analysis for semi-rigid frame design - lrfd.pdf

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LRFD Analysis for Semi Rigid Frame DesignWON-SUN KING AND WAI-FAH CHEN

ABSTRACT

A practical LRFD-base d analysis method for the design ofsemi-rigid frames is proposed. The proposed method usesfirst-order elastic analysis with a notional lateral load for thesecond-order effects. In the proposed method, a simplifiedthree-parameter model describing the tangent rotational stiff-ness of semi-rigid connections is used.

1. INTRODUCTION

Although partially restrained (PR) construction is permitted

by the AISC Specification for Structural Stee l BuildingsLoad and Resistance Factor Design, no specific analysis ordesign guidance is given in the current LRFD and ASDspecifications for these partially restrained frames.

Recently, a simplified procedure for the analysis and design of semi-rigid frames was proposed by Barakat andChen,^ using the B and B2 amplification factors together withthe beam-line concept. However, the beam-line method cannot adequately predict the drift of unbraced frames and thecalculation of effective length factor is cumbersome andtime-consuming.

A simplified procedure to improve these drawbacks isintroduced in this paper. Here, as in the Barakat method, the

proposed method is based on first-order linear elastic analysis, but the second-order effect will be included with the useof notional lateral loads.

2. MODELING OF SEMI-RIGID CONNECTIONS

2.1 Connection ModelsMost existing connection models express the moment interms of rotation from which the tangent stiffness can bederived. This paper proposes a direct tangent-stiffness expression for flexible connec tions. This proposed tangent-stiffnessmodel is based on the concept that connection stiffness de

grades g radually from an initial stiffness, Kj,

to zero followinga nonlinear relationship of the simple form:

Won-Sun King is associate professor, Department of CivilEngineering, Chung Cheng Institute of Technology, Ta-Hsi,Tao-Yuan, Taiwan.Wai-Fah Chen is professor and head of Structural Engineering, School of Civil Engineering, Purdue University.

dMde/

K, = K,M^

V ^ V(1)

where

K^ = tangent stiffnessKj = initial connection stiffnessM^^ = ultimate bending moment capacityM = connection momentC = shape factor account for decay rate of K^, C>0

The moment-rotation (M - 0 behavior of bolted extendedend-plate beam-to-column connections tested by Yee andMelchers^ is compared with the proposed model in Figure 1and a good agreement is observed with C = 1.6. In Figure 1,the initial stiffness, K, = 546,666 in-kip/rad, is the tangent tothe starting point of the curve. The ultimate m oment capacityMj, = 3,539 in-kip, is determined by test. The value C is usedto control the shape of a convex curve. If C is equal to I, K^decreases linearly. When C is less than 1, K^ decreases morerapidly. If C is greater than 1, K^ decreases much slower. Thisis illustrated in Figure 1 with C = 1.0,1.6, and 2.2 respectively.

In the following, the proposed tangent stiffness connectionmodel w ill be applied to several types of connections, including the extended end-plate, top and seat angle with doubleweb angles, framing angles, and single-plate connections

4000 -1 Loading

M(in-kip)

3000

2000

1000 H

C=1.6 C=1.0

Extended End PlateKi = 546,666 in-kip/radMu = 3,539 in-kipC = 1.6

3 0

d, (rad)xlO-^

Fig. 1. Moment-rotation curves of Yee connection (1986).

4 0

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The connections ranged from very stiff to rather soft connections. The moment-rotation curve is obtained by numericalintegration of tangent-stiffness Equation 1.

a) Jenkins Bolted Extended End Plate

A comparison of the proposed model w ith one of the Jenkins,Tong, and Prescott extended end-plate connection test^ is

shown in Figure 2. In this test, the beam is 305x165 UB 54,the column is 254x254 UC132 (stiffened), bolts are M20grade 8.8, thickness of end plate is 20 mm and K^ = 786,732in.-kip/rad and M = 1,989 in.-kip. Good agreement is observed with C = 0.555.

(b) Bolted Top and Bottom Angles with Web AnglesAzizinamini, Bradbum, and Radziminski^ reported test results on bolted semi-rigid steel beam-to-column connections.These connections are comprised of top and bottom anglesconnected to the flanges along with web angles. ASTM A 36steel was used for the mem bers and the connection eleme nts.Eighteen specimens were tested. The beam tested was aW14x38, the bolt diameter is 22.2 mm, and the web anglesare 214x31/2X^4. The thickness of flange angles is 15.9 mm,and the length of the test beam is 203.2 mm . The test number14S8 with Ki = 677,025 in.-kip/rad and M, = 1,101 in.-kipcompares well with that of the proposed model in Figure 3with C = 0.34.

c) Bolted Framing Angles

Bolted double-web angles were tested by Lewitt, Chesson,and Munse at the University of Illinois. In 1987, Richard, etal proposed a four-parameter formula to describe these full-scale tests. Figure 4 compares the results of the proposedmodel with one of these tests using a five-bolt design withrivets in the angle-to-beam w eb connection with ^, = 206,667in.-kip/rad and M = 761 in.-kip.

d) Single Plate

A total of seven tests were made by Richard^ on single-plateconne ctions. The first set of two-, three-, five-, and seven-bolttests were run w ith the framing connection plate welded to aflange plate which was in turn bolted to the support column.A second set of tests was run on the two-, three-, and five-boltconnections with the framing connection plate welded to thesupport column. In these tests, three bolts were used toconnect the beam and the single plate. The bolts are A325V4-in. diameter, and the plate thickness is Vg-in. The moment-rotation curve of the proposed model compares w ell with oneof the tests as illustrated in Figure 5 with Ki = 51,000 in.-kip/rad, M = 137 in.-kip, and C = 0.22.

2.2 Initial Stiffness

For simplicity, researchers^'^'^ have been using the initialconnection stiffness, K^, for their semi-rigid frames analysis.The use of initial stiffness throughout the flexible frameanalysis results in a frame behavior that is generally too stiffwhen the frame is subjected to a normal loading condition.

Extensive studies of frames by Ackroyd^^ with nonlinearconnections indicate that the secant stiffness of beam-to-column connections near ultimate frame capacity was typically20 percent of the initial stiffness, ^ at leeward ends of girdersand 80 percent of Kj at the windward ends of girders, whenthe frame is subjected to combined gravity and wind loading.It seems, therefore, reasonable to use an average connectionstiffness of 0.5AT, when com puting the design mome nts. Thisis adopted in the present analysis.

3 . DESIGN FORMULA IN AISC-LRFD

The equation for the maximum strength of beam-columns isgiven by AISC-LRFD as

M(in-kip)

2000

1500

1000

500 H

2000

Extended End-PlateKi = 786,732 in-kip/rad

Mu= 1,989 in-kipC = 0.555

r-1 0

I

2 0 I

3 0

Gr (rad)xlO--

Fig. 2. Moment-rotation curves o f Jenkins connection (1986).

M(in-kip)

1500 H

1000 H

500 H

Proposed

Top-Bottom Flange Angles with Web Angles

Ki = 677,025 in-kip/radMu= 1707 in-kipC = 0.34

r10 20

I ' 130 40

Br (rad)xlO

I

50

3

Fig. 3. Moment-rotation curves of Azizinaminiconnection (1987).

FOURTH QUARTER /1993 131


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f o r ^ > 0 . 2 ,

Pu , 8

for < 0 .2 ,

2tP \Mux

where

P = ultima te com

^bKy^

pression c

< 1 . 0 (2)

< 1 . 0 (3)

loaded columnM ^ , M^y = u l t i m a t e m o m e n t - r e s i s t i n g c a p a c i t y o f a

la teral ly unsupported be am abou t x and y axes,respect ively

j = co lumn r e s i s t ance f ac to r (= 0 .85 )(|) = beam re s i s t ance f ac to r (= 0.9 )

/^ = des i gn axia l forc e

Mj^, M^y = m e m b e r d e s i g n m o m e n t a b o u t x and y axes ,respect ively, wi th :

M, = B,M,, + B,M, (4)

M^t = f i r s t- o r d e r m o m e n t i n t h e m e m b e r a s s u m i n g n ola tera l t rans la t ion i n the f rame

Mif = f i r s t -o rde r m om en t i n the m e m b e r a s a r e su l t o fla tera l t rans la t ion o f the f rame

B = P-5 momen t amp l i f i ca t i on f ac to r

B,=

'4> 1 (5)

B2 = P- A moment amplif icat ion factor

1B,

1I.HL

(6)

C^ = 0.6 - 0 .4Mi /M 2, where M^ /M is the ratio of the

smaller to the larger end moment of a memberP, =TeEI/{KLf

Z i = axial loads on all colum ns in a story

A^ = first-order translationa l deflection of the story undconsiderat ion

JJi = sum of all story horizon tal forces prod ucing A^L = story heightK = e f f e c t i v e l e n g t h f a c t o r d e t e r m i n e d f r o m t h

al ignment char t

The second-order effects are taken into account approxmately by the mome nt amplif icat ion factors B^ and B2 on thnonsw ay and sway m ome nts obtained from f i rs t -order e las t

analyses, respectively. It usually leads to conservative result

4 . B E A M - C O L U M N S T I F F N E S S I NS E C O N D - O R D E R E L A S T I C A N A LY S I S

For second-order e las t ic analysis , we use the usual e lemegeometric stiffness matrix combined with the update of thelement ge ometry during the analysis . The f i rs t three terms the Taylor series expansion of the elastic stability functioare retained for the axial compressive force P to increase theaccuracy of the element s t i ffness . The corresp onding terms stiffness matrix were obtained by Goto and Chen^^ as

_ 4 7 2PL AAP^V

L ^ 15 ^25 , 000^ /

K,=2EI PL 26P^V

30 25 ,000 /

(7)

(8)

800M

(in-kip)

600

400

200

200 n

1 0

Double Web Angles

Ki = 206,667 in-kip/rad

Mu = 761 in-kip

C = 0.525

I

2 0 I

3 0 I

4 0

Or (rad)xlO-3

Fig. 4. Moment-rotation curves ofLew ittconnection (Richard, 1987).

M(in-kip)

150 H

100

50

r-20

Single Plate

Ki = 51,000 in-kip/rad

Mu =137 in-kip

C = 0.22

I

4 0 60 8

Or (rad)xlO-5

Fig. 5. Moment-rotation curves of Richard connection (1980).

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where

K,2EI PL 26P^U

30 25,000/

_ 4 7 2PL UP^V^ L ^ 15 ^25,000^/

(9)

(10)

P = axial force in memberA = area of a cross sectionr, = internal reactions at both ends of a member =1 ,6

di = displacements at both ends of a member =1 ,6

The beam stiffness matrix in Equation 11 can be simplifiedby recognizing that the axial force in beams of rectangularframes is usually negligible. That is, by setting K^ = Kjj =4EI/L, Kij = Kji = 2EI/L, and P = 0.

The stiffness matrix of a beam-column can be modified to

include the effect of semi-rigid con nections by comb ining themember stiffness with the connection stiffness using a staticconden sation. Details of this procedu re are given in Chen andLui,^^ and the resulting mem ber stiffness m atrix has the form:

AE

L

0

0

0

{Kii + lKij + Kjj) P

L 2 ' L

{Kii + Kij)

L

0

{K u + Kjd

L

Kii

-AE

L

0

0

0

-(Kii + 2Kij + Kij) p

L L

-iKji + Kij)

L

0

(Kij + Kjj)

L

KiJ

-AE

L

-jKii + lKij + Kjj) p

(Kji + Kji)

L

0

-JKii + Kjt)

L

AEL

iKii + 2Kij + Kjj) ^ P

-(Kji + Kij)

L

-(Kji + Kjj)

L

(11)

AEL

0

0

-AE

L

0

0

0

{Kil + lKiJ + KjJ) , P

L 2 ' L

L

0

-(Kif + 2Ki/ + Kj/) P

L L

(Kjf + Kjf)

L

0

(Kii+K/)

L

Ku

0

-(Ku+K/)

L

^ /

-AE

L

0

0

AEL

0

0

0

-{Ktf + 2Kt/ + KiJ) P

L L

-JKu+Kif)

L

0

iKif + 2Kif + Kj/) P

L 2 ' L

L

0

(^ii + f^jj)L

Kij

0

-(Ki/ + Kj/)

L

KJJ

(12)



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where or

^7 ~

^JJ ~

^ // = ^ / =SI

(13)

(14)

(15)

The coefficients R/ and i? in Equations 13 and 14 are theinstantaneous tangent stiffness coefficients of the connec tionsat ends i and7 of the member respe ctively. These coefficientsare obtained from Equation 1 when the connection is in thestate of loading, and are set equal to K^ when the connectionis in the state of unloading. Also, the parameter i?* is givenby

A: = B.K

/?*1 +

SIR,

S 'S 'ij ij (16)

(17)

(18)

Sj- - Sj/ - K - Kjj

in which P is negative for compressive force, and is small orzero for beam elements and can be neglected.

5. THE PROPOSED METHOD

Several simplifications are made in the present formulation.The moments of beam-column joints must be less than theultimate moment M^^ of semi-rigid connections or the plastic

moment capacity M^ . of beam-column s. The combined axialload and end mom ents in any mem ber must satisfy the AISC -LRFD bilinear interaction equations.

5.1 Rigid Fram e Analysis1. Perform the first-order elastic rigid frame analysis.2. Compute notional lateral loads, S//', using the relation

ship

5 , -^A ~ A

(19)

where A^ is the first-order trans lational deflec tion of the story,

and A is the second-order translationa l deflection of the storyunder consideration. From Equation 19, we have

(22)

J.HL

The A is the second-order lateral deflection due to P-Aeffect, and the notional lateral load is defined as

i : // ' = E / / -h i :P A/L (23)

3. U se ILH and original gravity loads to perform first-ordeelastic rigid frame analysis. The results of this stepinclude the second-order effect.

4. Calculate B factor with the effective length factor K =1.0 for each column and multiply the coiTcsponding endmoments.

5. Check the A ISC-LRFD bilinear interaction equations.

5.2 Semi-Rigid Fram e Analysis

1. Select connections from the maximu m beam-co lumnjoint moments in rigid frame analysis.

2. Determine the initial stiffness, AT, of connections fromtest results or any other available methods.

3. Substitute 0.5 ^, of connection stiffness for the semirigid joint. The average connection stiffness 0.5/^, assuggested by Ackroyd^^ is adopted h ere.

4. Use 0.5^, for semi-rigid connection stiffness with thenotional lateral loads S//' to carry out the first-orderelastic analysis.

5. Calculate B factor with effective length factor = 1.0for P^ of each column and multiply the correspondinglarger end moments.

6. Check the AISC-LRFD bilinear interaction equations.The effective length factor, K, for the column strength,i^, has to be modified in the case of semi-rigid frames.For beams connected to columns with semi-rigid connections rotational stiffness, i^ at both ends, a simplemodification of the relative stiffness, G, factors with themodified m oment inertia of beam is (Chen and Lui^''):

ri-h 2EI

KL

(24)

The / is used in G factors for the determination of the

effective length factor, K, for the v alue of ^ which is thecritical column stress.

A = (E // + Z / ^ ^ A / L ) ^ : 1 + -

from which we obtain

A

ILHL\ ,

Z/ZL

(20)

(21)

6. NUMERICAL EXAMPLESThe proposed method will now be illustrated by numericalexamples. Comparisons are made between results using direct second-order elastic analysis, Barakat's method,' and theproposed method. The semi-rigid frame examples includesingle-story and multi-story frames. All examples are ana-

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lyzed with a personal computer. All beams subjected to uniformly distributed loads are divided into two equal elements.

6.1 Two-Story One-Bay Frame with ConcentratedLoadsThe two-story one-bay frame as shown in Figure 6 is analyzedwith both rigid and sem i-rigid con nections. Two lateral loads,//, and four constant concentrated gravity loads, P, of 100 kipsare applied at the beam-column joints of the frame. Theflexible co nnections used are shown in Figure 2 where M isless than the plastic m oment Mp of beams and columns. The0.5Ki of Jenkins connection is 393,366 in-kip/rad. The second-order lateral displacement at Joint 5 is

A / = -1.512

^P,A ^ ^1.69 in.

Z//L

2 x 1 0 0 x 1 . 5 1 22 10 144

The second-order lateral load at Joint 5 is

E//5' = I// 5 + SP AsVL = 10 + 2 X 100 X 1.69/144 = 12.35 kips

The second-order lateral displacement at Joint 3 is

12 ft

12 ft

^ 20ft *^Columns:W12X96Beams:W14X48P = 100 kipsH = 10 kips

Fig. 6. Two-Story one-bay frame with concentrated loads.

Table 1.Maximum Moments in Elastic Rigid

Frame A nalysis (in.-kips)(Two-story one-bay frame, Figure 6)

ElementNo.

1

2

3

4

5

6

(1)

First-Order(Exact)

1449

712

1443

1437

711

712

(2)

Second-Order(Exact)

1649

794

1670

1664

794

794

(3)

Proposed

1839

894

1847

1839

893

894

(4)

(3)/(2)

1.12

1.13

1.11

1.11

1.13

1.13

A / :1.01

= 1.17 in.

ILHL4 x 1 0 0 x 1 . 0 12 x 1 0 x 1 4 4

The notional lateral load at Joint 3 is

SZ/g' = i:/f3 + ZP A37L = 10 + 4 X 100 X 1.17/144 = 13.25 kips

All moments of beams and columns predicted by theproposed method are normalized with respect to that ofsecond-order elastic analysis and are summarized in Tables 1and 2. The lateral displacements at windward beam-columnjoints are shown in Table 3. The mean values are the sum ofnormalized values of each mem ber divided by the number oftotal members. All the results predicted by the proposedmethod are close to the exact solutions. It is found that themaximum m oment and lateral displacement can be predictedwell by the proposed method.

6.2 Two-Story One-Bay Frame with UniformlyDistributed LoadsA two-story one-bay frame used by Barakat,^ et al as shownin Figure 7(a) is employed here for com parison of the maximum moments in members. The semi-rigid connection labeled III-17 is shown in Figure 8 and compared with the

proposed connection model. The moments predicted by second-order elastic analysis the Barakat method, and the proposed method are compared in Table 4. The average value ofColumn 3 in Table 4 is 0.98, while the average value ofColumn 5 is 0.97. The Baraka t method is slightly less conservative in this exa mple.

To verify the validity of the proposed method for softsemi-rigid connections, the bolted framing angles tested byLewitt^ are used. The lateral loads and uniformly distributed

FOURTH QUARTER / 1993 135


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Table 2.Maximum Moments in Elastic Semi-Rigid

Frame Analysis (in.-kips)(Two-story o ne-bay frame, Figure 6)

ElementNo.

1

2

3

4

5

6

(1)

Second-Orde r

(Exact)

1560

1116

1837

1834

1116

1116

(2)

Proposed

1696

1037

1847

1839

1037

1037

(3)

(2 ) / (1 )

1.09

0.93

1.01

1.00

0.93

0.93

loads are reduced as shown in Figure 7(b), so the maximummoments in members of the two-story one-bay frame are lessthan the ultimate moment, M^^, semi-rigid connections. Theresults predicted by the proposed method are compared withthat of second-order elastic analysis in Table 4(b) and thelateral displacements are shown in Table 4(c). It can be

0.155 kips/in2.88 kips

5.76 kips-^i^

6) H ] W 16x 31 [a]O I

CO

[3]255 kips/in

t 1 t M I W'@ [U W2 1x44 [e]

m

288 in

O Node No. [ I : Elem ent No.

Table 3.Lateral Displacements at W indward

Beam-Column Joints (in.)(Two-story one-bay frame, Figure 6)

NodeNo.

3

5

Rigid Frame

(1)

S e c o n d -Order

(Exact)

1.16

1.73

(2)

Proposed

1.21

1.82

(3)

(2 ) / (1 )

1.04

1.05

Semi-Rigid Frame

(4)

S e c o n d -Order

(Exact)

2.02

3.26

(5)

Proposed

1.85

2.98

(6)

(5 ) / (4 )

0.92

0.91

concluded that the proposed method is valid for soft connections, although the second-order lateral loads in the proposedmethod are determined from a rigid frame.

6.3 Three-Story One-Bay Frame with UniformlyDistributed Loads

The three-story one-ba y frame shown in Figure 9 is analyzedwith semi-rigid connections labeled III-17. Three beam-column joints are subjected to concentrated lateral loads. All thebeams are subjected to uniformly distributed gravity loadsThe results by the Barakat method^ are compared with thoseresults of the proposed method (Table 5). The average valueof Column 3 in Table 5 is 1.00, while the average value of

0.07 kips/in1.5 kips

mt t t t t Hl 6x31 I s j

3.0 kips0.10 kips/in

t t I t T T t@ H ] W21x44 [6]

m

^

77T7T

E ^

1 ' ^

^ 5:

_ i

288 in

Fig. 7(a). Two-story one-bay frame withuniformly distributed loads.

Fig. 7(b). Two-story one-bay frame withunifonnly distributed loads.

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Table 4(a).Maximum Moments in Elastic Semi-Rigid

Frame Analysis (in.-kips)(Two-story one-bay frame , Figure 7a)

ElementNo.

1

2

3

4

5

6

7

8

(1)

Second-

Order(Exact)

257

547

526

813

1497

1497

940

940

(2)

Proposed

220

588

552

818

1431

1431

922

922

(3)

(2)/(1)

0.86

1.07

1.05

1.00

0.96

0.96

0.98

0.98

(4)

Barakat

201

576

557

811

1434

1434

923

923

(5)

(4 ) / ( I )

0.78

1.05

1.06

0.99

0.96

0.96

0.98

0.98

Table 4(b).Maximum Moments in Elastic Semi-Rigid

Frame An alysis (in.-kips)(Two-story one-bay frame, Figure 7b)

ElementNo.

1

2

3

4

5

6

7

8

(1)

Linear-

ElasticRigid

88

324

283

400

562

673

384

400

(2)

Second-Order

ElasticSemi-Rigid

176

278

188

343

728

728

461

461

(3)

Proposed

168

276

187

338

718

718

463

463

(4)

(3)/(2)

0.95

0.99

0.99

0.99

0.99

0.99

1.00

1.00

Column 5 is 0.97. It can be seen that the Barakat method isless conservative in this case.

6.4 Four-Story Two-Bay Fram e with UniformlyDistributed LoadsA four-story two-bay frame as shown in Figure 10 is investigated here for the maximum column moments in both rigidand semi-rigid frames. The sem i-rigid connection of Jenkinsis utilized. The average v alue of Colum n 3 in Table 6 is 1.06,while the average value of Column 6 in Table 6 is 1.01. The

2000M

(in-kip)

1500

1000

500

Proposed

0 - ^

Ki = 684,600 in-kip/radMu = 1,630 in-kipC = 0.35

1 0I

20- n 30

I4 0

Br (rad)xlO-^

Fig. 8. Experimental III-17 connection curves (Barakat, 1991

Table 4(c).Lateral Displacements at Windward

Beam-Column Joints (In.)(Two-story one-bay fram e, Figure 7b)

Node

No.

3

6

Rigid Frame

(1)

LinearElastic(Exact)

0.14

0.23

Semi-Rigid Frame

(2)

Second-Order

(Exact)

0.25

0.50

(3)

Proposed

0.24

0.47

(4)

(3)/(2)

0.96

0.94

proposed method represents reasonably well the second-order effect for rigid and semi-rigid frames. The lateral displacements at windward beam-column joints are summarizedin Table 7. The lateral displacements predicted by the proposed method are less than those of the second-order elasticsemi-rigid frame analysis. However, the lateral displacementspredicted by the proposed method are larger than that of thesecond-order elastic rigid frame analysis.

7. SUMMARY AND CONCLUSIONSSeveral conclusions can be drawn from the present studies:

1. The mo men t-rotation relationships of semi-rigid co nnections as represented by a simple tangent stiffness



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expression are convenient and can lead to a close moment-rotation curve by numerical integration whencompared with test result. Note that only the tangentstiffness is needed in an incremental nonlinear frameanalysis.

2. The proposed method gives close results to that of second-order elastic analysis. It can handle both the uni

formly distributed gravity loads and concentrated loads,and predicts well the drift of unbraced frames.

3. All mean values of the normalized moment ratios arefound close to or slightly greater than one in the proposed method. This shows that the proposed method ismore accurate w hen compared with that of the Barakatmethod. The proposed method gives a reasonable procedure for estimating the approximate P -A column m oments for both rigid and semi-rigid frames.

4. T he notional lateral loads calculation is relatively simpleand straightforward because the tedious determination

2.976 kips

5 456 kips

5 456 kips

7 T 7

. C \ -

biC ^ .

sC ,1

m

TT

0.1088 kips/in

t ,[H W14x30 M

0.1488 kips/in

Bl W21X44 M

0.1488 kips/in

T T y ' ' ^[j\ W21x44 [11

COX

0 0

IT)CO

XCO

0

CO

X00

$

rrf

i

~ r

> 0

300 in

O Node No. 1 I : Ele me nt N o.

of the effective length factor, K, can be avoided. It is asimple and practical method for semi-rigid frame design

REFERENCES1. Barakat, M. and Chen, W. R, Design Analysis of Semi-

Rigid Frames: Evaluation and Implementation, AISC,Engineering Journal, 2nd Qtr., 1991, pp. 55-64.

2. Yee, Y. L. and Melchers, R. E., Mom ent-Rotation Curvesfor Bolted Connections, ASCE, J Struct Eng., 112(3),1986, pp. 615-634

3. Jenkins, W. M., Tong, C. S. and Prescott, A. T., Mom ent-Transmitting End-Plate Connections in Steel Construc-

75 k ips / in3-5 ^ 'p^ i i i i i i i i i i i i n m I

19

7.0 kips14

7.0 kips

7.0 kips

20 21

i i i i i i i i i i i i i i i n15 16]I D

i i i i i i i i i i i i m T10 11

0H I M I I I T T T T

JZ 22 23

o 0.1 5 kips /in1111111111111 i n17 18

J '

0.1 5 kips /inH i i i i i i i i i i i i n T

12 13

0.15 kips / in111111111111 i i m

360 360

1 inIto

Xl o j

îwn| < o

Xl O J

5nto

\xeg

ên

X

\^

W 1 4 X 3 0

W 1 6 X 4 0

W 1 6 X 4 0

W 1 6 X 4 0

360

r^XOJ

$at

XCJ

âtf^Xeg

5

Xeg

$

W 1 4 X 3 0

W 1 6 X 4 0

W 1 6 X 4 0

W 1 6 X 4 0

360

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5

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5

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10/11

Table 5.Maximum Moments in Elastic Semi-Rigid

Frame An alysis (in.-kips)(Three-story one-bay fram e, Figure 9)

ElementNo.

1

2

3

4

5

6

7

8

9

10

11

12

(1)

Second-Order

(Exact)

599

845

152

618

349

659

1181

1212

1082

1082

722

722

(2)

Proposed

575

879

123

669

356

663

1075

1386

1023

1148

714

714

(3)

(2 ) / ( I )

0.96

1.04

0.81

1.08

1.02

1.01

0.91

1.14

0.95

1.06

0.99

0.99

(4)

Barakat

533

836

115

659

367

651

1076

1322

1025

1101

715

715

(5)

(4)/(1)

0.89

0.99

0.75

1.07

1.05

0.99

0.91

1.09

0.95

1.02

0.99

0.99

Table 6.Maximum Column Moments in Elastic

Frame A nalysis (in.-kips)(Four-story two-bay frame, Figure 10)

ElementNo.

1

2

3

4

5

6

7

8

9

10

11

12

Rigid Frame

(1)

Second-Order

(Exact)

534

958

1202

455

656

1101

615

473

1029

702

200

818

(2)

Proposed

632

1066

1296

421

729

1142

603

525

1061

703

221

829

(3)

(2)/(1)

1.18

1.12

1.08

0.93

1.11

1.04

0.98

1.11

1.03

1.00

1.11

1.01

Semi-Rigid

(4)

Second-Order

(Exact)

843

1170

1397

291

596

1044

559

542

996

705

313

846

(5)

Proposed

799

1182

1398

322

698

1061

558

562

1012

672

269

812

(6)

(5)/(4)

0.95

1.01

1.00

1.11

1.17

1.02

0.99

1.04

1.02

0.95

0.86

0.96

tion, and a Proposed Basis for Flush End-Plate Design,Struct. Engrg., 64A(5), 1986, pp. 1 21-132.

4. A zizinamini, A., Bradbum, J. H., and Radziminski, J. B.,Initial Stiffness of Semi-Rigid Steel Beam-to-Column

Connections, /. Construct. Steel Research 5, 1987, pp.71-90

5. Richard, R. M., Hsia, W. K. and Chmielowiec, M., Moment Rotation Curves for Doub le Framing Angles,Materials and Member B ehavior, 1987, 107-121.

6. Richard, R. M., Gillett, P E., Kriegh, J. D. and Lewis, B.A., The Analysis and Design of Single-Plate FramingConnections, AISC, Engineering Journal, 2nd Qtr.,1980, pp. 38-52.

7. Frye, M. J. and Morris, G. A., Analysis of FlexiblyConnected Steel Frames, Can. J. Civ. Eng., 1975, 2(3),pp . 280-291.

8. Ang, K. M. and Morris, G. A., Analysis of Three-Dimen-sional Frames with Flexible Beam-Column Connections, Can. J. Civ Eng., 11, 1984, pp. 245-2 54.

9. Rom stad, K. M. and Subramanian , C. V., Ana lysis ofFrames with Partial Connection Rigidity, ASCE , J.Struct. Div, 96(11), 1970, pp. 2283-2300.

10. Ackroyd, M. H., SimpHfied Frame D esign of Type PR

Table 7.

Lateral Displacments at WindwardBeam-column Joints (in.)(Four-story two-bay frame, Figure 10)

NodeNo.

4

9

14

19

Rigid Frame

(1)

Second-Order

(Exact)

0.27

0.66

0.94

1.11

(2)

Proposed

0.30

0.73

1.04

1.23

(3)

(2)/(1)

1.11

1.11

1.11

1.11

Semi-Rigid Frame

(4)

Second-Order

(Exact)

0.40

1.07

1.61

1.95

(5)

Proposed

0.37

0.95

1.40

1.68

(6)

(5)/(4)

0.93

0.89

0.87

0.86

Construction, AISC, Engineering Journal, 4th Qtr.,1987, pp. 141-46.

11. Goto, Y. and Chen, W. R, Second-Order Elastic Analysis

FOURTH QUARTER/1993 139


11/11

for Frame Design, ASCE , Journal of Structural Engi- 13. Chen, W. F. and Lui, E. M., Stability Design Criteria forneering, Vol. 113, No. 7, 1987, pp. 1501-151 9. Steel Mem bers and Frames in the United States,J. of

12. Chen, W. F. and Lui, E. M., Structural Stability: Theory Constr Steel Research, 5, Great Britain, 1985, pp. 31-74.and Implementation, Elsevier, New York, 1987.

140 ENGINEERING JOU RNA L/AM ERIC AN INSTITUTE OF STEEL CONSTRUCTI

analysis for semi-rigid frame design - lrfd.pdf

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