general design method for bolted connections - soeren stephan et al

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Soeren STEPHAN et al: General Method for the Design of Bolted Connections for Space Frames, Space Structures 5, Telford Publishing, London 2002, p. 759-773(Corrected Edition)

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S. Stephan, Page 1

A General Method for the Design of Bolted Connections for Space FramesS. 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): (1) ( y , z ) = z y y z + o

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: lim ; E (y, z ) lim (y, z ) = E (y, z ); (E (y, z ) > lim ) (E (y, z ) 0 ) (2) 0 ; E ( y, z ) > 0 Further the tensile stress (positive) for any bolt can be calculated from the following formula: b lim ; E (y, z ) > b lim b(y, z ) = E (y, z ) ; (E (y, z ) > 0 ) (E (y, z ) b lim ) (3) 0 ; E ( y, z ) 0 Finally the normal force and the bending moments at the connection can be determined from: N = Ab v b(y v , z v ) + A (y, z ) dA (4a)

My = Ab v b(y v , z v ) z v A (y, z ) z dA

v

(4b) (4c)

Mz = Ab v b(y v , z v ) y v + A (y, z ) y dAv

v

Thus a certain combination of strain parameters y, z and o 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, Z k) and a particular thickness tJ which is constant over the length of the element (Fig. 1). v Any bolt (index v) has a particular thread diameter dv dv which determines the J accompanying stress area Abv.i

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 and moments Norig, Myorig, Mzorig.

tJ

LJ

O X

k

Y

Fig. 1: General scheme of a multi-bolt connection

This combination of normal force and bending moments will be used as initial values for the iteration parameters (iteration index it):

it = 0 ,

n

it

= N orig ,

my

it

= My orig ,

mz

it

= Mz orig

(5)

S. Stephan, Page 3

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

=

mz , E Jz

it

y

it

=

my , E Jy

it

o

it

=

n EA

it

(6)

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

(y, z ) it = z

it

y y

it

z + o

it

(7)

Figure 2 shows the strain distribution in the connection for an arbitrary combination of internal forces and moments.

Z

>0

The surfaces of the adjacent connection profiles will contact each other only within areas with negative strain ( < 0). The contact zone is limited by the zero strain line ( = 0) =0 Compression forces will be transferred only within the contact zone. Therefore only 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:LJ =

(y i,J y k ,J )2 + (z i,J z k ,J )2it

(8)it

Additionally the node strains are needed for the further calculation:

i = (y i,J , z i,J ) , k = (y k ,J , z k ,J ) (9) If there is a point with zero strain along the line element, the distance from node i must be determined: E i u0J = L J , u 0 J [0 K L J ] (10a) E i E k 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: E i lim u lim J = L J , u lim J [0 K L J ] (10b) E i E k

S. Stephan, Page 4

Thus the coordinates of the point with zero strain can be calculated: u0 u0 y0 J = y i,J + J (y k ,J y i,J ) , z0 J = z i,J + J (z k ,J z i,J ) LJ LJ In a similar way the coordinates of the point with limit stress will be determined: u lim J u limJ y limJ = y i,J + (y k ,J y i,J ) , z limJ = z i,J + (z k ,J z i,J ) LJ LJElements 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.X Z

(11a)

(11b)

i J z0 J zlim J

O y0 J ylim J

k

Y

limu0 J ulim J u LJ u

The splitting operation into sub-elements has to Fig. 3: Strain and stress distribution for a selected line element be executed as follows: y y0 J y limJ y k ,J ; u 0 J u limJ i ,J (12a) YJ = y i ,J y limJ y0 J y k ,J ; u 0 J > u lim J z z0 J z lim J z k ,J ; u 0 J u limJ i,J (12b) ZJ = z limJ z0 J z k ,J ; u 0 J > u limJ z i,J The length of a sub-element (Fig. 4) with its two nodes (index i, k) must be calculated from:

[ [ [ [

] ] ] ]

L sub j =

(Yi, j Yk , j )2 + (Z i, j Z k, j )2(

(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: lim ; E i, j lim i , j = E i, j ; E i, j > lim E i, j 0 (14) 0 ; E i, j > 0

) (

)

S. Stephan, Page 5

The same formula can be applied for k,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 i,j at node i and k,j at node k.

Z

j

i

tj

MZsub j k

Hence the arising normal force MYsub j Y will be obtained from the O product of the stress trapezoid X ij area and the element thickness. Nsubj With that the bending moments Lj kj 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: 1 N sub j = i, j + k , j t j L sub j (15a) 2 i, j + 2 k , j My sub j = N sub j Z i, j + Z k , j Z i, j (15b) 3 i, j + k , j i, j + 2 k , j Mz sub j = N sub j Yi, j + Yk , j Yi, j (15c) 3 i, j + k , j

(

)

(

)(

)

(

)(

)

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 tensi