Assumptions

X.H.1 Assumptions

We make several assumptions to simplify the developments that follow:

All the developements are done considering Cartesian coordinate systems. If the finite element model produces results in other type of coordinate systems, these results (forces, moments and coordinates) must be expressed in a Cartesian coordinate system before performing the bolt group calculations.
The global force and moment transferred by the group of connections is noted by $F$ and $M$ respectively. These vectors are defined wrt to bolt group center of gravity that will be defined later.
the two parts assembled by a group of connections are assumed infinitely stiff compared to the connections. The only source of flexibility of the assembly is the set of connections.
Each connection of the group is characterized by two scalar stiffnesses: a translational one and a rotational one. These stiffnesses are noted $k_{T}^{i}$ and $k_{R}^{i}$ respectively. We will see below that this assumption is a simplification of a general case.
The behaviour of each connection in the group is linear. The same is then true for the group of connections global behaviour.
Each connection is also characterized by its location in bolt group local coordinate system. Bolt coordinates are noted $x_{j}^{i}$ in which index $i$ corresponds to the three position components and index $j$ loops on the $N$ connections of the group ( $j = 1 . . . N$ ).
Bolt group local coordinate system is parallel to connection loads extraction coordinate system, but its origin is located as the bolt group center of gravity.

The calculation of bolt group center of gravity is made easier by the fact that connection translational stifnesses are scalar. If connection coordinates in the initial extraction Cartesian coordinate system are noted ${x_{j}^{i}}^{'}$ , center of gravity is calculated as follows:

x_{j}^{C o G} = \frac{\sum_{i = 1}^{N} k_{T}^{i} {x_{j}^{i}}^{'}}{\sum_{i = 1}^{N} k_{T}^{i}} .

(X.H.1)

Note that this simplification is possible only because connection translational stiffnesses are scalar (isotropy assumption). Once the center of gravity has been calculated, a new Cartesian coordinate system parallel to the initial one and with origin as center of gravity is defined. In this translated coordinate system, connection locations are simply calculated as:

x_{j}^{i} = {x_{j}^{i}}^{'} - x_{j}^{C o G}

(X.H.2)

and all the following calculations will be done using these coordinates.

Note that the definition of a bolt group center of gravity is possible only because connection stiffnesses are scalar. Otherwsie, a different center of gravity would be calculated for each direction. Actually, it would be even worse that that: the deformation of the assembly in one direction could lead to a global force with a different direction. This means that no center of gravity could be calculated.

The total force and moment transmitted by an interface are calculated as follows:

F = \sum_{i = 1}^{N} f_{F E M}^{i},

(X.H.3)

M = \sum_{i = 1}^{N} x^{i} \times f_{F E M}^{i} + \sum_{i = 1}^{N} m_{F E M}^{i},

(X.H.4)

in which $f_{F E M}^{i}$ and $m_{F E M}^{i}$ are the connection loads extracted from FEM results before bolt group redistribution.

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