optimal screw joint design of a bolt and … · optimal screw joint design of a bolt ... thin plate...

machine design, Vol.8(2016) No.2, ISSN 1821-1259 pp. 37-52

*Correspondence Author’s Address: HeikkiMartikka, HimtechOy Engineering, Ollintie 4, SF 54100 Joutseno, Finland, [email protected]

Preliminary note

OPTIMAL SCREW JOINT DESIGN OF A BOLT AND PLATES WITH APPLICATION TO EVALUATION OF A JOINT FAILURE CASE Heikki MARTIKKA1,* - Jouko KATAJISTO2 1 Himtech Oy Engineering, Joutseno, Finland 2 Lappeenranta University of Technology, Finland

Received (06.05.2016); Revised (27.06.2016); Accepted (29.06.2016) Abstract:In this study the results of developing a methodology for optimum design of screw joints is described. The chosen method is fuzzy optimum design for obtaining maximal trade off customer satisfaction on all desired properties. This goal is achieved by incorporating ten essential properties as decision variables into the model. Many variations of geometry, many materials and loadings can be considered. By this method the goals and constraints are formulated in a unified way using one flexible fuzzy standard function. This method can be extended to consider simultaneously properties like costs, complex loads, material properties, creep and relaxation and fatigue behaviour, tightness and pressure bearing capacity. The generality and feasibility of the program is tested by applying it to extremes of a thick plate and thin plate joint. Comparison of the safety factors gave the following results. The safety factor against decrease of prestress due to setting is 1.6 times higher with thick plate joint compared with thin plate joint and against safety factors against fatigue failures more than 1.5 times higher with thick plate joint. Other factors of safety are nearly the same. This methodology is combined with failure criticality analysis to evaluate causes of a motor vehicle failure when the variator motor shaft nut become loose shortly after overhauling. The probable cause is failure of locking glue of screws due faulty overhaul. The loosening mechanism is probably due to ttransverse bending of the nut causing opening torque impulses to the nut. The future goal is to extend the methods to optimize joints for jointing plates with many screws and loosening mechanisms.

Key words: Screw joints, endurance, optimisation, dynamic loosening and tightening,

1. INTRODUCTION

The aim in the introduction is describe the background and the design goal methodology. In machine construction the preloaded fasteners widely are used to join and tighten machine elements. The basic design is considered in textbooks like by [1] (Norton, 2006) and [2] (Decker, Kabus, 1981). Optimal level of utility of joints is obtained only with optimal level of prestress. Relaxation decreases prestress and promotes risks of separation, fatigue, leak and creep fracture. Creep is considered by [3] (Evans, Harrison, 1976) and by [4] (Dowling, 1998). These can be also included when needed but are beyond the scope of this study. [5] (Hess, Sudhirkashyap, 1997) consider dynamic loosening and tightening of a single-bolt assembly. The agreements of simulations and measurements is qualitative.[6](Kerley, 1987) observed that loosening occurred at low frequencies but not at high. It is probable that this phenomenon involves the nonlinear dynamic interaction of vibration and friction and the

resulting patterns of momentary sliding sticking and separation between threaded components. Conventionally screw joint design is made by choosingappropriate formulas and data and the results are often not optimal since iterative approaches with all essential endurance constraints are not.(Varga, Baratosy, 1995) [7] have discussed dynamic optimal prestressing of bolted flanges. Their method of calculation predicted an optimal prestressing resulting in minimum operating load and the same time saving the gasket. Optimum design is most useful at the initial concept design stage where the most design decisions are to be made. Then FEM is useful tool for designing simultaneously the overall structure and its details. Martikka et al have used this combined optimization approach in industrial design work in the following studies :[8](Martikka, Taitokari ,2006) , [9](Martikka, Pöllänen,2008), [10](Martikka, ,Pöllänen, 2009), [11](Martikka,Pöllänen,2011),[12](Martikka, Taitokari, 2011), [13](Martikka, Taitokari,2012), [14](Martikka, Pöllänen,2012) and[15] (Martikka, Taitokari,2013) .

Heikki Martikka, Jouko Katajisto: Optimal Screw Joint Design of a Bolt and Plates with Application to Evaluation of a Joint Failure Case; Machine Design, Vol.8(2016) No.2, ISSN 1821-1259; pp. 37-52

38

The aim in this study is to use the fuzzy optimum design with customer satisfaction goal.End-user goals are often vague and fuzzy. They can be defined fuzzily as maximisation of total customer satisfaction on it (Diaz, 1988)[16].Now ten decision variable based on design variables are formulated. Satisfactions on each of them are multiplied to get total satisfaction product which is maximized.

2. FUZZY GOAL FORMULATION USING DECISION VARIABLES

The aim in this section is give an overview of the optimization formulation.

2.1. Principle of goal formulation

The total design cycle starts when someone in the markets has recognised an unfulfilled need in the markets. The task of the optimum design, manufacturing and the supply chain is to satisfy the needs of the customer and also the designers and producers and auxiliaries. There are several optimization tools for performing this task. Now the design method of fuzzy optimization goal formulation is used e.g. by [16](Diaz, 1988) and [15] (Martikka, Taitokari, 2013). The design variable vector x = (load functions, geometry, materials) is

, , … (1) The decision variable eventsare formed using this x vector

s , , … , 1,2. . Decision variables are typically s1= “cost should be within a desired range: min ← cost < max” s2= “Technical performance should be within the desired range” The total event is decision variable s and it isintersection of partial decision variables sk s = (cost, factors of safety, corrosion resistance, aesthetic appearance ...) = s(x).

654321 sssssss xs (2)

The design goal is maximisation of the total satisfaction of the customer on the product

PQsPsPsPP max,..... n21 s (3)

Now all goals and constraints are formulated consistently by one standard flexible fuzzy function. Some typical decision variables sk and their satisfaction functions P (sk) are shown in Figure 1.In this method no separate constraints need to be defined. They are already built in the sk.

2.2. The fuzzy goal and the pseudocode of the optimum search algorithm

The method of exhaustive search (ES) is used. The ES method is satisfactorily effective in preliminary explorative small studies which are highly nonlinear cases. Disadvantage of the ES method slows with large number of choices. Parameters are given constant data. The design variable options are shown in Table 2.

2.3. Detailed description of the algorithm for optimisation

In engineering optimisation at concept stage most tasks are highly non-linear and also the design variables are few and discrete. User can select the materials from the list of available selections by index. Total satisfaction is first initialised to a low valuePsbest = .0000001. Material classimselection is made first. Loops FOR each k = 1, 2...15 decision variable s(k), The event of design is s = s1 AND s2 AND s3 AND s4 AND s5 AND s6

Each P(si) is obtained by a call CALL pzz(s1, s2, p1, p2, s, P(s)). The output is P(s) =Ps The total satisfaction P(s) is product

P(s) = P(s1)·P(s2)·P(s3)·P(s4)·P(s5)·P(s6)

Ps = 1 , the initialisation first, before next loop FOR i = 1 TO NPs = number of satisfaction functions Ps = Ps·Ps(i) NEXT i Pg=Ps IF Ps>Psbest THEN Psbest= Ps, new optimum with best design variables is obtained ELSE search is continued. END IF NEXT k IIIIIIIIIIIIIIIIIIIII

Fig.1. Some typical decision variables sk and their satisfaction functions P(sk).

xkx2

PK

K=s1

P2

P3

PN

s2

s3

N= s4

low s1 =K isdesired Indifference to s2 values

Medium s3 values are desired Narrow range of N = s4 is desired, Large s5 values are desired

b2

P5 s5

1


39

3. SCREW JOINT FREE BODY MODELS The aim in this section is to describe the principle of the functioning of the joint and to show some applications.

3.1. FBD models for screw joints

These are essential for understanding the functioning principle of the joint.

Fig.2. Typical screw joint to one bolt and two plates

Free body models are shown in Figure 3.

Fig.3. a)…f) Use of FBD modelling to explain the functioning of the screw joint g) Screw geometry, dr is

minor diameter, α is full thread angle , p is pitch

3.2. Geometry of the joint

The following approximate relationships are useful in conceptual design, (Norton, 2006)[1].

dD

ddqdddqdd

15.1

,84.0,92.0

km

rrpp2

(4)

Here d= outside diameter, now, d= 10 mm, dp is pitch diameter, dr is minor diameter, p is thread pitch, p= 1.5mm, = 600 is full thread angle , not half as by (Shigley, Mischke, 1989). Dkm is the diameter on which the friction line force result act between the nut and the plate. The tensile stress area is

88.02

,22

rpt

2

2

rp

422

t4

2

rp

4t

qqq

dqq

dqdd

A

(5)

Pitch angle for the present nut case is

048.092.0010.0

0015.0tan

pd

p (6)

Friction angle

17147.096.59½cos

15.0tan

½cos n1

o

(7)

Here the coefficient of friction in threads is 015. ,

friction between nut and plate k 0 15.

3.3 Basic geometry of a typical flange joint

Optimal bending stiffness is one desirable property of flange connections shown in Figure4. Flanges for general applications have been standardised.

kp

force applicationpoints ks

½FA = ½Fi ½FA = ½Fi

dr

dp d

p

FA

FV FV

FSA FPA

kPe= kx1

FA

kSe

Fse

FseFA

k =kPe+kx1

FSV FPV

kS kp

kS kp

FA

x

FSV

lSV lPV

FPV

ls0kS lp0kP

FA

FA

½kx2

Fx1 x

Fse

kS kx1 Fx1

Fx1 ½kx2Fse

FA Fx1

½x2

½FA = ½Fi ½FA = ½Fi

x1= nlp

lp=l

½x2

½FA = ½Fi

x2= (1-n)lp ½FA = ½Fi


40

a)

b)

Fig.4. a) A flange connection under bending and normal

axial load.

b) Typical application of screw joints using a gasket

sealing and a screw.

But for special applications new design may be needed.

Some heuristic design rules are useful in the optimization

program. The individual bolts have different loadings and

they need to be checked for endurance.

The radial distance T from bolt centre to outer tube radius

is

ironcaststeelK

KdTdssdT

.,7.1,,5.1

,,,21

(8)

The pitch diameter D0 depends on the flange geometry.

The outer diameter Du depends on dimension s

z

DusDDTtDD u

,2,22 0ini0 (9)

Here Di is inner tube diameter and tin is its wall thickness,

u is spacing of bolts on the pitch circle D0, and z is

number of bolts on the pitch diameter.

3.4 General loading of flange joints

The aim in this section is to review some types of external

loads to an assembly of joints.

Individualbolt loads can be calculated from it.

Load stresses

tensile stress due to inner pressure

bending stress at a flange joint under bending moment

direct shear stresses

Stresses acting on a bolt are due to tension, bending

and shear loads.

These are shown in Figure. 5.

Fig.5. Stresses acting on a bolt due to tension, bending

and shear loads

The total normal tensile stress on a bolt is due to normal

force and flange bending force

b i n FA i nt

b i P bend n

P

A

M

W

C C

1

, ,

(10)

Here C = n is the joint stiffness factor, P is the is load

force on the most stressed bolt, At is stress area and i is

initial prestress. Here M is bending moment on the

flange,W1 is bending resistance of the bolt distribution.

The effect of direct shear load on a bolt may be taken

into account using the Tresca equivalent stress model

b,eq b b bt

2 21

24 ,V

A

(11)

Dividing this by the yield strength Sy and using the

interaction methodology gives the safety factor against

yielding

11

eqb,

y

2

y21

b

2

y

b

y

eqb,

S

NNSSS

(12)

Here the strength forces are

F S A V A Sy y t y y t y y , , 12

(13)

d u=Bd

Di

Do

t1

r

dA = rdt1

y’

H

Du

s s

tin T

s = ad

Vx

Vy

L

b bg

gasket

ds Dgin = Din,tube

D D Dg 12 g,max gmin,effective

D D bgg,max g,min 2

M

P b

V P bend


41

Thus the endurance condition according to Tresca can be

written using forces

F

F

V

Vb

y

b

y

2 2

1 (14)

3.5. Material options

Steel design variables shown in Table 1.Metric

specifications and strengths of steel bolts (Norton, 2006,

page 834, Table 14-7)[1]

Table 1. Steel options for material design variable. Minimum proof strength Sp(MPa), Minimum yield strength Sy.(MPa),Minimum tensile strength .Sut(MPa)

class number

outside diameters (mm)

Sp (MPa)

Sy (MPa)

Sut (MPa)

8.8 M3-M36 600 660 830

10.9 M5-M36 830 940 1040

12.9 M1.6-M36 970 1100 1220

S-590 200

Material class Material

8.8 medium carbon . Q&T

10.9 low carbon martensite, Q&T

12.9 alloy, quenched and tempered

S-590 High temperature alloy

4. MODELS FOR THE JOINT STIFFNESS

FACTOR C

The aim in this section is to review the basics of the joint stiffness factor since it is important for the optimality of the joint. The stiffness coefficient of the platekpfor the compressed part of the plate [18] (Shigley, Mischke, 1989) is

p

rp

ln

tan

k

PfE

V

U

dE

l

Ak k

(15) Here

ddD

dDdDlV

dDdDlU

w

k

k

5.1

,tan2

,tan2

(16) Here f is deflection of the plate under force P, E is elastic modulus, d= outside diameter

dw the cone model diameter

k is half apex angle. According to ( Little,1967) [17]

this k = 45 overestimates the clamping stiffness.

(Shigley ,Mischke, 1989) [18] suggestk= 30. (Osgood,1979) [19] reports a range of suitable combinations

ok

o 3325

Ar the area of the surrogate rod spring, l is the length of the surrogate rod. One possible choice is the sum thickness of the clamped plates l = lp, .

b)

Fig.6. a) Cone model for the stiffness coefficient of the platekp for the compressed part of the plate.

b) Freeebody models

The effective spring stiffness of the parts with decreasing loading kPe The stiffness coefficient of the more stressed plate part 1 and 2 areas. Now n = 0.9 is chosen

)1(

,

P

2

PPx2

p

PPpPe

P

1

PPx1

n

k

x

AEk

l

AEkk

n

k

x

AEk

(17)

The effective spring stiffness of the parts with increasing loading kSe

FA

FA

kS kx1 kx2

0kx2

FA

FA

kS

kx2

kSe

fx2

fs

f

dw

½l t1

lp

k

l ½l

t2

D2


42

These springs are connected in series

x2SSe

x2

A

S

A

Se

A2s

111

kkk

k

F

k

F

k

Ffff x

(18)

The spring stiffness of the bolt is ks when it is loaded at both ends

k1 iGM S

1111

kkkk

m

(19)

Here kGM is the combined spring stiffness of the nut and the screw part inside it. kk is the spring stiffness of the top part of the screw ki is the spring stiffness of the ith cylindrical part of the screw A model for the kGM is by standard (SFS 2067) [20]

tm AAdAAE

l

AE

d

k 3

24N

Ns3SGM

,,5.01

(20)

Here the definitions for the symbols are

t

t

d

dd

d

hq

q

qRRdl

5.0

,65.01

4.53.0,

3

3m

(21)

Spring stiffness of the head of the screw for standard hexagonal nut and ki are

iS

i

iNSk

1,

4.01

AE

l

kAE

d

k

(22)

The external load force FA = P is sum of two resisting forces. It causes a displacement x , Figure 7

xkkF

FxkxkF

FxkF

FFFFF

)(

,

PeSeA

x1x1PePA

SeSeSA

x1SePASAA

(23)

From these the common displacement x can be solved and used to calculate the forces . The force addition is

F k xk

k kF FSA Se

Se

Se PeA n A

(24)

The joint stiffness factor C

s

PPS

Sn

1

1,

k

kCnC

kk

knC

(25)

5. INPUT DATA TO PROGRAM AND MAIN OUTPUTS

The aim in this section is to describe the main input and output of the program.

5.1. Main input data One geometrical input is the outer diameter of one bolt nut with thread M10. Now d= 10mm Nominal area A and true areaAt are

10 78 , , 0.88(26)

Coefficient of friction in threads is µ = 0.15

Material of screws is selected from steel class options

Material im = 7 with strength close to 8.8 class is chosen with slight modifications. The strength values are UTS Sut = 840MPa, and yield strength Sy = 660 =.8 Sut. Some other properties for im= 7 material are used in more comprehensive optimisation: Cost cm(im) = 5 units, density rho(im) = 8000, ecological value (0..1) eco(im)= 0.5, corrosion resistance value corres(im) = 0.5, elastic modulus E(im) = 2.11 1011 (Pa) Threshold stress intensity Kth (im) =190 -144 Rs, stress ratio Rs =0 now.Initial crack length a0 (im) = 0.2 mm

Design variable options

Three dimensionless variables were chosen as fundamental variables. The design variables can be all free or all fixed or desired fixity combination. Tightening stress ratio z is

, 0.4…1.15(27)

The load stress per prestress ratio V is

, 0.4…1.15 (28)

The joint relative thickness W is

, 1. .10(29)

The total plate thickness is the sum of two plate thicknesses

(30)


43

Table 2. Discrete design variable options z, V, W

iz, iV, iW

z(iz) V(iV) W(iW)

1 0.4 0.4 1 2 0.5 0.5 2 3 0.6 0.6 3 4 0.7 0.65 4 5 0.75 0.7 5 6 0.8 0.75 6 7 0.85 0.8 7 8 0.9 0.85 8 9 0.95 0.9 9

10 1 0.95 10 11 1.05 1 12 1.1 1.05 13 1.15 1.1 14 1.15

The output data are calculated by the following logic. First the inputs are given and design variable the design variable vector

, , … → , , (31) Then the total design event is decision variable s and it isintersection of decision variables sk

654321 sssssss xs (32)

Then the design goal is obtained as maximisation of the total satisfaction of the customer on the product and the optimal design variable vector

opt

n21 ,max,.....

xx

s

PQsPsPsPP

(33)

Extension of the model with more goals Many more goals can be added as needed. New decision variables necessitate availability of new design variables. Cost of joint = s11 = the 11th decision variable and satisfaction on it P11 are

4

, , (34)

Desired range for the load P (iP) = s12 and satisfaction P12 on it

, (35)

5.2. Main output data

The optimal joint will be maximally trade –off satisfying to the customer’s wishes

→ , → , → (36)

The output of optimal design values

The prestress is

, (37)

The load force stress is

, (38)

The allowable load force P is

∙ (39)

The optimal setting fZ is obtained using a linear model depending on W

minmax

minmax

111

110

3.37

,5.3,

WW

WfWfk

mfWWkff

ZZ

zzZ

(40)

Or

→ , (41)

Here the joint stiffness factor C with two notations is

s

p1

1

k

kC

ps

pK

(42)

6. FACTOR OF SAFETY FOR MINIMISING THE LOSS OF PRETIGHTENING FORCE DUE TO SETTING OF ROUGHNESS The decision variable is the factor of safety for minimizing the loss of pretightening force due to setting of roughness’s of nut to bolt. The partial goal is defined as satisfaction on it

max, 1setting1 sPNs (43)

Fig.7. The screw joint diagram of the spring deformations

of the screw and plate

Force F

FSA=Pb FSA= CP

FA= P

kP= km

FV= Fi

FPA= (1-C)P

kS= kb

FKR=Fm FZ

displacementx fZ x

lSV=ls-ls0 lPV=lP0-lP


44

Using the similar triangles in Figure 7 gives the ratio of

the loss in prestress force due to setting FZ to the initial

prestress force FV . This ratio is the relative prestress force

loss at the joint

ps

ZfF

k

F

k

Ff

ff

f

F

F

Z

p

V

s

V

Z

ps

Z

V

Z (43)

Using typical values by (Decker, Kabus, 2009)[2], p. 260

gives a model for the loss in prestress force due to setting

FZ

p

ZK

p

Z

fff

Fps

p

ps

ZZ

(44)

Here the joint stiffness factor C with two notations of the

literature is

s

p1

1

k

kC

ps

pK

(45)

The setting fz depends on the ratio and thickness of two

plates

, (46)

hered is the major diameter, t1 and t2 are plate thicknesses.

The setting can be roughly approximated by a linear

model of equation (40) suitable for

calculations .The decrease or loss of the tightening force

due to setting is

pZVloss kfCf

CFp

ZK

(47)

Relative prestress force loss at the joint and the allowed

loss are

04.0

,

allowedloss,Vrel,

V

pZ

V

ZlossVrel,

F

F

kfC

F

FCF

(48)

This may be written as

22t4

p

y

p

y

p

s

p

plossVrel,

V

Zp

V

ZplossVrel,

1

,,

dqd

lba

S

k

z

C

AzSd

lba

k

k

kF

F

fCk

F

FzkCF

t

(49)

Factor of safety against exceeding the allowed relative

setting 0.04 is

d

lbaCk

dqzSs

F

FF

FCNs

pp

22t4y

1

V

ZlossVrel,

allowedloss,Vrel,setting1

04.0

04.0

(50)

This model predicts that the safety factor can be increased

by increasing z, by increasing the yield strength Sy and by

decreasing the joint stiffness factor C.

7. OPTIMAL RANGES OF TIGHTENING

TORQUE FOR OPTIMAL PRELOAD

The aim in this section is to describe two models for

estimating the minimum torque and the sufficient

tightening torque.

s2 = NtightAThe factor of safety to avoid too small torque

tightening

s3 = NtightBThe factor of safety for sufficient tightening

torque s3 by Freebody model.

7.1. The factor of safety to avoid too small

tightening torque s2

The decision variable is the factor of safety for

minimizing the loss of pretightening force due to setting

of roughness of the threads of nut and bolt. The partial

goal is defined as satisfaction on it

max, 2tightA2 sPNs (51)

mintightA T

TN (52)

Here T is the applied tightening torque and Tmin is the

minimum recommended


45

Fig.8. The tightening torque T needed to obtain a desired

preload force Fv in lubricated threads.

dFKT V (53)

The prestress force is

VtV AF (54)

Here K = 0.2 is the torque coefficient. K varies very little over the entire range of thread sizes. The equation applies on the average best at friction coefficients of 0.5 according to (Norton,2006)[1]. Using these gives

dDdd

dK

DdKd

kmp 15.1,92.0

2.015.015.1215.092.0

tan

21

21

kkm21

p21

(55)

The maximum prestress and the prestress force are

tytV

tightpytight

p

AaSAF

AvaSA

Sv

maxmaxV

y

maxV 25.1,75.0,

(56)

A typical maximum prestress is now calculated as

0.6 0.6640 384 (57)

The maximum and minimum recommended torques are

max1

minmaxmax ,2.0 TTdFTAV (58)

HereA = 1.5 is tightening variability factor. The maximum tightening torque is

0.2 .

. (59)

The minimum tightening torque is

(60)

The applied torque should be within allowed limits

(61)

Here

(62)

The factor of safety to avoid too small tightening torque is

. ∙ .

.2.5 (63)

7.2. Factor of safety for sufficient tightening torque s3

This factor of safety based on the free body model, s3 = NtightB The tightening torque is

tan1

tan,tan 2

1max2

1,

dFKdFT Vztightz

(64)

Here Fz is the axial force on the screw is

(65)

Factor of safety for sufficient tightening torque is

tan1

tan,tan

min

21

min

,ightB3

T

daS

T

TNs ytightz

t (66)

This means that loss of friction between threads decreases much the factor of safety. The loss of friction from µ from 0.15 to 0.1 decreases the safety factor. It can be estimated by setting tanγ = 0.05

.

.

.

.

. .

. .

.

.1.33 (67)

Now the simulation gave 0.15 =0.76. Thus

0.10.761.33

0.57

Then the factor of safety can become too low.

Fs

Fu+

v

Fu+

su tan FF

dM+g Fs

Fs

Fu-

Fu-

v

Fs dM-g

su tan FF


46

7.3. Factor of safety to prevent plastic yielding s4

The decision variable is the factor of safety for acceptable plastic yielding. The partial goal is defined as satisfaction on it

max, 4y4 sPNs (68)

The safety factor against yielding is

VCznVCz

s

CV

S

CzS

S

SCz

CzNsV

y

y

y

,,1

1

1,

4

VP

y

Py4

(69)

Here

VPy

V , V

Sz (70)

8. FACTOR OF SAFETY AGAINST JOINT SEPARATION OF A NON- ECCENTRICALLY LOADED JOINT S5

The decision variable is the factor of safety against joint separation of a non-eccentrically loaded joint, Figure 9. The partial goal is defined as satisfaction on it

max, 5noeccsepar,5 sPNs (71)

This depends on two design variables, the tightening stress ratio z and the joint stiffness factor C

yVPVy

V ,, SVzVzSS

z y (72)

Using these one obtains

CVs

CCS

zCzNsP

V

1

1

11

1,

5

y

Pccsepar.none5

(73)

9. FACTOR OF SAFETY AGAINST JOINT SEPARATION OF AN ECCENTRICALLY LOADED JOINT The decision variable is the factor of safety against joint separation of an eccentrically loaded joint,Figure 10. The partial goal is defined as satisfaction on it

max, 6sepecc6 sPNs (74)

This calculation is based on models by [21](Thomala, 1986). Linear stress distribution is assumed to act on the surface between plates.

xI

M

A

Fx

BT

B

D

(75)

The eccentrically loaded joint is shown in Figure 10.

Fig.10. Eccentrically loaded joint. Typical dimensions are u =0.02,v=0.02,B =0.05,a =0.8u,

s =0.1u, H =u+v

Here AD = BH –Abolt, A = BH = total area , Abolt = bolt area, F = resultant load due to of bending stress distribution on are AD, MB is bending moment. Force and moment equilibrium of the upper FBD of Figure 10 are

0 SA FFFF (76)

0 ASBz aFsFMM (77)

From these the results force F can be solved

AVASAV

AS

FCFFFFF

FFF

)1(

(78)

Thus the moment is solved as

ASB aFsFM (79)

VAAVAB sFsCaFCFFsaFM (80)

The loaded area is now typically

A

IkBHI

mvuH

BHA

,

040.0020.0020.0

102004.005.0

3121

4

(81)

The stress distribution may be written as

x

k

MF

Ax

1 (82)

or

l2

σ(x)

a

s

l1

FA FS

x

lK

MB

FA F

FS

v u

H


47

xk

sFsCaFFCF

Ax VA

AV )1(1

(83)

The separation starts at outer edge x = u when the compressive stress is zero there

uk

sFsCaFFCF

Au VA

AV

*)1(*

10

(84)

From this the critically low force FV at which separation occurs can be solved

AV Fu

suk

saCF

1* (85)

This critical prestressing force is sum of two forces, Figure 7

PFFFF

Fsuk

usaFCF

AsepeccV,PAV

AAV

,*

1* (86)

Here FPA is plate compression force due to P. The minimum separation prevention force is

usuas

saQ

suk

usaQQPF

uk

1.0,8.0,

,sepeccV,

(87)

Safety factor in eccentric loading against separation is

VQF

FVQ

F

PQ

F

FNs

V

V

VV

sepeccV, sepecc6

(88)

This does not depend on the pre stress ratio z and the joint stiffness factor C. 10. FACTOR OF SAFETY OF ENSURING SELF-LOCKING S7 The decision variable is factor of safety for ensuring self-locking of nut to bolt. The partial goal is defined as satisfaction on it

max, 7lock7 sPNs (89)

The risk is that the nut will loosen with too small a small opening torque Mg

-.

n21

n

s221

gg

costan

½cosATN

0)1(tan

FdMM

(90)

The decision variable may be derived from this condition. Factor of safety against opening depend on three design variables

""

""

costan

,,tan

217

7

stress

strengths

Ns

n

nlock

(91)

It is possible that a critical frequency of axial and transverse vibrations causes the nut to rotate and become loose. 11. FACTOR OF SAFETY AGAINST CRACK INITIATION BY GOODMAN DIAGRAM MODEL S8 The decision variable is the factor of safety against crack initiation using the Goodman diagram. The partial goal is defined as satisfaction on it

max, 8Good8 sPNs (92)

The safety factor is by (Norton, 2006)

yVP VzSVS

z

k

KK

SC

zS

SK

CzN

,,

,

1

,

y

V

fafm

ut

P21

ut

yfm

Good (93)

This depends on two main design variables, the tightening stress ratio z and the joint stiffness factor C. It may be also expressed in more useful dimensionless formulation

,

1

,,,fa

fmut

y

21

ut

yfm

Good8

k

KK

S

SCVz

zS

SK

S

SzVCNs

ut

y (94)

Here factor k = 0.5 is fatigue strength lowering factor. It is product of all other factors than the notch effect. The notch factors are conservatively for mean stress Kfm= 1.1 and for amplitude stress Kfa = 3 by [1]( Norton, 2006) The Goodman diagram applied for torsion can be transformed to diagram for tension components as shown in Figure 11.

Fig.11. Goodman fatigue diagram applied to rods (Norton, 2006) [1]

The transformation from shear stress to normal stress diagram is

Alternating stressamplitude 1. Shear a, 2. normala

Ses Se

Sa

aa

0.5Sew

ii

m(Sa)

Ses

Sus Sut

load line Stress amplitude vs.

Goodman line

mean stress mm(a)

m

45

0.5Sew


48

e

a

ut

im

ut

i

f

es

a

us

im

us

i

fs

11

SS

SN

SS

SN

(95)

12. FACTOR OF SAFETY AGAINST FATIGUE FAILURE USING TECHNICAL DEFINITION FOR QUENCHED AND TEMPERED SCREWS S9 The decision variable is the factor of safety for acceptable fatigue life. The partial goal is defined as satisfaction on it

max, 9fatigue9 sPNs (96)

The allowable strength amplitude at which fatigue life is acceptable is by [22] (VDI 2230, 1986)

MPa52

5275.0)(52)mm(

18075.0

ASV

10180

ASV

MPad (97)

The stress amplitude and the factor of safety are typically

8.218

52

,

,

183561.0

9

a

ASVfatigue9

21

P21

a

amplitudeloading

amplitudealloweds

Ns

C

(98)

13. FACTOR OF SAFETY FOR OBTAINING SUFFICIENT CRACK PROPAGATION LIFE TIME S10 The decision variable is the factor of safety for sufficient crack propagation life based on Paris model. The partial goal is defined as satisfaction on it

max,10

, 106

parislife10 sP

NNs (99)

The model includes crack growth factor Cp. Now it is estimated according to (Gurney, 1978)[23]. It is based on a statistical analysis of some crack propagation data for steel subjected to pulsating tension loading. In this model the crack growth exponent m depends on the yield strength approximately as

pcorrmp264.0

y Pa600 CCCB

ACSm

(100)

Here A = 131.510-6, B = 895.4 at the stress ratio In the program the m and Cp parameters are calculated as

Re = Sy = Re(im), Rm = Sut= Rm(im) FOR im = 6 TO 7

m(im) = 600 * Re(im)-0.264 A = .0001315: B = 895.4 Cp(im) = a / bm(im)

NEXT Stress ratio Rs = min/max = 0. Now the corrosion factor Ccorr=1. The fatigue life in cycles from initial to final crack length is

MPa0

11

1

1)(

Pmaxminmax

1f

10

m

21

p

paris21

21

C

aaYmCCN

mm

(101)

Now units are (MPa). The initial crack length a0 =0.2mm and the final af =10 mm are in (mm) units. At very low K values Ccorr is 20 and at high K values it is about 3.

Fig.12. Definition of fracture mechanics and connection

to SN curve

Fig.13. Basic relationships of fracture mechanics. The notch factor is often set to Y=3 at the root. Now the model (Kloos, Thomala, 1979)[24] is used since it applies explicitly to screws

4.5 (102)

The stress in the bolt b is sum on nominal initial prestressV and a fraction C of the load P stress P

,,t

PPnomi,b A

PC (103)

a, N a+da, N+dN

max

min

a

m

Kt,incl =2

N

C

m

da

dNC K m

K K k Kt h I I

A

B

C

x log( )

y N logRe , yield strength

m


49

The life does depend on the tightening stress ratio z and on the joint stiffness factor C. The load stress is determined by the choice of factor V

9.0,18.0

,,VP

opt

yV

zzoptimum

zSV (104)

The factor of safety for achieving 1 million cycles is

6paris

life10 10, megamega

CNCNs (105)

A simplified SN type illustrative model can be made to illustrate the dependencies

mmparisSzVC

MMNy

11

11

(106)

This shows the main factors which influence the Paris fatigue life of a screw in a screw joint. Low joint stiffness C increases N .It is obtained by high ratio kp/ks or stiff plate and not stiff screws. Low stresses with low bolt yield strength increase N.

14. RESULTS

Results are summarized in Tables 3, 4 and 5 and in Figure 6 and in Appendix 1. The feasibility of the program was tested with a thick plate joint and thin plate joint. Summary of the comparison of their safety factors gave the following results. The safety factor against decrease of prestress due to

setting was 1.6 times higher with thick plate joint compared with thin plate joint

The safety factor against fatigue failure was 1.5 times higher with thick plate joint.

Other factors of safety were nearly the same.

14.1 .Results for a thick plate joint

All design variables z, V, W are allowed to be free The optimal joint was thick plate screw joint with W=8. This gave for plate thickness

lp = W d = 8 d

Main input data One bolt nut, thread M10, d = 10mm outer diameter. Nominal area A and true area At

10 78 , , 0.8 (107)

friction µ = 0.15. strength class is 8.8, Sut = 840MPa, Sy = 660 =0.8 Sut Main output data Design variable values obtained by the program

0.95, 0.65, 8, 80 (108)

The obtained design property values The prestress is

0.95 ∙ 660 627, (109)

Load force stress is 0.65 ∙ 627 408, (110)


∙ 0.650.950.88 0.478 ∙ (111)

or

0.478 ∙ 0.478 ∙ 660 ∙ 78

24.6 (112)

The setting can be roughly approximated by a line

minmax

minmax

111

110

3.37

,5.3,

WW

WfWfk

mfWWkff

ZZ

zzZ

(113)

This gave

fZ = 6.18· 10-6 m


054.0

1

1

s

p

k

kC

ps

pK

(114)

Table 3. Results for a thick plate joint with optima W =8

Decision variable s(k) name s1=Nsetting ,decrease of prestress s2=NtightA ,handbook model s3=NtightB ,byFreebody model s4=Ny ,plastic yielding s5=Nsepnoecc ,separation by non-eccentric load s6=Nsepecc,separation by eccentric loading s7=Nlock,self locking s8= NGood,Goodman diagram s9=Nfatigue,fatigue, handbook s10 =Nparis/M, crack propagation fatigue life P(s) total satisfaction of design event s = s1 AND…s10

Decision variable s(k)

W=8 DecV sk

W=8 P(sk)

Range desired

s1=Nsetting 1.70 0.9905 1-2s2=NtightA 2.38 0.9993 2-3s3=NtightB 0.763 0.878 1-3s4=Ny 1.05 0.4 0.9-1.1 s5=Nsepnoecc 1.63 0.9876 1-2 s6=Nsepecc 1.05 0.9229 1-2s7=Nlock 3.34 0.6815 2-3s8= NGood 1.5 0.9514 1-2s9=Nfatigue 4.75 0.5949 2-5s10 =Nparis/M 0.236 0.6552 1-2 P(s) - 0.08

Results for a thin plate joint with fixed choice W =3 are shown in Table 4 C= joint stiffness factor


50

Table 4. Results for a thin plate joint with fixed choice W =3


DecV sk

P(sk)

Range s

s1=Nsetting 1.02 0.72 1-2 s2=NtightA 2.25 0.997 2-3 s3=NtightB 0.763 0.88 1-3 s4=Ny 1.11 0.24 0.9-1.1 s5=Nsepnoecc 1.87 0.9987 1-2 s6=Nsepecc 0.97 0.8924 1-2 s7=Nlock 3.34 0.6815 2-3 s8= NGood 1.07 0.215 1-2 s9=Nfatigue 2.72 0.984 2-5 s10 =Nparis/M 0.049 0.01 1-2 P(s) - 0.0002

14.2. Comparison of thick plate and thin plate joint satisfactions

Comparison of the decision variables and satisfactions of the thin plate W=3 and thick plate W=8 joints is shown in Table 5. Table 5. Comparison of the decision variables and satisfactions of the thin plate W=3 and thick plate W=8 joints

14.3. Results for a thin plate joint

Now W = 3 was fixed to get a thin plate joint. Other design variable z, V were free This joint not as optimal as the W=8 joint. The plate thickness is

lp = W d = 3 d.

Main input data One bolt nut.thread M10, d = 10mm outer diameter. Nominal area A and true area

4 410 78 ,

, 0.88 (115) friction is µ = 0.15, strength class is 8.8, Sut = 840MPa, Sy = 660 =0.8 Sut Main output data Design variable values obtained by the program

0.9, 0.60, : 3,

3 ∙ 10 30 (116)

The obtained design values The prestress is

0.9 ∙ 660 594, (117)

Load force stress is

0.60 ∙ 594 356, (118)


∙ 0.600.900.88 0.418 ∙ (119)

or

0.478 ∙ 0.418 ∙ 660 ∙ 78

21.5 (120)

The setting can be roughly approximated by a linear model of equation (110)

fZ = 4.12· 10-6 m


1085.0

1

1

s

p

k

kC

ps

pK

(121)

14.4. Comparison of the safety factors of the thick plate joint W=8 with the thin plate joint W=3

The comparison is shown in Table 6 The differences and similarities The safety factor against decrease of prestress due to

setting was 1.6 times higher with thick plate joint compared with thin plate joint

The safety factors against fatigue failure were more than 1.5 times higher with thick plate joint than with thick plate joint.

Other factors of safety were nearly the same.

Table 6. Comparison of the safety factors of the thick plate joint W=8 with the thin plate joint W=3 C = joint stiffness factor


W=3 Thin plates

W=8 Thick plates

Comparison W=8 vs.W=3

s1=Nsetting 1.02 1.70 W8 1.6x better s2=NtightA 2.25 2.38 about same s3=NtightB 0.763 0.763 about same s4=Ny 1.11 1.05 about same s5=Nsepnoecc 1.87 1.63 about same s6=Nsepecc 0.97 1.05 about same s7=Nlock 3.34 3.34 about same s8= NGood 1.07 1.5 W8 1.5x better s9=Nfatigue 2.72 4.75 W8 1.7x better s10 =Nparis/M

0.049 0.236 W8 4.8x better

C C=.108 C=.054 C=0.054


W=3 DecV sk

W=3 P(sk)

W=8 DecV sk

W=8 P(sk)

Good Range s

s1=Nsetting 1.02 0.72 1.70 0.9905 1-2 s2=NtightA 2.25 0.997 2.38 0.9993 2-3 s3=NtightB 0.763 0.88 0.763 0.878 1-3 s4=Ny 1.11 0.24 1.05 0.4 0.9-1 s5=Nsepnoecc 1.87 0.9987 1.63 0.9876 1-2 s6=Nsepecc 0.97 0.8924 1.05 0.9229 1-2 s7=Nlock 3.34 0.6815 3.34 0.6815 2-3 s8= NGood 1.07 0.215 1.5 0.9514 1-2 s9=Nfatigue 2.72 0.984 4.75 0.5949 2-5 s10 =Nparis/M

0.049

0.01 0.236 0.6552 1-2

P(s) - 0.0002 - 0.0800 C 0.108 0.054


51

14.5. Feasibility test of the methodology in a nut loosening case study

In Appendix 1 the mechanisms of loosening of nut in a typical joint of a motor vehicle component are analysed. The top event is union of three technical events involving also human error is all stages TOP = A1 and A2 and A3. A1 = Manufacturing failure, geometry failure, material defects A2 = Assembly failure, joint too loose or too tight, dirty screws, glue failure

A2 is union of sub events B1 =Theprestress was too high or too low B2 = Setting too high, low friction B3 = Locking glue of screws with additives fails

A3 = Loading failure, overloading The conclusion is that the overhaul assembly A2 has dominantly caused the event B3 or failure of locking glue of screws .It’s total probability is 0.55. The two others have 0.17 and 0.17. Transverse force bends the nut with nonlinear thread stiffness’ dynamic decrease of the friction torque acts as an impulse making the nut rotate somewhat. With some repetitions it may become loose. 15. CONCLUSIONS The following conclusions can be drawn.

1) The first goal of this study was to develop and test feasibility the methodology for fuzzy optimum design of a screw joint. The chosen method was fuzzy optimum design for obtaining maximal trade off customer satisfaction on all desired properties. This was well achieved using ten essential properties as decision variables of screw joint design into the model. Using the fuzzy optimum design methodology it is possible to take into consideration all wishes of the customer and then translate them into technical definitions of desired properties, or decision variable and obtain an optimally satisfying trade off design.

Many variations of geometry and many materials and loadings can be considered.

The aim in any design task is to satisfy all goals and constraints. This task can be done more easily by formulating all goals and constraints by one flexible unified fuzzy standard function.

This method can be extended to consider rationally also many not commeasurable properties like costs, complex loads, material properties creep and relaxation and fatigue behaviour, tightness and pressure bearing capacity and geometry.

2) The second goal was to test the feasibility of the program with two extreme case of screw joints. One is a thick plate joint and the other a thin plate joint. This goal was achieved satisfactorily since the results were reasonable and similar as in literature.

Comparison of the safety factors gave the following results

The safety factor against decrease of prestress due to setting was 1.6 times higher with thick plate joint compared with thin plate joint

The safety factors against fatigue failure were more than 1.5 times higher with thick plate joint

Other factors of safety were nearly the same. 3) The third goal was to estimate risks and mechanisms of loosening nut in a typical joint of a motor vehicle component. This goal was reasonably well achieved using failure criticality analysis and the decision variable models for screw joint. The conclusion is that the overhaul assembly A2 has caused the event B3 or failure of locking glue of screws .It’s total probability is 0.55. The two others have 0.17 and 0.17 total probabilities Transverse force bends the nut with nonlinear thread stiffness’ dynamic decrease of the friction torque acts as an impulse making the nut rotate somewhat. With load repetitions it may become loose. Dynamic interaction of axial vibration and tensional vibration may synergically combine with bending load and cause loss of effective friction causing rotation or loosening of the nut. The possible mechanisms of decrease of friction are little known, the dynamic loading is assumed to cause it. 4) The fourth goal in the future is to extend the methods to optimize joints of plates with many screws. This goal is feasible to achieve since basic models are already available. New desires like cost and load options can be easily added to the present methodology. ACKNOWLEDGEMENT The authors are grateful for the support to this research given by the companies HimtechOy Engineering and MSc JuhaKärki for expert program technical advice. REFERENCES [1] Norton, R., L.(2006) Machine Design.An Integrated

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Carl HanserVerlag. [3] Evans, W.J.,Harrison,G.F.(1976) The development of

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[4] Dowling, N.E.(1998) Mechanical behavior of materials, Prentice Hall..

[5] Hess D.P.,Sudhirkashyap,S.V. (1997) Dynamic loosening and tightening of a dingle-Bolt assembly, Journal of Vibration and acoustics, Vol.119 pp. 311-316

[6] Kerley,J.J.,(1987) An application of retroduction to Analyzing and testing the backing off of nuts and bolts during dynamic loading, NASA Technical Memorandum 4001, Goddard Space Center, Maryland.

[7] Varga, L.; Baratosy, J.(1995) ,The International Journal of Pressure Vessels and Piping, Volume 63, Number 1, 1 January 1995, pp.

[8] Martikka,H.,Taitokari,E.(2006 ). Optimal design of fibre reinforced tubular structures. High Performance structures and materials III, HPSM2006, pp.593-604, http://dx.doi.org/10.2495/HPSM06058


52

[9] Martikka,H., Pöllänen,I. (2008 ) Comparison of optimisation tools for design of welded beams ,Machine design fundamentals, The Monograph of Faculty of Technical Sciences,Novi Sad, Serbia.

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[12] Martikka, H., Taitokari,E.(2011) Optimum Fuzzy Design of Ecological Pressurised Containers,Hindawi Publishing Corporation.Advances in Fuzzy Systems, Volume 2011, Article ID 791687,9 pages, doi:10.1155/2011/791687.

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[15] Martikka,H., Taitokari,E.,(2013) Bridge concept design using heuristic fuzzy optimum design and FEM.Mechanical Engineering Research, Vol 3, No.1 2013,ISSN 1927-0607 E-ISSN 1927-0615,pp. 44-67,doi:10.5539/merv3n1p44, URL:http://dx.doi.org/10.5539/mer.v3n1p44.

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[17] Little R. E. (1967) Bolted joints: How much give?,Machine Design Nov.9.

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[20] Standard SFS 2067. Finnish Standard. [21] Thomala, W. ErläuterungzurRichlinie VDI 2230

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propagation data for steel subjected to pulsating tension loading, The Welding Institute 59/1978/E.

[24] Kloos,K.-H,Thomala,W.(1979) SpannungsverteilungimSchraubengewinde, VDI-Z 121 ,12, pp. 127...135.

[25] Variator motor. Ref.[25]

https://encrypted-tbn3.gstatic.com/images?q=tbn:ANd9GcS7k-FudBqC4_QFtlQ7sYcOXYFzybEDXNfbyt1Fm9Tcf7koiUoM

[26] Engine variator and clutch variator.Ref. [26

https://encrypted-tbn0.gstatic.com/images?q=tbn:ANd9GcTUxhtHPvqeHFB3nJUuuCtNIc7B8Ni8z3m9tv8TdcwcoP9S6wotxg

[27] Figure A1-3. The engine variatornut.Ref. [27]

https://encrypted-tbn0.gstatic.com/images?q=tbn:ANd9GcQsftiFLkePsxN-m7Jd99z-WFIom8BkGyPeLXmi_NLSVpA_BUQx

[28] Leitch R.D. (1988) Basic reliability engineering analysis, Butterworths, London

[29] Dhillon,B.S.,Singh,C. (1981) Engineering reliability, Wiley-Interscience.

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