Technical Note

Simplified Procedure to Calculate by Hand the NaturalPeriods of Semirigid Steel Frames

Hameed Shakir Al-Aasam1 and Parthasarathi Mandal2

Abstract: Current seismic design codes provide empirical equations for the calculation of the fundamental period of a framed structure pri-marily as a function of height, and do not consider the effect of beam-column joint stiffness. In the literature, far too little attention has been paidto incorporate this effect in approximate formulas to calculate by hand the natural periods of semirigid steel frames, which is the aim of thispaper. The proposed formulas have been developed after modifications of existing formulas in the literature for rigid-jointed plane steel frames.Then, the accuracy of these approximate formulaswas verified by a finite element analysis usingABAQUS software. Finally, a parametric studywas conducted to highlight the effect of semirigid connections on the natural periods of vibration of plane steel frames.DOI: 10.1061/(ASCE)ST.1943-541X.0000695. © 2013 American Society of Civil Engineers.

CE Database subject headings: Semi-rigid connection; Steel frames; Computation; Seismic design.

Author keywords: Natural period; Semirigid connection; Steel frames.

Introduction

The estimation of the fundamental period of vibration of a steelstructure is an essential step in earthquake design. Traditionally, thebeam-to-column connection is assumed to be either completelypinned or ideally rigid. However, a number of previous experimentalinvestigations clearly demonstrated that almost all types of con-nections of a steel frame are semirigid with various degrees offlexibility (Davison et al. 1987; Chen and Kishi 1989; Kishi et al.1997). Conventionally, frames with semirigid connections wereconsidered to be unsuitable for use in seismic areas because of theirflexibility. Subsequently, from experimental studies it was foundthat semirigid frames have considerable potential in resisting seismicloads owing to their higher energy dissipation capacities (Astanehet al. 1989; Elnashai andElghazouli 1994;Nader andAstaneh 1996).Also, analytical studies by Rosales (1991) and Sekulovic et al.(2002) showed that the increase in flexibility of semirigid framesmay significantly reduce the vibration frequencies, especially thefundamental frequency, and thus semirigid frames attract lesserinertial forces (Nader and Astaneh 1991). This may result in a moresatisfactory earthquake-resistant structure, even in areas ofmoderateto high peak ground motion.

Previous research, including design codes, provide empiricalformulas to estimate the fundamental period for buildings. Theseformulas are usually dependent on the building material (steel,reinforced concrete, etc.), building type (frame, shear wall, etc.), andoverall dimensions [Goel and Chopra 1997; European Committee

for Standardization (CEN) 2004] (BS EN 1998-1:2004). All of theseempirical formulas do not take into account joint behavior. Theperiod of a semirigid frame, which is the category that most realframes will fall into, can be twice that of a rigid frame (Smith andCrowe 1986). It must be mentioned that the actual vibration periodsof real buildings are most likely to be affected by many factors,including nonstructural members and fixtures.

In this study, a simple hand-calculation procedure is proposed tocalculate the first three natural periods of steel frames with semirigidconnections, which is not available in the literature. The proposedprocedure is based on a shear-flexure cantilever model of rigidframes (Smith and Crowe 1986), which in turn was based onSkattum (1971), Heidebrecht and Smith (1973) and Rutenberg(1975). The accuracy of the proposed simplified procedure has beenverified by finite element analysis of a plane steel frame withsemirigid connections using ABAQUS software (ABAQUS 6.10.1).The procedure is limited to plane frameswith uniform geometric andmaterial properties along its height. Finally, a parametric study wasconducted to quantify the effects of the flexibility of connectionson the natural frequencies of vibrations of plane steel frames withsemirigid connections.

Characteristic Equation for Shear-Flexural Motionof Cantilever Beam

Heidebrecht and Smith (1973) idealized the vibration behavior ofa plane frame structure similar to the behavior of a shear-flexuralcantilever beam. The characteristic equation was later modified byRutenberg (1975) after incorporating the effect of axial rigidity, andwas represented as

2þ��

l1l2

�2

þ�l2l1

�2�cos l1 H cosh l1H

þ�l2l1

2l1l2

�sin l2 H sinh l2H ¼ 0 ð1Þ

The eigenvalues l1 and l2 are written as follows:

1Ph.D. Student, School of Mechanical Aerospace and Civil Engi-neering, Univ. of Manchester, Manchester M13 9PL, U.K. E-mail:Hameed.Al-Aasam@postgrad.manchester.ac.uk

2Lecturer in Structural Engineering, School of Mechanical Aerospaceand Civil Engineering, Univ. of Manchester, Manchester M13 9PL, U.K.(corresponding author). E-mail: P.Mandal@manchester.ac.uk

Note. This manuscript was submitted on January 23, 2012; approvedon July 26, 2012; published online on August 11, 2012. Discussion periodopen until November 1, 2013; separate discussions must be submitted forindividual papers. This technical note is part of the Journal of StructuralEngineering, Vol. 139, No. 6, June 1, 2013. ©ASCE, ISSN 0733-9445/2013/6-1082–1087/$25.00.

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l21 ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðakÞ44

þ v2mEI

s2

ðakÞ22

; l22 ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðakÞ44

þ v2mEI

sþ ðakÞ2

2;

and v ¼ l2ffiffiffiffiffiEIm

rð2Þ

where E 5 Young’s modulus of elasticity, and m 5 mass per unitheight. The parameters a and k consider the effect of shear rigidity(GA) and axial rigidity as follows:

a2 ¼ GAEI

and k2 ¼ 1þ IAc2

ð3Þ

where I5 secondmoment of area,A5 cross-sectional area, and c5distance parameter that will be explained in the context of a framesubsequently.

Purely Flexural Motion (a2 50, v5vf , and l5lf)

For purely flexural motion, a2 5 0. Then Eq. (2) reduces tol1 5 l2 5 lf , and the eigenvalues can be obtained from Eq. (1) as

�lf�n ¼

anH

0 an ¼ 1:875 for n ¼ 1

ðn2 0:5Þp for n ¼ 2, 3, 4, . . .ð4Þ

Coupled Shear-Flexural Motion (v5vsf , and l5lsf )

For values of (kaH$ 6), Skattum (1971) proposed the followingequation:

l2sf ¼ðn2 0:5Þpð1þ kaHÞ

H2 ð5Þ

For a practical range of (kaH, 6), Heidebrecht and Smith (1973)obtained that l1 ≅ lf and substituting in Eq. (2) gives

l2sf ¼ l2f

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi�1þ

�kaHlf H

�2�sð6Þ

Combined Effect of Pure-Flexural andShear-Flexural Motion

Using the Southwell-Dunkerley approximation for isolated compo-nents, the period of vibration could be computed (Rutenberg 1975) as

T ¼ 2pl2

ffiffiffiffiffimEI

rwhere 1

l2�

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1k2

1l4sf

þ12 1

k2

1l4f

sð7Þ

Approximate Formulas for Natural Periods

Using Eqs. (5)–(7), and after rearranging the parameters, the naturalperiod of vibration can be written as

Tn ¼ Tf

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ 1

k2

�1Rn

2 1

�sð8Þ

Tf ¼ 2pH2

ka2n

ffiffiffiffiffimEI

rand Rn ¼ 1þ

�kaHan

�2

for kaH, 6

�ðn2 0:5Þp

�1þ kaH

a2n

��2for kaH$ 6

ð9Þ

Evaluation of Characteristic Parametersof Plane Frame

The concept of modeling the plane-framed structures as shear-flexuralcantilever beams is applicable for any type of uniform frame structure

Fig. 1. Plots of normalized fundamental natural frequency versus joint stiffness ratio; I 5 second moment of area of beam cross section

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provided the centers of mass and resistance are coincident (Smithand Crowe 1986). The characteristic parameters of a frame can beexpressed in terms of geometrical and material parameters as

GA ¼ 12E=h1Pnj¼1

Icjh

þ 1Pmi¼1

Ibilbi

, a2 ¼ GAPnj¼1EIcj

, k2 ¼ 1þ

Pnj¼1

Icj

Pnj¼1

Acjc2j

ð10Þ

where Acj and Icj 5 cross-sectional area and second moment of areaof the jth column; h5 story height; and Ibi and lbi 5 secondmoment

of area and span of the beam in the ith bay. The term cj 5 distance ofthe jth column from the centroid of the column assembly.

Effect of Connection Flexibility on Natural Frequencyof Steel Frames

To investigate the effect of the flexibility of the connection on thenatural frequency of a steel frame, a simple portal steel frame wasused. The beam had a universal beam (UB) 2543 1463 37 sectionand the columns were of universal column (UC) 2033 2033 60sections. A commercial finite-element software program (ABAQUS6.10.1), was used to find the natural frequencies. Rotational springelements were attached to the beam ends to simulate the effect of

Fig. 2. Three-bay 6-story steel frame used in the numerical calculation

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connection flexibility of the beam-to-column connections. Thestiffness of the springs ranged from those close to perfect-pinned toalmost perfect-rigid connections. The results of frequency analysisare nondimensionalized using the pinned case as zero and the naturalfrequency of the rigid case as 1 as shown in Fig. 1. The springstiffness ratio, ks=ðEI=LÞ, is used as the abscissa because the effect ofthe flexibility of the connection is highly affected by the flexuralstiffness of the beam which it is attached to. A parametric study wascarried out for various ratios of the second moment of area of thebeam (0.25, 0.5, 2, 4, and 8) from the original (I5 5,537 mm4) toshow the effect of flexural stiffness of a beam on the natural fre-quency of a steel frame. It is evident from Fig. 1 that the flexibility ofthe connection must be taken into account in any frequency analysisof steel frames. This means that the original equations to calculate thecharacteristic parameters, a2 and k2, of the steel frame should bemodified to take into account the flexibility of the connections.

Modification of the Approximate Formulas forSemirigid Frames

The approximate formulas for determining the natural periods forrigid frames cannot be used directly for frames with semirigidconnections. The difference is appreciable in the range of the stiff-ness ratio between 0.01 and 100 (Fig. 1). The flexural stiffness ofthe beams needs to be modified to take into account the effect of theflexibility of the connections. There are many procedures in theliterature on modifying the stiffness of a beam to consider the effectof the flexibility of end connections, such as the procedures de-veloped by Chen and Lui (1985), Chui and Chan (1997), andWonget al. (2007). It was decided to adopt for this work the proceduresuggested by Chen and Lui (1985) for the stiffness of beams becauseit is applicable for both braced and unbraced conditions. The mod-ified second moment of area of beam Ib can be calculated using theequation

Ib ¼ Ib�1þ bEIb

lbks

� ð11Þ

whereb5 6 and 2 for unbraced and braced frames, respectively, andks 5 rotational stiffness of the semirigid connections. This modifiedsecond moment of inertia of beam Ib should be used in Eq. (10).

To investigate the accuracy of the proposed procedure and theeffect of joint stiffness, a three-bay, 6-story plane steel frame (Fig. 2)was analyzed for a wide range of flexibility of beam-to-column con-nections. The Young’s modulus of elasticity was taken as 200 GPa.The results from the hand calculations [Eqs. (8)–(10)] were com-pared with the numerical results from ABAQUS (Figs. 3 and 4). Itis observed that the fundamental natural frequency calculated bythe proposed procedure is very similar to the ABAQUS results(Fig. 3). The formula given in Eurocode 8 [BS EN 1998-1:2004(CEN 2004)] is in between the perfectly rigid and pinned cases, andthe error could be as much as 200%, and may underestimate thedesign-base shear force, which will be highly unsafe. The results forthe second and third natural frequencies are also close to theABAQUS results and the proposed procedures with a maximumdifference of ∼6%.

ParametricStudyon theEffectofDifferentkaHValues

The parametric study involved evaluating the effects of the fol-lowing factors: the height of the frame, H; the flexibility of theconnections; and the axial, flexural, and shear rigidity of the frame. Itis clear from Eq. (10) that the parameter k considers the effect of therelative ratio of flexural stiffness to axial stiffness, and its value willapproach unity for axially rigid columns. The parameter a incor-porates the effect of the relative ratio of shear stiffness to thesummation of the flexural stiffness of the columns and beams. Using

Fig. 3. Joint stiffness ratio versus fundamental natural frequency of steel frame in Fig. 2

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Eqs. (5)–(9), the ratio of the shear-flexural period to the total periodcan be expressed as

TsfTn

� 1ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1k2

ð12RnÞ þ Rn

r ð12Þ

For the fundamental period (i.e., n5 1 and an 5 1:875) and forkaH5 6

TsfT

¼ 1ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi9:7822 8:782 1

k2

r

For k2 5 1 (i.e., flexural stiffness/axial stiffness ratio approaches 0),Tsf =T5 1, and for k2 5‘ (i.e., flexural stiffness/axial stiffnessratio approaches ‘), Tsf =T 5 0:319.

For any given values of a and k, as the height of a frame (H)increases, the percentage of the pure-flexural period will increaseproportionally and the ratio of the shear-flexural to total period(Tsf =T) will decrease as shown in Fig. 5. In summary, the percentageof the shear-flexural period to the total period is inversely pro-portional to the height of the frame, the connection flexibility, andthe flexural to shear rigidity ratio.

Summary and Conclusions

It is evident from Figs. 1 and 3 that the flexibility of the connectionhas a considerable effect on the natural frequencies (or natural

Fig. 4. Joint stiffness ratio versus second and third natural frequencies of steel frame in Fig. 2

Fig. 5. Effect of frame parameters on shear-flexural to total periods

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periods) of a steel frame. Also, the ideal-rigid assumption for beam-to-column connections may lead to an overestimation of the naturalfrequency or underestimate the natural periods of a steel frame.Inaccurate values of a fundamental period may result in unsafedesign because the design value of a seismic base-shear forcedepends on the period [BS EN 1998-1:2004, Clause 4.3.3.2.2.(1);CEN 2004]. In addition, Figs. 3 and 4 show clearly that the proposedprocedure can predict the natural periods of steel frames for a widerange of flexibility of beam-to-column connections. The validity ofthe proposed procedure is confirmed by comparing it with the resultsfromABAQUS and good agreement between them are demonstrated.

References

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