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Analysis of Residual Stresses Induced by Riveting Process and Fatigue Life
Prediction
Article in Journal of Aircraft · August 2016
DOI: 10.2514/1.C033715
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JOURNAL OF AIRCRAFT
Vol. 53, No. 5, September–October 2016
Analysis of Residual Stresses Induced by Riveting Process
and Fatigue Life Prediction
Bin Zheng,∗ Haidong Yu,† Xinmin Lai,‡ and Zhongqin Lin§
Shanghai Jiao Tong University, 200240 Shanghai, People’s Republic of China
Downloaded by SHANGHAI JIAO TONG UNIVERSITY on April 22, 2017 | http://arc.aiaa.org | DOI: 10.2514/1.C033715
DOI: 10.2514/1.C033715
Residual stress induced by riveting process is critical to the fatigue life of riveted structures. A residual stress model
is proposed to predict the residual stress distribution after riveting process. In the residual stress model, the radial
pressure at the hole boundary is obtained, and the strain hardening effect of the rivet material is taken into
consideration. The other characteristic of the model is that the solution of radial and circumferential residual stresses
is extended to include an unfree springback process by using a springback coefficient. The residual stresses on the
faying surface with various parameters, such as the height of the rivet drive head, the hole diameter, and the material
property, are predicted with the residual stress model. The residual stresses calculated by the model are employed in
the fatigue life prediction using the multi-axial fatigue criterion. The predicted results of the fatigue life show a good
agreement with the experimental data. The investigation in this paper can help the residual stress prediction of riveted
joints and improve the awareness of the effect of the residual stress on the fatigue behavior of riveted structures.
σa , σm
Nomenclature
D
dz
=
=
dθ
E, υ
=
=
f−1 , f−1p
=
J2;a , J2;m
=
H, n
=
h
h0
=
=
k
kt , kf
=
=
p1
r
r1
unload
rload
1 , r1
=
=
=
=
r2
rp
r∞
ur1 α, β
λ
=
=
=
=
=
=
diameter of the rivet drive head
thickness of the equivalent thick-walled
cylinder
central angle of the fan-shaped element
Young’s modulus and Poisson’s ratio, respectively
fatigue limit in fully reversed bending and in
fully reversed tension, respectively
amplitude and mean value of second invariant
of deviatoric stress tensor, respectively
strength coefficient and strain hardening
exponent
height of the rivet drive head
initial length of protruding portion of the rivet
shank
spingback coefficient
stress concentration factor and fatigue notch
factor, respectively
radial pressure on the hole surface
distance from the axis of rivet
radius of the rivet hole
hole radius under the squeeze force and after
unloading, respectively
radius of the region beneath the rivet drive head
radius of plastic region
radius of the equivalent thick-walled cylinder
displacement of the hole surface
parameters in Marin fatigue criterion
parameter of the fatigue notch factor
=
σ ar
σ eq
1a ,
σ eq
1m
=
=
σ eq
2a ,
σ eq
2m
=
σ eq
3a ,
σ eq
3m
=
σr, σθ , σz
=
σ sheet
, σ sheet
r
θ
=
σ sheet
, σu
s
=
σ sheet
z
τ
τsheet
rz
ψ
=
=
=
=
amplitude and mean value of the applied load,
respectively
equivalent stress amplitude
amplitude and mean value of equivalent stress
in the loading direction, respectively
amplitude and mean value of equivalent stress
in the transverse direction, respectively
amplitude and mean value of equivalent stress
in the axial direction, respectively
radial stress, circumferential stress, and axial
stress of the rivet drive head
radial stress and circumferential stress of the
sheet, respectively
yield stress of the sheet and ultimate stress of
material, respectively
axial stress of the sheet
shear stress of the rivet drive head
shear stress of the sheet in r-z plane
displacement function
I.
R
Introduction
ESIDUAL stresses induced by the riveting process significantly
affect the fatigue behavior of riveted joints. Some analytical
models for residual stresses induced by the riveting process were
proposed to be employed in the residual stress analysis [1,2].
Shishkin [3] derived expressions for the stress distribution in the
sheets based on the Malinin small plastic-elastic deformation model.
Some discrepancies of the residual stress in the sheets occurred near
the rivet hole because the strain hardening effect is not considered for
calculating the radial pressure on the hole surface. The riveting
process is analogous to the cold expansion process where the fastener
holes are expanded to a desired level before inserting a fastener. Thus,
the analytical models for the cold expression process [4–6] were also
employed in the analysis of the stress distribution of the riveting
process. Park and Atluri [7] presented formulas for the residual stress
field with the Tresca yield criterion, where the residual stresses along
the radial direction are given for various radial pressures on the
surface of the rivet hole. Nevertheless, all of these models are limited
in their applicability to the riveting process. First, the radial pressure
on the hole surface is calculated without consideration of the strain
hardening effect of the rivet. Second, the unloading after the riveting
process is not a free springback procedure for the rivet hole resulted
from the limitation of the rivet.
Residual stresses are affected by a number of parameters involving
the geometric dimensions and the material property parameters. In
the geometric parameters, the height of the rivet drive head and the
hole diameter directly influence the residual stress field of riveted
Received 12 September 2015; revision received 6 December 2015;
accepted for publication 16 December 2015; published online 17 August
2016. Copyright © 2015 by the American Institute of Aeronautics and
Astronautics, Inc. All rights reserved. Copies of this paper may be made for
personal and internal use, on condition that the copier pay the per-copy fee to
the Copyright Clearance Center (CCC). All requests for copying and
permission to reprint should be submitted to CCC at www.copyright.com;
employ the ISSN 0021-8669 (print) or 1533-3868 (online) to initiate your
request.
*Ph.D. Student, School of Mechanical Engineering, Key Laboratory of
Mechanical System and Vibration; 19890118bln@sjtu.edu.cn.
†
Associate Professor, School of Mechanical Engineering, Shanghai Key
Laboratory of Digital Manufacture for Thin-Walled Structures; hdyu@sjtu.
edu.cn (Corresponding Author).
‡
Professor, School of Mechanical Engineering, Key Laboratory of
Mechanical System and Vibration; xmlai@sjtu.edu.cn.
§
Professor, School of Mechanical Engineering, Shanghai Key Laboratory
of Digital Manufacture for Thin-Walled Structures; zqlin@sjtu.edu.cn.
1431
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1432
ZHENG ET AL.
joints [8,9]. Li et al. [8,10] conducted the riveting process with strain
gauges mounted on the sheet surface to capture the strain variation of
riveted specimens with different heights of the rivet drive head. Rans
et al. [11] employed a three-dimensional finite element model to
obtain the residual stress field with various heights of the rivet drive
head. Aman et al. [12] carried out finite element analysis to study the
effect of some controllable process parameters on the quality of
riveted lap joints. Yoon and Kim [13] conducted a riveting simulation
of laminated composites with the consideration of the washer inner
fillet. The mechanical behaviors of the sheet material also affect the
residual stress significantly. As the new lightweight materials, such as
the Al-Li alloy, are generally applied to the aircraft structures, the
residual stress distribution is different from that in the traditional
aluminum alloy due to the improved material properties. Manes et al.
[9] and Viganò et al. [14] investigated the residual stresses of the
riveted structure with 8090 Ai-Li alloy sheets by the numerical
method. In this paper, the parameters of the height of the rivet drive
head, the hole diameter, and the material property of the sheet are
considered to investigate the residual stresses induced by the riveting
process.
The fatigue behavior of riveted joints is influenced by the residual
stress in the vicinity of the rivet hole. The fatigue life of a riveted joint
was investigated with the experimental and analytical methods
[15,16]. Skorupa et al. [17] conducted an experimental research on
the influence of several production-related factors on the fatigue
behavior of riveted lap joint specimens. The fatigue experiments of
riveted joints are time-consuming. Therefore, the numerical
simulation and the fatigue criterion are used in the fatigue life
prediction of the riveted joint. Fung and Smart [18] conducted the
fatigue life estimation employing both the strain-based Coffin–
Manson method and a stress-based method. However, the residual
stress induced by the riveting process is not involved in the fatigue life
estimation. Viganò et al. [14] studied the local stress of a T-joint from
the riveting process through the tensile loading stage. A multi-axial
fatigue criterion was applied to predict the fatigue life of the T-joint.
The tensile load was coupled with the residual stress induced by the
riveting process, and the combination of the stresses was used in
the fatigue life prediction. The effect of the residual stress on the
fatigue life is not presented separately, which is unfavorable of the
optimization of the parameters in the riveting process.
In this study, a residual stress model is established. There are two
main characteristics in the residual stress model. First, the radial
pressure between the rivet and the hole is obtained with the
consideration of the plastic behavior of the rivet material. Second, the
springback coefficient is applied to describe the sheet springback
degree after expansion. The effects of various parameters on residual
stresses of the faying surface are predicted with the proposed model.
The residual stresses calculated by the model are also employed in the
fatigue prediction of the riveted lap joint.
II.
Residual Stress of Riveted Sheets
A residual stress model is proposed to predict the residual stress
induced by the riveting process. The radial pressure on the hole
surface is obtained first. Then, the stress components in the riveting
process and unloading process are calculated. The unfree springback
of the rivet hole in the unloading process is taken into consideration.
Fig. 1
Stress components of the rivet drive head.
A fan-shaped element of the rivet drive head is shown in Fig. 1. The
height of the rivet drive head is h, and the hole radius is r1 . The force
equilibrium equation can be expressed as
dθ
σ r hrdθ − σ r dσ r hr drdθ 2σ θ hdr sin
2
− 2τrdθdr 0
(1)
Because of the small value of the central angle, it can be obtained
that sindθ∕2 ≈ dθ∕2. The compression of the rivet shank is an
axisymmetric process that σ r is equal to σ θ [19]. Neglecting the
second-order infinitesimal, Eq. (1) can be simplified as
dσ r −
2τ
dr
h
(2)
The shear stress increases to τmax and is then kept constant. The
maximum shear stress is τmax 0.5σ z [20]. Thus, the shear stress of
the chosen element in Eq. (1) is τ τmax 0.5σ z . Substituting the
shear stress into Eq. (2) gives
dσ r −
σz
dr
h
(3)
On the contact surface of the rivet drive head and the inner sheet,
the radial stress σ r is equal to zero at r r1 b. With this boundary
condition, the radial stress in Eq. (3) can be solved as
σr σz
r b − r
h 1
(4)
Let 2r1 2b D, where D is the diameter of the rivet drive head.
Thus, the radial stress σ r becomes
σr σz
D − 2r
2h
(5)
A. Radial Pressure on the Hole Surface
The cylindrical coordinate is employed to facilitate the analysis of
stress components. The z direction is along the axial direction of the
rivet, and the radial direction is in accordance with the direction of the
radius of the rivet hole. The radial pressure on the hole surface is
caused by the plastic flow of the rivet. The protruding portion of the
rivet shank is squeezed in the riveting process, which results in the
plastic flow of the rivet. The compression of the protruding portion of
the rivet shank in the riveting process is analogous to the upsetting
process. The principle stress method is widely used in the analysis of
the upsetting process [19]. Here, the principle stress method is
employed to solve the radial pressure induced by the riveting process.
The plastic behavior of the rivet material influences the stress
components induced by the riveting process directly. Here, the
Ramberg–Osgood relationship is chosen to describe the stress-strain
relationship with the consideration of the plastic behavior of the rivet
material. The Ramberg–Osgood relationship is simple, in which the
correlation between the stress and the plastic strain is established.
Besides, the Ramberg–Osgood model is shown to provide a realistic
treatment of the plastic behavior of the material [21]. The axial
compression stress σ z on the contact surface of the rivet drive head
and the inner sheet can be given according to the Ramberg–Osgood
model as
1433
ZHENG ET AL.
σ z Hεpz n
The elastic strain in the riveting process is ignored for the small
value compared to the plastic strain. The average strain in the axial
direction can be written as
Z
εpz h
ho
dh
h
ln
h
h0
(7)
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Substituting Eq. (7) into Eq. (6), the axial compression stress on the
contact surface of the rivet drive head and the inner sheet is given by
n
h
(8)
σ z H ln
h0
The radial stress at the bottom of the hole surface, which is denoted
as “A” in Fig. 1, can be calculated by substituting Eq. (8) into Eq. (5)
and replacing the variation r with the initial radius of the hole r1 .
Thus, the radial pressure at point A can be calculated with geometric
parameters and material properties, which is given by
n
D − 2r1
h
ln
(9)
p1 −σ r −H
h0
2h
Here, the radial pressure on the hole surface is assumed to be
uniform, and the radial pressure at point A is employed to calculate
the residual stress. Actually, the radial pressure is not uniform
through the sheet thickness. The radial pressure at point A, p1 , is
larger than those at other locations of hole surface through the
thickness direction. The peak tensile stress of the circumferential
residual stress predicted with the radial pressure, p1 , will also be
larger than the real value. However, the larger peak tensile stress can
result in a conservative fatigue life result, which is beneficial for the
safety of the riveted structure in service. In addition, the prediction of
the residual stress can be simplified with the uniform distribution
assumption of the radial pressure.
B. Stress Components of Sheets in Riveting Process
The sheets are subjected to the radial pressure on the hole surface
and the axial stress beneath the rivet drive head. Because of the small
ratio of the hole diameter to the dimension of the sheets, the region
containing a rivet hole in the sheets can be equivalent to a thickwalled cylinder. A thin layer of the sheet is chosen to analyze the
stress components. The equivalent thick-walled cylinder with inner
pressure and axial stress is shown in Fig. 2.
The sheets deform elastically at first in the riveting process. The
general formation of the displacement function for the spatial
axisymmetric problem is given as [22]
ψr; z A1 z4 A2 r4 A3 z3 A4 z2 r2 A5 z2 ln r
A6 zr2 A7 r2 ln r A8 z ln r
(10)
The equivalent thick-walled cylinder in Fig. 2 is subjected to the
uniform stresses on the hole surface and beneath the rivet drive head.
Thus, the radial stress and the circumferential stress are independent
of the z direction. The displacement function can be simplified for the
equivalent thick-walled cylinder as
z
r2
r1
p1
dz
Fig. 2
ψr; z α1 z3 α2 zr2
(6)
Thick-walled cylinder of the sheet.
r
(11)
The Love method is employed to solve the correlation between the
stress components and the displacement function [23]. The stress
components that satisfy the biharmonic function are given as
∂
∂2 ψ
∂
2 ψ − 1 ∂ψ
υ∇2 ψ − 2 ; σ sheet
υ∇
θ
∂z
∂z
r ∂r
∂r
2
∂
∂ ψ
∂
∂2 ψ
sheet
2
sheet
2
σz 2−υ∇ ψ − 2 ; τrz 1−υ∇ ψ − 2 (12)
∂z
∂z
∂r
∂z
σ sheet
r
where the Laplacian is given by ∇2 ∂2 ∕∂x2 ∂2 ∕∂y2 ∂2 ∕∂z2 .
The boundary conditions derived in Sec. II.A are
τsheet
0
rz
r r1 : σ sheet
−p1
r
r r∞ : σ sheet
0
r
z 0;
z dz: σ sheet
σ z r ≤ r2 ;
z
σ sheet
0r > r2 z
(13)
Combining Eqs. (12) and (13), the stress components in the elastic
stage are solved as
2
r21
r∞
−
1
r2∞ − r21 r2
2
r2
r
p1 2 1 2 ∞2 − 1
r∞ − r1 r
−p1
σ sheet
r
σ sheet
θ
σ sheet
σ z r ≤ r2 ;
z
σ sheet
0r > r2 z
τsheet
0
rz
(14)
As far as the actual riveted joint is concerned, r2∞ ≫ r21 .
Equation (14) can be simplified as
−p1
σ sheet
r
r21
r2
r21
r2
σ z r ≤ r2 ;
σ sheet
p1
θ
σ sheet
z
τsheet
0
rz
σ sheet
0r > r2 z
(15)
The plastic deformation will appear in the vicinity of the hole
bound by r1 ≤ r ≤ rp as the radial pressure is increased. For the
thick-walled cylinder, the axial stress σ sheet
is the middle stress
z
among the principle stresses [24]. Based on the Tresca yield criterion,
it can be obtained that σ sheet
− σ sheet
σ sheet
. The equilibrium
r
s
θ
equation for this axisymmetric problem is dσ sheet
∕dr
r
σ sheet
− σ sheet
∕r 0. Thus, the radial and the circumferential
r
θ
stresses can be written with the yield stress of sheet material, σ sheet
.
s
The boundary condition σ sheet
−p1 at r r1 is also employed in
r
solving the stress components. The radial and the circumferential
stresses in the plastic stage can be expressed as
r
sheet 1ln r −p r ≤ r −p1 ; σ sheet
σ
1
p
s
θ
r1
r1
rp 2
rp 2
1
1
− σ sheet
; σ sheet
σ sheet
r > rp (16)
σ sheet
r
θ
2 s
2 s
r
r
σ sheet
σ sheet
ln
r
s
where the radius of the plastic region can be solved with the stress
continuity condition at r rp .
1434
ZHENG ET AL.
C. Limited Unloading Process
The residual stresses after the removal of the squeeze force are
obtained by subtracting the elastic stresses from the plastic stresses
[7]. However, the sheets are not free to spring back after the expansion
due to the rivet shank remaining in the rivet hole [2]. Here, a
springback coefficient is defined, which is used to describe the
springback magnitude of the rivet hole. The springback of the rivet
hole after the removal of the squeeze force is an elastic process. The
springback magnitude of sheets can be described by the ratio of the
displacement of the hole surface. Thus, the springback coefficient k is
defined as
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k1−
runload
− r1
1
rload
− r1
1
It can be seen from Eq. (17) that the spingback coefficient is zero
when runload
rload
1
1 , which represents the condition that there is no
springback of the rivet hole. The spingback coefficient is equal to 1
when runload
r1 , which represents the complete springback of the
1
rivet hole.
The hole radius under the squeeze force can be calculated as
rload
r1 ur1 1
(18)
The displacement of the hole surface, ur1 , is solved in [24] as
ur1 1 − 2υ1 υ sheet r1 r2p
r1 r1
σs
r
ln
−
1
E
2 r2∞
rp 2
r2p
1 − υ2 σ sheet
s
E
r1
(19)
Substituting Eq. (19) into Eq. (18), the hole radius under the
squeeze force can be presented as
rload
r1 1
1 − 2υ1 υ sheet r1 r2p
r1 r1
σs
r
ln
−
1
E
2 r2∞
rp 2
r2p
1 − υ2 σ sheet
s
E
r1
(20)
, is an important parameter
The hole radius after unloading, runload
1
to describe the quality of the riveted joints. It is generally measured by
the experimental testing [25] or by the numerical simulation [11].
Here, the hole diameter after unloading is obtained by the
measurement of riveted joints.
The residual stresses of sheets can be written as
r
r2
− p1 − k −p1 12 ;
r1
r
r
r21
sheet 1 ln
σ
−
k
p
σ re
−
p
r ≤ rp 1
1
s
θ
r1
r2
1 sheet rp 2
r2
σ re
− k −p1 12 ;
r − σs
2
r
r
2
1 sheet rp 2
r
− k p1 12 r > rp σ re
θ σs
2
r
r
Specimen with riveted lap joints.
The sheet materials of specimens include 2060-T8 Al-Li alloy and
2024-T3 aluminum alloy. There are three heights of the rivet drive
head for riveted joints with the sheet material of 2060-T8 Al-Li alloy,
which are 1.8, 2.1, and 2.5 mm, respectively. The height of the rivet
drive head is 2.1 mm for riveted joints with the sheet material of 2024T3 aluminum alloy. There are also two radii of the rivet hole for the
riveted joints with the sheet material of 2060-T8 Al-Li alloy. The
specific parameters of riveted joints are listed in Table 1. Different
conditions of riveted joints are denoted as numbers 1–5.
The geometrical dimensions are measured, such as the diameter of
the rivet drive head (D), the height of the rivet drive head, and the hole
radius after unloading (runload
). The diameter and the height of the
1
rivet drive head are measured directly on the riveted joints. The hole
radius of the riveted joint, runload
, is measured after splitting the
1
riveted joint along the axial direction of the rivet. The toolmaker
microscope is employed to measure the hole radius after the rivet
process, as proposed in [26]. The hole radius at the faying surface is
measured, where cracks are easy to initiate [8,17]. The measurement
results of the diameter of the rivet drive head and the hole radius after
unloading are summarized in the Table 2. The numbers in Table 2
represent the same riveted joints with those in Table 1.
The mechanical parameters of materials are also indispensable to
calculate the residual stress. Two different materials of sheets are
considered to study the effect of material property on the residual
stress. The sheet materials of 2060-T8 Al-Li alloy and 2024-T3
aluminum alloy are chosen in the investigation. The mechanical
parameters of 2117-T4 aluminum alloy and 2060-T8 Al-Li alloy are
obtained by the uniaxial tension experiments. For 2024-T3 aluminum
alloy, the mechanical parameters in [11] are applied. The Ramberg–
Osgood model is employed to describe the stress–strain relationship
of the materials. The mechanical parameters of materials are given in
Table 3.
IV.
sheet ln
σ re
r σs
III.
Fig. 3
(17)
Effects of Parameters on the Residual Stress
Riveting process parameters and material properties have a
significant influence on the residual stress of the riveted structure.
Table 1
(21)
Number
1
2
3
4
5
Parameters employed for the investigation
Height of rivet
drive head, mm
1.8
2.1
2.5
2.1
2.1
Hole radius, mm
2.45
2.45
2.45
2.40
2.45
Sheet material
2060-T8
2060-T8
2060-T8
2060-T8
2024-T3
Experiments for Parameters in the Residual
Stress Model
The geometric parameters involved in the residual stress model are
measured from the riveted lap joints. The lap joints are riveted by
using the multi panel assembly cell automatic riveting machine. The
riveted specimen consists of two 2.0-mm-thick sheets and 4.76-mmdiam 2117-T4 aluminum rivets, which are countersunk-type
NAS1097AD6-6 rivets. The configuration of the riveted specimen is
presented in Fig. 3.
Table 2
Measurement results of D and runload
1
Parameters
Diameter of rivet drive head (D)
Hole radius runload
1
1
8.20
2.460
2
7.68
2.458
Number
3
4
7.22
7.74
2.455 2.408
5
7.58
2.467
1435
ZHENG ET AL.
Table 3
Mechanical parameters of sheets and the rivet
2060-T8
(sheets)
75.5
485
0.33
714
0.069
Parameter
Young’s modulus E, GPa
Yield stress σ s , MPa
Poisson’s ratio υ
Strength coefficient H, MPa
Hardening exponent n
Table 4
2117-T4
(rivet)
71.7
165
0.33
598
0.222
B. Effect of the Hole Diameter
Relevant parameter values for joints with different heights
of rivet drive head
Parameter
h0
h
Value
5.525 mm
1.8, 2.1, 2.5 mm
Parameter
σ sheet
(for 2060-T8)
s
D
r1
2.45 mm
runload
1
r∞
H (for 2117-T4)
n (for 2117-T4)
12.50 mm
598 MPa
0.222
E (for 2060-T8)
υ (for 2060-T8)
— —
Value
485 MPa
8.20, 7.68,
7.22 mm
2.460, 2.458,
2.455 mm
75.5 GPa
0.33
— —
Here, the influences of the height of the rivet drive head, the hole
diameter, and the material property on the residual stress are studied
by the proposed model.
Fatigue cracks tend to nucleate on the faying surface in the vicinity
of the hole [27]. Therefore, it is important to investigate the residual
stresses on the faying surface. Here, the residual stresses on the faying
surface are predicted with various riveting parameters listed in
Table 1.
There are some parameters involved in the equations to calculate
the residual stress on the faying surface. To be brief, only the relevant
parameter values for joints with different heights of rivet drive head
are summarized in Table 4. The other values of relevant parameters
for joints with different hole diameters and material properties can be
found in Tables 2 and 3.
The hole diameter is another important parameter in the riveting
process. Here, the hole radius varies from 2.40 to 2.45 mm. The radii
in this research are in the tolerance range of the rivet hole, as
suggested in [26]. The circumferential and radial residual stresses on
the faying surface are shown in Figs. 5a and 5b, respectively. The gray
regions in the figures are enlarged and presented at the right side. In
Fig. 5a, the magnitude of the peak tensile stresses are approximate to
130 MPa for both conditions. The location of the peak tensile stress
moves a short distance from the hole edge for the hole radius of
2.40 mm. As shown in Fig. 5b, the radial residual stresses are in a
compressive state. The compressive residual stress slightly increases
with the decrease of the hole diameter.
C. Effect of the Material Property
The 2024-T3 aluminum alloy is commonly used in the fuselage of
metallic aircraft. In addition, Al-Li alloys are generally used in the
aircraft for weight reduction. Here, sheets of 2024-T3 aluminum and
2060-T8 Al-Li alloy are chosen to investigate the influence of the
material properties of sheets on the residual stress. The circumferential
residual stresses on the faying surface are illustrated in Fig. 6a. A larger
compressive residual stress on the hole edge and a farther location of
the peak tensile stress away from the hole edge appear on the faying
surface for the 2024-T3 aluminum sheet. As shown in Table 3, the
elastic moduli of two sheet materials are close to each other. However,
the yield stress of 2024-T3 aluminum is approximately 35% lower than
the 2060-T8 Al-Li alloy. With the same height of the rivet drive head,
the compressive residual stress with a wider range and a larger
magnitude tends to be induced in the sheet with lower yield stress. The
radial residual stresses of sheets are shown in Fig. 6b. Larger
compressive residual stresses are also presented in the sheet of 2024-T3
aluminum.
A. Effect of the Height of Rivet Drive Head
The height of the rivet drive head has a remarkable effect on the
residual stress around the rivet hole. The circumferential residual
stresses on the outer-sheet faying surface are illustrated in Fig. 4a.
The compressive residual stress on the hole edge increases with the
decrease of the height of the rivet drive head. Meanwhile, the peak
tensile stress moves away from the rivet hole edge and slightly
decreases with the decrease of the height of the rivet drive head. The
peak tensile stresses are close to the value of 120 MPa for joints with
heights of the rivet drive heads of 1.8 and 2.1 mm and 130 MPa for the
joint with the height of the rivet drive head of 2.5 mm.
The radial residual stresses on the faying surface are shown in
Fig. 4b. The radial residual stresses are in a state of compressive
stress. It can be seen that larger compressive stresses appear on the
faying surface for the joint with lower height of the rivet drive head.
V.
Fatigue Life Prediction
Riveted joints are the critical elements for the fatigue behavior of
the airframe [17]. The residual stress and the cyclic load lead to the
failure of the riveted structure. Here, the residual stress calculated by
the proposed model is employed in the fatigue life prediction. The
predicted results are validated by the experimental data.
A. Prediction Procedure
The residual stress and the load applied on the component are
considered in the fatigue life prediction of riveted structures. The
notch effect exists in the vicinity of the rivet hole, which increases the
value of the applied load. The stress concentration factor kt along
the radial direction can be calculated as [28]
150
0
0
h=1.8 mm
h=2.1 mm
h=2.5 mm
-150
-300
Faying surface
Residual stress(Mpa)
Residual stress(Mpa)
Downloaded by SHANGHAI JIAO TONG UNIVERSITY on April 22, 2017 | http://arc.aiaa.org | DOI: 10.2514/1.C033715
2024-T3
(sheets)
72.4
310
0.33
530
0.100
The residual stress comparison of riveted joints with various heights
demonstrates that the decrease of the height of the rivet drive head
increases the compressive stress in both radial and circumferential
directions, and the increase of the compressive stress will be
beneficial to prolong the fatigue life of the riveted joint.
-40
h=1.8mm
h=2.1mm
h=2.5mm
-80
-120
Faying surface
-160
-450
1
2
3
X/r1
Fig. 4
4
-200
1
2
3
4
X/r1
a)
b)
Effect of the height of the rivet drive head on the residual stress of the faying surface: a) circumferential residual stress and b) radial residual stress.
1436
ZHENG ET AL.
150
90
0
r1=2.40mm
r1=2.45mm
Faying surface
-90
-180
1
2
3
Residual stress(Mpa)
Residual stress(Mpa)
180
125
100
75
1.2
4
1.4
1.6
X/r1
1.8
X/r1
a)
r1=2.40mm
r1=2.45mm
-80
-120
Faying surface
-160
Residual stress(Mpa)
Residual stress(Mpa)
-60
-40
-200
-90
-120
-150
-180
1
2
3
1.2
4
1.5
X/r1
1.8
2.1
X/r1
b)
Fig. 5 Effect of the hole diameter on the residual stress of the faying surface: a) circumferential residual stress and b) radial residual stress.
150
0
75
0
2024-T3
2060-T8
-75
-150
Faying surface
-225
-300
1
Fig. 6
2
3
X/r1
4
2060-T8
2024-T3
-80
-120
Faying surface
-160
-200
1
2
3
4
a)
b)
Effect of material properties on the residual stress of the faying surface: a) circumferential residual stress and b) radial residual stress.
(22)
The fatigue notch factor kf is defined to represent the life reduction
caused by the notch effect [22]. The correlation between kt and kf is
kf 1 kt − 1
q
1 rλ1
(23)
where λ is a parameter related to the ultimate stress σ u . Here, λ is 0.52
for 2060-T8 Al-Li alloy and 0.63 for 2024-T3 aluminum alloy.
The classical approach to predict the fatigue life of specimen with
the residual stress is to linearly superimpose the applied load with the
residual stress. The residual stress is equivalent to the mean stress in
the fatigue life prediction [29–31]. Thus, the equivalent stresses in the
uniaxial tension can be calculated as
1a
-40
X/r1
1 r1 2 3 r1 4
kt 1 2 r
2 r
σ eq
Residual stress(Mpa)
Residual stress(Mpa)
Downloaded by SHANGHAI JIAO TONG UNIVERSITY on April 22, 2017 | http://arc.aiaa.org | DOI: 10.2514/1.C033715
0
kf σ a ;
σ eq1m σ re
θ
σ eq
kf σ m ;
2a
0;
σ eq
2m
σ eq3a 0;
σ re
r ;
σ eq
3m
0
(24)
where σ eq 1a and σ eq 1m are the amplitude and the mean value of
equivalent stress in the loading direction. σ a and σ m are the amplitude
and the mean value of the applied load. σ re
θ is the circumferential
residual stress in the loading direction, which is located on the faying
surface. σ eq 2a and σ eq 2m are the amplitude and the mean value of
equivalent stress in the transverse direction. The stress amplitude in
the transverse direction is equal to zero, and the mean stress in the
transverse direction is the radial residual stress. The stress in the
thickness direction is ignored for the small value compared to the
stresses in the loading and the transverse directions [14].
The Marin fatigue criterion [32] is employed in the fatigue life
prediction. It is a valid multi-axial fatigue approach, which is
suggested in [33]. This criterion is established based on the amplitude
and the mean value of the second invariant of deviatoric stress tensor.
The Marin model and its parameters can be written as
q
α2 J2;a β2 J2;m ≤ f−1p
α
p f−1p
3
;
f−1
β3
f−1p
(25)
σu
Here, α is 2.65 for 2060-T8 Al-Li alloy and 2024-T3 aluminum
alloy. β is 0.67 for 2060-T8 Al-Li alloy and 0.82 for 2024-T3
aluminum alloy.
In the fatigue life prediction, the equivalent stress amplitude can be
written in the form of the Marin model as
σ ar q
α2 J2;a β2 J2;m
(26)
where the equivalent stress amplitude is the maximum in the vicinity
of the rivet hole.
1437
ZHENG ET AL.
Table 5
Downloaded by SHANGHAI JIAO TONG UNIVERSITY on April 22, 2017 | http://arc.aiaa.org | DOI: 10.2514/1.C033715
Number
1
2
3
4
5
Replicate specimen A
220,813
123,652
98,137
181,958
215,084
Fatigue life results of riveted joints
Experimental results, cycles
Replicate specimen B Replicate specimen C
203,474
112,802
182,085
175,212
113,089
99,222
160,317
174,541
215,671
209,883
The fatigue testing to obtain the stress versus life (S-N) curve of the
2060-T8 Al-Li alloy is carried out according to the American Society
for Testing and Materials Standard number E466. The S-N data of the
2024-T3 aluminum alloy are obtained from [34]. The data of the S-N
curve of the 2060-T8 Al-Li alloy and the 2024-T3 aluminum alloy for
the stress ratio R −1 and the stress concentration factor kt 1 can
be written as
log10 N 9.38 − 2.52 log10 σ ar − 61.32060 − T8
log10 N 10.4 − 3.06 log10 σ ar − 53.12024 − T3
(27)
The equivalent stress amplitude calculated with Eq. (26) is
substituted into Eq. (27) to solve the fatigue life of riveted joints. The
predicted results of fatigue life in Table 5 are calculated with Eq. (27).
A good description of the fatigue cycles is given by Eq. (27) from
3.46 × 104 cycles to 107 cycles for 2060-T8 Al-Li alloy and from
4.62 × 103 cycles to 107 cycles for 2024-T3 aluminum alloy. The
correlation coefficient R2 is equal to 0.91 and 0.82, respectively.
B. Comparison of the Fatigue Life Results
Riveted lap joints are prepared to conduct the fatigue experiments
with the dimensions of length L 230 mm and width W 60 mm.
Tabs with dimensions of 60 × 60 × 2 mm are clamped on the ends of
sheets to eliminate the second bending moment induced by the
loading procedure. The geometry of the riveted lap joint used in
fatigue experiments is presented in Fig. 7. Each specimen consists
of three 2117-T4 aluminum alloy countersunk-type rivets and two
2.0-mm-thick sheets.
The lap joint specimen is loaded in tension with an MTS Landmark
fatigue machine. A sinusoidal load with a load ratio R 0.1 is used
to test the specimens. A maximum remote stress of 77.11 MPa is
applied. To reduce the frequency effect in the fatigue testing, a low
frequency of 10 Hz is chosen. Four fatigue experiments of riveted
specimens are conducted for each condition listed in Table 1, and the
riveted specimens are denoted as replicate specimens A–D. The
results of fatigue life of riveted lap joints are summarized in Table 5.
2
Tab
2
Replicate specimen D
200,469
165,279
129,176
241,423
258,287
Average value, cycles
184,389
161,557
109,906
189,559
224,731
Predicted result, cycles
210,311
150,075
83,321
175,512
262,984
Compared with the experimental data of the fatigue life, the
agreement is appropriate for the fatigue life prediction with the
residual stress model and the Marin criterion. The fatigue life results
also confirm the effect of the height of the rivet drive head, the hole
diameter, and the material property on the fatigue life of riveted joints.
The increase of the fatigue cycles of specimens with the reduction of
the height of the rivet drive head and the hole diameter can be
attributed to the movement of the peak tensile stresses on the faying
surface. However, the hole diameter shows a slight influence on the
fatigue life due to the limited tolerance range of the rivet hole. The
traditional 2024-T3 aluminum alloy has advantages to prolong the
fatigue life of riveted joints, compared with the lightweight 2060-T8
Al-Li alloy, due to the reduction and the movement of the peak tensile
residual stress on the faying surface.
The residual stress is equivalent to the mean stress in the fatigue life
prediction. The influence of residual stresses on the fatigue life can be
investigated with this prediction method. It is beneficial to obtain the
correlation between the riveting parameters and the fatigue life of
riveted joints because the residual stress is tightly related to the
riveting parameters. The prediction method can be used in the design
stage of the riveted structure.
VI.
Conclusions
The residual stress induced by the riveting process significantly
influences the fatigue life of the riveted joints. A residual stress model
is established to analyze the residual stress distribution around the
rivet hole. This model is employed to investigate the effects of various
parameters on the residual stress distribution. The fatigue life of
riveted lap joints is predicted with the consideration of the residual
stress and the applied load. In this paper, the following conclusions
can be drawn.
1) The radial pressure between the hole and the rivet is solved
where the plastic behavior of the rivet material is also considered. In
the unloading process, the rivet hole is not free to spring back. Thus, a
springback coefficient is defined to describe the springback degree of
the rivet hole rather than employing the complete springback.
2) The residual stress model is employed to investigate the effects
of parameters, such as the height of the rivet drive head, the hole
diameter, and the sheet material property, on the residual stress
distribution of the sheet faying surface. The decrease of the height of
the rivet drive head and the hole diameter result in the peak tensile
stress moving a short distance from the hole edge. The compressive
residual stress in the traditional 2024-T3 aluminum sheets is larger
than that in the lightweight 2060-T8 Al-Li alloy.
3) The residual stresses calculated by the residual stress model are
used in the fatigue life prediction of riveted joints. The predicted
results are in good agreement with the results obtained by the fatigue
experiments. The prediction procedure can be used in the design
stage of the riveted structure.
Tab
Acknowledgments
a)
Fig. 7
b)
Fatigue testing and the specimen.
The present study was supported by the National Basic Research
Program of China (grant number 2014CB046600), the National
Natural Science Foundation of China (grant number 51275292), and
the Fund of National Engineering and Research Center for
Commercial Aircraft Manufacturing, China (SAMC13-JS-15-026).
1438
ZHENG ET AL.
Downloaded by SHANGHAI JIAO TONG UNIVERSITY on April 22, 2017 | http://arc.aiaa.org | DOI: 10.2514/1.C033715
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