S. Stephan, Page 1

A General Method for the Design of Bolted Connections for Space Frames S. STEPHAN, C. STUTZKI 1. INTRODUCTION The key problem of the design of space frames is the layout of the connections. With regard to the space frame concept, bolted connections are the prefered choice. For single layer structures, which are the most appropriate for glazed roofs and facades, the members often have to be connected by means of more than one bolt in order to increase the bending capacity of the connection. The design procedure of single bolt connections is well described in [2] [3] [4] [5], however, a general method for the design of multi-bolt connections is not available. This paper will present a general design method for single or multi-bolt connections of beams with arbitrary thin-walled cross sections, suitable for application in computer programs. The design method is based on the classical strain iteration algorithm for cross sections, which is described in [6]. In this method, the ultimate capacity of bolted connections will be obtained using an iterative numerical determination of the elastic-plastic stress distribution in the connection elements. The numerical method will be derived in two steps – the numerical determination of the stress distribution in the connection for a given combination of internal forces and – the determination of the ultimate capacity of the connection. Furthermore analytical design formulas for a multi-bolt tube connection will be derived. Finally results of numerical and analytical calculations will be compared with corresponding test results. 2. BASES AND ASSUMPTIONS For the derivation of both, numerical and analytical design methods, some general restrictive assumptions have to be made. Basis of the discussion is the bolted connection of a prismatic beam with a longitudinal axis x and cross section axes y and z. The cross section axes of the beam do not need to be central axes. Only normal forces and bending moments will be considered, lateral forces and torsion moment will be ignored. The connection profile, i.e. the cross section of the beam at the bolted connection, must be thin-walled. The outline of the connection profile can be open or closed. The wall of the connection profile is modelled as a set of line elements and curve elements. In the contact zone of the connection profile only compression forces can be transferred. The bolts are modelled as a set of point elements. Only tensile forces can be transferred through the bolts. The connection profile together with the bolts remains planar under full load (hypothesis of planar cross sections). Thus the strain equation for any point (y,z) of the planar connection can be written as follows (for explanation of symbols refer to the notation list at the end of this paper):

( ) ozyz,y yz ε+⋅κ−⋅κ=ε (1)

S. Stephan, Page 2

Hence the compression stress (negative) for any point (y,z) of the connection profile can be obtained using the following formula:

( )( )

( ) ( )( ) ( )( )( )

>ε⋅≤ε⋅∧σ>ε⋅ε⋅

σ≤ε⋅σ=σ

0z,yE;0

0z,yEz,yE;z,yE

z,yE;

z,y lim

limlim

(2)

Further the tensile stress (positive) for any bolt can be calculated from the following formula:

( )( )

( ) ( )( ) ( )( )( )

≤ε⋅σ≤ε⋅∧>ε⋅ε⋅

σ>ε⋅σ=σ

0z,yE;0

bz,yE0z,yE;z,yE

bz,yE;b

z,yb lim

limlim

(3)

Finally the normal force and the bending moments at the connection can be determined from: ( ) ( )∫ σ+∑ σ⋅= A

vvvv dAz,yz,ybAbN (4a)

( ) ( )∫ ⋅σ−∑ ⋅σ⋅−= Av

vvvv dAzz,yzz,ybAbMy (4b)

( ) ( )∫ ⋅σ+∑ ⋅σ⋅= Av

vvvv dAyz,yyz,ybAbMz (4c)

Thus a certain combination of strain parameters κy, κz and εo corresponds with a combination of internal forces and moments. This interdependence will be used further. 3. NUMERICAL CALCULATION OF MULTI-BOLT CONNECTIONS For the numerical design method some additional restrictive assumptions should be made. The wall of the connection profile is modelled as a set of only line elements. Curved sections must be adequately approximated by polygonal line elements. Any line element (index J) is defined by two nodes (index i, k) and a particular thickness tJ which is constant over the length of the element (Fig. 1). Any bolt (index v) has a particular thread diameter dv which determines the accompanying stress area Abv. 3.1. Iterative Determination

of Stress Distribution in the Bolted Connection

First of all it is necessary to derive an algorithm for the determination of the stress distribution in the connection elements for an arbitrary combination of internal forces Fig. 1: General scheme of a multi-bolt connection and moments Norig, Myorig, Mzorig. This combination of normal force and bending moments will be used as initial values for the iteration parameters (iteration index it):

origit

origit

origit Mzmz,Mymy,Nn,0it ==== (5)

O Y

Z

X

v

dv

k

L J

t J i

J

S. Stephan, Page 3

With those iteration parameters the strain parameters can be calculated from:

AE

no,

JyE

my,

JzE

mz itit

itit

y

itit

z ⋅=ε

⋅=κ

⋅=κ (6)

Hence the strain distribution of the planar connection (Fig. 2) for the current iteration is determinable as follows:

( ) itity

itz

it ozyz,y ε+⋅κ−⋅κ=ε (7)

Figure 2 shows the strain distribution in the connection for an arbitrary combination of internal forces and moments. The surfaces of the adjacent connection profiles will contact each other only within areas with negative strain (ε < 0). The contact zone is limited by the zero strain line (ε = 0) Compression forces will be transferred only within the contact zone. Therefore only line elements with negative strain at least in one node must be regarded for the determination of compression stress Fig. 2: Strain distribution in the connection in the connection profile. The bolts will be activated only within areas with positive strain (ε > 0). Within those areas there is a gap between the surfaces of the adjacent connection profiles. 3.1.1. Determination of Compression Stress in the Connection Profile Figure 3 shows the strain and the stress distribution for a selected line element within the contact zone of the connection profile. First the element length must be ascertained:

( ) ( )2J,kJ,i

2J,kJ,iJ zzyyL −+−= (8)

Additionally the node strains are needed for the further calculation:

( ) ( ) itJ,kJ,kk

itJ,iJ,ii z,y,z,y ε=εε=ε (9)

If there is a point with zero strain along the line element, the distance from node i must be determined:

[ ]JJJki

iJ L00u,L

EE

E0u K∈⋅

ε⋅−ε⋅ε⋅

= (10a)

If the limit stress is reached in one point along the line element, the distance from node i can be obtained in a similar way:

[ ]JlimJki

limilim L0u,L

EE

Eu

JJK∈⋅

ε⋅−ε⋅σ−ε⋅

= (10b)

O

Y

Z

X

εεεε = = = = 0000

�

�

εεεε > > > > 0000

εεεε <<<< 0000

S. Stephan, Page 4

Thus the coordinates of the point with zero strain can be calculated:

( ) ( )J,iJ,kJ

JJ,iJJ,iJ,k

J

JJ,iJ zz

L

0uz0z,yy

L

0uy0y −⋅+=−⋅+= (11a)

In a similar way the coordinates of the point with limit stress will be determined:

( ) ( )J,iJ,kJ

limJ,ilimJ,iJ,k

J

limJ,ilim zz

L

uzz,yy

L

uyy J

J

J

J−⋅+=−⋅+= (11b)

Elements with at least one of those intermediate points (with zero strain or with limit stress) must be split into sub-elements to simplify the calculation of the resulting normal force and bending moments. During this the position of the intermediate points must be considered. The arising sub-elements are line elements (index j) with two nodes and a constant thickness tj (Fig. 4). For further calculations only the sub-elements will be used. The splitting operation into sub-elements has to Fig. 3: Strain and stress distribution for a selected line element be executed as follows:

[ ][ ]

>≤

=JJ

JJ

limJJ,kJlimJ,i

limJJ,klimJJ,iJ u0u;y0yyy

u0u;yy0yyY (12a)

[ ][ ]

>≤

=JJ

JJ

limJJ,kJlimJ,i

limJJ,klimJJ,iJ u0u;z0zzz

u0u;zz0zzZ (12b)

The length of a sub-element (Fig. 4) with its two nodes (index i, k) must be calculated from:

( ) ( )2j,kj,i2

j,kj,isub ZZYYLj

−+−= (13)

The strain calculation for nodes of sub-elements is similar to (9). On that basis the node stress can be obtained using the following formula:

( ) ( )

>ε⋅≤ε⋅∧σ>ε⋅ε⋅

σ≤ε⋅σ=σ

0E;0

0EE;E

E;

j,i

j,ilimj,ij,i

limj,ilim

j,i (14)

O

Z

�

�

J

i

k

� y0 J ylim J

ulim J

LJ

z0 J

zlim J

u

u

u0 J

�

σ

Y

X

σlim

S. Stephan, Page 5

The same formula can be applied for σk,j accordingly. Thus the resulting normal force and bending moments of a sub-element for the current iteration can be determined. This will be done using a linear stress distribution between σi,j at node i and σk,j at node k. Hence the arising normal force will be obtained from the product of the stress trapezoid area and the element thickness. With that the bending moments can be determined as the product of the normal force and the corresponding coordinate of Fig. 4: Stress distribution for a sub-element the centre of gravity of the stress trapezoid:

( )jj subjj,kj,isub Lt

2

1N ⋅⋅σ+σ⋅= (15a)

( ) ( )

−⋅

σ+σ⋅σ⋅+σ

+⋅= j,ij,kj,kj,i

j,kj,ij,isubsub ZZ

3

2ZNMy

jj (15b)

( ) ( )

−⋅

σ+σ⋅σ⋅+σ

+⋅= j,ij,kj,kj,i

j,kj,ij,isubsub YY

3

2YNMz

jj (15c)

3.1.2. Determination of Tensile

Stress in the Bolts Figure 5 shows a selected bolt (index v) with its thread diameter dv and its stress area Abv. That bolt is located at the position (yv, zv) within the tension zone of the connection. The bolt strain for the current iteration will be calculated as follows:

( ) itvvv z,yε=ε (16)

With that the tension stress in the bolt is determinable from:

( )( )

≤ε⋅σ≤ε⋅

∧>ε⋅ε⋅

σ>ε⋅σ

=σ

0E;0bE

...0E;E

bE;b

b

v

limv

vv

limvlim

v

(17) Fig. 5: Internal forces and moments of a bolt

O

Y

Z

X

� Lj

Nsubj

tj

j

k

i

�

� MZsub j

MYsub j

σ kj

σ ij σ

O

Z

MZbv

MYbv

Nbv

σ

v Abv

�

�

�

X

Y

S. Stephan, Page 6

3.1.3. Determination of Internal Forces and Moments The resulting normal force in the bolts must be obtained from the product of the tensile stress and the stress area of the bolt. The bending moments will then be determined as the product of the normal force and the corresponding coordinate of the bolt position. Now it is possible to calculate the normal force and bending moments in the connection, which results from the combination of strain parameters (6) for the current iteration step ⟨it⟩:

∑∑ +σ⋅=j

subv

vvit

jNbAbN (18a)

∑∑ −⋅σ⋅−=j

subv

vvvit

jMyzbAbMy (18b)

∑∑ +⋅σ⋅=j

subv

vvvit

jMzybAbMz (18c)

3.1.4. Check of Convergence To verify whether convergence is reached in the current iteration, it is necessary to determine the deviation of the normal force and bending moments from the given original combination of internal forces and moments Norig, Myorig, Mzorig:

itorig

it NNN −=∆ (19a) it

origit MyMyMy −=∆ (19b)

itorig

it MzMzMz −=∆ (19c)

Convergence will be reached, if the following condition is fulfilled:

( ) ( ) ( )0Mz0My0N ititit ≅∆∧≅∆∧≅∆ (20) If condition (20) is not fulfilled, then it will be necessary to execute a new iteration step. To begin with the new iteration the following iteration parameters must be adjusted:

Mzmzmz,Mymymy,Nnn,1itit 1itit1itit1itit ∆+=∆+=∆+=+= −−− (21) Afterwards all equations from (6) to (19) has to be calculated again and condition (20) has to be checked. The iteration must be repeated until condition (20) is fulfilled. The stress and strain distribution in the connection elements, which is caused by the given combination of internal forces and moments Norig, Myorig, Mzorig, is thus determined. 3.2. Iterative Determination of the Ultimate Connection Capacity The above described determination of the stress distribution in the connection elements for an arbitrary combination of normal force and bending moments will be used now for the calculation of the ultimate connection capacity. Two of three parameters N, My and Mz (normal force and bending moments) must be given, the limit value of the third parameter will be determined. Beginning with zero, the third parameter has to be increased in steady steps. For each step the procedure described in pragraph 3.1 has to be carried out. If the convergence condition (20) cannot be fulfilled within a reasonable number of iterations, then the limit value of the third parameter was obviously exceeded. Hence, the previous increase of the third parameter has to be cancelled, the parameter step size has to be halved and the third parameter will be increased with reduced step size.

S. Stephan, Page 7

After that procedure 3.1 has to be repeated. If the convergence condition (20) can be fulfilled now, then the third parameter will be continually increased with reduced step size and procedure 3.1 will be carried out for each step. If, however, the convergence condition (20) cannot be fulfilled within a reasonable number of iterations, then the previous increase of the third parameter has to be cancelled once more, the parameter step size has to be halved again and the third parameter should be increased with the smaller step size. Then procedure 3.1. has to be carried out once more. This process must be repeated until the parameter step size is lower than a small predefined value. If that condition is met, the limit value of the third parameter is found. The current combination of all three parameters N, My and Mz is the ultimate capacity of the connection. The advantage of the presented numerical method is that it can be easily adapted to arbitrary thin-walled connection profiles with any bolt scheme. 4. ANALYTICAL CALCULATION OF MULTI-BOLT TUBE CONNECTIO NS In this paragraph an analytical calculation method will be presented for a bolted tube connection as shown in Fig. 6, which has a simple and continuous connection profile. This analytical method will be used to check the results of the above described numerical method. The idea of the above numerical method was the iterative modification of strain parameters until the resulting internal forces are equal to the given combination of internal forces (refer to [6]). The numerical calculation of the ultimate connection capacity was finally a stepwise increase of a given internal force or moment until the equality of the resulting Fig. 6: Multi-bolt tube connection internal forces with the given combination cannot be achieved anymore. Thus, the ultimate capacity of a connection can be determined only as a result of a double iteration process. In contrast, the analytical method does not require any iteration. Ultimate strain conditions have to be defined for the connection profile and for the bolts. Together with assumed coordinates of the zero strain line this is sufficient to calculate all strain parameters and hence the resulting combination of normal force and bending moments That combination represents simultaneously the ultimate capacity of the connection. Modifying the coordinates of the zero strain line, a limit state interaction of normal force and bending moments can be determined. Obviously this is an implicit calculation method, since it is impossible to determine the combination of normal force and bending moments directly - knowing two of three parameters N, My and Mz. The combination of normal force and bending moments can be ascertained only from a set of parametric equations based on the strain parameters.

Z

Y �

t

�

�

�

�

Z

R

σ

Z

S. Stephan, Page 8

It is assumed again that the connection profile together with the bolts remains planar under full load (hypothesis of planar cross sections). In order to simplify the derivation it is assumed further that the bending moment Mz is zero. Hence it follows that only two strain parameters are necessary: the coordinate of the zero strain line z0 and a certain strain value ε1 with its corresponding coordinate z1. Strain ε1 is either the ultimate strain of the outermost tube wall in the compression zone or the ultimate strain of the outermost bolt in the tension zone, depending on which of the two is the decisive criterion. This procedure is described below in paragraph 4.4. Consequently, the strain distribution for the tube connection (Fig. 6) can be determined:

( )0z1z

0zz1z

−−⋅ε=ε (22)

4.1. Determination of Compression Stress

in the Tube Wall Considering equations (2) and (22), the stress distribution in the tube wall can be obtained from:

( )( )

( ) ( )( )( )( )

( )

>ε⋅σ>ε⋅

∧≤ε⋅ε⋅

σ≤ε⋅σ

=σ

0zE;0zE

...0zE;zE

zE;

zlim

limlim

(23)

The corresponding stress distribution Fig. 7: Elastic-plastic tube material is shown in figure 8. Figure 7 shows the elastic behaviour of the tube material until the yield strain εyield is reached and the plastic behaviour until ultimate strain εlim is reached. According to Hooke's law the yield strain can be determined from the following equation:

Elim

yieldσ=ε (24)

Provided that zlim is the coordinate with the yield strain, equation (22) can be transformed to:

0z1z

0zz1 lim

yield −−⋅ε=ε (25)

Considering equations (24) and (25), the coordinate with the yield strain zlim can be determined for (-R ≤ zlim ≤ R) from: Fig. 8: Stress and strain distribution for the tube wall

( ) 0z0z1z1E

z limlim +−⋅

ε⋅σ= (26)

The geometrical characteristics in figure 8 will be obtained from:

=α

=α

=α

=βR

zarcsin,

R

1zarcsin1,

R

0zarcsin0,

R

zarcsin lim

lim (27)

O ��lim

σ

�yield

σlim

dA

Zlim ZO

Z1

�1

σlim Y

�

t

�

�

�

Z Z

Z

R

σ

dß

∝O

ß

∝lim

�yield

Z

S. Stephan, Page 9

As can be seen from figure 8, the area differential of the tube wall is defined as follows: β⋅⋅= dtRdA (28)

The tube wall can be divided in a section with no contact and hence no stress (-R ≤ z < z0), a section with elastic stress (z0 ≤ z < zlim) and a section with plastic stress (zlim ≤ z ≤ R). Thus the normal force of the section with only elastic stress will be obtained from:

( ) ( )∫ β⋅⋅⋅

α−αα−βε⋅⋅=∫ ⋅σ⋅=

α

α

limlim

0

z

0ze dtR

)0sin()1sin(

)0sin(sin1E2dAz2N (29a)

( ) ( ) ( ) ( ))0sin()1sin(

0sin0cos0costR1E2N limlim

e α−αα⋅α−α+α−α

⋅⋅⋅ε⋅⋅= (29b)

The normal force of the section with only plastic stress will be derived accordingly:

( ) ∫∫π

αβ⋅⋅⋅σ⋅=⋅σ⋅=

2/

lim

R

zp

limlim

dtR2dAz2N (30a)

( ) tR2N limlimp ⋅⋅σ⋅α⋅−π= (30b)

The bending moment of the section with elastic stress will be derived in a similar way:

( )∫ ⋅⋅σ⋅=limz

0ze dAzz2My (31a)

( ) ( )∫α

αβ⋅β⋅⋅⋅

α−αα−βε⋅⋅=

lim

0

2e dsintR

)0sin()1sin(

)0sin(sin1E2My (31b)

( ) ( ) ( ) ( ) ( )

)0sin()1sin(2

02sin

2

2sin0cos0sin2

tR1EMy

limlimlim

2e α−α

α⋅−α⋅

−α−α+α⋅α⋅⋅⋅⋅ε⋅= (31c)

The bending moment of the section with plastic stress will be obtained from:

( ) ( )∫∫π

αβ⋅β⋅⋅⋅σ⋅=⋅⋅σ⋅=

2/2

lim

R

zp

limlim

dsintR2dAzz2My (32a)

( ) tRcos2My 2limlimp ⋅⋅σ⋅α⋅⋅= (32b)

4.2. Determination of Tensile Stress in the Bolts Considering equations (2) and (22), the stress distribution in the bolts can be calculated as follows:

( )( )

( ) ( )( )( )( )

( )

≤ε⋅σ>ε⋅

∧>ε⋅ε⋅

σ>ε⋅σ

=σ

0zE;0bzE

...0zE;zE

bzE;b

zblim

limlim

(33)

The corresponding stress distribution is shown in figure 10. Figure 9 shows the elastic behaviour Fig. 9: Elastic-plastic bolt material of the tube material until the yield strain εbyield is reached and further the ideal plastic behaviour until ultimate strain εblim is reached. According to Hooke's law the yield strain can be determined from the following equation:

E

bb lim

yieldσ=ε (34)

Provided that zblim is the coordinate with the yield strain, equation (22) can be transformed to:

0z1z

0zzb1b lim

yield −−⋅ε=ε (35)

O ��blim

σ

�byield

σblim

S. Stephan, Page 10

Considering equations (34) and (35), the coordinate with the yield strain zblim can be calculated from the following equation (-R ≤ zblim ≤ R):

( ) 0z0z1z1E

bzb lim

lim +−⋅ε⋅

σ= (36)

The bolt area can be divided in a section with no stress due to contact in the tube wall (z0 ≤ z < R), a section with elastic stress (zblim ≤ z < z0) and a section with plastic stress (-R ≤ z ≤ zblim). Thus the normal force of all bolts (index v) will be determined as follows: Fig. 10: Stress and strain distribution for the bolts

( ) ( )∑

>

≤∧≥⋅−−

⋅ε⋅

<⋅σ

=v

v

vlimvvv

limvvlim

b

0zz;0

0zzzbz;Ab0z1z

0zz1E

zbz;Abb

N (37)

The bending moment of all bolts will be calculated accordingly:

( ) ( )∑

>

≤∧≥⋅⋅−−

⋅ε⋅

<⋅⋅σ

=v

v

vlimvvvv

limvvvlim

b

0zz;0

0zzzbz;zAb0z1z

0zz1E

zbz;zAbb

My (38)

4.3. Determination of Ultimate Normal Force and Bending Moment The normal force and bending moment represent the ultimate capacity of the connection, since the calculation is based on ultimate strain conditions. The ultimate normal force and ultimate bending moment of the multi-bolt tube connection will be determined from the following equations:

bpe NNNN ++= (39a)

bpe MyMyMyMy −−−= (39b)

The limit state interaction of the normal force and the bending moment can be determined by a stepwise increase of the zero strain coordinate z0 from –R to R. 4.4. Determination of Ultimate Strain Conditions For the above described analytical method, a certain strain value ε1 with its corresponding coordinate z1 has to be assumed. The strain value ε1 is either the ultimate strain of the outermost tube wall in the compression zone or the ultimate strain of the outermost bolt in the tension zone, depending on the critical strain conditions. According to [1], the ultimate strain for mild steel can be assumed as:

1.0lim =ε (40a) Obviously, the corresponding coordinate of the outermost tube wall is:

R.z R = (40b)

ZO

Y

�

�

Z Z Z

σ

�byield

σblim

�1 �Z1

Zblim

S. Stephan, Page 11

Likewise, according to [1] the ultimate strain of high strength bolts (grade 8.8 and 10.9) can be calculated as follows:

( )

⋅⋅+−⋅⋅+

⋅⋅+

⋅⋅

⋅⋅⋅⋅

=22

lim2

lim 446.1

4 do

hnknLsLbko

d

Ls

dLbE

bdob

πππσπε (41a)

The corresponding coordinate of the outermost bolt in the tension zone can be obtained from: ( )vB zmin.z = (41b)

The thread strain coefficient has to be determined from the following formula:

+

+=

9.10grade;013.0

1.09.0

8.8grade;021.0

2.08.0

ko (42)

The nut strain coefficient has to be calculated as follows:

+

+=

9.10grade;013.0

06.054.0

8.8grade;021.0

12.048.0

kn (43)

The resulting ultimate strain of the bolts usually lies between 0.02 and 0.06, depending on the geometric parameters of the bolts (diameter, shaft length and thread length). Providing that the specific ultimate strain appears at the outermost bolt in the tension zone, the resulting strain at the outermost tube wall in compression zone has to be calculated as follows:

limB

RR b

0zz

0zz ε⋅−−=ε (44)

Providing, however, that the specific ultimate strain appears at the outermost tube wall in the compression zone, Fig. 11: Two possibilities of ultimate strain distribution the resulting strain at the outermost bolt in the tension zone has to be calculated as follows:

limR

BB 0zz

0zz ε⋅−−=ε (45)

The strain value ε1 depends on which of the two strain distributions is permissible:

( ) ( )( ) ( )

ε≤ε∧ε>εεε>ε∧ε≤εε

=εlimRlimBlim

limRlimBlim

b;b

b;1 (46a)

Hence the corresponding coordinate z1 has to be calculated accordingly:

( ) ( )( ) ( )

ε≤ε∧ε>εε>ε∧ε≤ε

=limRlimBB

limRlimBR

b;z

b;z1z (46b)

Those values are sufficient for the analytical calculation according to 4.1, 4.2 and 4.3.

ZO

�blim

Z

Y

�

Z Z

�

O O

ZB

ZR

�B

�lim �R

O

S. Stephan, Page 12

5. TENSILE AND BENDING TESTS The calculated ultimate connection capacity can be verified only by experimental tests. Usually, these tests will be carried out as single axial load tests either with pure tensile load or with pure bending load. Multiaxial load tests, i.e. with interaction of normal force and bending moments, are difficult to realize and very expensive. Every specific test has to be carried out at least three times. The test results must be evaluated according to relevant codes or regulations, e.g. BS 5950-1. The objective of the tests is the determination of the connection capacity for a given load. Another objective is the determination of the corresponding connection stiffness, which has to be used in the structural model to reflect the real semi-rigid behaviour of the bolted connection. In order to determine the connection stiffness, the applied load and the corresponding deflection must be continuously recorded. The applied load will be gradually increased from zero to the ultimate load (connection failure or extreme deflections).

Fig. 12: Typical tensile test arrangement Figure 12 shows a typical tensile test of a multi-bolt tube connection. The tensile force will be measured directly at the testing machine. The tensile deflection of the connection will be measured with inductive path-measuring instruments (WA2, WA3 in Fig. 12).

Fig. 13: Typical bending test arrangement Figure 13 shows a typical bending test of a multi-bolt tube connection. The bending force will be measured with load cells underneath the crossbeam of the testing machine. The bending deflection of the connection will be measured with inductive path-measuring instruments (WA1, WA2, WA3, WA4 in Fig. 13). The applied bending moment is constant between the two points of force introduction and can be easily calculated due to the symmetric four-point test arrangement.

S. Stephan, Page 13

6. COMPARISON OF NUMERICAL, ANALYTICAL AND TEST RESULT S The above described numerical calculation method was converted to a computer program by means of the programming language FORTRAN 90. The data model of the connection profile and the bolt scheme is very flexible and can be adopted to any configuration. The analytical calculation method for tube connections was programmed by means of the mathematical software MATHCAD 2000.

Fig. 14: Longitudinal section of the tested 4-bolt connection for Glasgow Science Center Figure 14 shows a longitudinal section through the 4-bolt connection of the Exploratorium roof structure of the Glasgow Science Center. Figure 15 shows the corresponding cross section. This connection was designed according to the paragraphs 3, 4 and 5. The 4-bolt connection of the above structure has the following parameters: tube diameter 323.9 mm tube wall thickness 12 mm tube material S355 J2H bolt size, grade 4 x M27 – 10.9 pitch diameter 180 mm The results of the numerical and analytical calculations as well as the test results are shown in figure 16. As can be seen from the diagram the analytical and numerical results are close together. These calculations have been confirmed by test results Fig. 15: Cross section of the tested 4-bolt connection for the relevant load combinations. The evaluation of the test results has been done according to BS 5950-1.

S. Stephan, Page 14

0

50

100

150

200

250

-5000 -4000 -3000 -2000 -1000 0 1000 2000

Normal force, kN

Ben

din

g m

om

ent,

kNm

Analytical Calculation Numerical Calculation Test Results

Fig. 16: Analytical & numerical calculation and test results for Glasgow Science Center Although the node connection for the roof structure of the Eden Project is a single bolt connection, it can be designed according to the above described methods. The results of calculations and tests are shown in figure 17. The connection parameters are: tube diameter 193.7 mm tube wall thickness 10 mm tube material S355 J2H bolt size, grade 1 x M33 – 10.9 As can be seen from the diagram the analytical and numerical results are close together and have been confirmed by the test results for the relevant load combinations.

0

10

20

30

40

50

60

-2500 -2000 -1500 -1000 -500 0 500 1000

Normal force, kN

Ben

din

g m

om

ent,

kNm

Analytical Calculation Numerical Calculation Test Results Fig. 17: Analytical & numerical calculation and test results for Eden Project

S. Stephan, Page 15

7. REFERENCES [1] STEURER A., Das Tragverhalten und Rotationsvermögen geschraubter Stirnplatten-

verbindungen, ETH Zürich: IBK Bericht Nr. 247, Birkhäuser Verlag, Basel 1999, pp 242 – 244 & 255 – 256

[2] Allgemeine bauaufsichtliche Zulassung Z-14.4-10: MERO Raumfachwerk, DIBt Berlin [3] KLIMKE H., Developing a Space Frame System, IASS Singapore 1997, pp 439 – 446 [4] STUTZKI C., MERO Plus – Handbuch, Würzburg 1990, pp II.10.34 – II.10.48 [5] KLIMKE H., How Space Frames Are Connected, IASS Madrid 1999, pp B4.13 – B4.19 [6] KINDMANN R. & FRICKEL J., Elastische und plastische Querschnittstragfähigkeit,

Ernst & Sohn, Berlin 2002, pp 411 – 412 & 485 – 499

8. NOTATION LIST 8.1. Symbols ∝0 angle with zero strain (z0) kn nut strain coefficient ∝1 angle with given strain (z1) ko thread strain coefficent ∝lim angle with limit stress (zlim) L, Lsub element length ß current angle (z) Lb clamping length of bolt σ compression stress (profile) Ls shaft length of bolt σlim ultimate compression stress My, Mz bending moments σb tensile stress (bolt) Mysub, Mysub bending moments σblim

ultimate tensile stress (bolt) my, mz iteration parameters ∆My, ∆Mz deviation of bending moment Myb bending moment of bolts ∆N deviation of normal force Mye, Mze bend. moments of elastic area κy, κz strain parameters Myp, Mzp bend. moments of plastic area ε strain in x-direction Myorig, Mzorig given bending moments εlim ultimate strain N, Nsub normal force εyield yield strain n iteration parameter εblim ultimate strain of bolt Nb normal force of bolts εbyield yield strain of bolt Ne normal force of elastic area εB strain at outermost bolt Np normal force of plastic area εR strain at outermost tube wall Norig given normal force εo strain parameter R tube radius

ε1 given strain t wall thickness A cross section area u0 offset of zero strain point Ab stress area of bolt ulim offset of limit stress point d thread diameter of bolt y, z node coordinates do diameter of stress area of bolt Y, Z node coordinates dA area differential y0, z0 coordinates of zero strain point dß angle differential ylim, zlim coordinates of limit stress point E modulus of elasticity zB coordinate of outermost bolt hn height of nut zR coord. of outermost tube wall Jy, Jz moments of inertia z1 coordinate with given strain 8.2. Indices i, k node index it iteration index J, j element index v bolt index

S. Stephan, Page 16

8.3. Addendum 03.10.2002 Figure 18 shows a longitudinal section through the 6-bolt connection of the foyer roof structure of the Scottish Parliament in Edinburgh. Figure 19 shows the corresponding cross section. This connection was designed according to the paragraphs 3, 4 and 5. Fig. 18: Longitudinal section of the 6-bolt connection for Scottish Parliament Edinburgh The 6-bolt connection of the above structure has the following parameters: tube diameter 406.4 mm tube wall thickness 25 mm tube material S355 J2H bolt size, grade 6 x M36 – 10.9 pitch diameter 220 mm The results of the numerical and analytical calculations as well as the test results are shown in figure 20. As can be seen from the diagram the analytical and numerical results are close together. These calculations have been confirmed by test results for the relevant load combinations. The evaluation of the test results has been done according to BS 5950-1. Fig. 19: Cross section of the 6-bolt connection

S. Stephan, Page 17

0

100

200

300

400

500

600

700

800

-12000 -10000 -8000 -6000 -4000 -2000 0 2000 4000 6000

Normal force, kN

Ben

din

g m

om

ent,

kNm

Analytical Numerical Test Results

Fig. 20: Calculation and test results for Scottish Parliament Edinburgh, foyer roof