Bolt–plate contact assemblies with prestress and external loads: Solved with super element technique

doi:10.1016/j.compstruc.2009.07.004

Computers & Structures

Volume 87, Issues 21–22, November 2009, Pages 1374-1383

https://doi.org/10.1016/j.compstruc.2009.07.004 Get rights and content

Abstract

In dynamically loaded bolted assemblies fatigue is the designing factor. The primary quantity that controls the fatigue is the part of the external load that is transferred to the bolt, as described by the ratio between transferred load to external load. In an idealized setting this ratio can be expressed by a stiffness ratio between the plate members and the bolt. The ratio of transferred load is, however, highly dependent on the size and position of the external loading.

The bolt–plate/member connection is modelled using the finite element technique. The modelling includes contact between bolt and members and between the clamped member plates. The modelling includes the prestress that is described by a single scalar value. This prestressed state is followed by an arbitrary external load case.

In this paper a finite element technique with super elements is applied to solve the contact problem of the combined prestress and external load in an overall non-iterative way. The conclusions are that it is difficult to make general design rules but in most cases the standard calculation methods will underestimate the strength of the connections. However for some specific loading situations this might not be the case.

Introduction

Bolt–plate assemblies, hereafter termed bolt-member assemblies, plays a significant role in many applications in relation to civil and mechanical designs. Fatigue is usually the controlling factor in the dimensioning of bolts. The simplest design rules can be found in many textbooks on machine elements such as [1], [2]. When the design becomes more complex in relation to multiple bolts in the connection and different locations of the external load, which also includes eccentrically loading, one is usually referred to the use of [3]. From the VDI it is known that the standard calculation of the load transfer ratio, as presented in many textbooks, is only valid in the case where a single bolt is concentrically clamped.

The design theory for bolted assemblies cannot be based solely on linear stiffness analysis, because external loads after the prestress change the problem into a nonlinear one. Both the position and the size of the external load have influence, especially the size of the contact area between the members has a large influence. This aspect is studied in [4], [5], [6] and specific design guidance can be found in [3] for specific cases. These presentations are mainly given as modification to the linear theory through nonlinear axial stiffness characteristics, obtained with FE models. In the present paper a more direct alternative presentation is given, with focus directly on the change in the bolt force due to the external loads.

Ref. [4] give a very informative presentation of the problem from a physical point of view, and then analyze a specific assembly, using the ANSYS program with special orthotropic thread elements and standard two-dimensional axisymmetric gap elements for the interfaces. Reference is given to the earlier research by [7], which research contain FE analysis (including plasticity) as well as experimental results, especially related to bolted connections in buildings.

Gerbert and Båstedt [5] similarly analyzed a FE model and compared with performed physical experiments. The authors found substantially lower force transfer to the bolt than the one suggested by [8]. For actual cases they state that the load fraction is fairly independent of the location of external load, this is in contrast to the findings of the present study.

Lehnhoff and Wistehuff [6] use a FE model with the I-DEAS software and gap elements at contact surfaces. Friction is not considered and threads are not modelled. Decreasing member stiffness with increasing external loads is reported, but the definition and use of this stiffness may be debated. [6] recommend experimental verification and future work on the subject. The present study gives such additional insight.

The FE solutions in [4], [7], [5], [6] apply fictitious gap/interface elements, without detailed specifications of these and of the necessary iterations. The direct approach of the present study does not apply such elements, and thus solves the contact problems without this approximation. Furthermore, the present approach directly determines the bolt force with combined prestress and external loads, i.e., without involving the bolt and member stiffness, as it is possible when the external loads ideally act at the contact zone.

The research area is still very active as seen in, e.g. [9], [10] related to specific bolted configurations or with a more general but also narrow subject of finding the stiffness of the bolt as presented in [11]. In [12] the subject was also finding the stiffness of the bolt and of the more involved part, i.e., the stiffness of the member. The method of analysis was extended in [13] to include material dependency through the Poisson’s ratio. In the later paper also possible design improvements are discussed.

The layout of the paper is as follows: first the problem definitions and assumptions are presented. The difference between distributed force analysis and single force analysis is commented together with an introduction of the main notations. In Sections 3 FE analysis for the stiffness of the bolt part, 4 FE analysis for the stiffness of the member part the independent analysis of bolt and member is presented with focus on the relation between elastic energy and stiffnesses. Then the pure prestress problem is formulated and solved in Section 5 applying the super element technique for the combined FE analysis. This problem must be solved only once for an assembly problem, independently of the external loads, and the only resulting quantity for the further calculations is the displacement of mutual compression, e, that is proportional to a given total prestress force $F_{p}$ . The distribution of this contact prestress force ${F_{cp}}$ is also determined, but these forces are not used in the further calculations.

Formulation and solution with external loads on the prestress state is presented in Section 6, and the main result of this analysis is the distribution of new contact forces ${F_{c}}$ that account for external loads as well as for the prestress state. With the force vector ${F_{c}}$ , bolt and member can be analyzed independently as done in a traditional FE approach, giving all the necessary details. It is important to note that in many cases the boundary condition between members is different with and without external loads, this is the source to the nonlinearity of the problem. Finally the bolt and member analysis in Section 7 focus on the change in bolt load as a function of the size and the position of the external loads. This includes comparison with results that follows from a pure prestress state with the important ratio of member stiffness to bolt stiffness.

The examples play a major role, although conclusions are given in rather general terms. For certain assemblies with small member width, the contact zone between members is not changed by the external loads. From this follows a linear dependence on the size of the external loads, and examples of this case are presented in Section 8, that also includes weakly nonlinear cases. The nonlinear case of changing contact zone between the members is exemplified in Section 9 and here a few iterations are necessary to locate the node where the members separate. However the computations are still so moderate that interactive solutions are performed.

Section snippets

Problem description and assumptions

In the present paper it is assumed that we can use an axisymmetric model and a symmetry plane for the axial direction. The washer is treated as an integrated part of the bolt, i.e., no contact surface between bolt and washer. Although the bolt and the member, as defined in Fig. 1, are analyzed separately, it is possible to solve the contact problems between washer (bolt head) and member and between the members. Thus the correct distribution of contact forces (stresses) are determined not only

FE analysis for the stiffness of the bolt part

The stiffness $k_{b}$ of the bolt part is often determined by simple calculations, approximating the deformation of the bolt head and of the nut. Assuming uniform tensile stress $σ$ and uniform strain $ϵ = σ / E$ for a total contact pressure force $F_{p}$ with a uniform bolt cross sectional area $A_{b}$ , we get for the total elastic energy $U_{b}$ (sum of the elastic strain energy $U_{ϵ}$ and the elastic stress energy $U_{σ}$ in the bolt and for linear elasticity $U_{b} = 2 U_{ϵ} = 2 U_{σ}$ ). $U_{b} = U_{ϵ} + U_{σ} = σ ϵ \tilde{V} = \frac{σ^{2} \tilde{V}}{E} = \frac{F_{p}^{2} A_{b} \tilde{L}}{A_{b}^{2} E} = \frac{F_{p}^{2} \tilde{L}}{{EA}_{b}}$ where $\tilde{V}$ is the

FE analysis for the stiffness of the member part

The stiffness of the members is termed $k_{m}$ and is defined similar to (3) $k_{m} = \frac{F_{p}}{v_{m}} = \frac{F_{p}^{2}}{F_{p} v_{m}} = \frac{F_{p}^{2}}{U_{m}}$ but without the simple approximation of the total elastic energy (2), as discussed in detail in [12]. However, from a finite element calculation $U_{m}$ is available.

The contact problem where the two members meet needs to be solved iteratively, but this is a relatively simple contact problem. A single parameter ( $d_{c}$ in Fig. 1) must be determined such that only compressive contact forces act between the

Contact pressure distribution with prestress only

The distribution of the contact pressure has a not negligible influence even on the global stiffness quantities, especially for the members. We therefore describe how this distribution can be obtained in a direct procedure, i.e., without iteration.

The bolt analysis can be carried out as a super element FE procedure $[S_{bp}] {D_{bp}} = {F_{cp}} \Rightarrow {D_{bp}} = [S_{bp}]^{- 1} {F_{cp}}$ where $[S_{bp}]$ is the bolt super element stiffness matrix of order equal to the number of nodes at the bolt-member contact surface, with degrees of

Super element approach with prestress and external loads

External loads can act on the bolt (not the usual case), on the member, or on both. They can be a single force, multiple forces or even distributed forces. When calculating the super element stiffness matrices, the equivalent loads on the contact nodal degrees of freedom are determined for the external load on the member as well as for eventual external load on the bolt. They correspond to the negative reactions at the contact nodal degrees of freedom when these are given fixed support in the

Final bolt and member analyses

When the contact forces ${F_{c}}$ are known for the problem of prestress followed by external loads, then the bolt and the member can be analyzed independently, and detailed FE results are available.

The goal is here to concentrate on the total bolt force without detailed analysis of stress concentrations. Thus the total reaction force at the symmetry plane orthogonal to the z-axis, indicated in Fig. 2, is the main quantity. For the bolt the reaction forces in the z-direction are described by the

Examples with small member width

The chosen total prestress force is a primary quantity and therefore specifically discussed. The stress area of the bolt is approximated by $A_{s} = \frac{π}{4} {(\frac{d_{p} + d_{m}}{s})}^{2}$ where the pitch diameter $d_{p}$ and the minor diameter $d_{m}$ , assuming ISO standard thread, are given by $d_{p} = d_{nom} - \frac{3}{8 \tan (\frac{π}{6})} P, d_{m} = d_{nom} - \frac{17}{24 \tan (\frac{π}{6})} P$ where P is the thread pitch. For a M10 bolt the standard thread pitch is, $P = 1.5 mm$ , and for a M20 bolt we have that $P = 2.5 mm$ . Applying this we find that $A_{s} = 0.58 d_{nom}^{2}$ for the M10 bolt and $A_{s} = 0.61 d_{nom}^{2}$ for the M20 bolt.

Examples with large member width

The examples to follow are based on a M10 bolt with prestress $F_{p} = 2 \cdot 10^{4} N$ and the geometry parameters are $\begin{matrix} L = 60 mm, d = 10 mm, α = 1.1, β = 1.5, γ + ζ = 0.9 \\ d_{a} = 71 mm, d_{c} to be determined for the specific problem \end{matrix}$ These cases with nonlinear dependence show three different aspects. First how external load position influences the extension of contact region between members and the important ratio $ϕ$ , followed by studying the influence from the size of the external loads. Then influence from radial member constraints as an

Conclusion

External loads on a prestressed bolt-member assembly are analyzed with focus on the part of the external loads to be taken by the bolt, which is the important part for fatigue strength. This problem is often tackled with rather rough assumptions and the present paper shows how a detailed axisymmetric analysis is possible.

The performed FE analysis uses super element technique that solves the involved contact problem without iteration and incrementation. Primarily the pure prestress problem is

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