Phân tích dao động tấm composite lớp gấp nếp có gân gia cường bằng cách sử dụng phần tử tứ giác đăng tham số tám nút

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VIBRATION ANALYSIS OF STIFFENED FOLDED COMPOSITE 
PLATES USING EIGHT NODDED ISOPARAMETRIC 
QUADRILATERAL ELEMENTS 
PHÂN TÍCH DAO ĐỘNG TẤM COMPOSITE LỚP GẤP NẾP 
CÓ GÂN GIA CƯỜNG BẰNG CÁCH SỬ DỤNG PHẦN TỬ TỨ GIÁC 
ĐĂNG THAM SỐ TÁM NÚT 
Bui Van Binh 
Electric Power University 
Tóm tắt: Bài báo trình bày một số kết quả tính tần số dao động riêng, phân tích 
đáp ứng tức thời của chuyển vị, phân tích dạng dao động riêng của tấm 
composite lớp gấp nếp có và không có gân gia cường bằng phương pháp 
phần tử hữu hạn. Ảnh hưởng của góc gấp nếp, góc sợi, cách sắp xếp gân, 
số gân của tấm được làm rõ qua các kết quả số. Chương trình tính bằng 
Matlab được thiết lập dựa trên lý thuyết tấm bậc nhất có kể đến biến dạng 
cắt ngang của Mindlin. Các kết quả số thu được có tính tương đồng cao 
khi so sánh với các kết quả của các tác giả khác đã công bố trên các tạp 
chí có uy tín. 
Từ khóa: Phân tích dao động, đáp ứng động lực học, tấm composite gấp nếp có 
gân gia cường, phương pháp phần tử hữu hạn. 
Abstract: This paper presents several numerical results of natural frequencies, 
transient displacement responses, and mode shape analysis of unstiffened 
and stiffened folded laminated composite plates using finite element 
method. The effects of folding angle, fiber orientations, stiffeners, and 
position of stiffeners of the plates are illustrated. The program is 
computed by Matlab using isoparametric rectangular plate elements with 
five degree of freedom per node based on Mindlin plate theory. The 
calculated results are correlative in comparison with other authors’ 
outcomes published in prestigious journals. 
Keywords: Vibration analysis, dynamic response; stiffeners, stiffened folded laminated 
composite plates, finite element method. 
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INTRODUCTION 
Folded laminate composite plates have 
been found almost everywhere in 
various branches of engineering, such 
as in roofs, ship hulls, sandwich plate 
cores and cooling towers, etc. Because 
of their high strength-to-weight ratio, 
easy to form, economical, and have 
much higher load carrying capacities 
than fat plates, which ensures their 
popularity and has attracted constant 
research interest since they were 
introduced. Because the laminated 
plates with stiffeners become more and 
more important in the aerospace 
industry and other modern engineering 
fields, wide attention has been paid on 
the experimental, theoretical and 
numerical analysis for the static and 
dynamic problems of such structures in 
recent years. 
The flat plate with stiffeners based on 
the finite element model and were 
presented in [1, 2, 3, 5, 6, 7, 8]. In 
those studies, the Kirchhoff, Mindlin 
and higher-order plate theories are 
used. Those researches used the 
assumption of eccentricity (or 
concentricity) between plate and 
stiffeners: a stiffened plate is divided 
into plate element and beam element. 
Behavior of unstiffened isotropic 
folded plates has been studied 
previously by a host of investigators 
using a variety of approaches. Goldberg 
and Leve [9] developed a method based 
on elasticity. According to this 
method, there are four components of 
displacements at each point along the 
joints: two components of translation 
and a rotation, all lying in the plane 
normal to the joint, and a translation in 
the direction of the joint. The stiffness 
matrix is derived from equilibrium 
equations at the joints, while expanding 
the displacements and loadings into the 
Fourier series considering boundary 
conditions. Bar-Yoseph and Herscovitz 
[10] formulated an approximate 
solution for folded plates based on 
Vlassov’s theory of thin-walled beams. 
According to this work, the structure is 
divided into longitudinal beams 
connected to a monolithic structure. 
Cheung [11] was the first author 
developed the finite strip method for 
analyzing isotropic folded plates. 
Additional works in the finite strip 
method have been presented. The 
difficulties encountered with the 
intermediate supports in the finite strip 
method [12] were overcome and 
subsequently Maleki [13] proposed a 
new method, known as compound strip 
method. Irie et al. in [14] used Ritz 
method for the analysis of free 
vibration of an isotropic cantilever 
folded plate. Perry et al. in [15] 
presented a rectangular hybrid stress 
element for analyzing a isotropic folded 
plate structures in bending cases. In 
this, they used a four-node element, 
which is based on the classical hybrid 
stress method, is called the hybrid 
coupling element and is generated by a 
combination of a hybrid plane stress 
element and a hybrid plate bending 
element. Darılmaz et al. in [16] 
presented an 8-node quadrilateral 
assumed-stress hybrid shell element. 
Their formulation is based on Hellinger 
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- Reissner variational principle for 
bending and free vibration analyses of 
structures, which have isotropic 
material properties. Haldar and Sheikh 
[17] presented a free vibration analysis 
of isotropic and composite folded plate 
by using a sixteen nodes triangular 
element. Suresh and Malhotra [18] 
studied the free vibration of damped 
composite box beams using four node 
plate elements with five degrees of 
freedom per node. Niyogi et al. in [19] 
reported the analysis of unstiffened and 
stiffened symmetric cross-ply laminate 
composite folded plates using first-
order transverse shear deformation 
theory and nine nodes elements. In 
their works, only in axis symmetric 
cross-ply laminated plates were 
considered. So that, there is uncoupling 
between the normal and shear forces, 
and also between the bending and 
twisting moments, then besi ... 
1 63.3 63.6 68.7 71.49 66.4 73.5 
2 69.7 69.8 75.6 73.18 69.5 73.9 
3 150.5 152.7 155.3 157.8 149.9 146.1 
4 156.7 158.3 159.5 161.2 156.3 156.1 
900 
5 203.9 201.9 183.5 183.6 190.8 194.6 
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[00/00/00] 
Present: 
Angle-ply off axis 
Present: 
Cross-ply in axis α ωi 
Present [19] [450/-450]s [45
0/-450]ns [90
0/00]s [90
0/00]ns 
1 59.5 59.3 56.2 57.1 56.8 57.7 
2 63.1 63.4 73.3 72.7 66.1 73.1 
3 150.3 152.5 154.0 157.1 149.7 146.1 
4 153.9 155.0 156.1 158.0 153.1 152.2 
1200 
5 193.5 190.9 167.4 168.1 175.2 176.0 
1 42.3 42.3 40.2 40.7 39.7 38.9 
2 60.7 60.8 66.5 66.4 62.3 67.5 
3 133.2 131.5 119.0 119.1 122.5 125.1 
4 144.9 145.6 143.0 144.2 142.9 138.7 
1500 
5 149.9 151.8 153.9 157.2 149.3 145.9 
Table 3. First three natural frequencies of stiffened two folded composite plate for 
folding angle α=900,1200,1500, fiber orientation of [900/900/900]. 
Case 2 Case 3 Case 4 
α ωi 
Present [19] Present [19] Present [19] 
1 69.54 69.6 72.73 72.2 95.12 95.6 
2 73.98 73.9 81.55 81.1 119.36 122.5 900 
3 183.82 181.4 173.19 171.0 195.42 199.1 
1 65.36 65.0 74.28 73.8 67.63 67.3 
2 69.80 69.9 77.04 76.2 112.11 109.6 1200 
3 176.95 174.7 161.28 160.4 180.36 182.5 
1 52.86 52.4 66.29 65.3 42.27 42.5 
2 68.54 68.5 76.27 75.7 93.15 93.5 1500 
3 125.16 123.5 133.12 131.4 148.21 147.9 
The first five mode shapes of the 
unstiffened and three cases of stiffened 
composite plate are plotted in Fig. 3 for 
folding angle α=1200, fiber orientation 
of [450,-450/450]. 
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Fig.3. First five mode shapes of the unstiffened and three cases of stiffened composite 
plate, for folding angle α=1200; fiber orientation of [450,-450/450]. 
a- Folding angle α=900, b- Folding angle α=1500 
Fig.4. Effects of fiber orientation θ on the first five natural frequencies for folding angle 
α=900 and α=1500, [θ0/θ0/θ0], thickness t=0.02L. 
0 10 20 30 40 50 60 70 80 90
60
80
100
120
140
160
180
200
220
Fiber Orientaions(deg)
N
a
tu
ra
l 
F
re
q
u
e
n
c
ie
s
(H
z
)
_1Mode
_ 2Mode
_ 3Mode
_ 4Mode
_ 5Mode
0 10 20 30 40 50 60 70 80 90
40
60
80
100
120
140
160
Fiber Orientaions(deg)
N
a
tu
ra
l 
F
re
q
u
e
n
c
ie
s
(H
z
)
_1Mode
_ 2Mode
_ 3Mode
_ 4Mode
_ 5Mode
f1= 60.17(Hz) f2= 117.62(Hz) f4= 186.21(Hz) f5= 201.80(Hz) f3= 163.84(Hz) 
f1= 66.81(Hz) f2= 74.92(Hz) f4= 170.58(Hz) f5= 263.54(Hz) f3= 162.41(Hz) 
f1= 58.49(Hz) f2= 76.83(Hz) f4= 179.81(Hz) f5= 203.02(Hz) f3= 153.74(Hz) 
f1= 55.67(Hz) f2= 73.21(Hz) f4= 154.40(Hz) f5= 156.98(Hz) f3= 151.23(Hz) 
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Fig.3 shows that the stiffeners do not 
make any change in getting mode 
shapes of presented plates (mode 
shapes make this study interesting, 
useful in dynamic analysis of the 
plates, but any generalized 
recommendation is very difficult 
without undergoing numerical 
experiments). 
* The effects of fiber orientations on 
natural frequencies: 
Secondly, the effects of fiber 
orientations on the first five natural 
frequencies of two folded composite 
plate made of [θ0/θ0/θ0] has been carried 
out for various folding angle α. The 
results are plotted in Fig. 4a and Fig.4b 
for folding angle α = 900 and α = 1500, 
respectively. 
3.2 Transient analysis. 
We consider a cantilever two folded 
composite plate with the same 
dimension and material properties of 
section 3.1 for unstiffened and three 
cases of stiffened composite plates. The 
folded plates subjected to a uniformly 
distributed step loading of intensity 
q0 = 10kN/m
2 on face (1) for all cases. 
The location of point A (central point 
of top face) is shown in Fig.5a, analysis 
time step of 0.0005t ms, duration 
time of T = 0.025 (sec). The loading 
condition scheme is shown in Fig.5b 
with t1 = 1ms, t2 = 2ms, t3 = 25ms. 
(b)- Triangular step loading scheme. 
Time (s) 
t1 
q(t) 
t2 t3 
q0 
0 
(a)- Two folded composite plate. 
L 
L/3 
L/3 
L/3 α 
x 
z 
y 
Face (1) 
q0 
Point A 
Fig.5. Two folded composite plates with folding angle α subjected 
to uniformly step loading 
Fig.6a, 6b, 6c and 6d plotted the effect 
of folding angle α on displacement 
responses measurement at point A of 
the plate which having the fiber 
orientation [450/-450/450/-450] for case 
1, case2, case3 and case 4, respectively. 
From Fig.6, it can be observed that the 
displacement responses of folding 
angle α =900 and α =1200 are closed to 
each other, the displacement response 
of α =1500 is extremely higher than the 
others. The different become more 
rapidly for Case 1. The displacement 
amplitude and wave of Case 4 change 
more dramatic in the early time. 
Furthermore, there is a significant 
increase of vibration frequencies when 
the plates having clamped at edges. 
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0 0.005 0.01 0.015 0.02 0.025
-1.5
-1
-0.5
0
0.5
1
1.5
x 10
-4
Time (sec)
D
e
fl
e
c
ti
o
n
s
(m
)
090 
0120 
0150 (a) 
0 0.005 0.01 0.015 0.02 0.025
-1.5
-1
-0.5
0
0.5
1
1.5
x 10
-4
Time (sec)
D
e
fl
e
c
ti
o
n
s
(m
)
090 
0120 
0150 (b) 
0 0.005 0.01 0.015 0.02 0.025
-1.5
-1
-0.5
0
0.5
1
1.5
x 10
-4
Time (sec)
D
e
fl
e
c
ti
o
n
s
(m
)
090 
0120 
0150 (c) 
0 0.005 0.01 0.015 0.02 0.025
-1.5
-1
-0.5
0
0.5
1
1.5
x 10
-4
Time (sec)
D
e
fl
e
c
ti
o
n
s
(m
)
090 
0120 
0150 (d) 
Fig.6. Effect of folding angle α on transient response, [450/-450/450/-450]. 
0 0.005 0.01 0.015 0.02 0.025
-8
-6
-4
-2
0
2
4
6
8
x 10
-5
Time (sec)
D
e
fl
e
c
ti
o
n
s
(m
)
Case 1 
Case 2 
Case 3 
Case 4 
(a) 
0 0.005 0.01 0.015 0.02 0.025
-1.5
-1
-0.5
0
0.5
1
1.5
x 10
-4
Time (sec)
D
e
fl
e
c
ti
o
n
s
(m
)
Case 1 
Case 2 
Case 3 
Case 4 
(b) 
(a)- Folding angle α=900; (b)- Folding angle α=1500 
Fig.7.Comparision of transient response for different stiffener conditions 
of composite folded plate, [450/-450/450/-450]
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Fig. 7a and Fig. 7b plotted comparison 
of transient response of the composite 
folded plates for different stiffener 
conditions for α = 900 and α = 1500, 
respectively. It is revealed that the 
stiffness of the structure gradually 
reduces such as case1→ case2→ 
case3→ case4. With stiffener 
conditions, the deflection reduces and 
smallest amplitude in Case 3. 
To observe effect of fiber orientation 
on transient response of the plates, we 
compared the response of two fiber 
orientation ([450/-450/450/-450] and 
[900/00/900/00]) for four cases: Case 1-
Case4. The result is given in Fig. 8. In 
which: Fig.11a, 11b, 11c and 11d 
plotted the displacement responses 
measurement at point A of the plates 
(which have the folding angle α = 1200 
for: Case 1, Case2, Case3 and Case 4, 
respectively. 
0 0.005 0.01 0.015 0.02 0.025
-8
-6
-4
-2
0
2
4
6
8
x 10
-5
Time(sec)
D
e
fl
e
c
ti
o
n
s
(m
)
0 0 0 0[45 / 45 / 45 / 45 ] 
0 0 0 0[90 / 0 / 90 / 0 ]
(a) 
0 0.005 0.01 0.015 0.02 0.025
-6
-4
-2
0
2
4
6
x 10
-5
Time(sec)
D
e
fl
e
c
ti
o
n
s
(m
)
0 0 0 0[45 / 45 / 45 / 45 ] 
0 0 0 0[90 / 0 / 90 / 0 ]
(b) 
Fig.8 (a, b). Comparing effect of fiber orientation on transient response of the plate 
 for different stiffener condition: Case 1 and Case 2, folding angle α =1200 
0 0.005 0.01 0.015 0.02 0.025
-6
-4
-2
0
2
4
6
x 10
-5
Time(sec)
D
e
fl
e
c
ti
o
n
s
(m
)
0 0 0 0[45 / 45 / 45 / 45 ] 
0 0 0 0[90 / 0 / 90 / 0 ]
(c) 
0 0.005 0.01 0.015 0.02 0.025
-5
-4
-3
-2
-1
0
1
2
3
4
5
x 10
-5
Time(sec)
D
e
fl
e
c
ti
o
n
s
(m
)
0 0 0 0[45 / 45 / 45 / 45 ] 
0 0 0 0[90 / 0 /90 / 0 ]
(d) 
Fig.8 (c, d). Comparing effect of fiber orientation on transient response of the plate 
 for different stiffener condition: Case 3 and Case 4, folding angle α =1200
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Fig.8 shows that the transient response 
of the laminate plates does not change 
in significant for angle-ply off axis and 
cross-ply in axis fiber orientation. 
4. CONCLUSION 
In the present study, a finite element 
method using an eight nodded 
isoparametric plate elements, based on 
the first order shear deformation theory 
were investigated for analysis of free 
vibration and the transient response of 
the unstiffened and stiffened folded 
laminate composite plate. 
Good agreement is found between the 
results of this technique and other 
published results available in the 
literature. 
The effects of various parameters as 
folding angle, fiber orientation on 
natural frequencies, dynamic responses 
and mode shapes of unstiffened; 
stiffened folded laminate composite 
plates were indicated by some 
numerical results. 
The applicability of the present 
approach covers a wide range of forced 
vibration problems, geometric features, 
and boundary conditions. 
The results of this study will serve as a 
benchmark for future research for 
designing folded composite structures 
and sandwich structures made of 
composite materials, as it was 
extremely quick and reliable in 
producing design results. 
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Giới thiệu tác giả: 
Tác giả Bùi Văn Bình hiện đang công tác tại Khoa Công nghệ cơ 
khí - Trường Đại học Điện lực. 
Hướng nghiên cứu chính trong 5 năm gần đây: mô hình hoá và tính 
toán số kết cấu composite lớp. 
 97