Optimal stiffening of square plates subjected to air-blast loading

Optimal stiffening of square plates subjected to air-blast loading

Compurers & Sfrucfures Vol. 36,No.5.pp.891-899, I990 004%7949/90 s3.00+0.00 Q 1990Pergamon Press plc Printed in Great Britain. OPTIMAL STIFFENING ...

727KB Sizes 0 Downloads 28 Views

Compurers & Sfrucfures Vol. 36,No.5.pp.891-899, I990

004%7949/90 s3.00+0.00 Q 1990Pergamon Press plc

Printed in Great Britain.

OPTIMAL

STIFFENING OF SQUARE PLATES SUBJECTED TO AIR-BLAST LOADING M. V. DHARANEEPATHY and K. G. SUDHESH Structural Engineering Research Centre, Madras 600 113, India (Received

1 May 1989)

Abstract-Studies carried out to determine the free and forced vibration responses of a square clamped plate with three different stiffener patterns are described. The aim is to arrive at an optimal lay-out of the stiffeners for a given mass. To achieve this objective, the response of an equivalent solid (unstiffened) plate are compared with those of the three stiffened plates. At the outset, it is shown that blast loads induce only the fundamental symmetric mode. Then, the eight lowest natural frequencies and mode shapes, as well as the blast responses, are determined for stiffened and unstiffened plates. For a given mass the plate, which has maximum fundamental natural frequency and minimum blast response, is deemed to be superior to the rest. T’he results obtained facilitate the conclusion that a plate with a rhombus rib is superior to one with conventional bi-directional orthogonal stiffener for the resistance of blast loads.

INTRODUCTION Stiffened plates are lightweight, high-strength structural elements, commonly used in buildings, bridges, ships, offshore structures and aircrafts. In modem warfare, aircraft and naval ships are subjected to high-intensity air blast and underwater shock loads. A knowledge of the response of stiffened plates under such loading conditions is essential, not only to assess their vulnerability and survivability, but also to improve their design. The lay-out of the stiffeners should be consistent with the natural modes likely to be excited by the service loads, so as to arrive at a design having a high strength-to-weight ratio. The functional requirement of a stiffened plate is to resist either buckling or bending, and the orientation of the stiffeners may be in one or more directions to satisfy such requirements. Hofmeister and Felton [l] investigated the concept of ‘pyramiding’ of stiffeners for the least-weight design of waffle plate structures. They examined a variety of stiffener patterns in an attempt to improve their buckling strength, and conclude that two levels of stiffeners are significantly more efficient than conventional waffle plates. The concept of multiple stiffener sizes in plate and cyclinder design is devised to provide an alternative to honeycomb construction, whose costly fabrication and inspection procedures have rendered it undesirable for many applications. Fabricational constraints and functional requirements generally dictate the spacing and eccentricity of the stiffeners. Plates designed to resist bending are usually stiffened with unidirectional or bi-directional stiffeners, running parallel to their edges. Long [2] investigated the effect of eccentricity on the natural frequencies and mode shapes of a unidirectionally stiffened, simply supported rectangular plate. He analysed the

plate with one, two and three stiffeners, uniformly placed parallel to the short side. It is interesting to note that the fundamental frequency is unaffected by the number of stiffeners, until the ratio of stiffener eccentricity to plate thickness exceeds a value of about 2.5. Obviously, the effect of such unidirectional shallow stiffening on the response to forces likely to excite the fundamental modes would be insignificant. The aim, in this paper, is to arrive at an optimal layout of stiffeners to improve the flexural strength of clamped square plates of the same mass. Taking into consideration the fabricational constraints and the modes likely to be excited by blast loads, only three types of stiffener configurations are analysed for blast response. The scope of this study is limited to only linear analysis, as the aim is to arrive at an efficient stiffener configuration. Blast-resistant design, being outside the perview of this study, is not carried out here. PROBLEMSTATEMENT A square plate measuring 508 x 508 mm, of thickness 3.4 mm, clamped on all sides, is considered. The experimental and theoretical results for the blast response of this plate, in an unstiffened condition, have heen compared elsewhere [3]. The aim here is to study the improvements in its response obtained by different stiffener orientations. The parameters under investigation are the natural frequencies and blast response in terms of central displacement and element stresses. FINITE ELEMENTMODEL The plate (Fig. 1) is discretised into a 12 x 12 mesh of 144 rectangular (square in this case), four-noded 891

M. V.

892

DHARANEZEPATHYand

K. G.

SUDHESH

where W is the equivalent charge weight and x is the actual distance of the plate from the charge.

CHOICE

I-

so&WI

-4

Fig. 1. Finite element model. plate elements, developed and detailed in [+6]. Stiffener elements developed by Karve [7] for rectangular plate elements are used to model the stiffeners, running parallel to the X- and Ycoordinate axes. The stiffeners running skew to these coordinate axes are modelled as rectangular plate-shell elements [8].

BLAST LOAD MODEL

The reflected overpressure on plates due to air blast is known, from experiments [3], to be uniform. Its decay with time is generally governed by the modified Friedlander exponential function P(r) = P,[l

- t/fp]exp(-a’f/tp],

(1)

where Pm is the peak pressure, a’ is the pressure decay-rate correction coefficient and tPis the duration of the positive phase of the pulse. The reflected impulse, I,, on the surface of the plate is given by [9]

OF STIFFENER

PATTERN

As already stated in the Introduction, the stiffener layout pattern should be designed taking into consideration the modes of natural vibrations likely to be excited by the service loads. Depending on the nature of the loads, the dynamic response of the plates may involve either symmetric modes only, or both symmetric and antisymmetric modes. To demonstrate this feature, the square plate under consideration is subjected to a unit point load of pink-noise, with a frequency band of O-500 Hz, at the centre of one quarter of the plate. The asymmetry of this load is seen to induce symmetric as well as asymmetric modes (Fig. 2). If the same load is applied symmetrically, at the centres of all four quarters as well as at the centre of the plate, then the response spectrum (Fig. 3) contains only the symmetric modes, the fundamental being the most dominant. Air blast loads, being symmetrically distributed pressures [3], would hence excite only the fundamental mode of the plate. From an examination of this mode shape (Fig. 4), one is inclined to investigate the effectiveness of centrally concentrated stiffener patterns, in comparison with the conventional bidirectional (waffle) patterns. There need not be any objection to such central concentration of stiffeners, even from the inertial point of view, as the stiffener contributes a lot more to stiffness than to mass. Based on these considerations, three types of stiffener patterns (Fig. 5), two of diamond shape and one of the conventional waffle pattern, are chosen for their performance assessment in relation to the unstiffened plate.

‘P I, =

P(t) dt.

(2)

I0 Substituting results in

the expression

I, =

for P(r)

from eqn (1)

P,tp[u' - 1 + e-o’]/(a’)*.

(3)

Equation (3) is useful for determining the decay-rate coefficient, a’, from a knowledge of the parameters I,, P,,, and tp, which, in turn, are taken from standard tables [lo, 111. To use these tables, the scaled distance, x0, of the plate from the charge location is required to be computed by the cube-root scaling law [12] x0 = x/W”‘,

(4)

0.00

1.61

FREQUENCY

3.22 I HZ

1

4.83 x102

8.45

Fig. 2. Response spectrum of unstiffened plate subjected to eccentric pink-noise quarter-point load.

Optimal stiffening of square plates

893

I

0.00

1.81

4.83

3.22

FRECUENCY

(HZ1

8.45

x102

Fig. 3. Response spectrum of unstiffened plate subjected to central pink-n&e load.

PATTERN -

B

PwEnPEf

COMPUTATIONALALGORITHM Free vibration

The natural frequencies and mode shapes of the plate, with and without stiffeners, are determined by solving the generalised eigenvalue problem

where K and M are the structural stiffness and lumped mass matrices, while A and 4 are the matrices of eigenvalue and mode shapes. A subspace iteration algorithm [13], implemented incore, is used for this purpose.

PAYIERN-c PlATETYPE

Fig. 5. Stiffener patterns.

direct numerical integration algorithm of Newmark 1131.The average acceleration scheme is used.

Forced vibration

The dynamic equilibrium subjected to the exponential

equation for the plate blast pressure, P(t), is

Mlj+Czj+fq=P,

(61

where C is the damping matrix and q is the displacement vector. This is solved in the time-domain by the

Fig. 4. Fundamental mode of a fixed square plate.

NUMERICALEXAMPLES The following examples are considered in this study for free and forced vibration response. 1. Unstiffened plate. 2. Stiffened plate with type A pattern, with stiffener eccentricity to plate thickness (e/t) ratios of 1, 2, 3 and 4. 3. Stiffened plate with type B pattern, with (e/t) ratios of 1.12,2.36,3.60 and 4.84 so adjusted that the total mass remains the same as for the four cases of type A pattern. 4. Stiffened plate with type C pattern, of four different (e/t) ratios in the range 1.38-9.09, (Table 1) so as to have the same mass as the type A pattern. 5. Equivalent unstiffened plates of same mass as the four cases of type A pattern. In the case of type C pattern, the depth of the skew stiffeners is kept as less than that of the orthogonal stiffeners for the test cases 3 and 4 (Table 1). This is done in order to investigate the concept of

M. V. DHARANFEPATHY and K. G. SUDHESH

894

Table I. Plate stiffener data

Stiffener pattern

Plate tYPe

Plate thickness

Stiffener eccentricity (mm) Skew Orthogonal -

Effective depth (mm) Orthoaonal Skew

Case -

Total weight (N) . ,

1

66.8592

3.4

1 2 3 4

69.5169 74.8333 80.1487 85.4641

3.4 3.4 3.4 3.4

1

69.5169 74.8333 80.1487 85.4641

3.4 3.4 3.4 3.4

5.9 14.3 22.8 31.2

5.9 14.3 22.8 31.2

3.8 8.0 12.2 16.4

3.8 8.0 12.2 16.4

1.12 2.36 3.60 4.84

1.12 2.36 3.60 4.84

2 3 4

69.5169 74.8333 80.1487 85.4641

3.4 3.4 ;:t

7.7 19.5 19.5 19.5

7.7 19.5 39.9 60.1

4.7 10.6 10.6 10.6

4.7 10.6 20.8 30.9

1.38 3.12 3.12 3.12

1.38 3.12 6.12 9.09

1 2 3 4

69.5169 74.8333 80.1487 85.4641

3.5 3.8 4.1 4.3

2 3 4

1

(mm) . ,

5.1 11.9 18.7 25.5

‘pyramiding’ of stiffeners [ 11, with multiple eccentricities, for out-of-plane loads. The particulars of the plate-stiffener data, such as mass, thickness, stiffener

e/r Skew Orthogonal

3.4 6.8 10.2 13.6

I

2 3 4

depth, eccentricity, etc. for the four cases of e/t ratios are presented in Table 1. The material properties for steel, such as an elastic modulus of 207 kN/mm*,

Table 2. Free vibration results

Plate type

Natural frequency (Hz) Case

S, 115.57

AS, 240.71

s2

S,

240.73

349.44

443.40

445.77

537.36

537.40

2 3 4

125.54 224.28 375.75 546.69

263.89 461.86 756.58 1073.69

263.91 461.87 756.61 1073.70

381.09 650.84 1046.95 1445.59

485.54 831.27 1544.00 1925.12

488.74 838.47 1769.08 1989.20

587.04 996.28 1822.23 2015.54

775.48 1349.39 1948.79 2082.88

3.7 6.6 11.0 16.0

1 2 3 4

172.86 291.92 458.95 635.43

352.88 590.22 862.86 1041.32

369.87 609.67 879.14 1050.56

538.23 755.35 985.08 1112.75

610.81 913.30 1124.81 1186.36

651.54 967.76 1133.07 1188.44

733.63 1013.44 1138.35 1190.70

767.61 1156.39 1168.23 1211.02

5.0 8.5 13.4 18.7

1 2 3 4

181.64 374.39 526.62 534.43

326.01 500.18 532.39 537.81

326.27 500.21 533.95 537.85

528.97 524.02 533.98 538.35

617.84 593.58 1064.87 1071.10

622.14 856.62 1066.66 1071.39

624.37 1054.44 1152.34 1175.50

654.55 1062.58 1162.25 1178.87

5.3 11.0 15.4 15.7

1 2 3 4

119.01 129.60 139.82 146.91

247.75 269.20 290.41 304.69

247.78 269.23 290.44 304.72

359.66 390.62 421.36 441.96

456.39 495.54 534.54 560.58

458.85 498.19 537.38 563.56

553.07 600.46 647.67 679.19

553.80 600.71 648.53 679.54

I 1 2

3

4

5

Frequencyequivalent solid-plate thickness (mm)

AS2

AS,

A&

A&

-

Note: S, = ith symmetric mode; AS, = ith antisymmetric mode. This classification of mode shape is based on the free. response of isotropic plates (types 1 and 5). The mode shapes of orthotropic plates (types 24) deviate from this classification with increasing e/r ratio.

895

Optimal stiffening of square plates RE3JLTS

I-

*

ah 0 .

.

0’

it IL

,I’

,

V’ /’

#’

. ..’

,..

_:’ “’ ,..’

,..’

5 ___--______

,..’ ~~~__-____---

100.

107.

MASS

121.

121.

114.

RATIO

(w/w0

I

x10-a

Fig. 6. Influence of stiffener pattern on the fundamental (S, ) natural frequency.

a Poisson ratio of 0.3, a mass density of 7.6 x IO-’ N/mm3 and a damping factor of 0.01, are used. For forced response computations, the blast load time history is generated as per eqn (l), using the values of P,,, = 0.0676 N/mm2, lP = 2.0 m set and a’ = 1.0. Response time history is obtained by direct numerical integration using 1000 steps of 0.1 msec. Taking advantage of the symmetry of load and geometry, only a quarter of the plate is analysed with a 6 x 6 mesh. However, in the case of free vibration analyses, the entire plate is analysed with a 12 x 12 mesh for the lowest eight modes, in order not to miss the antisymmetric modes. The aim of the free vibration analysis is to study the influence of stiffener patterns and eccentricities on all of the modes.

The lowest eight natural frequencies of the five types of plate problems, listed earlier, are given in Table 2. In all, 17 computer runs have been taken, one for the type 1 plate and four each for the rest. In the last column of this table, the thickness of equivalent unstiffened plate with the same fundamental frequency as the stiffened plate is given. This is arrived at by trial and error, i.e. by adjusting the thickness until the frequency nearly matches with that of the stiffened plate. Figure 6 shows the influence of stiffener pattern on the fundamental (S,) frequency. The frequency ratio, f&, is plotted against the mass ratio, w/w,, for the three stiffener patterns A, B and C. The frequency, fO, and the mass, w,,, are those of the unstiffened plate, whereas f and w are the corresponding values of the stiffened plates. The results of blast-induced responses such as standard deviations of maximum shears and moments anywhere in the plate, central and quarter point displacements, are presented in Table 3. The time-histories and the power spectral densities for the central displacement are plotted in Figs 7.1-7.5, respectively, for plate types 1-5. For plate types 2-5 only case 4 results are plotted. Figures 8 and 9 show the influence of stiffener pattern on the standard deviation of quarter-point and central displacements. The percentage reduction in central displacement, with reference to the unstiffened plate (type l), is plotted against the mass ratio. DISCUSSION

The art of optimal stiffening of dynamically loaded plates lies essentially in laying the stiffeners so that the

Table 3. Forced response results

Plate type

Case

1 2

3

4

5

Standard deviation of maximum stresses anywhere in the plate Iuxx MYY MXY (Nmm) (Nmm) (G (Nmm) -

-

-

Standard deviation of displacement (mm) Quarter Centre point point

1575.2003 - 1601.6792

74.0625

0.672

2.177

1310.7546 - 1332.0162 -478.3658 471.9227 154.0778 - 156.7551 76.5240 -78.6914

67.5016 - 35.3444 14.1711 10.4543

0.561 0.204 0.064 0.022

1.827 0.673 0.211 0.072

1 2 3 4

518.6295 1008.7068 734.2383 443.195

517.9194 1008.7363 734.2079 443.2087

166.1365 336.6566 249.0821 151.2504

1 2 3 4

-301.5662 531.0108 489.1339 358.0997

301.7535 530.5754 489.1486 358.1516

142.5840 264.2692 212.3157 147.9004

-684.2344 272.5365 135.6602 72.3364

- 730.4254 293.1410 - 161.5311 -81.6139

- 106.2882 80.6822 43.6313 - 25.6257

0.180 0.047 0.015 0.005

1.20 0.400 0.130 0.046

1 2 3 4

1182.8997 1195.5302 341.5386 167.6968

1186.0439 1195.6263 341.5719 167.7978

406.0628 405.6450 112.8845 55.1350

1103.366 295.0828 295.4751 276.8908

- 1131.6592 -297.0835 -309.3912 -289.3850

-61.2447 45.2495 95.6967 91.8720

0.392 0.142 0.114 0.104

0.954 0.193 0.019 0.005

1604.1800 1675.1435 1752.5403 1788.5222

-

75.4551 78.8483 82.6338 84.3696

0.628 0.512 0.427 0.377

2.033 1.658 1.383 I .224

1 2 3 4

-

-

-

1631.7475 1704.2605 1784.570 1822.170

M. V. DHARANEEPATHY and K. G. SUDHESH

(a) (b)

c 0.00

2.48

4.88 TIMEISECSI

7.43 Xl 0-a

5.98

u.3

I

I

I

1.ae

3.00 FREQUENCYLHZ)

4.03 x102

5.00

Fig. 7.1. Plate type I. (a) Time history of central deflection. (b) Spectra of central deflection.

(a) :

(b)

@-I

01 -_ X

*-

d

I-

I

m -0

i;

EO

0

!!o

.

1

ml

a rnh -

.

O-L l

$4 0

0.00

2.48

4.98 TIME(SECSI

7.48 x10-a

e.55

,

0.86

3.17 FRECUENCY

5.40 I HZ

I

7.82 x10*

8.85

Fig. 7.2. Plate type 2. Case 4. (a) Time history of central deflection. (b) Spectra of central deflection (b)

(a)

FJ, 0.00

2.48

4.88 TIMEISECS)

7.48 x10-2

8.98

0.85

5.40 3.17 FREQUENCYIHZI

7.52 x102

Fig. 7.3. Plate type 3. Case 4. (a) Time history of central deflection. (b) Spectra of central deflection.

5.5s

897

Optimal stiffening of square plates (b)

(D : 0

f

ii

s

0.00

2. so

1.00 TIMEfSECSI

10.0

7.60 x10-=

0.86

1.40

3.17 FRECUENCY

I HZ

1

7.82 x10=

a. 88

Fig. 7.4. Plate type 4. Case 4. (a) Time history of central deflection. (b) Spectra of central deflection.

\ 2. so

6.00 TIMEISECSI

0.0

I

es

1.

I

3.00

FRECUENCY

IHZ

I

4.03 x10=

6.01

Fig. 7.5. Plate type 5. Case 4. (a) Time history of central deflection. (b) Spectra of central deflection.

TYPE

100.

Fig. 8. Influence of stiffener pattern on quarter-point placement.

dis-

107. MASS

114. RATIClw/*,

121. I

120. x10-a

Fig. 9. Influence of stiffener pattern on central displacement.

I

M. V. DHARANFEPATHY and

K. G. SULMESH

CASE - 1 161 Hz

CASE 374

- 2 Hz

(d)

CASE 526

- 3 Hz

CASE 534

- 4 Hz

Fig. 10. Fundamental (S,) modes of type 4 plate. consequent increase in stiffness is far greater than the increase in inertia induced by the modes of significance As the fundamental (S,) mode is the only mode of significance in the case of blast-loaded plates, the discussion here is essentially confined to the influence of stiffener pattern on this mode. Examining the free-vibration results (Table 2 and Fig. 6) in the context, one notices the radical changes in the fundamental frequency caused by the three stiffener patterns. The gradual reduction in the degree of such changes towards the higher order frequencies is not of any relevance to the problem on hand. The exercise here is basically to investigate the aspect of how a given mass, when distributed in different ways, has vastly differing natural frequencies. If one may define the relative superiority on the basis of the frequencies ratio (Fig. 6), among the three stiffener patterns, the pattern C (Fig. 5) stiffener seems to be superior for e/r ratios of up to 6.12 (Case 3). As the eccentricity increases further, the pattern B stiffener

has better frequency ratio than the rest. For small eccentricities, the conventional waffle plate (pattern A stiffener) shows poor performance. The plate type 5, with the same mass as the stiffened plate, is obviously far inferior when compared with stiffened plates. In this case, the improvement in frequency ratio is Seen to be of almost the same order as the mass ratio. From the results in the last column of Table 2, which gives the thicknesses of frequency-equivalent solid plate, certain interesting observations may be made. For instance, as noted in case 3 of plate type 2, for a 225% increase in S, frequency, an identical magnitude of increases in mass, and hence thickness, is required if a solid unstiffened plate is used. The same effect may be produced with just 20% increase in mass with a pattern A stiffener, and only 12% mass increase with a pattern C stiffener (case 3 of type 4 plate). From the forced vibration results presented in Table 3 and Figs 8 and 9, it is noticed that plate types

899

Optimal stiffening of square plates

3 and 4 perform better than type 2. Type 4 plate undergoes less displacements than the type 3 plate until the e/t ratio reaches 3.12. Thereafter, the fundamental mode of the type 4 plate changes (Fig. 10). Consequently, the quarter-point displacements are more than the central displacements. Hence the type 3 plate behaves better than the type 4 plate with increasing eccentricity. The plate moments are seen (Table 3) to reduce significantly when stiffened. Here also, plate types 3 and 4 perform better than the type 2 plate. However, the plate shears caused by the orthotropy of the ribbed plates are quite significant and must be given due consideration while arriving at the minimum thickness of the plate. CONCLUSIONS

The process of arriving at an optimal stiffener layout is rather tedious. There may be many solutions. In this study, from the considerations of ease of fabrication and the modes induced by blast load, two novel patterns of stiffening are investigated and compared with the conventional waffle plate design. From this limited study, it may broadly be concluded that, for a blast-loaded plate, the combination of skew and orthogonal stiffeners is a better proposition than the conventional bi-directional waffle stiffening. The type C pattern of stiffening (plate type 4) seems to be an ideal choice for small eccentricities. The type B stiffener (plate type 3) may be used if large eccentricities are needed. Care should be taken in providing adequate plate thickness to withstand the shear stresses introduced by the orthotropy of the stiffened plate. Although the example plate here is steel, these observations are equally valid for reinforced concrete plates. The relative simplicity of formwork, in adopting type B or C patterns, is obvious. Acknon~ledgemenr-The authors thank the Director, Structural Engineering Research Centre, Madras, for providing the opportunity to carry out this investigation.

REFERENCES

1. L. D. Hofmeister and L. P. Felton, Design examples of plates with multiple rib-sixes. AIAA/ASME 1Ith Structures, Structural Dynamics, and Materials Conference, Denver, CO, 22-24 April. pp. 65-74, (1970). 2. B. R. Long, Vibration of eccentrically stiffened plates. Shock Vibr. Bull. 38, 45-53 (1968).

3. R. Houlston and C. G. DesRochers, Nonlinear structural response of ship panels subjected to air blast loading. Comput. Srruct. 26, I-15 (1987). 4. V. Ahamed, Theoretical and experimental investigations on shear wall structures. Ph.D. thesis, Department of Civil Engineering, IIT, Madras (1987). Analytical study of core5. S. Gomathinayagam, wall structures with arbitrary plan forms. M.S. thesis, Department of Civil Engineering, IIT, Madras (1982). 6. T. P. Ganesan and S. Gomathinayagam, A special purpose program ‘CORE-3D’ for shear cores of arbitrary plan forms. Second International Conference on Computer-Aided Analysis and Design in Civil Engineering, Department of Civil Engineering, University of Roorkee, India, 29 January-2 February,_ __ pp. B-45-11-51 (1985). I. S. R. Karve, Analysis of stiffened plate systems by finite element method. Ph.D. thesis, Deoartment of Civil Engineering, IIT, Bombay (1974). 8. M. N. Keshava Rao, M. V. Dharaneepathy, S. Gomathinayagan, K. Rama Raju and K. G. Sudhesh, Stochastic mechanics and reliability analysis of offshore structures. Project Report No. RD-I/OST-RR-88-I. Structural Engineering Research Centre, Madras (1988). 9. A. D. Gupta, F. H. Gregory, R. L. Bitting and S. Bhattacharya, Dynamic analysis of an explosively loaded hinged rectangular plate. Compuf. Strucr. 26, 339-344 (1987).

10. H. J. Goodman, Compiled free-air blast data on bare spherical pentolite. BRL-R-1092, U.S. Army Ballistic Research Laboratory, APG, MD (1960). 11. IS Code 1334991-1968,Criteria for blast resistant design of structures for explosions above ground. Indian Standards Institution, New Delhi (1969). 12. Engineering Design Handbook, Explosion in air-part 1.AMC Pamphlet, AMCP-706-I 8 1,Head- quarters, US Army Material Command, Alexandria, VA (1974). 13. K. J. Bathe and E. L. Wilson, Numerical Methods in Finite Element Analysis. Prentice-Hall, Englewood Cliffs, NJ (1976).