薄壁钣金零件铆接过程变形建模与仿真-[中国航空学报英文版

Chinese Journal of Aeronautics 24(2011) 369-377

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Chinese Journal of Aeronautics

journal homepage: www.elsevier.com/locate/cja

Riveting Process Modeling and Simulating for Deformation

Analysis of Aircraft’s Thin-walled Sheet-metal Parts

ZHANG Kaifu*, CHENG Hui, LI Yuan

The Key Laboratory of Contemporary Design and Integrated Manufacturing Technology, Ministry of Education,

Northwestern Polytechnical University, Xi’an 710072, China

Received 7 July 2010; revised 19 August 2010; accepted 29 October 2010

Abstract

The riveting joint is one of the important joint methods to permanently fasten two thin-walled sheet-metal parts. It is most ba-sic to efficiently analyze and estimate the deformation of the riveting joint for the performance, fatigue durability and damage of the riveting structure in the aircraft. This paper researches the riveting process mathematics modeling and simulating to more accurately analyze deformation of thin-walled sheet-metal parts. First, the mathematics and mechanics models for the elastic deformation, plastic deformation and springback of the rivet are built by mechanics theory. Second, on the basis of ABAQUS system, a finite element system, an instance made up of the rivet and two thin-walled sheet-metal parts of aluminum alloy is used to analyze and simulate the stress and deformation. What’s more, a comparison is made between the results obtained by the mathematics and mechanics models and those by finite element method (FEM). The models are proved true by the calculating and simulation results of the instance.

Keywords : riveting; deformation; mechanics modeling; finite element method; thin-walled structure; aircraft

1. Introduction

The riveting joint is one of the important ways to keep fastening the thin-walled sheet-metal parts per-manently. There are many advantages, such as simple jointing process, reliable jointing intensity, high work-ing efficiency, and so on, so it is used diffusely in the aircraft industry. However, the riveting can cause the deformation of rivets and the sheet-metal parts. The performance of the riveting structure and its fatigue durability and damage tolerance may be affected by the deformation. So it is very important to effectively

*Corresponding author. Tel.: +86-29-88493303.

E-mail address: [email protected]

Foundation items: National Natural Science Foundation of China (50805119); Aeronautical Science Foundation in China (2010ZE53049); Fund of National Engineering and Research Center for Commercial Aircraft Manufacturing (SAMC11-JS-07-200)

analyze, estimate and control the deformation of rivets and sheet-metal parts for riveting process.

A key step for exactly expressing deformation be-havior of the riveting joint is to build the models that explain the riveting process. Bedaira and Eastaugh[1] presented a numerical procedure for the analysis of riveted splice joints, taking into account the effect of the secondary out-of-plane bending and plates/rivet interaction. Blanchot and Daidie[2] presented adjust-ment of a numerical model simulating a riveted link using a number of different approaches. The analytical results improved knowledge about the riveting process and behavior of riveted links. Linde and de Boer[3] researched the modeling of hybrid composites and focused on a detailed simulation of the inter-rivet buckling behavior in a stiffened fuselage shell. These researches gained perfect results by simulation and experiment. But in order to effectively analyze and estimate the deformation of the riveting joint, the stress, strain and plastic/elastic deformation of the riv-

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eting joint are formulated by mathematics or mechan-ics method to precisely calculate and estimate the de-formation on the basis of the model of the riveting process [4-10]. Analysis and estimation for fatigue dura-bility and damage tolerance of the riveting joint be-tween the sheet-metal parts of aluminum alloy are very important for manufacturing products to work safely and reliably[11-15]. In addition, self-piercing riveted joint is an active and important research field, includ-ing its quality and behavior, fatigue, etc[16-21].

In order to estimate and control the deformation of rivets and sheet-metal parts for riveting process, it is necessary to build the riveting model to analyze the deformation of the riveting joint. The riveting process includes three different and representative phases: the elastic deformation of rivet, the plastic deformation of rivet and springback of rivet after the hammer departs from the rivet. The main objective of this paper is to build the mechanics models of each phase and analyze the deformation of the riveting joint in each phase. At the end of this paper, the simulation of the riveting process by finite element method (FEM) is used to validate the models.

2. Riveting Process Modeling and Deformation Analysis Generally, there are several typical riveting joints, such as single-riveted lap joint, double-riveted lap joint, single-riveted single strap butt joint, single-rive- ted double strap butt joint, double-riveted single strap butt joint, double-riveted double strap butt joint. Fig.1 shows a typical riveting joint, which includes two sheet-

metal parts and many rivets with countersunk head. All of the symbols in Fig.1 are defined by Table 1.

Table 1 Definition of symbols in Fig.1

Symbol Definition l 1 l 2

Length of rivet

Length of rivet except countersunk head Angle of countersunk head of rivet

Diameter of rivet

Diameter of riveting hole on the sheet-metal parts

Thickness of two sheet-metal parts

Height of cone-shaped hole on the sheet-metal part1

Distance between two rivets Length of two sheet-metal parts

α D 1 D 2 t 1, t 2 t 3 2d L 1, L 2

Denotation of subscript: 1—the first rivet, 2—the second rivet.

The riveting process related to riveting deformation generally consists of three phases: elastic deformation, plastic deformation and springback of the rivet. For each phase, there is its own riveting process model because each deforming behavior is different. Fig.2

shows the rivet deformation during riveting process.

Fig.2 Rivet deformation during riveting process.

2.1. Elastic deformation of rivet

For the phase of elastic deformation of rivet, the hammer begins touching the rivet, then continues moving and makes the rivet deform elastically. The riveting force will increase along with increased dis-placement of the hammer. In the process, the rivet be-comes shorter and bigger. Fig.3 shows elastic defor-

mation of rivet.

Fig. 1 A typical riveting joint.

Fig.3 Elastic deformation of rivet.

The elastic deformation of the rivet satisfies Hooke

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law. Let countersunk head of rivet be a metabolic. For X direction, the direct stress is formulated by

σe x =

F e 4F e

= (1) S πD 12

becomes bigger and extrudes the riveting holes in two sheet-metal parts. And the rivet tail begins touching the sheet-metal Part 2. This will bring fastening force between the sheet-metal parts and the rivet tail. Fig.4

shows the plastic deformation of rivet.

where F e is the riveting force at one moment, and S the cross-sectional area of the rivet. Then the direct strain along X axis can be defined as

σe x 4F e ⎧

==εe x ⎪E E πD 12⎪

(2) ⎨

′l Δl l −⎪ε=22=−e

e x ⎪l 2l 2⎩

where E is elastic modulus of the rivet. Then the rivet’s

elastic deformation along X axis is

Δl e =−εe x l 2=−

4F e l 2E πD 12

Fig.4 Plastic deformation of rivet.

(3)

where Δl e

4μF e ⎧

==εμεe e y x ⎪E πD 12⎪

(4) ⎨

′−ΔD D D 11⎪ε=1

=e y

⎪D 1D 1⎩

For the area I in Fig.4(b), there is elastic deforma-tion. The forces between the rivet and the sheet-metal

parts are balanced state. So the diameter of the rivet is D 1+ΔD 1. According to Hooke law, the direct stress in the area I is

σp x =

F p S p

4F p π(D 1+ΔD 1) 2

4F p

⎛4μF e ⎞π⎜D 1+⎟

πE D ⎝1⎠

(7)

And for the yield deformation of the rivet, the direct

stress along X axis is

σp x =σ0.2 (8)

where μ is Poisson’s ratio, and D 1′ the diameter before the deformation of D 1. Then the rivet’s deformation along Y axis and Z axis is

ΔD 1=εe y D 1=

4μF e

Let s represent the shift of the hammer from the point touching the rivet to the stopping point. Then the plastic deformation of rivet can be represented by

Δl p =s −Δl e =s −

4F e l 2E πD 12

E πD 12

D 1=

4μF e

(5) E πD 1

(9)

When the rivet deforms because of the riveting force F e , the rivet will extrude the riveting holes in the sheet-metal parts. The riveting holes will become wider. The increment of the diameter of the riveting holes can be formulated by

4μF e

(6) ΔD 2=D 1+ΔD 1−D 2=D 1−D 2+

E πD 1

Assume the volume of the rivet is invariable during the riveting process. As the head of the rivet does not deform, the volume of the rivet not including the head of the rivet is

⎧V =π(D 1/2)2l 2⎪⎪2

⎨V ≈π[(D 2+ΔD 2) /2](t 1+t 2−t 3) + (10)

⎪

π(D p /2)2][l 2−s −(t 1+t 2−t 3)]⎪⎩

2.2. Plastic deformation of rivet

When the hammer continues moving and the rivet

reaches the yield limit, the rivet’s area between the hammer and the sheet-metal Part 2 will bring the plas-tic deformation under the riveting force F p . Here σy ≥σ0.2, where σ0.2 is the yield limit of the rivet. The area of the plastic deformation of the rivet is named by the rivet tail. When the hammer reaches the target po-sition, it will stop moving. At the same time, the plastic deformation of the rivet stops, too. During this proc-ess, under the function of the riveting force, the rivet

So the diameter of the rivet tail is approximately

D p ≈

(11)

2.3. Springback of rivet

When the hammer departs from the rivet, the rivet-ing force becomes zero, i.e., the springback force

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F s =0. Because there is residual stress in the rivet, the rivet will spring back. At the moment of spring-back, σs x I =σp x I for area I in Fig.4(b); σs x II =σp x II for area II in Fig.4(c).

For area I in Fig.4(b), there is elastic springback.

E Δl s I

σs x I =E εs x I = (12)

t 1+t 2−t 3Then

4F p

Table 3 Mechanics parameters of instance

Aluminum ρ / σ 0.2 /

Load typeE /MPa μ

alloy MPa ( kg·m−3) 2A12 Tension 69 500 0.33 2.77 368

In Table 3, ρ is the density. 3.1. Riveting force analysis

When the hammer moves and touches the rivet, the riveting force exists. Assume that the relationship be-tween the time and shift for the moving hammer satis-fies the law of sines, s =2.5sin (πΤ ), where T repre-sents time, shown in Fig.5[2]

⎛4μF e ⎞π⎜D 1+⎟

E πD 1⎠⎝

So the elastic springback of area I is

Δl s I =

E Δl s I t 1+t 2−t 3

(13)

4F p (t 1+t 2−t 3) ⎛4μF e ⎞E π⎜D 1+⎟

E πD 1⎠⎝

(14)

For area II in Fig.4(c), there is plastic springback.

Δl s II E

σs x II =E ′εs x II =⋅ (15)

1−μ2l 2−(t 1+t 2−t 3) −s and the plastic where E ′=E/(1−μ2) and σs x II =σp x II ,

springback of the area II is

Δl s II =σ0.2(1−μ2)[l 2−(t 1+t 2−t 3) −s ]/E (16) According to Eq.(14) and Eq.(16), the total spring-back of the rivet after removing the hammer is

Fig.5 Curve of s -T of moving hammer.

In order to analyze the riveting force, this paper se-lects the point in the surface touching the rivet and the

hammer as a testing object. Based on the ABAQUS system, the riveting force F

can be gained by Fig.6.

Δl s =Δl s I +Δl s II =

4F p (t 1+t 2−t 3)

⎛4μF e ⎞E π⎜D 1+⎟

E πD 1⎠⎝

σ0.2(1−μ2)[l 2−(t 1+t 2−t 3) −s ]/E (17)

3. Instance Research

This paper uses the riveting joint assembly, made up

of rivets and two sheet-metal parts with aluminum alloy 2A12, as the instance for analyzing the riveting deformation. The geometrical parameters and the me-chanics parameters of the riveting joint structure are shown in Table 2 and Table 3 respectively. This paper uses ABAQUS system to analyze and simulate the riveting process.

Table 2 Geometrical parameters of instance

Parameter Value

Length of rivet l 1/mm 9 Length of rivet except countersunk head l2/mm 8 Angle of countersunk head of rivet α/(°)

120

Fig.6 Riveting force during riveting process.

3.2. Analysis and simulation of riveting deformation

and springback Ten points (P 1, P 2, …, P 10), shown as Fig.7, are signed along X axis for analyzing the deformation and springback of the rivet. The point P 5 is just located on the undersurface of sheet-metal Part 2.

Diameter of rivet D 1/mm 4 Diameter of riveting hole on sheet-metal parts D 2/mm 4.1 Thickness of two sheet-metal parts t 1, t 2/mm 2.5 Height of cone-shaped hole on sheet-metal Part 1 t 3/mm 1

Fig.7 Ten points of rivet.

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The riveting deformations of the rivet for ten points during the riveting process are shown as Fig.8. During 0-0.5 ms, the hammer still moves along –X direction and the deformation values along X axis and Y axis for

every point increase continuously. After the hammer

Fig.8 Deformation of ten points during riveting process.

departs from the rivet at 0.5 ms, the deformation val-ues along X axis and Y axis for every point decrease continuously and there is springback of the rivet.

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Table 4 and Table 5 show the deformation values along X axis and Y axis for the nodes of rivet. When T =0.05 ms, the deformation value along X axis of the point P 5 is 0.051 0 mm. The diameter D 1 of the rivet becomes 4.12 mm. After this moment, D 1>D 2. The

rivet begins extruding the holes of two sheet-metal

parts, but the holes of two sheet-metal parts prevent the rivet from extruding. So the plastic deformation of the rivet between P 5 and P 10 appears.

Table 4 Deformation values along X axis for ten points of rivet

T /ms

P 1

P 2

P 3

P 4

Deformation/mm P 5

P 6

P 7

P 8

P 9

P 10

0 0 0 0 0 0 0 0 0 0 0 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50 0.55 0.60

0 0.004 6 0.044 5 0.083 0.129 0.182 0.240 0.294 0.341 0.391 0 0.005 9 0.056 4 0.100 0.153 0.216 0.364 0.488 0.608 0.773 0 0.006 0 0.057 1 0.101 0.155 0.220 0.482 0.666 0.843 1.135 0 0.006 2 0.057 9 0.103 0.157 0.223 0.602 0.836 1.059 1.471 0 0.006 5 0.059 4 0.105 0.160 0.227 0.724 1.004 1.263 1.769 0 0.006 8 0.061 1 0.108 0.164 0.232 0.841 1.160 1.447 2.024 0 0.007 3 0.063 7 0.112 0.170 0.238 0.960 1.325 1.636 2.279 0 0.007 6 0.065 2 0.115 0.173 0.242 1.002 1.388 1.710 2.379 0 0.007 9 0.067 1 0.118 0.178 0.247 1.013 1.443 1.775 2.470 0 0.008 0 0.067 7 0.119 0.180 0.249 1.013 1.460 1.795 2.500 0 0.004 6 0.060 6 0.109 0.165 0.229 0.993 1.435 1.769 2.470 0 0.002 5 0.054 7 0.100 0.153 0.216 0.979 1.421 1.756 2.455

Table 5 Deformation values along Y axis for ten points of rivet

T /ms

P 1

P 2

P 3

P 4

Deformation/mm P 5

P 6

P 7

P 8

P 9

P 10

0 0 0 0 0 0 0 0 0 0 0 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50 0.55 0.60

0.004 1 0.006 4 0.006 4 0.006 4 0.006 7 0.007 1 0.007 6 0.008 0 0.008 4 0.008 6 0.006 6 0.005 3

0.005 7 0.009 4 0.009 5 0.009 7 0.010 2 0.010 7 0.011 6 0.012 1 0.012 8 0.013 1 0.011 2 0.010 2

0.035 0 0.045 7 0.046 4 0.047 1 0.048 4 0.049 8 0.051 7 0.052 7 0.053 9 0.054 2 0.051 8 0.050 0

0.048 2 0.053 2 0.053 5 0.054 0 0.054 7 0.055 5 0.057 0 0.057 8 0.058 8 0.059 2 0.056 2 0.054 1

0.051 0 0.053 7 0.054 2 0.054 8 0.055 3 0.056 0 0.057 1 0.057 9 0.059 0 0.059 4 0.057 1 0.055 2

0.053 6 0.058 7 0.059 7 0.060 6 0.061 7 0.063 7 0.068 0 0.070 0 0.072 3 0.072 9 0.072 1 0.070 8

0.061 2 0.155 0 0.269 0 0.386 0 0.502 0 0.612 0 0.737 0 0.794 0 0.832 0 0.846 0 0.846 0 0.844 0

0.061 8 0.195 0 0.349 0 0.508 0 0.668 0 0.825 0 1.004 8 1.085 2 1.164 1 1.192 0 1.188 1 1.186 4

0.046 5 0.165 0 0.302 0 0.452 0 0.611 0 0.771 0 0.956 0 1.036 0 1.113 0 1.139 6 1.136 8 1.136 1

0.017 6 0.056 8 0.079 0 0.124 0 0.207 0 0.312 0 0.451 0 0.513 0 0.572 0 0.591 0 0.570 0 0.558 0

3.3. Analysis and simulation of riveting stress The riveting stress during the riveting process is

shown as Fig.9, where Fig.9(a) shows the riveting stress when the hammer just touches the rivet; Fig.9(b) shows the riveting stress when the rivet has the elastic deformation; Figs.9(c)-9(k) show the riveting stress during the process of the plastic deformation of the rivet; Figs.9(l)-9(m) show the riveting stress during the springback of the rivet after the hammer departs the

rivet at 0.50 ms.

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According to Eq.(17), the total springback of the rivet along X axis by the analytical models is

Δl s =Δl s I +Δl s II ≈0.04776mm

According to Table 4, the total springback of the rivet along X axis by FEM is

Δl P 10=0.045mm

Error between the result of the analytical models and that by FEM is

δΔl |T =0.60ms =

Δl P 10−Δl s

Δl s

≈5.78%

It can be seen that the two are very approximate. 4. Conclusions

Fig.9 Riveting stress during riveting process.

3.4. Comparison between results of analytical models and those by FEM (1) The hammer moves to the limit position when T =0.50 ms.

When the hammer moves to the limit position, the shift of the hammer s =2.5 mm and the deflection of the rivet Δl =2.5 mm. According to Table 4, the shift of the 10th point calculated by ABAQUS system s P 10 is 2.500 mm. Then s P 10=Δl . The results obtained by the analytical models is the same as that by FEM.

For the area II in Fig.4(c), according to Eq.(11),

D p ≈

≈6.36mm

This paper researches riveting process mathematics modeling and simulating to more accurately analyze deformation of thin-walled sheet-metal parts. Firstly, the mathematics and mechanics models for the elastic deformation, plastic deformation and springback of the rivet are built. Secondly, an instance made up of the rivet and two thin-walled sheet-metal parts of alumi-num alloy is used to analyze and simulate the stress and deformation based on ABAQUS system. The models are proved by the simulation results. The ex-perimental research will be implemented to compare with the models, simulation results and revising the models. References

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Biography:

ZHANG Kaifu Born in 1977, he received B.S., M.S. and Ph.D. degrees from Northwestern Polytechnical University in 2000, 2003 and 2006 respectively, and then became a teacher there. He visited University of Michigan as a visiting scholar from 2007 to 2008. Now he is an associate professor. His main research interests are advanced assembly and joint, computer aided design, and so on. E-mail: [email protected]