STABILITY OF PITCHED-ROOF FRAMES WITH SEMI-RIGID CONNECTIONS

IN-PLANE BUCKLING BEHAVIOROF PITCHED ROOF STEEL FRAMESWITH SEMI-RIGID CONNECTIONS

N. Silvestre, A. Mesquita D. Camotim1, L. Silva2

INTRODUCTION

Traditionally, the design and analysis of steel frames assumes either rigid or pinned connections. However, a large number of experimental studies have shown that this hypothesis is not valid for the vast majority of the connections employed in practical situations (1, 2(. In fact, the most commonly used frame connections display a behavior which lies somewhere between rigid and pinned. Such connections are named semi-rigid and influence significantly the frame structural behavior, namely the aspects related to in-plane buckling.

This paper presents and discusses some results concerning the in-plane elastic stability and non-linear buckling behavior of unbraced one bay pitched roof frames with semi-rigid connections (figure 1). Initially, typical connection configurations are displayed and a methodology to evaluate their stiffness values is described and illustrated. Then, earlier

K

j,init

M

M

j,Rd

M

j,Sd

f

Cd

f

Xd

f

Ed

f

Figure 1: Pitched roof frame with semi-rigid connections

(a) geometry (b) loading (c) stability model

results related to the in-plane stability of pitched roof frames (3( are extended to incorporate the influence of the connections semi-rigidity. In particular, analytical expressions are developed to estimate the relevant bifurcation loads. Finally, the paper deals with the consideration of second-order effects (P- type). It is shown that an indirect amplification method, previously proposed and validated in the context of rigid frames (4(, remains valid for semi-rigid frames.

CONNECTIONS IN PITCHED ROOF FRAMES

Introduction. The need to transmit high bending moments and the tight constraints related to the column and rafter layout make the commonly used pitched roof frame connections quite standard. Figure 2(a)-(b) shows typical apex and eaves connection configurations. The presence of extended end-plates (towards the tension zone - dominant load case) and haunches increases the bolts lever arm and, therefore, improves both the steel section capacity and the bolt force distribution.

As the column axial forces are usually small, most column base connections are unstiffened, typically consisting of a thick plate welded to the column section and anchored to a rigid concrete block, by two or more bolts (figure 2(c)). Concerning the column base stiffness, one should mention that it highly depends on the geotechnical conditions and frame span-to-height ratio. As a result, one may have column base connections ranging from nominally pinned to nominally fixed.

Figure 2: Typical connection configurations

(a) apex (b) eaves (c) column bases

In order to determine the in-plane elastic buckling behavior of semi-rigid frames, it is essential to assess the moment-rotation characteristics of every connection, namely the initial stiffness values Kj,init (figure 3). It is well known that an accurate evaluation of a connection stiffness is a rather difficult problem, requiring lengthy and complex finite element calculations. In order to avoid such procedures, a simpler methodology, known in the literature as “component method” [5] and providing the background for Annex J of Eurocode 3 (EC3) [6], is employed here. The joints are replaced by mechanical models, made of extensional springs and rigid links, which adequately incorporate all the relevant aspects concerning the connections deformability (figures 4 and 5).

Figure 3: Connection moment-rotation curve and initial stiffness Kj,init

M

f

k

1

k

2

k

eq

k

3

k

4

k

eff,1

k

eff,2

k

k

6

5

Figure 4: Mechanical model for the eaves and apex connections

Eaves and Apex Connection Stiffness. The mechanical model shown in figure 4 illustrates the application of the “component method” to the eaves and apex connections. All the different connection components contributing to its overall deformability are displayed. Eaves and apex connections are acted by bending moment M, axial force N and shear force V and their stiffness stems from a several individual component behaviors combined. It should be outright mentioned that only the connection M- behavior is dealt with here. Moreover, the (weak) influence of the deformations due to axial and shear forces is neglected.

In eaves connections one identifies the following relevant component behaviors (haunch and rafter flange-web assumed undeformable in compression): (i) unstiffened column web shear k1, (ii) unstiffened column web compression k2, (iii) column web tension k3, (iv) column flange flexure k4, (v) end-plate flexure k5 and (vi) bolt tension k6. Then, the initial stiffness value Kj,init may be evaluated using the equations

c

2

eq

init

,

j

k

z

E

K

×

×

=

(1)

eq

2

1

c

k

1

k

1

k

1

k

1

+

+

=

(2)

eq

n

1

r

r

r

,

eff

eq

z

h

k

k

å

×

=

=

(3)

å

å

=

r

r

r

,

eff

r

2

r

r

,

eff

eq

h

k

h

k

z

(4)

å

=

i

r

,

i

r

,

eff

k

1

1

k

, (5)

where hr is the distance between bolt-row r and the connection center of compression, and the equivalent stiffness coefficient keq takes into account the combined influence of the different component behaviors which are grouped in the various bolt-rows in tension. The effective stiffness coefficient for bolt-row r, keff,r, takes into account the stiffness coefficients of the corresponding i relevant components (k3-k6 in this case) and the distance zeq represents an equivalent lever arm.

A similar reasoning may be applied to apex connections. The haunch and rafter flanges-webs are assumed undeformable in compression (small pitches) and the component behaviors are: (i) extended end-plate flexure k5 and (ii) bolt tension k6. Equations (1)-(5) also remain valid.

Column Base Connection Stiffness. Due to the substantial influence of the axial force, the same column base may show rather different rotational characteristics. Two phenomena are involved, namely the deformability of the links (i) between the steel column and the concrete base and (ii) between the concrete foundation and the soil [7]. In the first case, the rotation behavior is represented by a M- curve, which depends on the connection bending moment-axial force ratio. In the second case, however, the M- curve (rotation of the concrete block within the soil) must take into account the soil settlement due to the column compression.

Since no available information exists concerning the concrete-to-soil link, this paper deals solely with the deformability taking place between the steel column and the concrete foundation. Figure 5 shows the corresponding mechanical model, which accounts for the interaction between bending moment and axial force.

The model components include (i) extensional springs to simulate the column cross-section deformation (tension and compression), (ii) extensional springs to simulate the anchor bolts and base plate deformation (tension), (iii) extensional springs to simulate the concrete under the base plate (compression) and (iv) rotational springs to simulate the plastic deformation associated to an eventual development of a yield line in the extended part of the plate (connection compressive zone). It should also be mentioned that the use of this mechanical model involves the performance of an iterative numerical procedure.

Yield

line

Anchors

Plate to

concrete

contact

Undeformed

position

Column

deformation

M

N

Figure 5: Mechanical model of the column base connection

Illustrative Numerical Results. In order to illustrate the application of the “component method”, the initial stiffness values of typical pitched roof frame connections were evaluated. The results obtained are shown in tables 1(a) (apex – 2 bolts without haunch and 3 bolts with haunch), 1(b) (eaves) and 1(c) (column base - 2 and 4 bolts (7() and correspond to IPE 180-330 rafters and a HEB160 column.

Table 1: Connections initial stiffness values

(a) apex (b) eaves (c) column base

(a)

RaftersBolt rows

Kj,init

(kNm/rad)

Bolt rowsKj,init

(kNm/rad)

IPE 180

2 M 16

16380

3 M 16

46294

IPE 200

2 M 16

21770

3 M 16

59954

IPE 220

2 M 16

27912

3 M 16

70378

IPE 240

2 M 16

35509

3 M 16

87774

IPE 270

2 M 16

45545

3 M 16

112894

IPE 300

2 M 16

60023

3 M 16

138193

IPE 330

2 M 16

75641

3 M 16

173496

(b)

Column / RafterBolt rowsKj,init

(kNm/rad)

HEB160 / IPE 1803 M 16

7963

HEB160 / IPE 200

3 M 16

9865

HEB160 / IPE 220

3 M 16

13220

HEB160 / IPE 240

3 M 16

15259

HEB160 / IPE 270

3 M 16

19154

HEB160 / IPE 300

3 M 16

24407

HEB160 / IPE 330

3 M 16

28808

(c)

Column

Bolts

Kj,init

(kNm/rad)

HEB160

2 M 16

750 - 5000

HEB160

4 M 16

3500 - 8000

Concerning the apex and eaves connections, it is observed that, depending on the rafter, Kj,init varies from 16380 to 173496 kNm/rad (apex) and from 7963 to 32525 kNm/rad (eaves). The column base results show the substantial influence of the axial force on the stiffness. In fact, Kj,init varies from 750 to 5000 kNm/rad (2 bolts) and from 3500 to 8000 kNm/rad (4 bolts) when N changes from 100 to 1000 kN.

FRAME STABILITY BEHAVIOR

Introduction. It is assumed that the frame loading consists only of compressive forces acting on the columns and rafters, related by the parameter (figure 1(c)). It was shown before (3( that the in-plane elastic stability behavior of rigid frames is conditioned by two buckling modes, one anti-symmetric (ASM) and the other symmetric (SM), both involving lateral (sway) displacements at the column tops (figure 6(a)).

ASM

SM

ASM

N

SM

c.0

N

ASM

c.0

N

ASM

r.0

N

SM

r.0

N

c

N

r

1

g

a

1

a

2

0

a

3

SM

a

3

>

a

2

>

a

1

>0

(a) (b)

Figure 6: (a) Buckling modes (b) Variation of (Nc-Nr)b with and

An extensive parametric study unveiled the essential aspects of the in-plane elastic stability of one bay symmetric pitched roof rigid frames (fixed and pinned-base), which are shown in figure 6(b) (8(. Concerning the critical load cr (scaled to the columns Euler load PEc), the variation of the combination (Nc-Nr) at bifurcation with the inclination and the loading ratio , for given RI and RL, was shown to display the following characteristics (N EQ \o(\s\up4(ASM),\s\do2(c.0 )), N EQ \o(\s\up4(SM),\s\do2(c.0)), N EQ \o(\s\up4(ASM),\s\do2(r.0 )) and N EQ \o(\s\up4(SM),\s\do2(r.0)) - bifurcation loads for =0,():

(i)The ASM behavior (solid line) does not depend on the value of .

(ii)The SM behavior (dashed lines) depends on the value of , as the frame stability is controlled essentially by the rafters.

(iii)For low values, cr= EQ \o(\s\up4(ASM),\s\do2(b )) (no dependence on ).

(iv)For high values, cr= EQ \o(\s\up4(ASM),\s\do2(b )) or cr= EQ \o(\s\up4(SM),\s\do2(b )) (dependence on

(v)The variation of cr with involves either both the ASM and SM (=0(; 1; 2) or only the ASM (=3).

Semi-Rigid Connections. When performing a linear stability analysis, one looks for equilibrium configurations in the vicinity of the fundamental path. Therefore, in the case of semi-rigid frames, such an analysis only requires the knowledge of the connections initial stiffness values Kj,init (the shape of the full M- curve is not involved).

Let us now consider the pitched roof frame depicted in figure 1, with symmetrically located semi-rigid connections in all joints (figure 1(a)). The initial stiffness values of the three connection types are designated as K1 (column base), K2 (eaves) and K3 (apex) and, out of convenience, one defines the non dimensional parameters S1, S2 and S3 (Sj=KjLc/EIc). The frame stability behavior continues to be conditioned by two buckling modes (ASM and SM), with shapes similar to the ones shown in figure 6(a), and it still exhibits the characteristics displayed in figure 6(b). The conclusions drawn with respect to rigid frames remain qualitatively valid in the presence of semi-rigid connections. As the frame overall stiffness is reduced, the bifurcation loads EQ \o(\s\up4(ASM),\s\do2(b )) and EQ \o(\s\up4(SM),\s\do2(b )) obviously decrease and, moreover, the nature of the critical mode may vary. The aim of this study is to estimate the order of magnitude of these changes and also to appraise the relative importance of each connection type.

Naming the bifurcation loads of the rigid (Sj=() and semi-rigid (at least one Sj(() frames as ( and S, table 2 shows, for a frame with a commonly used geometry (RI=1.5, RL=3 ,=6() and values produced by a single uniformly distributed vertical span load (figures 1(b)-(c)), the variation of the ratios EQ \o(\s\up4(ASM),\s\do2(S ))/ EQ \o(\s\up4(ASM),\s\do2(( )), EQ \o(\s\up4(SM),\s\do2(S ))/ EQ \o(\s\up4(SM),\s\do2(( )) and EQ \o(\s\up4(ASM),\s\do2(S ))/ EQ \o(\s\up4(SM),\s\do2(S )) with the Sj values. Assuming a test frame with HEB160 columns, the connection stiffness numerical results presented earlier correspond to the Sj ranges: 0.9(S1(9.2, 9.1(S2(37.3 and 18.8(S3(198.9. On the basis of these results, the following realistic Sj values were considered: S1=0, 1, 5, (; S2=10, ( and S3=20, (. It is possible to observe that:

Table 2: Stability results of semi-rigid frames: variation with Sj

EQ \f(l\o(\s\up4(ASM),\s\do2(S )),l\o(\s\up4(ASM),\s\do2(( )))

EQ \f(l\o(\s\up4(SM),\s\do2(S )),l\o(\s\up4(SM),\s\do2(( )))

EQ \f(l\o(\s\up4(ASM),\s\do2(S )),l\o(\s\up4(SM),\s\do2(S )))

S1

S2

S3=20

S3=(

S3=20

S3=(

S3=20

S3=(

S3=20

S3=(

0

10

0.879

0.876

0.544

0.545

1.327

1.355

0.440

0.431

(

0.891

0.889

0.555

0.555

1.358

1.386

0.439

0.430

1

10

1.078

1.075

1.028

1.030

1.161

1.185

0.950

0.933

(

1.085

1.082

1.024

1.025

1.195

1.219

0.920

0.903

5

10

1.339

1.335

1.071

1.074

1.024

1.044

1.123

1.104

(

1.338

1.335

1.081

1.083

1.058

1.079

1.096

1.077

(

10

1.563

1.559

0.981

0.984

0.949

0.967

1.110

1.092

(

1.557

1.553

0.998

1.000

0.981

1.000

1.092

1.074

(i)As the results for S3=20 and S3=( are practically identical, an apex connection with S3=20 may be treated as rigid.

(ii) As the results for S2=10 and S2=( differ only slightly (maximum difference of (3%, for EQ \o(\s\up4(SM),\s\do2(S ))/ EQ \o(\s\up4(SM),\s\do2(( ))), eaves connections with S2=10 may be treated as almost rigid.

(iii) As S1 varies from 0 to (:

(iii.1) increases substantially (higher rafter compression) and, as a result, the S/( ratios reach values higher than 1 (i.e., semi-rigid column bases may improve the frame stability).

(iii.2) EQ \o(\s\up4(SM),\s\do2(S ))/ EQ \o(\s\up4(SM),\s\do2(( )) steadily decreases (30%.

(iii.3) EQ \o(\s\up4(ASM),\s\do2(S ))/ EQ \o(\s\up4(ASM),\s\do2(( )) increases (100% for 0(S1(5 ((90% for 0(S1(1) and then decreases (10% (5(S1(().

(iii.4)As a result of (iii.2-3) , EQ \o(\s\up4(ASM),\s\do2(S ))/ EQ \o(\s\up4(SM),\s\do2(S )) varies, for S2=S3=(, between 0.430 (S1=0) or 0.903 (S1=1) and 1.077 (S1=5). This means that the critical mode changes from AS to S.

In order to assess the influence of the column and rafter geometry, the variation of EQ \o(\s\up4(ASM),\s\do2(S ))/ EQ \o(\s\up4(ASM),\s\do2(b* )), EQ \o(\s\up4(SM),\s\do2(b ))/ EQ \o(\s\up4(SM),\s\do2(b*))and EQ \o(\s\up4(ASM),\s\do2(b ))/ EQ \o(\s\up4(SM),\s\do2(b )) with RI and RL is now investigated. Designating the reference frame (RI=1.5, RL=3 ,=6() bifurcation loads as b*, table 3 shows, for two of the previous connection stiffness combinations and similarly varying values, the variation of the three bifurcation ratios.

Table 3: Stability results of semi-rigid frames: variation with RI and RL

S1=S2=S3=(

S1=1 S2=10 S3=20

RI

RL

EQ \f(l\o(\s\up4(ASM),\s\do2(b )),l\o(\s\up4(ASM),\s\do2(b( )))

EQ \f(l\o(\s\up4(SM),\s\do2(b )),l\o(\s\up4(SM),\s\do2(b( )))

EQ \f(l\o(\s\up4(ASM),\s\do2(b )),l\o(\s\up4(SM),\s\do2(b )))

EQ \f(l\o(\s\up4(ASM),\s\do2(b )),l\o(\s\up4(ASM),\s\do2(b( )))

EQ \f(l\o(\s\up4(SM),\s\do2(b )),l\o(\s\up4(SM),\s\do2(b( )))

EQ \f(l\o(\s\up4(ASM),\s\do2(b )),l\o(\s\up4(SM),\s\do2(b )))

2

0.956

3.732

4.126

0.975

0.675

2.340

3.854

0.577

1

3

1.459

1.436

1.496

1.035

1.027

1.356

1.435

0.898

4

1.945

0.670

0.754

0.958

1.370

0.726

0.714

0.967

2

1.010

2.752

2.830

1.048

0.712

1.876

2.791

0.639

1.5

3

1.553

1.000

1.000

1.074

1.078

1.000

1.000

0.950

4

2.083

0.454

0.504

0.973

1.440

0.515

0.488

1.004

2

1.052

2.203

2.132

1.114

0.736

1.597

2.177

0.697

2

3

1.634

0.761

0.745

1.100

1.115

0.806

0.763

1.004

4

2.202

0.340

0.377

0.973

1.497

0.403

0.368

1.039

The following set of geometrical parameter values was considered: RI=1, 1.5, 2; RL=2, 3, 4. One observes that:

(i) The influence of RL clearly surpasses that of RI.

(ii) As RI varies from 1 to 2:

(ii.1) increases 9-13% in a fairly proportional fashion.

(ii.2) EQ \o(\s\up4(ASM),\s\do2(b ))/ EQ \o(\s\up4(ASM),\s\do2(b* )) decreases 40-50% (S1=() and 30-45% (S1=1).

(ii.3) EQ \o(\s\up4(SM),\s\do2(b ))/ EQ \o(\s\up4(SM),\s\do2(b*)) decreases 45-50%.

(ii.4) EQ \o(\s\up4(ASM),\s\do2(b ))/ EQ \o(\s\up4(SM),\s\do2(b )) increases 2-14% (S1=() and 7-20% (S1=1). The variation magnitude decreases when RL varies from 2 to 4.

(iii) As RL varies from 2 to 4:

(iii.1) increases (100% in a fairly proportional fashion.

(iii.2) EQ \o(\s\up4(ASM),\s\do2(b ))/ EQ \o(\s\up4(ASM),\s\do2(b* )) decreases (85% (S1=() and 70-75% (S1=1).

(iii.3) EQ \o(\s\up4(SM),\s\do2(b ))/ EQ \o(\s\up4(SM),\s\do2(b*)) decreases 81-83%.

(iii.4) EQ \o(\s\up4(ASM),\s\do2(b ))/ EQ \o(\s\up4(SM),\s\do2(b )) increases 50-70% (S1=1) and varies 8-15% without a consistent pattern (S1=().

Overall, the EQ \o(\s\up4(ASM),\s\do2(b ))/ EQ \o(\s\up4(SM),\s\do2(b )) values range from 0.58 (S1=1; RI=1; RL=2) to 1.11 (S1=(; RI=2; RL=2). Although no clear pattern can be identified concerning the variation of the critical mode nature, the results show that the SM is more relevant in fixed-base frames.

Approximate Analytical Expressions. In order to avoid the need to perform linear stability analyses, easy-to-use analytical expressions to estimate EQ \o(\s\up4(ASM),\s\do2(b )) and EQ \o(\s\up4(SM),\s\do2(b )) were previously developed, in the context of rigid frames (3(. The bifurcation loads are obtained from the equation

2

/

1

2

0

.

r

L

2

0

.

c

b

RR

1

C

-

l

ú

ú

û

ù

ê

ê

ë

é

÷

÷

ø

ö

ç

ç

è

æ

r

g

+

÷

÷

ø

ö

ç

ç

è

æ

r

=

l

, (6)

where R=RIRL, c.0 and r.0 are related to the bifurcation loads when only the columns or rafters are compressed (expressions for their estimation are also available) and b is scaled to the columns Euler load PEc. The form of equation (6) stems from the fact that all the curves shown in figure 6(b) are almost elliptic, C being an adjustment factor (0.9(C(1.1). An extensive parametric study (8( showed the formulas to be valid and to lead to accurate results for all commonly built frames.

Similar analytical expressions, accounting for semi-rigid connections, are now presented. It was found that the form of equation (6) may be retained, provided that C, c.0 and r.0 are evaluated by the formulas given in table 4 (RH=RLsin), which contain the stiffness values Sj. Making Sj(( or 0, they are also valid for rigid or pinned connections.

Table 4: Approximate analytical expressions to evaluate EQ \o(\s\up4(ASM),\s\do2(b )) and EQ \o(\s\up4(SM),\s\do2(b ))

ASM

(

)

(

)

(

)

(

)

2

2

1

2

1

2

1

2

1

0

.

r

S

S

6

6

R

4

.

1

S

S

S

S

6

6

R

2

S

S

÷

÷

ø

ö

ç

ç

è

æ

+

+

+

+

+

+

=

r

SM

1

C

=

l

(

)

(

)

(

)

(

)

(

)

(

)

(

)

(

)

(

)

2

2

1

1

3

2

3

2

H

3

2

1

1

3

2

3

2

H

3

2

1

0

.

r

R

S

S

12

5

48

S

16

S

S

16

R

24

S

S

4

R

R

8

S

S

S

24

S

8

S

S

8

R

24

S

S

2

R

R

8

S

S

S

÷

÷

ø

ö

ç

ç

è

æ

+

+

+

+

+

+

+

+

-

+

+

+

+

+

+

+

=

r

In order to assess the accuracy and validity of the proposed formulas, the exact and approximate bifurcation loads were compared for all the 32 frames considered in tables 2-3. The results yielded by the formulas were found to be quite accurate and mostly conservative (53 out of 64). Concerning the errors, one has (4%(MAS((1% and (6%(MS((2%.

SECOND-ORDER EFFECTS

Let us now turn our attention to the non-linear buckling behavior of semi-rigid unbraced pitched roof frames acted by horizontal and vertical loads (figures 1(a)-(b)), namely the consideration of second-order effects of the P- type.

As symmetric vertical distributed loads acting on symmetric pitched roof frames lead to significant displacements at the column tops, three first-order displacement(u)/moment(M) components may be identified: (i) non-sway (NS) and (ii) symmetric sway (SS), both due to the vertical load, and (iii) anti-symmetric sway (AS), due to the horizontal load. The similarity between (i) the first-order deformed configurations u EQ \o(\s\up4(I ),\s\do2(SS)) and u EQ \o(\s\up4(I ),\s\do2(AS)) and (ii) the S and AS buckling mode shapes led to the formulation, in the context of rigid pitched roof frames, of an indirect amplification method to incorporate the P- effects (4(. Two amplifications are involved and the second-order displacements and moments are calculated by means of

I

AS

1

ASM

Sd

I

SS

1

SM

Sd

I

NS

II

ap

u

V

V

1

u

V

V

1

u

u

-

-

÷

÷

ø

ö

ç

ç

è

æ

-

+

÷

÷

ø

ö

ç

ç

è

æ

-

+

=

(7)

I

AS

1

ASM

Sd

I

SS

1

SM

Sd

I

NS

II

ap

M

V

V

1

M

V

V

1

M

M

-

-

÷

÷

ø

ö

ç

ç

è

æ

-

+

÷

÷

ø

ö

ç

ç

è

æ

-

+

=

, (8)

where VSd, VASM e VSM are the design and bifurcation values of the vertical load. The relative magnitude of the two amplified terms depends on the combined values of the ratios (i) (VSM/VASM) and (ii) (u EQ \o(\s\up4(I ),\s\do2(SS))/u EQ \o(\s\up4(I ),\s\do2(AS))) or (M EQ \o(\s\up4(I ),\s\do2(SS))/M EQ \o(\s\up4(I ),\s\do2(AS))). The method was shown to yield accurate results for a wide range of fixed and pinned-base frames (8(.

Provided that the connections display a linear behavior (Mj.Sd inside the linear portion of the M- curve - figure 3), the fundamentals of the above method apply also for semi-rigid frames. It is, therefore, natural to expect similarly accurate results in this case. This statement is confirmed by the results shown in table 5, which consist of the ratios between the approximate and exact maximum second-order moments and displacements (top of right column) and correspond to VSd=0.25Vcr and HSd=0.05VSd. It is observed that, for all the 20 frames considered (S1=0, 1, 5, (; S2=S3=(; RI=1, 1.5, 2; RL=2, 3, 4), the approximate results are rather accurate (max(4%) and always conservative.

Table 5: Relation between approximate and exact second-order results

M EQ \o(\s\up4(II),\s\do2(ap))/M EQ \o(\s\up4(II),\s\do2(ex))

u EQ \o(\s\up4(II),\s\do2(ap))/u EQ \o(\s\up4(II),\s\do2(ex))

RI

RL

S1=0

S1=1

S1=5

S1=(

S1=0

S1=1

S1=5

S1=(

1.5

3

1.02

1.01

1.01

1.01

1.02

1.01

1.02

1.02

1.5

2

1.00

1.00

1.01

1.00

1.01

1.02

1.02

1.02

1.5

4

1.03

1.02

1.01

1.01

1.04

1.03

1.02

1.03

1

3

1.01

1.00

1.01

1.01

1.02

1.02

1.02

1.03

2

3

1.02

1.01

1.01

1.01

1.02

1.02

1.01

1.02

CONCLUDING REMARKS

Some results concerning the in-plane elastic stability and non-linear buckling behavior of unbraced one bay pitched roof frames with semi-rigid connections were presented and discussed.

First, typical connection configurations were shown and a simple methodology, proposed in the revised version of Annex J of EC3, was used to evaluate their initial stiffness values. While most commonly employed apex and eaves connections were, naturally, found to be almost rigid, the column bases displayed a wide stiffness value range.

Next, it was investigated how the connections semi-rigidity influences the frame elastic in-plane stability. Different geometries were dealt with and it was found that the commonly used apex and eaves connections may be treated as rigid and that the column base stiffness strongly affects the frame stability behavior. In particular, even small stiffness changes may alter the nature of the critical buckling mode. In order to enable an easy estimate of the two relevant frame buckling loads, approximate formulas were developed, which yield rather accurate and mostly conservative results.

Finally, it was shown that, for linear semi-rigid connections, an existing indirect amplification method accurately takes into account the second-order P- effects. The incorporation of the buckling load formulas into this method will certainly lead to an efficient design tool. One last word to mention that the authors are presently working on the buckling behavior of semi-rigid frames with non-linear connections.

REFERENCES

(1(Chen, W.F., Goto, Y. and Liew, J. – Stability Design of Semi-Rigid Frames, John Wiley, 1996.

(2(Semi-Rigid Connections in Steel Frames, Council on Tall Buildings and Urban Habitat, McGraw-Hill, 1993.

(3(Silvestre, N., Camotim, D. and Corrêa, M. - “On the Design and Safety Checking of Unbraced Pitched Roof Steel Frames”, Journal of Constructional Steel Research, Vol. 46, nº1-3, 1998, 328-329 (full paper in the CD-ROM – paper # 188).

(4(Silvestre, N. and Camotim, D. – “Second-Order Effects in Pitched Roof Steel Frames”, Proceedings of the SSRC Annual Technical Session and Meeting, Toronto, 1997, 85-98.

(5(Weynand, K., Jaspart, J-P. and Steenhuis, M. - “The Stiffness Model of the Revised Annex J of Eurocode 3”, Connections in Steel Structures III - Proceedings of the 3rd International Workshop on Connections, Eds. R. Bjorhovde, A. Colson, R. Zandonini, Pergamon, 1996, 441-452.

(6(“Revised Annex J of Eurocode 3”, Document CEN/TC 250/SC 3 - N 419 E, CEN, 1994.

(7(Guisse, S., Vandegans, D. and Jaspart, J-P. - “Application of the Component Method to Column Bases – Experimentation and Development of a Mechanical Model for Characterization”, Report MT 195, CRIF, Liège, 1996.

(8(Silvestre, N. – “Stability and Second-Order Effects in Pitched Roof Steel Frames”, M.A.Sc. Thesis (in portuguese), 1997.

Ic, Lc

Ic, Lc

Ir, Lr

Ir, Lr

1

2

3

2

1

=Nr/Nc

RL=Lr/Lc

RI=Ic/Ir

Nr

Nc

Nr

Nr

Nr

Nc

Nc

Nc

Semi-rigid

connection

(a)

(c)

(b)

VSd/L

(c)

(b)

(a)

HSd

L

HSd

� Civil Engineering Department, Technical University of Lisbon,

Instituto Superior Técnico, Av. Rovisco Pais, 1096 Lisboa Codex, Portugal

� Civil Engineering Department, University of Coimbra, 3049 Coimbra Codex, Portugal

_967361104.unknown

_967366619.unknown

_967366710.doc

k

k

k

3

k

4

k

eff,1

k

eff,2

M

k

1

k

2

k

eq

6

5

_967366765.doc

N

M

Column

deformation

Undeformed

position

Plate to

concrete

contact

Anchors

Yield

line

_967361138.unknown

_967361150.unknown

_967361120.unknown

_966955499.doc

SM

ASM

_967196448.unknown

_967290979.doc

ASM

� EQ \o(\s\up(SM),\s\do(c.0 ))�

� EQ \o(\s\up(ASM),\s\do(c.0 ))�

� EQ \o(\s\up(ASM),\s\do(r.0 ))�

� EQ \o(\s\up(SM),\s\do(r.0 ))�

c

r

SM

3>2>1>0

_967196492.unknown

_967184413.unknown

_967196019.unknown

_967184321.unknown

_966758470.unknown

_966950761.doc

Kj,init

M

Mj,Rd

Mj,Sd

(Cd

(Xd

(Ed

(

_966760012.unknown

_966758469.unknown

_966758466.unknown