Step pressure and blast responses of clamped orthotropic hemispherical shells

Int. J. Impact Engng Vol. 8, No. 3, pp. 191-207, 1989

0734-743X/89 $3.00+0.00 © 1989 Pergamon Press pie

Printed in Great Britain

STEP PRESSURE AND CLAMPED ORTHOTROPIC

BLAST RESPONSES OF HEMISPHERICAL SHELLS*

C. C. CHAOand T. P. TUNG Department of Power Mechanical Engineering, National Tsing Hua University, Hsinchu, Taiwan, R.O.C. (Received 24 August 1988; and in revised form 10 January 1989)

Summary--This work is concerned with studies on the axisymmetric free vibration and'dynamic response of polar orthotropic clamped hemispherical shells subjected to suddenly applied loads, including idealized step pressure, a rectangular pressure impulse of finite duration and realistic blast pressure over the total or partial area. The mathematical model is formulated in terms of the mid-surface displacements and cross-section rotation with effects of transverse shear strain and rotary inertia taken into account. The Rayleigh-Ritz method is employed to solve the hemispherical shell vibration problem by assumed modes in Legendre polynomials, and the normal mode superposition is used in the analysis of its dynamic response. Numerical results for the natural frequencies, mode shapes, structural deformation and dynamic stresses are obtained with the fiber orientation parallel or perpendicular to the meridian direction as a design guide.

NOTATION Aij, Dij E~, E 2

E F G~z h I1, Is I,J K

[K], [M]

K,,M, P~ Qo r,O,~ R td tv

T, Tmax

{u}

U, Uma, V~ W

v'., wn, q'. {x}~ 2"

A K

4o P,~ ao, ep¢, Zepz to, [~

extension and bending stiffness, respectively longitudinal and transverse Young's moduli, respectively = s~E1+ ~sE~, quasi-isotropic modulus applied surface traction transverse shear modulus thickness integrals of loading half angle and time, respectively second and fourth mass moments of inertia, respectively number of layers stiffness and mass matrices, respectively modal stiffness and modal mass, respectively magnitude of pressure intensity Legendre polynomials of order n reduced stiffness constants spherical coordinates mid-surface radius of hemispherical shell duration of rectangular pressure impulse positive phase duration of blast pressure kinetic energy and its maximum during a cycle mode displacement vector strain energy and its maximum during a cycle meridian and radial mid-surface displacements, respectively generalized coordinates = ({Xb} T, {Xc} r, {Xd}V)n, eigenvector of mode n radial distance of a point from mid-surface transverse shear strain = 1 - vo,~v~

normal strain components transverse shear correction factor half angle of surface loading rotation of cross-section mass density per unit volume and surface area, respectively normal stress components transverse shear stress natural frequency and its nondimensional form

* Paper presented at the XVIIth International Congress of Theoretical and Applied Mechanics, 21-27 August 1988, Grenoble, France. 191

192

C . C . CHAO and T. P. TUNG

subscripts and superscripts k n, r

layer number of laminate mode number

Symbol

i )

e( ) =.-

1. I N T R O D U C T I O N

The free vibration and forced motion of spherical shells have been under vast investigation for many years because of their important role in structural engineering applications. A number of different cases of complete and open spherical shells have been solved in closed form by means of the Legendre polynomials and Legendre functions, as given in Kraus' [1] and Soedel's [2] books according to the traditional membrane and bending theories of thin shells, in which the Kirchhoff-Love hypothesis is observed. For the case of moderately increased thickness, effects of transverse shear and rotatory inertia were taken into account by Kalnins and Kraus when studying the vibration of spherical shells [3]. These effects were also considered in non-symmetric dynamic problems by Wilkinson and Kalnins, introducing cross-section rotation and other auxiliary functions [4]. For dynamic buckling, the studies of Jones and Ahn and of Song and Jones on elastic-plastic complete spherical shells [5, 6] are noted. In general, the existing literature is centered on spherical shells constructed of isotropic materials. Regarding composites, extensional vibrations of fiber-reinforced closed spherical shells were studied by Naghieh and Hayek for shells in vacuo and fluid-filled [7, 8]. Rath and Das treated the natural frequencies of axisymmetric vibration of layered complete spherical shells based on continuity of interlaminar transverse shear stresses for a general shell [9]. Ganapathi and Varadan analyzed the problem of dynamic buckling for orthotropic shallow spherical shells by using assumed modes [10]. Recently, Chao et al. conducted a series of research projects in these areas. Static and dynamic snap-through of thin orthotropic spherical caps was studied by taking moderate rotation and initial imperfection into account [11]. In Refs [12] and [13], and also in part of this work, the thick orthotropic shell theory is established in Fliigge's version in an investigation of buckling and vibration of complete spherical and hemispherical shells with the effects of transverse shear deformation and rotary inertia considered. The first part of this work deals with solution of axisymmetric free vibration of polar orthotropic hemispherical shells by means of the Rayleigh-Ritz method. Legendre polynomials of appropriate order are selected as admissible functions to satisfy the clamped edge conditions. In the second part, dynamic response of the orthotropic shell is investigated for suddenly applied loads including idealized step pressure, rectangular pressure impulse, and realistic blast loading over the total or partial area. Only in recent years has the blast response of structures been reported--by Houlston et al. and Gupta on isotropic plates [ 14, 15], and by Birman and Bert on laminated plates [ 16]. Dynamic studies of the current orthotropic hemispherical shell problem are carried out for all the loadings mentioned above by use of mode superposition [17].

2. VIBRATION ANALYSIS 2.1. Background Consider a spherical shell element of mid-surface radius R, as shown in Fig. 1. The radial, latitudinal and longitudinal coordinates of an arbitrary point in it are denoted by r, 0, ~ respectively. For the axisymmetric vibration, the displacements are 0-independent. Let w, v denote the mid-surface displacements in the directions of r and ~, and ~ denote the cross-section rotation at the parallel circle. Displacements of a point at distance z from

Step pressure and blast responses of hemisphericalshells

no~r

193

!

I FIG. 1. A spherical shell elementunder axisymmetricdeformation. the mid-surface can be written as

v,,(q~, z, t) = v(¢, t) + zO(4~, t) WA(~b,z, t) = w(q~, t).

(1)

Within the linear elastic limit of small deformation, the axisymmetric strain components are t0 = R ~ z (v cot ~b+ w + z~Ocot 4)) 1 ~* = r + z--(v + w + zO) 1 ? , ~ = - - ( -+v +z

(2)

w +

where

According to the generalized Hooke's law, we can put the stress-strain relation for the orthotropic spherical shell in the form

(3) "C,#z

Q44 7

with Q 11 = E o / A , Q22 = E , / A ,

Q12 = v eaoEo/A = Voc,E , / A Q** = G4~z,

A = 1 - Vo4~V¢,o.

2.2. R a y l e i g h - R i t z solution In dealing with a problem as complicated as vibration of an orthotropic hemispherical shell, it is unlikely that one can get an exact solution, as one can in the isotropic case [1]. A semi-analytical solution is sought by use of the Rayleigh-Ritz method, following Refs [12] and [13]. The shell is assumed to be in harmonic motion in time, with axisymmetry all over the surface. Series expansion is made longitudinally along the meridian in terms of spherical harmonics (Legendre polynomials of the first kind), since the hemispherical shell is closed at the apex. At the clamped edge, the fixed boundary conditions v=w=qJ=O

at

q~ = r~/2

(4)

194

C.C. CHAOand T. P. TUNG

are met by proper choice of the Legendre polynomials of the right order. We can therefore write the displacements and rotation as follows: v(q~, t) = ~ bfi(cos ~b) sin oJt

l even

w(~b, t) = ~ c,.P,.(cos ~b) sin ~ot

rn odd

~,(~b, t) = ~. d.P.(cos 4') sin ~ot

n even,

l

m n

(5)

where bz, c,., d. are the undetermined coefficients for the series expansion. From the theory of anisotropic elasticity, we have

T=2 ~" fvk pk(O2A't+ W]t)kdV'

(6)

1~, U=2k=l

(7)

fv k (Q11e2+2Qt2e°E4~+Q22e~+ ~cQ44'~z)kdV'

where K is the number of layers of the laminated hemispherical shell. During a complete cycle of the vibratory motion of the hemispherical shell, the maximum kinetic energy and the maximum strain energy are obtained by use of equations (1), (2) and (5)-(7) as Tmax = ~ZW2

(R2/5 +/-)

b2iP21 + Czi-iP2i-1

dO

+4R b2iP'2i d2jP2j+ (RI +J) d2iP'2i

sin q~d~b

(8)

J

Um.x=~/Z{Al,[(~i b2iP'2icotdp)2+2(~ib2iP'2i)(~ic2j-lP2j-x)cot~

+2A12[(~b2,P2,)(~jb2jP2j)c°t~+(~i b2iP2i)(~c2j-xP2J-~)c°tdP

2.+

2

)2

+D22(~d2,P~,) A4't[(~b2,P'2,) +(~c2,-1P2,-1

i , j = 1,2,3 . . . . . N,

195

Step pressure and blast responses of hemispherical shells where N is the number of terms for series truncation in equation (5), and

(~, I, J) =

Ok

(1, Z2, Z4) dz

k=l

k-I

(Aij, Dij) = ~

tk) 2)dz Qij(1,z

k=l

.

i,j=l,2

k-1

A44 = ~c

~44 ' ~ ,

k=l

k-1

where ~c= x2/12 is the transverse shear correction factor.

2.3. Frequency equations It is seen that the Tn~,x and Ureax are dependent on the undetermined coefficients b2n , c2.-1 and d2.. Using the Rayleigh-Ritz principle, i.e. ,

--

,

(Tmax - -

Umax)

=

n = 1,2, 3 . . . .

0

N,

(10)

t~b2. c9c2n- 1 ~

we obtain the following set of frequency or characteristic equations for the free vibration of the clamped orthotropic hemispherical shell:

(A 11 - A 22)Fe(i)b2i + ~ (A 11 -- A 22)Fe(n)b2j i

d

+ [AaxGe(n) + 2Ax2H¢(n ) + A22K~(n)

+ A44Le(n)]b2n

+ ~ [(Aaa + A12)X(i, n) + (A12 + A22)S(i, n ) - A44Y(i, n)]c2i_ 1 -- RA4.Le(n)d2. i

(ll)

= fo2[(R2/5 + -OLe(n)b2. + 2RrL~(n)d2,] [(All + Al2)X(n, i) + (A12 + A22)S(n, i ) - A44Y(n, i)]bzi i

+ [(All + 2A12 + Azz)To(n) + A44Lo(n)]c2n- 1 + E RA44Y(n, i)d21 i

= ¢°2(R2/5 + l)To(n)c2.- 1 --RA44Le(n)b2.

(12)

+ ~ RA44 Y (i, n)c2i_ 1 + ~ (Dxl - D22)Fc(i)d2i i

i

+ ~. (Dll -- D22)F~(n)d2j + [R2A44Le(n) + D11Ge(n) + 2D12/-/~(n) + D22K¢(n)]d2. J

= ¢o2[2Ri-L¢(n)b2, + ( R 2 I + f)Le(n)d2.], i = 1,2,3 . . . . . n - 1 ; j = n +

(13)

l , n + 2 . . . . . N.

Symbolically, equations (11)-(13) can be put in the matrix form of an eigenproblem: ([K] - co2[M]){X}, = {0}

r = 1, 2 . . . . , 3 N ,

(14)

where [ K ] and [ M ] are the 3N x 3N stiffness and mass matrices, respectively, the eigenvalue f O r2 is the squared natural frequency of an arbitrary mode r, and the eigenvector {X}~= (b2 b,.-"bzN]Cl C a ' ' "C2N-~ ]d2 d4.. "d2N), = ({xb} T { x o V

196

C . C . CHAO and T. P. TUN(;-

is the corresponding normal mode. A number of special integrals, through the energy formulation in equations (8) and (9) for Tmaxand U. . . . are obtained by using orthogonality and some integral relations [18, 19] (see Appendix). 3. D Y N A M I C

RESPONSE

3.1. Normal mode superposition

The normal modes of the hemispherical shell have been established in the vibration analysis. If there are any repeated frequencies, the associated modes are assumed to have been orthogonalized so that the orthogonality conditions { X } r [ M ] { X } s = {X}r[K]{X}s = 0

(15)

are satisfied for all r ~ s. When r = s, the modal mass and modal stiffness are obtained respectively as follows: M, = {x}T[M]{X},

N = ~ {[(RZt3 + l--)b~. + 4R-i-bz.d2, + (R2/-+ Y)d~.]Le(n) + (RZ~ + T)c~._l To(n)} n=l _

(16)

2

Kr - cot Mr, where Le(n) and To(n) are given in the Appendix. The hemispherical shell is now subjected to various time-dependent surface tractions acting inward in the radial direction, in the form of on external pressure F(4~, t). The system response can be approximated by N

v(4,, t)= E P2.(cos 4,)v.(t) n=l N

w(qS, t) = ~ P2.- x(COSqb)W.(t) n=l N

q/(~b, t ) = ~ P2.(cos ~b)tP.(t),

(17)

n=l

where the Legendre polynomials are once again used as the admissible functions for the satisfaction of the clamped edge conditions. Going through Lagrange's equations, we obtain the following equations of motion of a linear 3N degree-of-freedom system: [M]{u},tt + [K]{u} = {f(t)},

(18)

where [M] and [K] are the same mass and stiffness matrices of the orthotropic hemispherical shell as obtained before in the vibration analysis. The generalized coordinates are partitioned in groups following equations (14) and (17) such that {u(t)} T = (V, V2. . . VNI W 1 W2. . . WNllY~1 kI/2'' "LPN) = ( { g } T { W } T {kI/}T)

and the generalized forces are non-vanishing for the radial components only: {f(t)} T = (0 0 . . . 0 [ p 1 P2"" "Pt¢l0 0 - . . 0 ) = ({0}T {p}T {0}T),

where p.=R 2

f n/2 F(c~,t)P2._l(cosc~)sin~d#6, dO

n - - 1 , 2 . . . . . N.

By the use of the normal mode superposition method, the set of coupled equations can

Step pressure and blast responses of hemispherical shells

197

be transformed into a set of uncoupled equations by introducing the coordinate transformation [17] 3N

{u(t)} = Z

{X}rqr(t)'

(19)

r=l

where {X}r is the rth normal mode of free vibration and q,(t) is referred to as the corresponding principal coordinate. By substitution of equation (19) into (18), the resulting equations, after being premultiplied by {X},T and divided through by M r = Kr/O22,become uncoupled in the principal coordinates 2 gtr+e)rqr=Q,

r = l , 2 . . . . . 3N,

(20)

where

F(c~,t)

Qr(t) =

C2n_IP2n_I(COS~) sin t~ d~b.

General solution of equation (20) is obtained by use of the convolution integral and the specified initial conditions

qr(t)=qr(O)cosco,t +lor(O)sine)rt + co,

3.2.

lfo

Qr(z) sine)r(t-z)dz.

(21)

Suddenly applied loads

The three types of suddenly applied load considered in this study are uniform external pressures exerted axisymmetrically with respect to the apex, presumably over the midsurface for simplicity, subtending a total of 2~bo in spherical angle, with different forms of time dependence. The forcing function can be expressed as F(~b, t) = -p[H(~b) - H(~b - q~o)]F(t),

(22)

i0 being the magnitude of the pressure intensity. The hemispherical shell is assumed to be initially at rest. Substituting equation (22) and the zero initial conditions qr(0)= Or(O)= 0 into equation (21) gives

q,(t) = l~r)((ao)l~')(t),

(23)

l~')(4~o)= f ? [ ~ c2,_ lP2,_ l(cos c~)] sinc~d4~

(24)

where

Pzr)(t)=

- Mr~°rR2~fo F(z) sin ~or(t - z)dr.

(25)

For a specific mode r, the first integral I]r) which is to be evaluated through numerical integration of the expanded series depends on the loading half angle ~bo. The case ~bo < n/2 represents partial area and the case q~o = rr/2 represents total area for uniform pressure distribution over the hemispherical shell. The second integral I~r) is a function of time which in turn depends on the form of the force-time relationship. The three types of suddenly applied load are treated as follows. 3.2.1. Step loadin9. A step pressure of intensity /~ is suddenly applied onto the hemispherical shell for infinite duration such that

F(t) = -~H(t) R2[~ l~zr)(t)- - -

(26) (1 - cos ~ort).

(27)

198

C . C . CHAO a n d T. P. T U N G

3.2.2. Rectanoular impulse. A step pressure of magnitude/? is suddenly applied onto the system for a finite duration t d and then released. In this case, we have F(t) = - fi[H(t) - H(t - ta)] lt2')(t) -

RZP

(28)

{[cos ~o,(t - ta) - cos ¢o,t]H(t - ta)}.

(29)

M,(Dr 3.3.3. Blast pressure. As the last example, we consider the case of an orthotropic hemispherical shell under explosive blast. The peak pressure is reached instantly in general, and uniform distribution of the pressure can be assumed over the structure [14]. The overpressure-time history is written in the form of modified Friedlander exponential decay [15, 16]: F(t) = --/3[(1 - t/tp) e-"(t/tp)], (30) where p is the peak reflected pressure in excess of the ambient pressure, tp is the positive phase duration of the pressure impulse and a is a coefficient for the blast lt2r)(t) =

R2p { 1 M,[(a/tp)2 + ¢02] (1 - t/tp)e -"~'/'°) + - - (a/tp) sin %t - cos (D,t % x 2(a/tp) e-"~'/'p) -~

1 [(a/tp) 2 + w,2]tp

sin co,t - 2(a/tp) cos (D,t

.

(31)

(Dr

3.3. Deformation and dynamic stresses Suppose that the clamped orthotropic hemispherical shell is subjected to one of the three types of suddenly applied load as we discussed above. The modal integrals lt[)(~bo) and lt2")(t) can be obtained from equations (24) and (27), (29) or (31) respectively, and the mode displacements from equation (19) in partitioned form as W(t)~= ~

~

I~"(dPo)l~2"(t).

(32)

The deformation of the hemispherical shell mid-surface is described by N

e2.(cos 4))v.(t)

t)= rim1 N

w(q~, t ) = ~ P2,_,(cos ~b)W.(t)

(17)

n=l N

~J(q~, t ) = ~ PE.(COS ~)V.(t), n=l

and the dynamic stresses anywhere in the shell are given as N

as(z, dp, t) -

Q1, ~ {PE.(COS ~)[V.(t) + z~.(t)] cot 4~ + P2._x(cos ~)W.(t)} R+z.=l N

+ Qx2 ~ {P2.(cos ~b)[V,(t) + zV.(t)] + P2.-,(c°s ~b)W.(t)} R+z.=l N

Q12 ~ {P2.(cos q~)[V.(t) + zW.(t)] cot q~+ P2.- 1(cos 4~)W.(t)} %(z, 4), t) - R + z . = 1 N + Q22 ~ {P~/.(cos ~)[V.(t) + zW.(t)] + P2.-I(cos 4~)W.(t)} R+z.=I z,z(z, q~, t) - ~cQ44 ~ {P2.(cos 4 0 [ - V.(t) + RW.(t)] + P~n_l(COS R+z.=l

~b)W.(t)}.

(33)

Step pressure and blast responses of hemispherical shells

199

4. RESULTS AND DISCUSSION Numerical verification of the consistency and versatility of the orthotropic theory has already been made for complete spherical shells in the simplified case of construction from isotropic and transversely isotropic materials for buckling and free vibration analysis [12, 13]. The theory should also apply to the present case of clamped hemispherical shells, so far as the specific edge conditions are satisfied. The hemispherical shell under consideration is made of polar orthotropic graphite/epoxy lamination, with mechanical properties as follows: E 2 -----4.8 G P a ,

G12=G13=O.5E2,

E 1 = 25E2, G23 = 0.2E2,

v12 = 0.25, p = 2547 kg m -3 .

To show the effects of transverse shear strain and rotary inertia, predictions are made for thickness-to-radius ratios of up to 0.1.

4.1. Natural frequencies and mode shapes The problem of free vibration of clamped orthotropic hemispherical shells is treated first. A convergence test is carried out for the series truncation of Legendre polynomials in expressing the mid-surface displacements v and w and the cross-section rotation ~b. A total of 3N = 36 terms is enough to guarantee a sequential difference as low as 0.010% for the fundamental mode and 0.024% for the second mode. Numerical results for the relationship between the natural frequencies and shell thickness are presented in non-dimensional form as ~=ogR(p/E) 1/2 versus h/R in the range 0 <<.h/R <<.0.1, as shown in Fig. 2(a), (b) with two different orthogonal fiber orientations. An average representative stiffness g = 3E 1 + ~E 2 is used for the composite material on the basis of a quasi-isotropic equivalence after Tsai and Pagano [20]. In each case, the first six modes starting from the fundamental at the bottom are all of the transverse type, as shown in the figures, and the non-dimensional natural frequency increases with the increase in the shell thickness ratio as a general rule. For thin orthotropic hemispherical shells, higher frequencies are found when the fibers or E1 are oriented perpendicular to the meridians. For the thick shells, on the other hand, the frequencies are higher when the fibers are oriented parallel to the meridian. In so doing, higher stiffness is offered in a global sense with the aid of the inherent fixity of the clamped edge. Transition of this kind of frequency behavior takes place in the range 0.023 ~
200

C.C. CHAO and T. P. TUNG

(a) E~ ± meridian E= = 4.8 GPa ,

E, = 25 E2

G,= = G~ = 0.5 Ez ,

~

2

P,z = 0 . 2 5

,

Gu = 0.2 Ez = 2 5 4 7 kg/m 3

p

&

_________

1i [

,

I

,

I

,

2

I

,

I

,

4

1

,

I

,

6

1

,

l

,

10

8

(b) E, / / m e r i d i a n

.~

2

T,

,~

T,

"3

T,

II

1

T=

_

0

T,

I

0

,

i

2

,

I

,

J

,

i

,

4

I

,

I

6

,

I

8

~

I

,

10

h/R ( I 0 -z) FIG. 2. Non-dimensional natural frequencies versus thickness-to-radius r a t i ~ l a m p e d ortho-

tropic hemisphericalshells. Mode

Steel

E, ± m e r .

E, d mer.

FIG. 3. Mode shapes of free vibration--clamped isotropic and orthotropic hemispherical shells.

histories of the significant dynamic stresses, i.e. the hoop stresses a e in Fig. 5(a) and the longitudinal stresses a, in Fig. 5(b), with the fiber orientation E 1 perpendicular to and parallel to the meridians respectively at the critical stressed locations, where these local stresses will be important in the prediction of 'impulsive strength' and failure of the orthotropic hemispherical shells. All results shown in Figs 4 and 5 pertain to a thick orthotropic hemispherical shell of h / R = O. 1.

Step pressure and blast responses of hemispherical shells

201

2O

1o!

E, .L mer . E,//

|

io

.

td =4ms . .

t , =4ms

.......................

mer

i

td = Go . .

.

"

r

~.

f'

,:=-

-10

-20 5

0

10

15

20

t ~lflS

FIG. 4. Time history of apex displacements---clamped orthotropic hemispherical shells subjected to total area pressures.

150

_~

(a) E,I mer.//~

100 -,,

50

~,

o

M

': : ' / / \

\"//

~\\_!,¢", \ i \ / I

\, ; \/ ',, \ ,,'i ',>,i,; ""

-50

v

%.'

\ /

t~=co

-100 -150

l

i

I

I

i

i

i

~

t~=4ms G=4ms I

5

Xi

I

I

I

I

i

10

I

I

I

i

15

20

100 I^

(b) E, llmer.

CM

-5o _

,J td = co

-lOO

,

0

,

,

I

5

,

,

,

,

f..,, =4as

I

10

,

,

x

tp =4as

~

I

15

,

,

,

,

20

~lm8

FIG. 5. Time history of significant dynamic stresses at critical points--clamped orthotropic hemispherical shells subjected to total area pressures.

202

C . C . CHAO and T. P. TUNG

4.3. Dynamic stresses--partial pressure The same types of pressure loading as above are considered over a partial area of the orthotropic hemispherical shell for axisymmetric uniform distribution. Cases of loading half angle q~o = 10°, 30°, 60° and 90 ° are studied at different times for t = 1, 2, 5 and 10 ms. It is noted that the shell deformation and dynamic stress distributions for total pressure, i.e. ~bo--90 °, are nearly the same as for q~o = 60° with no need for separate presentation under all three types of load. This seems only reasonable, since most of the loading applied beyond q~o = 60° is taken by the clamped edge support directly. In all cases, the shell stress distributions are positive and negative dominated alternately in time. The hemispherical shells under investigation are also of the thick type with h / R = O. 1. In all cases, only the mid-surface deformation and dynamic stress distributions of primary concern are presented. This is done by outward normal vectors drawn to scale from around the mid-surface, whose maximum is clearly marked wherever applicable for reference. Near the clamped edge, the stresses are all so small that they can hardly be seen at the scale of the diagrams owing to the small change in curvature of the deformed shell configuration. The radial displacements have maxima at the apex most of the time for the various loading types and loading angles, and are shown with exaggerated scales to demonstrate the deformed configuration of the hemispherical shell. In Figs 6-8, the fiber orientation E~ is considered to be perpendicular to the meridians. Firstly, we take the loading case of unit step pressure. Distributions of the radial displacements w and the most significant hoop stresses are shown at around the hemispherical mid-surface in Fig. 6(a) and (b) respectively for the specific moments. Secondly, cases of the idealized rectangular pressure impulse and the realistic blast pressure are examined in comparison in Figs 7 and 8 for the same force duration of 4 ms. For the rectangular pressure impulse, the shell dynamic response is exactly the same as those of the

~ =I MPa

qD0 = I 0 °

t = For

I ms

5 ms

2 ms .

.

.

.

.

.

.

.

.

10 ms

.

. ~-~..

~F~-~. ~,',

,'g X~, /

t

'\

,:Y I

rl

~#

~0 = 30o

I

\

t

I

~o = 60 +

(a) Radlal d i s p . , w

(b} Hoop stress, if0

FIG. 6. Shell deformation and dynamic hoop stresses due to step pressure for E 1 oriented perpendicular to meridians.

Step pressure and blast responses of hemispherical shells

,/,

,

t =

~ =1 MPB

I ms

5 ms

2 ms

~OP

/

203

10 ms

..........

~

'l

\

/z

',

,'

(Do

=

x

300

o

77

I

(Do = 60 °

~

i

o

-~'-"~O*~

N o

(a) (re - Rect. impulse

(b) fie- Blast

FIG. 7. Dynamic hoop stresses due to rectangular impulse and blast pressures for E 1 oriented perpendicular to meridians.

r

r

~" =1MPa (Do = 10 °

D

35MPa 1"~ I

i,

(Do = 30o

44MPa

(Do = 600

(a) £*z- Rect.

impulse

(b) £ez- B]ast

FIG. 8. Dynamic transverse shear stresses due to rectangular impulse and blast pressures for E 1 oriented perpendicular to meridians.

204

C . C . CHAO and T. P. TUNG

~- =1MPa

~o = 10°

o

t

=

1 ms

2 ms

5 ms

10 ms

~III

~* = 30*

o

(a) Radial d i a p . , w

(b) Longitudinal ,if.

Fro. 9. Shell deformation and dynamic stresses due to step pressure for E, oriented parallel to meridians.

unit step pressure before the end of the specified duration. More fluctuations are noted in the period of residual vibration as a result of removal of the applied load. As to the blast pressure, the shell dynamic responses are much milder, about half of those of the rectangular pressure impulse, owing to the load relief of modified Friedlander exponential decay. Under such an arrangement for fiber orientation E 1 perpendicular to the meridians, the hoop stresses a o are of vital importance, and the transverse shear stresses z~= also have considerable magnitudes. Distribution of these dynamic stresses around the hemispherical mid-surface is demonstrated in Figs 7(a), (b) and 8(a), (b) for the purposes of comparison between the two types of rectangular impulse and blast pressure. In Figs 9 and 10 are shown respectively the dynamic responses of the hemispherical shell subjected to unit step pressure and the rectangular pressure impulse along with blast pressure, for the fibers or E1 oriented parallel to the meridians. As in the analysis of free vibration, higher stiffness is obtained for the orthotropic hemispherical shell as a whole with parallel fiber orientation since no further reinforcement is needed edgewise in addition to the inherent stiffness of the clamped edge. Under the action of the unit step pressure, the maximum radial displacement, which also occurs at the apex, as shown in Fig. 9(a), is reduced to about 1/10 of that shown in Fig. 6(a) for a hemispherical shell similar except that the fibers are otherwise oriented. In the mean time, the most significant stresses turn out to be the longitudinal a~ alone, as shown in Fig. 9(b). No joint contribution is required of the transverse shear stresses T~z in resisting the applied external load. As a result of the rectangular pressure impulse and the unit step pressure, the same argument holds for the longitudinal stresses tr~ around the hemispherical mid-surface in Figs 10(a) and 9(b) for the present meridianwise fiber orientation as for the hoop stresses tr0 in Figs 7(a) and 6(b) where the fibers were oriented perpendicularly. As to the blast pressure loading, the longitudinal stresses tr~ are now increased by 24.5% in Fig. 10(b) as compared to the

205

Step pressure and blast responses of hemisphericalshells ~ =I MPa

(:I)0 = I 0°

t =

| ms

2 ms

5 ms

10 ms

..........

For

~* = 30*

o

o

(a} fro-

Rect. impulse

(b) if,- Blast

FIG. 10. Longitudinaldynamic stresses due to rectangular impulse and blast pressures for E1 oriented parallel to meridians. previous hoop stresses in Fig. 7(b) for lack of support from the transverse shear stresses as in the previous case. 5. CONCLUSION By using orthogonal Legendre polynomials, semianalytic solutions are obtained for free vibration of clamped edge polar orthotropic hemispherical shells. Also found is the dynamic response to suddenly applied loads consisting of idealized step pressure, rectangular pressure impulse of finite duration and realistic blast-type pressure over the total or partial area of the shell surface. Effects of transverse shear strain and rotary inertia are taken into account for high thickness-to-radius ratios. In the free vibration analysis of clamped orthotropic hemispherical shells, natural frequencies of all modes increase with increase in shell thickness. Of the two orthogonal fiber orientations, frequencies are higher for thin shells when E1 is oriented perpendicular to the meridians; for thick shells, the frequencies are higher when E1 is parallel thereto. The mode shapes are slimmer with E~ perpendicular to the meridians and are fatter with E 1 parallel. Dynamic study indicates that for the same peak pressure and force duration, the blast pressure is milder because of its exponential decay in load intensity as compared to the rectangular pressure impulse. When E: is perpendicular to the meridians, the hoop stresses play a leading role in resisting the dynamic loads, with the transverse shear stresses as a significant supplement. The combined higher stresses imply an earlier failure of the orthotropic hemispherical shell. As a result of increased stiffness with the meridianwise fiber orientation arrangements, the apex displacements reduced to 10% of those in the first case. The longitudinal stresses alone in the resistance to the applied loads are significantly reduced. In conformity with the clamped edge effect, arrangements for meridianwise fiber orientation are desirable.

206

C.C. CHAO and T. P. TUNG

Acknowledgements- -The authors are grateful to the National Science Council for support of this research under Projects NSC 76-0405-E007-04 and 77-0405-E007-09. Special thanks are also due to the Chung Shan Institute of Science and Technology for support of the second author's Ph.D. program at this University.

REFERENCES 1. H. KRAUS, Thin Elastic Shells, pp. 333-394. John Wiley, New York (1967). 2. W. SOEDEL,Vibration of Shells and Plates, pp. 124-132. Marcel Dekker, New York (1981). 3. A. KALNINS and H. KRAUS, Effects of transverse shear and rotary inertia on vibration of spherical shells. Proc. 5th U.S. Nat. Cong. Appl. Mech., p. 134. ASME (1966). 4. J. P. WILKINSON and A. KAENINS,On nonsymmetric dynamic problems of elastic spherical shells. J. appl. Mech. 32, 525-531 (1965). 5. N. JONES and C. S. AHN, Dynamic elastic and plastic buckling of complete spherical shells. Int. J. Solids Struct. 10, 1357-1374 (1974). 6. B. SONG and N. JONES, Dynamic buckling of elastic-plastic complete spherical shells under step loading. Int. J. Impact Engn# 1, 51-71 (1983). 7. M. NAGHIEH and S. I. HAYEK, Vibration of fiber-reinforced spherical shells. J. Fiber Sci. and Tech. 4, 115-135 (1971). 8. M. NAGHIEH and S. I. HAYEK, Vibration of fluid-filled fiber-reinforced spherical shells. J. Sound Vib. 19, 153-166 0971). 9. B. K. RATH and Y. C. DAS, Axisymmetric vibration of closed layered spherical shells. J. Sound Vib. 37, 123-136 (1974). 10. M. GANAPATHI and T. K. VARADAN,Dynamic buckling of orthotropic shallow spherical shells. Comput. Struct. 15, 517-520 (1982). 11. C. C. CHAO and I. S. LIN, Snap-through buckling of orthotropic shallow spherical shells. Proc. 11th ROC Nat. Conf. Theor. Appl. Mech., pp. 517-528 0987). 12. C. C. CHAO,T. P. TUNG and Y. C. CHERN, Buckling of thick orthotropic spherical shells. Composite Struct. 9, 113-137 (1988). 13. C. C. CHAO and Y. C. CHERN, Axisymmetric free vibration of orthotropic complete spherical shells, d. Composite MatiN 22, 1116-1130 (1989). 14. R. HOULSTON, J. E. SEATER, N. PEGG and C. G. DES ROCHERS, On the analysis of structural response of ship panels subjected to air blast loading. Comput. Struct. 21,273-289 0985). 15. A. D. GUPTA, Dynamic analysis of a flat plate subjected to an explosive blast. Proc. ASME Int. Comput. Engng Conf. 1, pp. 491-496 (1985). 16. V. BIRMAN and C. W. BERT, Behavior of laminated plates subjected to conventional blast. Int. J. Impact Engng 6, 145-155 (1987). 17. R. R. CRAIG, JR, Structural Dynamics, pp. 189-379. John Wiley, New York (1981). 18. O. J. FARREEE and B. Ross, Solved Problems: Gamma and Beta Functions, Legendre Polynomials, Bessel Functions, pp. 118-234. Macmillan, New York (1963). 19. L. C. ANDREW, Special Functions for Engineers and Applied Mathematicians, pp. 116-165. Macmillan, New York 0985). 20. S. W. TSA! and N. J. PAGANO, Invariant properties of composite materials. In: Composite Materials Workshop (Edited by S. W. TSAI, J. C. HALPIN and N. J. PAGANO), pp. 233--253. Technomic, Stamford, Connecticut (1968).

APPENDIX INTEGRAL

FORMULAS

In the following integrals, the Legendre polynomials are written for brevity as P, = P.(cos ~b), P~ = e~(cos 95). . . . . etc. For the hemispherical shells, we have

-

F~(n) =

P~.kP2.cot 2 ~bsin q~dq~= n(2n + 1)

Fc(n)

P~kP~n sin ~bd~b

Ge(n) =

(P:2n)2 cot 2 ~ sin ~ d~b = n(2n + 1)(4n - 1)/(4n + 1)

H.(n) =

(P2~)(P~) cot 4~sin ~ dq~ = n(2n + 1)/(4n + 1)

Step pressure and blast responses of hemispherical shells

['n/2 K¢(n) = Jo

(p~.)2 sin q~d¢ = n(2n + 1)(8n 2 - 1)/(4n + 1) n/2

Le(n) = Jo

(p2~)z sin ~d~b = 2n(2n + 1)/(4n + 1)

~n/2

L°(n) = Jo

(P2n- 1)z sin q~dqb = 2n(2n - 1)/(4n - 1)

~' n/2

T°(n) = Jo S(m, n) =

(P2"-1)2 sin q~d~b = 1/(4n - 1)

f(2

P~i~P2,~1 sin ~ d4~

X(m, n) = ;f/2 P2.P2m- 1 cot 4~ sin ~d4~ Y(m, n) =

P2.P2m- ~ sin ~ d4~ re, n = 1 , 2 , 3 . . . . .

N;k>n.

207