ANALYSIS OF STRESS DISTRIBUTION IN ROOTS OF BOLT THREADS Gennady Aryassov, Andres Petritshenko Tallinn University of Technology Department of Mechatronics

1. Motivation When threaded joints are subjected to an axial load a stress concentration in thread roots take place (Fig.1). Under ideal conditions, the tension in the screw (bolt) and the compression in the nut (such scheme of loading is designated as Bolt-Nut I (Fig.1a) should be reduced uniformly starting from full load at the first contact between screw and nut threads. The same condition is required for other schemes of loading when the screw is compressed and the nut is tensioned (Bolt-Nut II) (Fig.1b) or both screw and nut are tensioned (Tightener) (Fig.1c) or compressed (Post) (Fig.1d). F F/2 c) d) b) a) q(z) Fig. 1 F(i)

It is well known that axial load acting on a bolt is distributed between threads of threaded joints extremely nonuniformly. So for standard metric, trapezoidal, rectangular and buttress threads, having nominal diameter 10mm with five working threads, the loads on the most loaded threads are 33,1; 48,1 and 51,9 percents of axial load on a bolt accordingly. For threaded joints, working within dynamic loads or made of sensitive materials to stress concentration, stresses in the root of the first working threads can considerably exceed nominal tensile stresses in a cross section of bolt shank, and this fact considerably reduces fatigue life of the threaded joints. Russian scientist N. Zhukovski first proposed solution of nonuniform distribution of axial load between threads of threaded connections in 1902. He idealized that roots of threads are independent to each other and nonuniform distribution of axial load between independent roots of thread occur due to shear only.

More accurate solution of nonuniform distribution of axial load between roots of real threaded connections was reported by (Birger & Iosilevich, 1973). Roots of threads were considered as continuous and actual stress distribution in thread roots was defined by force flows, which cause tension of the bolt body, as well as bending and shearing of the threads. In this case, compatibility of displacements between roots of threads of threaded connections was used as basic assumption for solution. Solution of nonuniform distribution of axial load in some idealized threaded connections by finite element method was introduced by (Marujama, 1973, 1974, 1976). Later in their studies (Strizhak et al.,1999 and Aryassov et al., 2000) have shown that , that stress distribution and stress concentration in threaded connections by existing calculation schemes depends upon loading schemes mainly and so can not be extended to every kind of thread connections.

As it has seen in limited overview, a lot of contribution is made to develop precise calculation schemes for stress distribution in thread connections. Most of existing calculation schemes is idealized and don’t taking into account thread fit or tolerances in thread connections, which plays a vital role in precision of calculations. Due to thread fit or tolerances between the bolt and nut threads, solution of real stress distribution and stress concentration in thread connection is more complicated. In this paper more general calculation scheme of stress distribution and stress concentration is proposed. Proposed calculation scheme takes into account thread fit in thread connections and based on more exact models of finite element method. Theory to use extended approach in calculations is given as well.

2. Basis for calculations As far as threads in threads connections can be considered as a group of small recesses, in which circular stresses can be ignored in comparison with the axial and radial stresses, a task of calculation of threaded joints can be reduced to axisymmetric task of the elasticity theory. In process of calculation by the finite element method, a detail is divided into a suitable number of the finite element, connected to each other at nodal points. Relation between displacements of nodal points and acting forces is described by [K] {u} = {f} (1) where [K] – stiffness matrix, {u} and {f} – vector-columns of nodal displacements and forces (external and contact forces).

Thread connection is meshed by square element with 16 degrees of freedom (Fig.2). Every nodal point has two linear and two angular generalized displacements. Stiffness matrix of square element is obtained by variation of potential energy of body deformation, which is described by b a z q5 r q6 q7 q8 q2 q3 q4 q1 q15 q16 q14 q13 q11 q12 q10 q9 Fig.2 where v and w - components of displacements on the y and z - axes, μ – Poisson’s ratio, G – shear modulus, and - non-dimensional coefficients

Displacements inside the square element are to be approximated by interpolation polynomials as follows (3) where  - unknown coefficients of interpolation polynomials, z and r - local coordinate axes. Displacements inside the elements with respect of generalized coordinates are (4) and - interpolation shape functions, according to kinematical relations by theory of elasticity. Elements of stiffness matrix can be found by substituting equation (4) to expression of potential energy (2) and following variation (2) with respect of generalized coordinates .

Mathematical model for determination of load distribution in a contact zone between threads is represented in Fig. 3.   Fig. 3. Interaction between bolt and nut threads For determination of load distribution compatibility equations of displacements and equilibrium are used. The equation of equilibrium for one thread component can be expressed as (5) where - contact pressure, i - number of thread root in contact, - area of contact zone. a)

Compatibility equation of displacement can be expressed by contact points C1 and C2 between bolt and nut thread as represented in Fig. 3 and are (6) where and - kinematical displacements in global coordinate system, and - displacements in local coordinate systems. For determination of the contact pressure in finite element model, delta function is used.

The kinematical displacements represented in Eq The kinematical displacements represented in Eq. (6) are projected to the normal of the working plane of roots of thread using delta function can be expressed as (7) where - influence function of contact pressure for bolt and nut threads, - influence function of external forces for bolt and nut threads, - width of k’s block of shape function of contact pressure,  - angle between roots of thread.

3. Calculation results Thread M10 was used as calculation model and it was subjected to uniformly distributed stress 1=10 МPа on the bolt cross-section. Modulus of elasticity Е = 200 GPа, Poisson's ratio  = 0.3 and influence of temperature and gravity were not considered. Model assumes unmovable hinged nodes on an axis r, passing within of free bolt edge, which corresponds to constrained deformation conditions appearing, when there are free threads of the bolt over the nut.

Results of calculation of load distribution on threads are given in Table 1 and are compared with calculation results according (Birger & Iosilevich, 1973), (Marujama, 1976) and to the ANSYS program, which used a standard triangular element with six nodal points. Table 1. Contact load (pressure) in roots of thread

Results of the present calculation of stress distribution (MPa) in roots of thread are represented in Fig. 4. M10 Fig. 4. Stress distribution in roots of thread

Fig.5. Rectangular three-dimensional element of plate 3-D model of the finite element with 20 degrees of freedom (in every nodal point are 5 displacements –two in-plane, one transverse and two angular) is also used. This model was not used up to day for the calculations of the thread joints. In this work achieved results with this model have a preliminary character. 14 8 7 6 10 9 5 4 3 1 2 17 16 18 19 20 15 12 11 13 z y x Fig.5. Rectangular three-dimensional element of plate

4. Conclusion The calculations results of load distribution coincide with results which were made by (Marujama, 1973,1974) and (Strizhak et al., 1999) and with results of experimental researches (Marujama, 1974, 1976). Analysis of results shows, that loads on the first roots of thread calculated by the more exact model are higher as compared to the simplified ones for 10-15. Calculation results of stress distribution according to calculated load distribution between roots of thread show differences in stress state rate in comparison to stress state rate determined by theoretical coefficients of stress concentration. In future calculated theoretical results will be compared to theoretical results calculated by more complex 3D finite element models and will be done experimental researches of thread joints for ladder frame construction.

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