Columns With End Restraint And Bending In Load And .

Plastic Stability—NumericalAs pointed out earlier, real steel columns not only exhibit inelasticity due to the presence of residual stresses,but also they possess initial crookedness. The analysisof columns with residual stresses and initial crookednessis rather complicated. The eigenvalue approach, whichis valid only for perfectly straight columns, can not beused here. Instead, a different approach known as thestability approach must be utilized. In the stability approach, the load-deflection behavior of the column istraced from the start of loading to failure. The procedureis often carried out numerically using the computer because the differential equation governing the behavior ofinelastic-crooked columns are often intractable, so closedform solutions are very difficult, if not impossible, toobtain. Various methods to obtain numerical solutionsare presented in Refs. 10 and 11.In addition to inelasticity and initial crookedness, theend conditions of a column also play an important rolein affecting its behavior. The analyses of columns takinginto consideration inelasticity, initial crookedness and endrestraint were reported by a number of researchers in thepast few years. The results are summarized in Ref. 12.structures loaded into the inelastic range can also be performed for certain types of structures.The continued development in computer hardware andsoftware has made it possible for engineers and designers to predict structural behavior rather accurately. Theadvancement in structural analysis techniques coupledwith the increased understanding of structural behaviorhas made it possible for engineers to adopt the limit statedesign philosophy. A limit state is defined as a conditionat which a structural member or its component ceases toperform its intended function under normal condition(serviceability limit state) or failure under severe condition (ultimate limit state). Load and Resistance FactorDesign is based on the limit state philosophy and thus itrepresents a more rational approach to the design ofstructures.This paper attempts to summarize the state-of-the-artmethods in the analysis and design of columns as individual members and as members of a structure. A second objective is to introduce to engineers the stabilitydesign criteria of members and frames in LRFD. Highlights of recent research as well as directions of furtherresearch will be discussed.Structural Stability—EngineeringColumns in real structures seldom exist alone. The behavior of a column as an integral part of a structure isaffected by the behavior of other structural members. Inparticular, in addition to carrying axial force, the columnmust be able to resist bending moments induced by thebeam, so the column in reality behaves as a beam-column resisting both axial load and bending moments. Themoment transfer mechanism between beams and columns is different depending on whether the connectionis rigid or flexible. In other words, the behavior of theframe and its structural members is dependent on the rigidity of the connections. The stability analysis offrameworks with flexible connections has been a popularresearch topic in recent years. In particular, the recentlypublished Load and Resistance Factor Design (LRFD)Specification designates two types of construction in itsprovision: Type FR (fully restrained) and Type PR (partially restrained) constructions. Type PR construction requires explicit consideration of connection flexibility inproportioning structural members.The stability analysis of flexibly connected frames requires connection modeling. Since connection momentrotation behavior is usually nonlinear, the inclusion of aconnection as a structural element in a limit state analysis requires the use of nonlinear structural theory. Withthe advent of computer technology, great advancementhas been made in computer-aided analysis and design ofstructures. At the present time, first- and second-orderelastic analyses of structures can conveniently be performed for nearly all types of structures. Analysis ofTHIRD QUARTER / 19852. PIN-ENDED COLUMNA pin-ended column is the most fundamental case of acolumn. The behavior of a pin-ended column representsan anchorpoint for the study of all other columns. Forcolumns with long slendemess ratio, the Euler formula(Eq. 1) will provide a good estimate of their behavior.For intermediate or short columns, the Euler formula hasto be modified according to the reduced modulus concept or the tangent modulus concept (Eqs. 2 and 3) toaccount for yielding (or plastification) over the cross section due to the presence of residual stresses. As mentioned earlier, the tangent modulus theory gives a betterprediction of inelastic column behavior and hence it isadopted for design purposes.CRC CurveBased on the study of idealized columns with linear andparabolic residual stress distribution, as well as the testresults of a number of small and medium-size, hot-rolled,wide-flange shapes of mild structural steel, the ColumnResearch Council recommended in the first edition of theGuide " a parabola of the formFy- ' - )(4)to represent column strength in the inelastic range. Thisparabola was chosen because it represented an approximate median between the tangent modulus strength ofa W column buckled in the strong and weak directions.The column strength in the elastic range, however, is109

represented by the Euler formula. The point of demarcation between inelastic and elastic behavior was chosento be F 0.5 Fy. The number 0.5 was chosen as aconservative measure of the maximum value of compressive residual stress present in hot-rolled wide-flangeshapes which is about 0.3 Fy. To obtain a smooth transition from the parabola to the Euler curve, the constantB in Eq. 4 was chosen to be Fy/4 ir E. The slendernessratio that corresponds to F . 0.5 Fy is designated asCr in whichC ITT E(5)Thus, for columns with slenderness ratios less than orequal to Q , the CRC curve assumes the shape of a parabola and for slenderness ratio exceeding C , the CRCcurve takes the shape of a hyperbola, i.e.F. 1(KL/rf—Fcr IT E\(K L/r?KLsCrFor comparison purposes, Eq. 6 is rewritten in its loadform in terms of the nondimensional quantities P/Py andX in which Py is the yield load given by Py AFy andkc is the slenderness parameter given by X (KL/r)VFTJTT t x;'X, \ / 2(7)X, V 2The CRC curve is plotted in Fig. 5 in its nondimensional form (Eq. 7).AISC/ASD CurveThe CRC curve divided by a variable factor of safety of53 [KL/r\1 (KL/r\ 38\C, /8\C,/53 / \, \1 3 8 VV j " 8 VV .(6)KLr c.1 - 0.25X?pin tiie inelastic range and a constant factor of safety of23/12 in the elastic range gives the AISC Allowable Stress1.0Elasticregime0.8J0.60.4AISCAllowable StressDesign Curve0.2h00.250.500.751.001.251.501.752.00KFig. 5.110Column design curvesENGINEERING JOURNAL/AMERICAN INSTITUTE OF STEEL CONSTRUCTION

Design (ASD) curve. The factors of safety are employedto account for geometrical imperfections and load eccentricities which are unavoidable in real columns. TheAISC/ASD curve is also plotted in Fig. 5. The ASDcolumn curve is used in conjunction with the ASD format given byF.S.J Qni(8)whereRn nominal resistance(for column design, RJF.S. is represented by the ASDcolumn curve)Qn service loadsAISC/PD CurveThe ASD curve multiplied by a factor of 1.7 forms theAISC Plastic Design (PD) curve (Fig. 5). In plastic design only the inelastic regime of the curve is utilized because of the slenderness requirement. The design formatfor plastic design of columns is thus- l QnTo provide a compromise between the CRC curve (developed based on the tangent modulus concept) and theSSRC curves (developed based on the stability concept),the 1985 AISC/LRFD Specification' adopted a curve ofthe formP J exp -0.419X?) X, 1.5K 1.50.877X:(10)to represent basic column strength.The LRFD curve is plotted on Fig. 5, together withthe other curves described above. Note the LRFD curvelies between the CRC curve and the SSRC curve 2.The LRFD format is(11)(9)where 7 is the load factor used in the present AISC/PDSpecification. The values for 7 are: 7 1.7 for live anddead loads only and 7 1.3 for live and dead loadsacting in conjunction with wind or earthquake loads.SSRC CurvesBefore proceeding any further, it should be stated thatboth the ASD curve and PD curve are originated fromthe CRC curve which was developed based on the bifurcation concept which postulates that the column isperfectly straight. Although residual stress is explicitlyaccounted for, the effect of geometrical imperfections isonly accounted for implicitly by applying a variable factor of safety to the basic curve. Analysis of columns whichexplicitly take into consideration the effects of both residual stresses and initial crookedness was reported. Thestability approach was used in the analysis and a set ofthree curves referred to as the SSRC multiple columncurves was developed. Detailed expressions for thesecurves are given in Ref. 16. Approximate formulas forthese curves based on physical reasoning which are useful for design are also reported. " ' For comparison purposes, the three SSRC curves areplotted with the CRC, ASD and PD curves in Fig. 5.These curves belly down in the intermediate slendernessrange (0.75 \ 1.25) due to the combined maximumdetrimental effects of both residual stresses and initialcrookedness on column strength in the numerical analysis. Tests of real columns have demonstrated the det-THIRD QUARTER / 1985AISC/LRFD Curve Rn S QnI.IRF.S.rimental effects of residual stresses and initial crookedness are not always synergistic, so the SSRC curves whichbelly down in the intermediate slenderness range will betoo conservative for most columns in building frames.whereRn Qn (j) 7 nominal resistancenominal load effectsresistance factorload factorNote the LRFD format has the features of both theASD and PD formats in that factors of safety are appliedto both the load and resistance terms to account for thevariabilities and uncertainties in predicting these values.Furthermore, these load and resistance factors (( ), 7) areevaluated based on first order probabilistic approach. Sincedifferent types of loads have different degrees of uncertainties, different load factors are used for different typesof loads (e.g. 1.6 for live load, 1.2 for dead load, etc.).Therefore, the LRFD format represents a more rationaldesign approach.The expressions for various column curves describedabove and the three state-of-the-art design formats (ASD,PD, LRFD) are summarized in Tables 1 and 2.3.COLUMNS WITH END RESTRAINTEigenvalue AnalysisIn addition to residual stresses and initial crookedness,the end conditions of a column have a significant influence on column behavior. For perfectly straight elasticcolumns with idealized end conditions (ideally pinned orfully rigid), an eigenvalue analysis can be carried out todetermine the critical load P r- The effective length fac-111

Table 1. Summary of Column CurvesColumn CurvesColumn EquationsCRC CurvePKPy 4K V2P 1X, V 2X;1AISCX, V21 / X,38 VV2/8 \V2Allowable Stress DesignCurveP12 1P,23 X;X, V2AISCPlastic Design1.7I1- 1.0CurveP.X, V 25 /K38 Vy2/8 VV2, exp (-0.419X:)AISCX, 1.5P,LRFD CurveP(1985 Version)0.877X, 1.5X;Table 3. Theoretical and Recommended K Valuesfor Idealized ColumnsTable 2. Summary of Design FormatsAllowable Stress Design(ASD)Pn0F.S.IXLPlastic Design(PD)P n l lLoad and ResistanceFactor Design(LRFD)Qn,Buckled shape of columnis shown by dashed lineK (12)r\\\\1\where P is the Euler load given by P TT EI/L inwhich L is the length of the column.The effective length factor multiplied by the true lengthL of the column gives the effective length of the columnwhich can be used for design. Table 3 gives the theoretical and recommended K values for columns with112//t11m n-mtor K for the column with the particular set of end conditions can be obtained by\JU»( R,, 1 7, Q„.(I\\'Hieoretica] K value0 50 7Recommended designvalue when ideal ccmdi1 tioDS are approximated0 650 80"T(c)(e)(cI)1rJL1\111mw1 \(f)(1 111111111111'V rr1Tfim1 01 02 02.01 21 02 102.0Rotation fixed and translation fixedRotation free and translation fixedEnd condition codeRotation fixed and translation freeTRotation free and translation freevarious types of idealized end conditions. Since fully rigidsupports are seldomly realized in real life, the recommended K values for cases with fixed support idealization are slightly higher than their theoretical values.ENGINEERING JOURNAL/AMERICAN INSTITUTE OF STEEL CONSTRUCTION

Numerical AnalysisIt should be remembered that eigenvalue analysis canonly be carried out for perfectly straight columns. Forcolumns with initial crookedness, the stability or loaddeflection approach must be used. In the load-deflectionapproach, the load-deflection behavior of the column istraced from the start of loading to collapse. The maximum load the column can carry is the peak point of theload-deflection curve. The analyses of non-sway columns with residual stresses, initial crookedness and smallend restraint using the load-deflection approach have beenreported by a number of researchers. The important results are summarized by the authors. Some of the important findings are:1. Comparing with pin-ended columns, the maximumload-carrying capacity of end-restrained columns increases as the degree of end restraint (as measuredby the rotational stiffness of the connections connecting beams and columns) increases.2. The increase in load-carrying capacity of end-restrained columns is more pronounced for slender columns when stability is the limit state than for shortcolumns when yielding is the limit state.3. The end-restraining effect on column strength is morenoticeable for columns bent about their weak axesthan for columns bent about their strong axes.4. While residual stresses and initial crookedness havea destabilizing effect on columns strength, end restraint will provide a stabilizing effect which counteracts the detrimental effects of residual stresses andinitial crookedness. However, the strengthening ef-fect of end restraint is highly dependent on the slenderness of the column.Practical Design of Initially Crooked Column withEnd RestraintsFor design purposes, it is convenient to use the effectivelength factor approach in which the actual column withend restraints is converted to an equivalent pin-endedcolumn by multiplying the actual unbraced length of thecolumn by the effective length factor K, so the pin-endedcolumn curves described in the preceding section can beutilized directly. The procedure to determine the effective length factor for initially crooked end-restrained columns with residual stresses is more involved than thatof perfectly straight elastic columns with idealized endconditions. Equation 12 is not applicable anymore forthe determination of the effective length factor K. Instead, a number of load-deflection curves, each corresponding to a specific slenderness ratio L/r (or slenderness parameter X), are generated numerically. The peakpoints of these load-deflection curves are then plottedwith the associated slenderness ratios (or slenderness parameters) to form a column curve (see Fig. 6). Each column curve is unique for a specific value of initial crookedness, a specific distribution of residual stress and aspecific end restraint characteristic. To get the effectivelength factor, the end-restrained column curve (Fig. 6b)is compared with the corresponding pin-ended columncurve and the K factor at any load level is given (Fig. 7),K (13)where X ,, X c i* depicted in the figure.Curve FittingThrough D a t a Pointsa.o100Deflection(a)( 8 )L o a d - D e f l e c t i o n CurveFig. 6.THIRD QUARTER / 1985Slenderness Ratio(b)(—)Column CurveDetermination of column-strength curve from load-deflectioncurves for an initially crooked end-restrained column113

1,00 I " —0.80H0.60Restrained End0.40H0.20H0.000.00T"T"0.250.50— I0.751.001.251.501.7512.002.25XFig. 7.Determination of ejfective length factor, KUpon investigations of 83 end-restrained columns, the values of K for each curve do not vary significantlyover the load levels. Thus, a relationship between the Kfactor and the magnitude of end restraint can be established. In particular, the expression/ - 1.0 - 0 . 0 1 7 d 0 . 6(14)where2EL(MXL,2ELanalysis assuming perfectly straight columns with endrestraints provided by linear elastic rotational springshaving spring stiffness Rkj at the ends. Such comparisonis shown in Fig. 9. The dotted line is a plot of K versusd whereas the solid lines are plots of K i versus d. Ascan be seen, K i gives a conservative estimate of columnstrength provided that X is relatively low and d is relatively high.(15)MOMENTMin whichIg moment of inertia of the girder connected tothe columnLg length of the girder(Mp) plastic moment capacity of the columnRki initial connection stiffness of the connectionjoining the beam to the column (Fig. 8)was proposed for non-sway columns with initial crookedness, residual stresses and small end restraints, takinginto account the effect of beam flexibility. Proceduresfor the design of such columns have been reported inRefs. 12, 21 and discussed in Ref. 22.At this point, it is interesting to compare the effectivelength factor K as described by Eq. 14 with the elasticeffective length factor K i determined by an eigenvalue114ROTATIONFig. 8.9 Determination of R,,iENGINEERING JOURNAL/AMERICAN INSTITUTE OF STEEL CONSTRUCTION

XLUOLJU.U.bJOCFig. 9.Comparison of K i and K4. COLUMNS IN FRAMESAs mentioned earlier, columns in real structures usuallyexist as part of a frame. A column in a frame is usuallysubjected to the combined action of bending momentsand axial thrust. As a result, part of the strength of themember is required to resist the bending moment andonly the remaining part of the strength is available toresist the axial force. Thus, most columns in frames mustbe treated as beam-columns.M.t I \ \ \ 15 ---J 7Columns in Braced Frames—B FactorA phenomenon associated with a beam-column is thesecondary effect. When a braced member is subjected toboth bending moments and axial force, the axial forceacts through the deflection caused by the primary moments (moments arised from transverse loads and endmoments acting on the member) to produce additionalmoment referred to as secondary or P-8 moment. Figure10 shows schematically these two types of moments. Themoment acting along the member is thus the algebraicsum of the primary and secondary moments. To obtainthe exact value of this moment, a second-order analysisof the member is necessary. However, in lieu of suchanalysis, a simplified approach to obtain the total moment can be used.Using the assumptions that1. The deflection is smallTHIRD QUARTER / 1985M,Mj f(M,,M2,Q,q,x)Mjj PyFig. 10P-8 effect115

2. The secondary moment Mu is in the form of a halfsine wave3. The maximum deflection 8 occurs at midspan4. The maximum primary moment M/ occurs at ornear midspanan approximate expression for the maximum moment canbe derived.Because of assumptions 1 to 3, we can relate the curvature caused by the secondary moment to the maximumdeflection asMuP87T;C(16)sin —EIEILIntegrating Eq. 16 twice and enforcing the boundaryconditions y{0) 0 and y(L) 0 it can easily be shownthat the secondary deflection (deflection caused by theP-8 effect) can be written asy'li Pbyii2L'EI U /sm(17)LEquation 22 shows the maximum moment in the member can be obtained by multiplying the maximum primary moment M/ by an amplification factor Bi (thefactor in parenthesis). Note this amplification factor mustbe greater than unity if it is of any importance. This isbecause if this factor is less than unity, then from Eq.22 it is clear that Mj a.x max and the designer will useMj, rather than M in proportioning the members. Thecondition that Bi must be greater than unity is adoptedin the present AISC/LRFD Specification which was notthe case for the AISC/ASD Specification.Figure 11 shows the value of \\f and C for severaldifferent load cases. It is important to point out that because of assumption 4, Eq. 24, which is derived fromEq. 21, is only applicable to the two simply supportedcases (Cases 1 and 4). For the other cases in which themaximum primary moment Mj x occurs at the end(s)(Cases 2, 3, 5) or occurs at midspan as well as at theends (Case 6), the exact values of the maximum moments are first evaluated; the values for ij; are then obtained from calibration. from which the secondary deflection at midspan is(18)8// yii kCaseSince the total deflection at midspan is the sum of theprimary and secondary deflections, i.e.8 8; 8;;(19)8/Cm01 .O'-1we can eliminate 8// by substituting Eq. 18 into Eq. 19.The result is8 0.3 -0.3P/Pg-0.2l-0.2P/Pg13P.From assumption 4, we can write fTTTrTTTnTiiiniiml(21)If we substitute Eq. 20 into Eq. 21 and rearrange, wecan writeM C1 - P/P.B M„M„(22)4f1t5whereCB -1- 1P/P,(23)-i— —\-C„ 1 it;P/P,in whichili -1161(o(24)Fig. 11.Values of if/for beam-columns undertransverse loadingsENGINEERING JOURNA

steel column, the nonlinearity of the average stress-strain behavior of the cross-section is due to the presence of residual stress. Residual stresses arise as a result of the manufacturing process. When a compressive axial force is applied to a stub column (very short column), the fi bers that have compressive residual stresses will yield first.