Composite Structures 94 (2012) 2465–2473

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Composite Structures
journal homepage: www.elsevier.com/locate/compstruct

Nonlinear dynamical analysis of eccentrically stiffened functionally graded
cylindrical panels
Dao Huy Bich a, Dao Van Dung a, Vu Hoai Nam b,⇑
a
b

Vietnam National University, Hanoi, Viet Nam
Faculty of Civil Engineering, University of Transport Technology, Ha Noi, Viet Nam

a r t i c l e

i n f o

Article history:
Available online 28 March 2012
Keywords:
Functionally graded material (FGM)
Dynamical analysis
Critical dynamic buckling load
Vibration
Cylindrical panel
Stiffeners

a b s t r a c t

E-mail address: (V.H. Nam).
0263-8223/$ - see front matter Ó 2012 Elsevier Ltd. All rights reserved.
/>
shells under complex combinations of thermo–electro-mechanical
loads [10]. Huang and Han [11] presented nonlinear dynamic buckling problems of functionally graded cylindrical shells subjected to
time-dependent axial load by using Budiansky–Roth dynamic
buckling criterion [12]. Various effects of the inhomogeneous
parameter, loading speed, dimension parameters; environmental
temperature rise and initial geometrical imperfection on nonlinear
dynamic buckling were discussed. Liew et al. [13] presented the
nonlinear vibration analysis for layered cylindrical panels containing FGMs and subjected to a temperature gradient arising from
steady heat conduction through the panel thickness.
Ganapathi [14] studied the dynamic stability behavior of a
clamped FGMs spherical shell structural element subjected to
external pressure load. He solved the governing equations employing the Newmark’s integration technique coupled with a modified
Newton–Raphson iteration scheme. Sofiyev [15–17] studied the
vibration and buckling of the FGM truncated conical shells under
dynamic axial loading. Based on first-order shear deformation theory, the dynamic thermal buckling behavior of functionally graded
spherical caps is studied by Prakash et al. [18]. Dynamic buckling of
functionally graded materials truncated conical shells subjected to
normal impact loads is discussed by Zang and Li [19].
For FGM shallow shells, Alijani et al. [20], Chorfi and Houmat
[21] and Matsunaga [22] investigated nonlinear forced vibrations
of FGM doubly curved shallow shells with a rectangular base. Nonlinear dynamical analysis of imperfect functionally graded material
shallow shells subjected to axial compressive load and transverse
load was studied by Bich and Long [23] and Dung and Nam [24].
The motion, stability and compatibility equations of these


2466

@2w
¼
À wþ
; v2 ¼ 2 ;
2 @x2
@x2 R
@x2

e01 ¼
e02

c012 ¼

@u
@ v @w @w
þ
þ
;
@x2 @x1 @x1 @x2

where R is radius of the cylindrical shell.
The strains across the shell thickness at a distance z from the
mid-surface are

e1 ¼ e01 À zv1 ; e2 ¼ e02 À zv2 ; c12 ¼ c012 À 2zv12 ;

@ 2 e01 @ 2 e02
@ 2 c012
þ 2 À
¼

s1
(i) Longitudinal Stiffeners

z1

À

@2w @2w 1 @2w
À
:
@x21 @x22 R @x21

ð4Þ

ð5aÞ

and for stiffeners
st
1
st
2

r ¼ E0 e1 ;
r ¼ E0 e2

ð5bÞ

where E0 is Young’s modulus of ring and stringer stiffeners
Taking into account the contribution of stiffeners by the
smeared stiffeners technique and omitting the twist of stiffeners

M 2 ¼ B12 e01 þ ðB22 þ C 2 Þe02 À D12 v1 À D22 þ
s2
M 12 ¼ B66 c012 À 2D66 v12 ;

a
x2

!2

EðzÞ
ðe1 þ me2 Þ;
1 À m2
EðzÞ
¼
ðe2 þ me1 Þ;
1 À m2
EðzÞ
¼
c ;
2ð1 þ mÞ 12

b
0
d1

@2w
@x1 @x2

rsh
1 ¼


2. Eccentrically stiffened FGM cylindrical panels (ES-FGM
cylindrical panels)


k
2z þ h
;
2h

k
2z þ h
qðzÞ ¼ qm V m þ qc V c ¼ qm þ ðqc À qm Þ
;
2h

ð2Þ

x1

0
h2
z

d2

s2
(ii) Transversal Stiffeners

Fig. 1. Configuration of an eccentrically stiffened cylindrical panel.

2
2
1 À m2
1 À m2
Àh=2 1 À m
Àh=2 1 À m
Z h=2
EðzÞ
E1
A66 ¼
dz ¼
;
2ð1 þ mÞ
Àh=2 2ð1 þ mÞ
Z h=2
Z h=2
zEðzÞ
E2
zEðzÞm
E2 m
B11 ¼ B22 ¼
dz
¼
;
B
¼
dz ¼
;
12
2

z EðzÞ
E3
z EðzÞm
E3 m
D11 ¼ D22 ¼
dz
¼
;
D
¼
dz ¼
;
12
2
2
2
1
À
m
1
À
m
1
À
m
1
À m2
Àh=2
Àh=2
Z h=2 2

3
h ;
E3 ¼
À
þ
þ ðEc À Em Þ
k þ 3 k þ 2 4k þ 4
12
3
3
d1 h1
d2 h2
I1 ¼
þ A1 z21 ; I2 ¼
þ A2 z22 :
12
12

Substituting Eq. (10) into Eq. (7) yields

M 1 ¼ BÃ11 N1 þ BÃ21 N2 À DÃ11 v1 À DÃ12 v2 ;
M 2 ¼ BÃ12 N1 þ BÃ22 N2 À DÃ21 v1 À DÃ22 v2 ;
M 12 ¼ BÃ66 N12 À 2DÃ66 v12 ;
where

E0 I1
À ðB11 þ C 1 ÞBÃ11 À B12 BÃ21 ;
s1
E0 I2
¼ D22 þ

@2w
þ2
þ
þ N 1 2 þ 2N12
@x1 @x2
@x1 @x2
@x21
@x22
@x1
þ N2

E 0 A1 z 1
E0 A2 z2
C1 ¼
; C2 ¼
;
s1
s2
h1 þ h
h2 þ h
; z2 ¼
:
z1 ¼
2
2

ð10Þ

q1 ¼



with

q0 = qm for metal stiffener,
q0 = qc for ceramic stiffener.
The first two of Eq. (14) are satisfied automatically by choosing
a stress function u as

N1 ¼

@2u
;
@x22

N2 ¼

@2u
;
@x21

N12 ¼ À

@2u
:
@x1 @x2

Á @4/
@4u À Ã
@4u
@4w


The substitution of Eq. (10) into the compatibility Eqs. (4) and
(12) into the third of Eq. (14), taking into account expressions (2)
and (15), yields a system of equations

AÃ11 ¼

BÃ22 ¼ AÃ11 ðB22 þ C 2 Þ À AÃ12 B12 ;
BÃ12 ¼ AÃ22 B12 À AÃ12 ðB22 þ C 2 Þ;

Z

Àh=2

AÃ11

where

BÃ11 ¼ AÃ22 ðB11 þ C 1 Þ À AÃ12 B12 ;

@2w 1
@2w
þ N2 þ q0 ¼ q1 2 ;
2
@x2 R
@t

where

In above relations (6), (7) and (9) the quantityE0 is the Young

;
A66
D



E0 A1
E 0 A2
A22 þ
À A212 ;
D ¼ A11 þ
s1
s2

ð13Þ

DÃ21 ¼ D12 À B12 BÃ11 À ðB22 þ C 2 ÞBÃ21 ;

and

e01 ¼ AÃ22 N1 À AÃ12 N2 þ BÃ11 v1 þ BÃ12 v2 ;
e02 ¼ AÃ11 N2 À AÃ12 N1 þ BÃ21 v1 þ BÃ22 v2 ;
c012 ¼ AÃ66 þ 2BÃ66 v12 ;

ð12Þ

ð16Þ

Á @4w
@2w

@x1
@x21 @x22
@x42 R @x21

q1

À

@2u @2w
@2u
@2w
@2u @2w
þ2
À 2
¼ q0 ;
2
2
@x1 @x2 @x1 @x2 @x1 @x22
@x2 @x1

ð17Þ


2468

D.H. Bich et al. / Composite Structures 94 (2012) 2465–2473

For initial imperfection ES-FGM panels: The initial imperfection of the
panel considered here can be seen as a small deviation of the panel
middle surface from the perfect shape, also seen as an initial deflection which is very small compared with the panel dimensions, but

2mpx1
2npx2
mpx1
npx2
þ u2 cos
À u3 sin
sin
a
b
a
b

x22
; ð23Þ
2

where denote

u1 ¼

n2 k2 f 2
;
32m2 AÃ11
m2 f 2

u2 ¼

;
32n2 k2 AÃ22
h

w
@
w
@
w
@
w
@
w
@
w
0
0
0
5þ4
5
À4
À 2
À
@x1 @x2
@x1 @x2
@x21 @x22
@x1 @x22
þ BÃ12

þ

u ¼ u1 cos

Substituting the expressions (21)–(23) into Eq. (19) and applying Galerkin method to the resulting equation yield

4
@x1
@x21 @x22
@t
þ DÃ22

2

2

2

2

2

2

a4

p4

q1 ;

À
Á
A ¼ AÃ11 m4 þ AÃ66 À 2AÃ12 m2 n2 k2 þ AÃ22 n4 k4 ;

Á @4u
@ 4 ðw À w0 Þ

À
Á
a2 1
B ¼ BÃ21 m4 þ BÃ11 þ BÃ22 À 2BÃ66 m2 n2 k2 þ BÃ12 n4 k4 À 2 m2 ;
p R
À
Á
D ¼ DÃ11 m4 þ DÃ12 þ DÃ21 þ 4DÃ66 m2 n2 k2 þ DÃ22 n4 k4 ;
"
#


2mnk2 BÃ21 BÃ12
a2 n2 k2 1
d1 d2 ;
À
þ
H¼
3p2
AÃ11 AÃ22
6p4 mn AÃ11 R
!
1 m4 n4 k4
a
; k¼ ;
þ Ã
K¼
16 AÃ22
b
A11


M 2 ¼ 0;

N2 ¼ 0;

N 12 ¼ 0;

N 12 ¼ 0;

at

at

x1 ¼ 0;

x2 ¼ 0;

b:

a;

3.2. Vibration analysis

ð20Þ

where a and b are the lengths of in-plane edges of the panel.
The mentioned conditions (20) can be satisfied identically if the
buckling mode shape is represented by

w ¼ f ðtÞ sin

uniformly distributed excited transverse load q0 = Q sin Xt and
r0 = 0, the non-linear Eq. (25) has of the form

!
À
Á
B2
8mnk2 B
€
ðf À f0 Þ þ
d1 d2 ðf À f0 Þf þ H f 2 À f02
Mf þ D þ
A
3p2 A
À
Á
4a4
þ K f 2 À f02 f ¼ 6
d d Q sin Xt:
p mn 1 2

ð27Þ

By using Eq. (27), three aspects are taken into consideration:
fundamental frequencies of natural vibration of ES-FGM panel
and FGM panel without stiffeners, frequency–amplitude relation
of non-linear free vibration and non-linear response of ES-FGM panel. The non-linear dynamical responses of ES-FGM panels can be
obtained by solving this equation combined with initial conditions
to be assumed as f ð0Þ ¼ 0; f_ ð0Þ ¼ 0 by using the Runge–Kutta iteration schema.


:
Dþ
xL ¼
M
A

ð29Þ

rupper ¼

where

!
1
B2
;
Dþ
M
A

xNL ¼ xL 1 þ

8H2
3H
g þ 32 g2
3px2L
4xL

ð35Þ


non-linear free vibration is obtained



8mnk2 B
2
d1 d2 þ H f þ Kf :
3p2 A

2
b ¼ 8mnk B d1 d2 þ H:
H
2
3p A

where denoting

!
1 8mnk2 B
d
þ
H
;
d
1 2
M
3p 2 A

þ


!

From Eq. (35), the upper static buckling load can be determined
by putting f = 0

The equation of non-linear free vibration of a perfect panel can
be obtained from (27)

ð30Þ

r0 ¼

!

12
;

ð32Þ

where xNL is the non-linear vibration frequency and g is the amplitude of non-linear vibration.

D¼

D
3

h
f
n¼ ;
h

r0
ct
¼
;
r scr r scr

ð38Þ

where rscr = minrupper vs. (m, n).
The non-dimension form of Eq. (33) is written as
2

1 d n
þ
S1 ds2

"
Dþ

À
Á i
þk n2 À n20 n

3.3. Nonlinear dynamic buckling analysis

B2

!

A

large value of loading speed, the amplitude-time curve of obtained
displacement response increases sharply depending on time and
this curve obtains a maximum by passing from the slope point
and at the corresponding time t = tcr the stability loss occurs. Here
t = tcr is called critical time and the load corresponding to this critical time is called dynamic critical buckling load.
Consider an ES-FGM cylindrical panel subjected to axial load
r0(t). In this case q0 = 0, Eq. (25) gives

M€f þ D þ

!
À
Á
B2
8mnk2 B
ðf À f0 Þ þ
d1 d2 ðf À f0 Þf þ H f 2 À f02
A
3p 2 A

À
Á
m2 a2 h
r0 ðtÞ f ¼ 0
þ K f 2 À f02 f À
2

p

ð33Þ

:
2
b c2 q1

ð40Þ

Solving Eq. (39) by Runge–Kutta method and applying Budiansky–Roth criterion, the critical value sdcr, the dynamic critical time
tdcr ¼ sdcrcrscr and dynamical buckling load rdcr = ctdcr respectively are
obtained.
4. Numerical results and discussions
4.1. Validation of the present formulation
In this section, first of all, the
qffiffiffifficomparison on the fundamental
~ ¼ xL h qE c (xL is calculated from Eq. (29))
frequency parameter x
c
given by the present analysis with the results of Alijani et al.[20]
based on the Donnell’s nonlinear shallow-shell theory, Chorfi and
Houmat [21] based on the first-order shear deformation theory
and Matsunaga [22] based on the two-dimensional (2D) higher-order theory for the perfect unreinforced FGM cylindrical panel
Àa
Á
¼ 1; ha ¼ 0:1 with simply supported movable edges is suggested.
b
The material properties in Refs. [20–22] are aluminium and alumina, i.e. Em = 70.109 N/m2, qm = 2702 kg/m3 and Ec = 380.109
N/m2, qc = 3800 kg/m3 respectively. The Poisson’s ratio is chosen
to be 0.3. As can be observed in Table 1, a very good agreement
is obtained in this comparison study.
Next, the present frequency xL (in Eq. (29)) is compared with
the result of Szilard [29] and Troitsky [30] based on the classical

0.0597
0.0506
0.0456
0.0396
0.0380

0.0577
0.0490
0.0442
0.0383
0.0366

0.0588
0.0492
0.0430
0.0381
0.0364

FGM cylindrical panel
0.5
0
0.0648
0.5
0.0553
1
0.0501
4
0.0430
10
0.0409


k

Unreinforced (m, n)

Reinforced (m, n)

0.2
1
5
10

1172.51
982.14
822.19
783.56

(1,
(1,
(1,
(1,

3)
3)
3)
3)

1571.27
1435.02
1266.54

2)

1192.51
1128.40
1011.97
924.63

(1,
(1,
(1,
(1,

2)
2)
1)
1)

0.2
1
5
10

622.96
524.39
435.45
413.06

(1,
(1,
(1,

325.48

(1,
(1,
(1,
(1,

1)
2)
1)
1)

551.26
494.97
427.52
411.30

(1,
(1,
(1,
(1,

1)
1)
1)
1)

0.2
1
5

1)

1.5

3

5

10

0.6m

0.41 m

0.0127 m

0.02222 m

1 (plates)

E=211GPa
=0.3
=7830 kg/m3

0.00633 m
0.02222 m

0.0127 m
Fig. 2. Configuration of an eccentrically stiffened plate.


1356.1
1512.4
3066.7
5029.2
5589.8

4.2. Vibration results
To illustrate the proposed approach to eccentrically stiffened
FGM cylindrical panels, the panels considered here are cylindrical
panels and plates with in-plane edges a = b = 1.5 m; h = 0.008 m;
f0 = 0. The panels are simply supported at all its edges. The combination of materials consists of aluminum Em = 70 Â 109 N/m2;
qm = 2702 kg/m3 and alumina Ec = 380 Â 109 N/m2, qc = 3800 kg/
m3. The Poisson’s ratio is chosen to be 0.3 for simplicity. Material
of reinforced stiffeners has elastic modulus E = 380.109 N/m2;
q = 3800 kg/m3. The height of stiffeners is equal to 30 mm, its
width 3 mm, the spacing of stiffeners s1 = s2 = 0.15 m, the eccentricities of stiffeners with respect to the middle surface of panel
z1 = z2 = 0.019 m.
4.2.1. Results of fundamental frequencies of natural vibration
The obtained results in Table 3 show that the effect of stiffeners
on fundamental frequencies of natural vibration xL (xL is

calculated from Eq. (29)) is considerable. Obviously the natural frequencies of unreinforced and reinforced FGM cylindrical panels observed to be dependent on the constituent volume fractions, they
decrease when increasing the power index k, furthermore with
greater value k the effect of stiffeners is observed to be stronger.
This is completely reasonable because the lower value is the elasticity modulus of the metal constituent.
4.2.2. Results of frequency–amplitude of non-linear free vibration
Fig. 3 shows the relation frequency–amplitude of non-linear
free vibration of reinforced and unreinforced panel (calculated
from Eq. (32)) with m = 1, n = 1. As expected the non-linear vibration frequencies of reinforced panels are greater than ones of unreinforced panels.
4.2.3. Non-linear response results


NL

R=3m, k=5, q 0=5000sin(500t)

f(m)

R=10m, k=0.2

6.0E-4

1.2E+3

4.0E-4

Reinforced
9.0E+2

Unreinforced

2.0E-4
0.0E+0

6.0E+2

-2.0E-4
R=10m, k=5

3.0E+2



0.03

0.045

t(s)

0.06

Fig. 6. Influence of initial imperfection on non-linear responses.

Fig. 3. Frequency–amplitude relation.

1.2E-2

Perfect

Ω=950(rad/s)

ξ

Unreinforced Panel

0.5

8.0E-3

0.4

4.0E-3


f(m)

Ω=500 (rad/s)

τ

9.50E-01

1.00E+00

1.05E+00

1.10E+00

0.4
Fig. 7. Effect of buckling mode shapes on load–deflection curve of unreinforced
panel.

Fig. 4. Nonlinear response of ES-FGM cylindrical panel.

9.0E-4

0
9.00E-01

1.2

Ω=600 (rad/s)


-9.0E-4
0

0.025

0.05

0.075

t (s)

0.1

Fig. 5. Nonlinear response of FGM cylindrical panel.

4.3. Nonlinear dynamic buckling results
To evaluate the effectiveness of the reinforcement of stiffener in
the nonlinear dynamic buckling problem, we consider the case of
imperfect ES-FGM cylindrical panel subjected to an axial compressive load. The critical dynamic buckling loads is determined by
solving Eq. (39) and applying Budiansky–Roth criterion.
Materials and structures used in this section are the same in the
previous section.
Figs. 7 and 8 show the effect of buckling mode shapes on load –
deflection curve of reinforced and unreinforced FGM cylindrical
panel subjected to an axial compressive load with the power law index k = 1, R = 3 m and compressive load r0 = 1.5 Â 109 t. Clearly, the
smallest critical dynamic buckling load corresponds to the buckling
mode shape m = 5, n = 2 in the case of unreinforced panel and m = 2,
n = 2 in the case of reinforced panel. This figure also shows that
there is no definite point of instability as in static analysis. Rather,


4 also considers the effect of loading speed to the dynamic buckling
load; the results show that the dynamic buckling loads increases
when the loading speed increases.
Fig. 9 shows the influence of initial imperfection amplitude f0 on
the non-linear buckling of ES–FGM Cylindrical panel. Clearly, the
initial imperfection strongly influences on the critical dynamic
buckling loads of ES-FGM cylindrical panel subjected to an axial
compressive load.


2472

D.H. Bich et al. / Composite Structures 94 (2012) 2465–2473

Table 4
Nonlinear critical buckling loads of the cylindrical panels subjected to an axial compressive load (Â108 N/m2).
R (m)

k

Unreinforced
Static (m, n)

Reinforced
Dynamic (m, n)

Static (m, n)

c = 1.5 Â 109


2.0700 (5,
1.7895 (5,

2)
2)
2)
1)

5.2125
3.4016
2.0960
1.8160

(5,
(5,
(5,
(5,

2)
2)
2)
1)

9.5082
7.1505
5.0807
4.6866

(2,
(2,

(2,
(2,

2)
2)
2)
2)

0.2
1
5
10

3.0985 (3,
2.0070 (3,
1.1942 (3,
1.0243 (4,

2)
2)
2)
1)

3.1800
2.1120
1.3109
1.1483

(4,
(4,

(1,
(1,

2)
2)
1)
1)

6.7395
5.4255
3.6690
3.1575

(2,
(2,
(1,
(1,

2)
2)
1)
1)

6.7942
5.4763
3.7804
3.2803

(2,
(2,

0.7968
0.7176

(3,
(3,
(3,
(3,

1)
1)
1)
1)

1.7615
1.2218
0.8488
0.7672

(3,
(3,
(3,
(3,

1)
1)
1)
1)

2.9007
2.1341

2.0288
1.9180

(1,
(1,
(1,
(1,

1)
1)
1)
1)

0.2
1
5
10

0.3204
0.1948
0.1285
0.1171

(1,
(1,
(1,
(1,

1)
1)

1)
1)

1.3503
1.1552
1.0309
1.0236

(1,
(1,
(1,
(1,

1)
1)
1)
1)

1.8405
1.6575
1.5315
1.5255

(1,
(1,
(1,
(1,

1)
1)


0.8
0.6
0.4
0.2

R=3m, k=1,
m=2, n=2.
ξ0 =1e-5/h
ξ =2e-5/h
0

ξ =3e-5/h
0

0
0.65

0.75

τ
0.85

0.95

1.05

Fig. 9. Influence of initial imperfection on critical dynamic buckling load of
reinforced panel.


[6] Pradhan SC, Loy CT, Lam KY, Reddy JN. Vibration characteristics of functionally
graded cylindrical shells under various boundary conditions. Appl Acoust
2000;61:111–29.
[7] Sofiyev AH. The stability of compositionally graded ceramic–metal cylindrical
shells under aperiodic axial impulsive loading. Compos Struct
2005;69:247–57.
[8] Sofiyev AH, Schnack E. The stability of functionally graded cylindrical shells
under linearly increasing dynamic torsional loading. Eng Struct
2004;26:1321–31.
[9] Shariyat M. Dynamic thermal buckling of suddenly heated temperaturedependent FGM cylindrical shells under combined axial compression and
external pressure. Int J Solids Struct 2008;45:2598–612.
[10] Shariyat M. Dynamic buckling of suddenly loaded imperfect hybrid FGM
cylindrical with temperature-dependent material properties under thermoelectro-mechanical loads. Int J Mech Sci 2008;50:1561–71.
[11] Huang H, Han Q. Nonlinear dynamic buckling of functionally graded
cylindrical shells subjected to a time-dependent axial load. Compos Struct
2010;92:593–8.
[12] Budiansky B, Roth RS. Axisymmetric dynamic buckling of clamped shallow
spherical shells. NASA technical note D_510; 1962. p. 597–609.
[13] Liew KM, Yang J, Wu YF. Nonlinear vibration of a coating-FGM-substrate
cylindrical panel subjected to a temperature gradient. Comput Methods Appl
Mech Eng 2006;195:1007–26.
[14] Ganapathi M. Dynamic stability characteristics of functionally graded
materials shallow spherical shells. Compos Struct 2007;79:338–43.
[15] Sofiyev AH. The stability of functionally graded truncated conical shells
subjected to aperiodic impulsive loading. Int Solids Struct 2004;41:3411–24.
[16] Sofiyev AH. The vibration and stability behavior of freely supported FGM
conical shells subjected to external pressure. Compos Struct 2009;89:356–66.
[17] Sofiyev AH. The buckling of functionally graded truncated conical shells under
dynamic axial loading. J Sound Vib 2007;305(4-5):808–26.
[18] Prakash T, Sundararajan N, Ganapathi M. On the nonlinear axisymmetric

Mech 2011;33(3):132–47.
[27] Volmir AS. Non-linear dynamics of plates and shells. Science ed. M; 1972 [in
Russian].
[28] Brush DD, Almroth BO. Buckling of bars, plates and shells. Mc. Graw-Hill;
1975.
[29] Szilard R. Theory and analysis of plates. Prentice-Hall; 1974.
[30] Troitsky MS. Stiffened plates. Elsevier; 1976.