BEAM MODEL FOR THE CALCULATION OF THE MULTIPLE BOLT CONNECTIONS

By P. Agatonovic, D-85244 Röhrmoos, Germany

Abstract: Different non-linear interactions in a bolted flange connection do not allow

accurate evaluation of bolt loading using current calculation methods, which are

based on linear relationships. An algorithm that compensates for non-linear

interactions has been developed, allowing accurate evaluation of all significant

parameters of the proper MBC design The algorithm is proved based on

experimentally and numerically obtained results.

Introduction

The bolt connections occurring in the practice are mostly the connections with more than

one bolt. For such connections a non-linear model, which could guarantee the reliable design, has

to be developed. The basic idea of the model is shown in Figure 1. It is assumed, that each

multiple bolt connection can be decomposed in a number of single bolt connections, which are

essentially constructed from plate segments or beams and a connecting element. The connection

of the single model to the remaining structure is defined by the influence coefficients depending

on support geometry and stiffness.

Figure 1 : Beam model of the bolt connection

Under eccentric external load, the reaction force shifts with increasing load from the

position of the connection axis balancing at the same time the moment caused by external force.

B A K KF a M F s (1)

Usually, during assembly of the connection, the both forces, bolt preload FV and the

reaction force on the separation surface FK, act along the bolt axis (Fig. 2, a). The clamped parts

are pressed together where the compliance is P. Under the external load this effect separates

(Fig. 2, b), so that the compliance is divided to the effect based on the bolt force and effect based

on the force in the separation surface.

P PS PF (2)

Fig. 2: Compliances in connection under loading

1. Determination of the Clamping force

Differential equation of the beam has the following form

2

K S K2

d yEJ F x F x s

dx

(3)

This equation applies to the two fields of the beam, if the expression in brackets, if not

greater than zero, is not taken into consideration. The integration yields:

22

K

K S 1

x sdy xEJ F F C

dx 2 2

(4)

33

K

K S 1 2

x sxEJ y F F C x C

6 6

(5)

A currently unknown rotation angle of the beam o can emerge in lieu of the clamping

force

x 0 O 1

dy 1C

dx EJ

Dissolved according to C1

1 OC EJ

The beam can be only displaced at this position, as the elastic flexibility in the connection

this allows. Consequently

x 0 K pfy F

and

2 K pfC EJ F

Under the bolt (x = sK) the deflection of the beam equals the amount between initial

deformation after preloading of the connection and the bolt extension by the additional force FSA :

Kx s

S V S PS V pf S S PS Vy F F F F F

Or respectively, after the introduction of the equilibrium condition: FS = FK + FA

3

K KK S PS A S PS V O K K pf

F s1F F F EJ s EJ F

EJ 6

( 6)

Dissolved against FK this yields:

s psV O KA

V A S ps O K A

K 3 3

K KS PS PF

F sF

F F s FF

s s1

6 EJ 6 EJ

(7)

The expression in the first brackets is based on the significant relationship, which

determines the loading conditions in a connection: the ratio between preload and working force

and the ratio of the resilience of the parts of the connection. After inserting

S ps SE

and

V SE

A

F

F

the relationship (7) simplifies in:

KA O

K 3

K

sF

Fs

16 EJ

(7’)

2. The support stiffness influence

The conditions at the connection of the model to the rest of the structure (x = sK + a) depend on the force relationships at this point and can be written in general:

i i F A M A PL F M p....... (8)

We consider nearby the connection point, which is in equilibrium under the influence of

external forces Li, where under the "force" also a moment (MA ) or pressure (p) is to be

understood. Under the assumption of linear elastic behaviour, the rotation angle of the connection

is determined by superposition of the rotating parts originating from the individual load

components.

The angular position of the beam (see (2)) is for x = sK + a:

22

K

K S O

s ady 1 aF F EJ

dx EJ 2 2

and because FS = FK + FA

2 2

K K K A O

dy 1F s 2 s a F a

dx 2 EJ (9)

The introduction of relations (8) in this equation results in:

2 2

F A M A P K K K A O

1F M p F s 2 s a F a

2 EJ .

The solution of this equation for MA:

p2 2 O F

A K K K A A

M M M M

1 M F s 2 s a F a F p

2 EJ

(10)

leads, after the consideration of (1) and the simplifications in the form of influence numbers:

M Mb 2 EJ

FFb 2 EJ

a

PPb 2 EJ

a

to the second relationship for the clamping force

pF OA

M M M A M

KK

K K

M

bb 2 EJa 11 a F

b b b F bF

Ss s 2 a

b

that after the multiplication with bM

bM

M F p A O

A

K

K K M

1a b b b a F 2 EJ

FF

s s 2 a b

and simplification B = (a + bM + bF + bp.1

FA ) can be written

A OK

K M K

B a F 2 EJF

s b 2 a s

(10')

3. Solution of the system

The solution is possible if both conditions (6') and (10')) become fulfilled. After equating

both relationships for clamping force:

KA O

A OK 3

KK K M

sF

B a F 2 EJF

ss 2 a s b1

6 EJ

the slope of the beam at the place of the clamping force could be determined:

3

KK K M

O AK

3

K K K M

B a

ss 2 a s b1

6 EJ .Fs

2 EJ

s s 2 a s b1

6 EJ

(11)

or

O K A(s ) F (11’)

The simplest conditions for the solution apply, if the position of the clamping force is far

enough from the edge, so that the edge influences are not to be expected. It can be assumed, under

these conditions, that the angle o equals zero, leading to:

3

KK K M

B a

ss 2 a s b1

6 EJ

and after rearranging according sK

3

2KK M K

ss b 2 a s 1 0

6 EJ B a B a

(12)

or after introducing of Ksxa

as the new unknown to the characteristic equation of the

connection:

C1.x3 + C2.x

2 + C3.x +C4 = 0 (13)

Here are

3

1

aC

6 EJ

2

aC

B

M

3

b 2 aC

B

4C 1

For two flanges (also of different thickness) counts

1

EoJo =

1

2.(

1

E1J1 +

1

E2J2)

S = S1 + S2 (symbolically)

P = P1 + P2

A reasonable solution to the equation above presumes that the sK distance is not

approaching the edge. If the position of the clamping force to the edge is so close that the pressure

distribution in the joint is not symmetric, the beam tilts on at the edge and the conditions o = 0 become increasingly inaccurate. It must be pointed out, that the so-called lever principle must not

be valid for this case, because the clamping is not free and when turning around the edge, it

cannot happen without the influence of the restraint of the remaining structure. Thus, concerning

the multi-bolted connection is the lever principle in its primitive form, a rough simplification,

which is also on the unsafe side, and therefore really should not be used.

Nevertheless, a reasonable solution of the system is still possible. Putting in the

relationship for O (11) for an effective clamping force eccentricity the value that edge

approaches (for example R0.8 S ) O may be determined. Adopting this value in (10 ') results in

FK evaluation that can be used for the determination of other forces in the connection.

However, shifting the bolt axis position and approaching the edge has an additional effect -

the reduction of the effective preloading force due to the additional embedding at the new loaded

separation surfaces caused by the change in the position of the clamping force. For the present, the

effective reduction of the initial preload can be approximated by the following relationship:

zV

f 1F tanh

(14) Therefore, the bolt additional force is to be calculated based on

FZ = FK + FA - FV + FV (15)

4. Determination of the influence numbers “b“

The influence numbers for a series of the typical connection forms could be determined (or

approximated) according to the table:

In this way, when the form of the connection is approaching the tabular forms, the

calculation of the connection or the estimation of the influence numbers necessary for the

calculation simplifies. The joining stiffness may be also taken from a FE-calculation. The

significance of this possibility is commonly underestimated. Compared to the usual FE -

calculations with non-adapted boundary conditions (at the joining) and linear behaviour, a

combined analysis delivers more and more exact information about the effect of the preloading of

the connection, eccentricity of the force introduction and the separation in the joining surfaces. A

non-consideration of the non-linear effects can particularly in the case of the FE-analysis leads to

inaccurate results.

The demonstration of the above method is given by two typical examples given in the

Annexe.

November 2010

Dr. P.Agatonovic

Pappelweg 11

D-85244 Roehrmoos

Germany

Tel.: ++8138-8160

Email: P.Agatonovic@t-online.de

Annexe:

AN EXAMPLE OF THE CALCULATION OF THE ECCENTRICALLY LOADED SINGLE

BOLT CONNECTION (USING MATHCAD)

Input Data for calculation:

Axial loading

Material data: Bolts Plates

Geometry:

Bolt:

Calculation of stiffness:

F 5000 N F V 11344.21 N

Es 2.0105 N

mm2

Ep 2.0105 N

mm2

0.3

H F 16 mm S R 18 mm B 30 mm a 40 mm

W 13 mm d N 8 mm i 1 2L

i

5 mm

13 mm

Di

8.5 mm

8 mm

H K 6 mm P 1 mm L K 16 mm A S 40.5 mm2 a 40 mm

sgP

E s A S

0.9d N

P

0.14d N

P

2

sg 4.963107 mm

N

sk0.15

H K Es

sk 1.25 107 mm

N

s1

E s1

2

i

Li

4D

i

2=

sg sk s 2.355106 mm

N

p1

E p d N Keg

ln W d N

W L K Keg d N

W d N W L K Keg d N

Keg .5

p 2.518107 mm

N

ps 0.6 p pf 0.4 p

s p 2.607106 mm

N

SE s ps

For the calculation the connection loading is scaled by the factor

Reduction of preload force

Using the plate thickness

the equation of the connection can be defined

and solved using MATHCAD procedure

with the edge correction

Based on the relationship

The forces in the connection can be determined

Or using so called "lever principle":

K 0.05 0.1 1.

F Bmax F F K( ) K F Bmax

F V K( )

f z tanh F K( )

F V

f z 0.005mm

K( )F V F V K( )

F K( )

SE

I1

12B H F

3 I 1.024 104

mm4

I m I

C 1a

3

3 E p I m

( )C 2 K( ) 0 C 3 K( ) K( ) C4 K( ) 1

x 1 t K( ) wurzel C 1 x3 C 2 K( ) x

2 C 3 K( ) x C 4 K( ) x

S s K( ) t K( ) a S ef K( ) wenn t K( ) a S RE t K( ) a S RE

S s K( )

m m

S ef K( )

m m

F K( )

0 1000 2000 3000 4000 50000

5

10

15

20

25

B K( )S s K( )

3

6 E p I m

F K K( ) K( ) F K( )

12 B K( )

F S K( ) F K K( ) F K( ) F Z K( ) F S K( ) F V F V K( )

The results are shown in the following diagram:

The comparison of the above calculation with FEM Results confirms the achieved results:

F ZH K( ) F K( ) 1a

S RE

F V F V K( )

F Z K( )

F ZH K( )

F V K( )

F K( )

0 1000 2000 3000 4000 50000

1000

2000

3000

4000

5000

6000

7000

8000

THE CALCULATION OF THE COMPLEX MBC USING MATHCAD

Numerous experimental investigations have shown recently, that the eccentrically loaded bolt

connections are essentially higher stressed in comparison to the centrically loaded. These stresses,

due to the non-linear relationships, cannot be determined using traditional calculation methods for

centrically loaded bolt connections.

Using the developed method with the help of MATHCAD, one can come to a simple and very

flexible solution.

Input data for the calculation

All data for the calculation are summarized according to the Fig. 1.

Fig. 1: Flange parameters

Bolt: Nom. Diameter

Number of screws

Bolt shaft

Thread pitch: Thread section Head diameter Head height

d N 10 mm Z 72

i 1 3L

i

3.5 mm

12.5 mm

4 mm

Di

10 mm

8 mm

7.181 mm

L3

R a

hsh

D1

D2

D3

Do

L1

L2 L

K

hF

H

W

Preload force

Flange height

Shell diameter Shall thickens

Bolt position diameter

Edge distance of the bolts

Main Connection data:

Clamping length

Bolt eccentricity

Material data:

Bolt Flange

Max. loading:

Axial force

Flange Moment

Complaisance of the components:

The total complaisance of the bolt is determined through the addition of the complaisance of its

individual elements. The complaisance of the screwed thread part

and of the bolt head

increase particularly the complaisance of short screws and should be considered. Accordingly, it

follows for the complaisance of the screw under installation conditions (cold):

For the calculation of the complaisance of the clamped parts the solution according to I. A. Birger

with an assumed cone can be used (Keg = tan )

P 2 mm A S 40.5 mm2 W 16 mm H K 5 mm

F P 25000 N

H F 10 mm

R 295 mm

H sh 5 mm

D o 610 mm

S R 15 mm

L K 20 mm a 10 mm

Es 2.05105 N

mm2

E p 2.15105 N

mm2

Poissonscoeff 0.3

F 1000000N

M 55000N m

sgP

E sk A S

0.9d N

P 0.14

d N

P

2

sk0.15

H K E sk

so1

E sk1

3

i

Li

4D

i

2=

sg sk

Keg .5

Under the influence the working loading a part of the flange facing the screw is pressed against

the screw and its loading arises, whereas the parts near to the separation surface are relieved. In

the first approach we assume, that the complaisance of the clamped parts is distributed according

to the.

so that follows

Entire complaisance of the connection equals the sum of all complaisance:

Attention: All values relate to the complete connection, i.e. two beams of equal thickness presses

to each other. For further calculation it is necessary to divide the values with 2.

Preload of the connection The installation preload of one connection is exposed to the different negative influences.

Due to the scatter of the values between individual screws the nominal preloading force should be

valued at the level of the minimum expected preload force value.

Furthermore, under the operational conditions additional displacements appear due to the

Shear displacements (rotation, tension)

Embedding

so that the remaining working preload force only

is.

Clamping complaisance of the connection segment

The twisting of the segment at the fixation of the connection with the remaining structure (in this

case a shell) is proportional to the section loading, clamping moment and the working force, i.e.:

= M*MB + F.FB

Co2

E Ck d N Keg

ln W d N

W L K Keg d N

W d N W L K Keg d N

ps 0.6 p pf 0.4 p

s 2.74 106 mm

N p 3.88110

7 mm

N pf 1.55310

7 mm

N ps 2.32910

7 mm

N

s p SE s ps

3.128106 mm

N

F Vm F V 10.4

Z

F Vm 2.382104

N

f Q 0 mm

f s 0.007 mm

F VR F Vm

f s

F VR 2.15810

4N

It follows for the shell and flange geometry

whereby for two equal flanges

From it, the factors necessary for the calculation of the complaisance of the clamping emerge

Connection forces

Assuming a linear load distribution the load conditions for the maximally loaded segments can be

determined

This force is calculated using 100 steps:

Influence of the bolt preload is considered through the factor

It follows for so called preload factor:

For the calculation of the clamping force eccentricity in the dependence on the working load the

characteristic equation of the connection with the constants

has to be solved (Mathcad procedure):

DE p H sh

3

12 1 2

4

3 1 2

R2

H sh2

I1

12

D o

Z H F

3

MZ

4 D R F 0 I 2.21810

3mm

4

I m I

b M 2 E p I M

b F 2E p I F

a

b M 224.887mm b F 0 mm

F Bmax1

Z

2 M

RF F Bmax 1.90710

4N

K 0.01 0.02 1.

F K( ) K F Bmax

f z 0.005mm

F V K( )

f z tanh F K( )

F V

K( )F VR F V K( )

F K( )

SE

C 1a

3

3 E p I

C 2 K( ) K( ) a

a b M b F

C 3 K( )2 a b M

a b M b F

K( ) C 4 1

x 1

It follows for

or after correction for edge (flange overhang) effect

Fig.2: Clamping force eccentricity

On the basis of this solution for the bending complaisance of the segment beam of the connection

we become:

The forces in the connection could be calculated based on the equilibrium relationships (with and

without edge correction)

Clamping force

Bolt force

Additional bolt force

If the position of the clamping force to the edge is so close that the pressure distribution in the

joint is not symmetric, the beam tilts on at the edge and the conditions o = 0 become

increasingly inaccurate. However, the so-called lever principle must not be valid for this case,

because the clamping is not free and when turning around the edge, it cannot happen without the

t K( ) wurzel C 1 x3 C 2 K( ) x

2 C 3 K( ) x C 4 x

S s K( ) t K( ) a

S RE 0.85S R

S ef K( ) wenn t K( ) a S RE t K( ) a S RE

S s K( )

mm

S ef K( )

mm

F K( )0 5000 1 10

41.5 10

42 10

40

4

8

12

16

20

B K( )S s K( )

3

6 E p I

F K K( ) K( ) F K( )

12 B K( )

F KR K( )a b M b F a F K( )

S ef K( ) 2 a S ef K( ) b M

F S K( ) F K K( ) F K( ) F SR K( ) F KR K( ) F K( )

F Z K( ) F S K( ) F VR F V K( ) F ZR K( ) F SR K( ) F VR F V K( )

influence of the restraint of the remaining structure. In this case we have first to calculate the

angle o:

Based on this calculation

Clamping force

Bolt force

Additional bolt force

The results comparison is shown in Fig. 3 and 4.

Fig. 3: Additional bolt force in the dependence on the segment loading

B a b M b F D 1 pf

2 S RE 3

6 E p I m

B1 b M S RE 2 a

IME K( ) K( )

D

B a

S RE B1

NEN

2 S RE

D

2 E p I m

S RE B1

O K( )IME K( ) F K( )

NEN

F KA K( )B a F K( ) 2 E p

I m O K( )

S RE B1

F SA K( ) F KA K( ) F K( )

F ZA K( ) F SA K( ) F VR F V K( )

F Z K( )

F ZR K( )

F ZA K( )

F K( )0 5000 1 10

41.5 10

42 10

4

0

2000

4000

6000

8000

1 104

Fig. 4:

Clamping force calculation

Knowing the forces the stress condition in the screw can also be calculated. The bending stresses

of the screw could be determined with the help of

For bending diameter of the bolt:

so that the total stresses are (Fig. 5):

Fig. 6: Stresses in the bolt

F K K( )

F K( )0 5000 1 10

41.5 10

42 10

41 10

4

1.2 104

1.4 104

1.6 104

1.8 104

2 104

2.2 104

K( )1

E p IF KR K( )

S ef K( )2

2

D b 8 mm I s

D b4

64

B K( ) K( ) E s

D b

L K

M S K( )F KR K( ) S ef K( )

2 I s

I m L K

K( )F S K( )

A S

B K( )

K( )m m

2

N

F K( )0 5000 1 10

41.5 10

42 10

4500

600

700

800

900

1000

Stresses at full load

with bending

Stresses in the flange

Two cross-sections at the flange become critical:

at the clamping to the remain structure

at the bolt holes

Fig. 7: Flange stresses

As can be seen, the tensile stresses at the flange transition are prevailing bending stresses.

However, an essentially higher stress concentration is expected in the bolt.

After arrangement of these basic subroutines, through the variation of parameters, sensitivity

analysis of the different influences can be easily performed.

November 2010

Dr. Ing. Petar Agatonovic

1( ) 941.058N

mm2

B 1( ) 190.564N

mm2

T1 K( ) Z F K( )

2 R H sh

6 Z F KR K( ) S ef K( ) 6 Z a F K( )

2 R H sh2

T2 K( ) Z F K( )

2 R H sh

6 Z F KR K( ) S ef K( ) 6 Z a F K( )

2 R H sh2

SB K( )6 Z F KR K( ) S ef K( )

2 R Z d N H F

2

T1 K( )m m

2

N

T2 K( )m m

2

N

SB K( )m m

2

N

F K( )0 5000 1 10

41.5 10

42 10

4200

0

200

400

600

800