Parametric instability of edge cracked plates

Thin-Walled Structures 40 (2002) 29–44 www.elsevier.com/locate/tws

Parametric instability of edge cracked plates A. Vafai*, M. Javidruzi, H.E. Estekanchi Department of Civil Engineering, Sharif University of Technology, Tehran, Iran Received 6 February 2001; accepted 17 August 2001

Abstract The presented paper deals with the parametric instability behavior of a simply supported rectangular plate with a crack emanating from one edge, subjected to in-plane compressive periodic edge loading. The problem is reduced to computing the free vibration frequencies and the corresponding mode shapes and substituting them into an integral equation based formula, which leads to a compact matrix form. Once the components of this matrix are found, the rest of the computation, i.e., establishing regions of instability, buckling loads and modified frequencies, is straightforward and fast. Several plates, each with a different dimension and crack length size are analyzed using this approach. The comparison of results with those of finite element models is found to be in close agreement.  2001 Elsevier Science Ltd. All rights reserved. Keywords: Cracked plates; Vibration; Buckling; Parametric instability

1. Introduction Plates are widely used in the fields of civil, mechanical, aerospace and marine engineering (e.g., plate-girders, automotive, aircraft and submarine). Sharp corners of openings, cutouts and welding have great potential for initiation and propagation of cracks especially when subjected to dynamic loads. The behavior of thin plates with a crack has been the subject of numerous investigations [1–5]. In this type of problem a plate carries all or some of the in-plane load. A variety of modes of static and dynamic behavior is possible and failures often result from the development of fatigue cracks, which propagate from a stress * Corresponding author. Tel.: +98-021-600-5419; fax: +98-021-601-2983. E-mail address: [email protected] (A. Vafai). 0263-8231/02/$ - see front matter  2001 Elsevier Science Ltd. All rights reserved. PII: S 0 2 6 3 - 8 2 3 1 ( 0 1 ) 0 0 0 5 0 - 7

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Nomenclature a¯, b¯ Actual plate length and half of width, respectively b Half of plate width scaled by p/a¯ c Crack size (length) scaled by p/a¯ c¯ Actual crack size (length) D Plate rigidity E Module of elasticity of plate h Plate thickness I Unit matrix k(x,y,x,h) An analytical function for the deflection of the plate at point x,y due to a unit load applied at a point x,h ith critical static buckling load N∗s,i 1th critical buckling load of plate without crack N∗us,1 Static portion of parametric loading Ns Dynamic portion of parametric loading Nt Parametric loading along x direction Nx(t) w(x,y,t) Plate deflection at point x and y and time t a,b Static and dynamic factors which are percentages of the fundamental compressive critical load Dirac delta dik g Width to length ratio of plate jk(x,y) Normalized eigenfunctions of kth mode of free vibration m Mass of plate per unit area n Poisson’s ration of plate q Frequency of parametric loading Natural frequency of ith mode of free vibration wi Fundamental frequency of free vibration of plate without crack wu1

concentration point at the crack tip. A problem of interest in static analysis is associated with out-of-plane deflections, which accompany buckling, and post-buckling. Dynamic behavior studies on this class of problems usually involve bending vibration in the absence or presence of an initial in-plane stress. Vibration and/or compressive buckling behavior of plates with a crack has been examined by many investigators [6–10]; however, there is not much work available in the literature on the dynamic stability of cracked plates. Considering dynamic stability problems, the study of parametric instability is a great concern from both theoretical and practical points of view. In parametric instability study, the transverse vibrations may be induced in structural elements when subjected to in-plane periodic forces. These vibrations may be resonant for combinations of natural frequency of transverse vibration, the frequency of the in-plane forcing functions and the magnitude of in-plane loading. The spectrum

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of values of parameters causing unstable motion is referred to as the region of dynamic instability or parametric resonance. A comprehensive analysis of parametric instability can be found in the literature [11,12]. In order to study the parametric instability behavior of a structure, both analytical [11] and numerical [13,14] methods can be utilized, however, analytical methods are more interesting from the theoretical point of view. In the present work, the problem of parametric instability in rectangular plates with edge crack is studied using the integral equation method. The individual effects of parameters, such as static load factor, crack size and plate dimension, on parametric instability behavior, are investigated.

2. Formulation The deflections w(x,y,t) of a plate subjected to a parametric loading, Nx(t) = Ns + Nt cosqt (positive when Nx(t) produces tension), leads to the integration of the equation ⵜ4w(x,y,t) ⫽

冉

冊

1 ∂2w ∂2w Nx(t) 2 ⫺m 2 , D ∂x ∂t

(1)

where D is flexural rigidity, D⫽

Eh3 , 12(1⫺v2)

(2)

with h the plate thickness, E the plate module of elasticity and n Poisson’s ratio, and m is the mass of plate per unit area. Introducing the deflection influence function k(x,y,x,h), i.e., an analytical function for the deflection of the plate at point x,y due to a unit load applied at a point x,h, the solution of Eq. (1) can be represented in the following integro-differential equation form:

冕冕 冕冕

w(x,y,t) ⫹ m ⫺Nx(t)

∂2w(x,h,t) k(x,y,x,h) dxdh ∂t2 ∂2w(x,h,t) k(x,y,x,h) dxdh ⫽ 0. ∂x2

(3)

The solution of Eq. (3) is sought in the form of a series [11]

冘 ⬁

w(x,y,t) ⫽

fk(t)jk(x,y),

(4)

k⫽1

where jk(x,y) are the normalized eigenfunctions for the free vibration problem or

冕冕

m

ji(x,y)jk(x,y)dxdy ⫽ dik,

(5)

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(d␫␬ is dirac delta) and fk(t) are time functions, which must be determined. The convergence of series (4) follows from a well-known theorem in the theory of linear integral equations. Let us substitute series (4) into Eq. (3). Taking into account that

冕冕

ji(x,y) w2i

k(x,y,x,h)ji(x,h)dxdh ⫽

(6)

(note that Eq. (6) represents the problem of free vibration) and using the expansion

冕冕

Nx(t)

∂2jk(x,h) k(x,y,x,h) dxdh ⫽ ∂x2

冘 ⬁

aik(t)ji(x,y),

(7)

i⫽1

we arrive at a system of differential equations for the coefficients of series (4):

冘 ⬁

fi⬙ ⫹ w2i [fi⫺

Fik(t)fk] ⫽ 0, i ⫽ 1,2,….

(8)

k⫽1

Functions Fik(t) are defined by the formula 1 w2i

Fik(t) ⫽

冕冕 冉

冊

∂2 j k ji Nx(t) 2 dxdy, ∂x

(9)

where the primes denote differentiation with respect to time. Eq. (8) can be rewritten in the following matrix form: C

d 2f ⫹ [I⫺F(t)]f ⫽ 0, dt2

(10)

in which

冤

1/w21

C⫽

0

0 1/w

0

0

⯗

⯗

0 2 2

…

0

% 3 3

1/w % ⯗

哻

冥

,

(11)

and

F(t) ⫽

冤

F11(t) F12(t) F13(t) … F21(t) F22(t) F23(t) % F31(t) F32(t) F33(t) % ⯗

⯗

⯗

哻

冥

where Fik(t) are presented in Eq. (9).

,

(12)

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33

3. Plate with crack emanating from one edge In the case of simply supported plates with a crack emanating from one edge, the eigenfunctions for the free vibration problem have the following form [10]:

冘 ⬁

wi(x,y) ⫽

Yim(y)sin(mx),

(13)

m⫽1

where 1 [A sinh(r1iy) ⫹ Bim cosh(r1iy) ⫹ Cim sinh(r2iy) D im ⫹ Dim cosh(r2iy)],

Yim(y) ⫽

(14)

r1i ⫽ √kwi ⫹ m2

(15)

with r2i ⫽ √kwi⫺m2 k⫽

冪D. m

In the following, one should employ the normalized form of eigenfunctions (13), using Eq. (5); ji(x,y) ⫽

wi(x,y)

冪冕冕w (x,y)dxdy

.

(16)

2 i

After some simplifications, Eq. (16) can be rewritten as ji(x,y) ⫽

冪pm 2

wi(x,y)

冪冘冕Y ⬁

b

m⫽1

0

2 im

(17) (y)dy

and substituting Eq. (17) into Eq. (9), Fik(t) can be rewritten as (taking into account that, here, Nx(t) is compressive):

冘 冕 册冋 冘 冪冋 冘 ⬁

Fik(t) ⫽ Nx(t)aik, aik ⫽

1 mw2i

b

m2 Yim(y)Ykm(y)dy 0

m⫽1 ⬁

Y (x)dx

p⫽1

q⫽1

册

.

⬁

2 ip

2 iq

(18)

Y (x)dx

In the first approximation, the boundaries of the principal regions of instability can be obtained from the following equation:

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|

|

1 I⫺lA⫺ q2C ⫽ 0, 4

(19)

1 l ⫽ Ns± Nt, 2

(20)

where

Now, some important properties of Eq. (19) are mentioned. If q and Nt are equal to zero, Eq. (19) is reduced to the following eigen value problem:

|

|

I⫺N∗s A ⫽ 0,

(21)

in which Ns∗’s are the critical static buckling loads. Another property of Eq. (19) emerges when Nt is set to zero

|

|

1 I⫺NsA⫺ q2C ⫽ 0, 4

(22)

therefore, Eq. (22) leads to the natural frequencies of a plate, subjected to in-plane compressive (|Ns|⬍|Ns∗|) or tensile (Ns⬍0) loading. Finally, when l is equal to zero, the solution of Eq. (19) gives the natural frequencies w, scaled by two.

4. Computation The general geometrical configuration of a plate with edge crack is illustrated in Fig. 1. In this figure, the coordinates and dimensions of the plate geometry are scaled by the factor p/a¯ , where a¯ is the actual plate length. Actual (barred) coordinates and dimensions are obtained by use of x¯ = a¯ x/p, y¯ = a¯ y/p, b¯ = a¯ b/p and c¯ = a¯ c/p. The

Fig. 1. Geometry and loading.

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35

first five natural frequencies and their corresponding mode shapes are computed for plates with different dimensions g = 2b/p, and crack length size c/p (for details of this procedure, see [10]). Fig. 2 shows the values of the first three natural frequencies, which are normalized by the fundamental frequency of free vibration of uncracked plate wu1, and a number of vibration mode shapes. The coefficients aik, in Eq. (18), are calculated using numerical integrations, e.g.,

Fig. 2. Effect of crack size on non-dimensionalized natural frequencies. (a) g = 2 and (b) g = 1.

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Fig. 3. Fundamental frequency parameters for a rectangular plate with an edge crack subjected to axial compressive loading (g = 2).

Fig. 4. Fundamental frequency parameters for a rectangular plate with an edge crack subjected to axial tensile loading (g = 2).

Simpson’s method. A computer code has been written to perform this procedure and control accuracy and numerical error. It is obvious that for the first five-mode shapes, F(t) leads to a 5×5 matrix; therefore, only 25 quantities are required to be saved for each particular case. In comparison with the finite element method, this results in considerable saving in computation effort and data storage. In order to check the accuracy of results the static stability (as a special case) is compared with calculated buckling loads, using other analytical or numerical methods. The equation for static stability is presented in Eq. (21). It is worth noting that this equation provides the exact amount of buckling loads, provided the buckling mode shapes exactly coincide with the free vibration mode shapes, which are taken

(a) g = 2 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 (b) g = 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

c/p

1.000 0.998 0.968 0.871 0.777 0.689 0.630 0.587 0.547 0.506 0.485

1.000 1.000 0.996 0.980 0.943 0.887 0.821 0.761 0.712 0.679 0.668

1.000 1.000 0.996 0.981 0.945 0.889 0.825 0.763 0.714 0.679 0.668

Mode #1

Mode #1

1.562 1.560 1.535 1.411 1.157 0.916 0.753 0.642 0.562 0.508 0.485

n=3

n=1

1.563 1.563 1.554 1.529 1.502 1.491 1.491 1.472 1.196 1.005 1.005

1.175 1.172 1.146 1.127 1.253 0.948 0.793 0.702 0.642 0.601 0.668

Mode #2

6.248 6.248 6.241 6.134 5.297 3.195 1.973 1.480 1.450 1.417 1.452

1.562 1.561 1.542 1.495 1.390 1.279 1.117 0.925 0.774 0.693 0.974

Mode #3

1.000 1.000 0.996 0.980 0.943 0.887 0.821 0.761 0.712 0.679 0.668

1.000 0.998 0.968 0.871 0.764 0.685 0.629 0.586 0.546 0.506 0.485

Mode #1

n=5

1.563 1.563 1.554 1.529 1.502 1.490 1.490 1.315 1.119 0.969 1.402

1.175 1.172 1.146 1.127 1.125 0.894 0.747 0.666 0.613 0.575 0.668

Mode #2

2.963 4.001 3.982 3.757 2.902 2.071 1.604 1.479 1.450 1.417 2.637

1.562 1.558 1.542 1.495 1.191 1.099 1.000 0.870 0.763 0.692 0.974

Mode #3

1.501 1.487 1.111 1.401

0.943 0.821 0.712 0.668

0.669

0.486

1.553

0.609

0.546

0.996

0.744

0.630

1.562

1.144 1.125 1.109

0.967 0.870 0.765

1.000

1.174

Mode #2

1.000

Mode #1

Target

Table 1 Effect of numbers of the base functions on the convergence of non-dimensionalized compressive critical loads (N∗s,i/N∗us,1)

2.632

1.447

1.574

2.670

2.756

2.775

0.974

0.762

0.996

1.513 1.476 1.123

1.562

Mode #3 A. Vafai et al. / Thin-Walled Structures 40 (2002) 29–44 37

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Fig. 5.

A. Vafai et al. / Thin-Walled Structures 40 (2002) 29–44

Effect of crack size on non-dimensionalized critical loads. (a) g = 2 and (b) g = 1.

into account in Eq. (18). In all cases, this can be assured through examining the effect of the number of vibration mode shapes on the convergence of buckling load values, as shown in Table 1, where N∗us,1 is the first critical buckling load of plate without crack. In this table, n denotes the number of vibration mode shapes, e.g., for n = 1 only the first mode of vibration is taken into consideration. Consequently, the first three buckling loads are derived (for n = 1 this can only be done for the first buckling load) and compared with the work of other investigators [10] (denoted by target values). It should be noted that the aforementioned procedure is essential for the parametric instability study, which is presented in the following section. In fact, for each plate,

A. Vafai et al. / Thin-Walled Structures 40 (2002) 29–44

Fig. 6.

39

Effect of crack size on fundamental regions of instability (a = 0.6 and g = 2).

the first step is to determine the least required number of vibration modes that lead to results with a satisfactory accuracy.

5. Results and discussion A simply supported rectangular plate with a crack emanating from one edge subjected to unidirectional uniform in-plane loading acting on a pair of opposite edges normal to the x-direction, is considered. The effects of the length to width ratio of plate g, and the crack size to length ratio c/p, the magnitude, nature and frequency of the in-plane forces on the vibration, buckling and parametric instability behavior of the plate are investigated, and the results are discussed in the following sections. 5.1. Vibration behavior The effect of applied in-plane compressive loading Ns, on the fundamental frequency of vibration w1, obtained from Eq. (22), is plotted in Fig. 3 and compared with the finite element results. In this figure, for simplicity, values of Ns and w1 are ∗ and the fundamental frequency of vibration scaled by the critical buckling load Ns,1 u of plate without crack w1, respectively. It can be seen that as the in-plane load increases, the frequency decreases and becomes zero at the respective values of the buckling load of the plate. Fig. 4 illustrates the vibration of fundamental frequencies with in-plane tensile load for plates with different crack size and, again, comparison with the finite element results is provided. It is evident from these figures that the method presented here and the finite element method provide results in quite close agreement. Nevertheless, obtaining each point using the presented method is much faster (with the magnitude of 104)

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Fig. 7.

First three regions of instability (c/p = 0.6). (a) g = 2, (b) g = 1 and (c) g = 0.5.

than the finite element method; therefore, the presented method offers significant advantages over the finite element method.

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41

Fig. 8. Effect of static load on the fundamental regions of instability (c/p = 0.6). (a) g = 2 and (b) g = 0.5.

5.2. Static buckling behavior Fig. 5 shows the effect of c/p and g on the compressive critical buckling load of plates at the first three modes. In this figure, the curved lines represent the results obtained using Eq. (21) and the target points are the finite element results. It is observed that except for the third mode of buckling in g = 1.0, when c/p⬍0.5, both methods provide the same results. The reason behind those cases mentioned where the presented method does not lead to the target results, underlies the fact that in Eq. (4), jk(x,y)’s must be orthogonal functions. However, as depicted in Fig. 2, for these cases, the eigenfunctions of modes two and three of vibration coincide with each other. In other words, the orthogonality of modes two and three is invalid and this phenomenon causes disagreement in results.

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A. Vafai et al. / Thin-Walled Structures 40 (2002) 29–44

Table 2 Primary regions of instability for plate with different crack sizes a

(a) g = 2 0.4

0.6

(b) g = 1 0.4

0.6

(c) g = 0.5 0.4

0.6

b

c/p = 0.2

c/p = 0.8

U

L

U

L

0 0.2 0.4 0.6 0.8 0 0.2 0.4 0.6 0.8

0.859 0.895 0.928 0.961 0.993 0.782 0.821 0.859 0.895 0.928

0.859 0.821 0.782 0.740 0.679 0.782 0.740 0.679 0.497 0.013

0.474 0.435 0.389 0.339 0.278 0.389 0.435 0.474 0.510 0.545

0.474 0.510 0.545 0.577 0.608 0.389 0.339 0.278 0.198 0.007

0 0.2 0.4 0.6 0.8 0 0.2 0.4 0.6 0.8

0.774 0.706 0.633 0.549 0.447 0.633 0.706 0.774 0.837 0.894

0.774 0.837 0.894 0.947 0.999 0.633 0.549 0.447 0.316

0.654 0.596 0.533 0.464 0.379 0.533 0.596 0.654 0.706 0.758

0.654 0.706 0.758 0.800 0.847 0.533 0.464 0.379 0.268 0.018

0 0.2 0.4 0.6 0.8 0 0.2 0.4 0.6 0.8

0.790 0.729 0.661 0.587 0.501 0.632 0.707 0.774 0.836 0.895

0.790 0.844 0.903 0.953 1.000 0.632 0.548 0.447 0.317 0.019

0.742 0.677 0.606 0.525 0.429 0.606 0.677 0.742 0.801 0.853

0.742 0.801 0.853 0.911 0.953 0.606 0.525 0.429 0.304 0.029

5.3. Parametric instability In this part, the parametric instability behavior of a simply supported plate with an edge crack subjected to compressive in-plane loading is examined. The static load factor a and the dynamic load factor b are percentages of the fundamental compressive critical load N∗s,1. Fig. 6 shows the effect of c/p on the first region of instability when all specimens have the same value for static load factor, a = 0.6. Fig. 7 illustrates the first three regions of instability for simply supported plates with different g, having static load factors of a = 0.4 and a = 0.6.

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43

The effect of static load factor, a, on the first region of instability is shown in Fig. 8 for plates with different g. It can be seen that as a increases, the central frequency of the first region of instability moves inward on the frequency ratio axis and the zone is also widened. The upper (U) and lower (L) boundaries of primary instability regions are presented in Table 2 for a = 0.4 and a = 0.6 and for different values of g and c/p.

6. Conclusions In this paper, the integral equation method is utilized for the static and dynamic analyses of plates with an edge crack subjected to uni-directional uniform periodic compressive loadings. The results of this study can be summarized as follows: 1. The method employed in this study has the advantage of high accuracy accompanied by great savings in computation time and data storage as compared with the finite element method. 2. For utilizing the integral equation method, special attention should be paid to the effect of those eigenfuncions that do not meet the orthogonality property. 3. The results of dynamic instability under compressive periodic loading demonstrate that, as the static load factor increases, the width of the instability region increases and the instability region moves inward on the frequency ratio axis, thus making the plate more susceptible to instability.

References [1] Anifantis NK, Actis RC, Diamarogonas AD. Vibration of cracked annular plates. Engng Fract Mech 1994;49:371–9. [2] Dyshel MS. Stress–intensity coefficient taking account of local buckling of plates with cracks. Soviet Appl Mech 1990;26:87–90 [English trans. of Prikladnaya Mekhanika]. [3] Gilbert A, Sibillot P, Sornette D, Vanneste C, Maugis D, Muttin F. Buckling instability and pattern around holes or cracks in thin plates under a tensile load. Eur J Mech A/Solids 1992;11:65–89. [4] Lee HP, Lim SP. Vibration of cracked rectangular plates including transverse shear deformations and rotary inertia. Comput Struct 1993;49:715–8. [5] Qiang GL, Gu SN, Jiang JS. A finite element model of cracked plates and application to vibration problems. Comput Struct 1991;39:483–7. [6] Quirk A, Bevitt E. Crack stability in centre cracked plates subjected to uniformly-distributed loadings. In: Fracture mechanics verifications by large-scale testing, IAEA Spec. Meet Large Scale Test, Bury St. Edmunds (UK): MEP; 1991:87–103. [7] Roy YA, Shastry BP, Rao GV. Stability of square plates with through transverse cracks. Comput. Struct. 1990;36:387–8. [8] Riks E, Rankin CC, Bargon FA. Buckling behavior of a central crack in a plate under tension. Engng Fract Mech 1992;43:529–48. [9] Shaw D, Huang YH. Buckling behavior of a central cracked thin plate under tension. Engng Fract Mech 1990;35:1019–27. [10] Stahl B, Keer M. Vibration and stability of cracked rectangular plates. Int J Solids Struct 1972;8:69–92.

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A. Vafai et al. / Thin-Walled Structures 40 (2002) 29–44

[11] Bolotin VV. The stability of elastic systems. San Francisco (CA): Holden-Day, 1964. [12] Kratzig WB, Niemann HJ. Dynamics of civil engineering structures. Netherlands: A.A. Balkema Publishers, 1996. [13] Hutt JM, Salam AE. Dynamic stability of plates by finite element method. J Engng Mech Div ASCE 1971;97:879–99. [14] Prabhakara DL, Datta PK. Vibration, buckling and parametric instability behavior of plates with centrally located cutouts subjected to in-plane edge loading (tension or compression). Thin-Walled Structures 1997;27:287–310.