The model of bent threaded connection in three segments/ Triju ruozu lenkiamos srigines jungties modelis.

Krenevicius, A. ; Juchnevicius, Z. ; Leonavicius, M.K. 等

1. Introduction

The analysis of load distribution in the threads is a prerequisite for determining stress concentration and fatigue durability of the threaded connection which can be subjected not only to axial force but also simultaneously to bending moment. In engineering practice, the loading is usually asymmetric or eccentric and this causes bending moments to be applied to structures or to it's elements such as rods and the threaded joints also [1-6].

The experimental study of the effect of bending directly on the distribution of stresses along the helix of the thread root by using photoelastic models is presented in [6]. However in analytical detailed calculations of the stud fatigue strength the turn loads and the loads in crosssections of stud/bolt core are determining primarily and then the stresses in turn roots [7]. Previous analytical analysis of the load distribution in threads due to multiple loading which includes bending moments and also axial and tangential forces is presented in [8]. However here this is performed without estimation of turn deflections influence on bolt core and nut wall displacements. In the article [9] the equation for the compatibility of bent threaded connection elements' displacement and analytical solution of this equation are obtained from the fundamental theory of elasticity. Here, the threaded connection presents one segment model of full profile turns which disregards runouts in the nut.

[FIGURE 1 OMITTED]

For more accurate analysis of load distribution along the thread due to bending of the threaded connection it is useful to estimate the influence of runouts also. The most interest occurs in the case of coarse-pitch thread. Then the both runouts are located in a great part of the nut. That is close to the third part in the standard nut length.

This paper describes a modification to the theory given in [9] and models the threaded connection by three longitudinal segments. The first and third segments here represent runouts where turns have partial engagment (Fig. 1).

2. Positions of threaded connection elements

In the present model a threaded connection is divided into three segments i (i = 1; 2; 3) which pliabilities of turn pairs are described by separate functions (Fig. 1). Pliability of the turns in the middle segment [H.sub.2] is constant ([gamma][z.sub.2]) = [gamma] = const). Contact depth of the stud and nut turns within the segments [H.sub.1] and [H.sub.3] (in runouts) varies (Fig. 1, b). Therefore the pliability of the turn pair here varies also ([gamma]([z.sub.1]) 4 const, [gamma]([z.sub.3]) 4 const). The length of these segments is equal to the thread pitch: [H.sub.1] = P and [H.sub.3] = P.

In the model the origin of any cross-section location coordinate z = [z.sub.i] (i = 1; 2; 3) is receded from the free end of the nut on a phase length [z.sub.f], which is designed to set a position of the threaded connection with respect to longitudinal axis thus with respect to bending plane also. So the coordinate z of any cross-section always is linked with its distance z from the free end of the nut by equality z = [z.sub.f] + z. Position for any thread helix point now can be expressed by turning angle [alpha] (which is equal to [[alpha].sup.*] shown in Fig. 1) in the following way

[alpha]a = 2[pi]/P z = cz (1)

Chanching of the phase length value [z.sub.f] gives a possibility to set position of the bottom runout origin with respect to bending plane by the value of [z.sub.03]. This is necessary because turn pairs near the bearing surface of the nut are mostly loaded in the area around z03 and can be in various positions. The four specific positions (positions I, II, III, IV where sin([cz.sub.03]) = 1, 0, 0, -1) of the threaded connection in respect to bending plane are shown in Fig. 2.

[FIGURE 2 OMITTED]

3. Differential equation for turn deflections

External load of the threaded joint can be schematized by two main components. It is axial load of tightening [F.sub.t] and external bending moment [M.sub.f] (Fig. 1, a). Therefore on turns of the stud and nut in opposite directions arise equal distributed longitudal load intensities [q.sub.t](z) and [q.sub.b](z) caused by tightening and bending respectively (Fig. 1). Due to the action of these loads the proportional turn pair deflections [[delta].sub.t](z) and [[delta].sub.b](z) occur (Fig. 3)

[[delta].sub.t] (z) = [gamma](z)[q.sub,t] (z), [[delta].sub.t](z) = [gamma](z)[q.sub.b] (z) (2)

The relations of these turn pair deflections to the stud and the nut cross-sections displacements are shown in Fig. 3.

[FIGURE 3 OMITTED]

Last-mentioned displacements are caused by internal axial force Q(z) and internal moment M(z) which even act to the stud core and to the nut wall but in opposite direction. The load intensity [q.sub.r](z) and turn pair deflection [[delta].sub.r] (z) caused by tightening can be calculated by the method given in [10]. Further there are analysed regularities of the load intensity [[delta].sub.b] (z) and the turn pair deflection [[delta].sub.b] (z) due to bending only as was desined to present in this paper.

It is seen in Fig. 3 that turn pair deflection has relation with inter-deviation of the stud and nut crosssections. The compatibility of displacements of these elements in any segment i of threaded connection can be expressed by the following equation

[[phi].sub.s] (z) + [[phi].sub.n] (z)/R sin(cz) - [[delta].sub.b] ([z.sub.0]/R sin([cz.sub.0]) (3)

where [[phi].sub.s](z) [approximately equal to] tan[[phi].sub.s](z) and [[phi].sub.n](z) [approximately equal to] tan[[phi].sub.n](z) are deviations of the stud and nut cross-sections, z = [z.sub.i], [z.sub.0] = [z.sub.0i] is coordinate of the segment i origin.

According to the theory of elasticity the deviations of the stud and nut cross-sections are

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (4)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (5)

where [E.sub.s] and [E.sub.n], [I.sub.s] and [I.sub.n] are modulus of elasticity and moments of inertia of the cross-sectional area for the stud core and the nut wall respectively.

Bending moment in cross-section of any segment by using Eq. (2) and certain designation could be expressed in the following forms

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (6)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (7)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (8)

where m(z) is local moment due to [q.sub.b](z) [8] and y(z) is the function which expresses variation of turn pair deflection amplitude.

Now by using Eqs. (4), (5), (7), (8) the Eq. (3) was rewrited in the other form and differentiated two times. Hereby were found

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (9)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (10)

where [lambda] = 1/([E.sub.s][I.sub.s]) + 1/([E.sub.n][I.sub.n]).

The next equation which can be used to express the boundary conditions for any segment i at z = [z.sub.0i] and z = [z.sub.Hi] is obtained from Eqs. (7), (9)

M (z) = 1/R[lambda] y' (z) (11)

Because the turns pliability in the middle segment [H.sub.2] is constant and in runouts (in segments [H.sub.1] and [H.sub.3]) it is variable the differential Eq. (10) must be solved separate.

4. Analytical solution for middle segment [H.sub.2]

In segment [H.sub.2] of the threaded connection [gamma]([z.sub.2]) = [gamma] = const. Therefore the differential Eq. (10) can be rewrited

y" (z)- by( z) sin( z) = 0 (12)

where b = [R.sup.2][lambda]/[gamma] is constant factor.

The approximate analytical solution of Eq. (12) was postulated in the next form

y([z.sub.2]) = [A.sub.2] sinh(n[z.sub.2]) + [B.sub.2] cosh(n[z.sub.2]) (13)

Further factor n of Eq. 13 must be find. At first after substitution of y ([z.sub.2]) in Eq. (11) the next expression for certain mean bending moment was got

[M.sub.m] ([z.sub.2] = [A.sub.2]n/R[lambda]cosh(n[z.sub.2]) + [B.sub.2]n/R[lambda]sinh(n[z.sub.2]) (14)

Another expression for bending moment was got after substituting of Eq. (13) into Eq. (7) and after integrating within it

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (15)

To determine expression for factor n was used equality of Eqs. (14) and (15), i.e., [M.sub.m] (c[z.sup.*]) = M (c[z.sup.*]). When cz = [k.sub.1](P/4), where [k.sub.1] = 0, 1, 3, 5.., and when c[z.sup.*] = [k.sub.2](P/2), where [k.sub.2] = 0, 1, 2, 3.. , from this equality were obtained two expressions respectively

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (16)

Both these expressions give practically the same value of the factor n. For example, in the case of connection M16x2 the values of this factor are n = 0.15640 and n = 0,15630. In the sixth chapter numerically it is shown that the value of n defined by using Eq. (16) is right for all values of z, not only for [z.sup.*].

Thus, the analytical solution for segment [H.sub.2] express Eq. (13) (to find [[delta].sub.b]([z.sub.2]) and [q.sub.b]([z.sub.2]) from Eq. (8)) and Eq. (15) to find M([z.sub.*]).

The factors [A.sub.2] and [B.sub.2] of Eqs. (13) and (15) can be found by using boundary conditions of all segments together.

5. Analytical solutions for runouts

In the runouts, i.e., in segments [H.sub.1] and [H.sub.3] of the threaded connection [gamma]([z.sub.i]) [not equal to] const (i = 1 or 3). The variation of the turn pair pliability in length of any runout was described by the following formula

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (18)

where [V.sub.i] and [u.sub.i] are constant factor and power exponent which can be defined according to the tensile test results of the turn' pairs, egaged over the incomplete profile [10]. They have been defined by using the known fully engaged turns' pliabilities in one edge of segment [H.sub.1] or [H.sub.3] where [gamma](z[H.sub.1]) = [gamma] and [gamma]([z.sub.03]) = [gamma], and also the experimental turns' pliability factors in the middle of these segments. By using data given in [10] the corresponding ratio was determined: [gamma](z[H.sub.1] - P/2)/[gamma] [approximate;y equal to] 1.67 or y([z.sub.03] + P/2)/[gamma] [approximately equal to] 1.67.

The approximate analytical solution of differential Eq. (10) for runouts was postulated in the following form

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (19)

where [n.sub.Ai], [W.sub.Ai], [n.sub.Bi] and [W.sub.Bi] are the factors which need to find, [f.sub.Ai] and [f.sub.Bi] are designations (further indexes [A.sub.i] = A, [B.sub.i] = B at i = 1 and i = 3).

At first after substitution of y (z) in Eq. (11) the next expression for certain fictional bending moment was obtained

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (20)

Another expression for bending moment for runouts was obtained after substituting of Eqs. (18) and (19) into Eq. (7) and after integrating within it

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (21)

where [F.sub.A]([z.sub.i]) and [F.sub.B]([z.sub.i]) are designations which can be expressed in the following common form

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] ((22)

where [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] where are dual signs the upper sign is valid at w=Ai and the under sign is valid in the case of [omega] = [B.sub.i].

To determine four unknown factors [n.sub.Ai], [W.sub.Ai], [n.sub.Bi[ and WBi was used equality of Eq. (20) to Eq. (21) i.e., [M.sup.*] (c[z.sup.*.sub.i]) = M (c[z.sup.*.sub.i]) at [z.sup.*.sub.i] = [z.sub.0i] + P/4 and [z.sup.*.sub.i] = [z.sub.Hi] - P /4. The factors [n.sub.Ai] and [W.sub.Ai] for any runout can be solved from the equations system

[f'.sub.A] ([z.sub.0i]+P/4)/R[lambda] = [F.sub.A] ([z.sub.0i] + P/4) (23)

[f'.sub.A] ([z.sub.Hi]+P/4)/R[lambda] = [F.sub.A] ([z.sub.Hi] + P/4) (24)

Further in analogous way the factors [n.sub.Bi] and [W.sub.Bi] can be solved from similar equations system also

[f'.sub.A] ([z.sub.0i]+P/4)/R[lambda] = [F.sub.B] ([z.sub.0i] + P/4) (25)

[f'.sub.A] ([z.sub.Hi]+P/4)/R[lambda] = [F.sub.B] ([z.sub.Hi] + P/4) (23)

These two systems authors solved numerically by using the suite of mathematical programs Maple-9. By numerical experiments authors persuaded that it is enough to have coincidence of the functions [M.sup.*] (c[z.sup.*.sub.i]) and M(c[z.sup.*.sub.i]) in two points only and that chosen coordinates [z.sup.*.sub.i] give the most calculation results accuracy of the load distrbution in runouts.

Thus, the analytical solution for segments [H.sub.1] and [H.sub.3] (where i = 1 or i = 3) express Eq. (19) (to find [[delta].sub.b]([z.sub.i]) and [q.sub.b]([z.sub.i]) from Eq. (8)) and Eq. (21) to find M([z.sub.i]).

Eventually unknown factors Ai and Bi for all three segments (now i = 1, 2, 3) can be found by using the system of equations, which expresses all segment's boundary conditions

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (27)

6. Calculation results

The calculation of load distributions along the threads first has been made by using the approximate analytical method given in the chapters 3, 4 and 5. The objects of this calculation were threaded connections M16x2, M16x1.5, M16x1, M52x4 and M110x6 with compressed nut (height of the nut H = 0.8d) - made from grade 25X1M [PHI] steel.

Then the differential Eq. (10) writed for both runouts and differential Eq. (12) for the middle segment of threaded connection were solved separate numerically by Runge-Kutta method. (It was realized by using the suite of mathematical programs Maple-9). For this the same boundary conditions M([z.sub.0i]) of every segment obtained in the analytical solution were used. The calculation results are presented with reference to the real coordinate z = z - [z.sub.f] of the cross-section of threaded connection in Figs. 4-7.

[FIGURE 4 OMITTED]

Average indices of mechanical properties of conections grade 25X1M[PHI] steel: proof strength [R.sub.p0.02] = 860 MPa, tensile strength [R.sub.m] = 1010 MPa, percentage area of reduction of tension specimen Z = 60.2%, module of elasticity E = 210 GPa.

Pliabilities for one turn pair (made from grade 25X1M[PHI] steel) for every thread were established experimentally by the technique given in [10]: [gamma] = 3.78 x [10.sup.-3] ; 4.37x[10.sup.-3] ; 7.31x[10.sup.-3] ; 3.54x[10.sup.-3] and 3.26x[10.sup.-3] mm/(kN/mm) for threads M16x2, M16x1.5, M16x1, M52x4 and M110x6 respectively.

The calculations of turn load amplitude function y(z)/[gamma](z), [q.sub.b](z) and M(z) for connections M16, M52 and M110 have been performed at external bending moment Mf applied to the studs which values caused the ratio of nominal maximal normal stresses in the stud with the proof strength to be [[delta].sub.bnommax]/[R.sub.p0.02] = 0.31. For threaded connections M16x2 the total turn loads and total local stresses in stud thread have been determined also. In this case the turn loads and local stresses due to tightening have been obtained at [[delta].sub.t,nom,m]/[R.sub.p0.02] = 0.6 from the method presented in [10], which gives the possibility to estimate the influence of runouts.

The calculation results given in Figs. 4-6 show the serviceability of the three segments analytical model to estimate position, pitch and size of threaded connection subjected to bending.

[FIGURE 5 OMITTED]

[FIGURE 6 OMITTED]

In Fig. 7 the variation of the internal bending moment M(z) in stud M16x2, which has been calculated by using three segments analytical model (Eqs. (15 and 21)) is shown also.

The values of turn load (due to bending) amplitude functions and M(z) obtained by Runge-Kutta method in Figs. 4, 5, 6, 7 are shown by criss-cross. These values differ from corresponding values obtained by analitical method less than by 1%.

[FIGURE 7 OMITTED]

As at the worst the maximal turn load location found after tightening in the stud (aproximately at z [approximately equal to] [z.sub.03]) is in it's bending plane and coincides with the location of the maximum turn load caused by bending. This occurs when the threaded connection is in position I. The ditribution of the total turn loads for this connection position is presented in Fig. 8.

[FIGURE 8 OMITTED]

In strength calculation norm for nuclear equipments [11] the fatigue durability is estimating according the local conditional elastic stresses [[sigma].sup.*]. These stresses for axial loaded stud thread which arise at tightening were calculated by using the following formula [1]

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (28)

here [K.sub.0,t], [K.sub.m,t] are concentration factors of stresses due to the axial force [Q.sub.t](z) and the stud turn load [q.sub.t](z) respectively; [A.sub.s] is cross-sectional area of the stud core; f is the turn's contact surface projection into the plane, perpendicular to the stud axis; P is the thread pitch. The values of elastic stresses concentration factors, defined in work [1] are: [K.sub.0,t] = 2 and [K.sub.m,t] = 1,95 , at the turns' root rounding up radius being R = 0.144P.

[FIGURE 9 OMITTED]

The conditional elastic stresses in stud thread due to bending were calculated on the analogy of Eq. (28)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (29)

here [K.sub.0,b] [approximately equal to] [k.sub.r] [K.sub.0,t] and [K.sub.m,b] [approximately equal to] [K.sub.m,t] are concentration factors of the stresses due to bending moment M(z) and the stud turn load [q.sub.b](z) respectively; [k.sub.r] is factor which estimates difference between the local stresses in the stud at bending and at it's tension; here was assumed [k.sub.r] [approximately equal to] 0.9 after analysis of concentration factors of the notched rods.

The total local stresses in the thread of the stud are

[[sigma].sup.*.sub.[SIGMA]], (z) = [[sigma].sup.*.sub.t] (z) + [[sigma].sup.*.sub.b] (z) (30)

Distribution of local conditional stresses along the thread in stud M16x2 for connection positions I, II, III, IV are shown in Fig. 9. The stress [[sigma].sup.*.sub.[SIGMA]](z) as [q.sub.sum] (z) have maximum at position I of connection (aproximately at z [approximately equal to] [z.sub.03]). At the position IV it has minimum value, which is by 8% less than that at connection position I.

For comparison purpose the local stresses [[sigma].sup.*.sub.[SIGMA]], (z) for the stud thread M16x2 have been calculated also by using one segment model where runouts are neglected (Fig. 10).

[FIGURE 10 OMITTED]

In this case the stud total local stresses [[sigma].sup.*.sub.[SIGMA]] (z) have maximum at the connection being in position I but at z = [z.sub.3] (on curve 1 in Fig. 10). This maximum value is by 21% greater than that in the case of three segments model (on curve 2 in Fig. 9, a). The maximum value of the stud local stresses due to bending [[sigma].sup.*.sub.B] (z) is by 22% greater in the case of one segment model also (at connection position I and at z = z[H.sub.3]) than that in the case of three segments model (at connection position I and at z [approximately equal to] [z.sub.03]).

7. Conclusions

1. The differential equation for the compatibility between deflections of the partly engaged turn pair and deviations of the stud and nut cross-sections in runouts of the threaded connection subjected to bending is derived. The approximate analytical solution of this equation is proposed also.

2. For the bent threaded connection the designed three segments model gives a possibility to estimate the influence of the runouts and of the connection position upon the load distribution along the thread. The turn load calculation results for the threaded connections M16, M52 and M110 obtained by using approximate analytical solutions and obtained by using numerical Runge-Kutta method differ very slight - less than 1%.

3. In dangerous cross-sections the maximum and minimum values of the greatest total local stresses in stud threads occur when threaded connection is in the positions I and IV respectively. These values differ by 8% in the case of threaded connection M16x2 considered by using three segments model.

4. The maximum thread local stresses [[sigma].sup.*.sub.[SIGMA]] and [[sigma].sup.*.sub.b] in the stud at its' beeing in the position I and determined by using three segments model are noticeably less than these stresses in the case of one segment model. The both differences of the maximum stresses of [[sigma].sup.*.sub.[SIGMA]], and of the maximum stresses of [[sigma].sup.*.sub.b] obtained for the threaded connection M16x2 exceed 20%.

Received March 27, 2010

Accepted July 2, 2010

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A. Krenevicius *, Z. Juchnevicius **, M. K. Leonavicius ***

* Vilnius Gediminas Technical University, Sauletekio al. 11, 10223 Vilnius, Lithuania, E-mail: kron@fm.vgtu.lt

** Vilnius Gediminas Technical University, Sauletekio al. 11, 10223 Vilnius, Lithuania, E-mail: ma@fm.vgtu.lt

*** Vilnius Gediminas Technical University, Sauletekio 11, 10223 Vilnius, Lithuania,

E-mail: mindaugas.leonavicius@vgtu.lt