Practical Advanced Analysis for Semi Rigid Frame Design

Practical Advanced Analysis forSemi-rigid Frame DesignSEUNG-EOCK KIM and WAI-FAH CHEN

ABSTRACT

This paper presents three practical advanced analysisprocedures for a two-dimensional semi-rigid steel framedesign. Herein, the nonlinear behavior of beam-to-columnconnections is discussed, and practical modeling of theseconnections is introduced. The proposed methods can predictaccurately the combined nonlinear effects of connection,geometry, and material on the behavior and strength of semi-rigid frames. The strengths predicted by these methods arecompared well with those available experiments. Analysisand design procedures using the proposed methods aredescribed in detail, and a case study is also given. Theproposed procedures can be used for the LRFD designwithout tedious separate member capacity checks, includingthe calculations of K-factor. The procedures are suitable foradoption in practice.

INTRODUCTION

Conventional analysis and design of steel framed structuresare usually carried out under the assumption that the beam-to-column connections are either fully rigid or ideally pinned.However, most connections used in current practice are semi-rigid type whose behavior lies between these two extremecases. In the AISC-LRFD Specification,1 two types ofconstructions are designated: Type FR (fully restrained)construction; and Type PR (partially restrained) construction.The LRFD Specification permits the evaluation of theflexibility of connections by rational means when theflexibility of connections is accounted for in the analysis anddesign of frames.

The semi-rigid connections influence the momentdistribution in beams and columns as well as the drift and P-D effect of the frame. One way to account for all these effectsin semi-rigid frame design is through the use of a directsecond-order inelastic frame analysis known as "Advanced

Seung-Eock Kim is graduate research assistant, structures area ofschool of civil engineering, Purdue University, West Lafayette,IN.

Wai-Fah Chen is professor and head of structures area of school ofcivil engineering, Purdue University, West Lafayette,IN.

Analysis." Advanced analysis indicates a method that cansufficiently capture the limit state strength and stability of astructural system and its individual members so that separatemember capacity checks are not required. Since the power ofpersonal computers and engineering workstations is rapidlyincreasing, it is feasible to employ advanced analysistechniques directly in engineering design office. Herein, weshall develop a practical advanced analysis/design method forplanar semi-rigid frames without the use of K-factor.

Since the study is limited to two-dimensional steelframes, the spatial behavior of frames is not considered hereand lateral torsional buckling of members is assumed to beprevented by adequate lateral braces. This study covers bothbraced as well as unbraced semi-rigid frames. A compact W-section is assumed so that the section can develop full plasticmoment capacity without local buckling.

BEHAVIOR OF SEMI-RIGID FRAMES

The important attributes which affect the behavior of semi-rigid steel framed structures may be grouped into threecategories: connection, geometric, and materialnonlinearities. The connection nonlinearity indicates thenonlinear moment-rotation relationship of semi-rigidconnections. The geometric nonlinearity includes second-order effects associated with the P-d and P-D effects andgeometric imperfections. The material nonlinearity includesgradual yielding associated with the influence of residualstresses on flexure behavior.

Nonlinear Behavior of Connections

The forces transmitted through beam-column connectionsconsist of axial force, shearing force, bending moment, andtorsion. The effect of axial force and shearing force isnegligible since their deformations are small compared withthe rotational deformation of connections, and torsion is alsoneglected since the present study is limited to planar frames.The deformation behavior of a connection may becustomarily described by moment-rotation relationship, andits typical behavior is nonlinear. The schematic moment-rotation curves of commonly used semi-rigid connections areshown in Figure 1. It may be observed that a relativelyflexible connection has a smaller ultimate moment capacityand a larger rotation, and vice versa. Herein, Kishi-Chenpower model shall be adopted to describe the

ENGINEERING JOURNAL / FOURTH QUARTER / 1996 129

moment-rotation relationship of semi-rigid connections.2

If the direction of incremental moment applied to aconnection is reversed, the connection will unload with theinitial slope of the moment-rotation curve. This loading andunloading behaviors of connections can be adequatelyaccounted for by the use of tangent stiffness and initialstiffness, respectively.3 Herein, these stiffnesses shall beobtained by simply differentiating Kishi-Chen power modelequation.

Geometric NonlinearityThe bending moments in a beam-column consist of two types:primary bending moment; and secondary bending moment.Primary bending moments are caused by applied endmoments and/or transverse loads on members. Secondarybending moments are from axial compressive force actingthrough the lateral displacements of a member. Thesecondary bending moments include the P-d and P-Dmoments. Herein, stability functions are used for eachmember to capture these second-order effects in a directmanner.

Geometric imperfections result from unavoidabletolerance during fabrication or erection, and they may beclassified as out-of-straightness and out-of-plumbness. Theseimperfections cause additional moments in column membersthat result in further degradation of members' bendingstiffness. In this paper, geometric imperfections will beconsidered by either an explicit imperfection modeling,

Fig. 1. Schematic moment-rotation curves ofvarious semi-rigid connections.

equivalent notional loads, or a further reduction of members'tangent modulus.4.5

Material Nonlinearity

Residual stresses result in a gradual axial stiffnessdegradation. The fibers that have the highest compressiveresidual stress will yield first under compressive force,followed by the fibers with a lower value of compressiveresidual stress. Due to this spread of yielding or plasticity,the axial and bending stiffnesses of a column segment isdegraded gradually along the length of a member. Thisstiffness degradation effect will be accounted for later by thetangent modulus concept.6

When a wide flange section is subjected to pure bending,the moment-curvature relationship of a section has a smoothtransition from elastic to fully plastic. This is because thesection yields gradually from extreme fibers which havehigher stresses than interior fibers. The gradual yieldingeffect leads to the concept of a hardening plastic hinge whichmay be represented simply by a parabolic stiffness reductionfunction of a plastic hinge.6 This will be described later.

PRACTICAL CONNECTION MODELING

The connection behavior is represented by its moment-rotation relationship. Extensive experimental works onconnections have been performed, and a large body ofmoment-rotation data has been collected.7-10 Using theseabundant data base, researchers have developed severalconnection models including: linear; polynomial; B-spline;power; and exponential models. Herein, the three parameterpower model proposed by Kishi and Chen2 is adopted. TheKishi-Chen power model contains three parameters: initialconnection stiffness Rki, ultimate connection moment capacityMu, and shape parameter n. The power model may be writtenas (Figure 2):

Fig. 2. Moment-rotation behavior of three parameter model.

130 ENGINEERING JOURNAL / FOURTH QUARTER / 1996

m mn n= +>

qq

q( ) /1

01 for > 0, (1)

where

m = M/Muq = qr/qoqo = reference plastic rotation, Mu/RkiMu = ultimate moment capacity of the connectionRki = initial connection stiffnessn = shape parameter.

When the connection is loaded, the connection tangentstiffness Rkt at an arbitrary rotation qr can be derived bysimply differentiating Equation 1 as:

nno

u

rkt

MddMR /11)1( +q-q

=q

= (2)

When the connection is unloaded, the tangent stiffness isequal to the initial stiffness as:

kio

u

rkt R

MddMR =

q=

q= (3)

It is observed that a small value of the power index n makesa smooth transition curve from the initial stiffness Rkt to theultimate moment Mu. On the contrary, a large value of theindex n makes the transition more abruptly. In the extremecase, when n is infinity, the curve becomes a bilinear lineconsisting of the initial stiffness Rki and the ultimate momentcapacity Mu.

An important task for practical use of the power model isto determine the three parameters for a given connectionconfiguration. Herein, the practical procedures fordetermining the three parameters are presented for thefollowing four types of connections with angels:single/double web-angle connections; and top and seat anglewith/without double web angle connections.

The values of Rki and Mu can be determined by a simple

Fig. 3. Mechanism of web angle connection of ultimate condition.

mechanical procedure with an assumed failure mechanism.2

For single/double web angle connections shown in Figure 3,the initial connection stiffness and the ultimate momentcapacity are given by:

R Gt

kia=

-

3

3a ab

ab ab abcosh( )

( ) cosh( ) sinh ( )(4)

MV V

dupu a

a=+2

62 (5)

whereG = shear modulita = thickness of web anglea = 4.2962 when Poisson's ratio is 0.3b = g1/dada = height of web angleg1 = gage distance from the fixed support line to free

edge lineVpu = minimum value of VpyVo = maximum value of VpyVpy = plastic shear force per unit lengthVo = shear force capacity per unit length in the absence

of bending of web angle.

For the top and seat angle connections shown in Figure 4, theinitial connection stiffness and the ultimate moment capacityare given by:

R EIt g

dg

kit

=+

31 0 78 2 1

212

13( . / )

(6)

Mu = Mos + Mp + Vpd2 (7)

where

El = bending stiffness of angle's leg adjacent tocolumn face

(a) (b)

Fig. 4. Top and seat angle connection (a) deflectedconfiguration at elastic condition; and (b) mechanism

of top angle at ultimate condition.


Table 1.Empirical Equations for Shape Parameter n (Kishi and Chen 1991)

Connection n

Single web-angle connection 0.520log10qo + 2.291 for log10qo > 3.073

0.695 for log10qo < 3.073

Double web-angle connection 1.322log10qo + 3.952 for log10qo > 2.582

0.573 for log10qo < 2.582

Top and seat angle connection 2.003log10qo + 6.070 for log10qo > 2.880

0.302 for log10qo < 2.880

Top and seat angle connection withdouble web angle

1.398log10qo + 4.631 for log10qo > 2.721

0.827 for log10qo < 2.721

d1 = distance between centers of legs of top andbottom angles

tt = thickness of top angleg1 = g1 D/2 tt/2D = db for rivet fastenerD = W for bolt fastenergt = gage distance from top angle's heel to center of

fastener holes in leg adjacent to column faceMos = plastic moment capacity at point CMp = plastic moment capacity at point H2 of top angleVp = shear forced2 = d + ts/2 + k

For top and seat angle connections with double web anglesshown in Figure 5, the initial connection stiffness and theultimate moment capacity are given by:

(a) (b)

Fig. 5. Top and seat angle with web angle connection(a) deflected configuration of elastic condition; and

(b) applied forces in ultimate state of connection.

REI d

g g tEI d

g g tki

t

t

a

a

=+

++

3078

60 78

12

1 12 2

32

3 32 2( . ( . ))

(8)

Mu = Mos + Mpt + Vptd2 + 2Vpad4 (9)

where

EIt, EIa = bending stiffness of legs adjacent to columnface of top angle and web angle

g3 = gc W/2 ta/2W = diameter of nutta = thickness of top angleMpt = ultimate moment capacity of top angleVpt = shearing force acting on plastic hingesVpa = resulting plastic shear force

d4 =2)(3

)2( sla

oapu

oapu tldVVVV

++++

Vpu = shearing force at upper edge of web angleVoa = shearing force at lower edge of web anglel1 = distance from bottom edge of web angle to

compression flange of beam.

As for the shape parameter n, the equations developed byKishi and Chen et al.11 are implemented here. Using astatistical technique for n values, empirical equations of n aredetermined as a linear function of log10qo shown in Table 1.

PRACTICAL ADVANCED ANALYSIS

During the past 20 years, research efforts have been devotedto the development and validation of several advancedanalysis methods. The advanced analysis methods may beclassified into two categories: (1) Plastic-zone method; and(2) Plastic hinge method. Whereas the plastic-zone solutionis known as the "exact solution," but can not be used forpractical design purposes.12 This is because the method is toointensive in computation and costly due to its complexity. Inrecent works by Liew6 and White,13 among others, the refined


plastic-hinge method has been proposed for two-dimensionalframe analysis. The refined plastic-hinge method isdeveloped by a simple modifications of the elastic-plastichinge analysis to account for the gradual degradation inmember's axial and bending stiffness. The refined plastic-hinge method is equivalent to the plastic-zone analysis in itsaccuracy but is much simpler than the plastic-zone method.Thus, the refined plastic hinge concept is implemented hereas a practical method, and its key considerations arediscussed in what follows:

Geometric Second-Order Effects

As mentioned previously, stability functions are used tocapture the second-order effects since they can account forthe effect of the axial force on the bending stiffness reductionof a member. The benefit of using stability functions is that itenables only one element to predict accurately the second-order effect of each framed member.

Stiffness Degradation Associated with Residual Stresses

The tangent modulus concept proposed by Liew6 is employedhere to account for gradual yielding effects due to residualstresses along the length of members under axial loadsbetween two plastic hinges. Herein, the CRC tangentmodulus is selected as the better choice between the LRFD Etand the CRC Et although the CRC Et accounts only for theresidual stress effect but not geometric imperfection effect.4

This is because different members with different residualstresses can be incorporated easily in this model, since theeffects of geometric imperfections are treated separately fromthe model. When this model incorporates appropriategeometrical imperfections, it may provide a very goodcomparison with the plastic-zone solutions.

Stiffness Degradation Associated with Flexure

A gradual stiffness degradation of a plastic hinge is requiredto represent the distributed plasticity effects associated withbending actions. Herein, we shall introduce the hardeningplastic hinge model to represent the gradual transition fromelastic stiffness to zero stiffness associated with a fullydeveloped plastic hinge. When the hardening plastic hingesare present at both ends of an element, the incremental

Fig. 6. Beam-column element with semi-rigid connections.

force-displacement relationship may be expressed as:6

hh

h--h

=

0

)1(

2

1

22

1

SSS

S

LIE

PMM

BA

BA

tB

A

)10(

/0

0)1(

0

1

22

1

2

qq

h--h

hh

eIASSS

S

B

A

AB

BA

where

PMM BA ,, = incremental end moments and axialforce, respectively

S1, S2 = stability functionsEt = tangent modulusI = moment of inertia of cross sectionL = length of elementA = area of cross sectionh hA B, = scalar parameter for gradual inelastic

stiffness reduction

BA qq , = incremental rotations at element ends Aand B

e = incremental axial deformation.

Effect of Semi-Rigid Connection ElementThe connection may be modeled as a rotational spring in themoment-rotation relationship represented by Equation 11.Figure 6 shows a beam-column element with semi-rigidconnections at both ends. If the effect of connectionflexibility is incorporated into the member stiffness, theincremental element force-displacement relationship ofEquation 10 is modified as:6

qq

=

eIASSSS

LIE

PMM

B

A

jjij

ijiit

B

A

/0000

**

**

(11)

where

*

2

*

R

LRISE

LRSISE

S

SktB

ijt

ktB

jjiitij

ii

-+

= (12a)

*

2

*

R

LRISE

LRSISE

S

SktA

ijt

ktA

jjiitjj

jj

-+

= (12b)

S S Rij ij* */= (12c)


ktBktA

ijt

ktB

jjt

ktA

iit

RRS

LIE

LRISE

LRISE

R22

* 11

-

+

+= (12d)

where

RktA, RktB = tangent stiffness of connections A and BSii, Sij = generalized stability functions

** , jjii SS = modified stability functions that account for

the presence of end connections.

PRACTICAL GEOMETRIC IMPERFECTIONMODELING

Geometric imperfection modelings combining with therefined plastic-hinge analysis are discussed in what follows.These include an explicit imperfection modeling method, anotional load method, and a reduced tangent modulus method.

Explicit Imperfection Modeling MethodThe AISC Code of Standard Practice1 limits an erection out-of-plumbness equal to Lc/500 in any story. This imperfectionvalue is conservative in taller frames since the maximumpermitted erection tolerance of 50 mm (2 in.) is much lessthan the accumulated geometric imperfection calculated by1500 times building height. In this study, however, Lc/500 is

used for geometric imperfection without any modifications.This is because the system strength is often governed by aweak story which has the out-of-plumbness equal to Lc/500,14

and the imperfection value may be easily implemented inpractical design use.

The frame out-of-plumbness may be used for unbracedframes but not for braced frames. This is because the PDeffect caused by out-of-plumbness is diminished by braces inbraced frames. As a result, the member out-of-straightnessinstead of the out-of-plumbness should be used to account

Fig. 7. Further reduced tangent modulus curve.

for geometric imperfections for braced frames. The AISCCode recommends a maximum fabrication tolerance ofLc/1,000 for member out-of-straightness. In this study, thesame geometric imperfection of Lc/1,000 is adopted by thecalibration with the plastic-zone solutions. The out-of-straightness may be reasonably assumed to vary sinusoidallywith a maximum in-plane deflection of Lc/1,000 at the mid-height. Ideally, many elements are necessary in order tomodel the sinusoidal out-of-straightness of a beam-columnmember. In this study, we find that two elements with amaximum deflection at the mid-height of a member arepractically adequate to capture the imperfection effects.

Equivalent Notional Load MethodThe geometric imperfections of a frame may be replaced byequivalent notional lateral loads expressed as a fraction ofthe gravity loads acting.15 In this study, the proposedequivalent notional load for practical use is to be 0.002Pu,where Pu is the total gravity load in a story.5 The notionalload should be applied laterally at the top of each story. Forbraced frames, the notional loads should be applied at mid-height of a column since the ends of the column are braced.In this study, appropriate notional load factor equal to 0.004is adopted for braced frame. It may be observed this value isequivalent to the geometric imperfection of Lc/1,000.

Further Reduced Tangent Modulus MethodThe idea of using the reduced tangent modulus method is tofurther reduce the tangent modulus Et to account for thegeometric imperfections. The degradation of memberstiffness due to geometric imperfections may be replaced byan equivalent reduction of member stiffness. This may beachieved by a further reduction factor fi of tangent modulusas:4

yiyy

t PPEPP

PPE 5.0for14 >f

-= (13a)

yit PPEE 5.0for f= (13b)

where

tE = reduced Et

fi = reduction factor for geometric imperfection

Figure 7 shows the further reduced tangent modulus curvesfor the CRC Et with geometric imperfections. When thetangent modulus Et is replaced by an appropriate reducedtangent modulus of 0.85Et in CRC column curve, themodified CRC column curve is compared well with the LRFDcolumn curves shown in Figure 8. The same reduction factorof 0.85 may be used for braced frames as well as forunbraced frames. The advantage of this method over the othertwo methods is its convenience for design use, because itcompletely eliminates the inconvenience of inputs of theexplicit


Table 2.Key Considerations of LRFD and Proposed Methods

Key Considerations LRFD Proposed Methods

Connection nonlinearity No procedure Power model/Rotational springSecond-order effects Column curve Stability function

B1, B2 factorGeometric imperfection Column curve Explicit imperfection modeling method

y = 1 /500 for unbraced frame dc = Lc/1,000 for braced frameEquivalent notional load method a = 0.002 for unbraced frame a = 0.004 for braced frameFurther reduced tangent modulus method tE = 0.85Et

Stiffness degradation Column curve CRC tangent modulusassociated with residual stresses

Stiffness degradation Column curve Parabolic degradation functionassociated with flexure Interaction equation

imperfection modeling or the equivalent notional loadmodeling.

Table 2 summarizes the proposed methods and their keyconsiderations as compared with the LRFD procedures.

VERIFICATION STUDY

In the open literature, no available benchmark problems ofsemi-rigid frames with geometric imperfections are availablefor verification study. One way to verify the proposedmethods is to make separate verifications for the effects ofsemirigid connections and of geometric imperfections.

Fig. 8. Comparison of strength curves by further reduced CRCtangent modulus method and LRFD method for axially loaded

pin-ended column.

Effect of Semi-Rigid Connections

Stelmack16 studied the experimental response of two flexibly-connected steel frames. A two-story, one-bay frame amonghis studies is selected as a benchmark frame in the presentstudy. The benchrnark frame was fabricated from the sameA36 W516 sections, and all column bases are pinnedsupports in Figure 9. The connections used in the frame werebolted top and seat angle connections of L44 made ofA36 with bolt fasteners of A325-in. D, and its experimentalmoment-rotation relationship is shown in Figure 10. Gravityloading of 10.7 kN (2.4 kips) was first applied at third pointsof the beam of the first floor, and then a lateral load wasapplied as the second loading sequence. The lateral load-displacement relationship was provided by the experimental

Fig. 9. Configuration of two-story semi-rigidframe for verification study.


work.Herein, the three parameters of the power model are

determined by a curve fitting with the experimentalconnection curve as well as by the Kishi-Chen Equations 4-9and Table 1. The curves by the experiment and by the curve-fitting result in a good agreement as shown in Figure 10. Theparameters by Kishi-Chen equations and by the experimentshow a difference to some degree as shown in Table 3 andFigure 10. In spite of this difference, Kishi-Chen equationsare preferably in practical design since experimentalmoment-rotation curves are not available in design stages, ingeneral. In the analysis, the gravity load is first applied,followed by the lateral load. The lateral displacements by theproposed methods and by the experiment compare well inFigure 11. As a result, the proposed method is adequate inpredicting the behavior and strength of semi-rigidconnections.

Effect of Geometric Imperfections

Kim and Chen4,5 performed a comprehensive verificationstudy of various geometric imperfection effects on framebehavior by comparing the results of the proposed methodswith those of the plastic-zone analysis and the conventionalLRFD method. Herein, some typical examples are presentedin what follows.

The AISC-LRFD column strength curve is used here fora verification of the column strength since it properlyaccounts for the second-order effect, residual stresses, andgeometric imperfections of an isolated column in a practicalmanner. In the explicit imperfection modeling, the two-element column is assumed to have an initial geometricimperfection equal to Lc/1,000 at mid-height. In theequivalent notional load method, the notional loads equal to0.004 times the gravity loads are applied at the mid-height ofthe column. In the residual stresses of 0.3Fy without

Fig. 10. Comparison of moment-rotation behavior by experimentand three parameter power model for verification study.

Table 3.Comparison of the Three Parameters of Power Model

for Verification StudyMethod Rki Mu n

Curve-Fitting 4.250 kN-m/rad. 24.9 kN-m/rad. 0.91(40,000 kip-in/rad. (220 kip-in./rad.)

Kishi-Chen 4,499 kN-m/rad. 23.5 kN-m/rad. 1.50(39,819 kip-in/rad.) (208 kip-in/rad.)

further reduced tangent modulus method, the reduced tangentmodulus factor equal to 0.85 is used. The proposed threemethods result in a good fit to the LRFD column strengthsshown in Figure 12.

Kanchanalai17 developed exact interaction curves usingthe plastic-zone analysis for sway frames. In his studies, themembers were assumed to have maximum compressivegeometric imperfections. Thus, the curves are adjusted hereto account for the effect of geometric imperfections. TheAISC-LRFD interaction curves are obtained based on theLeMessurier K-factor approach.18 The inelastic stiffnessreduction factor t is employed with the LeMessurierprocedure for K-factor calculations.1 Geometric imperfectionof Lc/500, the notional load factor of 0.002, and the reducedtangent modulus factor of 0.85 are used respectively for thethree proposed methods. The proposed methods predict wellthe strengths of the frame as shown in Figure 13.

ANALYSIS AND DESIGN RULES

Design Format

The proposed methods are based on the LRFD design format.The limit state format may be written as:

Fig. 11. Comparison of lateral displacements by experiment andproposed methods for verification study.


giQifRn (14)

where

gi = load factorsQi = nominal design loadsf = resistant factorsRn = nominal resistances.

The limit state format provides a uniform reliability forstructures.

Load CombinationsThe load combinations in the proposed methods are based onthe LRFD load combinations.1 The member sizes of thestructure are determined from an appropriate combination offactored loads.

Live Load ReductionThe live load reduction is based on the ASCE 7-88. It isimportant to carry out properly the application of the liveload reduction in analyzing a structural system. This isbecause the influence area for each beam and column isgenerally different and different influence areas result indifferent reduction factors. In the present study, the live loadreduction procedures follow the work of Ziemian andMcGuire.19

Resistance FactorsHere, as in LRFD, the resistance factors are selected to be0.85 for axial strength and 0.9 for flexural strength. Theseresistance factors are included in the computer program forpractical design purposes.

Fig. 12. Comparison of strength curves by explicit imperfection modeling method and LRFD method

for axially loaded pin-ended column.

Serviceability LimitThe LRFD Specification does not provide specific limitingdeflections and lateral drifts. Such limits depend on thefunction of a structure. Based on the studies by Ad HocCommittee20 and Ellingwood,21 the deflection limits of thegirder and the story are adopted as follows:

1) Floor girder live load deflection: L/3602) Roof girder deflection: L/2403) Lateral drift: H/400 for wind load4) Interstory drift: H/300 for wind load

At service load levels, no plastic hinges are allowed todevelop since permanent deformations should be preventedunder service loads.

Ductility Requirement

Adequate inelastic rotation capacity is required for beam-column members to develop their full plastic momentcapacity and to sustain their peak loads The required rotationcapacity, i.e., the ductility requirement may be achievedwhen members are adequately braced and their cross sectionsare compact. The limitations of the compact section are basedon the LRFD Specification. According to the limit on spacingof braces in the LRFD seismic provisions, the L / ry valueshould be less than 17,200 / Fy for beam-columns where Fy isin MPa.

Geometric ImperfectionThe magnitude of the geometric imperfections is summarizedin Table 2. Users can choose either the explicit imperfectionmodeling, or the equivalent notional load input, or the furtherreduced tangent modulus. For simplicity, the further reducedtangent modulus method is recommended.

Fig. 13. Comparison of strength curves by equivalent notionalload, plastic-zone, and LRFD method for portal frame.


RECOMMENDED ANALYSIS/DESIGNPROCEDURES

The analysis/design of semi-rigid frames is more involvedthan the analysis and design of rigidly jointed frames. Apossible design procedure for semi-rigid frames based on theproposed method is recommended as follows:Step 1Preliminary analysis/design assuming rigid frame. Thepreliminary member sizing is intrinsically dependent onengineer's experiences, the rule of thumb, or some simplifiedanalysis. For example, beam members are usually selectedassuming that beams are simply supported and subjected bygravity loads only. For the preliminary sizing of columnmembers, the overall drift requirements should be a goodguideline to determine preliminary member sizes rather thanthe tedious strength checks of the individual column.Step 2Preliminary selection of connection type and dimension. Theconnection types and dimensions should be determined byconsidering the overall flexibility of a structural system sincethe connection flexibility influences the second-order momentof a structural system. For illustration, relatively flexibleconnections such as single/double web angles may be usedfor a braced system, and conversely, relatively rigidconnections suchs as extended header plates may be used foran unbraced system. The dimensions of connections may be-selected from the resulting member forces and displacementsinformation in Step 1.

Fig. 14. Configuration of two-story semi-rigid framefor design example.

Step 3Determination of connection parameters. The power modelfor connections contains three parameters and they can bedetermined by Equations 4-9 and Table 1 for the connectiontypes and dimensions determined in Step 2.Step 4Analysis of semi-rigid structural system. Once thepreliminary member and connection sizes are determined inSteps 1-3, the refined plastic hinge analysis may beperformed to consider the effect of semi-rigid connectionsand geometric imperfections.Step 5Check for strength, serviceability, and ductility. Theadequacy of system and its component member strength canbe directly evaluated by comparing the predicted ultimateloads with the applied factored loads. The serviceability of astructural system should be also checked to ensure theadequacy of the system and member stiffness at serviceloads. Adequate ductility is required for members in order todevelop their full plastic moment capacity. The requiredductility may be achieved when members are adequatelybraced and their cross sections are compact.

Step 6Local strength checks of members and connections. Since theproposed analysis account for only the global behavioreffects, the independent local strength checks of members andconnections are required based on the LRFD Specification.Step 7Adjustment of member and connection sizes. If the conditionsof Steps 6-7 are not satisfied, appropriate adjustments ofmember and connection sizes should be made. The system

Fig. 15. Comparison of member sizes by proposed methods and LRFD method for design example.


behavior is influenced by the combined effects of membersand connections. As an illustration, if an excessive lateraldrift occurs in a structural system, the drift may be reducedby increasing member size or by using more rigidconnections. If the strength of a beam exceeds the requiredstrength, it may be adjusted by reducing the beam size orusing more flexible connections. Once the member andconnection sizes are adjusted, iteration of Steps 2-7 leads toan optimum design.

DESIGN EXAMPLEBarakat22 studied several frames with semi-rigid connections.In this study, the two-story, one-bay semi-rigid frame isselected for the present case study. Herein, the design of thissemi-rigid frame follows the analysis and design rules andprocedures described above. The member sizes determinedby the proposed methods are compared with those determinedby the modified LRFD method proposed by Barakat.22

Description of Frame

The height of each story is 3.7 m (12 ft) and the width is 7.4m (24 ft). The frame was subjected to vertical distributedloads and concentrated lateral loads according to the loadcombination of 1.2D + 1.3W + 0.5L shown in Figure 14. Allconnections are top and seat angle of L64.08.0 withdouble web angles of L43.58.5 made of A36 steel withbolt fasteners of A325 78 -in. D.

Analyses

The yield stress was selected as 250MPa (36 ksi), andYoung's modulus was 200,000MPa (29,000 ksi). Theanalyses were carried out by the proposed three methods. Theexplicit geometric imperfection was assumed to be y =2/1,000, the equivalent notional load was equal to 0.002Pu,and the further reduced tangent modulus was adopted as0.85Et. The connection parameters of the power model wascalculated as the initial connection stiffness Rki of 40,021kN-m/rad. (354,232 kip-in/rad.), the ultimate momentcapacity Mu of 128 kN-m (1,131 kip-in.) and the shapeparameter n of 1.14 by using Equations 4-9 and Table 1. Thevertical and lateral loads were applied to the frame at thesame time in an incremental manner. The resulting ultimateloads were compared with the factored applied loads, andmember sizes were adjusted.

Member Size

All three methods result in identical member sizes. They arecompared in Figure 15 with those determined by the modifiedLRFD method proposed by Barakat and Chen.22-24 The sizesof member sections are chosen such that the width of thebeam flange is less than that of the column flange. This is dueto the need of detailing of beam-to-column semi-rigidconnections. The member sizes by the proposed methods aregenerally one size smaller than those by the modified LRFDmethod, because the proposed methods possess the benefit of

inelastic moment redistribution that leads to a reduced steelweight for highly indeterminate steel structures.

ServiceabilityThe overall drift of the first-order analysis under wind load iscalculated as H/218 which does not meet the drift limit ofH/400. When the column sizes are increased from W824 toW1226. the overall drift limit is found to be H/408 whichsatisfies the drift limit. The deflections in the beams underlive load are smaller than the limit of L/360.

SUMMARY AND CONCLUSIONS

Three practical methods are developed for semi-rigid framedesign using the refined plastic hinge analysis. They are theexplicit imperfection modeling method, the notional loadmethod, and the reduced tangent modulus method. Thepractical procedures for determining connection parametersare provided for a given connection configuration. Theproposed methods can predict accurately the combinedeffects of connection, geometric, and material nonlinearitiesfor semi-rigid frames. The strengths predicted by thesemethods are compared well with those available experiments.The proposed methods provide a practical procedure for theLRFD design of semi-rigid frames. To this end, theanalysis/design rules and procedures for practical design ofsemi-rigid frames are described in details. The methods donot require separate member capacity checks including thecalculations of K-factor. Since the proposed methods strike abalance between the requirement for realistic representationof actual behavior and failure mode of a structural systemand the requirement for simplicity in use, it is considered thatin both these respects, all three methods are satisfactory, butthe further reduced tangent modulus method appears to be thesimplest and is therefore recommended for general use.

REFERENCES

1. American Institute of Steel Construction, Manual ofSteel Construction, Load and Resistance Factor Design,2nd Ed., Vol. 1 and 2, Chicago, IL, 1993.

2. Kish, N. and Chen, W. F., "Moment-rotation Relations ofSemi-rigid Connections with Angles," ASCE Journal ofStructural Engineering, 116, 7, 1990, pp. 1813-1834.

3. Chen, W. F. and Lui, E. M., Stability Design of SteelFrames, CRC Press, 1991, 380 pp.

4. Kim, S. E. and Chen, W. F., "Practical AdvancedAnalysis for Braced Steel Frame Design," StructuralEngineering Report No. CE-STR-95-11, School of CivilEngineering, Purdue University, West Lafayette, IN,1995, 36 pp.

5. Kim, S. E. and Chen, W. F., "Practical AdvancedAnalysis for Unbraced Steel Frame Design," StructuralEngineering Report No. CE-STR-95-15, School of CivilEngineering, Purdue University, West Lafayette, IN,1995, 36 pp.

6. Liew, J. Y. R., "Advanced Analysis for Frame Design,"


Ph.D. Thesis, School of Civil Engineering, PurdueUniversity, West Lafayette, IN, 1992, 392 pp.

7. Goverdhan, A. V., "A Collection of ExperimentalMoment-rotation Curves and Evaluation of PredictionEquations for Semi-rigid Connections," Master's Thesis,Vanderbilt University, Nashville, TN, 1983, 490 pp.

8. Nethercot, D. A., "Steel Beam-to-column Connectionsa Review of Test Data and its Applicability to theEvaluation of Joint Behavior in the Performance of SteelFrames," CIRIA Project Record, RP 338, 1985.

9. Kishi, N. and Chen, W. F., "Data Base of Steel Beam-to-column Connections," Structural Engineering ReportNo. CE-STR-86-26, School of Civil Engineering, PurdueUniversity, West Lafayette, IN, 1986, 653 pp.

10. Chen, W. F. and Kishi, N., "Semi-rigid Steel Beam-to-column Connections: Data Base And Modeling," ASCEJournal of Structural Engineering, 115, 1, 1989, pp.105-119.

11. Kishi, N., Chen, W. F., Goto, Y., and Matsuoka K. G.,"Applicability of Three-Parameter Power Model toStructural Analysis of Flexibly Jointed Frames," Proc.Mechanics Computing in 1990's and Beyond, Columbus,OH, May 20-22, 1991, pp. 233-237.

12. Chen, W. F. and Toma, S. Editors, Advanced Analysis ofSteel Frames, CRC Press, 1994, 383 pp.

13. White, D. W., "Plastic Hinge Methods for AdvancedAnalysis of Steel Frames," Journal of ConstructionalSteel Research, Vol. 24, No. 2, 1993, pp. 121-152.

14. Maleck, A. E., White, D. W., and Chen, W. F.,"Practical Application of Advanced Analysis in SteelDesign," Presented at the 4th Pacific Structural SteelConference, Singapore, October 1995.

15. Liew, J. Y. R., White, D. W., and Chen, W. F.,"Notionalload Plastic-hinge Method For Frame Design,"ASCE Journal of Structural Engineering, 120(5), 1994,pp. 1434-1454.

16. Stelmack T. W., "Analytical and Experimental Responseof Flexibly-connected Steel Frames," M.S. Thesis,Department of Civil, Environmental, and ArchitecturalEngineering, University of Colorado, 1982, 134 pp.

17. Kanchanalai, T., "The Design and Behavior of Beam-columns in Unbraced Steel Frames," Aisi Project No.189, Report No. 2, Civil Engineering/structures ResearchLab., University of Texas, Austin, TX, 1977, 300 pp.

18. Lemessurier. W. J., "A Practical Method of SecondOrder Analysis, Part 2Rigid Frames," AISCEngineering Journal, 2nd Qtr., Vol. 14, 1977, pp. 49-67.

19. Ziemian, R. D. and Mcguire, W., "A Method forIncorporating Live Load Reduction Provisions in FrameAnalysis," AISC Engineering Journal, 1st Qtr., Vol. 29,1992, pp. 1-3.

20. An Hoc Committee on Serviceability, "StructuralServiceability: a Critical Appraisal and ResearchNeeds," ASCE Journal of Structural Engineering,112(12), 1986, pp. 2646-2664.

21. Ellingwood, "Serviceabilitv Guidelines for SteelStructures," AISC Engineering Journal, 1st Qtr., Vol.26, 1989, pp. 1-8.

22. Barakat, M., "Simplified Design Analysis of Frames withSemi-rigid Connections," Ph.D. Dissertation, School ofCivil Engineering, Purdue University, West Lafayette,In, 1989, 211 pp.

23. Barakat, M. and Chen, W. F., "Practical Analysis ofSemi-rigid Frames," AISC Engineering Journal, 2ndQtr., Vol. 27, 1990, pp. 54-68.

24. Barakat, M. and Chen, W. F., "Design Analysis of Semi-rigid Frames, Evaluation and Implementation," AISCEngineering Journal, 2nd Qtr., Vol. 28, 1991, pp. 55-64.

LIST OF SYMBOLS

E modulus of elasticityEt tangent modulus

tE further reduced Et

Fy material yield stressG shear moduliGA, GB ratio of bending stiffness of columns versus

that of beams at beam-to-column joint,subscripts apply to respective ends of column

Ib, Ic moment of inertia of cross section, subscriptsb and c denote beam and column

L column lengthLb, Lc length, subscripts b and c denote beam and

columnm Nondimensional connection moment, M/MpMp plastic moment capacityMu ultimate moment capacity of connection

PMM BA ,, incremental end moments and axial forceP, M second-order axial force and bending momentn shape parameterPy squash loadQi nominal design loadsRki Initial connection stiffnessRkt tangent stiffness of connectionsRktA, RktB tangent stiffness of connections A and BRn nominal resistancesS1, S2 stability functionsSii, Sij generalized stability functions

*** ,, jjijii SSS modified stability functions that account for

the presence of end connectiona equivalent notional load factor in Table 2gi load factordc geometric imperfection at mid-height of

column in Table 2hA, hB scalar parameter for gradual inelastic

stiffness reductionq ratio of arbitrary rotation to reference plastic

rotation of connection


qo reference plastic rotation of connection, Mu/Rkiqr arbitrary rotation of connection

BA..

,qq incremental rotations at element ends A and Blc column slenderness parameter

f resistant factorfi reduction factor for geometric imperfectiony out-of-plumbness


Main Menu Search MenuABSTRACTINTRODUCTIONBEHAVIOR OF SEMI-RIGID FRAMESPRACTICAL CONNECTION MODELINGPRACTICAL ADVANCED ANALYSISPRACTICAL GEOMETRIC IMPERFECTION MODELINGVERIFICATION STUDYANALYSIS AND DESIGN RULESRECOMMENDED ANALYSIS/DESIGN PROCEDURESDESIGN EXAMPLESUMMARY AND CONCLUSIONSREFERENCESLIST OF SYMBOLS

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Practical Advanced Analysis for Semi Rigid Frame Design