An Analytical Method for Determining the Load Distribution

Research ArticleAn Analytical Method for Determining the Load Distribution ofSingle-Column Multibolt Connection

Nirut Konkong and Kitjapat Phuvoravan

Department of Civil Engineering Faculty of Engineering Kasetsart University Bangkok 10900 Thailand

Correspondence should be addressed to Kitjapat Phuvoravan fengkppkuacth

Received 23 March 2017 Accepted 28 May 2017 Published 6 July 2017

Academic Editor Dimitris Rizos

Copyright copy 2017 Nirut Konkong and Kitjapat PhuvoravanThis is an open access article distributed under the Creative CommonsAttribution License which permits unrestricted use distribution and reproduction in any medium provided the original work isproperly cited

The purpose of this research was to investigate the effect of geometric variables on the bolt load distributions of a cold-formedsteel bolt connection The study was conducted using an experimental test finite element analysis and an analytical methodThe experimental study was performed using single-lap shear testing of a concentrically loaded bolt connection fabricated fromG550 cold-formed steel Finite element analysis with shell elements was used to model the cold-formed steel plate while solidelements were used to model the bolt fastener for the purpose of studying the structural behavior of the bolt connections Materialnonlinearities contact problems and a geometric nonlinearity procedure were used to predict the failure behavior of the boltconnections The analytical method was generated using the spring model The bolt-plate interaction stiffness was newly proposedwhich was verified by the experiment and finite element model It was applied to examine the effect of geometric variables on thesingle-columnmultibolt connectionThe effects were studied of varying bolt diameter plate thickness and the plate thickness ratio(11990521199051) on the bolt load distribution The results of the parametric study showed that the 11990521199051 ratio controlled the efficiency of thebolt load distribution more than the other parameters studied

1 Introduction

A bolt connection is generally used in cold-formed steelstructures It can result in a high-stress concentration at con-tact regions especially with thin cold-formed steel membersThe shearing of cold-formed steel bearing on material infront of the bolt tearing of the cold-formed steel in thenet section and shearing of the bolt are the main failuremodes of a single bolt connection In multibolt connectionsthe failure modes are identified by numerical and analyticalmethods with the results showing combined bearing failureand net-section failure modes [1 2] The failure behaviorof a multibolt connection is considered in two steps as (a)the load distribution between bolts which is the process forconsidering the stress bypass in the multibolt connectionand (b) local stresses analysis as the bearing stress andnet-section stress around a critical hole which is used toinvestigate ensuring a safety margin of connection [3ndash5]Practically the lap shear connection is used for tension in astructural member such as the diagonal and tension chord

truss element It is designed using a multiple bolt connectionwith even bolt load distribution The bolts in a row in theloading direction are assumed to have an equal distributionof load among the bolts In an actual lap connection thebehavior of each bolt in a row is different due to an unequalload distribution that also results from the stiffness of thesingle bolt deformation and plate deformation between boltsas shown in Figure 1 where the evolution of the internal forcesversus the applied force (119875) is demonstrated with a five-boltconnection in which 120575 and 120575119887minus119894 are the connection elongationand bolt deformation

In the first step the connection evolves elastic behaviorwith the outer bolts being more loaded than the inner onesIn the second step the outer bolt rows reach their elastic limitstrength but the inner bolt rows are still in the elastic stage Inthe third step the plastic behavior is distributed to the innerbolt rows In the fourth step the plastic behavior is developedin all the bolt rows However the limitation of the cold-formed ductility between the bolt-plate in bearing may bereached before the connection develops full plastic strength

HindawiAdvances in Civil EngineeringVolume 2017 Article ID 1912724 19 pageshttpsdoiorg10115520171912724

2 Advances in Civil Engineering

P1 2 3 4 5

1 2 3 4 5

1 2 3 4 5

1 2 3 4 5

P PP

Bolt loadLoad deformation Load of bolt 5 Load of bolt 4 Load of bolt 3

P

P PP

PP

P

P

PP

P

P

Step 1

Step 2

Step 3

Step 4

P

P

Ppl

Pe

Pe

P

Ppl

Ppl

Ppl

Pe

P

Ppl

Pe

P

Ppl

Pe

P lt Pe

P = Pe

Pe lt P lt Ppl

P = Ppl

120575

120575

120575

120575

120575

120575

120575

120575

120575b-5

120575b-5

120575b-5

120575b-5

120575b-4

120575b-4

120575b-4

120575b-4

120575b-3

120575b-3

120575b-3

120575b-3

of connection

Figure 1 Stages of force distribution in bolt connections [6]

These failures result in a lower average bearing stress valueper bolt hole of the multibolt connection In this case theconnection would not develop the ultimate bearing capacityas recommended by the design method [6 7] Bolt load pre-diction is a complex procedure involving the establishmentof the bolt load distribution The stiffness components (theaxial stiffness of the plate and the interaction of the bolt-plate)must be included in the investigation of bolt load distributionHowever the interaction stiffness of the bolt-plate is noteasily defined in a closed-form solution because the analysisrequires knowledge of the influenced parameters such as theshear deformation of the bolt bending deformation of thebolt bearing deformation of the plate bearing deformationof the bolt and deformation of the plate between boltsThus the equation of the bolt-plate interaction stiffness inindustry today is derived by using a semiempirical approachbased on the testing data [8ndash10] The weakness of theseempirical equations is that they ignore the shear deformationassumption in the bolt component which is dominated by theshort bolt length

This study describes the method of analysis for deter-mining the load distribution among bolts in a single-columnmultibolt connection A bolt-plate interaction stiffness equa-tion is proposedwhich is verified by the experimental test andfinite element analysis (FEA)The experimental investigationwas performed using lap shear bolt connection testing andthe connections were fabricated from cold-formed steel

2 Material Properties and Model Description

The laboratory test was performed using G550 cold-formedsteel with a plate thickness of 10mm for the bolt connectiontest G550 is a hot-dipped zinc-coated structural steel and

has a minimum yield strength of 550MPa The materialproperties for G550 steels were specified by following ASTMStandards A370-07 [11] as shown in Table 1 Following theresearch objective and scope the possibility of bolting shearfailure was eliminated by using M5 Grade 88 steel boltsconforming to ASTM A325M [12] as shown in Table 2

The bolt connection specimen and the hole size are shownin Figure 2 The bolts and nuts were assembled with integralwashers that were tightened by hand which allowed for slipin the connection after the initial loading

The bolt connection specimens were tested using auniversal testingmachine (UTM) in the structural laboratoryof Kasetsart University Bangkok Thailand as shown inFigure 3(a) A typical bolted shear connection test setup isshown in Figure 3(b) The load and the axial deformationwere measured to describe the connection failure behaviorThe load was applied to the specimen using displacementcontrol with a speed of 1mmminute A linear variabledifferential transformer (LVDT) was installed to measure theelongation of a single bolt connection over a distance of200mm from the center line The load was gradually applieduntil either the specimen failed or the specimen entered intoplastic deformation Once the applied load began to dropeven though the specimen continued to elongate the test wasstopped

3 Finite Element Model and Validation

In the present study the ANSYS finite element package[13] was used to predict the failure behavior of the boltconnection as shown in Figure 4 Nonlinear finite elementanalysis was used to compare and validate the results of thesingle bolt connection test The nonlinear geometry material

Advances in Civil Engineering 3

F

F

1530

100

15

11

Bolt diameter = 5

Washer outside diameter = 15 and thinness = 2

Unit mm(a) Bolt connection geometries

F

F

Hole size

d

lt127

dℎ

1mm

1mm

Hole diameter dℎ (mm)Bolt diameter d (mm)d + 08d + 16ge127

(b) Hole size

Figure 2 Bolt connection specimens

Table 1 Material properties of cold-formed steel G550-1T

Test number SD COVCT-1 CT-2 CT-3 CT-4 CT-5 Mean

T (mm) 1045 1045 1044 1045 1045 104 00004 004W (mm) 1264 1265 1259 1258 1252 1259 00522 041119865119910 (MPa) 60847 60577 60590 61750 60646 60882 49715 081119865119906 (MPa) 62700 62250 62550 63000 62000 62500 38890 062119864 (GPa) 21351 21349 21349 21358 21358 21353 441757 002119865119906119865119910 103 102 103 102 102 102 00054 053Remark Fy is the yield stress Fu is the ultimate stress E is the modulus of elasticity T is the plate thicknessW is the width of coupon test specimen COV isthe coefficient of variation and SD is the standard deviation

Table 2 Material properties of bolts

Test number SD COVBT-1 BT-2 BT-3 BT-4 BT-5 Mean119889 (m) 493 497 497 497 494 495 0019 0393119865119910 (MPa) 93000 94900 92700 97600 87200 93080 38232 4107

119865119906 (MPa) 963 930 963 997 909 95240 33908 3560119864 (GPa) 20408 20408 20409 20410 20409 20408 7385 0003Remark d is the bolt diameter

nonlinearity and contact problem were modeled in the finiteelement model The cold-formed steel was modeled usingeight nodes with six degrees of freedom at each node shellelements (SHELL281) as shown in Figure 4(b)

Three-dimensional 20-node solid elements (SOLID186)with three degrees of freedom at each node were used tomodel the steel bolts and thewashers as shown in Figure 4(c)Contact between the steel plates steel bolts washers andwallof the holes was defined as the contact elements (CONTA173and TARGE170) The pure penalty contact algorithm wasused to solve the constrained optimization contact problemas shown in Figures 4(d)-4(e) The mesh size was controlledby the aspect ratios (length-to-width ratio) which wereselected to be close to 10 for all components Around thebolt holes fine mesh was utilized in order to transfer thestress from the bolt to the cold-formed steel plate The nodesalong the endplateswere restrainedwith geometric boundary

conditions as pinned and roller support The uniform loadwas applied along the end plate line 119861 with the displacementcontrol as shown in Figure 4(f) The static friction betweenthe steel plates was neglected due to the smooth coatingsurfacesThe actual stress-strain values from the cold-formedsteel tensile testing were converted to the true stress (120590true)and strain (120590true) which imposed a multilinear isotropichardening model as shown in Figure 4(g) It was convertedusing

120590true = 120590test (1 + 120576test) 120576true = ln (1 + 120576test) (1)

The large deflection optionwas activated to achieve nonlineargeometry behavior By turning on this option iterative solv-ing and updating of the stiffness matrix based on the incre-mental nodal displacements at each equilibrium iteration


UTM rod load cell

Upper grip

Lower grip

Specimen

UTM rod

Fixed gagelengths

Data

LVD

T

logger

(a) Test apparatus (b) Test specimen

Figure 3 Bolted connection test setup

Cold-formed steel (SHELL281)Washer (SOLID186)

Bolt (SOLID186)

x

z

(a) Geometry model

M

I

J

N

K

OL

P

1

2

3

6

5

8

4

7

1

2

3 4

56

z0y0

x0

(b) SHELL281

J

1

x

yz R

K

I

O

W

S

Q

I

TL

I

Y

M

XP

N

Z

VU

2

3

4

5

6

(c) SOLID186

x

yz I

M

J

N

K

OL

P

Contact element (CONTAC174)

Target element (TARGET170)

PI998400

J998400

N998400

M998400

L998400

K998400

O998400

(d) Contact element

x

z

CONTAC174

TARGET170

Steel bolt

Cold-formed steel

fp = kpxp

kp kpkpkp xp

fp = the contact forcexp = the contact penetrationkp = the contact stiffness

(e) Pure penalty algorithm

xz

y

(pined)

(roller)

A

B

Fine mesh size

dx = 0

dy = 0

dz = 0

dx = free

dy = 0

dz = 0

(f) Mesh and boundary condition

0

100

200

300

400

500

600

700

0 002 004 006 008

Axi

al st

ress

(MPa

)

Axial strain (mmmm)

Actual stress-strain (average)True stress-strain(g) Actual stress and plastic strain

Figure 4 Geometry element type and material model of FEA


0

2000

4000

6000

8000

10000

12000

0 1 2 3 4 5 6 7 8 9 10 11 12

Load

(N)

Axial deformation (mm)

S-1S-2S-3

S-4S-5S-FEA

(a) Load-deformation curves

Test FEA

Single bolt connection

(b) Failure mode

Figure 5 Load-deformation curves and failure mode of single bolted connection

Table 3 Summary of the lap shear connection results

Test Loading capacity 119875119899test (N) Failure mode 119875119899test119875119899FEAS-1 10812 Bearing 097S-2 10660 Bearing 095S-3 10707 Bearing 096S-4 10800 Bearing 097S-5 10758 Bearing 096Mean 10747 096FEA 11167 Bearing

were performed into the nonlinear geometry analysis Finallythe Newton-Raphson algorithm was activated to obtain thesolution after each displacement increment

4 Single Bolt Connection Test Results

The load capacity and failure mode of single bolt connectiontests are shown in Figure 5(a) The deformation shapefrom the experimental tests and the FEA showed that thespecimens were subjected to bearing failure as shown inFigure 5(b) The ratio of the ultimate loads between theexperiment and the FEA (119875119899test119875119899FEA) is shown in Table 3and this indicates that the ultimate load capacity of theexperimental results was in good correlation with the FEA

In order to identify the connection failure modes theVon-Mises stress contour was normalized by the ultimatestress of the cold-formed steel material as shown in Figure 6

Hart-Smith [14] and He and Wang [2] have suggesteda method for bearing failure prediction of the shear con-nection The monitoring strain point at 119862 119863 and 119864 is zerowhen the connection suffers from bearing failure and themonitoring stress normalization at point 119862 is equal to 1for net-section failure The stress-strain analysis of the FEAresults showed that the lap shear bolt connection sufferedfrom bearing failure

5 Bolt-Plate Interaction Stiffness Analysis

In multibolt connections the bolts in different rows carrydifferent amounts of the load which depend on the boltdiameter plate stiffness and other geometry configurations[15 16] The connection stiffness is fundamental to under-standing the load transfer and the load distribution of thebolt connection In accordance with the typical bolt thedeformation of the steel bolt illustrated in Figure 7 wassimplified as a short beamThe concept of bolt flexibility (119862119887)prediction can be defined by

119862119887 = 1119896119887 =120575119887119865 (2)

where 119896119887 is the stiffness of the steel bolt F is the applied forceand 120575119887 is the bolt deformation

An analytical beam theory with the shear deformationand rotational bending effects included is the theory knownas Timoshenko beam theory [17] For Timoshenko beamsplane cross-sections will rotate due to shear forces Therotation (120579) and deflection of a beam (119906) are written by

120579 (119909) = int1199090

119872(119904)119864119868 119889119904 + 1205790 (3a)

119906 (119909) = int1199090120579 (119904) 119889119904 minus int119909

0

119881 (119904)120581119860119866119889119904 + 1199060 (3b)

where 1198810 1198720 1205790 and 1199060 are constants of integration whichcan be determined using the boundary conditions for theparticular problem 119860 is area cross-section of the beam 119866 isa shear modulus of the beam and 120581 is the shear coefficientdue to the nonuniform distribution of shear stress along thecross-section Figure 8 shows the assumption of the loadcomponents of a steel bolt whichwas used to analyze the bolt-plate interaction stiffness Point 119860 point 119861 and point 119862 arethe reference points on the bolt that refer to the middle of theplate thickness and the bolt length

The bolt tolerated the bearing force shearing force andbending moment initiated by an eccentric load from the steel


A

BC

D

E

Stress normalization


Stre

ss n

orm

aliz

atio

n

Strain

Strain

A

BC

D

E

10004

089658

079278

068899

05852

04814

037761

027382

017003

0066233

180∘

0∘31592

28085

24578

21071

17564

14057

1055

070431

035361

00029019

120579 (degrees)

120579

13

10

08

05

03

000 20 40 60 80 100 120 140 160 180

0051152253

Stra

in (m

mm

m)

Figure 6 Stress normalization and strain distribution of bolt connection

F

Bolt

Plate 2

Plate 1 A

B

C

Bolt

Plate 2

Plate 1F = 0

t1

t2

t1

t2

120575b

Figure 7 Deformation of single bolt connection

F

F

M

L

A

B

C

1205962t2

1205961t1V1

V2

t1

t2

M

Figure 8 Forces in bolt


A

B

C

Fixed-free Fixed-free Fixed-free Fixed support

Guiding support

M

Case 3Case 2Case 1

= + +

1205962t2 1205962t2

1205961t11205961t1

Figure 9 Statically indeterminate system

L

a b

L

a b

x x

L

x

Load case 1 Load case 2 Load case 3

L

x

y

y y y120596 120596

MB

0 le x le a

a le x le L

120575(x) = minus120596x3

24EI(4x minus a)

120579B =120596a3

6EI

0 le x le a

120575 = minus120596x2b

12EI(3L minus 2x + 3a)

a le x le L

x = L

120575B =MBL

2

2EI

120579B =MBL

EI

120575B = deflection at end B of the beam

120579B = angle of rotation at end B of the beam

120575B120579B

120575 = minus120596

24EI(x4 minus 4Lx3 + 6L2x2 minus 4a3x + a4)

120575(x) = minus120596x2

24EI(6a2 minus 4ax + x2)

120579B =120596(L3 minus a3)

6EI

minus

Figure 10 Bending deformation of cantilever beam

plates The statically indeterminate system was divided inFigure 9

A cantilever beam with partially uniform loads and endmoments idealizes the load components of a steel boltThe superposition analysis method was used to analyze thebending deformation and shear deformation for a cantileverbeam carrying a uniform load of intensity over part of thespan as shown in Figures 10 and 11

The bending moment was introduced by eccentric forcefrom the plate It was derived by sum of the rotation of theneutral axis at the end of the bolt as shown in Figure 12 Thisrotation can be rewritten by

120601 = 1206011 + 1206012 + 1206013 = (1205791 + 1205741) + (1205792 + 1205742) + (1205793 + 1205743)= 0 (4)

where 1205791 1205792 and 1205793 are the bending rotation components dueto the loads 12059611199051 12059621199052 and119872 respectively 1205741 1205742 and 1205743 areshear rotation due to the loads1205961119905112059621199052 and119872 respectively

The bending rotation and shear rotation components canbe written as (5a)-(5b) which refer to Figures 10-11

1205791 = minus 1205961119905316119864119887119868119887 1205792 = 1205962 (1198713 minus 11990531)6119864119887119868119887 1205793 = 119872119861119871119864119887119868119887

(5a)

1205741 = 01205742 = 01205743 = 0 (5b)


a

b

L

x

La b

La b

x x

L

x


a

x

b

V

dx

x

x

x

y

y y y

120596

120596 120596

MB

0 le x le a

V1(x) = minus120596(x minus a)

1205741(x) = minus120596(x minus a)

120581AG0 le x le b

V2(x) = 0

1205742(x) = 0

0 le x le a

Vx(x) = minus120596b

1205743(x) = minus120596b

120581AG

0 le x le b

V4(x) = minus120596(x minus b)

1205744(x) = minus120596(x minus b)

120581AG

V(x) = 0

120574(x) = 0

120574

120575s(x) = 0

120574 =d120575sdx

d120575s

120575s

120596b

1205751(x) = intx

01205741(x)dx

1205752(x) = inta

01205741(x)dx + intx

01205742(x)dx

1205754(x) = inta

01205743(x)dx + intx

01205744(x)dx

1205753(x) = intx

01205743(x)dx

Figure 11 Shear deformation of cantilever beam

A

B

=C

Fixed support

Guiding support

Fixed-free Fixed-free Fixed-free

+ +

Case 1 Case 2 Case 3

1205962t2 1205962t2

1205961t11205961t1

MB

1206011 1206012 1206013

Figure 12 Rotation of the bolt

where 119864119887 is the modulus of elasticity of the bolt 119868119887 is themoment of inertia of the bolt 1199051 and 1199052 are the thickness ofthe plate 1205961 and 1205962 are the loads on the bolt and 119871 is thethickness of the connection The force in the bolt (119865119887) can bepresented by

119865119887 = 12059611199051 = 12059621199052 (6)

The relationship of the uniformed load in the connection canbe written by

1205961 = 1205962 (11990521199051) (7)

Thus (4) can be rewritten as

minus 1205961119905316119864119887119868119887 +1205962 (1198713 minus 11990531)6119864119887119868119887 + 119872119861119871119864119887119868119887 = 0 (8)

The bending moment in the bolt can be rewritten as

119872119861 = minus1205962 ((1199051 + 1199052)3 minus 11990531 minus 119905211199052)6 (1199051 + 1199052) (9)

The connection stiffness calculation was divided into twomain stiffness components being the bolt stiffness (119896119887) andthe contact stiffness (119896119888) as shown in Figure 13 The bolt


M

M

A

C

B1205962t2

1205961t1

kc

kb

kc

kb

Fixed support

Guiding support

Figure 13 Stiffness components of single bolt connection

stiffness was analyzed by considering the flexural stiffness(119896flex119887) and the shear stiffness (119896she119887) of the bolt The contactstiffness was the interaction stiffness between the plate andthe bolt which is referred to as the plate bearing stiffness(119896bea119901) and the bolt bearing stiffness (119896bea119887)

The bolt-plate interaction stiffness (119896bp) is associated withthe bending shearing and bearing stiffness of the assemblyparts It can be written as the spring components using (10a)-(10b)

1119896bp = 1119896119887 +1119896119888 (10a)

with

1119896119887 =1119896flex119887 +

1119896she119887 1119896119888 =

1119896bea119901 +1119896bea119887

(10b)

In the bending stiffness component the bolt deformationwas analyzed using (3b) which ignored the shear deformationterm as written by (11) 1205790 and 1199060 were set equal to zero for thecantilever beam boundary condition

119906 (119909) = int1199090120579 (119904) 119889119904 + 1199060

= int1199090(int1199090

119872(119904)119864119868 119889119904 + 1205790)119889119904 + 1199060(11)

The flexural deformation of the bolt was calculated bysum of deformation at point 119860 and point 119861 as shown inFigure 14

The flexural deformations of the bolt at point 119860 (120575flex119860)can be written as

120575flex119860 = 120575flex11986012059611199051

+ 120575flex11986012059621199052

+ 120575flex119860119872 (12)

where 120575flex11986012059611199051

120575flex11986012059621199052

and 120575flex119860119872 are the flexuraldeformation at point119860 due to1205961119905112059621199052 and119872 respectivelywhich are defined by

120575flex11986012059611199051

= minus 12059611199053124119864119887119868119887 (61198862 minus 4119886119909 + 1199092) (13a)

120575flex11986012059621199052

= 1205962119887119909212119864119887119868119887 (3119871 + 3119886 minus 2119909) (13b)

120575flex119860119872 = 119872(119909)22119864119887119868119887 (13c)

119886 = 1199051119887 = 1199052119871 = 1199051 + 1199052119909 = 11990512

(13d)

The deformations of the bolt at point 119861 (120575flex119861) can be writtenby

120575flex119861 = 120575flex11986112059611199051

+ 120575flex11986112059621199052

+ 120575flex119861119872 (14)

where 120575flex11986112059611199051

120575flex11986112059621199052

and 120575flex119861119872 are the flexuraldeformations at point119861due to1205961119905112059621199052 and119872 respectivelywhich are defined as

120575flex11986112059611199051

= minus 1205961119886324119864119887119868119887 (4119909 minus 119886) (15a)

120575flex11986112059621199052

= 120596224119864119887119868119887 (1199094 minus 41198711199093 + 611987121199092 minus 41198863119909 + 1198864) (15b)

120575flex119861119872 = 119872(119909)22119864119887119868119887 (15c)

119909 = 1199051 + 11990522 (15d)


F

F

Bolt

Plate 2

Plate 1A

B

C

A

BC

NA

t1

t2

120575flexA

120575flexB

120575flexC

120575bolt

120575bolt = (120575flexC minus 120575flexA)+ (120575flexB minus 120575flexC)120575flexB minus 120575flexA=

Figure 14 Flexural deformation of the bolt

The total bolt flexural deformation due to 12059611199051 12059621199052 and 119872was defined using

120575flex119887 = 120575flex119861 minus 120575flex119860= ( 1128

(1199051 + 1199052) (311990521 + 131199051199051199052 + 311990522)119864119887119868119887 )12059621199052 (16)

where119865119887 = 12059621199052Thus the total bolt flexural deformation canbe rewritten as

120575flex119887 = (311990521 + 16119905221199052 + 16119905111990522 + 311990532128119864119887119868119887 )119865119887 (17)

The bolt bending flexibility (119862flex119887) is defined as the ratioof the bending deformation due to the force of the bolt whichcan be calculated using

119862flex119887 = 1119896flex119887 =120575flex119887119865119887

= 311990521 + 16119905221199052 + 16119905111990522 + 311990532128119864119887119868119887 (18)

In the shear stiffness component the bolt shear deformationis analyzed using (3b) in which the flexural deformation termis ignored as written by (19) 1199060 is set equal to zero for thecantilever beam boundary condition

119906 (119909) = minusint1199090

119881 (119904)120581119860119866119889119904 + 1199060 (19)

The Timoshenko beam theory [17] proposed that theshear stress is uniform over the cross-section Actuallyit is not uniform over the cross-section Thus the shearcoefficient (120581) was introduced as a correction factor to allowthe nonuniform shear strain to be expressed as a constantHutchinson [18] proposed the shear coefficient equation of acircular cross-section using

120581 = 6 (1 + ])27 + 12] + 4]2 (20)

where ] is the Poisson ratio For ] = 03 for a steel bolt theshear coefficient is 093The shear deformation of the bolt was

calculated by sum of deformation at point 119860 and point 119861 asshown in Figure 15

The deformations of the bolt at point 119860 (120575119904119860) can bewritten using

120575119904119860 = 12057511990411986012059611199051

+ 12057511990411986012059621199052

+ 120575119904119860119872 (21)

where 12057511990411986012059611199051

12057511990411986012059621199052

and 120575119904119860119872 are the shear deformationsat point 119860 due to 12059611199051 12059621199052 and 119872 respectively which aredefined using

12057511990411986012059611199051

= int11990901205741 (119909) 119889119909 = minus 31205961119905218120581119866119887119860119887 (22a)

12057511990411986012059621199052

= int11990901205743 (119909) 119889119909 = 1205962119905111990522120581119866119887119860119887 (22b)

120575119904119860119872 = 0 (22c)

119909 = 11990512 (22d)

where 119860119887 is the cross-sectional area of the bolt and 119866119887 is theshear modulus of the bolt The deformations of the bolt atpoint 119861 (120575119861) can be written using

120575119904119861 = 12057511990411986112059611199051

+ 12057511990411986112059621199052

+ 120575119904119861119872 (23)

where 12057511990411986112059611199051

12057511990411986112059621199052

and 120575119904119861119872 are the shear deformationat point 119861 due to 12059611199051 12059621199052 and 119872 respectively which aredefined by

12057511990411986112059611199051

= int11988601205741 (119909) 119889119909 + int119909

01205742 (119909) 119889119909 = minus 1205961119905212120581119866119887119860119887 (24a)

12057511990411986112059621199052

= int11988601205743 (119909) 119889119909 + int119909

01205744 (119909) 119889119909

= 12059621199052 (81199051 + 31199052)8119866119887119860119887 (24b)

120575119904119861119872 = 0 (24c)

119909 = 11990522 (24d)


F

F

Bolt

A

B

C

A

BC

Plate 2

Plate 1

NA

t1

t2

120575bolt

120575sA

120575sB

120575sC 120575bolt = (120575sC minus 120575sA)(120575sB minus 120575sC)+

120575sB minus 120575sA=

Figure 15 Shear deformation of the bolt

The total bolt shear deformation due to1205961119905112059621199052 and119872wasdefined using

120575119904 = 120575119904119861 minus 120575119904119860 = 38 ( 1199051 + 1199052120581119860119887119866119887)12059621199052 (25)

where 1205961 = 1205962(11990521199051) Thus the total bolt shear deformationcan be rewritten by

120575119904 = 38 ( 1199051 + 1199052120581119860119887119866119887)12059621199052 (26)

The bolt shear flexibility (119862sd119887) is defined as the ratio of sheardeformation due to the force of the bolt which is calculatedusing

119862sd119887 = 1119896sd119887 =120575119904119865119887 =

38(1199051 + 1199052)120581119860119887119866119887 (27)

To calculate the bearing stiffness of the bolt and plate thebearing stiffness of connection was calculated by using theschematic diagram shown in Figure 16 The bearing stress(120590br) was defined by the applied load (119875) divided by the holediameter and the plate thickness which is written as (28a)The bearing strain (120576br) was defined by the plate bearingdisplacement divided by the hole diameter which is writtenusing (28b)

120590bea = 119875119889119905 (28a)

120576bea = 120575bea119901119889 (28b)

The bearing stiffness of bolt hole was calculated by usinglinear stress-strain relationship as presented using

120590bea119901 = 119864119901120576bea119901 (29a)

119896bea119901 = 119875120575bea119901 = 119905119889119864119901119889 = 119905119864119901 (29b)

Similarly the bearing stiffness of the steel bolt was calculatedin the same way using the samemanner which was written as

120590bea119887 = 119864119887120576bea119887 (30a)

119896bea119887 = 119875120575bea119887 =119905119889119864119887119889 = 119905119864119887 (30b)

Thus the bolt bearing flexibility (119888bea119887) and the plate bearingflexibility (119888bea119901) can be calculated using

119888bea119887 = 1119896bea119887 =11199051119864119887 +

11199052119864119887 (31)

119888bea119901 = 1119896bea119901 = 111990511198641199011 +111990521198641199012 (32)

where 1198641199011 and 1198641199012 are the modulus of elasticity of plate 1 andplate 2 respectively Finally the bolt-plate interaction stiff-ness was newly proposed by the equilibrium of interactionforce between the bolt and the plate which can be writtenusing

1119896bp = 311990521 + 16119905221199052 + 16119905111990522 + 311990532128119864119887119868119887 + 38(1199051 + 1199052)120581119860119887119866119887

+ ( 11199051119864119887 +11199052119864119887) + ( 111990511198641199011 +

111990521198641199012) (33)

A common method for modeling a connection stiffnessassociation of the single bolt connection is represented by thespring model as shown in Figure 17

Thematrix equation represented a single bolt connectionwhich can be written as

001198653

= [[[[

119896bp + 1198961199012 minus119896bp 0minus119896bp 119896bp + 1198961199011 minus11989611990110 minus1198961199011 1198961199011

]]]]119906111990621199063

(34)

The plate stiffness was calculated in two zones as shownin Figure 17(c) The plate zone represented the plate stiffness


Plate

Bolt hole

x

y

Compression zone

d

d

A

Deformation shape

120590beap

A998400

120575beap

(a) Bearing deformation in the plate

Bolt

x

zt

Compression zone

d

d

B

Deformation shape

120590beab

120590beab

120575beab

B998400

(b) Bearing deformation in the bolt

Figure 16 Schematic diagram of bearing deformation in the plate and bolt

F

Plate 2Plate 1

x

z

l1

l2

(a) Connection model

1

23F3

u3u2

kp1

kbp

u1 kp2

(b) Spring model

Plate 1

Plate 2

x

y(plate zone)

(Bolt zone)

(plate zone)

w1

l1 minus 05dℎdℎ

w2

l2 minus 05dℎ

k1-1

k1-2

k2-2k2-1

(c) Plate stiffness

Figure 17 The spring model of single bolt connection

in front of the bolt The bolt zone represented the platestiffness alongside the bolt The stiffness of the plate wasfound considering a plate subjected to uniform tensile loadwhich is presented in Table 4

The stiffness load vector and the node displacementmatrix have been implemented in the Maplesoft program[19] The load-deformation curve of node 3 was comparedwith a single-shear bolt connection test result which is shownin Figure 18(a) The results show that the analytical method

was in good correlation with the FEA and the experimentalresults Figure 18(b) shows the fastener stiffness predictionby several known semiempirical methods comparing withthe proposed equation The Grumman equation [8] is anempirically derived fastener stiffness formula that was pre-sented by the Grumman Aerospace Corporation as given in

1119896bp = (1199052 + 1199051)2119864119887119889 + 372( 111990511198641199011 +111990521198641199012) (35)


Table 4 Plate stiffness

Stiffness Stiffness equation Value (Nmm)1119896119901111198961-1 +

11198961-2 =1120596111990511198641199011 (1198971 minus 05119889ℎ) + 1

((1205961 minus 119889) 11990511198641199011) 05119889ℎ 163692

1119896119901211198962-1 +

11198962-2 =1(1205962 minus 119889) 1199052119864119901205119889ℎ +

1120596211990521198641199012 (1198972 minus 05119889ℎ) 163692

1119896bp Equation (33) 150660

Remark 1198961199011 and 1198961199012 are the plate stiffness and 1198641199011 and 1198641199012 are the modulus of elasticity of plate 1 and plate 2

0

1500

3000

4500

6000

7500

9000

10500

12000

0 1 2 3 4 5 6 7 8 9 10 11 12

Load

(N)


S-1S-2S-3S-4

S-5S-FEASpring model

(a) Comparison with tests and FEA

0

1500

3000

4500

6000

7500

0 01 02 03 04 05 06 07 08 09 1

Load

(N)


S-1S-2S-3S-4S-5

S-FEAJarfall [8]Huth [9]Swift [10]New proposed

(b) Comparison with other equations

Figure 18 Comparison of the load-deformation curve of the proposed spring model

The fitting equation to the load-displacement curves of thesingle steel bolt connection type was proposed by Huth [9]which is presented in

1119896bp = (1199052 + 11990512119889 )23

sdot 3 ( 111990511198641199011 +111990521198641199012 +

121199051119864119887 +121199052119864119887)

(36)

The Swift equation [10] is a flexibility empirical equationthat was established based on testing data of a single-shearmetallic connection as given by

1119896bp = 166605119889 (radic(119864119871 + 119864119879)plate-1 (radic119864119871 + 119864119879)plate-2)

+ 086( 1(radic119864119871 + 1198641198791199051)plate-1

+ 1(radic119864119871 + 1198641198791199052)plate-2)

(37)

The load comparison results showed that the stiffness valuesfrom the Grumman equation [8] and the Swift equation[10] were conservative which does not accurately reflectthe actual connection strength especially for a single boltconnection The Huth equation [9] matched the testing databut the shear deformation of the bolt and bending deforma-tion of the bolt were ignored

6 Bolt Load Distribution Analysis

An application of bolt-plate interaction stiffness analysisfor the multibolt connection is presented by the triple boltlap connection as in Figure 19 The structural model wasassembled using the 1mm thickness of cold-formed steeland a 5mm diameter steel bolt The analytical applicationinvestigated the load applied distribution among the boltrows due to relative deformation between the top plate (plate1) and bottom plate (plate 2)

The matrix equation representing the multibolt springmodel in Figure 20 can be written as (38) and the stiffnesscomponents are presented in Table 5


x

y

Unit mm30

Plate 2Plate 1

Bolt 1 Bolt 2 Bolt 3

Bolt diameter = 5 x

z

F = 5000 N t = 1

t = 1

l1 = 100 l2 = 3d l3 = 3d l4 = 100

Figure 19 Multibolt connection analysis

k8

k9 k7

k6

k5

k4

k3

k2

k1

u1

u2

u3

u4

u5

u6u7 67F

5

4

3

2

1

(a) Multibolt connection spring model

Plate 1Plate 2

(plate zone)(plate zone) (plate zone)

(Bolt zone)

la = l1 minus 05dℎ1

lb = l2 minus 05dℎ1 minus 05dℎ2

lc = l3 minus 05dℎ2 minus 05dℎ3

ld = l4 minus 05dℎ4

l1

la lb lc

l2 l3 l4

dℎ1 dℎ2 dℎ3

k92k91

k73 k72

k33

k42

k71 k41

k43

k63 k62k61

y k32

k31

k12

k11

ld

w2

w1

x

(b) Plate stiffness

Figure 20 The multibolt connection stiffness

Table 5 Calculation of stiffness

Stiffness equation Stiffness component11198961 =

111989611 +111989612 11989611 = 120596211990521198641199012119897119889 11989612 = (1205962 minus 119889ℎ3)1199052119864119901205119889ℎ3

11198963 =111989631 +

111989632 +111989633 11989631 = 120596211990521198641199012119897119888 11989632 = (1205962 minus 119889ℎ3)1199052119864119901205119889ℎ3 11989633 = (1205962 minus 119889ℎ2)1199052119864119901205119889ℎ2

11198964 =111989641 +

111989642 +111989643 11989641 = 120596111990511198641199011119897119888 11989642 = (1205961 minus 119889ℎ3)1199051119864119901105119889ℎ3 11989643 = (1205961 minus 119889ℎ2)1199051119864119901105119889ℎ2

11198966 =111989661 +

111989662 +111989663 11989661 = 120596211990521198641199012119897119887 11989662 = (1205962 minus 119889ℎ2)1199052119864119901205119889ℎ2 11989663 = (1205962 minus 119889ℎ1)1199052119864119901205119889ℎ1

11198967 =111989671 +

111989672 +111989673 11989671 = 120596111990511198641199011119897119887 11989672 = (1205961 minus 119889ℎ2)1199051119864119901105119889ℎ2 11989673 = (1205961 minus 119889ℎ1)1199051119864119901105119889ℎ1

11198969 =111989691 +

111989692 11989691 = 120596111990511198641199011119897119886 11989692 = (1205961 minus 119889ℎ1)1199051119864119901105119889ℎ111198962 11198965

11198968 Equation (33)

Remark 1198961 and 1198969 are the stiffness of the plate from the bolt to the ends of plates 1198963 1198964 1198966 and 1198967 are the stiffness of the plate between bolts and 1198962 1198965 and1198968 are the bolt-plate interaction stiffness


xz

yA

B

dx = 0

dy = 0

dz = 0

dx = freedy = 0

dz = 0

(a) Finite element model of triple bolt connection

Multibolt connection spring modelFEA

01 020Connection deformation (mm)

0

1000

2000

3000

4000

5000

Load

(N)

(b) Connection load

3472 36213056 2928

3472 3449

Bolt 1(spring)

Bolt 1(FEA)

Bolt 2(spring)

Bolt 2(FEA)

Bolt 3(spring)

Bolt 3(FEA)

05

10152025303540

Load

dist

ribut

ion

()

(c) Load distribution

Figure 21 Connection load and load distribution of analytical model and FEA

000000119865

=

[[[[[[[[[[[[[[[

1198961 + 1198962 + 1198963 minus1198962 minus1198963 0 0 0 0minus1198962 1198962 + 1198964 0 minus1198964 0 0 0minus1198963 0 1198963 + 1198965 + 1198966 minus1198965 minus1198966 0 00 minus1198964 minus1198965 1198964 + 1198965 + 1198967 0 minus1198967 00 0 minus1198966 0 1198966 + 1198968 minus1198968 00 0 0 minus1198967 minus1198968 1198967 + 1198968 + 1198969 minus11989690 0 0 0 0 minus1198969 1198969

]]]]]]]]]]]]]]]

1199061119906211990631199064119906511990661199067

(38)

The nodal deformation was calculated by inverting the stiff-ness matrix and multiplying by the load vector The carriedbolt load (119875) was calculated using

1198751 = 1198708 (1199066 minus 1199065) 1198752 = 1198705 (1199064 minus 1199063) 1198753 = 1198702 (1199062 minus 1199061)

(39)

To validate the analytical model the finite elementmodel wasgenerated for comparison with the analytical result as shownin Figure 21(a) An analysis was performed in which materialfailure was not involved in the study Thus the small appliedload was modeled using a finite element model to investigatethe load distribution behaviorThe validation of the analyticalmodel results is presented in Figure 21(b) The connection

load and load distribution indicated by the analytical modelwere in good agreement with the FEA The load distributionresults in Figure 21(c) have been expressed as a percentageof the total connection load (119865) It can be seen that the loadwas shared unevenly between each bolt The uneven loaddistribution in the triple bolt connection was analyzed usingthe parametric study

7 The Parametric Study ofMultibolt Connection

The analytical spring model as in the previous section wasused to examine the effects of various connection parametersThe parametric study calculated the load distribution of thebolt connection with variations in the bolt diameters platethickness (1199051 = 1199052) and the plate thickness ratio (11990521199051) The


000

050

100

150

200

000 050 100 150 200

Reference case

ddref ttref and (t2t1)(t2t1)ref

m = dydx

120595120595

ref

Low sensitivity levelHigh sensitivity level

Figure 22 Concept of the sensitivity analysis

geometry model and the load condition as in the previoussection were adapted in the parametric study The efficiency(120595) of the load distribution of multibolt connection wasproposed shown in

120595 = 119899119875minsum119899119894=1 119875119894 for 119894 = 1 2 3 119899 (40)

where 119875119894 is the bolt load in the 119894-th row 119875min is the smallestload in the bolts and 119899 is the number of rows in theconnection Sensitivity analysis investigated these potentialchanges and their impacts on inferences to be obtainedfrom the analytical model The conception of sensitivityanalysis is presented in Figure 22The slope of the normalizedefficiency graph with respect to changes in the geometricparameter was used tomeasure the sensitivity level A steepergraph indicates a higher sensitivity level while a gentlerslope indicates a lower sensitivity level In each case studythe efficiency graph was normalized by the reference valueThe reference cases involved bolt diameter of 8mm platethickness of 10mm and 11990521199051 of 30 which are used toanalyze the sensitivity of variation in the bolt diameter platethickness and 11990521199051 sensitivity respectively

Variation in the bolt diameter introduced efficiencyof load distribution shown in Table 6 and Figure 23(a)Increasing the bolt diameter had a significant effect on theload distributions in a multibolt connection The changein the bolt diameter affected the proportion of the bolt-plate interaction stiffness between the bolt and plates whichresulted in an even load distribution

The efficiency of bolt load distribution had small changeswith increased plate thickness as shown in Table 7 andFigure 24(a) The bolt-plate interaction stiffness and axialstiffness of the plates were related to the plated thickness as achange in the plate thickness produced an equal change in thebolt-plate interaction stiffness and the axial stiffness whichslightly changed the efficiency of load distribution

The efficiency of load distributions was changed suddenlywith changes to the 11990521199051 ratio as shown in Table 8 andFigure 25(a)

The efficiency of load distribution decreased when the11990521199051 ratio was greater than 1 but the carried load for thefirst bolt row increasingly changed Once the 11990521199051 ratio wasgreater than 1 (11990521199051 ≫ 1) the behavior of plate 2 wasestablished as a rigid bar as shown in Figure 26 The steelbolt was constrained by the axial stiffness of plate 2 which

Table 6 Bolt load distribution for variation in bolt diameter

119889 (mm) Bolt load distribution () Efficiency 120595 ()Bolt 1 Bolt 2 Bolt 3

4 3499 3003 3499 90096 3453 3095 3453 92848 3427 3146 3427 943710 3411 3178 3411 953312 3400 3200 3400 9599Remark 1199051 and 1199052 = 1mm and 1198971 and 1198974 = 100mm

was simulated as a semi-fixed support conditionThen bolt 1could not transfer the load before bearing failure or shearingof the bolt failure occurred

From the parametric study results it was clear that vary-ing the geometries influenced the connection stiffness whichsignificantly affected the load distribution The sensitivityanalysis results of the multibolt connection from variationin the bolt diameter plate thickness and 11990521199051 are shownin Figures 23(b) 24(b) and 25(b) The sensitivity value was00605 00136 and minus00646 for the bolt diameter platethickness and 11990521199051 sensitivity respectively The variation in11990521199051 showed high sensitivity with the triple bolt connectionwith sudden decay when 11990521199051 gt 1 Thus differences inthe plate thickness should be emphasized in the connectiondesign

8 Conclusion

The bolt load distribution efficiency of a cold-formed steelmultibolt connection was presented The analytical prece-dence with a spring model was used to simplify the stiffnessof the bolt connection A new bolt-plate interaction stiffnessequation was proposed which was verified by the FEA andexperimental tests The spring model with bolt-plate interac-tion stiffness was used to estimate the bolt load distributionof the multibolt connection The results showed that theanalytical model was accurate and had good correlationwith the FEA Furthermore the analytical procedure wasused to examine the efficiency of load distribution for themultibolt connection and these results showed an unevensharing of the load among bolts with one of them carryingthemajor component of the loadThus the stress distributionof the major load carrying hole resulted in high-stress


Bolt diameter (mm)4 6 8 10 12

Bolt 1Bolt 2Bolt 3

05

101520253035404550

Load

dist

ribut

ion

()

(a) Bolt load distribution

Reference case

y = 00605x + 09326

050 100 150 200000ddref

090

094

098

120595120595

ref

102

106

110

(b) Parameter sensitivity

Figure 23 Bolt load distribution and sensitivity analysis results of varying bolt diameters

Plate thickness (mm) 04 08 12 16 20

Bolt 1Bolt 2Bolt 3

05

101520253035404550

Load

dist

ribut

ion

()


Reference case

y = 00136x + 09884

090

094

098

102

120595120595

ref

106

110

050 100 150 200000ttref


Figure 24 Bolt load distribution and sensitivity analysis results of varying plate thickness

05

101520253035404550

Load

dist

ribut

ion

()

1 2 3 4 5t2t1

Bolt 1Bolt 2Bolt 3


Reference case

y = minus00646x + 10642

050 100 150 200000(t2t1)(t2t1)ref

090

094

120595120595

ref

098

102

106

110


Figure 25 Bolt load distribution and sensitivity analysis results of the plate thickness ratio (11990521199051)


Plate 2

Plate 1Bolt 1 Bolt 2 Bolt 3

F

t2 ≫ tt

Figure 26 Effect of the 11990521199051 ratio more than 1 (≫1) to the steel bolts

Table 7 Bolt load distribution for variation in plate thickness

1199051 and 1199052 (mm) Bolt load distribution () Efficiency 120595 ()Bolt 1 Bolt 2 Bolt 3

040 3475 3051 3475 9152060 3472 3056 3472 9169100 3467 3067 3467 9201140 3459 3083 3444 9262160 3448 3105 3448 9314Remark 1198971 and 1198974 = 100mm 1198972 and 1198973 = 15mm120596 = 30mm and 119889 = 5mm

Table 8 Bolt load distribution for variation in plate thickness ratio(11990521199051)11990521199051 Bolt load distribution () Efficiency 120595 ()

Bolt 1 Bolt 2 Bolt 31 3472 3056 3472 91692 3814 3065 3121 91953 3967 3075 2959 88764 4039 3086 2875 86255 4069 3098 2833 8498Remark 1199051 = 1mm 1198971 and 1198974 = 100mm 1198972 and 1198973 = 15mm and 119889 = 5mm

concentration which caused weakness in the mechanicallyfastened connectionMoreover increasing 11990521199051 had a greaterinfluence on the load distribution and efficiency of a multi-bolt connection than changing the bolt diameter and platethickness The results of the study indicate that the practicaldesign procedures with assuming equal load distributionamong the bolts may not be conservative because somecomponents may be failed before the connection developsthe full plastic strength Finally the proposed procedurecan be used together with the conventional design of thebolt connection to get insight into the more realistic loaddistribution

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

The authors wish to acknowledge the Kasetsart UniversityResearch and Development Institute (KURDI) for providingfunding support and BlueScope Lysaght (Thailand) Limitedfor material support

References

[1] Y Cai and B Young ldquoStructural behavior of cold-formedstainless steel bolted connectionsrdquo Thin-Walled Structures vol83 pp 147ndash156 2014

[2] Y C He and Y C Wang ldquoLoad-deflection behavior of thin-walled plates with multiple bolts in shearingrdquo Thin-WalledStructures vol 55 pp 51ndash63 2012

[3] F R Liu L B Zhao S Mehmood J Y Zhang and B J AFei ldquoA modified failure envelope method for failure predictionofmulti-bolt composite jointsrdquoComposites Science and Technol-ogy vol 83 pp 54ndash63 2013

[4] J Y Zhang F R Liu L B Zhao and B J Fei ldquoA novel character-istic curve for failure prediction of multi-bolt composite jointsrdquoComposite Structures vol 108 no 1 pp 129ndash136 2014

[5] J Zhang F Liu L Zhao Y Chen and B Fei ldquoA progressivedamage analysis based characteristic length method for multi-bolt composite jointsrdquo Composite Structures vol 108 no 1 pp915ndash923 2014

[6] J Henriques J-P Jaspart and L S da Silva ldquoDuctility require-ments for the design of bolted lap shear connections in bearingrdquoAdvanced Steel Construction vol 10 no 1 pp 33ndash52 2014

[7] G L Kulak J W Fisher and J H Struik A Guide to DesignCriteria for Bolted and Riveted Joints amp Sons New York NYUSA 1987

[8] L Jarfall ldquoShear loaded fastener installations Report KH R-3360rdquo Tech Rep Saab-Scania Sweden 1983

[9] H Huth ldquoInfluence of fastener flexibility on the prediction ofload transfer and fatigue life for multiple-row jointsrdquo SymposiaPapers STPs ASTM STP927-EB pp 221ndash250 1986

[10] T Swift ldquoRepair to damage tolerant aircraftrdquo in Proceedings ofthe International Symposium on the Structural Integrity of AgingAirplane Atlanta GA USA 1990

[11] ASTM A370-07b Standard Test Methods and Definitions forMechanical Testing of Steel Products American Society forTesting and Materials (ASTM) West Conshohocken PA USA2007

[12] ASTM A325M High strength bolts for structural steel joints[Metric] American Society for Testing and Materials (ASTM)West Conshohocken PA USA 1994

[13] ANSYS ANSYS userrsquos manual Revision 15 ANSYS IncCanonsburg PA USA 2014

[14] L J Hart-Smith ldquoBonded-bolted composite jointsrdquo Journal ofAircraft vol 22 no 11 pp 993ndash1000 1985

[15] V P LawlorM AMcCarthy andW F Stanley ldquoAn experimen-tal study of bolt-hole clearance effects in double-lap multi-boltcomposite jointsrdquo Composite Structures vol 71 no 2 pp 176ndash190 2005

[16] L Feo G Marra and A S Mosallam ldquoStress analysis ofmulti-bolted joints for FRP pultruded composite structuresrdquoComposite Structures vol 94 no 12 pp 3769ndash3780 2012


[17] S P Timoshenko ldquoOn the correction for shear of the differentialequation for transverse vibrations of prismatic barsrdquo Philosoph-ical Magazine vol 41 no 245 pp 744ndash746 1922

[18] J R Hutchinson ldquoShear coefficients for Timoshenko beamtheoryrdquo Journal of Applied Mechanics vol 68 no 1 pp 87ndash922000

[19] L Bernardin P Chin P DeMarco et al Maple ProgrammingGuide Maple Inc Waterloo Ontario Canada 2012

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Navigation and Observation



DistributedSensor Networks



P1 2 3 4 5

1 2 3 4 5

1 2 3 4 5

1 2 3 4 5

P PP

Bolt loadLoad deformation Load of bolt 5 Load of bolt 4 Load of bolt 3

P

P PP

PP

P

P

PP

P

P

Step 1

Step 2

Step 3

Step 4

P

P

Ppl

Pe

Pe

P

Ppl

Ppl

Ppl

Pe

P

Ppl

Pe

P

Ppl

Pe

P lt Pe

P = Pe

Pe lt P lt Ppl

P = Ppl

120575

120575

120575

120575

120575

120575

120575

120575

120575b-5

120575b-5

120575b-5

120575b-5

120575b-4

120575b-4

120575b-4

120575b-4

120575b-3

120575b-3

120575b-3

120575b-3

of connection

Figure 1 Stages of force distribution in bolt connections [6]

These failures result in a lower average bearing stress valueper bolt hole of the multibolt connection In this case theconnection would not develop the ultimate bearing capacityas recommended by the design method [6 7] Bolt load pre-diction is a complex procedure involving the establishmentof the bolt load distribution The stiffness components (theaxial stiffness of the plate and the interaction of the bolt-plate)must be included in the investigation of bolt load distributionHowever the interaction stiffness of the bolt-plate is noteasily defined in a closed-form solution because the analysisrequires knowledge of the influenced parameters such as theshear deformation of the bolt bending deformation of thebolt bearing deformation of the plate bearing deformationof the bolt and deformation of the plate between boltsThus the equation of the bolt-plate interaction stiffness inindustry today is derived by using a semiempirical approachbased on the testing data [8ndash10] The weakness of theseempirical equations is that they ignore the shear deformationassumption in the bolt component which is dominated by theshort bolt length

This study describes the method of analysis for deter-mining the load distribution among bolts in a single-columnmultibolt connection A bolt-plate interaction stiffness equa-tion is proposedwhich is verified by the experimental test andfinite element analysis (FEA)The experimental investigationwas performed using lap shear bolt connection testing andthe connections were fabricated from cold-formed steel

2 Material Properties and Model Description

The laboratory test was performed using G550 cold-formedsteel with a plate thickness of 10mm for the bolt connectiontest G550 is a hot-dipped zinc-coated structural steel and

has a minimum yield strength of 550MPa The materialproperties for G550 steels were specified by following ASTMStandards A370-07 [11] as shown in Table 1 Following theresearch objective and scope the possibility of bolting shearfailure was eliminated by using M5 Grade 88 steel boltsconforming to ASTM A325M [12] as shown in Table 2

The bolt connection specimen and the hole size are shownin Figure 2 The bolts and nuts were assembled with integralwashers that were tightened by hand which allowed for slipin the connection after the initial loading

The bolt connection specimens were tested using auniversal testingmachine (UTM) in the structural laboratoryof Kasetsart University Bangkok Thailand as shown inFigure 3(a) A typical bolted shear connection test setup isshown in Figure 3(b) The load and the axial deformationwere measured to describe the connection failure behaviorThe load was applied to the specimen using displacementcontrol with a speed of 1mmminute A linear variabledifferential transformer (LVDT) was installed to measure theelongation of a single bolt connection over a distance of200mm from the center line The load was gradually applieduntil either the specimen failed or the specimen entered intoplastic deformation Once the applied load began to dropeven though the specimen continued to elongate the test wasstopped

3 Finite Element Model and Validation

In the present study the ANSYS finite element package[13] was used to predict the failure behavior of the boltconnection as shown in Figure 4 Nonlinear finite elementanalysis was used to compare and validate the results of thesingle bolt connection test The nonlinear geometry material


F

F

1530

100

15

11

Bolt diameter = 5



F

F

Hole size

d

lt127

dℎ

1mm

1mm


(b) Hole size














UTM rod load cell

Upper grip

Lower grip

Specimen

UTM rod

Fixed gagelengths

Data

LVD

T

logger




Bolt (SOLID186)

x

z

(a) Geometry model

M

I

J

N

K

OL

P

1

2

3

6

5

8

4

7

1

2

3 4

56

z0y0

x0

(b) SHELL281

J

1

x

yz R

K

I

O

W

S

Q

I

TL

I

Y

M

XP

N

Z

VU

2

3

4

5

6

(c) SOLID186

x

yz I

M

J

N

K

OL

P



PI998400

J998400

N998400

M998400

L998400

K998400

O998400

(d) Contact element

x

z

CONTAC174

TARGET170

Steel bolt

Cold-formed steel

fp = kpxp

kp kpkpkp xp



xz

y

(pined)

(roller)

A

B

Fine mesh size

dx = 0

dy = 0

dz = 0

dx = free

dy = 0

dz = 0


0

100

200

300

400

500

600

700

0 002 004 006 008

Axi

al st

ress

(MPa

)

Axial strain (mmmm)




0

2000

4000

6000

8000

10000

12000

0 1 2 3 4 5 6 7 8 9 10 11 12

Load

(N)


S-1S-2S-3

S-4S-5S-FEA


Test FEA


(b) Failure mode











119862119887 = 1119896119887 =120575119887119865 (2)



120579 (119909) = int1199090

119872(119904)119864119868 119889119904 + 1205790 (3a)

119906 (119909) = int1199090120579 (119904) 119889119904 minus int119909

0

119881 (119904)120581119860119866119889119904 + 1199060 (3b)




A

BC

D

E



Stre

ss n

orm

aliz

atio

n

Strain

Strain

A

BC

D

E

10004

089658

079278

068899

05852

04814

037761

027382

017003

0066233

180∘

0∘31592

28085

24578

21071

17564

14057

1055

070431

035361

00029019

120579 (degrees)

120579

13

10

08

05

03

000 20 40 60 80 100 120 140 160 180

0051152253

Stra

in (m

mm

m)


F

Bolt

Plate 2

Plate 1 A

B

C

Bolt

Plate 2

Plate 1F = 0

t1

t2

t1

t2

120575b


F

F

M

L

A

B

C

1205962t2

1205961t1V1

V2

t1

t2

M



A

B

C


Guiding support

M

Case 3Case 2Case 1

= + +

1205962t2 1205962t2

1205961t11205961t1


L

a b

L

a b

x x

L

x


L

x

y

y y y120596 120596

MB

0 le x le a

a le x le L

120575(x) = minus120596x3

24EI(4x minus a)

120579B =120596a3

6EI

0 le x le a

120575 = minus120596x2b


a le x le L

x = L

120575B =MBL

2

2EI

120579B =MBL

EI



120575B120579B

120575 = minus120596


120575(x) = minus120596x2


120579B =120596(L3 minus a3)

6EI

minus





120601 = 1206011 + 1206012 + 1206013 = (1205791 + 1205741) + (1205792 + 1205742) + (1205793 + 1205743)= 0 (4)



1205791 = minus 1205961119905316119864119887119868119887 1205792 = 1205962 (1198713 minus 11990531)6119864119887119868119887 1205793 = 119872119861119871119864119887119868119887

(5a)

1205741 = 01205742 = 01205743 = 0 (5b)


a

b

L

x

La b

La b

x x

L

x


a

x

b

V

dx

x

x

x

y

y y y

120596

120596 120596

MB

0 le x le a


1205741(x) = minus120596(x minus a)

120581AG0 le x le b

V2(x) = 0

1205742(x) = 0

0 le x le a


1205743(x) = minus120596b

120581AG

0 le x le b


1205744(x) = minus120596(x minus b)

120581AG

V(x) = 0

120574(x) = 0

120574

120575s(x) = 0

120574 =d120575sdx

d120575s

120575s

120596b

1205751(x) = intx

01205741(x)dx

1205752(x) = inta

01205741(x)dx + intx

01205742(x)dx

1205754(x) = inta

01205743(x)dx + intx

01205744(x)dx

1205753(x) = intx

01205743(x)dx


A

B

=C

Fixed support

Guiding support


+ +


1205962t2 1205962t2

1205961t11205961t1

MB

1206011 1206012 1206013



119865119887 = 12059611199051 = 12059621199052 (6)


1205961 = 1205962 (11990521199051) (7)


minus 1205961119905316119864119887119868119887 +1205962 (1198713 minus 11990531)6119864119887119868119887 + 119872119861119871119864119887119868119887 = 0 (8)





M

M

A

C

B1205962t2

1205961t1

kc

kb

kc

kb

Fixed support

Guiding support




1119896bp = 1119896119887 +1119896119888 (10a)

with

1119896119887 =1119896flex119887 +

1119896she119887 1119896119888 =

1119896bea119901 +1119896bea119887

(10b)


119906 (119909) = int1199090120579 (119904) 119889119904 + 1199060

= int1199090(int1199090

119872(119904)119864119868 119889119904 + 1205790)119889119904 + 1199060(11)



120575flex119860 = 120575flex11986012059611199051

+ 120575flex11986012059621199052

+ 120575flex119860119872 (12)

where 120575flex11986012059611199051

120575flex11986012059621199052


120575flex11986012059611199051

= minus 12059611199053124119864119887119868119887 (61198862 minus 4119886119909 + 1199092) (13a)

120575flex11986012059621199052

= 1205962119887119909212119864119887119868119887 (3119871 + 3119886 minus 2119909) (13b)

120575flex119860119872 = 119872(119909)22119864119887119868119887 (13c)

119886 = 1199051119887 = 1199052119871 = 1199051 + 1199052119909 = 11990512

(13d)


120575flex119861 = 120575flex11986112059611199051

+ 120575flex11986112059621199052

+ 120575flex119861119872 (14)

where 120575flex11986112059611199051

120575flex11986112059621199052


120575flex11986112059611199051

= minus 1205961119886324119864119887119868119887 (4119909 minus 119886) (15a)

120575flex11986112059621199052

= 120596224119864119887119868119887 (1199094 minus 41198711199093 + 611987121199092 minus 41198863119909 + 1198864) (15b)

120575flex119861119872 = 119872(119909)22119864119887119868119887 (15c)

119909 = 1199051 + 11990522 (15d)


F

F

Bolt

Plate 2

Plate 1A

B

C

A

BC

NA

t1

t2

120575flexA

120575flexB

120575flexC

120575bolt





(1199051 + 1199052) (311990521 + 131199051199051199052 + 311990522)119864119887119868119887 )12059621199052 (16)


120575flex119887 = (311990521 + 16119905221199052 + 16119905111990522 + 311990532128119864119887119868119887 )119865119887 (17)


119862flex119887 = 1119896flex119887 =120575flex119887119865119887

= 311990521 + 16119905221199052 + 16119905111990522 + 311990532128119864119887119868119887 (18)


119906 (119909) = minusint1199090

119881 (119904)120581119860119866119889119904 + 1199060 (19)


120581 = 6 (1 + ])27 + 12] + 4]2 (20)




120575119904119860 = 12057511990411986012059611199051

+ 12057511990411986012059621199052

+ 120575119904119860119872 (21)

where 12057511990411986012059611199051

12057511990411986012059621199052


12057511990411986012059611199051

= int11990901205741 (119909) 119889119909 = minus 31205961119905218120581119866119887119860119887 (22a)

12057511990411986012059621199052

= int11990901205743 (119909) 119889119909 = 1205962119905111990522120581119866119887119860119887 (22b)

120575119904119860119872 = 0 (22c)

119909 = 11990512 (22d)


120575119904119861 = 12057511990411986112059611199051

+ 12057511990411986112059621199052

+ 120575119904119861119872 (23)

where 12057511990411986112059611199051

12057511990411986112059621199052


12057511990411986112059611199051

= int11988601205741 (119909) 119889119909 + int119909

01205742 (119909) 119889119909 = minus 1205961119905212120581119866119887119860119887 (24a)

12057511990411986112059621199052

= int11988601205743 (119909) 119889119909 + int119909

01205744 (119909) 119889119909

= 12059621199052 (81199051 + 31199052)8119866119887119860119887 (24b)

120575119904119861119872 = 0 (24c)

119909 = 11990522 (24d)


F

F

Bolt

A

B

C

A

BC

Plate 2

Plate 1

NA

t1

t2

120575bolt

120575sA

120575sB


120575sB minus 120575sA=



120575119904 = 120575119904119861 minus 120575119904119860 = 38 ( 1199051 + 1199052120581119860119887119866119887)12059621199052 (25)


120575119904 = 38 ( 1199051 + 1199052120581119860119887119866119887)12059621199052 (26)


119862sd119887 = 1119896sd119887 =120575119904119865119887 =

38(1199051 + 1199052)120581119860119887119866119887 (27)


120590bea = 119875119889119905 (28a)

120576bea = 120575bea119901119889 (28b)


120590bea119901 = 119864119901120576bea119901 (29a)

119896bea119901 = 119875120575bea119901 = 119905119889119864119901119889 = 119905119864119901 (29b)


120590bea119887 = 119864119887120576bea119887 (30a)

119896bea119887 = 119875120575bea119887 =119905119889119864119887119889 = 119905119864119887 (30b)


119888bea119887 = 1119896bea119887 =11199051119864119887 +

11199052119864119887 (31)

119888bea119901 = 1119896bea119901 = 111990511198641199011 +111990521198641199012 (32)


1119896bp = 311990521 + 16119905221199052 + 16119905111990522 + 311990532128119864119887119868119887 + 38(1199051 + 1199052)120581119860119887119866119887

+ ( 11199051119864119887 +11199052119864119887) + ( 111990511198641199011 +

111990521198641199012) (33)



001198653

= [[[[


]]]]119906111990621199063

(34)



Plate

Bolt hole

x

y

Compression zone

d

d

A

Deformation shape

120590beap

A998400

120575beap


Bolt

x

zt

Compression zone

d

d

B

Deformation shape

120590beab

120590beab

120575beab

B998400



F

Plate 2Plate 1

x

z

l1

l2


1

23F3

u3u2

kp1

kbp

u1 kp2

(b) Spring model

Plate 1

Plate 2

x

y(plate zone)

(Bolt zone)

(plate zone)

w1

l1 minus 05dℎdℎ

w2

l2 minus 05dℎ

k1-1

k1-2

k2-2k2-1

(c) Plate stiffness





1119896bp = (1199052 + 1199051)2119864119887119889 + 372( 111990511198641199011 +111990521198641199012) (35)




11198961-2 =1120596111990511198641199011 (1198971 minus 05119889ℎ) + 1

((1205961 minus 119889) 11990511198641199011) 05119889ℎ 163692

1119896119901211198962-1 +

11198962-2 =1(1205962 minus 119889) 1199052119864119901205119889ℎ +

1120596211990521198641199012 (1198972 minus 05119889ℎ) 163692

1119896bp Equation (33) 150660


0

1500

3000

4500

6000

7500

9000

10500

12000

0 1 2 3 4 5 6 7 8 9 10 11 12

Load

(N)


S-1S-2S-3S-4



0

1500

3000

4500

6000

7500

0 01 02 03 04 05 06 07 08 09 1

Load

(N)


S-1S-2S-3S-4S-5





1119896bp = (1199052 + 11990512119889 )23

sdot 3 ( 111990511198641199011 +111990521198641199012 +

121199051119864119887 +121199052119864119887)

(36)



+ 086( 1(radic119864119871 + 1198641198791199051)plate-1

+ 1(radic119864119871 + 1198641198791199052)plate-2)

(37)






x

y

Unit mm30

Plate 2Plate 1


Bolt diameter = 5 x

z

F = 5000 N t = 1

t = 1

l1 = 100 l2 = 3d l3 = 3d l4 = 100


k8

k9 k7

k6

k5

k4

k3

k2

k1

u1

u2

u3

u4

u5

u6u7 67F

5

4

3

2

1


Plate 1Plate 2


(Bolt zone)





l1

la lb lc

l2 l3 l4

dℎ1 dℎ2 dℎ3

k92k91

k73 k72

k33

k42

k71 k41

k43

k63 k62k61

y k32

k31

k12

k11

ld

w2

w1

x

(b) Plate stiffness




111989611 +111989612 11989611 = 120596211990521198641199012119897119889 11989612 = (1205962 minus 119889ℎ3)1199052119864119901205119889ℎ3

11198963 =111989631 +

111989632 +111989633 11989631 = 120596211990521198641199012119897119888 11989632 = (1205962 minus 119889ℎ3)1199052119864119901205119889ℎ3 11989633 = (1205962 minus 119889ℎ2)1199052119864119901205119889ℎ2

11198964 =111989641 +

111989642 +111989643 11989641 = 120596111990511198641199011119897119888 11989642 = (1205961 minus 119889ℎ3)1199051119864119901105119889ℎ3 11989643 = (1205961 minus 119889ℎ2)1199051119864119901105119889ℎ2

11198966 =111989661 +

111989662 +111989663 11989661 = 120596211990521198641199012119897119887 11989662 = (1205962 minus 119889ℎ2)1199052119864119901205119889ℎ2 11989663 = (1205962 minus 119889ℎ1)1199052119864119901205119889ℎ1

11198967 =111989671 +

111989672 +111989673 11989671 = 120596111990511198641199011119897119887 11989672 = (1205961 minus 119889ℎ2)1199051119864119901105119889ℎ2 11989673 = (1205961 minus 119889ℎ1)1199051119864119901105119889ℎ1

11198969 =111989691 +

111989692 11989691 = 120596111990511198641199011119897119886 11989692 = (1205961 minus 119889ℎ1)1199051119864119901105119889ℎ111198962 11198965

11198968 Equation (33)



xz

yA

B

dx = 0

dy = 0

dz = 0

dx = freedy = 0

dz = 0




0

1000

2000

3000

4000

5000

Load

(N)

(b) Connection load

3472 36213056 2928

3472 3449

Bolt 1(spring)

Bolt 1(FEA)

Bolt 2(spring)

Bolt 2(FEA)

Bolt 3(spring)

Bolt 3(FEA)

05

10152025303540

Load

dist

ribut

ion

()



000000119865

=

[[[[[[[[[[[[[[[


]]]]]]]]]]]]]]]

1199061119906211990631199064119906511990661199067

(38)



(39)






000

050

100

150

200

000 050 100 150 200

Reference case


m = dydx

120595120595

ref




120595 = 119899119875minsum119899119894=1 119875119894 for 119894 = 1 2 3 119899 (40)











8 Conclusion




Bolt 1Bolt 2Bolt 3

05

101520253035404550

Load

dist

ribut

ion

()


Reference case

y = 00605x + 09326

050 100 150 200000ddref

090

094

098

120595120595

ref

102

106

110




Bolt 1Bolt 2Bolt 3

05

101520253035404550

Load

dist

ribut

ion

()


Reference case

y = 00136x + 09884

090

094

098

102

120595120595

ref

106

110

050 100 150 200000ttref



05

101520253035404550

Load

dist

ribut

ion

()

1 2 3 4 5t2t1

Bolt 1Bolt 2Bolt 3


Reference case

y = minus00646x + 10642

050 100 150 200000(t2t1)(t2t1)ref

090

094

120595120595

ref

098

102

106

110




Plate 2


F

t2 ≫ tt










Acknowledgments


References





















RoboticsJournal of





Journal of



RotatingMachinery



Journal of

Volume 201


VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










F

F

1530

100

15

11

Bolt diameter = 5



F

F

Hole size

d

lt127

dℎ

1mm

1mm


(b) Hole size














UTM rod load cell

Upper grip

Lower grip

Specimen

UTM rod

Fixed gagelengths

Data

LVD

T

logger




Bolt (SOLID186)

x

z

(a) Geometry model

M

I

J

N

K

OL

P

1

2

3

6

5

8

4

7

1

2

3 4

56

z0y0

x0

(b) SHELL281

J

1

x

yz R

K

I

O

W

S

Q

I

TL

I

Y

M

XP

N

Z

VU

2

3

4

5

6

(c) SOLID186

x

yz I

M

J

N

K

OL

P



PI998400

J998400

N998400

M998400

L998400

K998400

O998400

(d) Contact element

x

z

CONTAC174

TARGET170

Steel bolt

Cold-formed steel

fp = kpxp

kp kpkpkp xp



xz

y

(pined)

(roller)

A

B

Fine mesh size

dx = 0

dy = 0

dz = 0

dx = free

dy = 0

dz = 0


0

100

200

300

400

500

600

700

0 002 004 006 008

Axi

al st

ress

(MPa

)

Axial strain (mmmm)




0

2000

4000

6000

8000

10000

12000

0 1 2 3 4 5 6 7 8 9 10 11 12

Load

(N)


S-1S-2S-3

S-4S-5S-FEA


Test FEA


(b) Failure mode











119862119887 = 1119896119887 =120575119887119865 (2)



120579 (119909) = int1199090

119872(119904)119864119868 119889119904 + 1205790 (3a)

119906 (119909) = int1199090120579 (119904) 119889119904 minus int119909

0

119881 (119904)120581119860119866119889119904 + 1199060 (3b)




A

BC

D

E



Stre

ss n

orm

aliz

atio

n

Strain

Strain

A

BC

D

E

10004

089658

079278

068899

05852

04814

037761

027382

017003

0066233

180∘

0∘31592

28085

24578

21071

17564

14057

1055

070431

035361

00029019

120579 (degrees)

120579

13

10

08

05

03

000 20 40 60 80 100 120 140 160 180

0051152253

Stra

in (m

mm

m)


F

Bolt

Plate 2

Plate 1 A

B

C

Bolt

Plate 2

Plate 1F = 0

t1

t2

t1

t2

120575b


F

F

M

L

A

B

C

1205962t2

1205961t1V1

V2

t1

t2

M



A

B

C


Guiding support

M

Case 3Case 2Case 1

= + +

1205962t2 1205962t2

1205961t11205961t1


L

a b

L

a b

x x

L

x


L

x

y

y y y120596 120596

MB

0 le x le a

a le x le L

120575(x) = minus120596x3

24EI(4x minus a)

120579B =120596a3

6EI

0 le x le a

120575 = minus120596x2b


a le x le L

x = L

120575B =MBL

2

2EI

120579B =MBL

EI



120575B120579B

120575 = minus120596


120575(x) = minus120596x2


120579B =120596(L3 minus a3)

6EI

minus





120601 = 1206011 + 1206012 + 1206013 = (1205791 + 1205741) + (1205792 + 1205742) + (1205793 + 1205743)= 0 (4)



1205791 = minus 1205961119905316119864119887119868119887 1205792 = 1205962 (1198713 minus 11990531)6119864119887119868119887 1205793 = 119872119861119871119864119887119868119887

(5a)

1205741 = 01205742 = 01205743 = 0 (5b)


a

b

L

x

La b

La b

x x

L

x


a

x

b

V

dx

x

x

x

y

y y y

120596

120596 120596

MB

0 le x le a


1205741(x) = minus120596(x minus a)

120581AG0 le x le b

V2(x) = 0

1205742(x) = 0

0 le x le a


1205743(x) = minus120596b

120581AG

0 le x le b


1205744(x) = minus120596(x minus b)

120581AG

V(x) = 0

120574(x) = 0

120574

120575s(x) = 0

120574 =d120575sdx

d120575s

120575s

120596b

1205751(x) = intx

01205741(x)dx

1205752(x) = inta

01205741(x)dx + intx

01205742(x)dx

1205754(x) = inta

01205743(x)dx + intx

01205744(x)dx

1205753(x) = intx

01205743(x)dx


A

B

=C

Fixed support

Guiding support


+ +


1205962t2 1205962t2

1205961t11205961t1

MB

1206011 1206012 1206013



119865119887 = 12059611199051 = 12059621199052 (6)


1205961 = 1205962 (11990521199051) (7)


minus 1205961119905316119864119887119868119887 +1205962 (1198713 minus 11990531)6119864119887119868119887 + 119872119861119871119864119887119868119887 = 0 (8)





M

M

A

C

B1205962t2

1205961t1

kc

kb

kc

kb

Fixed support

Guiding support




1119896bp = 1119896119887 +1119896119888 (10a)

with

1119896119887 =1119896flex119887 +

1119896she119887 1119896119888 =

1119896bea119901 +1119896bea119887

(10b)


119906 (119909) = int1199090120579 (119904) 119889119904 + 1199060

= int1199090(int1199090

119872(119904)119864119868 119889119904 + 1205790)119889119904 + 1199060(11)



120575flex119860 = 120575flex11986012059611199051

+ 120575flex11986012059621199052

+ 120575flex119860119872 (12)

where 120575flex11986012059611199051

120575flex11986012059621199052


120575flex11986012059611199051

= minus 12059611199053124119864119887119868119887 (61198862 minus 4119886119909 + 1199092) (13a)

120575flex11986012059621199052

= 1205962119887119909212119864119887119868119887 (3119871 + 3119886 minus 2119909) (13b)

120575flex119860119872 = 119872(119909)22119864119887119868119887 (13c)

119886 = 1199051119887 = 1199052119871 = 1199051 + 1199052119909 = 11990512

(13d)


120575flex119861 = 120575flex11986112059611199051

+ 120575flex11986112059621199052

+ 120575flex119861119872 (14)

where 120575flex11986112059611199051

120575flex11986112059621199052


120575flex11986112059611199051

= minus 1205961119886324119864119887119868119887 (4119909 minus 119886) (15a)

120575flex11986112059621199052

= 120596224119864119887119868119887 (1199094 minus 41198711199093 + 611987121199092 minus 41198863119909 + 1198864) (15b)

120575flex119861119872 = 119872(119909)22119864119887119868119887 (15c)

119909 = 1199051 + 11990522 (15d)


F

F

Bolt

Plate 2

Plate 1A

B

C

A

BC

NA

t1

t2

120575flexA

120575flexB

120575flexC

120575bolt





(1199051 + 1199052) (311990521 + 131199051199051199052 + 311990522)119864119887119868119887 )12059621199052 (16)


120575flex119887 = (311990521 + 16119905221199052 + 16119905111990522 + 311990532128119864119887119868119887 )119865119887 (17)


119862flex119887 = 1119896flex119887 =120575flex119887119865119887

= 311990521 + 16119905221199052 + 16119905111990522 + 311990532128119864119887119868119887 (18)


119906 (119909) = minusint1199090

119881 (119904)120581119860119866119889119904 + 1199060 (19)


120581 = 6 (1 + ])27 + 12] + 4]2 (20)




120575119904119860 = 12057511990411986012059611199051

+ 12057511990411986012059621199052

+ 120575119904119860119872 (21)

where 12057511990411986012059611199051

12057511990411986012059621199052


12057511990411986012059611199051

= int11990901205741 (119909) 119889119909 = minus 31205961119905218120581119866119887119860119887 (22a)

12057511990411986012059621199052

= int11990901205743 (119909) 119889119909 = 1205962119905111990522120581119866119887119860119887 (22b)

120575119904119860119872 = 0 (22c)

119909 = 11990512 (22d)


120575119904119861 = 12057511990411986112059611199051

+ 12057511990411986112059621199052

+ 120575119904119861119872 (23)

where 12057511990411986112059611199051

12057511990411986112059621199052


12057511990411986112059611199051

= int11988601205741 (119909) 119889119909 + int119909

01205742 (119909) 119889119909 = minus 1205961119905212120581119866119887119860119887 (24a)

12057511990411986112059621199052

= int11988601205743 (119909) 119889119909 + int119909

01205744 (119909) 119889119909

= 12059621199052 (81199051 + 31199052)8119866119887119860119887 (24b)

120575119904119861119872 = 0 (24c)

119909 = 11990522 (24d)


F

F

Bolt

A

B

C

A

BC

Plate 2

Plate 1

NA

t1

t2

120575bolt

120575sA

120575sB


120575sB minus 120575sA=



120575119904 = 120575119904119861 minus 120575119904119860 = 38 ( 1199051 + 1199052120581119860119887119866119887)12059621199052 (25)


120575119904 = 38 ( 1199051 + 1199052120581119860119887119866119887)12059621199052 (26)


119862sd119887 = 1119896sd119887 =120575119904119865119887 =

38(1199051 + 1199052)120581119860119887119866119887 (27)


120590bea = 119875119889119905 (28a)

120576bea = 120575bea119901119889 (28b)


120590bea119901 = 119864119901120576bea119901 (29a)

119896bea119901 = 119875120575bea119901 = 119905119889119864119901119889 = 119905119864119901 (29b)


120590bea119887 = 119864119887120576bea119887 (30a)

119896bea119887 = 119875120575bea119887 =119905119889119864119887119889 = 119905119864119887 (30b)


119888bea119887 = 1119896bea119887 =11199051119864119887 +

11199052119864119887 (31)

119888bea119901 = 1119896bea119901 = 111990511198641199011 +111990521198641199012 (32)


1119896bp = 311990521 + 16119905221199052 + 16119905111990522 + 311990532128119864119887119868119887 + 38(1199051 + 1199052)120581119860119887119866119887

+ ( 11199051119864119887 +11199052119864119887) + ( 111990511198641199011 +

111990521198641199012) (33)



001198653

= [[[[


]]]]119906111990621199063

(34)



Plate

Bolt hole

x

y

Compression zone

d

d

A

Deformation shape

120590beap

A998400

120575beap


Bolt

x

zt

Compression zone

d

d

B

Deformation shape

120590beab

120590beab

120575beab

B998400



F

Plate 2Plate 1

x

z

l1

l2


1

23F3

u3u2

kp1

kbp

u1 kp2

(b) Spring model

Plate 1

Plate 2

x

y(plate zone)

(Bolt zone)

(plate zone)

w1

l1 minus 05dℎdℎ

w2

l2 minus 05dℎ

k1-1

k1-2

k2-2k2-1

(c) Plate stiffness





1119896bp = (1199052 + 1199051)2119864119887119889 + 372( 111990511198641199011 +111990521198641199012) (35)




11198961-2 =1120596111990511198641199011 (1198971 minus 05119889ℎ) + 1

((1205961 minus 119889) 11990511198641199011) 05119889ℎ 163692

1119896119901211198962-1 +

11198962-2 =1(1205962 minus 119889) 1199052119864119901205119889ℎ +

1120596211990521198641199012 (1198972 minus 05119889ℎ) 163692

1119896bp Equation (33) 150660


0

1500

3000

4500

6000

7500

9000

10500

12000

0 1 2 3 4 5 6 7 8 9 10 11 12

Load

(N)


S-1S-2S-3S-4



0

1500

3000

4500

6000

7500

0 01 02 03 04 05 06 07 08 09 1

Load

(N)


S-1S-2S-3S-4S-5





1119896bp = (1199052 + 11990512119889 )23

sdot 3 ( 111990511198641199011 +111990521198641199012 +

121199051119864119887 +121199052119864119887)

(36)



+ 086( 1(radic119864119871 + 1198641198791199051)plate-1

+ 1(radic119864119871 + 1198641198791199052)plate-2)

(37)






x

y

Unit mm30

Plate 2Plate 1


Bolt diameter = 5 x

z

F = 5000 N t = 1

t = 1

l1 = 100 l2 = 3d l3 = 3d l4 = 100


k8

k9 k7

k6

k5

k4

k3

k2

k1

u1

u2

u3

u4

u5

u6u7 67F

5

4

3

2

1


Plate 1Plate 2


(Bolt zone)





l1

la lb lc

l2 l3 l4

dℎ1 dℎ2 dℎ3

k92k91

k73 k72

k33

k42

k71 k41

k43

k63 k62k61

y k32

k31

k12

k11

ld

w2

w1

x

(b) Plate stiffness




111989611 +111989612 11989611 = 120596211990521198641199012119897119889 11989612 = (1205962 minus 119889ℎ3)1199052119864119901205119889ℎ3

11198963 =111989631 +

111989632 +111989633 11989631 = 120596211990521198641199012119897119888 11989632 = (1205962 minus 119889ℎ3)1199052119864119901205119889ℎ3 11989633 = (1205962 minus 119889ℎ2)1199052119864119901205119889ℎ2

11198964 =111989641 +

111989642 +111989643 11989641 = 120596111990511198641199011119897119888 11989642 = (1205961 minus 119889ℎ3)1199051119864119901105119889ℎ3 11989643 = (1205961 minus 119889ℎ2)1199051119864119901105119889ℎ2

11198966 =111989661 +

111989662 +111989663 11989661 = 120596211990521198641199012119897119887 11989662 = (1205962 minus 119889ℎ2)1199052119864119901205119889ℎ2 11989663 = (1205962 minus 119889ℎ1)1199052119864119901205119889ℎ1

11198967 =111989671 +

111989672 +111989673 11989671 = 120596111990511198641199011119897119887 11989672 = (1205961 minus 119889ℎ2)1199051119864119901105119889ℎ2 11989673 = (1205961 minus 119889ℎ1)1199051119864119901105119889ℎ1

11198969 =111989691 +

111989692 11989691 = 120596111990511198641199011119897119886 11989692 = (1205961 minus 119889ℎ1)1199051119864119901105119889ℎ111198962 11198965

11198968 Equation (33)



xz

yA

B

dx = 0

dy = 0

dz = 0

dx = freedy = 0

dz = 0




0

1000

2000

3000

4000

5000

Load

(N)

(b) Connection load

3472 36213056 2928

3472 3449

Bolt 1(spring)

Bolt 1(FEA)

Bolt 2(spring)

Bolt 2(FEA)

Bolt 3(spring)

Bolt 3(FEA)

05

10152025303540

Load

dist

ribut

ion

()



000000119865

=

[[[[[[[[[[[[[[[


]]]]]]]]]]]]]]]

1199061119906211990631199064119906511990661199067

(38)



(39)






000

050

100

150

200

000 050 100 150 200

Reference case


m = dydx

120595120595

ref




120595 = 119899119875minsum119899119894=1 119875119894 for 119894 = 1 2 3 119899 (40)











8 Conclusion




Bolt 1Bolt 2Bolt 3

05

101520253035404550

Load

dist

ribut

ion

()


Reference case

y = 00605x + 09326

050 100 150 200000ddref

090

094

098

120595120595

ref

102

106

110




Bolt 1Bolt 2Bolt 3

05

101520253035404550

Load

dist

ribut

ion

()


Reference case

y = 00136x + 09884

090

094

098

102

120595120595

ref

106

110

050 100 150 200000ttref



05

101520253035404550

Load

dist

ribut

ion

()

1 2 3 4 5t2t1

Bolt 1Bolt 2Bolt 3


Reference case

y = minus00646x + 10642

050 100 150 200000(t2t1)(t2t1)ref

090

094

120595120595

ref

098

102

106

110




Plate 2


F

t2 ≫ tt










Acknowledgments


References





















RoboticsJournal of





Journal of



RotatingMachinery



Journal of

Volume 201


VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










UTM rod load cell

Upper grip

Lower grip

Specimen

UTM rod

Fixed gagelengths

Data

LVD

T

logger




Bolt (SOLID186)

x

z

(a) Geometry model

M

I

J

N

K

OL

P

1

2

3

6

5

8

4

7

1

2

3 4

56

z0y0

x0

(b) SHELL281

J

1

x

yz R

K

I

O

W

S

Q

I

TL

I

Y

M

XP

N

Z

VU

2

3

4

5

6

(c) SOLID186

x

yz I

M

J

N

K

OL

P



PI998400

J998400

N998400

M998400

L998400

K998400

O998400

(d) Contact element

x

z

CONTAC174

TARGET170

Steel bolt

Cold-formed steel

fp = kpxp

kp kpkpkp xp



xz

y

(pined)

(roller)

A

B

Fine mesh size

dx = 0

dy = 0

dz = 0

dx = free

dy = 0

dz = 0


0

100

200

300

400

500

600

700

0 002 004 006 008

Axi

al st

ress

(MPa

)

Axial strain (mmmm)




0

2000

4000

6000

8000

10000

12000

0 1 2 3 4 5 6 7 8 9 10 11 12

Load

(N)


S-1S-2S-3

S-4S-5S-FEA


Test FEA


(b) Failure mode











119862119887 = 1119896119887 =120575119887119865 (2)



120579 (119909) = int1199090

119872(119904)119864119868 119889119904 + 1205790 (3a)

119906 (119909) = int1199090120579 (119904) 119889119904 minus int119909

0

119881 (119904)120581119860119866119889119904 + 1199060 (3b)




A

BC

D

E



Stre

ss n

orm

aliz

atio

n

Strain

Strain

A

BC

D

E

10004

089658

079278

068899

05852

04814

037761

027382

017003

0066233

180∘

0∘31592

28085

24578

21071

17564

14057

1055

070431

035361

00029019

120579 (degrees)

120579

13

10

08

05

03

000 20 40 60 80 100 120 140 160 180

0051152253

Stra

in (m

mm

m)


F

Bolt

Plate 2

Plate 1 A

B

C

Bolt

Plate 2

Plate 1F = 0

t1

t2

t1

t2

120575b


F

F

M

L

A

B

C

1205962t2

1205961t1V1

V2

t1

t2

M



A

B

C


Guiding support

M

Case 3Case 2Case 1

= + +

1205962t2 1205962t2

1205961t11205961t1


L

a b

L

a b

x x

L

x


L

x

y

y y y120596 120596

MB

0 le x le a

a le x le L

120575(x) = minus120596x3

24EI(4x minus a)

120579B =120596a3

6EI

0 le x le a

120575 = minus120596x2b


a le x le L

x = L

120575B =MBL

2

2EI

120579B =MBL

EI



120575B120579B

120575 = minus120596


120575(x) = minus120596x2


120579B =120596(L3 minus a3)

6EI

minus





120601 = 1206011 + 1206012 + 1206013 = (1205791 + 1205741) + (1205792 + 1205742) + (1205793 + 1205743)= 0 (4)



1205791 = minus 1205961119905316119864119887119868119887 1205792 = 1205962 (1198713 minus 11990531)6119864119887119868119887 1205793 = 119872119861119871119864119887119868119887

(5a)

1205741 = 01205742 = 01205743 = 0 (5b)


a

b

L

x

La b

La b

x x

L

x


a

x

b

V

dx

x

x

x

y

y y y

120596

120596 120596

MB

0 le x le a


1205741(x) = minus120596(x minus a)

120581AG0 le x le b

V2(x) = 0

1205742(x) = 0

0 le x le a


1205743(x) = minus120596b

120581AG

0 le x le b


1205744(x) = minus120596(x minus b)

120581AG

V(x) = 0

120574(x) = 0

120574

120575s(x) = 0

120574 =d120575sdx

d120575s

120575s

120596b

1205751(x) = intx

01205741(x)dx

1205752(x) = inta

01205741(x)dx + intx

01205742(x)dx

1205754(x) = inta

01205743(x)dx + intx

01205744(x)dx

1205753(x) = intx

01205743(x)dx


A

B

=C

Fixed support

Guiding support


+ +


1205962t2 1205962t2

1205961t11205961t1

MB

1206011 1206012 1206013



119865119887 = 12059611199051 = 12059621199052 (6)


1205961 = 1205962 (11990521199051) (7)


minus 1205961119905316119864119887119868119887 +1205962 (1198713 minus 11990531)6119864119887119868119887 + 119872119861119871119864119887119868119887 = 0 (8)





M

M

A

C

B1205962t2

1205961t1

kc

kb

kc

kb

Fixed support

Guiding support




1119896bp = 1119896119887 +1119896119888 (10a)

with

1119896119887 =1119896flex119887 +

1119896she119887 1119896119888 =

1119896bea119901 +1119896bea119887

(10b)


119906 (119909) = int1199090120579 (119904) 119889119904 + 1199060

= int1199090(int1199090

119872(119904)119864119868 119889119904 + 1205790)119889119904 + 1199060(11)



120575flex119860 = 120575flex11986012059611199051

+ 120575flex11986012059621199052

+ 120575flex119860119872 (12)

where 120575flex11986012059611199051

120575flex11986012059621199052


120575flex11986012059611199051

= minus 12059611199053124119864119887119868119887 (61198862 minus 4119886119909 + 1199092) (13a)

120575flex11986012059621199052

= 1205962119887119909212119864119887119868119887 (3119871 + 3119886 minus 2119909) (13b)

120575flex119860119872 = 119872(119909)22119864119887119868119887 (13c)

119886 = 1199051119887 = 1199052119871 = 1199051 + 1199052119909 = 11990512

(13d)


120575flex119861 = 120575flex11986112059611199051

+ 120575flex11986112059621199052

+ 120575flex119861119872 (14)

where 120575flex11986112059611199051

120575flex11986112059621199052


120575flex11986112059611199051

= minus 1205961119886324119864119887119868119887 (4119909 minus 119886) (15a)

120575flex11986112059621199052

= 120596224119864119887119868119887 (1199094 minus 41198711199093 + 611987121199092 minus 41198863119909 + 1198864) (15b)

120575flex119861119872 = 119872(119909)22119864119887119868119887 (15c)

119909 = 1199051 + 11990522 (15d)


F

F

Bolt

Plate 2

Plate 1A

B

C

A

BC

NA

t1

t2

120575flexA

120575flexB

120575flexC

120575bolt





(1199051 + 1199052) (311990521 + 131199051199051199052 + 311990522)119864119887119868119887 )12059621199052 (16)


120575flex119887 = (311990521 + 16119905221199052 + 16119905111990522 + 311990532128119864119887119868119887 )119865119887 (17)


119862flex119887 = 1119896flex119887 =120575flex119887119865119887

= 311990521 + 16119905221199052 + 16119905111990522 + 311990532128119864119887119868119887 (18)


119906 (119909) = minusint1199090

119881 (119904)120581119860119866119889119904 + 1199060 (19)


120581 = 6 (1 + ])27 + 12] + 4]2 (20)




120575119904119860 = 12057511990411986012059611199051

+ 12057511990411986012059621199052

+ 120575119904119860119872 (21)

where 12057511990411986012059611199051

12057511990411986012059621199052


12057511990411986012059611199051

= int11990901205741 (119909) 119889119909 = minus 31205961119905218120581119866119887119860119887 (22a)

12057511990411986012059621199052

= int11990901205743 (119909) 119889119909 = 1205962119905111990522120581119866119887119860119887 (22b)

120575119904119860119872 = 0 (22c)

119909 = 11990512 (22d)


120575119904119861 = 12057511990411986112059611199051

+ 12057511990411986112059621199052

+ 120575119904119861119872 (23)

where 12057511990411986112059611199051

12057511990411986112059621199052


12057511990411986112059611199051

= int11988601205741 (119909) 119889119909 + int119909

01205742 (119909) 119889119909 = minus 1205961119905212120581119866119887119860119887 (24a)

12057511990411986112059621199052

= int11988601205743 (119909) 119889119909 + int119909

01205744 (119909) 119889119909

= 12059621199052 (81199051 + 31199052)8119866119887119860119887 (24b)

120575119904119861119872 = 0 (24c)

119909 = 11990522 (24d)


F

F

Bolt

A

B

C

A

BC

Plate 2

Plate 1

NA

t1

t2

120575bolt

120575sA

120575sB


120575sB minus 120575sA=



120575119904 = 120575119904119861 minus 120575119904119860 = 38 ( 1199051 + 1199052120581119860119887119866119887)12059621199052 (25)


120575119904 = 38 ( 1199051 + 1199052120581119860119887119866119887)12059621199052 (26)


119862sd119887 = 1119896sd119887 =120575119904119865119887 =

38(1199051 + 1199052)120581119860119887119866119887 (27)


120590bea = 119875119889119905 (28a)

120576bea = 120575bea119901119889 (28b)


120590bea119901 = 119864119901120576bea119901 (29a)

119896bea119901 = 119875120575bea119901 = 119905119889119864119901119889 = 119905119864119901 (29b)


120590bea119887 = 119864119887120576bea119887 (30a)

119896bea119887 = 119875120575bea119887 =119905119889119864119887119889 = 119905119864119887 (30b)


119888bea119887 = 1119896bea119887 =11199051119864119887 +

11199052119864119887 (31)

119888bea119901 = 1119896bea119901 = 111990511198641199011 +111990521198641199012 (32)


1119896bp = 311990521 + 16119905221199052 + 16119905111990522 + 311990532128119864119887119868119887 + 38(1199051 + 1199052)120581119860119887119866119887

+ ( 11199051119864119887 +11199052119864119887) + ( 111990511198641199011 +

111990521198641199012) (33)



001198653

= [[[[


]]]]119906111990621199063

(34)



Plate

Bolt hole

x

y

Compression zone

d

d

A

Deformation shape

120590beap

A998400

120575beap


Bolt

x

zt

Compression zone

d

d

B

Deformation shape

120590beab

120590beab

120575beab

B998400



F

Plate 2Plate 1

x

z

l1

l2


1

23F3

u3u2

kp1

kbp

u1 kp2

(b) Spring model

Plate 1

Plate 2

x

y(plate zone)

(Bolt zone)

(plate zone)

w1

l1 minus 05dℎdℎ

w2

l2 minus 05dℎ

k1-1

k1-2

k2-2k2-1

(c) Plate stiffness





1119896bp = (1199052 + 1199051)2119864119887119889 + 372( 111990511198641199011 +111990521198641199012) (35)




11198961-2 =1120596111990511198641199011 (1198971 minus 05119889ℎ) + 1

((1205961 minus 119889) 11990511198641199011) 05119889ℎ 163692

1119896119901211198962-1 +

11198962-2 =1(1205962 minus 119889) 1199052119864119901205119889ℎ +

1120596211990521198641199012 (1198972 minus 05119889ℎ) 163692

1119896bp Equation (33) 150660


0

1500

3000

4500

6000

7500

9000

10500

12000

0 1 2 3 4 5 6 7 8 9 10 11 12

Load

(N)


S-1S-2S-3S-4



0

1500

3000

4500

6000

7500

0 01 02 03 04 05 06 07 08 09 1

Load

(N)


S-1S-2S-3S-4S-5





1119896bp = (1199052 + 11990512119889 )23

sdot 3 ( 111990511198641199011 +111990521198641199012 +

121199051119864119887 +121199052119864119887)

(36)



+ 086( 1(radic119864119871 + 1198641198791199051)plate-1

+ 1(radic119864119871 + 1198641198791199052)plate-2)

(37)






x

y

Unit mm30

Plate 2Plate 1


Bolt diameter = 5 x

z

F = 5000 N t = 1

t = 1

l1 = 100 l2 = 3d l3 = 3d l4 = 100


k8

k9 k7

k6

k5

k4

k3

k2

k1

u1

u2

u3

u4

u5

u6u7 67F

5

4

3

2

1


Plate 1Plate 2


(Bolt zone)





l1

la lb lc

l2 l3 l4

dℎ1 dℎ2 dℎ3

k92k91

k73 k72

k33

k42

k71 k41

k43

k63 k62k61

y k32

k31

k12

k11

ld

w2

w1

x

(b) Plate stiffness




111989611 +111989612 11989611 = 120596211990521198641199012119897119889 11989612 = (1205962 minus 119889ℎ3)1199052119864119901205119889ℎ3

11198963 =111989631 +

111989632 +111989633 11989631 = 120596211990521198641199012119897119888 11989632 = (1205962 minus 119889ℎ3)1199052119864119901205119889ℎ3 11989633 = (1205962 minus 119889ℎ2)1199052119864119901205119889ℎ2

11198964 =111989641 +

111989642 +111989643 11989641 = 120596111990511198641199011119897119888 11989642 = (1205961 minus 119889ℎ3)1199051119864119901105119889ℎ3 11989643 = (1205961 minus 119889ℎ2)1199051119864119901105119889ℎ2

11198966 =111989661 +

111989662 +111989663 11989661 = 120596211990521198641199012119897119887 11989662 = (1205962 minus 119889ℎ2)1199052119864119901205119889ℎ2 11989663 = (1205962 minus 119889ℎ1)1199052119864119901205119889ℎ1

11198967 =111989671 +

111989672 +111989673 11989671 = 120596111990511198641199011119897119887 11989672 = (1205961 minus 119889ℎ2)1199051119864119901105119889ℎ2 11989673 = (1205961 minus 119889ℎ1)1199051119864119901105119889ℎ1

11198969 =111989691 +

111989692 11989691 = 120596111990511198641199011119897119886 11989692 = (1205961 minus 119889ℎ1)1199051119864119901105119889ℎ111198962 11198965

11198968 Equation (33)



xz

yA

B

dx = 0

dy = 0

dz = 0

dx = freedy = 0

dz = 0




0

1000

2000

3000

4000

5000

Load

(N)

(b) Connection load

3472 36213056 2928

3472 3449

Bolt 1(spring)

Bolt 1(FEA)

Bolt 2(spring)

Bolt 2(FEA)

Bolt 3(spring)

Bolt 3(FEA)

05

10152025303540

Load

dist

ribut

ion

()



000000119865

=

[[[[[[[[[[[[[[[


]]]]]]]]]]]]]]]

1199061119906211990631199064119906511990661199067

(38)



(39)






000

050

100

150

200

000 050 100 150 200

Reference case


m = dydx

120595120595

ref




120595 = 119899119875minsum119899119894=1 119875119894 for 119894 = 1 2 3 119899 (40)











8 Conclusion




Bolt 1Bolt 2Bolt 3

05

101520253035404550

Load

dist

ribut

ion

()


Reference case

y = 00605x + 09326

050 100 150 200000ddref

090

094

098

120595120595

ref

102

106

110




Bolt 1Bolt 2Bolt 3

05

101520253035404550

Load

dist

ribut

ion

()


Reference case

y = 00136x + 09884

090

094

098

102

120595120595

ref

106

110

050 100 150 200000ttref



05

101520253035404550

Load

dist

ribut

ion

()

1 2 3 4 5t2t1

Bolt 1Bolt 2Bolt 3


Reference case

y = minus00646x + 10642

050 100 150 200000(t2t1)(t2t1)ref

090

094

120595120595

ref

098

102

106

110




Plate 2


F

t2 ≫ tt










Acknowledgments


References





















RoboticsJournal of





Journal of



RotatingMachinery



Journal of

Volume 201


VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










0

2000

4000

6000

8000

10000

12000

0 1 2 3 4 5 6 7 8 9 10 11 12

Load

(N)


S-1S-2S-3

S-4S-5S-FEA


Test FEA


(b) Failure mode











119862119887 = 1119896119887 =120575119887119865 (2)



120579 (119909) = int1199090

119872(119904)119864119868 119889119904 + 1205790 (3a)

119906 (119909) = int1199090120579 (119904) 119889119904 minus int119909

0

119881 (119904)120581119860119866119889119904 + 1199060 (3b)




A

BC

D

E



Stre

ss n

orm

aliz

atio

n

Strain

Strain

A

BC

D

E

10004

089658

079278

068899

05852

04814

037761

027382

017003

0066233

180∘

0∘31592

28085

24578

21071

17564

14057

1055

070431

035361

00029019

120579 (degrees)

120579

13

10

08

05

03

000 20 40 60 80 100 120 140 160 180

0051152253

Stra

in (m

mm

m)


F

Bolt

Plate 2

Plate 1 A

B

C

Bolt

Plate 2

Plate 1F = 0

t1

t2

t1

t2

120575b


F

F

M

L

A

B

C

1205962t2

1205961t1V1

V2

t1

t2

M



A

B

C


Guiding support

M

Case 3Case 2Case 1

= + +

1205962t2 1205962t2

1205961t11205961t1


L

a b

L

a b

x x

L

x


L

x

y

y y y120596 120596

MB

0 le x le a

a le x le L

120575(x) = minus120596x3

24EI(4x minus a)

120579B =120596a3

6EI

0 le x le a

120575 = minus120596x2b


a le x le L

x = L

120575B =MBL

2

2EI

120579B =MBL

EI



120575B120579B

120575 = minus120596


120575(x) = minus120596x2


120579B =120596(L3 minus a3)

6EI

minus





120601 = 1206011 + 1206012 + 1206013 = (1205791 + 1205741) + (1205792 + 1205742) + (1205793 + 1205743)= 0 (4)



1205791 = minus 1205961119905316119864119887119868119887 1205792 = 1205962 (1198713 minus 11990531)6119864119887119868119887 1205793 = 119872119861119871119864119887119868119887

(5a)

1205741 = 01205742 = 01205743 = 0 (5b)


a

b

L

x

La b

La b

x x

L

x


a

x

b

V

dx

x

x

x

y

y y y

120596

120596 120596

MB

0 le x le a


1205741(x) = minus120596(x minus a)

120581AG0 le x le b

V2(x) = 0

1205742(x) = 0

0 le x le a


1205743(x) = minus120596b

120581AG

0 le x le b


1205744(x) = minus120596(x minus b)

120581AG

V(x) = 0

120574(x) = 0

120574

120575s(x) = 0

120574 =d120575sdx

d120575s

120575s

120596b

1205751(x) = intx

01205741(x)dx

1205752(x) = inta

01205741(x)dx + intx

01205742(x)dx

1205754(x) = inta

01205743(x)dx + intx

01205744(x)dx

1205753(x) = intx

01205743(x)dx


A

B

=C

Fixed support

Guiding support


+ +


1205962t2 1205962t2

1205961t11205961t1

MB

1206011 1206012 1206013



119865119887 = 12059611199051 = 12059621199052 (6)


1205961 = 1205962 (11990521199051) (7)


minus 1205961119905316119864119887119868119887 +1205962 (1198713 minus 11990531)6119864119887119868119887 + 119872119861119871119864119887119868119887 = 0 (8)





M

M

A

C

B1205962t2

1205961t1

kc

kb

kc

kb

Fixed support

Guiding support




1119896bp = 1119896119887 +1119896119888 (10a)

with

1119896119887 =1119896flex119887 +

1119896she119887 1119896119888 =

1119896bea119901 +1119896bea119887

(10b)


119906 (119909) = int1199090120579 (119904) 119889119904 + 1199060

= int1199090(int1199090

119872(119904)119864119868 119889119904 + 1205790)119889119904 + 1199060(11)



120575flex119860 = 120575flex11986012059611199051

+ 120575flex11986012059621199052

+ 120575flex119860119872 (12)

where 120575flex11986012059611199051

120575flex11986012059621199052


120575flex11986012059611199051

= minus 12059611199053124119864119887119868119887 (61198862 minus 4119886119909 + 1199092) (13a)

120575flex11986012059621199052

= 1205962119887119909212119864119887119868119887 (3119871 + 3119886 minus 2119909) (13b)

120575flex119860119872 = 119872(119909)22119864119887119868119887 (13c)

119886 = 1199051119887 = 1199052119871 = 1199051 + 1199052119909 = 11990512

(13d)


120575flex119861 = 120575flex11986112059611199051

+ 120575flex11986112059621199052

+ 120575flex119861119872 (14)

where 120575flex11986112059611199051

120575flex11986112059621199052


120575flex11986112059611199051

= minus 1205961119886324119864119887119868119887 (4119909 minus 119886) (15a)

120575flex11986112059621199052

= 120596224119864119887119868119887 (1199094 minus 41198711199093 + 611987121199092 minus 41198863119909 + 1198864) (15b)

120575flex119861119872 = 119872(119909)22119864119887119868119887 (15c)

119909 = 1199051 + 11990522 (15d)


F

F

Bolt

Plate 2

Plate 1A

B

C

A

BC

NA

t1

t2

120575flexA

120575flexB

120575flexC

120575bolt





(1199051 + 1199052) (311990521 + 131199051199051199052 + 311990522)119864119887119868119887 )12059621199052 (16)


120575flex119887 = (311990521 + 16119905221199052 + 16119905111990522 + 311990532128119864119887119868119887 )119865119887 (17)


119862flex119887 = 1119896flex119887 =120575flex119887119865119887

= 311990521 + 16119905221199052 + 16119905111990522 + 311990532128119864119887119868119887 (18)


119906 (119909) = minusint1199090

119881 (119904)120581119860119866119889119904 + 1199060 (19)


120581 = 6 (1 + ])27 + 12] + 4]2 (20)




120575119904119860 = 12057511990411986012059611199051

+ 12057511990411986012059621199052

+ 120575119904119860119872 (21)

where 12057511990411986012059611199051

12057511990411986012059621199052


12057511990411986012059611199051

= int11990901205741 (119909) 119889119909 = minus 31205961119905218120581119866119887119860119887 (22a)

12057511990411986012059621199052

= int11990901205743 (119909) 119889119909 = 1205962119905111990522120581119866119887119860119887 (22b)

120575119904119860119872 = 0 (22c)

119909 = 11990512 (22d)


120575119904119861 = 12057511990411986112059611199051

+ 12057511990411986112059621199052

+ 120575119904119861119872 (23)

where 12057511990411986112059611199051

12057511990411986112059621199052


12057511990411986112059611199051

= int11988601205741 (119909) 119889119909 + int119909

01205742 (119909) 119889119909 = minus 1205961119905212120581119866119887119860119887 (24a)

12057511990411986112059621199052

= int11988601205743 (119909) 119889119909 + int119909

01205744 (119909) 119889119909

= 12059621199052 (81199051 + 31199052)8119866119887119860119887 (24b)

120575119904119861119872 = 0 (24c)

119909 = 11990522 (24d)


F

F

Bolt

A

B

C

A

BC

Plate 2

Plate 1

NA

t1

t2

120575bolt

120575sA

120575sB


120575sB minus 120575sA=



120575119904 = 120575119904119861 minus 120575119904119860 = 38 ( 1199051 + 1199052120581119860119887119866119887)12059621199052 (25)


120575119904 = 38 ( 1199051 + 1199052120581119860119887119866119887)12059621199052 (26)


119862sd119887 = 1119896sd119887 =120575119904119865119887 =

38(1199051 + 1199052)120581119860119887119866119887 (27)


120590bea = 119875119889119905 (28a)

120576bea = 120575bea119901119889 (28b)


120590bea119901 = 119864119901120576bea119901 (29a)

119896bea119901 = 119875120575bea119901 = 119905119889119864119901119889 = 119905119864119901 (29b)


120590bea119887 = 119864119887120576bea119887 (30a)

119896bea119887 = 119875120575bea119887 =119905119889119864119887119889 = 119905119864119887 (30b)


119888bea119887 = 1119896bea119887 =11199051119864119887 +

11199052119864119887 (31)

119888bea119901 = 1119896bea119901 = 111990511198641199011 +111990521198641199012 (32)


1119896bp = 311990521 + 16119905221199052 + 16119905111990522 + 311990532128119864119887119868119887 + 38(1199051 + 1199052)120581119860119887119866119887

+ ( 11199051119864119887 +11199052119864119887) + ( 111990511198641199011 +

111990521198641199012) (33)



001198653

= [[[[


]]]]119906111990621199063

(34)



Plate

Bolt hole

x

y

Compression zone

d

d

A

Deformation shape

120590beap

A998400

120575beap


Bolt

x

zt

Compression zone

d

d

B

Deformation shape

120590beab

120590beab

120575beab

B998400



F

Plate 2Plate 1

x

z

l1

l2


1

23F3

u3u2

kp1

kbp

u1 kp2

(b) Spring model

Plate 1

Plate 2

x

y(plate zone)

(Bolt zone)

(plate zone)

w1

l1 minus 05dℎdℎ

w2

l2 minus 05dℎ

k1-1

k1-2

k2-2k2-1

(c) Plate stiffness





1119896bp = (1199052 + 1199051)2119864119887119889 + 372( 111990511198641199011 +111990521198641199012) (35)




11198961-2 =1120596111990511198641199011 (1198971 minus 05119889ℎ) + 1

((1205961 minus 119889) 11990511198641199011) 05119889ℎ 163692

1119896119901211198962-1 +

11198962-2 =1(1205962 minus 119889) 1199052119864119901205119889ℎ +

1120596211990521198641199012 (1198972 minus 05119889ℎ) 163692

1119896bp Equation (33) 150660


0

1500

3000

4500

6000

7500

9000

10500

12000

0 1 2 3 4 5 6 7 8 9 10 11 12

Load

(N)


S-1S-2S-3S-4



0

1500

3000

4500

6000

7500

0 01 02 03 04 05 06 07 08 09 1

Load

(N)


S-1S-2S-3S-4S-5





1119896bp = (1199052 + 11990512119889 )23

sdot 3 ( 111990511198641199011 +111990521198641199012 +

121199051119864119887 +121199052119864119887)

(36)



+ 086( 1(radic119864119871 + 1198641198791199051)plate-1

+ 1(radic119864119871 + 1198641198791199052)plate-2)

(37)






x

y

Unit mm30

Plate 2Plate 1


Bolt diameter = 5 x

z

F = 5000 N t = 1

t = 1

l1 = 100 l2 = 3d l3 = 3d l4 = 100


k8

k9 k7

k6

k5

k4

k3

k2

k1

u1

u2

u3

u4

u5

u6u7 67F

5

4

3

2

1


Plate 1Plate 2


(Bolt zone)





l1

la lb lc

l2 l3 l4

dℎ1 dℎ2 dℎ3

k92k91

k73 k72

k33

k42

k71 k41

k43

k63 k62k61

y k32

k31

k12

k11

ld

w2

w1

x

(b) Plate stiffness




111989611 +111989612 11989611 = 120596211990521198641199012119897119889 11989612 = (1205962 minus 119889ℎ3)1199052119864119901205119889ℎ3

11198963 =111989631 +

111989632 +111989633 11989631 = 120596211990521198641199012119897119888 11989632 = (1205962 minus 119889ℎ3)1199052119864119901205119889ℎ3 11989633 = (1205962 minus 119889ℎ2)1199052119864119901205119889ℎ2

11198964 =111989641 +

111989642 +111989643 11989641 = 120596111990511198641199011119897119888 11989642 = (1205961 minus 119889ℎ3)1199051119864119901105119889ℎ3 11989643 = (1205961 minus 119889ℎ2)1199051119864119901105119889ℎ2

11198966 =111989661 +

111989662 +111989663 11989661 = 120596211990521198641199012119897119887 11989662 = (1205962 minus 119889ℎ2)1199052119864119901205119889ℎ2 11989663 = (1205962 minus 119889ℎ1)1199052119864119901205119889ℎ1

11198967 =111989671 +

111989672 +111989673 11989671 = 120596111990511198641199011119897119887 11989672 = (1205961 minus 119889ℎ2)1199051119864119901105119889ℎ2 11989673 = (1205961 minus 119889ℎ1)1199051119864119901105119889ℎ1

11198969 =111989691 +

111989692 11989691 = 120596111990511198641199011119897119886 11989692 = (1205961 minus 119889ℎ1)1199051119864119901105119889ℎ111198962 11198965

11198968 Equation (33)



xz

yA

B

dx = 0

dy = 0

dz = 0

dx = freedy = 0

dz = 0




0

1000

2000

3000

4000

5000

Load

(N)

(b) Connection load

3472 36213056 2928

3472 3449

Bolt 1(spring)

Bolt 1(FEA)

Bolt 2(spring)

Bolt 2(FEA)

Bolt 3(spring)

Bolt 3(FEA)

05

10152025303540

Load

dist

ribut

ion

()



000000119865

=

[[[[[[[[[[[[[[[


]]]]]]]]]]]]]]]

1199061119906211990631199064119906511990661199067

(38)



(39)






000

050

100

150

200

000 050 100 150 200

Reference case


m = dydx

120595120595

ref




120595 = 119899119875minsum119899119894=1 119875119894 for 119894 = 1 2 3 119899 (40)











8 Conclusion




Bolt 1Bolt 2Bolt 3

05

101520253035404550

Load

dist

ribut

ion

()


Reference case

y = 00605x + 09326

050 100 150 200000ddref

090

094

098

120595120595

ref

102

106

110




Bolt 1Bolt 2Bolt 3

05

101520253035404550

Load

dist

ribut

ion

()


Reference case

y = 00136x + 09884

090

094

098

102

120595120595

ref

106

110

050 100 150 200000ttref



05

101520253035404550

Load

dist

ribut

ion

()

1 2 3 4 5t2t1

Bolt 1Bolt 2Bolt 3


Reference case

y = minus00646x + 10642

050 100 150 200000(t2t1)(t2t1)ref

090

094

120595120595

ref

098

102

106

110




Plate 2


F

t2 ≫ tt










Acknowledgments


References





















RoboticsJournal of





Journal of



RotatingMachinery



Journal of

Volume 201


VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










A

BC

D

E



Stre

ss n

orm

aliz

atio

n

Strain

Strain

A

BC

D

E

10004

089658

079278

068899

05852

04814

037761

027382

017003

0066233

180∘

0∘31592

28085

24578

21071

17564

14057

1055

070431

035361

00029019

120579 (degrees)

120579

13

10

08

05

03

000 20 40 60 80 100 120 140 160 180

0051152253

Stra

in (m

mm

m)


F

Bolt

Plate 2

Plate 1 A

B

C

Bolt

Plate 2

Plate 1F = 0

t1

t2

t1

t2

120575b


F

F

M

L

A

B

C

1205962t2

1205961t1V1

V2

t1

t2

M



A

B

C


Guiding support

M

Case 3Case 2Case 1

= + +

1205962t2 1205962t2

1205961t11205961t1


L

a b

L

a b

x x

L

x


L

x

y

y y y120596 120596

MB

0 le x le a

a le x le L

120575(x) = minus120596x3

24EI(4x minus a)

120579B =120596a3

6EI

0 le x le a

120575 = minus120596x2b


a le x le L

x = L

120575B =MBL

2

2EI

120579B =MBL

EI



120575B120579B

120575 = minus120596


120575(x) = minus120596x2


120579B =120596(L3 minus a3)

6EI

minus





120601 = 1206011 + 1206012 + 1206013 = (1205791 + 1205741) + (1205792 + 1205742) + (1205793 + 1205743)= 0 (4)



1205791 = minus 1205961119905316119864119887119868119887 1205792 = 1205962 (1198713 minus 11990531)6119864119887119868119887 1205793 = 119872119861119871119864119887119868119887

(5a)

1205741 = 01205742 = 01205743 = 0 (5b)


a

b

L

x

La b

La b

x x

L

x


a

x

b

V

dx

x

x

x

y

y y y

120596

120596 120596

MB

0 le x le a


1205741(x) = minus120596(x minus a)

120581AG0 le x le b

V2(x) = 0

1205742(x) = 0

0 le x le a


1205743(x) = minus120596b

120581AG

0 le x le b


1205744(x) = minus120596(x minus b)

120581AG

V(x) = 0

120574(x) = 0

120574

120575s(x) = 0

120574 =d120575sdx

d120575s

120575s

120596b

1205751(x) = intx

01205741(x)dx

1205752(x) = inta

01205741(x)dx + intx

01205742(x)dx

1205754(x) = inta

01205743(x)dx + intx

01205744(x)dx

1205753(x) = intx

01205743(x)dx


A

B

=C

Fixed support

Guiding support


+ +


1205962t2 1205962t2

1205961t11205961t1

MB

1206011 1206012 1206013



119865119887 = 12059611199051 = 12059621199052 (6)


1205961 = 1205962 (11990521199051) (7)


minus 1205961119905316119864119887119868119887 +1205962 (1198713 minus 11990531)6119864119887119868119887 + 119872119861119871119864119887119868119887 = 0 (8)





M

M

A

C

B1205962t2

1205961t1

kc

kb

kc

kb

Fixed support

Guiding support




1119896bp = 1119896119887 +1119896119888 (10a)

with

1119896119887 =1119896flex119887 +

1119896she119887 1119896119888 =

1119896bea119901 +1119896bea119887

(10b)


119906 (119909) = int1199090120579 (119904) 119889119904 + 1199060

= int1199090(int1199090

119872(119904)119864119868 119889119904 + 1205790)119889119904 + 1199060(11)



120575flex119860 = 120575flex11986012059611199051

+ 120575flex11986012059621199052

+ 120575flex119860119872 (12)

where 120575flex11986012059611199051

120575flex11986012059621199052


120575flex11986012059611199051

= minus 12059611199053124119864119887119868119887 (61198862 minus 4119886119909 + 1199092) (13a)

120575flex11986012059621199052

= 1205962119887119909212119864119887119868119887 (3119871 + 3119886 minus 2119909) (13b)

120575flex119860119872 = 119872(119909)22119864119887119868119887 (13c)

119886 = 1199051119887 = 1199052119871 = 1199051 + 1199052119909 = 11990512

(13d)


120575flex119861 = 120575flex11986112059611199051

+ 120575flex11986112059621199052

+ 120575flex119861119872 (14)

where 120575flex11986112059611199051

120575flex11986112059621199052


120575flex11986112059611199051

= minus 1205961119886324119864119887119868119887 (4119909 minus 119886) (15a)

120575flex11986112059621199052

= 120596224119864119887119868119887 (1199094 minus 41198711199093 + 611987121199092 minus 41198863119909 + 1198864) (15b)

120575flex119861119872 = 119872(119909)22119864119887119868119887 (15c)

119909 = 1199051 + 11990522 (15d)


F

F

Bolt

Plate 2

Plate 1A

B

C

A

BC

NA

t1

t2

120575flexA

120575flexB

120575flexC

120575bolt





(1199051 + 1199052) (311990521 + 131199051199051199052 + 311990522)119864119887119868119887 )12059621199052 (16)


120575flex119887 = (311990521 + 16119905221199052 + 16119905111990522 + 311990532128119864119887119868119887 )119865119887 (17)


119862flex119887 = 1119896flex119887 =120575flex119887119865119887

= 311990521 + 16119905221199052 + 16119905111990522 + 311990532128119864119887119868119887 (18)


119906 (119909) = minusint1199090

119881 (119904)120581119860119866119889119904 + 1199060 (19)


120581 = 6 (1 + ])27 + 12] + 4]2 (20)




120575119904119860 = 12057511990411986012059611199051

+ 12057511990411986012059621199052

+ 120575119904119860119872 (21)

where 12057511990411986012059611199051

12057511990411986012059621199052


12057511990411986012059611199051

= int11990901205741 (119909) 119889119909 = minus 31205961119905218120581119866119887119860119887 (22a)

12057511990411986012059621199052

= int11990901205743 (119909) 119889119909 = 1205962119905111990522120581119866119887119860119887 (22b)

120575119904119860119872 = 0 (22c)

119909 = 11990512 (22d)


120575119904119861 = 12057511990411986112059611199051

+ 12057511990411986112059621199052

+ 120575119904119861119872 (23)

where 12057511990411986112059611199051

12057511990411986112059621199052


12057511990411986112059611199051

= int11988601205741 (119909) 119889119909 + int119909

01205742 (119909) 119889119909 = minus 1205961119905212120581119866119887119860119887 (24a)

12057511990411986112059621199052

= int11988601205743 (119909) 119889119909 + int119909

01205744 (119909) 119889119909

= 12059621199052 (81199051 + 31199052)8119866119887119860119887 (24b)

120575119904119861119872 = 0 (24c)

119909 = 11990522 (24d)


F

F

Bolt

A

B

C

A

BC

Plate 2

Plate 1

NA

t1

t2

120575bolt

120575sA

120575sB


120575sB minus 120575sA=



120575119904 = 120575119904119861 minus 120575119904119860 = 38 ( 1199051 + 1199052120581119860119887119866119887)12059621199052 (25)


120575119904 = 38 ( 1199051 + 1199052120581119860119887119866119887)12059621199052 (26)


119862sd119887 = 1119896sd119887 =120575119904119865119887 =

38(1199051 + 1199052)120581119860119887119866119887 (27)


120590bea = 119875119889119905 (28a)

120576bea = 120575bea119901119889 (28b)


120590bea119901 = 119864119901120576bea119901 (29a)

119896bea119901 = 119875120575bea119901 = 119905119889119864119901119889 = 119905119864119901 (29b)


120590bea119887 = 119864119887120576bea119887 (30a)

119896bea119887 = 119875120575bea119887 =119905119889119864119887119889 = 119905119864119887 (30b)


119888bea119887 = 1119896bea119887 =11199051119864119887 +

11199052119864119887 (31)

119888bea119901 = 1119896bea119901 = 111990511198641199011 +111990521198641199012 (32)


1119896bp = 311990521 + 16119905221199052 + 16119905111990522 + 311990532128119864119887119868119887 + 38(1199051 + 1199052)120581119860119887119866119887

+ ( 11199051119864119887 +11199052119864119887) + ( 111990511198641199011 +

111990521198641199012) (33)



001198653

= [[[[


]]]]119906111990621199063

(34)



Plate

Bolt hole

x

y

Compression zone

d

d

A

Deformation shape

120590beap

A998400

120575beap


Bolt

x

zt

Compression zone

d

d

B

Deformation shape

120590beab

120590beab

120575beab

B998400



F

Plate 2Plate 1

x

z

l1

l2


1

23F3

u3u2

kp1

kbp

u1 kp2

(b) Spring model

Plate 1

Plate 2

x

y(plate zone)

(Bolt zone)

(plate zone)

w1

l1 minus 05dℎdℎ

w2

l2 minus 05dℎ

k1-1

k1-2

k2-2k2-1

(c) Plate stiffness





1119896bp = (1199052 + 1199051)2119864119887119889 + 372( 111990511198641199011 +111990521198641199012) (35)




11198961-2 =1120596111990511198641199011 (1198971 minus 05119889ℎ) + 1

((1205961 minus 119889) 11990511198641199011) 05119889ℎ 163692

1119896119901211198962-1 +

11198962-2 =1(1205962 minus 119889) 1199052119864119901205119889ℎ +

1120596211990521198641199012 (1198972 minus 05119889ℎ) 163692

1119896bp Equation (33) 150660


0

1500

3000

4500

6000

7500

9000

10500

12000

0 1 2 3 4 5 6 7 8 9 10 11 12

Load

(N)


S-1S-2S-3S-4



0

1500

3000

4500

6000

7500

0 01 02 03 04 05 06 07 08 09 1

Load

(N)


S-1S-2S-3S-4S-5





1119896bp = (1199052 + 11990512119889 )23

sdot 3 ( 111990511198641199011 +111990521198641199012 +

121199051119864119887 +121199052119864119887)

(36)



+ 086( 1(radic119864119871 + 1198641198791199051)plate-1

+ 1(radic119864119871 + 1198641198791199052)plate-2)

(37)






x

y

Unit mm30

Plate 2Plate 1


Bolt diameter = 5 x

z

F = 5000 N t = 1

t = 1

l1 = 100 l2 = 3d l3 = 3d l4 = 100


k8

k9 k7

k6

k5

k4

k3

k2

k1

u1

u2

u3

u4

u5

u6u7 67F

5

4

3

2

1


Plate 1Plate 2


(Bolt zone)





l1

la lb lc

l2 l3 l4

dℎ1 dℎ2 dℎ3

k92k91

k73 k72

k33

k42

k71 k41

k43

k63 k62k61

y k32

k31

k12

k11

ld

w2

w1

x

(b) Plate stiffness




111989611 +111989612 11989611 = 120596211990521198641199012119897119889 11989612 = (1205962 minus 119889ℎ3)1199052119864119901205119889ℎ3

11198963 =111989631 +

111989632 +111989633 11989631 = 120596211990521198641199012119897119888 11989632 = (1205962 minus 119889ℎ3)1199052119864119901205119889ℎ3 11989633 = (1205962 minus 119889ℎ2)1199052119864119901205119889ℎ2

11198964 =111989641 +

111989642 +111989643 11989641 = 120596111990511198641199011119897119888 11989642 = (1205961 minus 119889ℎ3)1199051119864119901105119889ℎ3 11989643 = (1205961 minus 119889ℎ2)1199051119864119901105119889ℎ2

11198966 =111989661 +

111989662 +111989663 11989661 = 120596211990521198641199012119897119887 11989662 = (1205962 minus 119889ℎ2)1199052119864119901205119889ℎ2 11989663 = (1205962 minus 119889ℎ1)1199052119864119901205119889ℎ1

11198967 =111989671 +

111989672 +111989673 11989671 = 120596111990511198641199011119897119887 11989672 = (1205961 minus 119889ℎ2)1199051119864119901105119889ℎ2 11989673 = (1205961 minus 119889ℎ1)1199051119864119901105119889ℎ1

11198969 =111989691 +

111989692 11989691 = 120596111990511198641199011119897119886 11989692 = (1205961 minus 119889ℎ1)1199051119864119901105119889ℎ111198962 11198965

11198968 Equation (33)



xz

yA

B

dx = 0

dy = 0

dz = 0

dx = freedy = 0

dz = 0




0

1000

2000

3000

4000

5000

Load

(N)

(b) Connection load

3472 36213056 2928

3472 3449

Bolt 1(spring)

Bolt 1(FEA)

Bolt 2(spring)

Bolt 2(FEA)

Bolt 3(spring)

Bolt 3(FEA)

05

10152025303540

Load

dist

ribut

ion

()



000000119865

=

[[[[[[[[[[[[[[[


]]]]]]]]]]]]]]]

1199061119906211990631199064119906511990661199067

(38)



(39)






000

050

100

150

200

000 050 100 150 200

Reference case


m = dydx

120595120595

ref




120595 = 119899119875minsum119899119894=1 119875119894 for 119894 = 1 2 3 119899 (40)











8 Conclusion




Bolt 1Bolt 2Bolt 3

05

101520253035404550

Load

dist

ribut

ion

()


Reference case

y = 00605x + 09326

050 100 150 200000ddref

090

094

098

120595120595

ref

102

106

110




Bolt 1Bolt 2Bolt 3

05

101520253035404550

Load

dist

ribut

ion

()


Reference case

y = 00136x + 09884

090

094

098

102

120595120595

ref

106

110

050 100 150 200000ttref



05

101520253035404550

Load

dist

ribut

ion

()

1 2 3 4 5t2t1

Bolt 1Bolt 2Bolt 3


Reference case

y = minus00646x + 10642

050 100 150 200000(t2t1)(t2t1)ref

090

094

120595120595

ref

098

102

106

110




Plate 2


F

t2 ≫ tt










Acknowledgments


References





















RoboticsJournal of





Journal of



RotatingMachinery



Journal of

Volume 201


VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










A

B

C


Guiding support

M

Case 3Case 2Case 1

= + +

1205962t2 1205962t2

1205961t11205961t1


L

a b

L

a b

x x

L

x


L

x

y

y y y120596 120596

MB

0 le x le a

a le x le L

120575(x) = minus120596x3

24EI(4x minus a)

120579B =120596a3

6EI

0 le x le a

120575 = minus120596x2b


a le x le L

x = L

120575B =MBL

2

2EI

120579B =MBL

EI



120575B120579B

120575 = minus120596


120575(x) = minus120596x2


120579B =120596(L3 minus a3)

6EI

minus





120601 = 1206011 + 1206012 + 1206013 = (1205791 + 1205741) + (1205792 + 1205742) + (1205793 + 1205743)= 0 (4)



1205791 = minus 1205961119905316119864119887119868119887 1205792 = 1205962 (1198713 minus 11990531)6119864119887119868119887 1205793 = 119872119861119871119864119887119868119887

(5a)

1205741 = 01205742 = 01205743 = 0 (5b)


a

b

L

x

La b

La b

x x

L

x


a

x

b

V

dx

x

x

x

y

y y y

120596

120596 120596

MB

0 le x le a


1205741(x) = minus120596(x minus a)

120581AG0 le x le b

V2(x) = 0

1205742(x) = 0

0 le x le a


1205743(x) = minus120596b

120581AG

0 le x le b


1205744(x) = minus120596(x minus b)

120581AG

V(x) = 0

120574(x) = 0

120574

120575s(x) = 0

120574 =d120575sdx

d120575s

120575s

120596b

1205751(x) = intx

01205741(x)dx

1205752(x) = inta

01205741(x)dx + intx

01205742(x)dx

1205754(x) = inta

01205743(x)dx + intx

01205744(x)dx

1205753(x) = intx

01205743(x)dx


A

B

=C

Fixed support

Guiding support


+ +


1205962t2 1205962t2

1205961t11205961t1

MB

1206011 1206012 1206013



119865119887 = 12059611199051 = 12059621199052 (6)


1205961 = 1205962 (11990521199051) (7)


minus 1205961119905316119864119887119868119887 +1205962 (1198713 minus 11990531)6119864119887119868119887 + 119872119861119871119864119887119868119887 = 0 (8)





M

M

A

C

B1205962t2

1205961t1

kc

kb

kc

kb

Fixed support

Guiding support




1119896bp = 1119896119887 +1119896119888 (10a)

with

1119896119887 =1119896flex119887 +

1119896she119887 1119896119888 =

1119896bea119901 +1119896bea119887

(10b)


119906 (119909) = int1199090120579 (119904) 119889119904 + 1199060

= int1199090(int1199090

119872(119904)119864119868 119889119904 + 1205790)119889119904 + 1199060(11)



120575flex119860 = 120575flex11986012059611199051

+ 120575flex11986012059621199052

+ 120575flex119860119872 (12)

where 120575flex11986012059611199051

120575flex11986012059621199052


120575flex11986012059611199051

= minus 12059611199053124119864119887119868119887 (61198862 minus 4119886119909 + 1199092) (13a)

120575flex11986012059621199052

= 1205962119887119909212119864119887119868119887 (3119871 + 3119886 minus 2119909) (13b)

120575flex119860119872 = 119872(119909)22119864119887119868119887 (13c)

119886 = 1199051119887 = 1199052119871 = 1199051 + 1199052119909 = 11990512

(13d)


120575flex119861 = 120575flex11986112059611199051

+ 120575flex11986112059621199052

+ 120575flex119861119872 (14)

where 120575flex11986112059611199051

120575flex11986112059621199052


120575flex11986112059611199051

= minus 1205961119886324119864119887119868119887 (4119909 minus 119886) (15a)

120575flex11986112059621199052

= 120596224119864119887119868119887 (1199094 minus 41198711199093 + 611987121199092 minus 41198863119909 + 1198864) (15b)

120575flex119861119872 = 119872(119909)22119864119887119868119887 (15c)

119909 = 1199051 + 11990522 (15d)


F

F

Bolt

Plate 2

Plate 1A

B

C

A

BC

NA

t1

t2

120575flexA

120575flexB

120575flexC

120575bolt





(1199051 + 1199052) (311990521 + 131199051199051199052 + 311990522)119864119887119868119887 )12059621199052 (16)


120575flex119887 = (311990521 + 16119905221199052 + 16119905111990522 + 311990532128119864119887119868119887 )119865119887 (17)


119862flex119887 = 1119896flex119887 =120575flex119887119865119887

= 311990521 + 16119905221199052 + 16119905111990522 + 311990532128119864119887119868119887 (18)


119906 (119909) = minusint1199090

119881 (119904)120581119860119866119889119904 + 1199060 (19)


120581 = 6 (1 + ])27 + 12] + 4]2 (20)




120575119904119860 = 12057511990411986012059611199051

+ 12057511990411986012059621199052

+ 120575119904119860119872 (21)

where 12057511990411986012059611199051

12057511990411986012059621199052


12057511990411986012059611199051

= int11990901205741 (119909) 119889119909 = minus 31205961119905218120581119866119887119860119887 (22a)

12057511990411986012059621199052

= int11990901205743 (119909) 119889119909 = 1205962119905111990522120581119866119887119860119887 (22b)

120575119904119860119872 = 0 (22c)

119909 = 11990512 (22d)


120575119904119861 = 12057511990411986112059611199051

+ 12057511990411986112059621199052

+ 120575119904119861119872 (23)

where 12057511990411986112059611199051

12057511990411986112059621199052


12057511990411986112059611199051

= int11988601205741 (119909) 119889119909 + int119909

01205742 (119909) 119889119909 = minus 1205961119905212120581119866119887119860119887 (24a)

12057511990411986112059621199052

= int11988601205743 (119909) 119889119909 + int119909

01205744 (119909) 119889119909

= 12059621199052 (81199051 + 31199052)8119866119887119860119887 (24b)

120575119904119861119872 = 0 (24c)

119909 = 11990522 (24d)


F

F

Bolt

A

B

C

A

BC

Plate 2

Plate 1

NA

t1

t2

120575bolt

120575sA

120575sB


120575sB minus 120575sA=



120575119904 = 120575119904119861 minus 120575119904119860 = 38 ( 1199051 + 1199052120581119860119887119866119887)12059621199052 (25)


120575119904 = 38 ( 1199051 + 1199052120581119860119887119866119887)12059621199052 (26)


119862sd119887 = 1119896sd119887 =120575119904119865119887 =

38(1199051 + 1199052)120581119860119887119866119887 (27)


120590bea = 119875119889119905 (28a)

120576bea = 120575bea119901119889 (28b)


120590bea119901 = 119864119901120576bea119901 (29a)

119896bea119901 = 119875120575bea119901 = 119905119889119864119901119889 = 119905119864119901 (29b)


120590bea119887 = 119864119887120576bea119887 (30a)

119896bea119887 = 119875120575bea119887 =119905119889119864119887119889 = 119905119864119887 (30b)


119888bea119887 = 1119896bea119887 =11199051119864119887 +

11199052119864119887 (31)

119888bea119901 = 1119896bea119901 = 111990511198641199011 +111990521198641199012 (32)


1119896bp = 311990521 + 16119905221199052 + 16119905111990522 + 311990532128119864119887119868119887 + 38(1199051 + 1199052)120581119860119887119866119887

+ ( 11199051119864119887 +11199052119864119887) + ( 111990511198641199011 +

111990521198641199012) (33)



001198653

= [[[[


]]]]119906111990621199063

(34)



Plate

Bolt hole

x

y

Compression zone

d

d

A

Deformation shape

120590beap

A998400

120575beap


Bolt

x

zt

Compression zone

d

d

B

Deformation shape

120590beab

120590beab

120575beab

B998400



F

Plate 2Plate 1

x

z

l1

l2


1

23F3

u3u2

kp1

kbp

u1 kp2

(b) Spring model

Plate 1

Plate 2

x

y(plate zone)

(Bolt zone)

(plate zone)

w1

l1 minus 05dℎdℎ

w2

l2 minus 05dℎ

k1-1

k1-2

k2-2k2-1

(c) Plate stiffness





1119896bp = (1199052 + 1199051)2119864119887119889 + 372( 111990511198641199011 +111990521198641199012) (35)




11198961-2 =1120596111990511198641199011 (1198971 minus 05119889ℎ) + 1

((1205961 minus 119889) 11990511198641199011) 05119889ℎ 163692

1119896119901211198962-1 +

11198962-2 =1(1205962 minus 119889) 1199052119864119901205119889ℎ +

1120596211990521198641199012 (1198972 minus 05119889ℎ) 163692

1119896bp Equation (33) 150660


0

1500

3000

4500

6000

7500

9000

10500

12000

0 1 2 3 4 5 6 7 8 9 10 11 12

Load

(N)


S-1S-2S-3S-4



0

1500

3000

4500

6000

7500

0 01 02 03 04 05 06 07 08 09 1

Load

(N)


S-1S-2S-3S-4S-5





1119896bp = (1199052 + 11990512119889 )23

sdot 3 ( 111990511198641199011 +111990521198641199012 +

121199051119864119887 +121199052119864119887)

(36)



+ 086( 1(radic119864119871 + 1198641198791199051)plate-1

+ 1(radic119864119871 + 1198641198791199052)plate-2)

(37)






x

y

Unit mm30

Plate 2Plate 1


Bolt diameter = 5 x

z

F = 5000 N t = 1

t = 1

l1 = 100 l2 = 3d l3 = 3d l4 = 100


k8

k9 k7

k6

k5

k4

k3

k2

k1

u1

u2

u3

u4

u5

u6u7 67F

5

4

3

2

1


Plate 1Plate 2


(Bolt zone)





l1

la lb lc

l2 l3 l4

dℎ1 dℎ2 dℎ3

k92k91

k73 k72

k33

k42

k71 k41

k43

k63 k62k61

y k32

k31

k12

k11

ld

w2

w1

x

(b) Plate stiffness




111989611 +111989612 11989611 = 120596211990521198641199012119897119889 11989612 = (1205962 minus 119889ℎ3)1199052119864119901205119889ℎ3

11198963 =111989631 +

111989632 +111989633 11989631 = 120596211990521198641199012119897119888 11989632 = (1205962 minus 119889ℎ3)1199052119864119901205119889ℎ3 11989633 = (1205962 minus 119889ℎ2)1199052119864119901205119889ℎ2

11198964 =111989641 +

111989642 +111989643 11989641 = 120596111990511198641199011119897119888 11989642 = (1205961 minus 119889ℎ3)1199051119864119901105119889ℎ3 11989643 = (1205961 minus 119889ℎ2)1199051119864119901105119889ℎ2

11198966 =111989661 +

111989662 +111989663 11989661 = 120596211990521198641199012119897119887 11989662 = (1205962 minus 119889ℎ2)1199052119864119901205119889ℎ2 11989663 = (1205962 minus 119889ℎ1)1199052119864119901205119889ℎ1

11198967 =111989671 +

111989672 +111989673 11989671 = 120596111990511198641199011119897119887 11989672 = (1205961 minus 119889ℎ2)1199051119864119901105119889ℎ2 11989673 = (1205961 minus 119889ℎ1)1199051119864119901105119889ℎ1

11198969 =111989691 +

111989692 11989691 = 120596111990511198641199011119897119886 11989692 = (1205961 minus 119889ℎ1)1199051119864119901105119889ℎ111198962 11198965

11198968 Equation (33)



xz

yA

B

dx = 0

dy = 0

dz = 0

dx = freedy = 0

dz = 0




0

1000

2000

3000

4000

5000

Load

(N)

(b) Connection load

3472 36213056 2928

3472 3449

Bolt 1(spring)

Bolt 1(FEA)

Bolt 2(spring)

Bolt 2(FEA)

Bolt 3(spring)

Bolt 3(FEA)

05

10152025303540

Load

dist

ribut

ion

()



000000119865

=

[[[[[[[[[[[[[[[


]]]]]]]]]]]]]]]

1199061119906211990631199064119906511990661199067

(38)



(39)






000

050

100

150

200

000 050 100 150 200

Reference case


m = dydx

120595120595

ref




120595 = 119899119875minsum119899119894=1 119875119894 for 119894 = 1 2 3 119899 (40)











8 Conclusion




Bolt 1Bolt 2Bolt 3

05

101520253035404550

Load

dist

ribut

ion

()


Reference case

y = 00605x + 09326

050 100 150 200000ddref

090

094

098

120595120595

ref

102

106

110




Bolt 1Bolt 2Bolt 3

05

101520253035404550

Load

dist

ribut

ion

()


Reference case

y = 00136x + 09884

090

094

098

102

120595120595

ref

106

110

050 100 150 200000ttref



05

101520253035404550

Load

dist

ribut

ion

()

1 2 3 4 5t2t1

Bolt 1Bolt 2Bolt 3


Reference case

y = minus00646x + 10642

050 100 150 200000(t2t1)(t2t1)ref

090

094

120595120595

ref

098

102

106

110




Plate 2


F

t2 ≫ tt










Acknowledgments


References





















RoboticsJournal of





Journal of



RotatingMachinery



Journal of

Volume 201


VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










a

b

L

x

La b

La b

x x

L

x


a

x

b

V

dx

x

x

x

y

y y y

120596

120596 120596

MB

0 le x le a


1205741(x) = minus120596(x minus a)

120581AG0 le x le b

V2(x) = 0

1205742(x) = 0

0 le x le a


1205743(x) = minus120596b

120581AG

0 le x le b


1205744(x) = minus120596(x minus b)

120581AG

V(x) = 0

120574(x) = 0

120574

120575s(x) = 0

120574 =d120575sdx

d120575s

120575s

120596b

1205751(x) = intx

01205741(x)dx

1205752(x) = inta

01205741(x)dx + intx

01205742(x)dx

1205754(x) = inta

01205743(x)dx + intx

01205744(x)dx

1205753(x) = intx

01205743(x)dx


A

B

=C

Fixed support

Guiding support


+ +


1205962t2 1205962t2

1205961t11205961t1

MB

1206011 1206012 1206013



119865119887 = 12059611199051 = 12059621199052 (6)


1205961 = 1205962 (11990521199051) (7)


minus 1205961119905316119864119887119868119887 +1205962 (1198713 minus 11990531)6119864119887119868119887 + 119872119861119871119864119887119868119887 = 0 (8)





M

M

A

C

B1205962t2

1205961t1

kc

kb

kc

kb

Fixed support

Guiding support




1119896bp = 1119896119887 +1119896119888 (10a)

with

1119896119887 =1119896flex119887 +

1119896she119887 1119896119888 =

1119896bea119901 +1119896bea119887

(10b)


119906 (119909) = int1199090120579 (119904) 119889119904 + 1199060

= int1199090(int1199090

119872(119904)119864119868 119889119904 + 1205790)119889119904 + 1199060(11)



120575flex119860 = 120575flex11986012059611199051

+ 120575flex11986012059621199052

+ 120575flex119860119872 (12)

where 120575flex11986012059611199051

120575flex11986012059621199052


120575flex11986012059611199051

= minus 12059611199053124119864119887119868119887 (61198862 minus 4119886119909 + 1199092) (13a)

120575flex11986012059621199052

= 1205962119887119909212119864119887119868119887 (3119871 + 3119886 minus 2119909) (13b)

120575flex119860119872 = 119872(119909)22119864119887119868119887 (13c)

119886 = 1199051119887 = 1199052119871 = 1199051 + 1199052119909 = 11990512

(13d)


120575flex119861 = 120575flex11986112059611199051

+ 120575flex11986112059621199052

+ 120575flex119861119872 (14)

where 120575flex11986112059611199051

120575flex11986112059621199052


120575flex11986112059611199051

= minus 1205961119886324119864119887119868119887 (4119909 minus 119886) (15a)

120575flex11986112059621199052

= 120596224119864119887119868119887 (1199094 minus 41198711199093 + 611987121199092 minus 41198863119909 + 1198864) (15b)

120575flex119861119872 = 119872(119909)22119864119887119868119887 (15c)

119909 = 1199051 + 11990522 (15d)


F

F

Bolt

Plate 2

Plate 1A

B

C

A

BC

NA

t1

t2

120575flexA

120575flexB

120575flexC

120575bolt





(1199051 + 1199052) (311990521 + 131199051199051199052 + 311990522)119864119887119868119887 )12059621199052 (16)


120575flex119887 = (311990521 + 16119905221199052 + 16119905111990522 + 311990532128119864119887119868119887 )119865119887 (17)


119862flex119887 = 1119896flex119887 =120575flex119887119865119887

= 311990521 + 16119905221199052 + 16119905111990522 + 311990532128119864119887119868119887 (18)


119906 (119909) = minusint1199090

119881 (119904)120581119860119866119889119904 + 1199060 (19)


120581 = 6 (1 + ])27 + 12] + 4]2 (20)




120575119904119860 = 12057511990411986012059611199051

+ 12057511990411986012059621199052

+ 120575119904119860119872 (21)

where 12057511990411986012059611199051

12057511990411986012059621199052


12057511990411986012059611199051

= int11990901205741 (119909) 119889119909 = minus 31205961119905218120581119866119887119860119887 (22a)

12057511990411986012059621199052

= int11990901205743 (119909) 119889119909 = 1205962119905111990522120581119866119887119860119887 (22b)

120575119904119860119872 = 0 (22c)

119909 = 11990512 (22d)


120575119904119861 = 12057511990411986112059611199051

+ 12057511990411986112059621199052

+ 120575119904119861119872 (23)

where 12057511990411986112059611199051

12057511990411986112059621199052


12057511990411986112059611199051

= int11988601205741 (119909) 119889119909 + int119909

01205742 (119909) 119889119909 = minus 1205961119905212120581119866119887119860119887 (24a)

12057511990411986112059621199052

= int11988601205743 (119909) 119889119909 + int119909

01205744 (119909) 119889119909

= 12059621199052 (81199051 + 31199052)8119866119887119860119887 (24b)

120575119904119861119872 = 0 (24c)

119909 = 11990522 (24d)


F

F

Bolt

A

B

C

A

BC

Plate 2

Plate 1

NA

t1

t2

120575bolt

120575sA

120575sB


120575sB minus 120575sA=



120575119904 = 120575119904119861 minus 120575119904119860 = 38 ( 1199051 + 1199052120581119860119887119866119887)12059621199052 (25)


120575119904 = 38 ( 1199051 + 1199052120581119860119887119866119887)12059621199052 (26)


119862sd119887 = 1119896sd119887 =120575119904119865119887 =

38(1199051 + 1199052)120581119860119887119866119887 (27)


120590bea = 119875119889119905 (28a)

120576bea = 120575bea119901119889 (28b)


120590bea119901 = 119864119901120576bea119901 (29a)

119896bea119901 = 119875120575bea119901 = 119905119889119864119901119889 = 119905119864119901 (29b)


120590bea119887 = 119864119887120576bea119887 (30a)

119896bea119887 = 119875120575bea119887 =119905119889119864119887119889 = 119905119864119887 (30b)


119888bea119887 = 1119896bea119887 =11199051119864119887 +

11199052119864119887 (31)

119888bea119901 = 1119896bea119901 = 111990511198641199011 +111990521198641199012 (32)


1119896bp = 311990521 + 16119905221199052 + 16119905111990522 + 311990532128119864119887119868119887 + 38(1199051 + 1199052)120581119860119887119866119887

+ ( 11199051119864119887 +11199052119864119887) + ( 111990511198641199011 +

111990521198641199012) (33)



001198653

= [[[[


]]]]119906111990621199063

(34)



Plate

Bolt hole

x

y

Compression zone

d

d

A

Deformation shape

120590beap

A998400

120575beap


Bolt

x

zt

Compression zone

d

d

B

Deformation shape

120590beab

120590beab

120575beab

B998400



F

Plate 2Plate 1

x

z

l1

l2


1

23F3

u3u2

kp1

kbp

u1 kp2

(b) Spring model

Plate 1

Plate 2

x

y(plate zone)

(Bolt zone)

(plate zone)

w1

l1 minus 05dℎdℎ

w2

l2 minus 05dℎ

k1-1

k1-2

k2-2k2-1

(c) Plate stiffness





1119896bp = (1199052 + 1199051)2119864119887119889 + 372( 111990511198641199011 +111990521198641199012) (35)




11198961-2 =1120596111990511198641199011 (1198971 minus 05119889ℎ) + 1

((1205961 minus 119889) 11990511198641199011) 05119889ℎ 163692

1119896119901211198962-1 +

11198962-2 =1(1205962 minus 119889) 1199052119864119901205119889ℎ +

1120596211990521198641199012 (1198972 minus 05119889ℎ) 163692

1119896bp Equation (33) 150660


0

1500

3000

4500

6000

7500

9000

10500

12000

0 1 2 3 4 5 6 7 8 9 10 11 12

Load

(N)


S-1S-2S-3S-4



0

1500

3000

4500

6000

7500

0 01 02 03 04 05 06 07 08 09 1

Load

(N)


S-1S-2S-3S-4S-5





1119896bp = (1199052 + 11990512119889 )23

sdot 3 ( 111990511198641199011 +111990521198641199012 +

121199051119864119887 +121199052119864119887)

(36)



+ 086( 1(radic119864119871 + 1198641198791199051)plate-1

+ 1(radic119864119871 + 1198641198791199052)plate-2)

(37)






x

y

Unit mm30

Plate 2Plate 1


Bolt diameter = 5 x

z

F = 5000 N t = 1

t = 1

l1 = 100 l2 = 3d l3 = 3d l4 = 100


k8

k9 k7

k6

k5

k4

k3

k2

k1

u1

u2

u3

u4

u5

u6u7 67F

5

4

3

2

1


Plate 1Plate 2


(Bolt zone)





l1

la lb lc

l2 l3 l4

dℎ1 dℎ2 dℎ3

k92k91

k73 k72

k33

k42

k71 k41

k43

k63 k62k61

y k32

k31

k12

k11

ld

w2

w1

x

(b) Plate stiffness




111989611 +111989612 11989611 = 120596211990521198641199012119897119889 11989612 = (1205962 minus 119889ℎ3)1199052119864119901205119889ℎ3

11198963 =111989631 +

111989632 +111989633 11989631 = 120596211990521198641199012119897119888 11989632 = (1205962 minus 119889ℎ3)1199052119864119901205119889ℎ3 11989633 = (1205962 minus 119889ℎ2)1199052119864119901205119889ℎ2

11198964 =111989641 +

111989642 +111989643 11989641 = 120596111990511198641199011119897119888 11989642 = (1205961 minus 119889ℎ3)1199051119864119901105119889ℎ3 11989643 = (1205961 minus 119889ℎ2)1199051119864119901105119889ℎ2

11198966 =111989661 +

111989662 +111989663 11989661 = 120596211990521198641199012119897119887 11989662 = (1205962 minus 119889ℎ2)1199052119864119901205119889ℎ2 11989663 = (1205962 minus 119889ℎ1)1199052119864119901205119889ℎ1

11198967 =111989671 +

111989672 +111989673 11989671 = 120596111990511198641199011119897119887 11989672 = (1205961 minus 119889ℎ2)1199051119864119901105119889ℎ2 11989673 = (1205961 minus 119889ℎ1)1199051119864119901105119889ℎ1

11198969 =111989691 +

111989692 11989691 = 120596111990511198641199011119897119886 11989692 = (1205961 minus 119889ℎ1)1199051119864119901105119889ℎ111198962 11198965

11198968 Equation (33)



xz

yA

B

dx = 0

dy = 0

dz = 0

dx = freedy = 0

dz = 0




0

1000

2000

3000

4000

5000

Load

(N)

(b) Connection load

3472 36213056 2928

3472 3449

Bolt 1(spring)

Bolt 1(FEA)

Bolt 2(spring)

Bolt 2(FEA)

Bolt 3(spring)

Bolt 3(FEA)

05

10152025303540

Load

dist

ribut

ion

()



000000119865

=

[[[[[[[[[[[[[[[


]]]]]]]]]]]]]]]

1199061119906211990631199064119906511990661199067

(38)



(39)






000

050

100

150

200

000 050 100 150 200

Reference case


m = dydx

120595120595

ref




120595 = 119899119875minsum119899119894=1 119875119894 for 119894 = 1 2 3 119899 (40)











8 Conclusion




Bolt 1Bolt 2Bolt 3

05

101520253035404550

Load

dist

ribut

ion

()


Reference case

y = 00605x + 09326

050 100 150 200000ddref

090

094

098

120595120595

ref

102

106

110




Bolt 1Bolt 2Bolt 3

05

101520253035404550

Load

dist

ribut

ion

()


Reference case

y = 00136x + 09884

090

094

098

102

120595120595

ref

106

110

050 100 150 200000ttref



05

101520253035404550

Load

dist

ribut

ion

()

1 2 3 4 5t2t1

Bolt 1Bolt 2Bolt 3


Reference case

y = minus00646x + 10642

050 100 150 200000(t2t1)(t2t1)ref

090

094

120595120595

ref

098

102

106

110




Plate 2


F

t2 ≫ tt










Acknowledgments


References





















RoboticsJournal of





Journal of



RotatingMachinery



Journal of

Volume 201


VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










M

M

A

C

B1205962t2

1205961t1

kc

kb

kc

kb

Fixed support

Guiding support




1119896bp = 1119896119887 +1119896119888 (10a)

with

1119896119887 =1119896flex119887 +

1119896she119887 1119896119888 =

1119896bea119901 +1119896bea119887

(10b)


119906 (119909) = int1199090120579 (119904) 119889119904 + 1199060

= int1199090(int1199090

119872(119904)119864119868 119889119904 + 1205790)119889119904 + 1199060(11)



120575flex119860 = 120575flex11986012059611199051

+ 120575flex11986012059621199052

+ 120575flex119860119872 (12)

where 120575flex11986012059611199051

120575flex11986012059621199052


120575flex11986012059611199051

= minus 12059611199053124119864119887119868119887 (61198862 minus 4119886119909 + 1199092) (13a)

120575flex11986012059621199052

= 1205962119887119909212119864119887119868119887 (3119871 + 3119886 minus 2119909) (13b)

120575flex119860119872 = 119872(119909)22119864119887119868119887 (13c)

119886 = 1199051119887 = 1199052119871 = 1199051 + 1199052119909 = 11990512

(13d)


120575flex119861 = 120575flex11986112059611199051

+ 120575flex11986112059621199052

+ 120575flex119861119872 (14)

where 120575flex11986112059611199051

120575flex11986112059621199052


120575flex11986112059611199051

= minus 1205961119886324119864119887119868119887 (4119909 minus 119886) (15a)

120575flex11986112059621199052

= 120596224119864119887119868119887 (1199094 minus 41198711199093 + 611987121199092 minus 41198863119909 + 1198864) (15b)

120575flex119861119872 = 119872(119909)22119864119887119868119887 (15c)

119909 = 1199051 + 11990522 (15d)


F

F

Bolt

Plate 2

Plate 1A

B

C

A

BC

NA

t1

t2

120575flexA

120575flexB

120575flexC

120575bolt





(1199051 + 1199052) (311990521 + 131199051199051199052 + 311990522)119864119887119868119887 )12059621199052 (16)


120575flex119887 = (311990521 + 16119905221199052 + 16119905111990522 + 311990532128119864119887119868119887 )119865119887 (17)


119862flex119887 = 1119896flex119887 =120575flex119887119865119887

= 311990521 + 16119905221199052 + 16119905111990522 + 311990532128119864119887119868119887 (18)


119906 (119909) = minusint1199090

119881 (119904)120581119860119866119889119904 + 1199060 (19)


120581 = 6 (1 + ])27 + 12] + 4]2 (20)




120575119904119860 = 12057511990411986012059611199051

+ 12057511990411986012059621199052

+ 120575119904119860119872 (21)

where 12057511990411986012059611199051

12057511990411986012059621199052


12057511990411986012059611199051

= int11990901205741 (119909) 119889119909 = minus 31205961119905218120581119866119887119860119887 (22a)

12057511990411986012059621199052

= int11990901205743 (119909) 119889119909 = 1205962119905111990522120581119866119887119860119887 (22b)

120575119904119860119872 = 0 (22c)

119909 = 11990512 (22d)


120575119904119861 = 12057511990411986112059611199051

+ 12057511990411986112059621199052

+ 120575119904119861119872 (23)

where 12057511990411986112059611199051

12057511990411986112059621199052


12057511990411986112059611199051

= int11988601205741 (119909) 119889119909 + int119909

01205742 (119909) 119889119909 = minus 1205961119905212120581119866119887119860119887 (24a)

12057511990411986112059621199052

= int11988601205743 (119909) 119889119909 + int119909

01205744 (119909) 119889119909

= 12059621199052 (81199051 + 31199052)8119866119887119860119887 (24b)

120575119904119861119872 = 0 (24c)

119909 = 11990522 (24d)


F

F

Bolt

A

B

C

A

BC

Plate 2

Plate 1

NA

t1

t2

120575bolt

120575sA

120575sB


120575sB minus 120575sA=



120575119904 = 120575119904119861 minus 120575119904119860 = 38 ( 1199051 + 1199052120581119860119887119866119887)12059621199052 (25)


120575119904 = 38 ( 1199051 + 1199052120581119860119887119866119887)12059621199052 (26)


119862sd119887 = 1119896sd119887 =120575119904119865119887 =

38(1199051 + 1199052)120581119860119887119866119887 (27)


120590bea = 119875119889119905 (28a)

120576bea = 120575bea119901119889 (28b)


120590bea119901 = 119864119901120576bea119901 (29a)

119896bea119901 = 119875120575bea119901 = 119905119889119864119901119889 = 119905119864119901 (29b)


120590bea119887 = 119864119887120576bea119887 (30a)

119896bea119887 = 119875120575bea119887 =119905119889119864119887119889 = 119905119864119887 (30b)


119888bea119887 = 1119896bea119887 =11199051119864119887 +

11199052119864119887 (31)

119888bea119901 = 1119896bea119901 = 111990511198641199011 +111990521198641199012 (32)


1119896bp = 311990521 + 16119905221199052 + 16119905111990522 + 311990532128119864119887119868119887 + 38(1199051 + 1199052)120581119860119887119866119887

+ ( 11199051119864119887 +11199052119864119887) + ( 111990511198641199011 +

111990521198641199012) (33)



001198653

= [[[[


]]]]119906111990621199063

(34)



Plate

Bolt hole

x

y

Compression zone

d

d

A

Deformation shape

120590beap

A998400

120575beap


Bolt

x

zt

Compression zone

d

d

B

Deformation shape

120590beab

120590beab

120575beab

B998400



F

Plate 2Plate 1

x

z

l1

l2


1

23F3

u3u2

kp1

kbp

u1 kp2

(b) Spring model

Plate 1

Plate 2

x

y(plate zone)

(Bolt zone)

(plate zone)

w1

l1 minus 05dℎdℎ

w2

l2 minus 05dℎ

k1-1

k1-2

k2-2k2-1

(c) Plate stiffness





1119896bp = (1199052 + 1199051)2119864119887119889 + 372( 111990511198641199011 +111990521198641199012) (35)




11198961-2 =1120596111990511198641199011 (1198971 minus 05119889ℎ) + 1

((1205961 minus 119889) 11990511198641199011) 05119889ℎ 163692

1119896119901211198962-1 +

11198962-2 =1(1205962 minus 119889) 1199052119864119901205119889ℎ +

1120596211990521198641199012 (1198972 minus 05119889ℎ) 163692

1119896bp Equation (33) 150660


0

1500

3000

4500

6000

7500

9000

10500

12000

0 1 2 3 4 5 6 7 8 9 10 11 12

Load

(N)


S-1S-2S-3S-4



0

1500

3000

4500

6000

7500

0 01 02 03 04 05 06 07 08 09 1

Load

(N)


S-1S-2S-3S-4S-5





1119896bp = (1199052 + 11990512119889 )23

sdot 3 ( 111990511198641199011 +111990521198641199012 +

121199051119864119887 +121199052119864119887)

(36)



+ 086( 1(radic119864119871 + 1198641198791199051)plate-1

+ 1(radic119864119871 + 1198641198791199052)plate-2)

(37)






x

y

Unit mm30

Plate 2Plate 1


Bolt diameter = 5 x

z

F = 5000 N t = 1

t = 1

l1 = 100 l2 = 3d l3 = 3d l4 = 100


k8

k9 k7

k6

k5

k4

k3

k2

k1

u1

u2

u3

u4

u5

u6u7 67F

5

4

3

2

1


Plate 1Plate 2


(Bolt zone)





l1

la lb lc

l2 l3 l4

dℎ1 dℎ2 dℎ3

k92k91

k73 k72

k33

k42

k71 k41

k43

k63 k62k61

y k32

k31

k12

k11

ld

w2

w1

x

(b) Plate stiffness




111989611 +111989612 11989611 = 120596211990521198641199012119897119889 11989612 = (1205962 minus 119889ℎ3)1199052119864119901205119889ℎ3

11198963 =111989631 +

111989632 +111989633 11989631 = 120596211990521198641199012119897119888 11989632 = (1205962 minus 119889ℎ3)1199052119864119901205119889ℎ3 11989633 = (1205962 minus 119889ℎ2)1199052119864119901205119889ℎ2

11198964 =111989641 +

111989642 +111989643 11989641 = 120596111990511198641199011119897119888 11989642 = (1205961 minus 119889ℎ3)1199051119864119901105119889ℎ3 11989643 = (1205961 minus 119889ℎ2)1199051119864119901105119889ℎ2

11198966 =111989661 +

111989662 +111989663 11989661 = 120596211990521198641199012119897119887 11989662 = (1205962 minus 119889ℎ2)1199052119864119901205119889ℎ2 11989663 = (1205962 minus 119889ℎ1)1199052119864119901205119889ℎ1

11198967 =111989671 +

111989672 +111989673 11989671 = 120596111990511198641199011119897119887 11989672 = (1205961 minus 119889ℎ2)1199051119864119901105119889ℎ2 11989673 = (1205961 minus 119889ℎ1)1199051119864119901105119889ℎ1

11198969 =111989691 +

111989692 11989691 = 120596111990511198641199011119897119886 11989692 = (1205961 minus 119889ℎ1)1199051119864119901105119889ℎ111198962 11198965

11198968 Equation (33)



xz

yA

B

dx = 0

dy = 0

dz = 0

dx = freedy = 0

dz = 0




0

1000

2000

3000

4000

5000

Load

(N)

(b) Connection load

3472 36213056 2928

3472 3449

Bolt 1(spring)

Bolt 1(FEA)

Bolt 2(spring)

Bolt 2(FEA)

Bolt 3(spring)

Bolt 3(FEA)

05

10152025303540

Load

dist

ribut

ion

()



000000119865

=

[[[[[[[[[[[[[[[


]]]]]]]]]]]]]]]

1199061119906211990631199064119906511990661199067

(38)



(39)






000

050

100

150

200

000 050 100 150 200

Reference case


m = dydx

120595120595

ref




120595 = 119899119875minsum119899119894=1 119875119894 for 119894 = 1 2 3 119899 (40)











8 Conclusion




Bolt 1Bolt 2Bolt 3

05

101520253035404550

Load

dist

ribut

ion

()


Reference case

y = 00605x + 09326

050 100 150 200000ddref

090

094

098

120595120595

ref

102

106

110




Bolt 1Bolt 2Bolt 3

05

101520253035404550

Load

dist

ribut

ion

()


Reference case

y = 00136x + 09884

090

094

098

102

120595120595

ref

106

110

050 100 150 200000ttref



05

101520253035404550

Load

dist

ribut

ion

()

1 2 3 4 5t2t1

Bolt 1Bolt 2Bolt 3


Reference case

y = minus00646x + 10642

050 100 150 200000(t2t1)(t2t1)ref

090

094

120595120595

ref

098

102

106

110




Plate 2


F

t2 ≫ tt










Acknowledgments


References





















RoboticsJournal of





Journal of



RotatingMachinery



Journal of

Volume 201


VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










F

F

Bolt

Plate 2

Plate 1A

B

C

A

BC

NA

t1

t2

120575flexA

120575flexB

120575flexC

120575bolt





(1199051 + 1199052) (311990521 + 131199051199051199052 + 311990522)119864119887119868119887 )12059621199052 (16)


120575flex119887 = (311990521 + 16119905221199052 + 16119905111990522 + 311990532128119864119887119868119887 )119865119887 (17)


119862flex119887 = 1119896flex119887 =120575flex119887119865119887

= 311990521 + 16119905221199052 + 16119905111990522 + 311990532128119864119887119868119887 (18)


119906 (119909) = minusint1199090

119881 (119904)120581119860119866119889119904 + 1199060 (19)


120581 = 6 (1 + ])27 + 12] + 4]2 (20)




120575119904119860 = 12057511990411986012059611199051

+ 12057511990411986012059621199052

+ 120575119904119860119872 (21)

where 12057511990411986012059611199051

12057511990411986012059621199052


12057511990411986012059611199051

= int11990901205741 (119909) 119889119909 = minus 31205961119905218120581119866119887119860119887 (22a)

12057511990411986012059621199052

= int11990901205743 (119909) 119889119909 = 1205962119905111990522120581119866119887119860119887 (22b)

120575119904119860119872 = 0 (22c)

119909 = 11990512 (22d)


120575119904119861 = 12057511990411986112059611199051

+ 12057511990411986112059621199052

+ 120575119904119861119872 (23)

where 12057511990411986112059611199051

12057511990411986112059621199052


12057511990411986112059611199051

= int11988601205741 (119909) 119889119909 + int119909

01205742 (119909) 119889119909 = minus 1205961119905212120581119866119887119860119887 (24a)

12057511990411986112059621199052

= int11988601205743 (119909) 119889119909 + int119909

01205744 (119909) 119889119909

= 12059621199052 (81199051 + 31199052)8119866119887119860119887 (24b)

120575119904119861119872 = 0 (24c)

119909 = 11990522 (24d)


F

F

Bolt

A

B

C

A

BC

Plate 2

Plate 1

NA

t1

t2

120575bolt

120575sA

120575sB


120575sB minus 120575sA=



120575119904 = 120575119904119861 minus 120575119904119860 = 38 ( 1199051 + 1199052120581119860119887119866119887)12059621199052 (25)


120575119904 = 38 ( 1199051 + 1199052120581119860119887119866119887)12059621199052 (26)


119862sd119887 = 1119896sd119887 =120575119904119865119887 =

38(1199051 + 1199052)120581119860119887119866119887 (27)


120590bea = 119875119889119905 (28a)

120576bea = 120575bea119901119889 (28b)


120590bea119901 = 119864119901120576bea119901 (29a)

119896bea119901 = 119875120575bea119901 = 119905119889119864119901119889 = 119905119864119901 (29b)


120590bea119887 = 119864119887120576bea119887 (30a)

119896bea119887 = 119875120575bea119887 =119905119889119864119887119889 = 119905119864119887 (30b)


119888bea119887 = 1119896bea119887 =11199051119864119887 +

11199052119864119887 (31)

119888bea119901 = 1119896bea119901 = 111990511198641199011 +111990521198641199012 (32)


1119896bp = 311990521 + 16119905221199052 + 16119905111990522 + 311990532128119864119887119868119887 + 38(1199051 + 1199052)120581119860119887119866119887

+ ( 11199051119864119887 +11199052119864119887) + ( 111990511198641199011 +

111990521198641199012) (33)



001198653

= [[[[


]]]]119906111990621199063

(34)



Plate

Bolt hole

x

y

Compression zone

d

d

A

Deformation shape

120590beap

A998400

120575beap


Bolt

x

zt

Compression zone

d

d

B

Deformation shape

120590beab

120590beab

120575beab

B998400



F

Plate 2Plate 1

x

z

l1

l2


1

23F3

u3u2

kp1

kbp

u1 kp2

(b) Spring model

Plate 1

Plate 2

x

y(plate zone)

(Bolt zone)

(plate zone)

w1

l1 minus 05dℎdℎ

w2

l2 minus 05dℎ

k1-1

k1-2

k2-2k2-1

(c) Plate stiffness





1119896bp = (1199052 + 1199051)2119864119887119889 + 372( 111990511198641199011 +111990521198641199012) (35)




11198961-2 =1120596111990511198641199011 (1198971 minus 05119889ℎ) + 1

((1205961 minus 119889) 11990511198641199011) 05119889ℎ 163692

1119896119901211198962-1 +

11198962-2 =1(1205962 minus 119889) 1199052119864119901205119889ℎ +

1120596211990521198641199012 (1198972 minus 05119889ℎ) 163692

1119896bp Equation (33) 150660


0

1500

3000

4500

6000

7500

9000

10500

12000

0 1 2 3 4 5 6 7 8 9 10 11 12

Load

(N)


S-1S-2S-3S-4



0

1500

3000

4500

6000

7500

0 01 02 03 04 05 06 07 08 09 1

Load

(N)


S-1S-2S-3S-4S-5





1119896bp = (1199052 + 11990512119889 )23

sdot 3 ( 111990511198641199011 +111990521198641199012 +

121199051119864119887 +121199052119864119887)

(36)



+ 086( 1(radic119864119871 + 1198641198791199051)plate-1

+ 1(radic119864119871 + 1198641198791199052)plate-2)

(37)






x

y

Unit mm30

Plate 2Plate 1


Bolt diameter = 5 x

z

F = 5000 N t = 1

t = 1

l1 = 100 l2 = 3d l3 = 3d l4 = 100


k8

k9 k7

k6

k5

k4

k3

k2

k1

u1

u2

u3

u4

u5

u6u7 67F

5

4

3

2

1


Plate 1Plate 2


(Bolt zone)





l1

la lb lc

l2 l3 l4

dℎ1 dℎ2 dℎ3

k92k91

k73 k72

k33

k42

k71 k41

k43

k63 k62k61

y k32

k31

k12

k11

ld

w2

w1

x

(b) Plate stiffness




111989611 +111989612 11989611 = 120596211990521198641199012119897119889 11989612 = (1205962 minus 119889ℎ3)1199052119864119901205119889ℎ3

11198963 =111989631 +

111989632 +111989633 11989631 = 120596211990521198641199012119897119888 11989632 = (1205962 minus 119889ℎ3)1199052119864119901205119889ℎ3 11989633 = (1205962 minus 119889ℎ2)1199052119864119901205119889ℎ2

11198964 =111989641 +

111989642 +111989643 11989641 = 120596111990511198641199011119897119888 11989642 = (1205961 minus 119889ℎ3)1199051119864119901105119889ℎ3 11989643 = (1205961 minus 119889ℎ2)1199051119864119901105119889ℎ2

11198966 =111989661 +

111989662 +111989663 11989661 = 120596211990521198641199012119897119887 11989662 = (1205962 minus 119889ℎ2)1199052119864119901205119889ℎ2 11989663 = (1205962 minus 119889ℎ1)1199052119864119901205119889ℎ1

11198967 =111989671 +

111989672 +111989673 11989671 = 120596111990511198641199011119897119887 11989672 = (1205961 minus 119889ℎ2)1199051119864119901105119889ℎ2 11989673 = (1205961 minus 119889ℎ1)1199051119864119901105119889ℎ1

11198969 =111989691 +

111989692 11989691 = 120596111990511198641199011119897119886 11989692 = (1205961 minus 119889ℎ1)1199051119864119901105119889ℎ111198962 11198965

11198968 Equation (33)



xz

yA

B

dx = 0

dy = 0

dz = 0

dx = freedy = 0

dz = 0




0

1000

2000

3000

4000

5000

Load

(N)

(b) Connection load

3472 36213056 2928

3472 3449

Bolt 1(spring)

Bolt 1(FEA)

Bolt 2(spring)

Bolt 2(FEA)

Bolt 3(spring)

Bolt 3(FEA)

05

10152025303540

Load

dist

ribut

ion

()



000000119865

=

[[[[[[[[[[[[[[[


]]]]]]]]]]]]]]]

1199061119906211990631199064119906511990661199067

(38)



(39)






000

050

100

150

200

000 050 100 150 200

Reference case


m = dydx

120595120595

ref




120595 = 119899119875minsum119899119894=1 119875119894 for 119894 = 1 2 3 119899 (40)











8 Conclusion




Bolt 1Bolt 2Bolt 3

05

101520253035404550

Load

dist

ribut

ion

()


Reference case

y = 00605x + 09326

050 100 150 200000ddref

090

094

098

120595120595

ref

102

106

110




Bolt 1Bolt 2Bolt 3

05

101520253035404550

Load

dist

ribut

ion

()


Reference case

y = 00136x + 09884

090

094

098

102

120595120595

ref

106

110

050 100 150 200000ttref



05

101520253035404550

Load

dist

ribut

ion

()

1 2 3 4 5t2t1

Bolt 1Bolt 2Bolt 3


Reference case

y = minus00646x + 10642

050 100 150 200000(t2t1)(t2t1)ref

090

094

120595120595

ref

098

102

106

110




Plate 2


F

t2 ≫ tt










Acknowledgments


References





















RoboticsJournal of





Journal of



RotatingMachinery



Journal of

Volume 201


VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










F

F

Bolt

A

B

C

A

BC

Plate 2

Plate 1

NA

t1

t2

120575bolt

120575sA

120575sB


120575sB minus 120575sA=



120575119904 = 120575119904119861 minus 120575119904119860 = 38 ( 1199051 + 1199052120581119860119887119866119887)12059621199052 (25)


120575119904 = 38 ( 1199051 + 1199052120581119860119887119866119887)12059621199052 (26)


119862sd119887 = 1119896sd119887 =120575119904119865119887 =

38(1199051 + 1199052)120581119860119887119866119887 (27)


120590bea = 119875119889119905 (28a)

120576bea = 120575bea119901119889 (28b)


120590bea119901 = 119864119901120576bea119901 (29a)

119896bea119901 = 119875120575bea119901 = 119905119889119864119901119889 = 119905119864119901 (29b)


120590bea119887 = 119864119887120576bea119887 (30a)

119896bea119887 = 119875120575bea119887 =119905119889119864119887119889 = 119905119864119887 (30b)


119888bea119887 = 1119896bea119887 =11199051119864119887 +

11199052119864119887 (31)

119888bea119901 = 1119896bea119901 = 111990511198641199011 +111990521198641199012 (32)


1119896bp = 311990521 + 16119905221199052 + 16119905111990522 + 311990532128119864119887119868119887 + 38(1199051 + 1199052)120581119860119887119866119887

+ ( 11199051119864119887 +11199052119864119887) + ( 111990511198641199011 +

111990521198641199012) (33)



001198653

= [[[[


]]]]119906111990621199063

(34)



Plate

Bolt hole

x

y

Compression zone

d

d

A

Deformation shape

120590beap

A998400

120575beap


Bolt

x

zt

Compression zone

d

d

B

Deformation shape

120590beab

120590beab

120575beab

B998400



F

Plate 2Plate 1

x

z

l1

l2


1

23F3

u3u2

kp1

kbp

u1 kp2

(b) Spring model

Plate 1

Plate 2

x

y(plate zone)

(Bolt zone)

(plate zone)

w1

l1 minus 05dℎdℎ

w2

l2 minus 05dℎ

k1-1

k1-2

k2-2k2-1

(c) Plate stiffness





1119896bp = (1199052 + 1199051)2119864119887119889 + 372( 111990511198641199011 +111990521198641199012) (35)




11198961-2 =1120596111990511198641199011 (1198971 minus 05119889ℎ) + 1

((1205961 minus 119889) 11990511198641199011) 05119889ℎ 163692

1119896119901211198962-1 +

11198962-2 =1(1205962 minus 119889) 1199052119864119901205119889ℎ +

1120596211990521198641199012 (1198972 minus 05119889ℎ) 163692

1119896bp Equation (33) 150660


0

1500

3000

4500

6000

7500

9000

10500

12000

0 1 2 3 4 5 6 7 8 9 10 11 12

Load

(N)


S-1S-2S-3S-4



0

1500

3000

4500

6000

7500

0 01 02 03 04 05 06 07 08 09 1

Load

(N)


S-1S-2S-3S-4S-5





1119896bp = (1199052 + 11990512119889 )23

sdot 3 ( 111990511198641199011 +111990521198641199012 +

121199051119864119887 +121199052119864119887)

(36)



+ 086( 1(radic119864119871 + 1198641198791199051)plate-1

+ 1(radic119864119871 + 1198641198791199052)plate-2)

(37)






x

y

Unit mm30

Plate 2Plate 1


Bolt diameter = 5 x

z

F = 5000 N t = 1

t = 1

l1 = 100 l2 = 3d l3 = 3d l4 = 100


k8

k9 k7

k6

k5

k4

k3

k2

k1

u1

u2

u3

u4

u5

u6u7 67F

5

4

3

2

1


Plate 1Plate 2


(Bolt zone)





l1

la lb lc

l2 l3 l4

dℎ1 dℎ2 dℎ3

k92k91

k73 k72

k33

k42

k71 k41

k43

k63 k62k61

y k32

k31

k12

k11

ld

w2

w1

x

(b) Plate stiffness




111989611 +111989612 11989611 = 120596211990521198641199012119897119889 11989612 = (1205962 minus 119889ℎ3)1199052119864119901205119889ℎ3

11198963 =111989631 +

111989632 +111989633 11989631 = 120596211990521198641199012119897119888 11989632 = (1205962 minus 119889ℎ3)1199052119864119901205119889ℎ3 11989633 = (1205962 minus 119889ℎ2)1199052119864119901205119889ℎ2

11198964 =111989641 +

111989642 +111989643 11989641 = 120596111990511198641199011119897119888 11989642 = (1205961 minus 119889ℎ3)1199051119864119901105119889ℎ3 11989643 = (1205961 minus 119889ℎ2)1199051119864119901105119889ℎ2

11198966 =111989661 +

111989662 +111989663 11989661 = 120596211990521198641199012119897119887 11989662 = (1205962 minus 119889ℎ2)1199052119864119901205119889ℎ2 11989663 = (1205962 minus 119889ℎ1)1199052119864119901205119889ℎ1

11198967 =111989671 +

111989672 +111989673 11989671 = 120596111990511198641199011119897119887 11989672 = (1205961 minus 119889ℎ2)1199051119864119901105119889ℎ2 11989673 = (1205961 minus 119889ℎ1)1199051119864119901105119889ℎ1

11198969 =111989691 +

111989692 11989691 = 120596111990511198641199011119897119886 11989692 = (1205961 minus 119889ℎ1)1199051119864119901105119889ℎ111198962 11198965

11198968 Equation (33)



xz

yA

B

dx = 0

dy = 0

dz = 0

dx = freedy = 0

dz = 0




0

1000

2000

3000

4000

5000

Load

(N)

(b) Connection load

3472 36213056 2928

3472 3449

Bolt 1(spring)

Bolt 1(FEA)

Bolt 2(spring)

Bolt 2(FEA)

Bolt 3(spring)

Bolt 3(FEA)

05

10152025303540

Load

dist

ribut

ion

()



000000119865

=

[[[[[[[[[[[[[[[


]]]]]]]]]]]]]]]

1199061119906211990631199064119906511990661199067

(38)



(39)






000

050

100

150

200

000 050 100 150 200

Reference case


m = dydx

120595120595

ref




120595 = 119899119875minsum119899119894=1 119875119894 for 119894 = 1 2 3 119899 (40)











8 Conclusion




Bolt 1Bolt 2Bolt 3

05

101520253035404550

Load

dist

ribut

ion

()


Reference case

y = 00605x + 09326

050 100 150 200000ddref

090

094

098

120595120595

ref

102

106

110




Bolt 1Bolt 2Bolt 3

05

101520253035404550

Load

dist

ribut

ion

()


Reference case

y = 00136x + 09884

090

094

098

102

120595120595

ref

106

110

050 100 150 200000ttref



05

101520253035404550

Load

dist

ribut

ion

()

1 2 3 4 5t2t1

Bolt 1Bolt 2Bolt 3


Reference case

y = minus00646x + 10642

050 100 150 200000(t2t1)(t2t1)ref

090

094

120595120595

ref

098

102

106

110




Plate 2


F

t2 ≫ tt










Acknowledgments


References





















RoboticsJournal of





Journal of



RotatingMachinery



Journal of

Volume 201


VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










Plate

Bolt hole

x

y

Compression zone

d

d

A

Deformation shape

120590beap

A998400

120575beap


Bolt

x

zt

Compression zone

d

d

B

Deformation shape

120590beab

120590beab

120575beab

B998400



F

Plate 2Plate 1

x

z

l1

l2


1

23F3

u3u2

kp1

kbp

u1 kp2

(b) Spring model

Plate 1

Plate 2

x

y(plate zone)

(Bolt zone)

(plate zone)

w1

l1 minus 05dℎdℎ

w2

l2 minus 05dℎ

k1-1

k1-2

k2-2k2-1

(c) Plate stiffness





1119896bp = (1199052 + 1199051)2119864119887119889 + 372( 111990511198641199011 +111990521198641199012) (35)




11198961-2 =1120596111990511198641199011 (1198971 minus 05119889ℎ) + 1

((1205961 minus 119889) 11990511198641199011) 05119889ℎ 163692

1119896119901211198962-1 +

11198962-2 =1(1205962 minus 119889) 1199052119864119901205119889ℎ +

1120596211990521198641199012 (1198972 minus 05119889ℎ) 163692

1119896bp Equation (33) 150660


0

1500

3000

4500

6000

7500

9000

10500

12000

0 1 2 3 4 5 6 7 8 9 10 11 12

Load

(N)


S-1S-2S-3S-4



0

1500

3000

4500

6000

7500

0 01 02 03 04 05 06 07 08 09 1

Load

(N)


S-1S-2S-3S-4S-5





1119896bp = (1199052 + 11990512119889 )23

sdot 3 ( 111990511198641199011 +111990521198641199012 +

121199051119864119887 +121199052119864119887)

(36)



+ 086( 1(radic119864119871 + 1198641198791199051)plate-1

+ 1(radic119864119871 + 1198641198791199052)plate-2)

(37)






x

y

Unit mm30

Plate 2Plate 1


Bolt diameter = 5 x

z

F = 5000 N t = 1

t = 1

l1 = 100 l2 = 3d l3 = 3d l4 = 100


k8

k9 k7

k6

k5

k4

k3

k2

k1

u1

u2

u3

u4

u5

u6u7 67F

5

4

3

2

1


Plate 1Plate 2


(Bolt zone)





l1

la lb lc

l2 l3 l4

dℎ1 dℎ2 dℎ3

k92k91

k73 k72

k33

k42

k71 k41

k43

k63 k62k61

y k32

k31

k12

k11

ld

w2

w1

x

(b) Plate stiffness




111989611 +111989612 11989611 = 120596211990521198641199012119897119889 11989612 = (1205962 minus 119889ℎ3)1199052119864119901205119889ℎ3

11198963 =111989631 +

111989632 +111989633 11989631 = 120596211990521198641199012119897119888 11989632 = (1205962 minus 119889ℎ3)1199052119864119901205119889ℎ3 11989633 = (1205962 minus 119889ℎ2)1199052119864119901205119889ℎ2

11198964 =111989641 +

111989642 +111989643 11989641 = 120596111990511198641199011119897119888 11989642 = (1205961 minus 119889ℎ3)1199051119864119901105119889ℎ3 11989643 = (1205961 minus 119889ℎ2)1199051119864119901105119889ℎ2

11198966 =111989661 +

111989662 +111989663 11989661 = 120596211990521198641199012119897119887 11989662 = (1205962 minus 119889ℎ2)1199052119864119901205119889ℎ2 11989663 = (1205962 minus 119889ℎ1)1199052119864119901205119889ℎ1

11198967 =111989671 +

111989672 +111989673 11989671 = 120596111990511198641199011119897119887 11989672 = (1205961 minus 119889ℎ2)1199051119864119901105119889ℎ2 11989673 = (1205961 minus 119889ℎ1)1199051119864119901105119889ℎ1

11198969 =111989691 +

111989692 11989691 = 120596111990511198641199011119897119886 11989692 = (1205961 minus 119889ℎ1)1199051119864119901105119889ℎ111198962 11198965

11198968 Equation (33)



xz

yA

B

dx = 0

dy = 0

dz = 0

dx = freedy = 0

dz = 0




0

1000

2000

3000

4000

5000

Load

(N)

(b) Connection load

3472 36213056 2928

3472 3449

Bolt 1(spring)

Bolt 1(FEA)

Bolt 2(spring)

Bolt 2(FEA)

Bolt 3(spring)

Bolt 3(FEA)

05

10152025303540

Load

dist

ribut

ion

()



000000119865

=

[[[[[[[[[[[[[[[


]]]]]]]]]]]]]]]

1199061119906211990631199064119906511990661199067

(38)



(39)






000

050

100

150

200

000 050 100 150 200

Reference case


m = dydx

120595120595

ref




120595 = 119899119875minsum119899119894=1 119875119894 for 119894 = 1 2 3 119899 (40)











8 Conclusion




Bolt 1Bolt 2Bolt 3

05

101520253035404550

Load

dist

ribut

ion

()


Reference case

y = 00605x + 09326

050 100 150 200000ddref

090

094

098

120595120595

ref

102

106

110




Bolt 1Bolt 2Bolt 3

05

101520253035404550

Load

dist

ribut

ion

()


Reference case

y = 00136x + 09884

090

094

098

102

120595120595

ref

106

110

050 100 150 200000ttref



05

101520253035404550

Load

dist

ribut

ion

()

1 2 3 4 5t2t1

Bolt 1Bolt 2Bolt 3


Reference case

y = minus00646x + 10642

050 100 150 200000(t2t1)(t2t1)ref

090

094

120595120595

ref

098

102

106

110




Plate 2


F

t2 ≫ tt










Acknowledgments


References





















RoboticsJournal of





Journal of



RotatingMachinery



Journal of

Volume 201


VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation












11198961-2 =1120596111990511198641199011 (1198971 minus 05119889ℎ) + 1

((1205961 minus 119889) 11990511198641199011) 05119889ℎ 163692

1119896119901211198962-1 +

11198962-2 =1(1205962 minus 119889) 1199052119864119901205119889ℎ +

1120596211990521198641199012 (1198972 minus 05119889ℎ) 163692

1119896bp Equation (33) 150660


0

1500

3000

4500

6000

7500

9000

10500

12000

0 1 2 3 4 5 6 7 8 9 10 11 12

Load

(N)


S-1S-2S-3S-4



0

1500

3000

4500

6000

7500

0 01 02 03 04 05 06 07 08 09 1

Load

(N)


S-1S-2S-3S-4S-5





1119896bp = (1199052 + 11990512119889 )23

sdot 3 ( 111990511198641199011 +111990521198641199012 +

121199051119864119887 +121199052119864119887)

(36)



+ 086( 1(radic119864119871 + 1198641198791199051)plate-1

+ 1(radic119864119871 + 1198641198791199052)plate-2)

(37)






x

y

Unit mm30

Plate 2Plate 1


Bolt diameter = 5 x

z

F = 5000 N t = 1

t = 1

l1 = 100 l2 = 3d l3 = 3d l4 = 100


k8

k9 k7

k6

k5

k4

k3

k2

k1

u1

u2

u3

u4

u5

u6u7 67F

5

4

3

2

1


Plate 1Plate 2


(Bolt zone)





l1

la lb lc

l2 l3 l4

dℎ1 dℎ2 dℎ3

k92k91

k73 k72

k33

k42

k71 k41

k43

k63 k62k61

y k32

k31

k12

k11

ld

w2

w1

x

(b) Plate stiffness




111989611 +111989612 11989611 = 120596211990521198641199012119897119889 11989612 = (1205962 minus 119889ℎ3)1199052119864119901205119889ℎ3

11198963 =111989631 +

111989632 +111989633 11989631 = 120596211990521198641199012119897119888 11989632 = (1205962 minus 119889ℎ3)1199052119864119901205119889ℎ3 11989633 = (1205962 minus 119889ℎ2)1199052119864119901205119889ℎ2

11198964 =111989641 +

111989642 +111989643 11989641 = 120596111990511198641199011119897119888 11989642 = (1205961 minus 119889ℎ3)1199051119864119901105119889ℎ3 11989643 = (1205961 minus 119889ℎ2)1199051119864119901105119889ℎ2

11198966 =111989661 +

111989662 +111989663 11989661 = 120596211990521198641199012119897119887 11989662 = (1205962 minus 119889ℎ2)1199052119864119901205119889ℎ2 11989663 = (1205962 minus 119889ℎ1)1199052119864119901205119889ℎ1

11198967 =111989671 +

111989672 +111989673 11989671 = 120596111990511198641199011119897119887 11989672 = (1205961 minus 119889ℎ2)1199051119864119901105119889ℎ2 11989673 = (1205961 minus 119889ℎ1)1199051119864119901105119889ℎ1

11198969 =111989691 +

111989692 11989691 = 120596111990511198641199011119897119886 11989692 = (1205961 minus 119889ℎ1)1199051119864119901105119889ℎ111198962 11198965

11198968 Equation (33)



xz

yA

B

dx = 0

dy = 0

dz = 0

dx = freedy = 0

dz = 0




0

1000

2000

3000

4000

5000

Load

(N)

(b) Connection load

3472 36213056 2928

3472 3449

Bolt 1(spring)

Bolt 1(FEA)

Bolt 2(spring)

Bolt 2(FEA)

Bolt 3(spring)

Bolt 3(FEA)

05

10152025303540

Load

dist

ribut

ion

()



000000119865

=

[[[[[[[[[[[[[[[


]]]]]]]]]]]]]]]

1199061119906211990631199064119906511990661199067

(38)



(39)






000

050

100

150

200

000 050 100 150 200

Reference case


m = dydx

120595120595

ref




120595 = 119899119875minsum119899119894=1 119875119894 for 119894 = 1 2 3 119899 (40)











8 Conclusion




Bolt 1Bolt 2Bolt 3

05

101520253035404550

Load

dist

ribut

ion

()


Reference case

y = 00605x + 09326

050 100 150 200000ddref

090

094

098

120595120595

ref

102

106

110




Bolt 1Bolt 2Bolt 3

05

101520253035404550

Load

dist

ribut

ion

()


Reference case

y = 00136x + 09884

090

094

098

102

120595120595

ref

106

110

050 100 150 200000ttref



05

101520253035404550

Load

dist

ribut

ion

()

1 2 3 4 5t2t1

Bolt 1Bolt 2Bolt 3


Reference case

y = minus00646x + 10642

050 100 150 200000(t2t1)(t2t1)ref

090

094

120595120595

ref

098

102

106

110




Plate 2


F

t2 ≫ tt










Acknowledgments


References





















RoboticsJournal of





Journal of



RotatingMachinery



Journal of

Volume 201


VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










x

y

Unit mm30

Plate 2Plate 1


Bolt diameter = 5 x

z

F = 5000 N t = 1

t = 1

l1 = 100 l2 = 3d l3 = 3d l4 = 100


k8

k9 k7

k6

k5

k4

k3

k2

k1

u1

u2

u3

u4

u5

u6u7 67F

5

4

3

2

1


Plate 1Plate 2


(Bolt zone)





l1

la lb lc

l2 l3 l4

dℎ1 dℎ2 dℎ3

k92k91

k73 k72

k33

k42

k71 k41

k43

k63 k62k61

y k32

k31

k12

k11

ld

w2

w1

x

(b) Plate stiffness




111989611 +111989612 11989611 = 120596211990521198641199012119897119889 11989612 = (1205962 minus 119889ℎ3)1199052119864119901205119889ℎ3

11198963 =111989631 +

111989632 +111989633 11989631 = 120596211990521198641199012119897119888 11989632 = (1205962 minus 119889ℎ3)1199052119864119901205119889ℎ3 11989633 = (1205962 minus 119889ℎ2)1199052119864119901205119889ℎ2

11198964 =111989641 +

111989642 +111989643 11989641 = 120596111990511198641199011119897119888 11989642 = (1205961 minus 119889ℎ3)1199051119864119901105119889ℎ3 11989643 = (1205961 minus 119889ℎ2)1199051119864119901105119889ℎ2

11198966 =111989661 +

111989662 +111989663 11989661 = 120596211990521198641199012119897119887 11989662 = (1205962 minus 119889ℎ2)1199052119864119901205119889ℎ2 11989663 = (1205962 minus 119889ℎ1)1199052119864119901205119889ℎ1

11198967 =111989671 +

111989672 +111989673 11989671 = 120596111990511198641199011119897119887 11989672 = (1205961 minus 119889ℎ2)1199051119864119901105119889ℎ2 11989673 = (1205961 minus 119889ℎ1)1199051119864119901105119889ℎ1

11198969 =111989691 +

111989692 11989691 = 120596111990511198641199011119897119886 11989692 = (1205961 minus 119889ℎ1)1199051119864119901105119889ℎ111198962 11198965

11198968 Equation (33)



xz

yA

B

dx = 0

dy = 0

dz = 0

dx = freedy = 0

dz = 0




0

1000

2000

3000

4000

5000

Load

(N)

(b) Connection load

3472 36213056 2928

3472 3449

Bolt 1(spring)

Bolt 1(FEA)

Bolt 2(spring)

Bolt 2(FEA)

Bolt 3(spring)

Bolt 3(FEA)

05

10152025303540

Load

dist

ribut

ion

()



000000119865

=

[[[[[[[[[[[[[[[


]]]]]]]]]]]]]]]

1199061119906211990631199064119906511990661199067

(38)



(39)






000

050

100

150

200

000 050 100 150 200

Reference case


m = dydx

120595120595

ref




120595 = 119899119875minsum119899119894=1 119875119894 for 119894 = 1 2 3 119899 (40)











8 Conclusion




Bolt 1Bolt 2Bolt 3

05

101520253035404550

Load

dist

ribut

ion

()


Reference case

y = 00605x + 09326

050 100 150 200000ddref

090

094

098

120595120595

ref

102

106

110




Bolt 1Bolt 2Bolt 3

05

101520253035404550

Load

dist

ribut

ion

()


Reference case

y = 00136x + 09884

090

094

098

102

120595120595

ref

106

110

050 100 150 200000ttref



05

101520253035404550

Load

dist

ribut

ion

()

1 2 3 4 5t2t1

Bolt 1Bolt 2Bolt 3


Reference case

y = minus00646x + 10642

050 100 150 200000(t2t1)(t2t1)ref

090

094

120595120595

ref

098

102

106

110




Plate 2


F

t2 ≫ tt










Acknowledgments


References





















RoboticsJournal of





Journal of



RotatingMachinery



Journal of

Volume 201


VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










xz

yA

B

dx = 0

dy = 0

dz = 0

dx = freedy = 0

dz = 0




0

1000

2000

3000

4000

5000

Load

(N)

(b) Connection load

3472 36213056 2928

3472 3449

Bolt 1(spring)

Bolt 1(FEA)

Bolt 2(spring)

Bolt 2(FEA)

Bolt 3(spring)

Bolt 3(FEA)

05

10152025303540

Load

dist

ribut

ion

()



000000119865

=

[[[[[[[[[[[[[[[


]]]]]]]]]]]]]]]

1199061119906211990631199064119906511990661199067

(38)



(39)






000

050

100

150

200

000 050 100 150 200

Reference case


m = dydx

120595120595

ref




120595 = 119899119875minsum119899119894=1 119875119894 for 119894 = 1 2 3 119899 (40)











8 Conclusion




Bolt 1Bolt 2Bolt 3

05

101520253035404550

Load

dist

ribut

ion

()


Reference case

y = 00605x + 09326

050 100 150 200000ddref

090

094

098

120595120595

ref

102

106

110




Bolt 1Bolt 2Bolt 3

05

101520253035404550

Load

dist

ribut

ion

()


Reference case

y = 00136x + 09884

090

094

098

102

120595120595

ref

106

110

050 100 150 200000ttref



05

101520253035404550

Load

dist

ribut

ion

()

1 2 3 4 5t2t1

Bolt 1Bolt 2Bolt 3


Reference case

y = minus00646x + 10642

050 100 150 200000(t2t1)(t2t1)ref

090

094

120595120595

ref

098

102

106

110




Plate 2


F

t2 ≫ tt










Acknowledgments


References





















RoboticsJournal of





Journal of



RotatingMachinery



Journal of

Volume 201


VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










000

050

100

150

200

000 050 100 150 200

Reference case


m = dydx

120595120595

ref




120595 = 119899119875minsum119899119894=1 119875119894 for 119894 = 1 2 3 119899 (40)











8 Conclusion




Bolt 1Bolt 2Bolt 3

05

101520253035404550

Load

dist

ribut

ion

()


Reference case

y = 00605x + 09326

050 100 150 200000ddref

090

094

098

120595120595

ref

102

106

110




Bolt 1Bolt 2Bolt 3

05

101520253035404550

Load

dist

ribut

ion

()


Reference case

y = 00136x + 09884

090

094

098

102

120595120595

ref

106

110

050 100 150 200000ttref



05

101520253035404550

Load

dist

ribut

ion

()

1 2 3 4 5t2t1

Bolt 1Bolt 2Bolt 3


Reference case

y = minus00646x + 10642

050 100 150 200000(t2t1)(t2t1)ref

090

094

120595120595

ref

098

102

106

110




Plate 2


F

t2 ≫ tt










Acknowledgments


References





















RoboticsJournal of





Journal of



RotatingMachinery



Journal of

Volume 201


VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











Bolt 1Bolt 2Bolt 3

05

101520253035404550

Load

dist

ribut

ion

()


Reference case

y = 00605x + 09326

050 100 150 200000ddref

090

094

098

120595120595

ref

102

106

110




Bolt 1Bolt 2Bolt 3

05

101520253035404550

Load

dist

ribut

ion

()


Reference case

y = 00136x + 09884

090

094

098

102

120595120595

ref

106

110

050 100 150 200000ttref



05

101520253035404550

Load

dist

ribut

ion

()

1 2 3 4 5t2t1

Bolt 1Bolt 2Bolt 3


Reference case

y = minus00646x + 10642

050 100 150 200000(t2t1)(t2t1)ref

090

094

120595120595

ref

098

102

106

110




Plate 2


F

t2 ≫ tt










Acknowledgments


References





















RoboticsJournal of





Journal of



RotatingMachinery



Journal of

Volume 201


VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










Plate 2


F

t2 ≫ tt










Acknowledgments


References





















RoboticsJournal of





Journal of



RotatingMachinery



Journal of

Volume 201


VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation













RoboticsJournal of





Journal of



RotatingMachinery



Journal of

Volume 201


VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation









RoboticsJournal of





Journal of



RotatingMachinery



Journal of

Volume 201


VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation









Documents

An Analytical Method for Determining the Load Distribution