1

PRE-DESIGN OF SEMI-RIGID JOINTSIN STEEL FRAMES

Martin Steenhuis (1), Nol Gresnigt (2), Klaus Weynand (3)

Keywords: steel, joints, frames, semi-rigid, pre-design, stiffness,classification

Abstract The response of steel frames is influenced by the mechanical properties of the joints(strength, stiffness, rotation capacity). In practice, the joints are usually considered as either rigid orpinned. Research has shown that frames with semi-rigid joints can be more economical than frameswith rigid or pinned joints. Computer programs are available to determine the joint properties. Theseprograms require a full description of the geometry of the joint. In a pre-design, these data are notyet available. Therefore, for the pre-design stage there is a need for simple rules to estimate themechanical properties of the joints. This paper presents such simple rules. Comparisons to theEurocode 3 are made. This paper also includes design examples.

1. Introduction

In various European countries, traditionally, two parties are responsible for the design of steelframes: the engineer designs the beams and columns and the steel fabricator designs the connections.In this design practice the engineer specifies the mechanical requirements of joints. The steelfabricator designs the joints to fulfil these requirements. The fabricator also considers manufacturingaspects.

Elastic global frame analysis is commonly used in Europe. In this analysis, pinned and rigiddesign of joints (see figure 1) leads to a limited data flow between engineer and steel fabricator. Incase of pinned joints the fabricator needs to design the joints like pins. In case of rigid joints, thefabricator needs to design the joints sufficiently rigid. In both cases, the joints should be capable oftransmitting the forces determined in the global frame analysis. Pinned and rigid joints, however, arenot necessarily economical [1].

On the contrary, the more economical semi-rigid design leads to intensive data flow betweenengineer and steel fabricator. The global frame analysis (task of engineer) requires the stiffnesses ofthe joints, which should be based on the actual geometrical layout of the joint (task of fabricator).This intensive communication (and the consequent mixture of responsibilities) between engineer andfabricator hinders the acceptance of semi-rigid design.

Pinned Joint Rigid Joint Semi Rigid Joint

figure 1: Examples of pinned, rigid and semi-rigid joints

Published in Proceedings of the Second State of the Art Workshop on Semi-Rigid Behaviour of CivilEngineering Structural Connections, ed. by F. Wald, COST C1, Prague, Czech Republic, 1994, pp 131-140.

(1) TNO Building and Construction Research, P.O. Box 49, 2600 AA Delft, The Netherlands(2) TU Delft, P.O. Box 5048, 2600 AA Delft, The Netherlands(3) RWTH Aachen, Mies-van-der-Rohe Straße 1, 52074 Aachen, Germany

2

This paper presents in chapter 2. a procedure for frame design with semi-rigid joints, whichrespects the existing share of responsibilities between engineer and steel fabricator. The advantagesof this procedure are that: 1) semi-rigid design can be applied without increase of communicationbetween designer and fabricator compared to the traditional situation, 2) the fabricator has a certainfreedom to design the joints in accordance with the available production technologies and materialsin stock. Of course, this procedure is also applicable when the engineer designs both frame and joints.Chapter 3. of this paper concerns the determination of the stiffness1 of joints in the pre-design phase.Chapter 4 treats the check of the stiffness requirements during the design of the joints. Chapters 5 and6 show some worked examples.

2. Design procedure

Traditionally, the design process of a steel frame consists of 7 logical steps (see table 1). Inthe modelling phase (step 1), the engineer models the joints as pinned or rigid. Pinned joints shouldbe capable of transmitting the forces calculated in design, without developing significant momentswhich might adversely affect the beams or columns in the frame. In elastic frame analysis, a rigidjoint has no influence on the distribution of internal forces and moments in the frame, nor on itsoverall deformation.

table 1: Design process of a steel frame with elastic global frame analysis, with rigid or pinned joints

1. Mechanical modelling of the frame in the building including modelling joints as pinned or rigid.2. Estimation of loads.3. Pre-design of beams and columns.4. Determination of forces and displacements in the frame.5. Check of beams and columns in limit state conditions.6. If required, adjustment of beams and columns (continue with step 4).7. Design of joints based on strength ( pinned or rigid).

The loads and the stiffness of beams and columns are input for the frame analysis (step 4).The deflections of the frame and the force distributions are output. The member sizes and the forceswhich should be transmitted by the joints are the starting point for the design of joints (step 7). Thepurpose of this design is to find a layout capable of transmitting the forces between the beam and thecolumn. Additionally, in case of pinned joints the fabricator should verify that no significantmoments develop in the joints. In case of rigid joints, the fabricator should verify that the joint issufficiently rigid.

Computer programs exist which support the design of joints [2, 3] (step 7) according toEurocode 3 [4]. These programs are essential in the design process because they enlightens the taskof the fabricator dramatically. They require the geometrical layout of the joint as input and give thestrength, stiffness and rotational capacity as output. A user of a program designs the joints in aninteractive way by trial and error. For example, the user first tries a simple solution. If this solutiondoesn't satisfy the strength criteria, the user will improve the design by adjusting the lay out of thejoint. The process ends when the design is satisfactory. Some programs are also capable to checkwhether a joint is rigid or not.

This paper proposes two modifications to make the procedure of table 1 suitable for semi-rigiddesign. This leads to a process as given in table 2.

1 When this paper speaks about stiffness of a joint, it is meant the elastic stiffness

3

table 2: Design process of a steel frame with elastic global frame analysis with semi-rigid joints

1. Mechanical modelling of the frame in the building.2. Estimation of loads.3. Pre-design of beams and columns and assessment of stiffness of joints.4. Determination of forces and displacements in the frame (stiffness of the joint included in the analysis)5. Check of beams and columns in limit state conditions.6. If required, adjustment of beams and columns (continue with step 4).7. Design of semi-rigid joints based on required strength and stiffness.

I Inclusion of stiffness in the frame analysis

Cunningham and Taylor proposed to assume a certain stiffness for the joints in the frame analysisrather than to take the joints as either rigid or pinned [6]. This stiffness is taken as a function of thebeam stiffness and the recommended fixity factor (the fixity factor is defined as the relation betweenthe stiffness of the joint and the beam). This stiffness is an estimate of the actual stiffness of thejoints to be designed in step 7.

This paper basically proposes, as an extension to the work of Cunningham and Taylor, toassess the stiffness of the joints relating to the beam and column properties and the type of the jointsin the pre-design phase. This is explained more in detail in chapter 3. These stiffnesses can be used inthe global frame analysis (step 4).

II Verification of stiffness in the design of joints.

It should be verified in step 7 that the stiffness of the joint designed by the fabricator is inreasonable agreement with the stiffness included in the global frame analysis. This replaces theverifications for rigid and pinned joints in traditional design. Chapter 4 gives some rules to carry outthis verification. These rules are based on the same philosophy as the classification diagrams ofEurocode 3. In combination with a computer program, these rules can easily be applied.

3. Prediction of the stiffness of joints in the pre-design stage

In the pre-design phase of a structure, it is difficult to assess the stiffnesses of the (semi-rigid)joints, because the joints have not been designed yet. To overcome this problem, some simplifiedformulae have been derived based on Eurocode 3 Annex J (revised) [5]. With help of these formulae,a designer can determine the stiffness of a joint by selecting the configuration.

These formulae assume some fixed choices for the connection design. These are for end-plated connections:• the connection has two bolt rows in the tension zone.• the bolt diameter is approximately 1.5 times the thickness of the column flange;• the location of the bolt is as close as possible to the root radius of the column flange, the beam

web and flange (about 1.5 times the thickness of the column flange);• the end-plate thickness is similar to the column flange thickness;For European I and H sections the following rules are valid:• the root radius is about the same size as flange thickness;• the web is about 0.6 times the flange thickness;• the clear depth of the web is about 15 times the flange thickness.

As an example, a simplified formula is derived for an un-stiffened extended end-plateconnection in a single sided joint configuration. In figure 2, first the stiffness factors according toAnnex J (revised) are calculated.

4

Column web in shear:

For simplicity we take hcht

equal to 0.8. This is a reasonable assumption in un-braced frames.

0.38 Av,c

ht ≈ 0.38

0.8 hc tw,cht

≈ 0.38 0.8 hc 0.6 tf,c

ht ≈ 0.18 tf,c

Column web in compression and tension:

We assume that in an un-stiffened joint with an extended end-plate, the deformation of the column web intension is similar to the deformation of the column web in compression. The effective width beff isapproximately:

beff = tf,b + 2te + 5 (rc + tf,c) ≈ 12 tf,c

0.7 beff tw,c

d ≈ 0.7 12 tf,c 0.6 tf,c

15 tf,c ≈ 0.33 tf,c

Column flange and end-plate in bending

0.85 leff t

3f,c

m3 ≈ 0.85 12 tf,c t

3f,c

1.53 t3f,c

≈ 3.0 tf,c

Two bolt rows in tension

2•1.6 Ab,s

lb ≈ 3.2

0.75 π/4 1.52t2f,c

3 tf,c ≈ 1.4 tf,c

mtf,b

tf,c tphc

ht

dn

The stiffness of the joint is:

Sj ≈ E h

2t tf,c

10.18 +

10.33 +

10.33+

13.0 +

13.0 +

11.4

≈ E h

2t tf,c

5.6 + 3 + 3+ 0.33 +0.33 + 0.7 ≈ E h

2t tf,c

13,0

In this example, the column web in shear contributes most to the flexibility of the joint.

figure 2: Derivation of a simplified stiffness formulae for an extended end plate connection

Table 3 contains formulae for different configurations. These formulae can be derived similarto the previous example. Stiffening plates have a great influence on the stiffness.

The formulae in table 3 contain only two parameters: ht and tf,c. Parameter ht is the distancebetween the point of compression and the centre of the tension zone. In an extended end-plateconnection with two bolt rows, this distance is approximately equal to the beam depth. For the samejoint with haunch, ht is equal to the sum of the beam depth and the haunch height. tf,c is the thicknessof the column flange.

5

table 3: Formulae for the approximation of the stiffness for beam-to-column configurationsConfiguration Sj

Extended end-plate, single sided andun-stiffened E h

2t tf,c

13,0

Extended end-plates, double sided,un-stiffened and symmetrically E h

2t tf,c

7,4

Extended end-plate, single sided,stiffened in tension and compression E h

2t tf,c

8,3

Extended end-plates, double sidedand stiffened in tension andcompression, symmetrically

E h2t tf,c

2,7

Extended end-plate, single sided andMorris stiffener E h

2t tf,c

2,7

Flush end-plate, single sided andcover plate E h

2t tf,c

11,5

Flush end-plates, double sided,cover plate and symmetrically E h

2t tf,c

6,1

Welded joint, single sided and un-stiffened E h

2t tf,c

11,6

Welded joints, double sided un-stiffened and symmetrically E h

2t tf,c

6,0

Welded joint, single sided, stiffenedin tension and compression E h

2t tf,c

5,6

Welded joints, double sidedstiffened in tension, compressionand symmetrically

∞

6

4. Required stiffness

Eurocode 3 gives two diagrams to classify joints according to their stiffness (pinned, semi-rigid, rigid): one for braced and one for un-braced frames. For braced frames Eurocode 3 says that ajoint may be regarded as rigid if:

Sj ≥ 8 E Ib

l .

The background of this rule for rigid joints is that the bearing capacity of the frame doesn't drop withmore than 5% due to the difference between the assumed joint stiffness in the frame analysis (Sj = ∞)and the 'actual' stiffness [7]. In the context of this paper, the 'actual' stiffness is the best value adesigner can obtain for the stiffness of a particular joint. This is, for example, a value obtained from atest or based on Eurocode 3.

If a difference between assumed (Sj = ∞) and 'actual' stiffness has a limited effect on theframe behaviour, then it is not required to perform a second frame analysis with the 'actual' stiffnessof the joint. The check whether a joint is rigid needs to be done in three steps, see figure 3.• Step a) shows the inclusion of the joint stiffness in the frame analysis (in step 3 of table 1).• Step b) shows the range in which the 'actual' stiffness should be (in step 7 of table 1).• Step c) shows the check that the 'actual' stiffness is in this range (in step 7 of table 1).

M

φFrame analysis based on the

M

φ

see Eurocode 3

Stiffness range for rigid joints

M

φCheck if 'actual' stiffness S

=j,app ∞assumption S

(the joint is rigid)

based on S =j,app ∞j,act

is in range

Sj,actSj,app

Step a) : engineer Step b): fabricator Step c): fabricator

figure 3: Check of stiffness requirement for a rigid joint

This concept can be generalized to a check whether a difference between assumed and 'actual'stiffness of semi-rigid joints has a significant influence to the frame behaviour, see figure 4. Thecorresponding formulae for the variance between the assumption used in the frame analysis and the'actual' stiffness of the joint are given in table 4. These criteria may be used to check whether adifference between assumed joint stiffness in the frame analysis and 'actual' stiffness have the abovementioned limited effect (5%) on the frame behaviour.

M

φ

Sj,app

φ

lower boundary

M

φ

j,act

Frame analysis based on the Stiffness range for semi-rigid joints Check if 'actual' stiffness S

j,appassumption S based on S j,app

j,actis in range

Step b): fabricatorStep a): engineer Step c): fabricator

M upper boundary

S

figure 4: Check of stiffness requirement of a semi-rigid joint

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table 4: Boundaries for variance between actual an approximated stiffnessFrame Lower boundary Upper boundaryBraced

Sj,act ≥ 8 Sj,app E Ib

10 E Ib + Sj,app l If Sj,app ≤ 8 E Ib

l then

Sj,act ≤ 10 Sj,app E Ib

8 E Ib - Sj,app lelseSj,act ≤ ∞

Un-bracedSj,act ≥

24 Sj,app E Ib30 E Ib + Sj,app l If Sj,app ≤

24 E Ibl then*

Sj,act ≤ 30 Sj,app E Ib

24 E Ib - Sj,app lelseSj,act ≤ ∞

in which:Sj,app = the assumed stiffness adopted in the frame analysis (this is an approximation of the 'actual'

stiffness)Sj,act = the 'actual' stiffness of a jointE = youngs modulesl = beam lengthIb = moment of inertia of the beam*For reasons of simplicity in the formulae for un-braced frames, the Eurocode 3 rigidity boundary is rounded

off from Sj ≥ 25 E Ib

l to Sj ≥ 24 E Ib

l

5. Comparison to the stiffness model of Annex J (revised)

The stiffness predictions with table 3 are compared to the stiffness according to Annex J(revised) (this is seen as the 'actual' stiffness) for 6 different joints. Wald & Steenhuis [8] give a fulldescription of the geometry of these 6 joints.

Tables 5 and 6 give for these 6 joints successively:• the approximation for the stiffness according to table 3 (step a)• the range in which the Eurocode 3 prediction should lie, if the frame analysis is performed with

the approximation for the stiffness according to table 3 (step b)• the 'actual' stiffness assuming this is predicted by Annex J (revised) and a check if the 'actual'

stiffness is in the range (step c).These tables show good agreement between prediction based on table 3 and Annex J (revised).

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table 5: Comparison of stiffnesses for a braced frame, beam span l = 20 hbJoint Approximation

for the stiffness,Range(step b)

'Actual' stiffnessaccording to

Check(step c)

(step a)see table 3

Lowerboundary

Upperboundary

Annex J(revised)

kNm/mrad* kNm/mrad* kNm/mrad* kNm/mrad*

83#9 44 20 591 53 Ok83#10 61** 24 ∞ 82 Ok83#11 86 46 259 59 Ok91#1 59** 23 ∞ 59 Ok91#2 59** 23 ∞ 60 Ok91#3 181** 36 ∞ 149 Ok*Kilonewton meter per milliradian **According to Eurocode 3 this joint is rigid

table 6: Comparison of stiffnesses for an un-braced frame, beam span l = 15 hb

Joint Approximationfor the stiffness,

Range(step b)

'actual' stiffnessaccording to

Check(step c)

(step a)see table 3

Lowerboundary

Upperboundary

Annex J(revised)

kNm/mrad kNm/mrad kNm/mrad kNm/mrad83#9 28 20 40 27 Ok83#10 38 26 59 35 Ok83#11 54 40 74 37 Not Ok*91#1 38 26 59 33 Ok91#2 38 26 59 33 Ok91#3 59 37 109 49 Ok*In this case, the 'actual' stiffness should be introduced in the frame analysis

6. Design example

The example in figure 5 shows the effect of the stiffness of a joint on the deformations andthe force distribution of an un-braced frame. First order elastic analysis is used. The span of the beamis 6 meters. The column height is 4 meters. Both columns and beams are IPE 360 sections. Loadsconsist of a horizontal load (F = 25 kN) and an uniformly distributed vertical load (q = 40 kN/m)

M

AM CMBM

6 m

4 m

q = 40 kNmF = 25 kN

EIb

EI c

j

G t

δ

Deformations

h

δ v

figure 5: Single storey single bay frame

Table 7 gives internal forces and deformations in the frame for different joints which may beconsidered as rigid according to Eurocode 3. A stiffness of Sj = 140 kNm/mrad is the lowest value tobe regarded as rigid. Compared to a theoretical rigid joint (Sj = ∞), vertical deflections vary 10% andhorizontal deflections vary 20%. The drop in bearing capacity is not more than 5%.

Table 8 shows the frame forces and the deformations for a semi-rigid joint with a stiffness Sj= 60 kNm/mrad. This table also gives deformations and forces for stiffnesses Sj = 35 kNm/mrad andSj = 130 kNm/mrad. These are the lower and upper boundaries for stiffness according to table 4 whenin the frame analysis a stiffness of Sj,app = 60 kNm/mrad is adopted. When comparing the case Sj =60 kNm/mrad to Sj = 35 kNm/mrad or Sj = 130 kNm/mrad, variations in deflections occur of 10% invertical deflections and 20% in horizontal deflections.

9

The variations in deflections between different rigid joints in table 7 are in close agreementto the variations between the different semi-rigid joints in table 8.

table 7: Comparison between frames withdifferent rigid jointsrigid joints

Sj kNm/mrad

140*lowerboundary

∞approximation

δv mm 9.7 8.8

δh mm 16.5 13.6

MA kNm 24.6 33.0MB kNm 105.3 96.9MC kNm 124.6 133.0*In accordance with Eurocode 3 Annex J (revised)half Sj is adopted in the frame analysis

stiffened web panel

table 8: Comparison between frames withdifferent rigidity for semi-rigid joints

semi-rigid jointsSj kNm/

mrad35*lowerboundary

60*approxi-mation

130*upperboundary

δv mm 12.0 10.8 9.7

δh mm 25.4 20.3 16.6

MA kNm 6.7 15.7 24.2MB kNm 123.2 114.2 105.7MC kNm 106.7 115.7 124.2*In accordance with Eurocode 3 Annex J (revised)half Sj is adopted in the frame analysis

unstiffened web panel

7. Conclusions

This paper shows that semi-rigid design of joints can fit in standard design practice byadopting an adjusted traditional design approach. In this approach, the share of responsibilitiesbetween engineer and steel fabricator are similar to those in traditional frame design with pinned orrigid joints. In the adjusted approach, a first approximation of the joint stiffness should be included inthe frame analysis. Simplified formulae based on Eurocode 3 Annex J (revised) help to make thisapproximation. The philosophy to check whether a joint is sufficiently stiff to be regarded as a rigidjoint in frame analysis can easily be extended to the application of semi-rigid joints. In thisapplication the difference between approximation and actual stiffness should fit within certain limits.These limits are dependent on the type of frame, the beam span, the beam stiffness and the jointstiffness. Examples show that the adjusted approach is feasible for application in practice.

We intend to compare the simplified formulae from table 3 to tests form the Aachen databank in the near future. As a possible result of this comparison, the check of the rigidity as describedin figure 4 could be omitted for certain joint geometry's, especially in braced frames.

10

References

[1] DOL C. & STEENHUIS C.M. Bolted end-plate connections (in Dutch), StaalbouwinstituutRotterdam, Bouwen met Staal nr 103, 1991.

[2] BROZZETTI, J., Design of Connections in the EUREKA "CIMSTEEL" project, Proceedings ofthe Second International Workshop on Connections in Steel Structures, edited by BjorhovdeR., Colson A., Haaijer G. and Stark J., AISC, Chicago, Illinois, USA, 1992;

[3] STEENHUIS, C.M., DOL C. & VAN GORP, L. Computerised calculation of forcedistributions in bolted end-plate connections according to Eurocode 3, Journal ofConstructional Steel Research Vol. 31, 1994;

[4] EUROCODE 3, ENV - 1993-1-1, Design of Steel Structures, Commission of the EuropeanCommunities, European Prenorm, Brussels, Belgium, April 1992;

[5] EUROCODE 3, ENV - 1993-1-1, Revised annex J, Design of Steel Structures, CEN, EuropeanCommittee for Standardization, Document CEN / TC 250 / SC 3 - N 419 E, Brussels, June1994.

[6] CUNNINGHAM R. & TAYLOR C., Practical design allowing for semi-rigid connections,Proceedings of the Second International Workshop on Connections in Steel Structures, editedby Bjorhovde R., Colson A., Haaijer G. and Stark J., AISC, Chicago, Illinois, USA, 1992;

[7] BIJLAARD F.S.K, STEENHUIS C.M. Prediction of the Influence of Connection Behaviour onthe Strength, Deformations and Stability of frames, by Classification of Connections,Proceedings of the Second International Workshop on Connections in Steel Structures, editedby Bjorhovde R., Colson A., Haaijer G. and Stark J., AISC, Chicago, Illinois, USA, 1992;

[8] WALD F., STEENHUIS C.M., The Beam-to-Column Bolted Joint Stiffness according toEurocode 3, proceedings of the first COST C1 workshop, Strasbourgh 1993.

11

ANNEX: COMPARISON WITH SPRINT DESING TABLES

Introduction

In the proceedings of the second state of the art workshop on Semi RigidBehaviour of Civil Engineering Structural Connections COST C1, held inPrague 26-28 October 1994, Martin Steenhuis, Nol Gresnigt and Klaus Weynandhave presented a paper concerning pre-design of semi-rigid connections.

They give simple formula to assess the stiffness of beam-to-columnconnections in the pre-design stage. This assessment should in a second stepbe verified against the 'actual' stiffness of the joint to be realised inthe structure. It should -of course- be in reasonable agreement with the'actual' stiffness.

The formula to assess the stiffness in the pre-design stage is as follows:

kt z E

= Sx

c.f2

app.j 1

where: E is the Youngs modulestf.c is the column flange thicknessz is the distance between centre of compression and tensionkx is a factor dependent from the type of joint (e.g. 13 for

extended unstiffened beam to column joints)Sj.app the 'good guess' for the initial stiffness

To see whether the estimation of the stiffness is in reasonable agreementwith the 'actual' stiffness of the joint, in the COST proceedings thefollowing formula are given:

For braced frames:

l S + I E 10I E S8

Sapp.jb

bapp.jact.j ≥ 2

but

l S - I E 8I E S10

Sapp.jb

bapp.jact.j ≤ 3

in case

lI E 8 S b

app.j ≤ 4

where: I is moment of inertiaSj.app is the approximation of the initial stiffnessSj.act is the actual initial stiffnessSj.low is the allowable lower bound for the initial stiffnessSj.upp is the allowable upper bound for the initial stiffnessl is the beam length

For unbraced frames:

12

S = l S + I E 30

I E S24 S low.j

app.jb

bapp.jact.j ≥ 5

but

S = l S - I E 24

I E S30 S upp.j

app.jb

bapp.jact.j ≤ 6

in case

lI E 24 S b

app.j ≤ 7

This document provides further comparisons between predictions based on the'good guess' formula with more accurate stiffness predictions based onEurocode 3 Annex J, as given in the "SPRINT Design Manual".

For this purpose, 6 series of beam-to-column connections have been analysed.It concerned:- two series with extended unstiffened end plate connections (series

"spr_ex1.dat" and "spr_ex2.dat"). In this case, distance z is calculatedas follows:z = hb - tf.b

- two series with flush end plate connections, with an extended part inthe bottom flange (series "spr_fl1.dat" and "spr_fl2.dat"). In thiscase, distance z is calculated as follows:z = hb - 0.5 tf.b - u1

where u1 is the distance between the top of the beam to the centre ofthe upper bolt row

- two series with flush end plate connections, with a non-extended part inthe bottom flange (series "spr_lfl1.dat" and "spr_lfl2.dat"). In thiscase, distance z is calculated as follows:z = hb - tf.b - u1

The differences between two series within one type concerns bolt spacing andbolt grade.

In all calculations, it is assumed that the beam span is 15 times the beamheight. This is taken as a lower bound of the beam span in practicalstructures. Higher values could also be adopted, but in that case, the frameis less sensitive for the stiffness variations of the joints.

It is assumed that the extended end plated joints will be adopted inunbraced frames, and the flush end plated joints in braced frames.

Example calculation

As an example, a flush end plate connection of chapter 3b of the "SPRINTdesign manual" is calculated.

Column: HEB140B, Beam: IPE220z = hb - 0.5 tf.b - u1 = 220 - 0.5 * 9.2 - 60 = 155.4 mm

13

kNm/rad 4346.8488 = 14

12 4155. 210.000 = k

t z E = S

2

x

c.f2

ini.j 8

l S + I E 10I E S8

5978 = Sapp.jb

bapp.jact.j ≥ 9

kNm/rad 2790.97 = 220*15 10 4346.8488 + 27720000 210000 10

27720000 210000 10 4346.8488 8 =6

6

10

l S - I E 8I E S10

5978 = Sapp.jb

bapp.jact.j ≤ 11

kNm/rad 7852.25 = 220*15 10 4346.8488 - 27720000 210000 8

27720000 210000 10 4346.8488 10 =6

6

12

Conclusion: The approximation is within the limits and can be used in theframe analysis.

Results

The figures herafter give the results to all 6 series in a graphical form on6 different sheets. Each sheet shows four lines:- 1 This line represents the 'actual' initial stiffness divided

by the 'actual' initial stiffness (Sj.act / Sj.act)- SJAPPSJ This line represents the approximated initial stiffness divided

by the 'actual' initial stiffness (Sj.app / Sj.act).- SJLOWSJ This line represents the lower bound of the stiffness divided by

the 'actual' stiffness (Sj.low / Sj.act).- SJUPPSJ This line represents the upper bound of the stiffness divided by

the 'actual' stiffness (Sj.upp / Sj.act).

It appears in all cases, that non of the upper or lower boundaries iscrossing the line '1', so all Sj.ini according to the SPRINT manual are withinthe predicted ranges based on the 'good guess' whenever the beam span is 15* hb or more.

Acknowledgement

This comparison was made in the frame of the ECSC project 'Design manual'.Thanks are given to the group preparing the 'SPRINT Design Manual' formaking their results available for this comparison.

14

0

0.51

1.52

2.53

050

100

150

200

250

300

350

400

450

join

t num

ber

spr_

ex1

1"S

JAP

PS

J""S

JLO

WS

J""S

JUP

PS

J"

15

0

0.51

1.52

2.53

050

100

150

200

250

300

350

400

450

join

t num

ber

spr_

ex2

1"S

JAP

PS

J""S

JLO

WS

J""S

JUP

PS

J"

16

0

0.51

1.52

2.53

050

100

150

200

250

300

350

400

450

join

t num

ber

spr_

fl1

1"S

JAP

PS

J""S

JLO

WS

J""S

JUP

PS

J"

17

0

0.51

1.52

2.53

050

100

150

200

250

300

350

400

450

join

t num

ber

spr_

fl2

1"S

JAP

PS

J""S

JLO

WS

J""S

JUP

PS

J"

18

0

0.51

1.52

2.53

050

100

150

200

250

300

350

400

450

join

t num

ber

spr_

lfl1

1"S

JAP

PS

J""S

JLO

WS

J""S

JUP

PS

J"

19

0

0.51

1.52

2.53

050

100

150

200

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