^e^ 
 
 + M„/M . 
 
 Q e 
 
 (b) 
 
 It should be noted that Eqs. a and b are valid for either elastic or inelastic conditions 
 
 The differential equations of the elastic deflection curve for the beam-column 
 shown in Fig. 4 may be expressed as 
 
 EI 
 
 dy 
 
 d7 
 
 Qd 
 
 P(e x - e 2 > 
 
 x - P(ej + y) 
 
 x L (S. - d) 
 
 (c) 
 
 EI- 
 
 dy 
 
 Q( It -d) P(ej - e 2 ) 
 
 x - P(e 1 +y) 
 
 Q( ft - d) . . . . x ^ (1 - d) 
 
 (d) 
 
 where E is the modulus of elasticity and the remaining symbols are as defined 
 
 2 
 previously. Using the notation /3 = P/EI and rearranging terms, the above equations 
 
 may be written as 
 
 Pe, 
 
 dy 2 
 dx 2 
 
 + „ 2 y=-A- 
 
 EI 
 
 Qd P(e i " e 2> 
 
 x - 
 
 
 ..... x ^ (i - d) 
 
 dy 2 
 
 dx 2 
 
 + 2 y -A. 
 EI 
 
 QU-d) , P(e l " e 2 } 
 
 
 T7 
 
 - T d > ...,x^({- 
 
 d). 
 
 EI 
 
 Pe, 
 
 EI 
 
 (e) 
 
 (f) 
 
 2 (Te l I 
 In the case of a rectangular section, for example, M = 
 
 CjA/I 
 
 jjt- so that y is equal to 6/h. 
 
The general solutions of Eqs. e and f are given by the following equations 
 
 y = C, cos fi x + C„ sin /3 x 
 . . . . . x L (£ - d) 
 
 Qd (e l - e 2 > 
 PI " 1 
 
 x - e. 
 
 y = Co cos (5x +C, sin /3 x + 
 
 QU_Ld> . . x A (£ . d) . 
 
 Q(I - d) (e l " e 2> 
 Pi + 1 
 
 - e. 
 
 T P 
 
 From the conditions at the ends of the beam-column (i. e. , y = for x = 
 and x = i), the constants of integration C, and C, are found to be 
 
 C l = e l ' C 3 = " C 4 tan ^ * + e 2 / cos B * • 
 
 (g) 
 
 (h) 
 
 (i) 
 
 The constants of integration CL and C are determined from the conditions at the 
 point of application of the transverse load Q. At this point, the two portions of the 
 elastic deflection curve, as defined by Eqs. g and h possess the same values of 
 deflection and slope. From these conditions, the values of C„ and C. are found 
 to be 
 
 C 2 = 
 
 e 2 
 
 - e, cos p H 
 
 
 sin/3£ 
 
 e 2 
 
 - e cos /3 9. 
 
 Q 
 
 P£ 
 
 sin p{i - d) 
 
 tan/3£ 
 
 cos p(i - d) 
 
 Q sin p(JL - d) 
 C 4 " sin/3£ " ~YJ tan 1 
 
 (J) 
 
 Substitution of the values of the constants C. . . . . C. into Eqs. g and h, 
 rearranging terms and simplifying leads to the following equations defining the 
 elastic deflection curve of the beam -column shown in Fig. 4, 
 
 y = e. 
 
 sin /3(l - x) 
 sin pi 
 
 ix 
 
 + e 
 
 sin p x x 
 
 2| sin p S. ' I 
 
 Q sin B d . _ Qd 
 + ts-s — ■ n /i sin fix- ^t x 
 P0sin/3£ H PS. 
 
 x £ (i - d) 
 
 (6a) 
 
y = e. 
 
 sin j3(l - x) 9. - x 
 sin £ /5 
 
 J sinjSx x 
 ' e 2 sin/3£ " £ 
 
 + g/if/iV d) sin^-x)- Q< £ D - d ^- x ) ....x^f-d) 
 
 (7) 
 
 (6b) 
 X* x) „ V//) 
 
 P/3 sin It smpi * * ; FT 
 
 The elastic curvatures may be obtained by differentiating Eqs. 6 twice, thus 
 
 j, = v „ = _ fl 2 sin /?(£ - x) _ 2 sin ff x 
 
 * y " e l P sin 4 e 2 P liifpT 
 
 " Q pt Sln // sin^x . . . . x^ (£-d) 
 
 P sin /3 £ K v ' 
 
 ri - v" ■■ P fl 2 sin ff (I - x) 2 sin fix 
 
 * " y " e l^ sin pi e 2 P ^TnTl 
 
 - Q Mi/fi\' d) sinMix)....xA (f -d). 
 
 The bending moment at any section along the span of the member is then given 
 by the equation 
 
 M = -El(Jp ). (8) 
 
 For a given member, the deflection, y , and curvature, ip , at the critical 
 section may be determined for any value of P from Eqs. 6 and 7, respectively. 
 Examination of the ratio ip /y revealed that, in general, this ratio did not vary 
 greatly as the axial load P increased from zero to its maximum elastic value. This 
 behavior led to the assumption that the ratio ip /y remains constant as the beam- 
 column is strained inelastically. Essentially, then, a relation between curvature 
 and deflection at the critical section in the inelastic range is assumed to be of the 
 form 
 
 ^ c =Cy c (9) 
 
 where the constant C is determined as the rat io of curvature to deflection at the 
 critical section when yielding is impending. 
 
 Eliminating the deflection, y , between Eqs. b and 9. and solving for P/P , 
 the required relation between axial load^moment, and curvature is obtained, namely 
 
 10 
 
M/M - (M /M ) 
 
 -^- (10) 
 
 p e Y[V c + e i-V 1(e i- e 2>] 
 
 where the subscript c designates conditions at the critical section. 
 3. Method of Solution 
 
 The purpose of this article is to show how the various relations established in 
 the preceeding article may be used to determine the moment -load curve or the load- 
 deflection curve as well as the load P that causes complete collapse of a beam-column 
 when it is subjected to a constant value of the transverse load Q. 
 
 The value of the axial load P that would initiate yielding in the member may be 
 determined, by trial and error, using Eqs. 7 and 8. The procedure consists of 
 first assuming a value for P and determining the quantity /3 = A P/EI. The value of 
 
 x locating the critical section is now determined by maximizing Eq. 7. The 
 maximum moment is then given by Eq. 8 and the maximum stress at the critical 
 section by the equation 
 
 (M ) c. P 
 
 = max 1 + - . (11) 
 
 max. j A 
 
 This process is repeated until the stress determined by Eq. 11 becomes equal to the 
 compressive yield stress of the material, a 
 
 With the value of P that initiates yielding known, the constant C may be easily 
 found from Eq. 9, in which, the values of y and ip are determined by the use of 
 Eqs. 6 and 7, respectively. Furthermore, this value of P may be used to locate 
 the intersection of the moment-load curve and the interaction curve defining the 
 beginning of yielding (k = 1), see point B in Fig. 3. 
 
 The solution is then carried into the inelastic range by means of the inter- 
 action curves along with Eqs. 5 and 10. 
 
 In applying Eq. 10, the assumption is made that the critical section does not 
 move as the member deforms inelastically. This is equivalent to saying that x 
 and (M„/M ) retain the same values they assumed at initiation of yielding. With 
 members which are symmetrically loaded (i. e. , d = 1/2 and e. = e 2 ), x does 
 retain the same value, namely H/2, in the inelastic as well as in the elastic range. 
 However, with members which are not symmetrically loaded (i. e. , d f 1/2 and/or 
 e 1 f e„), x does not, in general, retain the same value throughout the loading 
 process. Test data indicated that the above assumption did not introduce a serious 
 error into the theory. 
 
 11 
 
Now consider the problem of determining the value of P/P that would produce 
 a given depth of inelastic penetration (i. e, , k = k. = const), see Fig. 3. The 
 condition that has to be satisfied is that M/M be of such magnitude that the value of 
 P/P given by Eq. 10 be equal to that given by the relation P/P = f(M/M ) as expressed 
 by the interaction curves. To this end some value of M/M is assumed and the 
 curvature ip is determined by Eq. 5. Equation 10 is then used to obtain a value of 
 P/P which is compared to that given by the interaction curve corresponding to k = k. . 
 This process is repeated until the two values of P/P are the same or very nearly so. 
 This usually requires not more than three trials. Thus the problem is reduced to 
 the solution of two simultaneous equations in the unknowns M/M and P/P , The 
 deflection y , if needed, may now be obtained from Eq. 9. 
 
 Once the correct value of P/P is established for k = k 1 , a point is located on 
 
 the appropriate interaction curve. Other values of k are then considered and the 
 
 same process repeated to establish a set of points such as C, D, E, F, and G in 
 
 the inelastic domain, see Fig, 3. The location of point A on the M/M axis is 
 
 dictated by the value of the transverse load Q, A smooth curve is then constructed 
 
 through the points A, B, .... G to give the moment- load curve. The collapse load 
 
 for the beam-column may be obtained from the moment load curve and in dimensionless 
 
 form, is the maximum value of P/P . 
 
 e 
 
 If additional points are needed in the elastic range (i. e. , between points A and 
 B in Fig. 3) they may be easily determined from the elastic solution, using Eq. 6 
 and the equilibrium condition expressed by Eq. b. 
 4. Sample Problem 
 
 In this article, a sample problem is solved to illustrate the method used in 
 deriving the moment-load curve for a given beam-column. 
 
 (a) Given Data For a Rectangular SAE 1020 Steel Beam-Column: 
 A = b x h = 0. 500 x Q625 = 0. 3125 in 2 ; I = 0. 01017 in 4 ; it = 13. 52 in(4/r = 75) 
 M = 0. 5 M g ; d = 0. 25 i ; e = 0. 15h = 0. 09375 in 
 
 e =0; a = a =30, 300 psi; P = 9470 lb. 
 
 2 e l e 2 -4 e 
 
 e = € = 10. 1 x 10 in/in. 
 
 e l e 2 
 
 (b) Problem: To determine the moment -load curve for the 
 
 beam-column specified above. 
 
 12 
 
(c) Solution: 
 
 Q 16 " 
 
 81(7 
 
 a I 
 
 e 
 
 Q = 
 
 3 tc 
 
 = 194. 49 lb. 
 
 1 
 
 From the given conditions it is evident that the critical section occurs either under the 
 load Q or slightly to the left of this position Maximizing Eq. 7 for x ^ (£ - d) leads 
 to the condition that the location of the critical section is defined by the equation 
 
 tan fix = 
 
 r c 
 
 Q sin fl d - e. fl P cos fl I + e„ fl P 
 
 e, fl P sin fl £ 
 Since in this problem e„ = 0, Eq. k becomes 
 
 tan fix = ■ 
 
 H c e. fl P sin fl £ 
 
 (k) 
 
 Q sin fl d - e fl P cos fl £ 
 
 (i) 
 
 By Eqs. 7 and 8 and using the condition e = 0, the maximum moment, M , 
 is given by the equation 
 
 EI e, r 2 
 
 M 
 
 max 
 
 sin 
 
 jr sin m ' x c> 
 
 EIQflsinfld 
 
 + sinflx 
 
 Psinflf c 
 
 The maximum stress, a , will then be given by Eq. 11. 
 
 (m) 
 
 The computations for the axial load that would initiate yielding are given in 
 the following table. 
 
 Trial 
 
 P - lb 
 
 fl-l/in 
 
 x -in. 
 
 ^max 
 
 °max 
 
 
 
 
 
 lb-in. 
 
 psi 
 
 1 
 
 3000 
 
 0. 0992 
 
 10. 14 
 
 654. 12 
 
 29, 700 
 
 2 
 
 3200 
 
 0. 1024 
 
 10. 14 
 
 668.03 
 
 30, 770 
 
 3 
 
 3100 
 
 0. 1008 
 
 10. 14 
 
 661. 17 
 
 30, 240 
 
 4 
 
 3110 
 
 0.1010 
 
 10. 14 
 
 661.93 
 
 30, 290 
 
 13 
 
Thus, to initiate yielding P/P = 3110/9470 = 0. 328. 
 
 By Eq. 8 for x ^ (I - d), y = 0. 0308 in, for P = 3110 lb. 
 
 By Eq. 9 for x ^ (£ d), ip = 21. 69 x 10" 4 l/in,for P = 3110 lb. 
 
 C = if) /y =0. 0704 1/in 
 
 y = 6/h = 9.6 1/in 
 
 If) 14 
 At x c = 10. 14 = jj^fi ft = 0. 75 i , (M„/M e ) c = 0. 50 
 
 Therefore, the equations that apply in the inelastic range are 
 
 r c 
 
 32. 32 x 10 
 
 -4 
 
 M/M 
 
 M /M 
 u e 
 
 (5) 
 
 M/M - 0. 50 
 P/P e = 9. 6 ( 14.2 ip + 0.0234) - 
 
 (10) 
 
 and the functional relation P/P = f(M/M ) as expressed by the interaction curves 
 shown in Fig. 7. 
 
 A sample calculation is shown for k = 0. 7 in the following table. 
 
 Trial 
 
 M/M 
 (Assumed) 
 
 ^c 
 Eq. 5 
 
 P/P 
 e 
 
 Eq. 10 
 
 P/P 
 e 
 
 Fig. 7 
 
 y c 
 
 Eq. 9 
 
 1 
 2 
 3 
 
 0.830 
 0.832 
 0.831 
 
 34.21 x 10" 4 
 34. 30 x 10" 4 
 34. 25 x 10" 4 
 
 0.478 
 0.480 
 0.479 
 
 0.480 
 0.478 
 0.479 
 
 0. 0487 
 
 Thus to produce 0. 3 depth of inelastic penetration (k = 0. 7), P/P = 0. 479, which 
 locates a point on the moment-load curve. Other points, corresponding to other 
 values of k may be similarly located. 
 
 14 
 
III. MATERIALS AND TEST PROCEDURE 
 
 1. Materials and Specimens 
 
 In order to examine the validity of the analysis discussed in Section II, tests 
 were performed on beam-columns made of three materials, namely 2024-T4 
 aluminum alloy, SAE 1020 steel and 17-7PH stainless steel. The 2024-T4 aluminum 
 samples were tested in the as-received condition The SAE 1020 steel was annealed, 
 prior to machining, by soaking at 1600 F for 3 hours and furnace cooling. The 17-7PH 
 stainless specimens were precipitation hardened after machining. The SAE 1020 
 steel samples were machined from a 3 1/2 in. diameter bar, while the 2024-T4 
 aluminum and the 17-7PH stainless steel specimens were machined from 1/2 in. 
 plates. The various beam-columns tested in this program with their physical 
 characteristics are shown in Table I. Either one or two beam -columns per condition 
 were tested in this study. 
 
 Standard tensile and compressive tests were performed on the aluminum alloy 
 to determine the various properties needed for the analysis. However, only com- 
 pressive tests were performed on the mild and stainless steels. Results of previous 
 tests by Sidebottom and associates (21) on the last two materials have indicated that 
 the difference between tensile and compressive properties was negligible for all 
 practical purposes. The various properties used in the present analysis are shown, 
 for the three materials, in Table II. These values represent an average of at least 
 six tests. 
 
 Using the various properties indicated in Table II, and Eqs. 1 through 4, 
 interaction curves for the three materials and the two types of cross -section were 
 developed. These interaction curves are shown in Figs. 5 to 9. Figures 5 and 6 
 represent the 2024-T4 aluminum interaction curves for rectangular and T-sections, 
 respectively. Figures 7 and 8 show the SAE 1020 steel interaction curves for 
 rectangular and T-sections, respectively. And Fig. 9 illustrates the interaction 
 curves for rectangular sections made of 17-7PH stainless steel. In all instances, k 
 decreased in increments of one tenth to a minimum value of 0. 4. 
 
 This heat treatment consisted of soaking at 1400 F for 90 minutes, cooling to 
 60° F in 60 minutes, soaking at 60° F for 30 minutes, soaking at 1050° F for 
 90 minutes and finally air cooling. 
 
 15 
 
2. Apparatus and Test Procedure 
 
 The testing apparatus used in this program is shown in the photograph of Fig. 
 
 10. A Riehle testing machine having a capacity of 120, 000 lb. was used to apply the 
 
 2 
 axial load through fixtures provided with knife edges as illustrated in Fig. 11 The 
 
 position of the knife edges may be adjusted with respect to the centroid of the specimen 
 
 by means of set screws. The transverse load was applied by means of dead weights 
 
 through the pulley system shown in the photograph of Fig. 10. The deflection of the 
 
 specimen was measured by means of a 1/1000 in. dial indicator which may also be 
 
 seen in the photograph as well as in Fig. 11. 
 
 The specimen was properly placed in the test apparatus and a small value of 
 axial load was applied to maintain the specimen in the proper position. The pulley 
 system was then raised or lowered so as to apply the transverse load at the desired 
 position along the span of the beam-column. The dial indicator was subsequently 
 placed in positon and the desired transverse load applied. This transverse load was 
 maintained at a constant value throughout the test. 
 
 The axial load was subsequently increased, and readings of this load and the 
 corresponding deflection were taken at predetermined intervals of load until the 
 predicted value of the collapse load was approached. Beyond this point, judgment 
 was exercised in spacing the readings at reasonable intervals of deflection. Sufficient 
 readings were taken to make it possible to construct the moment-load curve well 
 beyond the collapse load. 
 
 Once the member was loaded beyond the elastic limit, sufficient time was 
 
 allowed for equilibrium conditions to be reached (i.e. , for the axial load to reach a 
 
 steady value). In the case of the aluminum and the stainless steel samples, a 
 
 period of approximately two minutes was sufficient for the axial load to stabilize. 
 
 In the case of the mild steel specimens, due to the presence of an upper yield point, 
 
 appreciably more time was required for steady state conditions to be reached. In 
 
 3 
 most instances, up to 20 minutes elapsed before a reading could be taken. It was 
 
 discovered, however, that this delay time could be shortened, without appreciably 
 
 2 
 
 A complete description of these fixtures may be found in a paper by Sidebottom 
 
 et al (22). The two end fixtures added a total of 1. 20 in. to the machined length 
 
 of the specimen. 
 
 3 
 
 This time-sensitivity of mild steels was investigated previously by Clark, 
 
 Corten, and Sidebottom (20). 
 
 16 
 
influencing the load-carrying capacity, by prick punching one row of shallow indentations 
 along the span of the member. These indentations were spaced approximately one- 
 half inch apart. In a few instances, two tests were performed for the same conditions, 
 one with and one without indentations. The results indicated the difference between 
 the load -carrying capacities of the two specimens to be insignificant. 
 
 17 
 
IV. RESULTS AND DISCUSSION 
 
 In this section, a comparison is made between the results of the proposed 
 approximate theory, and those of the experiments. The comparison is made on the 
 basis of moment Toad curves (i. e. , M/M vs. P/P ), This method of presentation 
 was selected over the more conventional load -deflection plots because it conveniently 
 indicates the value of M^/M to which the beam-column was initially subjected before 
 the axial load was applied. The various tests that were performed in this program 
 may be conveniently classified into two types as follows: 
 1. Symmetrically Loaded Beam -Columns 
 
 This class of members include those subjected to end loads with zero or equal 
 eccentricities and to a transverse load at midspan. They are characterized by the 
 fact that the maximum moment is located at midspan and possess symmetrical deflection 
 curves throughout the loading process. 
 
 In order to examine the degree of accuracy attainable by the approximate method, 
 a comparison was made between moment-load curves obtained by this method and those 
 developed by the so-called exact method (see Appendix A). In view of the fact that the 
 latter method involves rather lengthy computations, particularly with unsymmetrically 
 loaded members as indicated in Appendix A, the comparison was limited to two cases 
 only, namely one 2024-T4 aluminum member and one SAE 1020 steel member. Both 
 members were symmetrically loaded with zero end eccentricities, had a slenderness 
 ratio of 75 and were subjected to transverse loads at midspan such that M n /M was 
 equal to 0. 50. The results of both methods are illustrated in Fig, 12 together with 
 the values exhibited by test. In the case of 2024 -T4 aluminum, the two theoretical 
 moment -load curves were nearly identical and were found to compare very well with 
 the test values. In the case of SAE 1020 steel, however, a slight difference of about 
 5 per cent is observed between the collapse loads as predicted by the two methods. 
 While this difference was in favor of the exact method, it is too small to justify the 
 added labor required by this method. Furthermore the approximate theory was 
 conservative. 
 
 The actual moment -load curves for symmetrically loaded members are presented 
 in Figs. 13 to 21 in comparison to curves developed by the approximate method. Figures 
 13 to 16 illustrate the moment-load curves for members made of 2024-T4 aluminum 
 alloy; Figs. 17 to 19 show the moment-load curves for members made of SAE 1020 
 steel; and Figs. 20 and 21 indicate the moment-load curves for members made of 
 
 18 
 
17-7PH stainless steel. In the case of the 2024-T4 aluminum and SAE 1020 steel, both 
 rectangular and T- sections were examined. Furthermore, four different slenderness 
 ratios ranging between 50 and 100 were investigated. In addition, for each of these 
 two materials, two different values of M n /M were analyzed, namely 0. 25 and 0. 50 
 for 2024-T4 aluminum, and 0. 50 and 0. 75 for SAE 1020 steel. In the case of 17-7PH 
 stainless steel, however, testing was limited to members with rectangular-sections, 
 having slenderness ratios of 50 and 75, and subjected to values of M^/M equal to 
 0. 25 and 0. 50. 
 
 Examination of Figs. 13 to 21 leads to several interesting conclusions. It is 
 observed that, in all instances, the predicted curves compare very well with the test 
 data. In general, the discrepancy between the predicted collapse load and that 
 exhibited by test is less than about 7 per cent, as shown in Table HI by the ratio of 
 actual collapse load, P , to theoretical collapse load P . This discrepancy is not 
 excessive when considered with respect to the scatter that is normally encountered in 
 materials testing. 
 
 In general, when the beam column is symmetrically loaded, the theory always 
 predicts values of collapse load which are either equal to, or slightly lower than those 
 exhibited by test. In other words, the theory tends to predict values which are 
 conservative, as may be seen from Table III by the ratio P /P . 
 
 Ccl C L 
 
 It is also observed that, in every instance, there is a considerable increase in 
 the load carrying-capacity above that associated with the beginning of yielding. This 
 increase in the load carrying capacity is represented in Table III by the ratio of actual 
 collapse load, P , to the actual load that initiates yielding, P . Evidently, the 
 increase in the load carrying-capacity (i. e. , P /P - 1) depends upon several 
 
 Cci y^- 
 
 factors which include, among other things, the slenderness ratio, magnitude of the 
 
 transverse load, end eccentricities, material and type of cross -section. While the 
 
 tests that were performed are insufficient to evaluate the influence of all pertinent 
 
 factors, it is possible to determine qualitative trends regarding the effects of the 
 
 slenderness ratio, magnitude of transverse load, material and end eccentricities on 
 
 the ratio P /P . 
 ca ya 
 
 Examination of Table III indicates that for a given material and for zero 
 
 eccentricities the ratio P /P increases with increase in the magnitude of the trans - 
 
 ca ya & 
 
 verse load (i. e. , with increase in the ratio M n /M ), and with decrease in the slender- 
 ly e 
 
 ness ratio. Furthermore, a comparison of the data corresponding to M n /M = 0. 50 
 
 v e 
 
 and zero end eccentricites indicates that P /P is slightly, but consistently, higher 
 
 ca ya 
 
 19 
 
for SAE 1020 steel than for 2024-T4 aluminum. In addition, based on the two tests that 
 
 were performed on 17-7PH stainless steel with zero end eccentricities, the ratio P /P 
 
 * ca ya 
 
 for this material was found to be considerably less than that for the aluminum and mild 
 
 steel alloys. These trends lead to the conclusion that one of the factors influencing the 
 
 ratio P /P is the strain at yielding, namely e . Apparently, the larger the value of 
 ca ya e 
 
 e the less is the ratio P /P for a given slenderness ratio. 
 e ca ya ° 
 
 The influence of end eccentricities on the increase in the load-carrying capacity 
 
 may also be seen from Table III. A comparison of the values corresponding to M n /M 
 
 in the case of 2024-T4 aluminum, indicates that P /P attain higher values for 
 
 ca ya & 
 
 members provided with end eccentricities than for members subjected to concentric 
 
 axial loads. 
 
 2. Unsymmetrically Loaded Beam -Columns 
 
 This type of members include those subjected to unequal end eccentricities and/or 
 to a transverse load not at midspan. They are characterized by the fact that the 
 maximum moment may not occur at the same section of the member throughout the 
 loading process, and the deflection curve is, therefore, unsymmetrical. 
 
 The actual moment-load curves for beam-columns of the type described above 
 are presented in Figs. 22 to 27 in comparison to theoretical curves. Figures 22 and 
 
 23 illustrate the moment-load curves for members made of 2024-T4 aluminum; Figs. 
 
 24 to 26 show the moment -load curves for members made of SAE 1020 steel; and 
 Fig. 27 indicates the moment-load curve for a member made of 17-7PH stainless steel. 
 In the case of 2024-T4 aluminum and SAE 1020 steel, testing was performed to include 
 both rectangular and T-sections, four different values of slenderness ratio ranging 
 between 50 and 100, and two different values of the ratio M n /M , namely 0. 50 and 0. 75. 
 In the case of 17-7PH stainless steel, only one test was performed on a member of 
 rectangular cross-section having a slenderness ratio of 75 and subjected to a trans- 
 verse load such that M„/M was equal to 0. 25. 
 
 In this series of tests, as in the previous series, very good agreement is 
 
 observed between the predicted and the actual test values. The discrepancy between 
 
 the theoretical collapse load and that given by test, was found not to exceed about 
 
 5 per cent, as shown in Table IV by the ratio P /P . 
 
 ca ct 
 
 However, in this series, unlike the previous series, the tests did not consistently 
 yield values of collapse load which were either equal to, or slightly higher than that 
 predicted by theory. This lack of consistency may be seen in Table IV as the ratio 
 
 20 
 
P /P varied from a minimum of 0. 95 to a maximum of 1, 05 depending upon the 
 
 Ca c r 
 
 material, slenderness ratio and loading conditions. This inconsistency in the observed 
 trends may be explained as follows: 
 
 At the beginning of loading (i. e. , when only the transverse load, Q, is acting and 
 the axial load, P, is zero or very nearly so), the maximum moment occurs at the point 
 of application of the transverse load. When the axial load P reaches a certain value that; 
 depends upon the material, the slenderness ratio and the cross -sectional area, the 
 position of the maximum moment (critical section) begins to move away from the point 
 of application of the transverse load, Q . The axial load, P, at which movement of the 
 critical section begins, may occur either within the elastic or within the inelastic range 
 of the material depending upon the conditions stated previously. The conditions investi- 
 gated in this program were such that movement of the critical section was always from 
 the point of application of the transverse load Q towards midspan. In the elastic range, 
 the influence of the migration of the critical section can be easily included in the anslyses. 
 However, there is no simple method known by means of which the positon of the critical 
 section may be determined at any stage of inelastic action. This difficulty was circum- 
 vented in the present analyses by making the assumption that the critical section, during 
 inelastic deformation, retains the same position it assumed at initiation of yielding, as 
 explained previously. Obviously, this assumption cannot be expected to be equally 
 approached by members made of different materials, possessing different physical 
 characteristics and subjected to different loading conditions; hence, the lack of a con- 
 sistent trend observed in the actual behavior of members when compared to theoretical 
 predictions. However, this lack of consistency does not appear to be detrimental, 
 since for all cases examined in this program, the theory predicted values of collapse 
 load which were in close agreement with those exhibited by test. 
 
 As in the previous series of tests, an appreciable increase was observed in the 
 load-carrying capacity beyond that corresponding to initiation of yielding. This 
 increase may be seen in Table IV by the ratio P /P . Unfortunately., not all the para- 
 meters that may influence the increase in the load-carrying capacity can be examined 
 on the basis of the few tests that were performed. However, the remarks made 
 
 The dependence of the position of the critical section on the axial load, P, in the 
 elastic range may be seen from Eq. k. 
 
 21 
 
previously with regards to the influence of slenderness ratio, material and end 
 eccentricities on the ratio P ca /P ya apply equally well in the case of unsymmetrically 
 loaded beam-columns. 
 
 22 
 
V. SUMMARY AND CONCLUSIONS 
 
 In order to examine the inelastic behavior of beam -columns, two theories were 
 developed. An exact numerical procedure as outlined in Appendix A, would yield results 
 as accurate as desired. However, in view of the fact that this method may become 
 rather lengthy and sometimes prohibitive, an approximate, but relatively simple method 
 was developed. Essentially, the approximate method requires a trial -and-error 
 solution of two independent relations betwen the variables axial load, moment and curva- 
 ture. The first of the two P-M-ip relations is established from considerations of the 
 equilibrium of internal forces and moments, and is represented graphically by dimension- 
 less interaction curves along with a moment-curvature equation. The second P-M-$ 
 relation is developed by considering the equality between the internal and external 
 moments, as well as the deflection characteristics of the member. The proposed 
 method makes it possible to determine the magnitude of the axial load (concentric or 
 eccentric) at any stage of inelastic deformation including that at which collapse takes 
 place, for a beam -column which is initially subjected to a constant transverse load. 
 
 To examine the validity of the various assumptions made in developing the 
 approximate theory, tests were performed on members made of three different 
 materials, namely 2024 -T4 aluminum, SAE 1020 steel and 17-7PH stainless steel. 
 Several parameters were investigated within each material. These included the type 
 of cross section, slenderness ratio, position and magnitude of transverse load, and 
 eccentricity of the axial load. Dimensionless moment-load curves were constructed 
 from the test data and compared to those obtained analytically. Furthermore, in two 
 instances, a comparison was made between the approximate method and the so-called 
 exact method. The various results lead to a number of significant conclusions which 
 may be summarized as follows: 
 
 1. The results obtained by the proposed approximate method 
 compared well with those obtained by the exact method. The differences 
 observed between the results of the two methods are too small to justify the 
 added labor required by the exact method. 
 
 2, Very good agreement was observed, in all instances, 
 between the moment -load curves predicted by the approximate method and those 
 obtained by test, particularly up to the collapse load. While close agreement 
 was generally encountered beyond the collapse load, there were a few instances 
 in which the discrepancy between the predicted and actual curves became 
 pronounced. 
 
 23 
 
3. The difference between the predicted and actual values of 
 collapse loads was, in general; much less than 7 per cent. Thus the proposed 
 approximate method was found to be sufficiently adequate for all practical 
 purposes. 
 
 4. In general, inelastic action was found to induce a considerable 
 increase in the load-carrying capacity beyond that associated with initiation of 
 yielding. The increase in the load-carrying capacity was observed to range 
 from a maximum of 159 per cent for SAE 1020 steel, to a minimum of about 
 
 2 per cent for 17-7PH stainless steel. 
 
 5. The increase in the load-carrying capacity was found to vary 
 directly with M„/M , and inversely with £/r and e . 
 
 6. Members with end eccentricities of 15 per cent of the section 
 depth attained a much higher increase in load -carrying capacity than members 
 with zero end eccentricities. 
 
 24 
 
VI. BIBLIOGRAPHY 
 
 1. Seely, F. B. , and Smith, J. O. , Advanced Mechanics of Materials , John Wiley 
 and Sons, Inc., New York, 1952, pp. 511-581. 
 
 2. Sidebottom, O. M. , and Clark, M. E. , "The Effects of Inelastic Action on the 
 Resistance to Various Types of Loads of Ductile Members Made From Various 
 Classes of Metals, " WADC Technical Report 56-330, Pt, I, 1956. 
 
 3. Sidebottom, O. M. , and Clark, M. E. , "Theoretical and Experimental Analysis 
 of Members Loaded Eccentrically and Inelastically, " University of Illinois 
 Engineering Experiment Station, Bulletin No. 447, 1958, 
 
 4. Von Karman, T. , "Untersuchungen uber Knickfestigkeit, " Mitt, uber 
 Forschungsarbeiten des V. D. I. , H. 81, Berlin, 1910. 
 
 5. Chwalla, E. , "Die Stabilitat zentrisch und exzentrisch gedruckter Stabe aus 
 Baustahl, " Akad. d. Wissenschaften in Wien, math. -naturwiss. Klasse, 
 Sitzungsberichte Abt. II a, Bd. 137, 1928, pp. 469-512. 
 
 6. Chwalla, E. , "Theorie des aussermittig gedriickten Stabes aus Baustahl, " 
 Der Stahlbau, Vol. 7, 1934, pp. 161-165, 173-176, 180-184. 
 
 7. Osgood, W. R. , "The Double -Modulus Theory of Column Action, " Civil Eng. , 
 Vol. 5, No. 3, Mar. 1935, pp. 173-175. 
 
 8. Shanley. F. R. , "Inelastic Column Theory, " Jour, of Aero. Sci. , Vol. 14, 
 No. 5, May 1947, pp. 261-267. 
 
 9. Wang, C. T. , "Inelastic Column Theories and an Analysis of Experimental 
 Observations, " Jour, of Aero. Sci., Vol. 15, No. 5, May 1948, pp. 283 292. 
 
 10. Cicala, P. , "Column Buckling in the Elastoplastic Range, " Jour, of Aero. Sci. , 
 Vol. 17, No. 8, Aug. 1950, pp. 508-512. 
 
 11. Jordan, W. D , "Inelastic Behavior of Eccentrically Loaded Columns, " Doctor's 
 Thesis, Dept. of Theo. and Appl. Mech. , University of Illinois, 1952. 
 
 12. Bijlaard, P. P. , "Buckling of Columns with Equal and Unequal End Eccentricities 
 and Equal and Unequal Rotational End Strains, " Proc. 2nd U. S. National 
 Congress Appl. Mech. , 1954, pp. 555-562. 
 
 13. Galambos, T. V. , and Ketter, R. L , "Columns Under Combined Bending and 
 Thrust, " Proc. ASCE, Vol. 85, EM-2. 1959, p. 1990. 
 
 14. Costello, G. A. , "The Creep Buckling of Columns Made of Canvas Laminate. " 
 Doctor's Thesis, Dept. of Theo. and Appl. Mech. , University of Illinois, 1959. 
 
 15. Sidebottom, O. M. . Pocs, E. , and Costello, G. A. , "The Effect of End Conditions 
 on the Collapse Loads of Columns, " Dept. of Theo. and Appl Mech, Report 
 
 No. 585, University of Illinois, 1960, 
 
 25 
 
16. Timoshenko, S. , Theory of Elastic Stability, McGraw-Hill Book Co. , New York, 
 1936, pp. 1-63. 
 
 17. Osgood, W. R., "Beam -Columns, " Jour, of Aero. Sci. , Vol. 14, No. 3 Mar. 
 1947, pp 167-170. 
 
 18. Ketter, R. C. , "Stability of Beam-Columns Above the Elastic Limit, " Proc. ASCE, 
 Vol. 81, No. 692, Oct 1955. 
 
 19. Clark, M. E. , Corten, H. T. , and Sidebottom, O, M. , "Inelastic Behavior of 
 Ductile Members Under Dead Loading, " University of Illinois Engineering Experi- 
 ment Station, Bulletin No 426, 1954. 
 
 20. Morkovin, D. , and Sidebottom, O. M. , "The Effect of Non-uniform Distribution 
 of Stress on the Yield Strength of Steel, " University of Illinois Engineering 
 Experiment Station, Bulletin No. 372, 1947. 
 
 21. Sidebottom, O M. , Dharmarajan, S. , Gubser, J. L. , and Leasure, J. D. , "The 
 Effects of Inelastic Action on the Resistance to Various Types of Loads of Ductile 
 Members Made From Various Classes of Metals, " WADC Technical Report 56-330, 
 Pt. XIL 1959. 
 
 22. Sidebottom, O. M. , Clark, M. E. , and Dharmarajan, S. , "The Effects of Inelastic 
 Action on the Resistance to Various Types of Loads of Ductile Members Made From 
 Various Classes of Metals." WADC Technical Report 56-330, Pt. VIII, 1958. 
 
 26 
 
VII. APPENDIX A 
 
 In this appendix, a brief outline is given of another method of approach to the 
 problem of beam-columns. This method is capable of yielding extremely accurate 
 results, and is referred to here as the "exact method". Essentially, the exact method 
 differs from the approximate method described in this paper only in the way of relating 
 curvature to deflection in the inelastic range. For simplicity in presenting this 
 approach, a symmetrically loaded beam -column with zero end eccentricities will be 
 considered, as shown in Fig. 1A. 
 
 Fig. 1A 
 
 The method of relating curvature to deflection in the inelastic range is similar 
 in principle to that used by Chwalla (6) in his solutions of column problems. It consists 
 of subdividing the member into a number of equal segments, X , as shown in Fig. 1A. 
 The assumption is then made that within each segment, the curvature remains constant 
 (i. e. , within each segment, the deflection curve is a circular arc). The deflection at 
 any section of the. member may then be found by the relation 
 
 27 
 
A 2 3X 2 
 
 ^-1, N-l. 2, 3 etc. 
 
 z o 
 
 (1A) 
 
 where ip^, designates the curvature at the Nth section along the member, and y is 
 the deflection at midspan. 
 
 For the beam -column shown in Fig. 1A, Eq. b becomes 
 
 MP M 
 
 H- - y^)Y + m:^) (b) 
 
 e e e 
 
 where M is a factor varying linearly from unity at the center section to zero at the end 
 of the member. Eqs. 1A and b thus provide one P-M-^i relation. The second P-M-V 
 relation is obtained using Eq. 5 and interaction curves of the type shown in Fig. 3. 
 
 The solution of the problem is effected by means of the above relations in the 
 following manner. 
 
 As in the previous solution, point A (see Fig. 3) is determined by the magnitude 
 of the transverse load Q, and point B is determine from the elastic solution. For a 
 given depth of inelastic penetration (k = k.), some value of P/P is assumed and the 
 corresponding value of M/M determined from the appropriate interaction curve. With 
 M = 1, the value of y is then determined from Eq. b. Equation 5 is mow used to 
 determine the curvature, # , at the center of the member. The deflection y may 
 now be determined by Eq. 1A. Using the value of y and the appropriate value for 
 
 M in Eq. b, the dimensionless moment, M/M , corresponding to the section of the 
 
 1 e 
 
 member at y may be established. Then the curvature lj> . is found by Eq. 5. The 
 
 values of M /M and k for use in Eq. 5 are found from the interaction curves by inter- 
 polation. This process is continued until the deflection at the end of the member is 
 determined. The assumed value of P/P is correct if the computed value of deflection 
 at the end of the beam -column is zero or very nearly so. Thus to determine one point 
 on the moment-load curve, this numerical procedure may have to be repeated three or 
 four times until the boundary condition is satisfied. 
 
 In determining the value of M/M from Eq. b for any given section, the procedure 
 consisted of adding the maximum value due to P, to the average value due to Q. Thus 
 for example, (M/M^ = 7<P/P e ) y Q + 1/2 [(M Q /M e ) Q + (Mq/M^J 
 
 28 
 
Theoretically, very accurate results may be obtained by this method of solution 
 if the segments, X , are made extremely small. This, however, would entail very 
 lengthy if not prohibitive computational work, particularly if conditions other than those 
 shown in Fig. 1A are to be met. For example : when the transverse load Q is not 
 applied at midspan, one more unknown is introduced into the problem, namely the slope 
 of the deflection curve at the point of application of Q. Therefore, in applying the method 
 outlined above, it becomes necessary to assume a value for this slope in addition to the 
 value of P/P . The procedure is then applied to both portions of the beam -column until 
 the conditions at both ends of the member are satisfied (i. e. , until the determined values 
 of deflection are zero at both ends of the member). Obviously, this method can become 
 very laborious even when the number of segments considered is small. It would appear 
 that this method, compared to the approximate solution presented previously, can 
 become economically feasible only if a technique is developed to program the calculations 
 on a computing machine. 
 
 29 
 
Table I. Materials and Physical Characteristics of Specimens Tested in this Program 
 
 L/r = Slenderness Ratio 
 
 Material 
 
 Rectangular Sections 
 
 T-S 
 
 h- 
 
 tl , 
 
 ec 
 4t ] 
 
 tions 
 ■ 1 * 
 
 
 
 T 
 
 
 h 
 
 
 1 
 
 
 1 1/2 t^ 
 
 kt l 
 
 ■ 
 
 + b .> 
 
 
 
 
 
 i/r = 50 
 
 */r = 75 
 
 i/x = 100 
 
 i/r = 60 
 
 2024-T4 
 
 Aluminum 
 
 Alloy 
 
 b = 0. 500 
 h = 0. 625 
 
 b = 0. 500 
 h = 0.625 
 
 b = 0. 500 
 h = 0. 500 
 
 t 2 = 0. 100 
 
 SAE 1020 
 Steel 
 
 b = 0. 500 
 h = 0.625 
 
 b = 0. 500 
 h = 0.625 
 
 b = 0. 500 
 h = 500 
 
 t. = 0. 100 
 
 17-7PH 
 
 Stainless 
 
 Steel 
 
 b = 0. 350 
 b = 0. 500 
 
 b = 0. 400 
 h = 0. 420 
 
 
 
 30 
 
Table II. Properties of the Three Materials Used in this Investigation. 
 
 Material 
 
 E 
 psi x 10 6 
 
 a 
 e l 6 
 psi x 10 
 
 a 
 psi x 10 
 
 e 
 
 in/in x 10 
 
 e 
 
 e 2 
 
 -4 
 in/in x 10 
 
 a l 
 
 a 2 
 
 2024-T4 
 
 Aluminum 
 
 Alloy 
 
 10.8 
 
 37.5 
 
 51.5 
 
 34.7 
 
 47.7 
 
 0.189 
 
 
 
 SAE 1020 
 Steel 
 
 30.0 
 
 30.3 
 
 30.3 
 
 10. 1 
 
 10. 1 
 
 
 
 
 
 17-7PH 
 
 Stainless 
 
 Steel 
 
 28.0 
 
 170.0 
 
 170.0 
 
 60.7 
 
 60.7 
 
 0. Ill 
 
 0.111 
 
 31 
 
Table III. Values of P 
 
 ca 
 
 P /P, and P /P for the Various Symmetrically 
 ca' ct ca ya ' y 
 
 Loaded Beam -Columns Studied in this Program. P = Actual Collapse 
 
 Load; P . = Theoretical Collapse Load; and P = Actual Yield Load, 
 ct r ya 
 
 Material 
 
 Shape 
 
 I/r 
 
 M Q /M e 
 
 e l 
 
 e 2 
 
 P 
 ca 
 
 lb. 
 
 P /P ¥ 
 ca ct 
 
 P /P 
 
 ca ya 
 
 
 
 100 
 
 0.25 
 
 
 
 
 
 2,130 
 
 1.04 
 
 1.05 
 
 
 
 75 
 
 0.25 
 
 
 
 
 
 4,240 
 
 1.00 
 
 1.09 
 
 
 
 50 
 
 0.25 
 
 
 
 
 
 7,750 
 7,620 
 
 1.03 
 1.01 
 
 1.21 
 1.18 
 
 
 
 100 
 
 0.50 
 
 
 
 
 
 1,680 
 1.680 
 
 1.04 
 1.04 
 
 1.37 
 1.37 
 
 202 4 -T 4 
 Aluminum 
 
 Rect. 
 
 75 
 
 0.50 
 
 
 
 
 
 3,580 
 3,510 
 
 1.05 
 1.03 
 
 1.45 
 1.43 
 
 Alloy 
 
 
 50 
 
 0.50 
 
 
 
 
 
 6,360 
 6,330 
 
 1.06 
 1.05 
 
 1.70 
 1.69 
 
 
 
 100 
 
 0.25 
 
 0.15h 
 
 0.15h 
 
 1,690 
 
 1.03 
 
 1. 17 
 
 
 
 75 
 
 0.25 
 
 0.15h 
 
 0. 15h 
 
 3,250 
 
 1,06 
 
 1.28 
 
 
 
 50 
 
 0.25 
 
 0.15h 
 
 0.15h 
 
 5, 130 
 
 1.05 
 
 1.50 
 
 T- 
 
 60 
 
 0.50 
 
 
 
 
 
 1,850 
 
 1.01 
 
 1.27 
 
 
 Section 
 
 60 
 
 0.25 
 
 0.15h 
 
 0.15h 
 
 1,550 
 
 1.03 
 
 1.21 
 
 
 
 100 
 
 0.50 
 
 
 
 
 
 3,410 
 3,400 
 
 1.07 
 1.07 
 
 1.44 
 1.44 
 
 
 
 75 
 
 0.50 
 
 
 
 
 
 5,370 
 5,400 
 
 1.05 
 1.06 
 
 1.49 
 1.50 
 
 
 Rect. 
 
 50 
 
 0.50 
 
 
 
 
 
 6,400 
 6,450 
 
 1.02 
 1.03 
 
 1.52 
 1.54 
 
 SAE 1020 
 
 
 100 
 
 0.75 
 
 
 
 
 
 2, 460 
 
 1.06 
 
 2.24 
 
 Steel 
 
 
 75 
 
 0.75 
 
 
 
 
 
 3 980 
 
 1.04 
 
 2.36 
 
 
 
 50 
 
 0.75 
 
 
 
 
 
 5,260 
 5,100 
 
 1.07 
 1.04 
 
 2.59 
 2.54 
 
 T- 
 
 60 
 
 0.50 
 
 
 
 
 
 2,450 
 
 1.02 
 
 1. 17 
 
 
 Section 
 
 60 
 
 0.50 
 
 0.15h 
 
 0.15h 
 
 1,640 
 1,680 
 
 1.00 
 1.03 
 
 1.30 
 1.33 
 
 
 
 75 
 
 0.50 
 
 
 
 
 
 4,770 
 
 1.00 
 
 1.17 
 
 17-7PH 
 Stainless 
 
 Rect. 
 
 50 
 
 0.25 
 
 
 
 
 
 13,200 
 
 1.01 
 
 1.10 
 
 Steel 
 
 
 75 
 
 0.50 
 
 0.15h 
 
 0.15h 
 
 3,890 
 
 1.00 
 
 1.30 
 
 
 
 50 
 
 0.25 
 
 0.15h 
 
 0.15h 
 
 9,100 
 
 1.04 
 
 1.27 
 
 32 
 
Table IV. Values of P 
 
 P /P and P /P for the Various Unsymmetrically 
 ca ca' ct ca ya ' J 
 
 Loaded Beam -Columns Studied in this Program P = Actual Collapse 
 
 Ca 
 
 Load; P = Theoretical Collapse Load; and P = Actual Yield Load, 
 ct r ya 
 
 Material 
 
 Shape 
 
 f/r 
 
 M„/M 
 Q e 
 
 e l 
 
 e 2 
 
 P 
 ca 
 
 lb. 
 
 P /P . 
 ca' ct 
 
 P /P 
 
 ca ya 
 
 2024-T4 
 
 Aluminum 
 
 Alloy 
 
 Rect. 
 
 100 
 
 75 
 50 
 
 0.50 
 0.50 
 0.50 
 
 
 
 
 
 
 
 
 
 1,740 
 1,710 
 
 3,610 
 
 6,480 
 6,490 
 
 1.00 
 0.98 
 
 0,96 
 
 0.96 
 0.96 
 
 1.22 
 1.20 
 
 1.32 
 
 1.58 
 1.58 
 
 T- 
 
 Section 
 
 60 
 
 0.50 
 
 
 
 
 
 1,840 
 1,890 
 
 0.95 
 
 0.97 
 
 1.15 
 1.20 
 
 SAE 1020 
 Steel 
 
 Rect. 
 
 100 
 
 75 
 50 
 75 
 75 
 
 0.50 
 0.50 
 0.50 
 0.50 
 0.50 
 
 
 
 
 
 
 0. 15h 
 0. 15h 
 
 
 
 
 0. 15h 
 
 
 3,660 
 5,680 
 6, 530 
 3,590 
 4,400 
 
 1.05 
 1.04 
 1.01 
 1.00 
 0.96 
 
 1.41 
 1.47 
 1.52 
 
 1.69 
 
 1.47 
 
 T- 
 
 Section 
 
 60 
 
 0.75 
 
 
 
 
 
 2,210 
 
 1.01 
 
 1.28 
 
 17-7PH 
 
 Stainless 
 
 Steel 
 
 Rect. 
 
 75 
 
 0.25 
 
 
 
 
 
 6,480 
 
 1.00 
 
 1.02 
 
 33 
 
u 
 
 CD 
 
 1 
 
 s 
 
 X! 
 
 
 c 
 
 J3 
 
 CO 
 
 
 en 
 
 
 CD 
 
 
 tt 
 
 
 CO 
 
 
 cd 
 
 
 5 
 
 
 m 
 
 
 £ 
 
 
 o 
 
 
 X! 
 
 
 CO 
 
 
 J3 
 
 
 & 
 
 
 CD 
 
 
 J4 
 
 
 CO 
 
 
 (I) 
 
 
 X! 
 
 
 H 
 
 
 bj) 
 
 
 c 
 
 
 3 
 
 
 cd 
 
 
 o 
 
 
 J 
 
 
 T3 
 
 
 CD 
 
 
 .3 
 
 
 X] 
 
 . 
 
 s 
 
 o 
 
 •— < 
 
 O 
 
 4- 
 
 U 
 
 l/J 
 
 o 
 
 +-> 
 
 •a 
 
 <D 
 
 CD 
 
 >- 
 
 
 
 u 
 
 ♦j 
 
 CD 
 
 Jh 
 
 •T— > 
 
 
 ■a 
 
 CU 
 
 CO T3 
 
 (U 
 
 CO 
 
 XI 
 
 O 
 
 6 
 
 en 
 
 S3 
 
 2 
 
 cv 
 
 
 i| 
 
 C 
 
 
 O 
 
 >■ 
 
 
 
 
 M 
 
 O 
 
 M 
 
 cu 
 
 m 
 
 CO Pl, 
 
 H-2 
 
 EC 
 
 34 
 
CO 
 03 
 
 « e 
 
 
 
 
 
 e 2 
 
 
 
 
 
 J 
 
 i 
 
 
 e l 
 
 
 
 
 
 
 
 
 i 
 
 \ 
 
 
 
 
 
 
 
 
 / ^Slope = a,E (Compression) 
 
 
 
 
 
 
 — Slope = tt 2 E (Tension) 
 
 
 
 
 0) 
 
 b 
 
 
 
 
 b 
 
 
 
 
 
 _ Slope = E 
 
 
 r i 
 
 ' 
 
 
 Strain, e 
 
 Fig. 2. A Schematic Representation of Stress -Strain Relations in 
 Tension and Compression. 
 
 35 
 
1.0 
 
 M/M 
 
 k=0.5 
 
 0.6\. 
 
 
 ^0.8 N. \. ^\ 
 
 ^\ ^\ ^\ f\\ N, 
 
 v\\\l\ \ 
 
 >v \ \ J 
 
 
 N> \ D A 
 
 
 N. CV 
 
 
 ^V \ 
 
 
 
 
 Collapse 
 Load \ 
 
 3i 
 
 p/p 
 
 1.0 
 
 Fig. 3. A Schematic Representation of Interaction Curves for 
 Constant Depth of Yielding. 
 
 36 
 
" >•. 
 
 6 
 
 ed 
 
 o 
 
 W 
 
 T3 
 
 W 
 
 a 
 
 s 
 
 I 
 
 bO 
 
 .a 
 
 w 
 
 -G 
 
 B 
 <u 
 
 CO 
 
 
 37 
 
M/M 
 
 Fig 5. Dimensionless Interaction Curves for Rectangular Sections 
 Made of 2024-T4 Aluminum. 
 
 38 
 
1,8 
 
 M'M 
 
 1.2 
 
 Fig. 6. Dimensionless Interaction Curves for T-Sections Made 
 of 2024 T4 Aluminum. 
 
 39 
 
1.6 
 
 1.0 
 
 Fig. 7. Dimensionless Interaction Curves for Rectangular Section 
 Made of SAE 1020 Steel. 
 
 40 
 
2.0 
 
 P/P 
 
 ' e 
 
 Fig. 8. Dimensionless Interaction Curves for T-Sections 
 Made of SAE 1020 Steel. 
 
 41 
 
Fig. 9. Dimensionless Interaction Curves for Rectangular Sections 
 Made of 17-7PH Stainless Steel. 
 
 42 
 
Fig. 10. Test Apparatus Used to Apply Transverse and Axial Loads to Beam- 
 Columns. 
 
 43 
 
o 
 U 
 
 cti 
 
 -d 
 c« 
 o 
 
 J 
 
 CO 
 
 ■a 
 
 < 
 
 
 -a 
 
 CO 
 
 D 
 
 09 
 
 <D 
 (-1 
 3 
 
 IP 
 
 3 
 
 ■i-l 
 
 44 
 
1.6 
 
 1.4 
 
 1.2 
 
 1.0 
 
 0.8 
 
 0.6 
 
 0.4 
 
 0.2 — 
 
 
 
 1 
 
 
 
 1 
 
 
 
 
 
 
 
 
 Exact Method (20 Segments) 
 
 
 
 
 D _ 
 
 
 
 
 ° Ave. of Two Tests 
 
 
 
 
 3 1 
 
 
 
 
 ' 
 
 
 _ 2024-T4 Aluminum, £/r 
 
 = 75 
 
 
 
 
 
 \ \ 
 
 
 
 
 
 
 
 \o\ 
 
 
 
 
 
 \ \ .— SAE 1020 Steel, 
 
 
 
 
 
 \6<r L/t = 75 
 
 
 
 
 
 \ I 
 
 
 
 
 
 \ A 
 
 
 
 
 
 I 
 
 
 
 
 
 
 
 k 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 «- a/2 + 
 
 
 
 
 
 p 
 
 < 
 
 r 
 
 P 
 
 X 
 
 
 > 
 
 '. 
 
 k 
 
 
 1 
 
 y 
 
 1 
 
 _l 
 
 
 
 0. 
 
 2 
 
 
 
 
 .4 
 
 0. 
 
 6 0. 
 
 8 l.O 
 
 p/p 
 
 ' e 
 
 Fig. 12. Comparison of Moment- Load Curves for Symmetrically 
 
 Loaded 2024-T4 and SAE 1020 Steel Beam -Columns Determinec 
 by Two Theoretical Methods and By Test. 
 
 45 
 
1.6 
 
 0.2 
 
 0.4 
 
 0.6 
 
 0.8 
 
 1.0 
 
 P/P £ 
 
 Fig. 13. Moment- Load Curves for Symmetrically Loaded, Rectangular 
 2024-T4 Aluminum Beam-Columns. (Mq/M s = 25; 
 
 e x = e 2 = 0) 
 
 46 
 
1.6 
 
 1,0 
 
 Fig. 14. Moment-Load Curves for Symmetrically Loaded, Rectangular 
 2024-T4 Aluminum Beam -Columns. (M n /M = 0.50; 
 ei =e 2 =0) Q 
 
 47 
 
1.6 
 
 1.0 
 
 Fig. 15. Moment- Load Curves for Symmetrically Loaded, Rectangular 
 2024-T4 Aluminum Beam -Columns (M Q /M = 0. 25; 
 ej = e 2 = 0. 15h) ^ 
 
 48 
 
1.6 
 
 1.4 
 
 12 
 
 1.0 
 
 M/M 
 
 0.8 
 
 0.6 
 
 0.4 
 
 0.2 
 
 e i =e 2 = 
 
 M„/M = 0. 2 
 Q e 
 S./t = 60 
 
 Theoretical 
 Experimental 
 
 e x = e 2 = 0, 15h 
 
 M^ /M = 0. 50 
 Q- e 
 
 1/t = 60 
 
 \ 
 
 \ 
 
 X 
 
 
 
 Q 
 
 
 e l 
 
 
 «- St/2 -* 
 
 * IP 
 
 e 2 
 
 '■ 
 
 -« JE 
 
 i 
 ► 
 
 , t 
 
 \ 
 
 Initiation of 
 Yielding 
 
 ) 
 
 \ 
 
 "V 
 
 \ 
 
 \ 
 
 \ 
 
 _\j 
 
 0.2 
 
 0. 4 6 
 
 P/P 
 
 0.8 
 
 1.0 
 
 Fig. 16. Moment- Load Curves for Symmetrically Loaded T-Section 
 Beam-Columns of 2024-T4 Aluminum. 
 
 49 
 
0.2 
 
 p p 
 
 ' 
 
 Q 
 
 *- 1/2 -»- 
 
 1 
 
 - p 
 
 1 
 
 1 
 
 k 
 
 
 y 
 
 
 
 \ 
 
 V 
 
 V 
 
 \ 
 
 \ 
 
 \ 
 
 \J 
 
 0.2 
 
 0.4 
 
 6 
 
 0.8 
 
 1.0 
 
 P/P 
 
 Fig. 17 Moment- Load Curves for Symmetrically Loaded, Rectangular 
 SAE 1020 Steel Beam -Columns. <M Q /M = 0. 50; e^ = e 2 = 0) 
 
 50 
 
1.6 
 
 1.0 
 
 Fig. 18. Moment-Load Curves for Symmetrically Loaded, Rectangular 
 SAE 1020 Steel Beam -Columns. (M Q /M = 0. 75; e = e = 0) 
 
 51 
 
 UAUiXflSITY OF ILLINOIS 
 HBRAW 
 
1.6 
 
 0.2 
 
 4 
 
 0.6 
 
 U. b 
 
 1.0 
 
 p/p 
 
 Fig. 19. Moment- Load Curves for Symmetrically Loaded T Section 
 Beam Columns of SAE 1020 Steel. 
 
 52 
 
1.6 
 
 1.4 
 
 1.2 
 
 1.0 
 
 \ 
 
 8 
 
 0.6 
 
 0.2 
 
 \ 
 
 \ 
 
 \ 
 
 ,\ 
 
 Theoretical 
 
 O Experimental 
 
 M_/M = 0. 25 
 Q e 
 
 Jfc/r = 50 
 
 M_/M = 0. 50 
 Q e 
 
 i/T = 75 
 
 \ 
 x \ 
 
 Initiation 
 of Yielding 
 
 \ 
 
 \ 
 
 \ 
 
 \J 
 
 0. 2 
 
 0.4 
 
 0.6 
 
 0.8 
 
 1.0 
 
 P/P 
 
 Fig. 20. Moment -Load Curves for Symmetrically Loaded^ Rectangular 
 
 jel ] 
 
 53 
 
 17-7PH Stainless Steel Beam-Columns, (e. = e„ = 
 
1.6 
 
 Fig. 21. Moment-Load Curves for Symmetric aily Loaded Rectangular 
 17-7PH Stainless Steel Beam-Columns, (e = e = 0. 15h) 
 
 54 
 
1.6 
 
 Theoretical 
 O • Experimental 
 
 1.0 
 
 Fig. 22. Moment-Load Curves for Unsymmetrically Loaded, 
 Rectangular 2024-T4 Aluminum Beam-Columns. 
 (M Q /M g = 0.50; e x = e 2 = 0) 
 
 55 
 
1.6 
 
 0.2 
 
 0.4 
 
 0. 6 
 
 0. 8 
 
 1.0 
 
 P/P 
 
 Fig. 23. Moment-Load Curve for an Unsymmetrically Loaded T-Section 
 Beam-Column of 2024-T4 Aluminum (M„/M = 0. 50; 
 
 e x = e 2 = 0) 
 
 *Q' e 
 
1.6 
 
 1.0 
 
 P/P 
 
 Fig. 24. Moment -Load Curves for Unsymmetrically Lpaded, 
 
 Rectangular SAE 1020 Steel Beam-Columns (M^/M = 50; 
 
 e x = e 2 = 0) 
 
 V e 
 
 57 
 
1.0 
 
 Fig. 25. Moment- Load Curves for Unsymmetrically Loaded, 
 
 Rectangular SAE 1020 Steel Beam -Columns with Equal and 
 Unequal End Eccentricities. 
 
 58 
 
M/M 
 
 0.2 
 
 1.0 
 
 Fig. 26. Moment-Load Curve for an Unsymmen ically Loadeu 
 
 T-Section Beam-Column of SAE 1020 Steel. (M„/M = 0. 75; 
 e x = e 2 = 0) ^ e 
 
 59 
 
1.0 
 
 Fig. 27. Moment-Load Curve for Unsymme trie ally Loaded, Rectangular 
 17-7PH Stainless Steel Beam-Columns. (M n /M = 0. 25; 
 ei =e 2 =0) Q 
 
 60 
 
WADD TECHNICAL REPORT 60-580 
 Part V 
 
 INELASTIC DESIGN OF LOAD CARRYING MEMBERS 
 Part V. Theoretical and Experimental Creep Analyses 
 of Beam- Columns 
 
 D. L. Dewhirst 
 O, M. Sidebottom 
 
 Theoretical and Applied Mechanics Department 
 University of Illinois 
 
 September, 1961 
 
 Materials Centra] 
 
 Contract No. AF 33 (616)-7600 
 
 Project No. 7351 
 
 Wright Air Development Division 
 Air Research and Development Command 
 
 United States Air Force 
 Wright Patterson Air Force Base Ohio 
 
WADD TECHNICAL REPORT 60-580 
 Part V 
 
 INELASTIC DESIGN OF LOAD CARRYING MEMBERS 
 Part V. Theoretical and Experimental Creep Analyses 
 of Beam- Columns 
 
 D. L. Dewhirst 
 O, M. Sidebottom 
 
 Theoretical and Applied Mechanics Department 
 
 University of Illinois 
 
 September 1961 
 
 WRIGHT AIR DEVELOPMENT DIVISION 
 
„ Sf'*- ENGINEERING LIBRARY 
 
 FOREWORD 
 
 This report was prepared by the University of Illinois under USAF Contract No. 
 AF 33(616) -7600. This Contract was initiated under Project No. 7351, "Metallic 
 Materials", Task No. 73521, "Behavior of Metals". It was administered under the 
 direction of the Materials Central, Directorate of Advanced Systems Technology, 
 Wright Air Development Division with Mr. R. F. Klinger acting as the Project Engi- 
 neer. 
 
 This is a final report which covers work conducted from October 1, 1960 to 
 September 31, 1961. 
 
 The work was conducted in the Department of Theoretical and Applied Mechanics 
 in the Engineering Experiment Station, University of Illinois, Urbana, Illinois. Pro- 
 fessor O. M. Sidebottom was the Project Supervisor. 
 
 li 
 
ABSTRACT 
 
 A theory is presented for constructing load-creep deflection curves for beam- 
 columns at any specified time. The stress -strain-time relation for the material is 
 assumed to be represented by an isochronous stress -strain diagram approximated by 
 an arc hyperbolic sine curve (see Equation 16). A program has been written for the 
 IBM 650 digital computer to calculate points on moment -curvature curves for a general 
 I-section. The theory uses these curves and the successive approximations procedure 
 by Newmark (32). 
 
 The experimental part of the investigation included tests of rectangular- and 
 T-section beam-columns made of 17-7Ph stainless steel and tested at 972°F. Several 
 slenderness ratios were considered. The beam-columns were subjected to a constant 
 axial load located either at the centroid of the section or at an eccentricity of 15 per 
 cent of its depth and to a constant transverse load at midspan of sufficient magnitude 
 to produce a linear elastic bending stress when acting alone of 0. 50 cr . 
 
 PUBLICATION REVIEW 
 This report has been reviewed and is approved. 
 FOR THE COMMANDER: 
 
 W. J. Trapp 
 
 Chief, Strength and Dynamics Branch 
 Metals and Ceramics Laboratory 
 Materials Central 
 
 in 
 
TABLE OF CONTENTS 
 
 Page 
 
 I. INTRODUCTION 1 
 
 1. Preliminary Statement 1 
 
 2. Purpose and Scope 3 
 
 II. THEORY 5 
 
 1. Assumptions 5 
 
 2. Moment -Curvature Interaction Curves 5 
 
 3. Load- Deflection Curves 7 
 
 III. MATERIALS AND METHODS OF TESTING 10 
 
 1. Materials Used, Test Members, and Testing Procedure 10 
 
 2. Properties of Materials 10 
 
 3. Method of Loading Beam-Columns 11 
 
 IV. ANALYSIS OF RESULTS 12 
 
 V. SUMMARY AND CONCLUSIONS 14 
 
 1. Summary 14 
 
 2. Conclusions 14 
 
 BIBLIOGRAPHY 16 
 
 APPENDIX A 19 
 
 IV 
 
I. INTRODUCTION 
 
 1. Preliminary Statement 
 
 Since the heating of structural members is unavoidable in many present day en- 
 gineering applications such as nuclear reactors, missiles, and aircraft, the effect of 
 creep cannot be neglected. One structural element which is important in this respect 
 is the creep behavior of a beam-column. It is the purpose of this paper to present a 
 workable method for the design of beam-columns, whose dimensions and creep prop- 
 erties are known. The term beam-column is applied to a long, slender member which 
 is subjected to transverse as well as axial loads, that is to a member simultaneously 
 performing the function of a beam and column. 
 
 The principal difference between the method herein described and the methods 
 used by other authors is the form of the stress -strain-time law adopted. In previous 
 investigations (1-16, 18-22, 24-28)* including those on beams, columns, and beam- 
 columns, the stress-strain-time law has been represented by one of the following 
 forms: 
 
 e = Ba n (1) 
 
 e = Ae Ba (2) 
 
 € = B sinh - (3) 
 
 o 
 
 e = -g + \a (4) 
 
 e = Aa m cr + B(j n (5) 
 
 e = p + Act a + Ba (6) 
 
 a + / cr N n /0\m o /0 > 
 
 e= E ' ( X> + <"jL> fl < 8 > 
 
 I = | + XptP-V (9) 
 
 e = | + Ka n F(t) (10) 
 
 Numbers in parentheses refer to corresponding entries in the bibliography. 
 
 Manuscript released by authors September, 1961, for publication as a WADD 
 Technical Report. 
 
a . B<r k /, 1X 
 
 € = g + A e t (11) 
 
 e = (e o +mt n ) sinh -£ (12) 
 
 o 
 
 e= e +mt n (13) 
 
 o 
 
 e = | +Ka a (l-e" qt )+Ba n t (14) 
 
 a= Ke m e n (15) 
 
 In the above, ct denotes stress, e denotes strain, t denotes time and the dot indicates 
 differentation with respect to time. The remaining symbols are constants, em- 
 pirically obtained from constant stress creep curves. One of the difficulties in the 
 analysis of the beam column is that no beam fiber is subjected to a constant stress. 
 
 In order to overcome this difficulty, several investigators (14, 15, 16, 22, 28) 
 have attempted to follow the stress history of the various fibers of columns or beam 
 columns but have represented the stress -strain-time relation for the material by 
 equation 10 with the experimental constants obtained from constant stress creep 
 curves. Based on a study by Shanley (29), the stress-time curve for a given fiber of 
 the column is represented by infinitesimal steps. Each step consists of a constant 
 stress period of time followed by an instantaneous increment of stress. Libove 
 (14, 15) tested only pin ended columns, and assumed that the inelastic column deform- 
 ed into a sine curve. Higgins (22) treated only pin ended columns but did not assume 
 a configuration of the inelastic column. His analysis uses an iterative type of numer- 
 ical solution which requires six or seven 8 hour days on an IBM digital computer. Lin 
 (16, 28) has treated both the column and beam-column. His analysis uses a stepwise 
 method of calculation which is somewhat less involved than Higgins' method. Finnie 
 and Heller (4) have compared creep curves by Lin and Higgins with test data. The 
 agreement was not good. 
 
 Instead of predicting the creep curve for a given column, some investigators 
 (30, 33, 34, 35, 38) have considered the problem of predicting the behavior of the 
 column for a specified time. They have represented the stress -strain -time diagram 
 of the material by an isochronous stress -strain diagram of the material obtained by 
 plotting values of stress and strain at a specified time from constant stress creep 
 curves. Carlson and Manning (30) did not represent the isochronous stress -strain 
 diagram by a mathematical function so that a graphical solution is required for each 
 column. Sidebottom, Clark, Costello, Dharmarajan, and Pocs (33, 34, 35, 38) have 
 
assumed that the isochronous stress-strain diagram can be represented by an arc hy- 
 perbolic sine curve of the form 
 
 ct= a axe sinh — (16) 
 
 o e 
 
 o 
 
 in which <t and e are time dependent as well as temperature dependent constants. 
 It should be noted that Eq. 12 is identical with equation 16 for constant time. Further- 
 more, equation 16 is an integrated form of equation 3 which has been shown by Kauz- 
 mann (31) to have a theoretical basis. 
 
 Equation 16 has many advantages. It is readily integrated or differentiated. It 
 is characterized by only two material constants, while most of the equations used in 
 column analyses have three or more. Equation 16 shows extreme versatility. It has 
 been successfully applied to multiaxial state of stress problems (35, 39) in addition 
 to beams and columns. Equation 16 will be adopted in this investigation. 
 
 Theories based on equation 16 have the same limitations as all other theories in 
 that the experimental constants are obtained from constant stress creep curves while 
 stresses in columns and beam-columns are not independent of time. The theory must 
 be corrected if the behavior of the column or beam-column is to be predicted. In the 
 case of columns, several different variables have been investigated and the correc- 
 tion needed to predict the collapse load at any specified time has been found to be in- 
 dependent of the type of variable. The only correction needed was to increase a 
 by 10 per cent. It is believed that this correction will be satisfactory for the heam 
 column if the column action is predominant. 
 2. Purpose and Scope 
 
 The purpose of this investigation is to develop a theory to predict the load -creep 
 deformation curve and the collapse load at any specified time for columns and beam- 
 columns subjected to any given loading condition. The validity of the theory has been 
 checked by tests of 17-7PH stainless steel beam-columns at 972°F. 
 
 A program was written for the IBM 650 digital computer to determine points 
 on the moment -curvature diagram. This program is valid for the general I—section 
 which includes the rectangular and T-sections as special cases. Using the method 
 of successive approximations due to N. M. Newmark (32), the load -deflection curve 
 
for a given column or beam-column with any loading can be constructed as accurately 
 as desired. Furthermore, the resulting load -deflection curve is independent of the 
 properties of the material so that it can be used to predict the deflection at any load 
 and the collapse load at any time for which values of cr and e are known. 
 
 The experimental part of the investigation included tests of rectangular- and T - 
 section pivot ended beam-columns of 17-7PH stainless steel tested at an elevated tem- 
 perature of 972°F. These columns had slenderness ratios of 50.0, 60.2, 75.0, and 
 100.0. Each column was subjected to a constant transverse load at the center of the 
 column of sufficient magnitude to produce a maximum linear elastic bending stress 
 equal to a /2. The axial load was also held constant and was of sufficient magnitude 
 to cause collapse of each beam-column in 30 minutes. The axial load was applied 
 through the centroid of the cross section or with eccentricity equal to 15 per cent of 
 the depth. 
 
 Good agreement was found between theory and experiment. 
 
II. THEORY 
 
 1. 
 
 Assumptions 
 
 Theoretical load -deflection curves for columns and beam-columns are construct- 
 ed using a family of moment -curvature interaction curves for constant values of load. 
 The derivation of these curves are based on the following assumptions: 
 
 1. Plane cross-sections remain plane and do not warp with time. The validity 
 of this assumption for pure bending has been shown independently by at least two in- 
 vestigators, MacCollough (17) and Tapsell and Johnson (23), Presumably the assump- 
 tion is poor only for short deep beams when shear stresses are high. 
 
 2. The stress -strain-time relationship for the material is given by the iso- 
 chronous stress-strain relation represented by Equation 16. Since the beam fibers do 
 not experience a constant stress, this relationship is in error. Previous investigations 
 (34, 38) have shown that use of the arc hyperbolic sine relationship for eccentrically 
 loaded columns predicts a collapse load which is ten per cent too small. 
 
 3. The experimental constants a and e are assumed to be the same for 
 
 o o 
 
 tension and compression. If the creep properties are different for tension and com- 
 pression, the compression properties should generally be used, since compressive 
 stresses are appreciably greater than the tensile stresses in most columns and beam- 
 columns. If the stresses due to the axial load are small compared to the bending 
 stresses, <r and e can be based on the average isochronous stress -strain diagram. 
 
 2. 
 
 4. The material is assumed to be isotropic and homogeneous. 
 Moment -Curvature Interaction Curves 
 
 Consider the general I-section member shown in Figure 1 which is subjected to 
 an axial load P, and to a bending moment M locating the neutral axis at a distance qh 
 from the most strained fibers. With the strain distribution known, the stress distri- 
 bution is obtained using equation 16, and the magnitude of P and M can be determin- 
 ed from the equations of equilibrium: 
 
 P = 
 
 / 
 
 CTdA 
 
 M = f(y-d)adA 
 
 Equations 17 and 18 integrate into the following equations: 
 2 
 
 P = 
 
 M 
 
 a qh 
 o n 
 
 b, B„ - b„ B, 
 
 -b B + b n B 
 
 K L 1 K 4 1-q v 2 q-u^ 3 q-v 
 
 -K 
 
 ■K 
 
 K -I 
 
 a q h 
 o^ 
 
 b. C„ -b, C. -b„ C + 
 
 IK 4 1-q „ 2 q-u 
 
 K 
 
 K 
 
 b. C 1 
 
 3 ^L kJ 
 
 (17) 
 
 (18) 
 
 (19) 
 
 -(q-/3)Ph 
 
 (20) 
 
in which K = e,/e . In these equations, the functions B XT and C XT are defined as 
 1 o ^ N N 
 
 follows: 
 B. 
 
 N 
 
 N log e (N + V N + 1 ) 
 
 fi 
 
 /•J 
 
 N^+ 1 
 
 (21) 
 
 C N 4 
 
 (1 + 2N 2 ) log (N + VN 2 + 1 ) - N V N 2 + 1 J 
 
 (22) 
 
 N represents the various subscripts for B and C in equations 19 and 20. 
 
 If the deformations are considered small, the curvature can be written in terms 
 of the strain distribution, shown in Figure 1 to give : 
 
 A 2 
 
 d y 
 
 a 2 
 dx 
 
 4> = 
 
 R 
 
 1 ( 
 
 qh qh 
 
 (23) 
 
 It is convenient to define the cross sectional area A, by the relation: 
 
 A - Dh 2 (24) 
 
 and rewrite equations 19, 20 and 24 in dimensionless form as follows: 
 
 P q 
 
 Act 
 
 M 
 
 Act h 
 o 
 
 KD 
 
 DK' 
 
 h^ _ K 
 
 b, B -b„ B + b„ B 
 
 Ik 2 q-u K 3 q-.y K 
 
 B 
 
 2lL k J 
 
 b l C K 
 
 2 q-u 
 
 + b. 
 
 K 
 
 q-v 
 
 -b. 
 
 K 
 
 Vl 
 
 K 
 
 (25) 
 
 (q-/3)P 
 
 Act 
 o 
 
 (26) 
 (27) 
 
 It will be noted that the right sides of equations 25, 26 and 27 are functions of q and 
 K and are independent of the properties of the material and of the magnitude of the 
 relative dimensions of the cross -section. A trial and error solution is required if 
 a dimensionless moment-curvature curve for a specified load is required. A program 
 has been written for the IBM 650 digital computer to make the necessary calculations. 
 A description of this program is given in Appendix A. 
 
 The dimensionless moment -curvature curves for the rectangular-section 
 
 2 1 1 
 
 (b =b 4 =D,b =b „=0) are shown in Figure 2 and for the T- section (b.^j.b = -^-.b =0,b.=-g 
 
 u= -r, v=0) are shown in Figure 3. 
 
Although an IBM 650 digital computer program was written for the general I- 
 section, similar programs could be written for any cross-section made up of a finite 
 number or rectangular elements. 
 3. Load-Deflection Curves 
 
 The load-deflection curve for a given column or beam -column subjected to any 
 general loading can be computed by a method of successive approximations (32) pro- 
 vided moment -curvature curves for the given column cross -section are known. See 
 Figures 2, 3, and 4. In order to illustrate the procedure, a point on the load-deflec- 
 tion curve will be computed for the T- section beam -column used in this investigation. 
 
 Consider the T- section beam-column shown in Figure 4, For convenience in 
 
 the calculation of the deflection, the depth is assumed to be unity. The magnitudes of 
 
 2 1 11 
 
 the dimensionless coefficients in Figure 1 are b,= -^ , b = -y, ^>o = ^> ^4 = ~ft> u ~ 4' v= 0> 
 
 and J3 =0.3393. For a slenderness ratio of 60. 2, its length is 17. 46 inches. 
 
 The beam-column is subjected to a concentrated load Q at its midpoint and to 
 an axial load P at an eccentricity of 15 per cent of its depth. Since the deflections 
 are symmetric about the center, only half of the column need be considered. The 
 beam-column has been arbitrarily divided into eight equal segments A = ft/8, the end- 
 points of which specify station points a,b,c,d and f. The dimensionless moments due 
 to Q, M n /cr Ah, have been computed for the specified locations and are shown in Fig- 
 ure 4 For this example, P is taken of sufficient magnitude to make P/ffA = 0.7. 
 The moments due to P depend upon the unknown deflections. The first step in de- 
 termining the deflections is to assume a reasonable value for the deflection at each 
 station point . The initially assumed deflections shown in Figure 4 are calculated us- 
 ing the deflection formula for a linear elastic beam-column with E= u /e . The 
 dimensionless moments M p /cr Ah are computed and added to M„/cr Ah to obtain 
 the total moment M/cr Ah . For each dimensionless moment, the dimensionless 
 curvature, h^/e Q , is obtained from the curve for P/Aj = 0.7, Figure 3. Since the 
 curvature is equal to the angle change per unit length, the curvature diagram is some- 
 times called the angle -change diagram. Using the conjugate beam method , deflections 
 are calculated by loading the beam through stringers with the curvature diagram. The 
 number of stringers used corresponds to the number of segments into which the beam 
 is broken, in our case eight. By equations of statics, the concentrated curvature (i.e. 
 finite angle change) at each station point can be calculated. This is analogous to the 
 process of finding a reaction force on a stringer. Assuming the curvature to vary 
 parabolically between any three station points, this gives: 
 
 7 
 
€ A 
 # a = "H- < 7a + 6b - c) 
 
 and e X 
 
 ip h = -^-(a + lOb + c) 
 
 where ip is the equivalent concentrated curvature, or finite change of angle, at the 
 
 station point denoted by the subscript. 
 
 Since the member is assumed to be made up of infinitely stiff segments of length 
 a 
 X = o- with concentrated angle-changes at each station point, the slope of each segment 
 
 o 
 
 is easily computed. The slope is zero at the center so that the slope of the segment 
 from a to b is 22. 5 e A/12. The slope for the adjacent segment is equal to the slope 
 of segment a to b plus the angle change at b. In this way the slope for each segment 
 can be calculated. Beginning at the left hand reaction f where the deflection is zero, 
 the deflection of the adjacent station point d is seen to equal the product of the seg- 
 ment length and slope. The deflection of the next station point is equal to this deflec- 
 tion plus the product of its segment length and slope, i.e. 
 
 6 = 126. 4e A 2 /12 + X (100.0 c A/12) - 226. 4e A 2 /12 
 c o o o 
 
 Using the material properties listed in Figure 8, the deflections are computed and com- 
 pared to the assumed deflections. 
 
 The calculated deflections do not coincide with the assumed values so that another 
 set of assumptions must be made. The calculated deflections may be used as a closer 
 approximation to the real deflections and the computations repeated. However, the num- 
 ber of trials can be greatly reduced by doubling the correction. For instance the de- 
 flection at the center was 0. 017 inches greater than that assumed. For the next trial 
 assume the center deflection to be : 
 
 6 A = 0.125 + 2 x 0.017 = 0.159 in. 
 A 
 
 Let the other deflections be increased in the same proportion, i. e. : 
 
 132 
 6 b = 07T42 x0 - 159in - 
 
 The calculated deflections for the second trial are nearly identical with the assumed 
 
 deflections so that only two trials are required. The above calculations locate one point 
 
 on the dimensionless load-deflection curve shown in Figure 5. 
 
 There are some features to be noted for the convergence of the iteration procedure. 
 
 If the assumed deflection locates a point below the curve, the calculated deflection is 
 
less than the assumed. Above the curve, the calculated deflection is greater than the 
 assumed. Therefore, if the assumed deflection is below or to the left of the actual de- 
 flection curve, the numerical procedure will converge to the pre-collapse deflection. 
 The procedure always diverges from the post-collapse deflection. 
 
 Since the load-deflection curve can be presented in this dimensionless form, it 
 is independent of the material. It is valid for any material whose isochronous stress - 
 strain diagram is adequately represented by Equation 16. It can be used to determine 
 the collapse load for any magnitude of time for which values of o and e are known. 
 
III. MATERIALS AND METHOD OF TESTING 
 
 1. Materials Used, Test Members, and Testing Procedure 
 
 All test members were machined from a 1/2 inch by 2 inch bar of 17-7PH stain- 
 less steel. The compression specimens had square cross sections, 1/2 inch on a side. 
 The columns had rectangular or T -shaped cross sections with lengths and depths as 
 indicated in Table 1. 
 
 The T- section beam-columns were machined before being precipitation hardened, 
 while the rectangular section test members were machined after the heat treatment. 
 The specimens were heated to 1400°F. for 90 minutes, cooled to 60°F. in 60 minutes, 
 held at this temperature for 30 minutes, then heated to 1050°F. for 90 minutes, and 
 finally air cooled. 
 
 All of the test members were loaded in a Riehle testing machine of 120,000 
 pound capacity. The machine was equipped with a load holder to maintain any desired 
 load. 
 
 Temperatures were measured by vertically spaced thermocouples along the 
 test specimens. Two thermocouples were used on the short compression specimens, 
 while three were used on each of the beam-columns. A piece of asbestos was used to 
 cover each thermocouple as it was attached to the test member. An asbestos shield 
 was placed between the test member and the heating coils, and baffles were placed in 
 the furnace to prevent a chimney effect. Approximately one hour was necessary to 
 bring the furnace up to temperature. The temperature was manually controlled and 
 maintained at 972°F. + 2°F. Deformation readings were started as soon as the load 
 was applied, and were taken every minute thereafter. 
 
 2. Properties of Materials 
 
 Compression creep properties were obtained for the 17-7PH stainless steel at 
 972°F. The fixtures and furnace used in making the compression tests are shown in 
 Figure 6. The deformations were measured with a Riehle dial-type high temperature 
 creep extensometer with a 2 inch gage length. The compression creep curves for the 
 material are shown in Figure 7. From these creep curves, corresponding values of 
 stress and strain were obtained for zero time, for 30 minutes, and for 60 minutes to 
 give the isochronous stress-strain diagrams shown in Figure 8. An arc hyperbolic 
 sine curve (Equation 16) was fitted to the data for 30 minutes. The pertinent material 
 properties for the material are listed in Figure 8. 
 
 The heat treatment of the material used in this investigation was carried out at 
 a temperature 15 to 30°F. higher than in previous investigations (33, 34). This is be- 
 
 10 
 
lieved to account for a slightly higher value of u than before. Also, the data for zero 
 time in Figure 8 indicates that some inelastic deformation occurred before the test load 
 was reached, whereas in the previous tests on eccentrically loaded columns, the load 
 deformation curve remained linear for zero time. 
 3. Method of Loading Beam-Columns 
 
 The fixtures used in loading the beam-columns, the furnace and the 1/1000 inch 
 dial indicator are shown in Figure 9. The transverse load was applied by means of 
 dead weights attached to a wire, which passed over the pulley arrangement seen at the 
 right of the figure. The axial load was applied to the beam-columns through the knife 
 edge fixtures illustrated in Figure 10. The length of each knife edge was 2 inches. 
 This fixture added 0. 60 inches to each end of the beam-column. 
 
 In order to offset the effect of any initial crookedness in the beam column, the 
 eccentricity was measured with respect to its midpoint. The error in measuring the 
 eccentricity is believed to be less than plus or minus 0.002 inches. 
 
 UNiVERsiTy of 
 11 IUIN0/S LIBRARY 
 
IV. ANALYSIS OF RESULTS 
 
 A total of twenty-one beam-columns were subjected tq constant axial and lateral 
 loading at 972°F. Four different slenderness ratios, and two types of cross-section 
 were considered. 
 
 Eighteen of the beam-columns had rectangular cross sections. These were split 
 into three groups of six each with slenderness ratios of 50.0, 75.0, and 100.0. Half 
 of each group were tested with the axial load applied through the centroid of the section, 
 zero initial eccentricity, while the other half were subjected to an axial load with 15 
 per cent initial eccentricity. The deflection-time curves for these beam-columns are 
 shown in Figures 11 and 12. 
 
 The deflection-time curves for the three T- section beam-columns are shown in 
 Figure 13. Each T-section beam-column had a slenderness ratio of 60. 2. As noted 
 in Figure 15, two of these had zero initial eccentricity, and the other had an initial 
 eccentricity of 15 per cent of its depth. 
 
 The deflection of each beam-column is shown in dimensionless form in Figures 
 14 throughl7. The initial deflection, the maximum deflection, and one half maximum 
 deflection are given. The theoretical load- deflection curves for zero time shown in 
 these figures are based on the assumption that the material behaves elastically with 
 E = a /e (37). The actual test data falls to the right of these curves since the ma- 
 terial did not behave elastically while coming up to test load. 
 
 The theoretical load -deflection curves for 30 minutes, shown in Figures 
 14 through 17 were constructed using the arc hyperbolic sine theory presented in Sec- 
 tion II. Since the creep curves in Figures 11, 12 and 13 indicate that the deflection of 
 the beam-column immediately prior to collapse was independent of the time to collapse, 
 the experimental deflection may be compared with the theory for 30 minutes. It will 
 be noted that the deflections of the beam-columns corresponding to one half the max- 
 imum deflections are approximately equal to the deflections at the peaks of the theoreti- 
 cal load -deflection curves. The theoretical deflection at the collapse load is not high- 
 ly accurate while for loads appreciably below the collapse load the theoretical creep 
 deflections are small and quite close to the experimental values. This discrepancy 
 between the predicted and experimental values of the deflection at the collapse load is 
 not considered significant, since it is the buckling load which is of primary importance. 
 
 Experimental values of P/A are listed in Table 1 for all of the beam-columns 
 along with the time to collapse. Also listed in Table 1 are computed values of P/A 
 necessary to cause each beam-column to collapse in 30 minutes. The per cent differ- 
 ence between the theoretical collapse load at 30 minutes and the theoretical collapse 
 
 12 
 
load at other values of time was plotted as a function of time. The appropriate per 
 cent difference was applied to the test load to obtain the adjusted load. 
 
 Ratios of the theoretical to experimental collapse loads for 30 minutes are listed 
 in Table 1 for each beam-column. The theory ranged from 6 per cent conservative to 
 12 per cent nonconservative, with an average of 2. 6 per cent nonconservative. None of 
 the variables considered indicated any particular trend. The scatter was no more pro- 
 nounced for the beam-columns than for the creep specimens. (See Figure 7). 
 
 13 
 
V. SUMMARY AND CONCLUSIONS 
 
 1. Summary 
 
 A method is presented for predicting the load- deflection relationship and the 
 collapse load for beam-columns subject to creep. This method makes use of: 
 
 a) a stress -strain-time relation which is obtained by fitting the arc hyperbolic 
 sine equation to the isochronous stress-strain data. 
 
 b) a set of dimensionless moment -curvature interaction curves. The IBM 
 650 digital computer was used to facilitate the construction of these curves. 
 
 c) a method of successive approximations (32) for obtaining the load-deflec- 
 tion relationship. 
 
 To check the accuracy of the method developed, several tests were made on 
 beam-columns at elevated temperature. In all, twenty-one 17-7PH stainless steel 
 beam -columns were tested to collapse, at a constant temperature of 972°F. T-section 
 beam-columns having a slenderness ratio of 60.2 and rectangular- section beam-columns 
 having slenderness ratios of 50.0, 75.0, or 100.0 were used. Each beam-column was 
 subjected to a concentrated load at midspan and a constant axial load having an initial 
 eccentricity of zero or 15 per cent of its depth. 
 
 2. Conclusions 
 
 1. The arc hyperbolic sine function (Equation 16) adequately represents the 
 isochronous stress-strain diagram for 17-7PH stainless steel at 972°F for a time dur- 
 ation of thirty minutes. 
 
 2. Since the effect of stress redistribution with time is not directly considered 
 in the theory, it is recommended that the experimental constant o be increased by 
 ten per cent when calculating the collapse load or the load -deflection relationship. This 
 empirical correction is in agreement with the results of previous investigations on 
 eccentrically loaded columns (34, 36). 
 
 3. The IBM 650 digital computer can be conveniently used to calculate points 
 for dimensionless moment -curvature interaction curves. A program is included in the 
 appendix, which is valid for a general I-section. This includes the T-section and rec- 
 tangular-section as special cases. Approximately five to fifteen minutes was required 
 for the machine to make the computations for the particular cross- sections considered 
 in this investigation. 
 
 4. Since the dimensionless load-deflection relationship is independent of the 
 
 numerical values of a and e , these theoretical dimensionless load -deflection curves 
 
 o o 
 
 are independent of both time and material. 
 
 14 
 
5. Test data from the twenty-one beam- columns indicated that the theory ranged 
 from 6 per cent conservative to 12 per cent nonconservative in predicting the collapse 
 load for thirty minutes. The theory averaged 2. 6 per cent nonconservative. No trend 
 was noted for any of the variables investigated. The scatter was of approximately the 
 same magnitude as that exhibited by the compression creep data. 
 
 15 
 
BIBLIOGRAPHY 
 
 1. Bleich, H. H. & Dillon, D. W. Jr. , "Deformation of Columns of Rectangular 
 Cross Section", Journal of Applied Mechanics, December 1959, pages 517-525. 
 
 2. Davis, E. A. , "Creep of Metals at High Temperatures in Bending", Journal of 
 Applied Mechanics 5, A- 29. (1938). 
 
 3. Findley, W. N. and Poczatek, J. J. , "Prediction of Creep Deflection and Stress 
 Distribution in Beams from Creep in Tension", Journal of Applied Mechanics, 
 Vol. 77,. 1955, pages 165-171. 
 
 4. Finnie, I. and Heller, W. R. , "Creep of Engineering Materials", McGraw-Hill 
 Book Company, 1959. 
 
 5. Freudenthal, A. M. , "The Inelastic Behavior of Engineering Materials and Struc- 
 ture", John Wiley & Sons, 1950, First Edition, page 516. 
 
 6. Gerard, G. , "A Creep Buckling Hypothesis", Journal Aeronautical Science, Vol. 
 23, September 1956, page 879. 
 
 7. Hilton, H. H. , "Creep Collapse of Columns With Initial Curvature", Journal 
 Aeronautical Science, Vol. 19, December 1952, pages 844-846. 
 
 8. Hoff, N. J. , "A Survey of the Theories of Creep Buckling", Proceedings of the 
 Third U. S. National Congress of Applied Mechanics, pages 29-49. 
 
 9. Hoff, N. J., "Rapid Creep in Structures", Journal Aeronautical Science, October 
 1955. 
 
 10. Hu, L. W. and Triner, N. H. , "Bending Creep and Its Applications to Beam 
 Columns", Journal Applied Mechanics, Transactions A. S. M. E. , Vol. 78, 
 pages 35-42. 
 
 11. Kempner, J., "Creep Bending and Buckling of Nonlinearly Viscoelastic Columns", 
 N.A.C.A, Technical Note 3137, January 1954. 
 
 12. Kempner and Patel, "Creep Buckling of Columns", N.A.C.A. Technical Note 
 3138, 1954. 
 
 13. Kempner, J. , "Creep Bending and Buckling of Linearly Viscoelastic Columns", 
 N.A.C.A. Technical Note 3136, 1954. 
 
 14. Libove, C. , "Creep Buckling of Columns", Journal Aeronautical Science, July 
 1952, Vol. 19, No. 7. 
 
 15. Libove, C. , "Creep Buckling Analysis of Rectangular Section Columns", 
 N.A.C.A. Technical Note 2956, June 1953. 
 
 16. Lin, T. H. , "Creep Deflections and Stresses of Beam Columns", Journal Applied 
 Mechanics, March 1958, pages 75-77. 
 
 16 
 
17. MacCullough, G, H. , "An Experimental and Analytical Investigation of Creep in 
 Bending", Transactions A. S„ M. E, , Vol. 55, APM-55-9, 1933. 
 
 18. Marin, Joseph, "Creep Deflections in Columns", Journal of Applied Physics, 
 Vol. 18, No. 1, January 1947, pages 103-109. 
 
 19. Marin and Hu, "Deflection of Members Subjected to Bending Accompanied by 
 Creep", Proceedings of First National Congress of Applied Mechanics, 1952, 
 pages 613-618. 
 
 20. Pao, Yoh-Han and Marin, J. , "Deflections and Stresses in Beams Subject to 
 Bending and Creep", Journal Applied Mechanics, Vol. 74, 1952, pages 478-484. 
 
 21. Sanders, J. L. Jr. , "A Variational Theorem for Creep with Applications to 
 Plates and Columns", N.A.C.A. Technical Note 4003, 1957. 
 
 22. Shanley, F. R. , "Weight -Strength Analysis of Aircraft Structures", Dover Book 
 1960, pages 369-384. 
 
 23. Tapsell, H. J. and Johnson, A. E. , "An Investigation of the Nature of Creep 
 Under Stress Produced by Pure Flexure", Institute of Metals Journal, Vol. 57, 
 August 1935, page 121. 
 
 24. Johnson, A. E. , Mathur, V. D. and Henderson, J. , "The Creep Deflexion of 
 Magnesium Alloy Struts", Aircraft Eng. , 28: 419, 1956. 
 
 25. Odqvist, F. K. G. , "Influence of Primary Creep on Column Buckling", Journal 
 Applied Mechanics, 22: 295, 1954. 
 
 26. Hult, J. A. H. , "Critical Time in Creep Buckling", Journal Applied Mechanics, 
 22: 432, 1955. 
 
 27. Hoff, N. J., "Creep Buckling", Aeronautical Quarterly, 7:1, 1956. 
 
 28. Lin, T. H. , "Creep Stresses and Deflections in Columns", Journal of Applied 
 Mechanics, Transactions A.S.M. E. , Vol. 78, 1956, pages 214-218. 
 
 29. Shanley, F. R. , "Weight-Strength Analysis of Aircraft Structures", Dover Book 
 1960, pages 263-305. 
 
 30. Carlson, R. L.' and Manning, G. K. , "A Summary of Compressive-Creep Char- 
 acteristics of Metal Columns at Elevated Temperatures", WADC Technical 
 Report 57-96, 1958. 
 
 31. Kauzmann, W. , "Flow of Solid Metals from the Standpoint of the Chemical Rate 
 Theory", Transactions American Institute of Mining and Metallurgaical Engin- 
 eering, Institute of Metals Division, Vol. 143, 1941, page 57. 
 
 32. Newmark, N. M. "Numerical Procedure for Computing Deflection, Moments 
 and Buckling Loads", Proceedings of A.S.C.E., Vol. 68, 1942, pages 691-718. 
 
 33. Sidebottom, O. M. , Costello, G. A. , Dharmarajan, S. , "Theoretical and Ex- 
 perimental Analyses of Members Made of Materials that Creep", University of 
 Illinois Experiment Station Bulletin No. 460. 
 
 17 
 
34. Costello, G. A., Sidebottom, O. M. andPocs, E. , "Inelastic Design of Load 
 Carrying Members", Theoretical and Applied Mechanics Report 178, "Effect 
 of End Conditions on the Collapse of Load Carrying Members". 
 
 35. Dharmarajan, S. and Sidebottom, O. M. , "Inelastic Design of Load Carrying 
 Members", Theoretical and Applied Mechanics Report 174, "Theoretical and 
 Experimental Analyses of Circular Cross-Section Torsion-Tension Members 
 Made of Materials that Creep". 
 
 36. Gubser, J. L. , "Theoretical and Experimental Analysis of Eccentrically Loaded 
 17-7PH Stainless Steel Columns at 1000°F", Unpublished Master's Thesis, 
 University of Illinois, 1959. 
 
 37. Timoshenko, S. P. and Gere, J. M. , "Theory of Elastic Stability", Second 
 Edition, McGraw Hill Book Co. , 1961. 
 
 38. Sidebottom, O. M. , Clark, M. E. and Dharmarajan, S. , "The Effects of In- 
 elastic Action on the Resistance to Various Types of Loads of Ductile Members 
 Made from Various Classes of Metals -- Part VIII Eccentrically- Loaded Tension 
 Members Made of Two Stainless Steels Tested at Elevated Temperatures", 
 WADC Technical Report 56-330, April 1958. 
 
 39. Schweiker, J. W. , "Creep of Thick-Walled Cylinders Under Internal Pressure 
 and Axial Load", Unpublished Ph. D. Thesis, University of Illinois, 1961. 
 
 18 
 
APPENDIX A 
 
 The purpose of this appendix is to present a program for the IBM 650 digital 
 computer for computing points on constant load moment -curvature interaction curves. 
 Relations expressing dimensionless load, dimensionless moment, and dimensionless 
 curvature as functions of the cross section (see Figure 1) and of the variables K and 
 q are given by equations 25, 26 and 27. Before starting the program it was necessary 
 to decide on limits for K and q and the dimensionless load P/Aa . Dimensionless 
 moment- curvature interaction curves were constructed for dimensionless load incre- 
 ments of 0. 1 as the dimensionless load varied from 0. 2 to 2. 0. At each load, a mo- 
 ment and curvature were computed for each value of K which was applicable; the 
 values considered were 1, 1.5, 2, 2.5, 3, 4, 5, 7, 9, 12, and 15. The magnitude of 
 q was limited by considering only those values which would make 1/q greater than 
 0.60. 
 
 Figure 18 shows the flow diagram for the IBM 650 digital computer program. It 
 
 will be noted that there are three different dimensionless loads listed. P, /Aa is the 
 
 b' o 
 
 dimensionless load for which values of dimensionless moment and curvature are de- 
 sired: P /Aa is the dimensionless load calculated for assumed values of K and 
 a o 
 
 1/q; P /Aa is the dimensionless load calculated for previously assumed values of 
 K and 1/q. 
 
 The procedure followed in the computations can best be illustrated by consider- 
 ing Figure 19 which is a plot of P/Aa versus 1/q for a rectangular-section. Con- 
 sider the problem of calculating the first point on the moment -curvature interaction 
 curve for Pj 3 /Aa=0.7. The load P,/Aa is computed for initial values of K and 
 1/q of 1 and 0. 6 respectively. Since this load is below the desired load, the next 
 
 value of K is considered. This determines P„/Ao" . This load is above the desired 
 
 2 o 
 
 load so that increments of 0. 12 are added to 1/q to calculate P /Aa , P J Aa , and 
 
 /H 3' o 4' o 
 
 P c /Aa which is below the desired load. This value of load is placed in the location 
 
 5 o r 
 
 of P /Aa and an increment of 0. 04 is subtracted from 1/q to compute P,/Aer . 
 Since this load is above the desired load, a linear interpolation between P,- & P, is 
 used to calculate the desired 1/q. With K and 1/q known, the dimensionless mo- 
 ment and curvature are calculated. 
 
 Table 2 lists the entire program except for the square root (FLSQR) and log 
 (FLLNX) subroutines. To describe important steps in the program, notes have been 
 placed on the printed program. These notes correspond to the following commentary: 
 
 NOTE 1. Three cards are read into the immediate access storage to locate constants 
 
 19 
 
which are valid for all the cross sections. 
 
 Card 1 
 
 Card 2 
 
 Card 3 
 
 Location 
 
 Constant 
 
 Location 
 
 Constant 
 
 Location 
 
 Constant 
 
 9000 
 
 K = 1 
 
 9008 
 
 K = 
 
 9 
 
 9010 
 
 K= 15 
 
 9001 
 
 K- 1.5 
 
 9009 
 
 K = 
 
 12 
 
 9011 
 
 l/q = 0.6 
 
 9002 
 
 K= 2 
 
 
 
 
 9012 
 
 Al/q= 0.12 
 
 9003 
 
 K = 2. 5 
 
 
 
 
 9013 
 
 Al/q = 0.04 
 
 9004 
 
 K= 3 
 
 
 
 
 9014 
 
 P/Aa =0.2 
 o 
 
 9005 
 
 K = 4 
 
 
 
 
 9015 
 
 AP/Aa =0.1 
 
 9006 
 
 K = 5 
 
 
 
 
 
 
 9007 
 
 K = 7 
 
 
 
 
 
 
 NOTE 2. One card is read into the immediate access storage to locate constants 
 which are peculiar to a given cross section. 
 
 Card 4 
 
 Location 
 
 Constant 
 
 9020 
 
 -u 
 
 9021 
 
 -v 
 
 9022 
 
 b l 
 
 9023 
 
 b 2 
 
 9024 
 
 b 3 
 
 9025 
 
 b 4 
 
 9026 
 
 D 
 
 9027 
 
 P 
 
 NOTE 3. Initial values for load, 1/q and index registers A, B and C are set. 
 Index register A is used to keep track of the four values of EL, and C„ in Equations 
 25 and 26. Index register B specifies the particular value of K being used in the 
 calculations. Index register C is set to zero after each interpolation to indicate final 
 calculations. 
 
 NOTE 4. This is a subroutine to solve Equations 21 and 22 for the four values of N 
 in Equations 25 and 26. 
 
 20 
 
NOTE 5. Absolute values of N (except for N = K) are calculated and sent to the 
 subroutine to solve Equations 21 and 22. (See NOTE 4.). 
 
 NOTE 6. Equation 25 is solved. Calculations were made for the rectangular section 
 at the first location in the program and for the I- section at the second location. 
 
 NOTE 7. Index register C is checked for zero. If zero, goto NOTE 11. If not 
 
 zero, the difference between P„/Acr and P k /Act is computed. (Go to NOTE 8 .) . 
 
 a o do 
 
 NOTE 8. The upper accumulator is checked for negative. If P /Act <P, /Act , 
 
 the difference between P /Act and P,/Act is computed. (Go to NOTE 10. ). 
 
 co bo 
 
 NOTE 9. The upper accumulator is checked for zero. If zero, index register B 
 
 is increased by one to give new value of K and Equations 21 and 22 are solved for 
 
 four new values of N. If non-zero, 1/q is decreased by 0. 04, P /Act is set equal 
 
 to P /Act , index register A is set to -2 so as to solve Equations 21 and 22 for 
 
 three new values of N. 
 
 NOTE 10. The upper accumulator is checked for negative. If positive, 1/q is in- 
 creased by 0. 12, index register A is set to -2 so as to solve Equations 21 and 22 
 for three new values of N. If negative, interpolate to obtain correct value of 1/q, 
 increase index register C by 1, set index register A to -2 so as to solve Equations 
 21 and 22 for three new values of N. 
 
 NOTE 11. The dimensionless load, P /Act , is stored in a print location. The di- 
 
 3. O 
 
 mensionless curvature, hip/e , is computed and stored in a print location. The two 
 values of N (see Equations 21 and 22) may be negative. Each of these is checked for 
 sign and C , N| is made negative. The dimensionless moment, M/cr Ah, is com- 
 puted and stored in a print location. 
 
 NOTE 12. The computer is directed to print the results for one point on one inter- 
 action curve. Then index register B is increased by one. 
 
 NOTE 13. Index register B is checked for zero. If not zero, 1/q is increased by 
 0. 12 and the computations are repeated. If zero, the difference between Pj/Act 
 and 2.0 is computed. 
 
 NOTE 14. The upper accumulator is checked for negative. If negative, P,/Acr is 
 
 increased by 0. 1 and the computations are repeated. If not negative, the computer is 
 
 ordered to stop. If the stop is ignored, a new No. 4 card is read and computations 
 
 are repeated for a new cross -section. Computations will continue until all of the 
 
 No. 4 cards are used up. 
 
 21 
 
Table I 
 
 Comparison of Theoretical and Experimental Collapse Loads 
 for Beam -Columns at 972°F 
 
 Column 
 Number 
 
 Column 
 Depth 
 
 in. 
 
 I 
 
 r 
 
 e/h 
 
 % 
 
 Time 
 min. 
 
 a Ah 
 tion Be; 
 
 Experimental 
 Actual Adjusted 
 to 30 min 
 psi 
 
 P exp. /P theo. 
 
 
 
 Rectangul 
 
 ar-Sec 
 
 im-Columns 
 
 
 1 
 
 0.500 
 
 50.0 
 
 
 
 51 
 
 50 
 
 33,150 
 
 35,640 
 
 1.06 
 
 2 
 
 0.500 
 
 50.0 
 
 
 
 39 
 
 50 
 
 34,070 
 
 35,130 
 
 1.05 
 
 3 
 
 0.500 
 
 50.0 
 
 
 
 26 
 
 50 
 
 34,950 
 
 34,460 
 
 1.03 
 
 4 
 
 0.500 
 
 50.0 
 
 15 
 
 38 
 
 50 
 
 22, 120 
 
 22,770 
 
 0.97 
 
 5 
 
 0.500 
 
 50.0 
 
 15 
 
 23 
 
 50 
 
 23,450 
 
 22,850 
 
 0.97 
 
 6 
 
 0.500 
 
 50.0 
 
 15 
 
 16 
 
 50 
 
 23,450 
 
 21,650 
 
 0.92 
 
 7 
 
 0.420 
 
 75.0 
 
 
 
 34 
 
 50 
 
 19,880 
 
 20,120 
 
 1.01 
 
 8 
 
 0.420 
 
 75.0 
 
 
 
 34 
 
 50 
 
 19,880 
 
 20,120 
 
 1.01 
 
 9 
 
 0.420 
 
 75.0 
 
 
 
 14 
 
 50 
 
 20,950 
 
 19,000 
 
 0.95 
 
 10 
 
 0.420 
 
 75.0 
 
 15 
 
 30 
 
 50 
 
 14,400 
 
 14,400 
 
 0.98 
 
 11 
 
 0.420 
 
 75.0 
 
 15 
 
 50 
 
 50 
 
 14, 400 
 
 15,420 
 
 1.05 
 
 12 
 
 0.420 
 
 75.0 
 
 15 
 
 13 
 
 50 
 
 15,240 
 
 13,680 
 
 0.93 
 
 13 
 
 0.420 
 
 100.0 
 
 
 
 16 
 
 50 
 
 12,380 
 
 11,420 
 
 0.93 
 
 14 
 
 0.420 
 
 100.0 
 
 
 
 9 
 
 50 
 
 13,020 
 
 10,830 
 
 0.88 
 
 15 
 
 0.420 
 
 100.0 
 
 
 
 8 
 
 50 
 
 13,750 
 
 11,240 
 
 0.92 
 
 16 
 
 0.420 
 
 100.0 
 
 15 
 
 38 
 
 50 
 
 9,890 
 
 10,170 
 
 1.05 
 
 17 
 
 0.419 
 
 100.0 
 
 15 
 
 21 
 
 50 
 
 10,170 
 
 9,800 
 
 1.02 
 
 18 
 
 0.420 
 
 100.0 
 
 15 
 
 13 
 
 50 
 
 10,450 
 
 9,250 
 
 0.96 
 
 T- Section Beam -Columns 
 
 19 
 
 0.600 
 
 60.2 
 
 
 
 14 
 
 50 
 
 31,000 
 
 27,900 
 
 0.88 
 
 20 
 
 0.600 
 
 60.2 
 
 
 
 15 
 
 50 
 
 32,300 
 
 29,400 
 
 0.93 
 
 21 
 
 0.600 
 
 60.2 
 
 15 
 
 32 
 
 50 
 
 20,000 
 
 20,100 
 
 0.96 
 
 22 
 

 Table 2 
 
 
 D: 
 
 igital C 
 
 omputer Program 
 
 1 
 
 
 LOAD VS 
 
 CURVATURE 
 
 1 
 
 
 ARC HYPERBOLIC SINE 
 
 1 
 
 
 GENERAL 
 
 I SECTION 
 
 
 BLR 
 
 0000 
 
 0040 
 
 
 BLR 
 
 1900 
 
 1999 
 
 
 REG 
 
 L0041 
 
 0049 
 
 START 
 
 LDD 
 
 KEY1 
 
 
 
 STD 
 
 0022 
 
 Al 
 
 KEY1 
 
 00 
 
 0088 
 
 8888 
 
 Al 
 
 RD1 
 
 9000 
 
 NOTE 1 
 
 
 R01 
 
 9008 
 
 
 
 R01 
 
 9010 
 
 A2 
 
 A2 
 
 RD1 
 
 9020 
 
 NOTE 2 
 
 
 RSB 
 
 0011 
 
 NOTE 3 
 
 
 LOO 
 
 8006 
 
 
 
 STD 
 
 9033 
 
 
 
 RAU 
 
 9014 
 
 A9 
 
 A9 
 
 STU 
 
 9019 
 
 
 
 STU 
 
 9029 
 
 A3 
 
 A3 
 
 RAU 
 
 9011 
 
 A8 
 
 A8 
 
 STU 
 
 9018 
 
 
 
 RSC 
 
 0001 
 
 A10 
 
 A10 
 
 RSA 
 
 0003 
 
 
 
 RAU 
 
 9011 B 
 
 
 
 LDD 
 
 CI 
 
 Bl 
 
 Bl 
 
 STD 
 
 END 
 
 NOTE 4 
 
 
 STU 
 
 B2 
 
 
 
 FMP 
 
 8003 
 
 
 
 STU 
 
 B3 
 
 
 
 FAD 
 
 9000 
 
 
 
 STU 
 
 B4 
 
 
 
 FAD 
 
 B3 
 
 
 
 STU 
 
 B5 
 
 
 
 RAU 
 
 B4 
 
 
 
 LDD 
 
 B6 
 
 FLSQR 
 
 B6 
 
 STU 
 
 B7 
 
 
 
 FAD 
 
 B2 
 
 
 
 LDD 
 
 B8 
 
 FLLNX 
 
 B8 
 
 STU 
 
 89 
 
 
 
 FMP 
 
 B2 
 
 
 
 FSB 
 
 B7 
 
 
 
 STU 
 
 9055 A 
 
 
 
 RAU 
 
 B7 
 
 
 
 FMP 
 
 B2 
 
 
 
 STU 
 
 BIO 
 
 
 
 RAU 
 
 B9 
 
 
 
 FMP 
 
 B5 
 
 
 
 FSB 
 
 BIO 
 
 
 
 STU 
 
 9059 A 
 
 
 
 AXA 
 
 0001 
 
 END 
 
 CI 
 
 RAU 
 
 9000 
 
 NOTE 5 
 
 
 FSB 
 
 9018 
 
 
 23 
 
Table 2 (Continued) 
 
 
 BMI 
 
 
 C2 
 
 
 
 RSU 
 
 8003 
 
 C2 
 
 
 C2 
 
 FMP 
 
 9011 B 
 
 
 
 
 LOD 
 
 C3 
 
 Bl 
 
 
 C3 
 
 RAU 
 NZU 
 FMP 
 FAD 
 BMI 
 
 9020 
 
 9018 
 9000 
 
 C9 
 C5 
 
 
 
 RSU 
 
 8003 
 
 C5 
 
 
 C5 
 
 FMP 
 
 9011 B 
 
 
 
 
 LOD 
 
 C6 
 
 Bl 
 
 
 C9 
 
 AXA 
 
 0001 
 
 C6 
 
 
 C6 
 
 RAU 
 NZU 
 FMP 
 FAD 
 BMI 
 
 9021 
 
 9018 
 9000 
 
 Dl 
 C8 
 
 
 
 RSU 
 
 8003 
 
 C8 
 
 
 C8 
 
 FMP 
 
 9011 B 
 
 
 
 
 LDD 
 
 Dl 
 
 Bl 
 
 
 01 
 
 RAU 
 FMP 
 STU 
 RAU 
 NZU 
 RAU 
 
 9011 B 
 9018 
 9016 
 9023 
 El 
 9052 
 
 
 NOTE 6 
 
 
 FSB 
 
 9053 
 
 E6 
 
 
 E6 
 
 FDV 
 
 9016 
 
 
 
 
 NZC 
 
 
 D6 
 
 NOTE 7 
 
 
 STU 
 
 9028 
 
 
 
 
 FSB 
 
 9019 
 
 
 
 
 BMI 
 
 
 D3 
 
 NOTE 8 
 
 
 RAU 
 
 9018 
 
 
 
 
 FSB 
 
 9011 
 
 
 
 
 NZU 
 
 D4 
 
 
 NOTE 9 
 
 
 AXB 
 
 0001 
 
 
 
 
 LDD 
 
 8006 
 
 
 
 
 STD 
 
 9033 
 
 A10 
 
 
 D3 
 
 STU 
 RAU 
 FSB 
 
 9030 
 9029 
 9019 
 
 
 
 
 BMI 
 
 D5 
 
 
 NOTE 1C 
 
 
 RAU 
 
 9018 
 
 
 
 
 FAD 
 
 9012 
 
 
 
 
 STU 
 
 9018 
 
 
 
 
 RSA 
 
 0002 
 
 CI 
 
 
 OW 
 
 RAU 
 FSB 
 STU 
 LDD 
 STD 
 
 9018 
 9013 
 9016 
 9028 
 9029 
 
 
 
 
 RSA 
 
 0002 
 
 CI 
 
 
 24 
 
Table 2 (Continued) 
 
 05 
 
 RAU 
 
 9028 
 
 
 
 
 FSB 
 
 9029 
 
 
 
 
 STU 
 
 9031 
 
 
 
 
 RAU 
 
 9030 
 
 
 
 
 FOV 
 
 9031 
 
 
 
 
 FMP 
 
 9013 
 
 
 
 
 FAD 
 
 9018 
 
 
 
 
 STU 
 
 9018 
 
 
 
 
 AXC 
 
 0001 
 
 
 
 
 RSA 
 
 0002 
 
 CI 
 
 
 06 
 
 STU 
 
 0013 
 
 
 NOTE 11 
 
 
 LOO 
 
 9016 
 
 
 
 
 STO 
 
 0015 
 
 
 
 
 RAU 
 
 90*00 
 
 
 
 
 FDV 
 
 9018 
 
 
 
 
 STU 
 
 0014 
 
 
 
 
 FSB 
 
 9027 
 
 
 
 
 FMP 
 
 0013 
 
 
 
 
 STU 
 
 9017 
 
 
 
 
 RAU 
 
 9023 
 
 
 
 
 N2U 
 
 Fl 
 
 
 
 
 RAU 
 
 9000 
 
 
 
 
 FSB' 
 
 9018 
 
 
 
 
 BMI 
 
 
 D7 
 
 
 
 RSU 
 
 9057 
 
 
 
 
 STU 
 
 9057 
 
 D7 
 
 
 07 
 
 RAU 
 
 9056 
 
 
 
 
 FSB 
 
 9057 
 
 F6 
 
 
 F6 
 
 FDV 
 FDV 
 FDV 
 FSB 
 STU 
 
 9005 
 9016 
 9016 
 9017 
 0016 
 
 
 
 
 WR2 
 
 0013 
 
 
 NOTE 12 
 
 
 AXB 
 
 0001 
 
 
 
 
 N2B 
 
 
 D9 
 
 NOTE 13 
 
 
 LDD 
 
 9019 
 
 
 
 
 STD 
 
 9029 
 
 
 
 
 RAU 
 
 9018 
 
 
 
 
 FAD 
 
 9012 
 
 A8 
 
 
 09 
 
 RAU 
 FSB 
 
 9019 
 9002 
 
 
 
 
 N2U 
 
 
 D8 
 
 NOTE 14 
 
 
 RAB 
 
 9033 
 
 
 
 
 RAU 
 
 9019 
 
 
 
 
 FAD 
 
 9015 
 
 
 
 08 
 
 HLT 
 
 0000 
 
 A2 
 
 
 El 
 
 RAU 
 FMP 
 STU 
 RAU 
 FMP 
 STU 
 
 9055 
 9024 
 9055 
 9053 
 9025 
 9053 
 
 
 NOTE 6 
 
 25 
 
Table 2 (Continued) 
 
 E6 
 Fl RAU 9021 NOTE 11 
 
 F2 
 
 F3 
 
 F4 
 F5 
 
 RAU 
 
 9054 
 
 FMP 
 
 9023 
 
 STU 
 
 9054 
 
 RAU 
 
 9052 
 
 FMP 
 
 9022 
 
 FSB 
 
 9054 
 
 FAD 
 
 9055 
 
 FSB 
 
 9053 
 
 FDV 
 
 9026 
 
 RAU 
 
 9021 
 
 FMP 
 
 9018 
 
 FAO 
 
 9000 
 
 BMI 
 
 F2 
 
 RAU 
 
 9059 
 
 FMP 
 
 9024 
 
 STU 
 
 9059 
 
 RSU 
 
 9059 
 
 FMP 
 
 9024 
 
 STU 
 
 9059 
 
 RAU 
 
 9000 
 
 FSB 
 
 9018 
 
 BMI 
 
 F4 
 
 RAU 
 
 9057 
 
 FMP 
 
 9025 
 
 STU 
 
 9057 
 
 RSU 
 
 9057 
 
 FMP 
 
 9025 
 
 STU 
 
 9057 
 
 RAU 
 
 9058 
 
 FMP 
 
 9023 
 
 STU 
 
 9058 
 
 RAU 
 
 9056 
 
 FMP 
 
 9022 
 
 FSB 
 
 9058 
 
 FAO 
 
 9059 
 
 FSB 
 
 9057 
 
 FDV 
 
 9026 
 
 F3 
 F3 
 
 F5 
 
 F5 
 
 F6 
 
 26 
 

 i 
 
 , 
 
 
 
 
 
 
 
 
 
 cl 
 
 1 
 
 
 
 
 
 
 
 
 
 
 -^ 
 
 ■<— 
 
 
 P - 
 
 - A 
 
 
 *- 
 
 ^- 
 
 -*£ 
 
 
 
 
 qb 
 
 
 
 
 >- 
 
 qn 
 
 H 
 
 q£l 
 
 1}A 
 
 d 
 o 
 
 •M 
 
 3 
 
 -Q 
 
 ■rH 
 
 Sh 
 •M 
 CO 
 
 •rH 
 
 Q 
 
 •rH 
 
 3 
 u 
 
 C/3 
 
 T3 
 CJ 
 
 a 
 
 a 
 
 bD 
 
 C 
 
 BJ 
 
 Vh 
 J-i 
 
 < 
 s 
 
 ■rH 
 
 T3 
 cd 
 
 o 
 
 rt 
 o 
 
 ■rH 
 
 o 
 
 <D 
 C/3 
 
 CO 
 
 to 
 
 o 
 
 U 
 
 CO 
 
 C 
 o 
 
 CO 
 
 C 
 0) 
 
 1 
 
 + 
 
 CN 
 
 II 
 
 -C 
 X! 
 
 -Q 
 
 ^* 
 
 A 
 
 PI 
 
 ■rH 
 
 o 
 
 X! 
 W 
 
 1-4 
 
 § 
 
 •rH 
 
 0) 
 
 w 
 
 In 
 
 -rH 
 
 27 
 
1-4 
 
 s. 
 
 1-1 
 
CO 
 
 u 
 
 •v-C 
 

 > 
 
 -«- * — ■* 
 
 «— \ — ». 
 
 <- A — »- 
 
 A - 1/8 
 
 <— 
 e = 
 f 
 
 p 
 
 II ^ 
 
 " ~~~t^--_ 
 
 =r ^~^=rrr; 
 
 
 6 
 
 
 
 1 
 
 
 
 
 
 
 
 = . 15h 
 
 M Q /a o Ah 
 
 Assumed 6 
 
 A =5 + e 
 
 M p /a Q Ah 
 
 M/a Ah=(M„+M_)/a Ah 
 o P Q o 
 
 h ^ e o 
 
 Equiv. Concentrated ij) 
 
 Average Slopes 
 
 Calculated 6 
 
 Calculated 6 
 
 f 
 
 d 
 
 c 
 
 b 
 
 a 
 
 Common 
 Factors 
 
 
 
 ,0159 
 
 .0319 
 
 .0479 
 
 .0638 
 
 
 
 
 .050 
 
 .219 
 
 .113 
 
 .125 
 
 
 .150 
 
 .200 
 
 .219 
 
 .263 
 
 .275 
 
 
 .1050 
 
 .1400 
 
 .1673 
 
 .1841 
 
 .1925 
 
 
 .1050 
 
 .1559 
 
 .1992 
 
 .2320 
 
 .2563 
 
 
 1.49 
 
 2.22 
 
 2.93 
 
 3.52 
 
 3.96 
 
 
 
 26.4 
 
 35.0 
 
 42.1 
 
 22.9 
 
 6 A/12 
 
 
 
 126. 
 
 4 100.0 
 
 65.0 
 
 22.9 
 
 
 e A/ 12 
 
 
 
 
 
 126.4 
 
 226.4 
 
 291.4 
 
 314.3 
 
 e AY 12 
 o 
 
 
 
 .057 
 
 .103 
 
 .132 
 
 .142 
 
 
 Assumed 6 
 
 
 
 064 
 
 115 
 
 148 
 
 159 
 
 A = 6+ e 
 
 .150 
 
 .214 
 
 .265 
 
 .298 
 
 .309 
 
 
 M p /o- o Ah 
 
 .1050 
 
 .1497 
 
 .1855 
 
 .2085 
 
 .2165 
 
 
 M/a Ah 
 
 .1050 
 
 .1656 
 
 .2174 
 
 .2564 
 
 .2800 
 
 
 h$f€ 
 
 1.49 
 
 2.35 
 
 3.25 
 
 3.96 
 
 4.98 
 
 
 Equiv. Concentrated ip 
 
 
 28.2 
 
 38.8 
 
 47.3 
 
 25.9 
 
 e A/ 12 
 
 
 
 Average Slopes 
 
 140. 
 
 2 112 
 
 73.2 
 
 25.9 
 
 - 
 
 e A/ 12 
 o ' 
 
 Calculated 6 
 
 
 
 140.2 
 
 252.2 
 
 325.4 
 
 351.3 
 
 e A /12 
 
 
 
 Calculated 6 
 
 
 
 .064 
 
 .114 
 
 .142 
 
 .159 
 
 
 Figure 4. Sample Deflection Calculation for T-Section Beam -Column 
 
 30 
 
6/he 
 
 Figure 5 Dimensionless Load -Deflection Curve for T-Section Beam-Column 
 
 (£/r = 60. 2, Q = 2a I/ci, e/h = 0. 15) 
 
 31 
 
4) 
 O 
 
 0) 
 
 u 
 
 D 
 
 C 
 
 o 
 
 E 
 u 
 
 4) 
 Q. 
 
 uo 
 
 O- 
 0) 
 O 
 i_ 
 
 U 
 
 c 
 o 
 
 t/> 
 
 0) 
 
 o 
 U 
 
 c "g 
 
 « -£ 
 
 E u 
 
 u £ 
 
 Q. < 
 
 V5 _ 
 
 c 
 o 
 
 w 
 
 VI 
 
 01 
 
 a. 
 
 i x 
 
 c 
 
 01 
 
 
 a. 
 
 11 
 
 E 
 
 111 
 
 V 
 
 1- 
 
 1- 
 
 h. 
 
 -a 
 
 o 
 
 a> 
 
 *^- 
 
 
 
 o 
 
 01 
 
 > 
 
 3 
 
 01 
 
 X 
 
 LL! 
 
 
 
 UL 
 
 
 LL 
 
 32 
 
006 
 
 007 
 
 006 
 
 005 
 
 w 
 
 c .004 
 
 a 
 u 
 
 w 
 
 003 
 
 002 
 
 001 
 
 30 
 Time, min. 
 
 Figure 7. Compression Creep Curves for 17-7PH Stainless Steel at 972°F„ 
 
 33 
 
70,000 
 
 CO 
 
 CO 
 CO 
 0) 
 
 u 
 en 
 
 60,000 
 
 50,000 
 
 40,000 
 
 30,000 
 
 20,000 
 
 10,000 
 
 ;^ 
 
 cyb a 
 
 / 
 
 o 
 
 -o 
 
 A 
 
 A 
 A 
 
 Stress -Strain 
 
 Diagram 
 Representation 
 
 <j = Ee 
 
 a = a arc sinh — 
 
 o € 
 
 Test 
 Data 
 
 O 
 A 
 
 Experimental Constants 
 E = 22,000,000 psi 
 
 E = 22,000,000 psi 
 
 a = 25,100 psi 
 o 
 
 e -0.00114 
 o 
 
 006 
 
 008 
 
 .010 
 
 Figure 8. Compression Isochronous Stress -Strain Diagrams for 17 -PH 
 Stainless Steel at 97 2°F. 
 
 34 
 
Figure 9. Test Apparatus with Beam-Column in Furnace. 
 
 35 
 
a; 
 bo 
 ■a 
 
 W 
 
 0) 
 
 % 
 
 > 
 
 CD 
 O 
 
 o 
 U 
 
 6 
 
 <u 
 
 CQ 
 
 bO 
 
 C 
 
 ■rH 
 
 T3 
 
 (S3 
 
 O 
 J 
 
 Sh 
 O 
 
 <u 
 
 5-1 
 3 
 
 4-1 
 
 [-, 
 O 
 
 •t-i 
 Q 
 u 
 
 0) 
 
 o 
 
 0) 
 
 fc 
 
 36 
 
a 
 o 
 
 ■^^ 
 
 a 
 
 0.40 
 
 0.36 
 
 0.32 
 
 0.28 
 
 0.24 
 
 0.20 
 
 0.16 
 
 0.12 
 
 0.08 
 
 0.04 
 
 20 30 
 
 Time, min. 
 
 Figure 11. Defiection-Time Curves for Rectangular -Section Beam-Columns Having 
 Zero Initial Eccentricity, Made of 17-7PH Stainless Steel and Tested 
 at 972°F. 
 
 37 
 
0.40 
 
 0.36 
 
 0.32 
 
 0.28 
 
 0.24 
 
 0.20 
 
 0.16 
 
 0.12 
 
 0.08 
 
 0.04 
 
 Time, min. 
 
 Figure 12. Deflection-Time Curves for Rectangular-Section Beam-Columns Having 
 15 Per Cent Initial Eccentricity Made of 17-7PH Stainless Steel and Test- 
 edat972°F. 3g 
 
.040 
 
 036 
 
 032 
 
 028 
 
 024 
 
 020 
 
 c 
 o 
 
 jy 0.16 
 
 4-( 
 
 a 
 
 0.12 
 
 0.08 
 
 0.04 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 /~\ T1 
 
 51,000 psi, e 
 
 /h = 
 
 P P/A = 
 
 
 
 
 (. 
 
 1 
 
 A" ' 
 
 20,000 psi, 
 
 e/h = 0.15 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 Y P/A = 32 
 
 300 psi, e/h 
 
 = 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 10 
 
 20 
 
 30 40 
 
 Time, min. 
 
 50 
 
 60 
 
 Figure 13. Deflection-Time Curves for T-Section Beam-Columns Made of 17-7PH 
 Stainless Steel and Tested at 972°F (I/r = 60. 2). 
 
 39 
 
40 
 
 30 
 
 20 
 
 10 
 
 Theory for Zero Time 
 — Theory for 30 Minutes 
 Data for Zero Time 
 Data for Max. Deflection 
 Data for One Half Max. Deflection 
 
 0.2 
 
 0.4 
 A/h 
 
 0.6 
 
 0.8 
 
 Figure 14 Load-Deflection Curves for Rectangular -Section Beam-Columns 
 
 Made of 17-7PH Stainless Steel and Tested at 972°F. (£/r = 50.0), 
 
 40 
 
Figure 15. Load- Deflection Curves for Rectangular-Section Beam-Columns Made 
 of 17-7PH Stainless Steel and Tested at 972°F (j[/r = 75.0). 
 
 41 
 
Theory for Zero Time 
 
 Theory for 30 Minutes 
 
 O Data for Zero Time 
 
 • Data for Max. Deflection 
 
 A Data for One Half Max. Deflection 
 
 Figure 16. Load -Deflection Curves for Rectangular -Section Beam-Columns Made 
 of 17-7PH Stainless Steel and Tested at 972°F (£/r = 100.0). 
 
 42 
 
0.2 
 
 0.4 
 A/h 
 
 0.6 
 
 0.8 
 
 Figure 17. Load- Deflection Curves for T-Section Beam-Columns Made of 17-7PH 
 Stainless Steel and Tested at 972°F (£/r = 60. 2). 
 
 43 
 
START 
 
 HRead C 
 
 Cards 
 3 
 
 m 
 
 Read Card 
 4 
 
 Set Counter 
 B- -11 
 
 Store 
 Counter B 
 
 Store 
 P b/ Aff o 
 
 Compute and Store 
 Bj^ and 4 Cj^ 
 
 Set Counter 
 A- -3 
 
 Increase 
 Counter A by 1 
 
 Set Counter 
 C= -1 
 
 Store 1/q 
 
 Of 
 
 Set 
 P c /Aff Q -- P b / Aff o 
 
 U£- 
 
 Store 
 Counter B 
 
 Compute 
 |N| 
 
 Compute and Store 
 ! |N| 
 
 B| N i and 4 C i 
 
 3E 
 
 Set Counter 
 A--2 
 
 Increase 
 Counter B by 1 
 
 Compute 
 P a/ Aff o 
 
 Store 
 P a /Aa 
 
 Set 
 P c / Aff o- P a/ Aff o 
 
 l\ Yes 
 
 No 
 
 l/q= .6 ? 
 
 3 
 
 New 1/q = 
 Old 1/q-. 04 
 
 Yes 
 
 Yes A A No, 
 
 C-O? X-U P a / Aff o < P b/ Aff o ? 
 
 Compute 
 h^/ff Q 
 
 | No 
 
 Store 
 P a/^o- P b/ Aff o 
 
 Set 
 C -N---C N 
 
 P_/Aff < P b / Aff o? 
 
 No 
 
 Compute 
 M/a Q Ah 
 
 c 
 
 New 1/q = 
 Old l/q+-12 
 
 Store 1/q 
 
 Yes 
 
 Interpolate 
 
 to Calculate 
 
 Correct 1/q 
 
 Increase 
 Counter C by 1 
 
 Print 
 
 Increase 
 Counter B by 1 
 
 New 1/q = 
 Old l/q+-12 
 
 No A A Yes /" , . \ Yes 
 
 B= ? V— *4 P b /Aff o < 2 
 
 New P b /Aff Q = 
 Old P b /Aff +.1 
 
 No 
 
 Stop 
 
 Set Counter B 
 From Store 
 
 Figure 18. Flow Diagram for Digital Computer Program 
 
 44 
 
CN 
 
 * 
 
 
 
 
 CN 
 
 
 
 
 — H 
 
 
 
 
 o 
 
 
 
 
 f 
 
 A co 
 
 
 
 4 
 
 
 CN 
 
 
 
 
 ^H 
 
 
 
 
 ♦ 
 
 v^ < 
 
 * \ 
 
 
 v A 
 
 
 X '"" ' 
 
 
 
 
 X ° 
 
 *©• \ 
 
 
 \ f 
 
 
 L \f> \ 
 
 
 
 
 m \ 
 
 \ o 
 
 x o 
 
 
 • \ 
 
 \ 
 
 X t^ 
 
 
 — i \ 
 
 \ '— ' 
 
 X. II 
 
 
 II \ 
 
 \ " 
 
 x^ 
 
 
 14 \ 
 
 \w 
 
 
 
 
 
 
 
 
 
 
 o 
 
 CN 
 
 O l> 
 
 vO 
 
 
 o 
 
 
 
 4-> 
 
 
 CO 
 
 
 H 
 
 
 •rH 
 
 
 fc 
 
 00 
 
 0) 
 
 o" 
 
 ■a 
 
 
 s 
 
 
 o 
 
 
 4-J 
 
 
 u 
 
 
 0) 
 
 
 ■l-l 
 
 
 3 
 
 
 Oh 
 
 o 
 
 £ 
 
 ^H 
 
 o 
 
 
 u 
 
 
 >> 
 
 
 XI 
 
 
 T3 
 
 
 <U 
 
 
 £ 
 
 
 o 
 
 
 
 CN 
 
 I— 1 
 
 
 o 
 
 -H 
 
 b 
 
 
 (U 
 
 
 (H 
 
 cr 
 
 
 XJ 
 
 — 
 
 QJ 
 
 
 O 
 
 
 O 
 
 
 H 
 
 ■* 
 
 D-, 
 
 « 
 
 faO 
 
 
 C 
 
 
 •iH 
 
 
 £ 
 
 
 o 
 
 
 j= 
 
 
 w . 
 
 
 en P 
 
 
 o) r- 
 
 vO 
 
 So" 
 
 
 3 II 
 
 ^H 
 
 u o 
 
 
 ^ 
 
 
 ^£r 
 
 
 3* 
 
 00 
 
 03 
 
 n a 
 
 - 
 
 a> o 
 
 
 > u 
 
 
 cd 
 
 
 b^ 3 
 9 u 
 
 
 
 
 cu U 
 
 o 
 
 
 CN 
 
 ON 
 
 
 —1 
 
 
 <u 
 
 
 u 
 
 
 3, 
 
 
 •iH 
 
 
 b 
 
 flV/d 
 
 45