Regarding all the new constraints (environment, energy, material and transportation costs, …), designers work hard to optimize their structure. Given their known advantages, bolted connections are often used in mechanical structures. To optimize these components, designers need a simplified approach (analytical) to produce a first sketch before performing any FE Analysis or experimental testing. The existing analytical approach (VDI2230 [1] , NFE 25030 [2] , Cetim-Cobra [3] , EN13001 [4] , Guillot [5] [6] , Alkatan [7] , Massol [8] , …) provides formulas to calculate the behavior of a single joint (critical fastener). However, most industrial assemblies are defined with multiple connection points. To overcome this limitation, users use some assumptions, e.g., uniform distribution of loads and linear distribution of moments [1] [9] - [15] . For some specific applications, authors [1] [16] - [18] have developed specific analytic approaches to take into account the elasticity of loaded parts: Chakhari [17] used a hybrid approach (Simplified Finite Model) to treat an assembly with two bolts subjected to external axial load. He demonstrates that only the first preloaded bolt supports the full external load before any contact separation of its parts. Paroissien [18] developed a similar model to Chakhari’s model: in his model, he treats a hybrid assembly (bonded and riveted) subjected to a shearing load. In those hydride approaches, the adhesive and/or part-contact are considered as spring elements, and the parts are modeled by a series of beam elements, then a global FE matrix is built with appropriate boundary conditions. M. Sinthusiri & Nassar [16] and N. Konkong & K. Phuvoravan [19] have detailed another approach based on springs to represent loaded parts and joints of an assembly subjected to shearing load. Those cited works show very interesting results. On the other side, Geoffrey L. Kulak [20] provides experimental feedback of the load distribution of shared assemblies that confirms the limitation of the basic assumption (uniform and linear distribution of external loads on fastening points). P. Kawecki, A. Kozlowski [21] give a lot of experimental results that confirm previous observations. For slewing bearing, Z. Chaib and A. Daidie [22] and [23] have used in the first time FEA with a specific bearing modeling to calculate the load in each rolling element to study the load distribution on bolts in the second time. They have demonstrated that load distribution depends on a lot of geometric and physical parameters. It demonstrates the utility of using the “Krep” factor in the EN 13001-3-4 standard [2] to amplify loads obtained by the analytic distribution when flange supports of slewing bearings had some stiffeners.

Despite their accuracy, the above-listed models that take into account parts’ elasticities are not used in industrial applications for different reasons: the validity of the model is limited to a specific geometry of clamped parts, a specific load, or a specific type of joint (rivet, bolt, adhesive, hybrid…). Another factor that is frequently considered in industrial applications is the ease of use and verification of the approaches used. This criterion is missed by most authors.

The aim of this paper is to identify, using the simplest possible approach, the most critical joint, taking into account the main factors that influence the distribution of external loads.

To ensure a clear understanding of the proposed approach, it is essential to review some fundamental concepts, particularly in the context of preloaded joints. Preloaded joints exhibit complex mechanical behavior influenced by factors such as the level of applied preload, friction coefficients, and external loads acting on each joint. Specifically, in the case of a single-bolt assembly subjected to an axial load (F_a), the resulting load in the bolt (F_b) can be described by a bi-linear relationship, as expressed in Equation (1).

$\begin{array}{c}{F}_b=K_b\times \Delta L_b=\max \{\begin{array}{c}{F}_0+\lambda \times F_a\to Beforepartseparation\\ F_a\to Afterpartseparation\end{array}\end{array}$ (1)

According to Figure 1 , the part displacement (separation) at applied load points of each bolt could be defined by Equation (2).

$\begin{array}{c}\Delta L_{ass}=\Delta L_{b,Fa}+\left(1-\beta \right)\times {\delta }_P\times F_a\end{array}$ (2)

During assembly loading and before any contact opening, additional bolt elongation is done by: $\Delta L_{b,Fa}=\lambda \times F_a\times {\delta }_b$. Finally, the local assembly elongation could be expressed by Equation (3).

$\begin{array}{c}\Delta L_{ass}=\left(\lambda \times {\delta }_b+\left(1-\beta \right)\times {\delta }_P\right)\times F_a\end{array}$ (3)

From the last equation, local assembly resilience could be done by Equation (3).

$\begin{array}{c}{\delta }_{ass}^{F_a}=\lambda \times {\delta }_b+\left(1-\beta \right)\times {\delta }_P\end{array}$ (4)

After contact opening, only the load factor “ $\lambda$” should be set to 1.

For multi-joints, the displacement at loading point “ $\Delta$” is the same for all connections. So, for a joint “i”, the total displacement at the loading point is done by the addition of the part-elongation in concerned bolt “i” (due to F_a), and its deflection due to “F_a” and “M_e” as expressed by Equation (5).

Figure 2 , Equation (6) and Equation (7) provide an example of a part deflection in case of a prismatic part loaded by an excentric axial load “F_a” and by an external bending moment “M_e”. Other formulas could be used for other geometries.

$\begin{array}{c}\Delta =\Delta L_{P,i}=\Delta L_{ass,i}+\Delta L{B}_{Pi,Fa}+\Delta L{B}_{Pi,Me}\end{array}$ (5)

$\begin{array}{c}\Delta L{B}_{Pi,Fa}=\frac{L_i^3}{3\times E_{Pi}\times I_{Pi}}\times F_a=\frac{L_i^3}{3\times E_{Pi}\times \frac{w\times h_{}^3}{12}}\times F_a={\delta }_{B,Pi}^{Fa}\times F_a\end{array}$ (6)

$\begin{array}{c}\Delta L{B}_{Pi,Me}=\frac{L_i^2}{2\times E_{Pi}\times I_{Pi}}\times M_e=\frac{L_i^2}{2\times E_{Pi}\times \frac{w\times h_{}^3}{12}}\times M_e={\delta }_{B,Pi}^{Me}\times M_e\end{array}$ (7)

where:

To simplify formulas, the part elongation could be defined by Equation (8).

$\begin{array}{c}\Delta =\Delta L_{Pi}=\left({\delta }_{ass,i}^{Fa}+{\delta }_{B,Pi}^{Fa}\right)\times F_a+{\delta }_{B,Pi}^{Me}\times M_e\end{array}$ (8)

The same analysis could be done for any shearing load and sliding moment as shown by Figure 3 and Figure 4 . Some specific features should be considered for local behaviors of preloaded joints under any transversal load and bending moment. Indeed, due to preload and friction/adherence coefficient between the clamped parts and at the fasteners bearing surfaces, the assembly could have a complex behavior as illustrated by Figure 5 . Here, the total displacement/rotation at the loading point is the sum of the local joint movements and the loaded parts elongations. The parts elongations in the “X” and “Y” directions are done by Equation (9) and Equation (10). And the joint movements could be defined by Equation (11).

$\begin{array}{c}\Delta L_{P,X}=\Delta L_{b,X}+\Delta L_{P,F_{s,x}^{ext}}+{\theta }_b\times Y_{F_{s,x}^{ext}}\end{array}$ (9)

$\begin{array}{c}\Delta L_{P,Y}=\Delta L_{b,Y}+\Delta L_{P,F_{s,y}^{ext}}-{\theta }_b\times X_{F_{s,y}^{ext}}\end{array}$ (10)

Before any sliding, bolt deflections ( $\Delta L_{P,X}$ & $\Delta L_{P,Y}$) and joint rotation ( ${\theta }_b$) are ignored. Then after sliding those quantities could be calculated using basic notion of material strength. And for most industrial assembly with multi-joints points, the joint rotation is close to zero (0).

$\begin{array}{c}\Delta L_{P,F_{s,x}^{ext}}=\left(\frac{\left|X_{F_{s,x}^{ext}}\right|}{E_{P1}\times w_{P1}\times b_{P1}}+\frac{\left|X_{-F_{s,x}^{ext}}\right|}{E_{P2}\times w_{P2}\times b_{P2}}\right)\times F_{s,x}^{ext}\end{array}$ (11)

$\begin{array}{c}\Delta L_{P,F_{s,y}^{ext}}=\left(\frac{\left|Y_{F_{s,y}^{ext}}\right|}{E_{P1}\times w_{P1}\times L_{P1}}+\frac{\left|Y_{-F_{s,y}^{ext}}\right|}{E_{P2}\times w_{P2}\times L_{P2}}\right)\times F_{s,y}^{ext}\end{array}$ (12)

To ensure a good behaviors of preloaded connections (specially bolted connections), three main practical rules should be taken into account: no sliding between parts (micro or macro), no separation or significant opening at contact surfaces and anticipate or compensate any preload loosening [1] - [8] .

The proposed approach involves distributing external loads across connection points, based on a set of key assumptions:

By considering basic knowledge listed below, two main cases could happen during the application of external load:

Case 1 (Safe Industrial Design): all connection points are conformed to practical rules mentioned above (no sliding, no opening and no separation). In this case, preloaded joints have a linear behavior under any axial and/or any transversal loads. By consequence, internal joint loads could be defined as a function of loading point displacements (Equation (13)).

$\begin{array}{c}\left[\begin{array}{cccccc}& & & & & \\ & & & & & \\ & & & & & \\ & & & & & \\ & & & & & \\ & & & & & \end{array}\right]\times \left[\begin{array}{c}\Delta L_{P,x}\\ \Delta L_{P,y}\\ \Delta L_{P,z}\\ \Delta {\theta }_{P,x}\\ \Delta {\theta }_{P,y}\\ \Delta {\theta }_{P,z}\end{array}\right]=\left[\begin{array}{c}{F}_{b,x}\\ F_{b,y}\\ F_{b,z}\\ M_{b,x}\\ M_{b,y}\\ M_{b,z}\end{array}\right]\end{array}$ (13)

By writing the equilibrium principle for assembly at the loading point, all internal loads could be calculated as function of external loads.

$\begin{array}{c}\sum {\left\{\begin{array}{c}\begin{array}{c}{F}_{b,x}\\ F_{b,y}\end{array}\\ F_{b,z}\end{array}\text{|}\begin{array}{c}\begin{array}{c}{M}_{b,x}\\ M_{b,y}\end{array}\\ M_{b,z}\end{array}\right\}}_{\left(x_{Fe},y_{Fe},z_{Fe}\right)}={\left\{\begin{array}{c}\begin{array}{c}{F}_{e,x}\\ F_{e,y}\end{array}\\ F_{e,z}\end{array}\text{|}\begin{array}{c}\begin{array}{c}{M}_{e,x}\\ M_{e,y}\end{array}\\ M_{e,z}\end{array}\right\}}_{\left(x_{Fe},y_{Fe},z_{Fe}\right)}\end{array}$ (14)

$\begin{array}{c}\underset{i=1}{\overset{n}{{\displaystyle \sum }}}\left[\begin{array}{cccccc}& & & & & \\ & & & & & \\ & & & & & \\ & & & & & \\ & & & & & \\ & & & & & \end{array}\right]\times \left[\begin{array}{c}\Delta L_{P,x}\\ \Delta L_{P,y}\\ \Delta L_{P,z}\\ \Delta {\theta }_{P,x}\\ \Delta {\theta }_{P,y}\\ \Delta {\theta }_{P,z}\end{array}\right]=\left[\begin{array}{c}{F}_{e,x}\\ F_{e,y}\\ F_{e,z}\\ M_{e,x}\\ M_{e,y}\\ M_{e,z}\end{array}\right]\end{array}$ (15)

This approach could be divided into many simplified approaches: distribution of axial load, transversal load, bending moment and sliding moment.

For axial load (F_a ≠ 0, M_e = 0), by using Equation (14) and Equation (8), the axial displacement could be calculated by using Equation (16). And the axial load supported by each joint “i” could be estimated by Equation (17).

$\begin{array}{c}\Delta =\frac{F_a}{{{\displaystyle \sum }}_{i=\mathrm{1..}n}{K}_{eq}^{ass,i}}\end{array}$ (16)

$\begin{array}{c}{F}_a^{joint,i}=\frac{K_{eq}^{ass,i}}{{{\displaystyle \sum }}_{i=\mathrm{1..}n}{K}_{eq}^{ass,i}}{F}_a\end{array}$ (17)

where $K_{eq}^{ass,i}=\frac{1}{{\text{δ}}_{\text{ass},\text{i}}^{\text{Fa}}+{\text{δ}}_{\text{B},\text{Pi}}^{\text{Fa}}}$ is the equivalent stiffness between the joint “i” and the loading point.

Case 2 (Unsafe Industrial Design): The is a risk of sliding or joint opening. In this case the load distribution cannot be linear when external load is applied, and other phenomena may occur (settlement, spontaneous unscrewing, …) that leads to loosening preload after unloading the structure. Here, only iterative calculations could be applied, and in each iteration, stiffness will be updated regarding residual load or residual displacement at each joint.

This paper introduces a new analytical method designed to assist engineers in distributing external loads on preloaded bolted assemblies. The proposed approach takes into account key factors that influence load distribution, including geometric parameters (such as thickness, width, spacers, joint positions, and diameters), material properties of the loaded components (Young’s modulus), and the bolt preload applied to each joint. It’s mainly available for assemblies designed to prevent contact opening and sliding between loaded parts. An iterative approach could extend its validity area.

In order to validate the analytical formulas, experimental tests and finite element analysis (FEA) were performed on two representative industrial assemblies: a simple flat plate and welded plates. The results obtained from the analytical method were indirectly compared with those from FEA and experimental studies. These comparisons demonstrate a strong correlation, confirming that the method accurately identifies the most stressed joints.

In a future paper, a complementary study will present the application of this approach to assemblies subjected to shear loads.