Draft · 2019. 5. 21. · All stiffness relations are detailed in the references VDI 2230 BLATT 1....

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Modeling and analysis of a bolted joint under tension and shear loads

Journal: Transactions of the Canadian Society for Mechanical Engineering

Manuscript ID TCSME-2018-0042.R1

Manuscript Type: Article

Date Submitted by the Author: 02-Sep-2018

Complete List of Authors: Nasraoui, Mohamed Tahar; Laboratoire de Mécanique Appliquée et Ingénierie, Ecole Nationale d’Ingénieurs de Tunis, Université de Tunis El Manar, TunisiaCHAKHARI, JAMEL; Mechanical Engineering Department, College of Engineering, Prince Sattam bin Abdulaziz University, KSAKhalfi, Boubaker; CREPEC, Department of Mechanical Engineering, Ecole Polytechnique de Montréal, CanadaNasri, Mustapha; Laboratoire de Mécanique Appliquée et Ingénierie, Ecole Nationale d’Ingénieurs de Tunis, Université de Tunis El Manar, Tunisia

Keywords: Bolted joints, Numerical model, Stress analysis, FE Simulations, Experiments

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Transactions of the Canadian Society for Mechanical Engineering

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Mohamed Tahar Nasraoui 1, Jamel Chakhari 2*, Boubaker Khalfi 3 and Mustapha Nasri 11 Laboratoire de Mécanique Appliquée et Ingénierie, Ecole Nationale d’Ingénieurs de Tunis, Université de Tunis El Manar, Tunisia2 Mechanical Engineering Department, College of Engineering, Prince Sattam bin Abdulaziz University, KSA3 CREPEC, Department of Mechanical Engineering, Ecole Polytechnique de Montréal, Canada

*Corresponding author. Tel.: +966 537507669

E-mail address: [email protected]

Abstract: In this paper a prismatic bolted joint, subjected to tensile and shear loading, is studied. The two applied forces are in the same symmetry plane of the connection. A simplified numerical model is developed. It is constructed from unidirectional finite elements and contact elements. The elastic contact layer of connected parts is represented by a successive springs. An algorithm computing both the free structure stiffness matrix and the contact stiffness matrix is developed. Due to shear loading, static friction or kinetic friction can occur at contact surfaces between assembled parts. In each iteration, tangential contact forces are calculated and taken into account in problem solving. A program in C language is developed and used to calculate the model unknowns. 3D Finite Elements simulations are turned on ANSYS software in order to verify results obtained by the developed model. The model results are also compared to experimental tests data.

Keywords: Bolted joints, Stress analysis, Lateral loading, Numerical model, FE

simulations, Experiments.

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1. Introduction

Normally, Bolted joints are the most commonly used fasteners in mechanical industry

and metal structures. The dimensioning of such connections by conventional methods of

calculation is not satisfactory in some cases. Users are always looking for accurate

calculation tools of real assembly behavior.

In VDI 2230 recommendations VDI 2230 BLATT 1. (2003) are developed many

linear computational models of bolted joints mainly for the case of centered and slightly

eccentric load. The high eccentrically loaded bolted joint, is then studied by the authors

such as Agatonovic (1985), Guillot (1987), Bickford (1995) and Bulatovic et al. (2000).

Nonlinear models are developed for the two-bolted joints, by Daidié et al. (2007)

for the tension case and by Chakhari et al. (2008) for the compression one. All these

models are available for bolted joints in which loads are perpendicular to the contact area.

For zone in compression under bolt head, stiffness is formulated as in references

Rasmussen et al. (1978) and Alkatan et al. (2007).

Nasraoui et al. (2012) has developed an analytical model for tensile and shear

loaded bolted joint. In the case of pure shear loaded bolted joint case, the model is not

applicable. A theoretical approach to determine the allowable shear load in one-bolt joint

is presented by Kuenzi (1995). This model computes the bolt stresses, but it doesn’t show

if there is adhesion or sliding at contact surfaces.

Several three dimensional finite elements simulations are conducted to study

shear loaded bolted joints, Yongjie et al. (2009) and Ruiz et al. (2007). Numerical and

experimental study of lateral loaded bolted joint is also presented by Nasraoui et al.

(2013).

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In this paper we study a case of a shear and tension bolted joint where the tensile

loading is highly eccentric respect to the screw axis. A simplified numerical model is

established. It is based on unidirectional Finite Elements. Numerical simulations using

ANSYS software are done and experimental tests are conducted. Numerical and

experimental results are served to validate the simplified developed model and to analyze

the effect of different parameters on the bolted joint behavior.

2. Model Development

The modeling is for bolted joint of one bolt where the applied forces are situated in the

same symmetry plane. The joint is under a tensile force FE and a lateral or shear force FT.

These forces are applied at the end of the upper part (Fig. 1). A bending external moment

ME can also be applied at the same point of force application.

The main assumption is to suppose that the bending and compression stiffness of

connected parts remain constants during the loading. The behavior of such joint can be

studied using the model defined in Fig. 2. It is a simplified numerical model. Similar

models, Daidié et al. (2007) and Chakhari et al. (2008), were developed for the case two-

bolted joint under tension or compression only. In this time, we extend the previous

models in a manner that we take into account the existence of lateral force FT which

causes static or kinetic friction.

The model described in Fig. 2, is compound by unidirectional Finite Elements and

contact elements at joint interface. Contact elements compose a number of linear springs

simulating the elasticity of contact surfaces. The spring stiffness is proportional to the

relative contact surface represented by the spring. Surfaces contact or separation can be

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known when evaluating the relative spring displacement. For each spring is associated a

contact friction.

For comparison with experimental tests, we take at point A as boundary condition

(A= 0) around z-axis. This is to consider the symmetry of the experimental assembly

(Fig. 3).

The simplified numerical model is based on the following elements:

Beam elements (part AD): For every element the bending stiffness is calculated basing

on the cross section of the prismatic part AD. We take into account the presence of holes

on the assembled part, as in experimental parts. Each hole is considered as a succession

of beam elements of widths varying in scale form. The model represents the fourth of the

experimental connection due to the two planes of symmetry.

Beam element (CF): The element CF is the compressed zone of the assembled part that

is situated between the level of application of loads FE and FT (half height) and the bolt

head. This element has an equivalent section AP that can be determined using one of the

following models: VDI 2230 BLATT 1. (2003) or Rasmussen et al. (1978) or Guillot.

(1987). The two nodes of this beam element are kinematically coupled in z-rotation

because the bolt head follows the part deformation, Eq. (1).

(1)FC

Beam element (bolt FE): For the model the bolt is considered as a beam element of two

nodes (Fig. 4). Each node has three degree of freedom u, v, (axial displacement,

deflection and slope). The decomposition of the bolt in many beam elements is not

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necessary because the stresses remain constants along the bolt. Knowing the

displacements at the bolt extremities, nodal forces and stresses can be calculated.

The bolt element stiffness matrix is formulated function of the bolt material

modulus of elasticity Eb, the equivalent section Ab of the bolt, the equivalent second

moment Ib and the bolt length hb. All stiffness relations are detailed in the references VDI

2230 BLATT 1. (2003), Guillot (1987) and Alkatan et al. (2007).

The model presented in Fig. 2, corresponds to ¼ of the experimental assembly.

The calculation of the axial and bending stiffness correspond to ½ of the values obtained

and calculated for a bolted joint of height hb, Eq. (2).

(2)ℎ𝑏 = 2ℎ𝑃 + ℎ𝑠

where hp is the height of the assembled pat and hs is the height of support part.

For comparison with experimental results, the screw is subjected to simple

bending. The screw is free in x-displacement direction at node E; this is to satisfy

symmetry condition. The bending moment applied to the bolt is given by Eq. (3).

(3)𝑀𝐹𝐵𝑜𝑙𝑡 =2𝐸𝑏𝐼𝑏

ℎ𝑏𝜃𝑏

where b is the slope of the bolt head.

The bolt is preloaded by a force Q corresponding to a pre-stress . When the bolt 0

is loaded by an axial FBolt and a moment MFBolt there are supplements of force ( ) BoltF

and of bending moment ( ) on it. The supplements of loading are given by Eqs. BoltMF

(4)-(5).

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(4)QFΔF BoltBolt

(5)(Q)MF)(FMFΔMF BoltEBoltBolt

The alternating stress (a) in bolt is calculated function of supplements (FBolt)

and (MFBolt) as in Eq. (6), Junker (1986). (a) is compared to an admissible value S

fixed by the standard (European Standard E25-030. 1988).

(6)Sb

BoltBolta σ

I4dsΔMF

As2ΔFσ

where:

ds: Diameter of the threaded length of the bolt.

As: Cross-section of the bolt.

Ib: Second moment of the bolt.

S : Endurance limit of the bolt material for fatigue dimensioning.

Rigid body (B’D’): B’D’ is the base or support of the assembled part. It is

compound by non-deformable rigid elements.

Spring elements: These elements are modeling the elasticity of contact between

connected parts. A number of linear springs are considered along the parts contact. The

number of active springs changes function of the contact interface separation, so function

of external loading values. Spring stiffness is distributed relative to its contact surface.

Spring elements are associated to friction elements.

The spring stiffness ki of spring i and the total stiffness KT of the elastic

foundation are given by relations (7).

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; ; (7)N

T ii=1

K = k ii T

Sk = KS

i-1 i+1i

X -XS = .2a

2

where:

N: total number of springs

a: the half width of the assembled part

S: total area of contact surface.

Si: area of contact surface corresponding to spring i which is situated in abscise Xi.

ki: stiffness of spring i.

The normal stress on contact surface relative to a spring number i is function of

the nodal displacement of this spring, Eq. (8).

(8)i nini

i

k .u =

S

Tangential or shear stress on contact surface relative to spring number i is defined

by Eq. (9). This relation represents the static friction equilibrium condition on the surface

relative to an active spring i.

ti = tgi. ni (9)

uni , uti : normal and tangential displacements relative to spring element i.

ni , ti : normal and tangential stresses on contact surface relative to spring element i.

: Angle of static friction. f: limit coefficient of friction of parts contact.

Formulation of equivalent section AP: AP represents the equivalent section of the

compressed zone situated between the screw head and the support part. Its formulation is

developed by Rasmussen et al. (1978) as in relation (10). Assembly dimensions are

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defined in Fig. 5. In this relation dimensionless parameter are used, Eq. (11), where the

head diameter Da is taken as reference dimension.

Rasmussen model is applied to joint of Fig. 5. For the compressed zone a

dimensionless equivalent section is formulated, Eq. (10).

(10)

)DD2(

12L1L0.35tan1).D0.5()D(1

4πA

t2*

P2*

p2*

p*

1P

2*t

2*P

*

Where the dimensionless parameters are:

; ; ; (11)at

t* DD

D aPp*

DL

L a

PeqP

*

DD

D a

2P

P*

DAA

Da : bolt head diameter.

Dt : hole diameter.

Lp : total height of assembled parts: LP = 2hP + hs

Rasmussen model is applied to joint of Fig. 5. For the compressed zone a

dimensionless equivalent section is formulated, Eq. (10).

In case of prismatic joint physical diameter DP doesn’t exist. An equivalent fictive

diameter DPeq is defined. It takes into account local dimensions of parts. DPeq is function

of dimensions u, v, b and Da. To determine the diameter DPeq geometrical conditions are

considered:

if 3Da > b ; z = b ; else z = 3Da ;if 1.5Da > v ; x = 1.5Da + v ; else x = 3Da ;if 1.5Da > u ; x = 1.5Da + u ; else x = 3Da ;

where x and z are outer dimensions of intersection surface between the parts

contact interface and a circular surface of diameter (3Da) and the same axis as the screw

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(Fig. 5). DPeq is calculated by expression (12). The reduced dimensionless Rasmussen

section AP* is then obtained. Finally the equivalent section AP is found.

(12)2zxDPeq

Stiffness distribution factor: REP signifies the stiffness distribution factor

between the compressed zone under bolt head and the elastic foundation of parts contact.

It represents the stiffness distribution between the beam element (CF) and spring

elements. It is like in the modeling of bolted joints for very large diameter bearings

studied by Vadean (2000) and Vadean et al. (2006).

The stiffness KP of the compressed zone of the part of height hP and equivalent

section AP is given by relation (13). KT is the total stiffness of elastic foundation springs.

The factor REP is the ratio defined by Eq. (14).

(13)p

PpP h

.AEK

(14)KKR

P

TEP

REP is supposed to remain constant where geometry and material change

(REP=0.75). This assumption was verified experimentally in previous study, references

(Daidié et al. 2007), (Chakhari et al. 2008).

The total stiffness KT of elastic foundation is now deduced knowing the factor REP

and the stiffness KP and using Eq. (14). Then the stiffness ki of each spring i is calculated

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function of KP, Si and S using Eq. (7). The value of the stiffness ki depends on the

meshing of the upper part in beam elements.

3. Static and kinetic friction of assembled parts

To check if there is sliding or not at contact surface, it is necessary de know de

values of normal force NS and tangential force TS of contact for external loads FE, FT and

ME. The case of tensile force FE and shear force FT is shown in Fig. 6.

For parts contact the friction coefficient is equal to tg. The friction condition is

defined by Coulomb’s law, Eq. (15).

(15) TgNT ss

Preloading state: The normal contact force is equal to the bolt preload Q. The

maximum tangential contact force Ts Max is given by relation (16).

Ts Max = f Q (16)

Loading state: If q springs from N are in compression, the resultant normal

contact force Ns is the sum of normal forces applied to active springs, Eq. (17).

; (17)N

s j njj=1

N = k . u if unj > 0 then unj = 0

The resultant tangential contact force Ts is given by relation (18).

; (18)N

s j j njj=1

T = tg .k . u if unj > 0 then unj = 0

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Ts is the sum of tangential contact forces relative to active springs. This resultant is

maximal when all friction angles relative to active springs reach the limit value (tgj = f),

Eq. (19).

(19)nj

N

1jjMaxs u.kfT

When the bolted joint is subjected only to a shear force FT, a static friction happens if the

condition FT Ts Max is satisfied, Eq. (20).

(20)FT

Q ≤ f

When the assembly is under two loads (FE and FT), we apply equilibrium

condition and the contact law to the joint. A free body diagram is considered for the

structure (S) compound by the upper connected part (beam) and the bolt (Fig. 6). (S) is in

contact with the support. The beam is subjected to the forces FE, FT. The reaction in

anchor node E depends on the initial boundary conditions. In experimental assembly the

bolts are in pure bending, so the reaction force in X-axis direction is supposed null (F(X,

E) = 0). In this case the reaction force at node E is the same as the bolt force FBolt.

The numerical solving of the model is based on iterations of computation on the

vertical load FE (Fig. 7). In each iteration k a friction coefficient (tg)(k) is calculated to

check the possibility of contact sliding. Resultant reactions Ts and Ns, applied by the

support to the beam (Fig. 6), are calculated function of iteration k data and results of

iteration (k-1), Eqs. (21)-(22).

(21)Ts = FT + F(k ― 1)[X,E]

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(22)Ns = FE ― FBolt(k ― 1)

is the force applied in X-direction to node E calculated in iteration (k-1). This F(k ― 1)[X,E]

force is null if the node E is free in X-direction, as in experimental tests.

is the axial force applied to the bolt at node E, calculated in iteration (k-1).FBolt(k ― 1)

The friction coefficient (tg)(k) for an iteration k is given by Eq. (23):

(23)(tgφ)(k) =FT + F(k ― 1)[X,E]

FE ― FBolt(k ― 1)

There is no sliding for iteration k if:

(24)(k)(tg ) f

Sliding case: When sliding happens, loosening of mating and bolt tightening can

be obtained. To prevent this risk it is necessary to change assembly dimensions or bolt

pre-stresses. If not, don’t apply to the bolted joint forces that overload it and lead to

sliding and instability.

4. Solving Algorithm

Iterations (Fig.7): When there is static friction in preloading and under a shear

force FT applied to the assembled part at its extremity, the condition is satisfied.FT

Q ≤ f

Tj(k) : Tangential force applied to node j (nodes from B to D, Fig. 2) at iteration k. it is the

equivalent tangential force applied by the support to the upper part at the contact zone

modeled by the spring j.

Lc : length of contact interface of assembled parts (BD).

Lj : length of contact interface relative to spring j (Lj Lc)

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For a given shear load FT, we vary the tension force FE from iteration to the next.

The reason is that the bolt stresses are more influenced by the tension than by the shear

force. It is enough to take the variable FT as input data. To study FT effect on bolt stress

we use the same C program and we just vary FT.

Iteration (0): In first iteration tangential forces of contact are evaluated. In

preloading and for a given shear force FT it is assumed that there is static friction and

uniform contact along the contact interface, so the tangential force relative to spring j is:

(25)(0)

j

jT

C

LT =F

L

Iteration (k): Given the results of iteration (k-1), for iteration k we do as

following:

- Check the separation of the contact interface and update the stiffness matrix of contact

relative to the active springs.

- Calculate the overall friction coefficient for iteration (k) using relation (23), (k)

(tg )

serving results of iteration (k-1) and data of iteration k.

- The friction coefficient is assumed to be uniform along the active contact zone. This

value is used to calculate the tangential forces applied in iteration k.

- In iteration k, for each active spring element j apply a tangential force Tj(k)

as given by

relation (26). Then continue calculation of iteration k.

(26)(k-1) (k)(k)

j jT =N .(tg )

(27)N(k ― 1)j = kj.{Y(k ― 1)[j;v] ― Y(k ― 1)[(j + N);v]}

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(28)jj T

C

Lk =K .

L

: Normal force applied to spring j at iteration (k-1).(k-1)

jN

: Stiffness of spring j.jk

: Displacement of the upper node of spring j in Y-axis direction at iteration (k-Y(k ― 1)[j;v]

1).

5. 3D Finite Elements Model

An example of bolted joint is studied. Dimensions are defined in Fig. 5 and précised in

Table 1. Bolt M10 8.8 is considered. Assembled part are made from ordinary steel

(E=210000 MPa ; ν =0,3). The friction coefficient of contact interface between machined

parts is f=0,2 for the studied example. This value is determined experimentally. It

consists to determine the limit tangential force corresponding to beginning of sliding

between the two parts before fixation by bolt for a given normal force when applying

Coulomb’s friction law.

A tridimensional finite elements model is developed using the software ANSYS

Workbench (Fig. 8). Same boundary conditions are applied like by the simplified

numerical model previously presented. After geometry partition a mapped meshing is

done. The bolt is preloaded with a stress 0 = 200 MPa that corresponds to a preloading

force Q = 11600 N. Tensile and shear forces are applied at the end of the upper part. A

frictionless condition is applied to the bolt bottom face. The right extremity face is

constrained to zero rotation; this choice is to make possible the comparison with the

experimental tests results. The same value of friction coefficient is used for numerical

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models. In the 3D finite element model, contact elements are created on the interface

zone using the tools of the software ANSYS and the value of friction coefficient is

inserted.

3D Finite Elements simulations show that under two loads, joint deformation

occurs. The upper part is bent and the more important stresses are applied to the screw

(Fig. 9).

The bolted joint behaviour is studied for a pure shear force (FE = 0 and FT 0).

The contact interface pressure distribution is more important around the hole in

preloading and in shear loading (Figs. 10a and 10b). When a shear force is applied, the

pressure distribution changes. In the limit of static friction the shear force is FT = Ts Max

= f Q = 2320 N.

When the bolted joint is in addition subjected to a significant tension force a clear

bending appears on the upper part and the interface separation is expanded (Figs. 11a and

11b). The active contact surface is reduced under tension force and the separation can go

behind the screw. The presence of FT decreases the interface separation. There is no total

sliding while the shear force FT is not very high.

6. Experimental Tests

It is necessary to do experiments because results will be used to validate numerical

models for a bolted joint subjected to tension and shear loads. Experimental tests are

conducted on a symmetrical joint of two identical bolts H M10 quality 8.8. The bolted

joint is mounted on a rigid structure as shown in Fig. 12a. The two bolts are tightened in

the same manner and a 200 MPa pre-stress is installed.

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The lateral force is applied by a low-height hydraulic cylinder. The cylinder has a

piston diameter of 28.7mm. The tension force is also applied by a hydraulic cylinder of

annular piston (D=50mm, d=30mm). Each cylinder is supplied by a manual hydraulic

pump. The pumps pressure can go to 700 bars (Fig. 12b).

Normal stresses on each bolt are measured with strain gauges bonded on their

reduced diameter. Left and right gauges of bolt are placed in the symmetry plane of the

assembly that contains that applied forces. For a fixed value of shear force FT=2676N that

corresponds to a pump pressure PT= 55 bars, tension force FE is increased using the

second manual pump. Normal right and left stresses vary in time as described in Fig. 13.

7. Results Comparison and Model Validation

The simplified numerical model results are compared to results issued from 3D

finite elements simulations and to those obtained from experimental tests. For the three

methods, maximum normal stress is plotted function of tension force FE for a given value

of shear force FT = 2676 N (Fig. 14). Accepted and satisfying correlation between the

three results is obtained. Difference between results is smaller when tension load FE is

more important. This conclusion confirms that the developed model is validated and it

can be applied for dimensioning such bolted joints. Calculation and measurements are

done without exceed the real capacity of bolt. For a standard bolt M10 8.8 the yield

strength is Sy=640 MPa.

The value of shear force FT is then varied to see its effect on bolt stress. For a

fixed value of tension force FE the bolt normal stress decreases with the increasing of FT

(Fig. 15).

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If the surface finishing of contact surfaces of the assembled parts change the

obtained friction coefficient change also. Fig 16 shows how this parameter influences the

numerical response calculated by 3D finite element model. When only the friction

coefficient decreases the force on bolt increases. This variation is nonlinear. The source

of nonlinearity is due to sliding and separation conditions. If the friction coefficient is

less the bolt has to support more stress in order to maintain the stability of the joint.

In case of very high eccentricity between the point of application of the force FE

and the bolt axis, large deformation may happen under this load around the upper part

hole and plastic behaviour of material can be obtained. The proposed simplified model

cannot actually predict this situation. But it is possible to show large deformation in the

bolted joint using 3D finite element model such that used in ANSYS.

5. Conclusion

This paper presents a simplified numerical model for determining the stresses in

the screw of a bolted joint under tension and shear forces. The bolt stress is computed

after combining various parameters involving the material properties of the members

being joined and the joining bolt. The developed model serves also to locate the contact

zone, identify the limit of interfaces separation for any loading.

Coherence between the experimental and numerical results validates the current

model performance. Different configurations are investigated using an experimental

method to study the effect of joint design factors on the bolt stresses. The C code will be

associated to a VBA 6 interface to make an easy tool for dimensioning bolted joint of

such type. The advantage of this coded model is the rapid computation of these joints

compared to 3D finite elements calculations.

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References

Agatanovic, P. 1985. Beam model of bolted flanged connection. Engineering Computations. 2(1): 21-29.

Alkatan, F., Stéphan, P., Daidié, A., and Guillot, J. 2007. Equivalent axial stiffness of various components in bolted joints subjected to axial loading. Finite Elements in Analysis and Design. 43(8): 589-598. doi:10.1016/j.finel.2006.12.013

Bickford, J. 1995. Introduction to the design and behavior of bolted joints. CRC press, 97.

Bulatovic, R., and Jovanovic, J. 2000. An analysis of the Mathematical Models in Deformation Process of eccentrically Loaded Bolts. Mechanical Engineering. 1(7): 789-797.

Chakhari, J., Daidié, A., Chaib, Z., and Guillot, J. 2008. Numerical model for two-bolted joints subjected to compressive loading. Finite Elements in Analysis and Design. 44(4): 162-173. doi:10.1016/j.finel.2007.11.010

Daidié, A., Chakhari, J., and Zghal, A. 2007. Numerical model for bolted T-stubs with two bolt rows. Structural Engineering and Mechanics, An International Journal. 26(3): 343-361. doi:10.12989/sem.2007.26.3.343

European standard E25-030. 1988. Fasterners. Threaded connections. Design, calculation and mounting conditions, AFNOR Publications, 2nd version, February, ISSN 0335-3931.

Guillot, J. 1987. Assemblage par éléments filetés; modélisation et calculs”, Ed. Techniques de l’ingénieur, Volume 1 B5560 to B5562, 75006, Paris, 1-56.

Junker, G. H. 1986. Principe de calcul des assemblages boulonnés pour service sévère. Technical thesis. SPS-UNBARAKO S.A., Aulnay-sous-Bois.

Kuenzi, E.W. 1995. Theoretical design of nailed or bolted joint under lateral load. United States department of agriculture, Forest service, Forest products laboratory, Madison 5 in cooperation with the university of Wisconsin.

Nasraoui, M.T., Chakhari, J., Khalfi, B., and Nasri, M. 2012. Analytical model of tension and shear loaded bolted joint. International Journal of Research in Mechanical Engineering and Technology, 2(1): 40-44.

Nasraoui, M. T, Chakhari, J., Khalfi, B. and Nasri, M. 2013. Numerical and Experimental Study of Shear Loaded Bolted Joint”, International Journal of Current Engineering and Technology, 3(2): 239-243.

Rasmussen, J., Norgaard, I.B., Haastrup, O. and Haastrup, J. 1978. A two body contact problem with friction. Proceedings of Euromech Colloquium NR 110 Rimforsa, 115-120.

Ruiz, S., Daidié, A. and Chaib, Z. 2007. Modélisation du comportement d’un axe vissé chargé par un

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https://doi.org/10.1016/j.finel.2006.12.013

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effort transversal. Conception et Production Intégrées CPI’2007, 1-16.

Vadean A., Leray D., and Guillot J. 2006. Bolted joints for very large diameter bearings - Numerical model development. Finite Elements in Analysis and Design, 42(4): 298-313. doi: 10.1016/j.finel.2005.08.001

Vadean A. 2000. Modélisation et simulation du comportement des liaisons par éléments filetés de roulement de très grand diameter. Thèse N° 561, INSA de Toulouse France, 1-177.

VDI 2230 BLATT 1. 2003. Systematische berechnung hochbeanspruchter Schraubenverbindungen Zylindrische Einschraubenverbindungen, VDI Richtlinien. ICS 21.060.10, VDI-Gesellschaft Entwicklung Konstruktion Vertrieb, Fachberuch Konstruktion, Ausschuss Schraubenverbindungen.

Yongjie, Z. and Qin, S. 2009. Joint stiffness analysis of sheared bolt with preload. Second international conference on intelligent computation technology and automation. IEEE computer society.

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Tables

Table 1. Dimensions of bolted joint in mm.

u v w b h hs25 50 75 30 16 58

List of Figures

Figure 1. Bolted joint subjected to tension and shear loads.

Figure 2. Model components.

Figure 3. Geometrical symmetry of experimental assembly.

Figure 4. Bolt as beam element of two nodes. The beam length is hb.

Figure 5. Application of Rasmussen method for a prismatic bolted joint.

Figure 6. Structure under tensile loading (FE > 0).

Figure 7. Solving algorithm for a given shear force FT

Figure 8. 3D Finite Elements model.

Figure 9. Example of joint normal stress distribution for FE = 10000 N and FT = 5000 N.

Figure 10. Contact interface pressure distribution for FE = 0.

Figure 11. Contact interface pressure distribution for FE >0.

Figure 12. Experimental tests.

Figure 13. Experimental variation of normal right and left stresses on bolt.

Figure 14. Bolt normal stress variation function of force FE for a shear force FT=2676 N.

Figure 15. Bolt normal stress variation function of FE for different values of FT.

Figure 16. Friction coefficient effect on bolt stress for the same case study.

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Draft

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Draft

Figures

Figure 1. Bolted joint subjected to tension and shear loads.

Contact elementt

k

Rigid body

Beam elements

f

Q

FEBeam elements

D

F

C

Main nodeIintermediary node

D’

B

B’

E

A

x

y

(ME)

FT

Figure 2. Model components.

Figure 3. Geometrical symmetry of experimental assembly.

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Draft

hb

y

xo

u j

v i

v j j

ii

j

ui

Figure 4. Bolt as beam element of two nodes. The beam length is hb.

hp

hs/2 Symmetry plane

Screw

X

Y Zone in compression

3Da

z

x

b=2a

u v w

Figure 5. Application of Rasmussen method for a prismatic bolted joint.

E

FTFE

FBolt

F[X, E]

X

Ns

Ts

ME

Y

Figure 6. Structure under tensile loading (FE > 0).

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Draft

Data input :Q , FT , FE max, ME max

Calculus of friction coefficient in pre-loading and for a force FT:

yes

no

Pre-

load

ing

- Sliding happens

- Change bolt pre-loading

Loa

ding

tg f

Calculus of friction coefficient in loading

and for a force FT: ( tg )( k )

Equation (23)

- Sliding in interface

- Dimensioning the bolt for

shear force.

- Limit loads in sliding:

FE.slid , ME.slid

- Résults of iterations

(Forces, displacements, bolt

stresses …)

no

yes

(k)(tg ) f

No sliding in contact

interface

No separation test

Update the contact stiffness matrix

Results: Forces and

displacements of nodes

End

K-1<----k

FE<---FE+FE and ME<---ME+ME

Cal

culu

s of

iter

atio

n k

App

ly ta

ngen

tial

forc

es to

act

ive

spri

ng e

lem

ents

Figure 7. Solving algorithm for a given shear force FT

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Draft

Figure 8. 3D Finite Elements model.

Figure 9. Example of joint normal stress distribution for FE = 10000 N and FT = 5000 N.

(a) FE = FT = 0 (b) FE = 0 et FT = 2300 NFigure 10. Contact interface pressure distribution for FE = 0.

(a) FE = 10000 N et FT = 0 (b) FE = 10000 N et FT = 3000 NFigure 11. Contact interface pressure distribution for FE >0.

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Draft(a) Experimental bolted joint loaded by forces FE and FT (b) Manual hydraulic pumps supplying cylinders Figure 12. Experimental tests.

Figure 13. Experimental variation of normal right and left stresses on bolt.

Figure 14. Bolt normal stress variation function of force FE for a shear force FT=2676 N.

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DraftFigure 15. Bolt normal stress variation function of FE for different values of FT.

Figure 16. Friction coefficient effect on bolt stress for the same case study.

Page 28 of 28



Date post:	23-Mar-2021
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Draft · 2019. 5. 21. · All stiffness relations are detailed in the references VDI 2230 BLATT 1....

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