Large Deflection Behaviour of Plates under Uniform Transverse Loading A. Pradeep Kumar, K.N. Saha, D. Misra, P.K. De and S. Ghosal Department of Mechanical Engineering Jadavpur University, Calcutta-700 032, India
An experimental study on the deflection of thin isotropic rectangular plates under uniform loading with clamped edges has been carried out. The problem formulation is done using a variational method with orthogonal polynomial functions to describe the displacement fields. The resulting set of non-linear algebraic equations is solved using an iterative scheme. A comparative study between numerical results and measurements indicates good agreement up to the elastic limit of the material. Some experimental results in the post elastic region have also been furnished in the present work.
Nomenclature u , v, w U,V D q
-
displacements in x-, y- and z- directions respectively strain energy and potential energy flexural rigidity of the plate, Eh 3 / 12(1 − ν 2 ) uniformly distributed load on the plate
Q a, b, h E,ν φ , ϕ , α and β di
-
dimensionless load, qa 4 / Eh 4 plate dimensions along x-, y- and z- direction elastic modulus and poison’s ratio functions used to describe displacement fields unknown co-efficient in the assumed displacement fields
1. Introduction Engineering structures such as containers, ships, aeroplanes, etc. require complete enclosures that can be accomplished by plates. Such structures need stronger design and higher service life associated with saving in weight. Large deflection analysis of plates provides solution to such weight conscious design. Looking back into the history of ‘The Theory of Plates’ one can find that, Von Karman [1910] was the first to present the differential equation for large deflection. Later Timoshenko [1959] and Szilard [1974] presented methods for analytical and numerical analysis for the large deflection of plates. Iyenger [1986] addressed the elastic stability problems of rectangular plates with various complications. In recent times, Mukhopadhyay et al. [1993] investigated the behaviour of large deflection of stiffened plates using Finite Element method. A refined theory for the bending of plates
418
with moderately large deflection was proposed by Sarkani and Voyiadjis [1989]. A non-linear analysis of stiffened plates using super elements was done by Koko and Olson [1991]. A new rigid plastic analysis of stiffenedd plates subjected to uniformly distributed blast loads was developed by Anderson et al. [1993]. In the present work, a numerical method has been proposed to study the large deflection behaviour of isotropic plates considering geometric non-linearity. An experimental study is also carried out on large deflection of rectangular plates under uniform loading to compare the results with the analytical ones. Further, some experimental results in the post–elastic region are also provided.
2. Analysis
2.1. Problem formulation A clamped rectangular plate (a x b x h) when subjected to a uniformly distributed transverse load (q) produces deflection (w) with in-plane displacements (u and v). The various energies stored in the plate are the strain energy due to bending
Ub =
2 a b ⎧⎪⎛ ∂ 2 w ⎞ 2 ∂ 2 w ∂ 2 w ⎫⎪⎫⎪ D ⎧⎪⎛ ∂ 2 w ∂ 2 w ⎞ ⎟ − 2 ⎟ + 2(1 − υ )⎨⎜⎜ + ⎨⎜ ⎬⎬dxdy 2 ∫0 ∫0 ⎪⎜⎝ ∂x ∂x∂y ⎟⎠ ∂y ⎟⎠ ∂x ∂y 2 ⎪⎪ ⎪ ⎝ ⎩ ⎭⎭ ⎩
and the strain energy due to stretching a b
Eh US = 2(1 − υ ) ∫0 ∫0
⎧⎪⎡⎛ ∂u ⎞ 2 ∂u ⎛ ∂w ⎞ 2 ⎛ ∂v ⎞ 2 ∂v ⎛ ∂w ⎞ 2 ⎤ 1 ⎡⎛ ∂w ⎞ 2 ⎛ ∂w ⎞ 2 ⎤ ⎨⎢⎜ ⎟ + ⎜ ⎟ + ⎜⎜ ⎟⎟ + ⎜⎜ ⎟⎟ ⎥ + ⎢⎜ ⎟ + ⎜⎜ ⎟⎟ ⎥ ∂x ⎝ ∂x ⎠ ⎝ ∂y ⎠ ∂y ⎝ ∂y ⎠ ⎥ 4 ⎢⎝ ∂x ⎠ ⎝ ∂y ⎠ ⎥ ⎪⎩⎢⎣⎝ ∂x ⎠ ⎦ ⎣ ⎦
2
⎡ ∂u ∂v 1 ∂v ⎛ ∂w ⎞ 2 1 ∂u ⎛ ∂w ⎞ 2 ⎤ ⎜⎜ ⎟⎟ + + 2υ ⎢ + ⎜ ⎟ ⎥ 2 ∂x ⎝ ∂x ⎠ ⎥ ⎢⎣ ∂x ∂y 2 ∂y ⎝ ∂y ⎠ ⎦ 1−υ + 2
⎡⎛ ∂u ⎞ 2 ∂u ∂v ⎛ ∂v ⎞ 2 ∂u ∂w ∂w ∂v ∂w ∂w ⎤ ⎫⎪ +⎜ ⎟ +2 +2 ⎢⎜⎜ ⎟⎟ + ⎥ ⎬dxdy∗ ∂y ∂x ⎝ ∂x ⎠ ∂y ∂x ∂y ∂y ∂x ∂y ⎥ ⎪ ⎢⎣⎝ ∂y ⎠ ⎦⎭
Thus the total strain energy stored in the plate is U= U b + U s
(1)
The work function due to external forces is
419
a b
W= -V = ∫ ∫ (qw)dxdy
(2)
0 0
where, V is the total potential energy. From the principle of conservation of total energy of the system we have
δ (U + V ) = 0
(3)
which leads to the governing equation of the system. In Eqn. (3), mid-plane coordinates of the plate have been expressed in dimensionless form x y ξ = , η = , whereas deflection and load are kept in their dimensional forms. a b Substituting the expressions for the energies in Eqn. (3) the governing differential equation in semi non-dimensional form can be expressed as 2 1 1 ⎡ ⎡⎛ ∂ 2 w ⎞ 2 Dab ⎧⎪⎛ 1 ∂ 2 w 1 ∂ 2 w ⎞ 1 ∂ 2 w ∂ 2 w ⎤ ⎫⎪ ⎢ ⎜ ⎟ ⎜ ⎟ ⎢ ⎥ ⎬dξdη + + − − δ υ 2 ( 1 ) ⎨ ⎜ ∂ξ∂η ⎟ ⎢ 2 ∫0 ∫0 ⎪⎜⎝ a 2 ∂ξ 2 b 2 ∂η 2 ⎟⎠ a 2 b 2 ∂ξ 2 ∂η 2 ⎥ ⎪ ⎢ ⎝ ⎠ ⎣ ⎦⎭ ⎩ ⎣ 1 1 ⎧⎡ Ehab ⎪ 1 + ⎢ 2 ∫∫⎨ 2(1 − υ ) 0 0 ⎪⎢⎣ a 2 ⎩
1⎡ 1 + ⎢ 2 4 ⎢a ⎣
2 2 2 2 ⎛ ∂u ⎞ 1 ∂ u ⎛ ∂w ⎞ 1 ⎛ ∂v ⎞ 1 ∂v ⎛ ∂w ⎞ ⎤ ⎜⎜ ⎟⎟ + 3 ⎜ ⎟ + 2 ⎜⎜ ⎟⎟ + 3 ⎜ ⎟ ⎥ a ∂ξ ⎜⎝ ∂ξ ⎟⎠ b ⎝ ∂η ⎠ b ∂η ⎜⎝ ∂η ⎟⎠ ⎥⎦ ⎝ ∂ξ ⎠ 2
2 2 ⎤ ⎡ 1 ∂u ∂v 1 ∂v ⎛ ∂w ⎞ 1 ∂u ⎛ ∂w ⎞ ⎤ ⎜⎜ ⎟⎟ + ⎜ ⎟ ⎥ + 2 ⎥ + 2υ ⎢ 2a b 2 ∂ξ ⎜⎝ ∂η ⎟⎠ ⎥⎦ ⎥⎦ ⎢⎣ ab ∂ξ ∂η 2a b ∂η ⎝ ∂ξ ⎠ 2 2 1 − υ ⎡ 1 ⎛ ∂u ⎞ 2 ∂u ∂v 1 ⎛ ∂v ⎞ 2 ∂ u ∂w ∂w 2 ∂v ∂w ∂w ⎤ ⎫⎪ ⎜ ⎟ ⎜ ⎟ + + + + + ⎢ ⎥ ⎬dξdη 2 ab ∂η ∂ξ a 2 ⎜⎝ ∂ξ ⎟⎠ ∂η ∂ξ ∂η a 2 b ∂ξ ∂ξ ∂η ⎥ ⎪ 2 ⎢ a 2 ⎜⎝ ∂η ⎟⎠ ab ⎣ ⎦⎭ 2
⎛ ∂w ⎞ 1 ⎛ ∂w ⎞ ⎜⎜ ⎟⎟ + 2 ⎜⎜ ⎟⎟ b ⎝ ∂η ⎠ ⎝ ∂ξ ⎠
2
⎤ − ab ∫ ∫ qwdξdη ⎥ = 0 0 0 ⎦ Applying the δ operator in Eqn. (4), one can arrive at 1 1
(4)
⎤ 11⎡⎧ 1 ∂4w ∂4w ⎞⎟⎫⎪ ∂ 4w 2 ∂4w 1 ∂4w 2(1 −υ) ⎛⎜ ∂4w Dab∫ ∫ ⎢⎨ −2 + − + + ⎬dξdη⎥δw 4 4 2 2 2 2 4 4 2 2 2 2 2 2 ⎜ ⎟ ⎥ ξ η ξ η ∂ ∂ ∂ ∂ 00⎣⎢⎩a ∂ξ a b ∂ξ ∂η b ∂η a b ⎝ ∂ξ ∂η ∂η ∂ξ ⎠⎪⎭ ⎦ +
Ehab 11⎧⎪⎡⎧⎪ 1 ∂w ∂2u 1 ∂w ∂2v ⎫⎪ ⎡ 1 ∂v ∂2w 1 ∂u ∂2w⎤ + + ⎥ ∫ ∫ ⎨⎢⎨ ⎬ + υ⎢ (1 −υ 2 ) 00⎪⎩⎢⎣⎪⎩a3 ∂ξ ∂ξ 2 b3 ∂η ∂η2 ⎪⎭ ⎢⎣ a2b ∂η ∂ξ 2 a2b ∂ξ ∂η2 ⎥⎦
420
+
1 −υ ⎡ 1 ∂w ∂2u 1 ∂u ∂2w 1 ∂w ∂2v 1 ∂v ∂2w ⎤ + + + ⎢ ⎥ 2 ⎢⎣ ab2 ∂ξ ∂η2 a b2 ∂η ∂η∂ξ a2b ∂ξ ∂ξ 2 a2b2 ∂ξ ∂ξ∂η ⎥⎦
2 2 2 2 ⎤⎤ ⎡ 1 ∂2w ⎛ ∂w ⎞ 1 ∂2w ⎛ ∂w ⎞ ⎥⎥ 1 ⎢ 1 ⎛ ∂w ⎞ ∂2w 1 ⎛ ∂w ⎞ ∂2w ⎜ ⎟ (δw) ⎜ ⎟ + ⎜ ⎟ + + ⎜ ⎟ + 2 ⎢ a4 ⎜⎝ ∂ξ ⎟⎠ ∂ξ 2 b4 ⎜⎝ ∂η ⎟⎠ ∂η2 a2b2 ∂ξ 2 ⎜⎝ ∂η ⎟⎠ a2b2 ∂η2 ⎜⎝ ∂ξ ⎟⎠ ⎥⎥ ⎣ ⎦⎥⎦ ⎡⎧⎪ 1 ∂2u 1 ∂w ∂2u ⎫⎪ ⎧⎪ 1 ∂2v 1 ∂w ∂2w ⎫⎪ 1 −υ ⎧⎪ 1 ∂2u 1 ∂2v 1 ∂w ∂2w + υ⎨ + + + + + ⎢⎨ + ⎬ ⎨ ⎬ ⎢⎣⎪⎩a2 ∂ξ 2 a3 ∂ξ ∂ξ 2 ⎪⎭ ⎪⎩ab ∂ξ∂η a b2 ∂η ∂ξ∂η ⎪⎭ 2 ⎪⎩b2 ∂η2 ab ∂ξ∂η ab2 ∂η ∂ξ∂η
⎫⎪⎤ ⎬⎥(δu) ⎪⎭⎥⎦ ⎡⎧⎪ 1 ∂2v 1 ∂w ∂2v ⎫⎪ ⎧⎪ 1 ∂2u 1 ∂w ∂2w ⎫⎪ 1 −υ ⎧⎪ 1 ∂2w 1 ∂2u 1 ∂w ∂2w ⎫⎪ ⎫⎪ + + + + υ⎨ + + + ⎢⎨ ⎬(δv)⎬dξdη ⎨ ⎬ ⎬ ⎢⎣⎪⎩b2 ∂η2 b3 ∂η ∂η2 ⎪⎭ ⎪⎩ab ∂ξ∂η ab2 ∂ξ ∂ξ∂η ⎪⎭ 2 ⎪⎩a2 ∂ξ 2 ab ∂ξ∂η ab2 ∂ξ ∂ξ∂η ⎪⎭ ⎪⎭
11 − q ∫ ∫ δwdξdη = 0 00
(5)
Displacements u, v and w are assumed to be separable in ξ − η plane and are expressed through a linear combination of sets of one dimensional orthogonal polynomial functions φ (ξ ), ψ (η ), α (ξ ) and β (η ) . The starting function of the orthogonal sets satisfy the corresponding boundary conditions of the plate and Gram-Schmidt orthogonalisation procedure has been used to generate the higher functions. The displacements are given by w( ξ , η ) =
nw
∑d φψ i
i =1
u ( ξ ,η ) =
nu
∑d
i
6(i)
i
i + nwα iψ i
6(ii)
i =1
v( ξ , η ) =
nv
∑d
i + nw + nu φ i β i
6(iii)
i =1
Substituting the above series in Eqn. (5) and replacing δ by ∂/∂di, a set of non-linear algebraic equation is generated as follows
[K ]{d } = q{R}
where,
(7)
⎡[K 11 ] [K 12 ] [K 13 ] ⎤ [K ] = ⎢⎢[K 21 ] [K 22 ] K 23 ⎥⎥ ⎢[K 31 ] [K 32 ] [K 33 ]⎥ ⎣ ⎦
[ ]
is a symmetric square matrix of order ( nw+nu+nv).
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The details of the elements of each of the sub-matrices in [K] and vector{R} are given in the Appendix.
2.2. Solution methodology The unknown coefficients {d} can be obtained from
{d } = q[K ]−1 {R}
(8)
The coefficient matrix [K] is a function of unknown coefficients {d}. Solution of Eqn. (8) is obtained through an iterative scheme. In each iterative step the matrix [K] is computed using old values {d}old and the solution vector is updated using the relaxation parameter λ as follows.
{d } = λ{d }new + (1 − λ ){d }old
(9)
In each iteration the following error vector {ε } is computed
{ε } = q[K ]−1 {R} − {d }
(10)
If error is not within the permitted value of tolerance, the process is repeated with new
{d }old ={d} until {ε } becomes less than the tolerance. A tolerance value of 0.5% has been used in the computational work.
3. Experiment
3.1. Experimental setup The experimental setup for carrying out the large deflection analysis is shown in Fig. 1. The rectangular frame is made up of closed ISMC 75 channels with the bottom surface being closed by a 3.15 mm thick M.S plate. The test plate is mounted on the top and the clamped boundary condition is obtained by bolting the plate in-between the channel frame and an ISFB 40x6 mating flange with M10 bolts at 35 mm pitch. For effective sealing a soft rubber gasket is provided between the test plate and the frame. Three such setups are fabricated for different aspect ratios of the test plate. The clamping bolts are tightened with a torque-wrench (capacity of 7 kg-m at an interval of 0.2 kg-m) to ensure uniform clamping. Uniform distribution of loading is applied by supplying pressurised air into the enclosure from a compressor (45 liter at 100 psi). The pressure is regulated by the isolation and control valves. Once the enclosure pressure exceeds the desired value, supply is cut off with the isolation valve and then the system pressure is finally tuned by releasing excess air through the control valve. Pressure is noted by using two calibrated pressure gauges of 5 psi range, 0.05 psi accuracy and 15 psi range, 0.5 psi accuracy respectively. For small pressure range reading is taken from a suitable U-tube mercury
422
manometer. A movable suspension bridge is used for fixing the dial gauge (range 25 mm, accuracy 0.01 mm) at a position where deflection is to be measured.
4. Results and discussions
The numerical study is performed with a square plate under clamped boundary condition and results are furnished in Fig. 2. Graphs are plotted for non-dimensional pressure vs. non-dimensional deflection. Excellent agreement is observed between the present simulation and earlier works by Szilard [1974] and Aalami [1975]. The experimental study is conducted on square (400x400) and rectangular plates (360x450 and 300x500) with various thickness, ranging from 0.78 to 3.15. All dimensions indicated are in mm. The experiment is performed with a maximum test pressure of 40 psi. The experiment is carried out primarily to study the load-deflection behaviour under different thickness, aspect ratios and clamping torques. Some studies is also conducted to investigate the behaviour beyond elastic limit of the plate material. The load-deflection behaviour of the square plates is presented in normalized form in Fig.3 for different plate thickness (1.25, 1.6, 2.5, 3.15). The central deflection for plate thickness 3.15 shows better agreement with the theoretical results than those of the thinner plates. The deviation may be attributed to a pronounced membrane effect of these thin plates. Fig. 4 indicates the effect of aspect ratio (a/b) for a plate thickness of 1.25. The results are in good agreement with theory. The effect of clamping torque on deflection, is presented in Fig. 5. As the clamping torque is diminished, the central deflection increases indicating a deviation from the clamped boundary condition. Experimental observation of the load-deflection behaviour in the post-elastic region is shown in Figs. 6(a) and 6(b) for different maximum load levels. Measurements taken during loading as well as unloading conditions indicate a possible localized yielding. However, the phenomenon cannot be conclusively ascertained due to the presence of locked-up friction forces along the clamped boundaries of the plate. A similar experiment with a higher maximum loading is repeated for a number of times and the results are furnished in Fig. 7. The reduced deflection for subsequent loading cases may be ascribed to the strain-hardenening effect during the first loading.
5. Conclusions
In the present work, a numerical method is proposed to study the large deflection behaviour of clamped rectangular plates and excellent agreement is observed with earlier theoretical works. The experimental study is carried out primarily to validate the theoretical model. The effect of plate thickness, aspect ratios and clamping torques on the load-deflection behaviour are also investigated.
423
It is observed that thin plates have a pronounced membrane effect and hence in numerical study membrane boundary conditions must be appropriately modelled. The results due to variations in aspect ratio and clamping torque follow the theoretical predictions. Experimental observation in the post-elastic region during loading and unloading indicate possible localized yielding, but the phenomenon cannot be conclusively ascertained due to the presence of locked-up friction forces along the clamped boundaries of the plate. A similar experiment with higher loading shows strain-hardenening effect. It is felt that the actual membrane boundary condition could be ascertained more accurately by using suitable gasket materials. An experimental study on the deflection profile is further necessary to establish the theoretical model on stronger feet. Moreover, the study in the post elastic region is still in the nascent stage which need the support of appropriate theoretical model involving material non-linearity.
Acknowledgement
This work has been carried out under the financial support of AICTE, Ref : R&D Project , File No. 8017/RDII/R&D/DEG(743)/98-99 dated 27.3.1999.
References
1. Aalami, B., Williams, D.G., 1975 Thin Plate Design for Transverse Loading. Crosby Lockwood Staples, London. 2. Anderson, D.L., Olson, M.D. and Schubak, R.B., 1993 International Journal of Mechanics, Vol. 35, No. 3/4, 289-324. Rigid-Plastic Modelling of Blast Loaded Stiffened Plates. Part I&II. 3. Iyenger, N.G.R., 1986 Elastic Stability of Columns and Plates, Affiliated East-West Press, New Delhi. 4. Oslon, M.D., and Koko, T.S., 1991 International Journal of Non-Linear Mechanics Vol. 31, No. 2, 319-343. Non-Linear Analysis of Stiffened Plates Using Super Elements. 5. Mukhopadhyay, M., Sheikh, A.H., and Rao, V.D., 1993 Computers and Structures Vol. 47, No. 6, 987-993. A Finite Element Large Displacement Analysis of Stiffened Plates. 6. Sarkani, S. and Voyiadjis, G.Z., 1989 Journal of Engineering Mechanics Vol. 115, No. 5, 935-951. Engineering Large Deflection Theory for Thick Plates. 7. Timoshenko, S.P. and Woinowsky-Krieger, S., 1959 Theory of Plates and Shells. McGraw-Hill Book Co., London.
424
8. Szilard, R., 1974 Theory and Analysis of Plates. Prentice – Hall, Inc., N.J. 9. Von Karman,Th., 1910, Encycl. der Math. Wiss.,4, 334-351. Festigkeitsprobleme im Maschinenbau.
APPENDIX
The elements of each of the submatrices in [K] and vector {R} are given below nw nw
⎡
1 1
i =1 j =1
⎢⎣
0 0
2 2 ⎧⎛ 1 d 4φ i d 4ψ i 2 d φi d ψ i 1 ψ φ + + i i 4 4 a 2 b 2 dξ 2 dη 2 b 4 dξ 4 ⎪⎩⎝ a dξ
[K 11 ] = ∑ ∑ ⎢d i ∫ ∫ Dab⎪⎨⎜⎜
nu ⎧⎪⎡⎧ 1 dφ i d 2α k dψ i Ehab 1 d d ψ k + 4 φi ψ i ∫ ∫ ⎨⎢⎨ 4 i∑ k 2 2 dη dξ b (1 − υ ) k =1 ⎪⎩⎢⎣⎩ a dξ
+
nv
∑ d mφ m m =1
⎫ ⎞ ⎟φ jψ j ⎪⎬dξdη ⎟ ⎪⎭ ⎠
d 2βm ⎫ ⎬ dη 2 ⎭
nv ⎧⎪ 1 d 2φ i dβ k d 2ψ i nu d αm 1 ⎪⎫ d d ψ φ φ ψm⎬ +υ⎨ 2 + i∑ k k i 2 2 2 ∑ m dη ab dη m =1 dξ ⎪⎩ a b dξ ⎪⎭ k =1 (1 − υ ) ⎧⎪ 1 dφ i ψ nu d α d β k + 1 dφ i dψ i nu d α d ψ m + ⎨ ∑ m m dη i∑ k k 2 ⎪⎩ ab 2 dξ dη ab 2 dξ dη i m =1 k =1 d ψ i nv d 2φ m d φk 1 1 dφ i dψ i nv ⎪⎫ d β dk βk ⎬ + 2 φi + ∑ ∑ m m 2 2 a b dη m =1 dξ a b dξ dη i k =1 dξ ⎪⎭ 2 2 2 dφ ii dψ ii ⎞ ⎞ ⎛ nw d 2ψi ⎛ nw 1 ⎧⎪ 1 d φ i 1 ⎟ ⎜ ∑ d ii φ ii ψ i ⎜ ∑ d ii ψ ii ⎟⎟ + 4 φ i + ⎨ 4 2 ⎪ a d ξ 2 ⎜⎝ ii =1 dξ dη ⎟⎠ d η 2 ⎜⎝ ii =1 ⎠ b ⎩
dψ ii ⎞ d 2ψ i ⎛ nw 1 1 d ⎟ ⎜ ψ d φ φ + i ∑ ii ii i dη ⎟⎠ a 2 b 2 dξ 2 ⎜⎝ ii =1 a 2b 2 dη 2 2
+
dφ ii ⎞ ⎛ nw ⎜⎜ ∑ d ii ψ ii ⎟⎟ dξ ⎠ ⎝ ii =1
2
⎤ ⎫⎪⎤ ⎥ φ ψ ⎬ j j ⎥ dξ dη ⎥ ⎪⎭⎥⎦ ⎦
1 1 nw nw ⎡ Ehab ⎧ 1 dφ i [K 12 ] = ∑∑ d i + nw ⎢ 2 ∫ ∫ ⎨ 3 ψ i ∑ d ii dφ ii ψ ii + υ 2 φ i dψ i ∑ d ii dφii dψ ii dξ dη ii =1 dξ dη 2ab ⎣1 − υ 0 0 ⎩ a dξ i =1 j =1 ii =1 nw nu
+
1 − υ dψ i φi dη 2ab 2
nw
∑d ii =1
ii
⎤ dφiii dψ ii ⎫ ⎬(α jψ j )dξdηd ⎥ dξ dη ⎭ ⎦
⎡ Ehab 1 1 ⎧ 1 dψ i [K13 ] = ∑∑ d i + nw+ nu ⎢ φ 2 ∫∫⎨ 3 i dη i =1 j =1 ⎣ (1 − υ ) 0 0 ⎩ b nw nv
+
nw
∑ d ii ii =1
nw ⎤ dφ ii dψ ii ⎫ 1 − υ dφ i ψ ⎬(φ j β j )dξdη ⎥ i ∑ d ii 2 dξ dη ⎭ 2a b dξ ii =1 ⎦
425
d 2φ i dφ dψ ii υ d φ i nw ψ ψ i ∑ d ii ii + ii 2 2 dξ dη dξ 2a b dξ ii =1
[K 21 ] = 0 ⎡ Ehab 1 1 ⎧ 1 d 2α i ⎤ 1 − υ d 2ψ i ⎫ ( ) [K 22 ] = ∑∑ d i + nw ⎢ + ψ α α ψ d ξ d η ⎥ ⎨ ⎬ i i j j 2 ∫∫ 2 2 dη 2 ⎭ 2b 2 i =1 j =1 ⎣⎢ (1 − υ ) 0 0 ⎩ a dξ ⎦⎥ nu
nu
nu
nv
[K 23 ] = ∑ ∑ i =1 j =1
[K 31 ] = 0 nv
nv
[K 33 ] = ∑ ∑ i =1 j =1
⎡ Ehab 1 1 ⎧⎪⎛ υ 1 − υ ⎞ dα i ψ dψ i ⎫⎪ ⎤ ( ) + d i + nw+ nu ⎢ φ β d ξ d η ⎥ ⎜ ⎟ ⎨ ⎬ j j 2 ∫∫ ⎢⎣ (1 − υ ) 0 0 ⎪⎩⎝ ab 2ab ⎠ dξ dη ⎪⎭ ⎥⎦ [K 32 ] = 0
⎡ Ehab 1 1 ⎧ 1 ⎤ d 2 β i 1 − υ d 2φ i ⎫ + d i + nw+ nu ⎢ φ ψ i ⎬(φ j β j )dξdη ⎥ 2 ∫∫⎨ 2 i 2 2 e2 dη 2 a dξ ⎢⎣ (1 − υ ) 0 0 ⎩ b ⎥⎦ ⎭
⎧{R1 }⎫ {R} = ⎪⎨{R2 }⎪⎬ , where ⎪{R }⎪ ⎩ 3 ⎭
nw 1 1
{R1 } = ∑ ∫ ∫ (φ iψ i )dξdη , i =1 0 0
426
and
{R2 } = 0 . {R3 } = 0
Normalised central deflection (w/h)
2
1
Szilard [1974] Aalami [1975] Present numerical
0 0
200
400
Normalised pressure (qa4/Eh4
)
Fig. 2. Comparison of numerical results
3.0
Normalised central deflection (w/h)
2.5
2.0
1.5 Square plate (400x400) 1.0
Numerical (present) Experimental (1.25 mm) Experimental (1.6 mm)
0.5
Experimental (2.5 mm) Experimental (3.15 mm)
0.0 0
50
100
150
200
250
300
Normalised pressure (qa4/Eh4 )
Fig. 3: Variation of deflection with pressure
427
350
400
450
500
1.2
5 Plate thickness 1.25 mm
Plate : 400x400x0.78 Normalised central deflection (w/h)
Normalised central deflection (w/h)
4
3
2
Aspect ratio 1.0 (400x400) 0.8 (360x450)
1
1.0
0.8
0.6
Clamping condition bolts 60-deg released
0.4
bolts 45-deg released
0.6 (300x500)
bolts fully tightened 0.2
0 0
200
400 Normalised pressure (qa4/Eh4
600
0
)
Fig. 4. Variation of deflection with load for various aspect ratios
1000 Normalised pressure (qa4/Eh4 )
1500
Fig.5. Effect of clamping torque on plate deflection
4
5
Plate: 400x400x1.6 Normalised central deflection (w/h)
Plate: 400x400x1.6 Normalised central deflection (w/h)
500
4
3
2 Loading Unloading
1
3
2
Loading
1
Unloading
0
0 0
400
800 Normalised pressure (qa4/Eh4 )
0
1200
Fig.6 (b). Load-deflection behaviour during loading and unloading (beyond elastic limit)
200
400 Normalised pressure (qa4/Eh4 )
600
Fig.6 (a). Load-deflection behaviour during loading and unloading (beyond elastic limit)
428
5
Normalised central deflection (w/h)
Plate : 400 x 400 x 1.6 4
3
2
1st loading 2nd loading
1
3rd loading Unloading
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400 800 1200 Normalised pressure (qa4/Eh2)
FIG.7. Effect of residual stress
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1600
