Buckling analysis of tri-layer beams with enveloped delaminations

Composites: Part B 36 (2005) 33–39 www.elsevier.com/locate/compositesb

Buckling analysis of tri-layer beams with enveloped delaminations Parlapalli MS Rao, Tu Wenge, D. Shu* School of Mechanical and Production Engineering, Nanyang Technological University, 50 Nanyang Avenue, Singapore, Singapore 639798 Received 13 June 2003; revised 8 March 2004; accepted 4 April 2004 Available online 8 June 2004

Abstract By introducing new nondimensionalized axial and bending stiffnesses, a generalized analytical analysis using classical engineering solutions is proposed to study the buckling behavior of tri-layer beams having asymmetrical enveloped delaminations. Appropriate boundary and kinematical conditions are used to formulate the systematic homogeneous equations. The present analysis accurately predicts the buckling loads of tri-layer beams having double enveloped delaminations, which is verified with the previously published data for homogeneous beams having single and double delaminations. An excellent agreement between the results is observed. A transition region is observed as the normalized bending stiffness increases. Effective-slenderness ratio is defined and introduced, which gives a measure of local, mixed and global buckling phenomenon. The present analysis provides a useful reference for the tri-layer beams having enveloped delaminations. q 2004 Elsevier Ltd. All rights reserved. Keywords: A. Layered structures; B. Buckling; B. Delamination; Stiffness

1. Introduction Composites materials find wider applications in numerous fields like aerospace, transportation, marine, automobile, electronics, bio-medical and various manufacturing industries. The gamut of the composites includes design flexibility, high performance, light weight, good resistance to corrosion, high strength, low cost, low weight to density ratio, etc. The increasing usage of the composites in various industries demands better understanding of their structural behavior and failure modes. However, study of the behavior of such materials has shown that they are prone to environmental factors and/or localized defects. Of the defects, delaminations play a very important role and are common in multilayered materials, which arise due to manufacturing errors (e.g. by an imperfect curing process) or in-service accidents (e.g. low velocity impacts) [1]. Under compression, a delaminated composite beam may buckle and possibly undergo propagation of delaminations. This in turn can lead to degradation of the extensional or bending strength of the beam. Buckling analysis of delaminated composite beams and or plates has been a topic of researchers interest since Chai et al. * Corresponding author. Tel.: þ 65-6790-4440; fax: þ 65-6791-1859. E-mail address: [email protected] (D. Shu). 1359-8368/$ - see front matter q 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.compositesb.2004.04.014

[2] analyzed and developed a one-dimensional mathematical model. Later on Bottega and Maewal [3], Simitses et al. [4], Chai and Babcock [5], Yin et al. [6], Williams et al. [7], Kardomateas and Schumueser [8], Peck and Springer [9], Chen [10], Yeh and Tan [11], Shu and Mai [12], Short et al. [13] studied the buckling analysis of single delaminated beams considering various effects. Lim and Parsons [14] used the Rayleigh–Ritz method to analyze the buckling analysis of beams having double equal delaminations. Suemasu [15] investigated the compressive buckling stability of composite panels having through-the-width, equally spaced multiple delaminations. Adan et al. [16] developed an analytical model for buckling of multiply delaminated composite under cylindrical bending and studied their interactive effects. Kutlu and Chang [17,18] investigated the compression response of laminated composite panels containing multiple through-the-width delaminations by both nonlinear finite element method and experiments. Whitcomb [19] and Kim [20] studied the behavior of a post-buckled embedded delamination using a non-linear finite element analysis. Lee et al. [21] presented a one-dimensional finite element buckling and post-buckling analysis of cylindrically orthotropic circular plates containing single and multiple delaminations. Shu [22] identified ‘free mode’ and ‘constrained mode’ of buckling for a beam with multiple delaminations

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by an exact solution. Kyoung et al. [23] studied the buckling and post-buckling analysis of single and multiple delaminated orthotropic beams by nonlinear finite element analysis. Sekine et al. [24] investigated the buckling analysis of elliptically delaminated composite laminates by taking into account of partial closure of delamination. Yin and Jane [25,26] developed an analytical procedure based on the Rayleigh–Ritz method and von Karman’s non-linear plate theory to predict the buckling and post-buckling behavior. Gu and Chattopadhyay [27,28] developed a new higher order theory for delamination and post-buckling analysis and with classical laminated plate theory and conducted experimental analysis to verify the models. Haiying and Kardomateas [29] used a non-linear beam theory to study the multiple delaminations of orthotropic beams. Hu et al. [30] conducted the buckling analysis of laminates with embedded elliptical delaminations by employing finite-element method based on the Mindlin plate theory. Moradi and Taheri [31] studied the buckling analysis of delaminated composite plates by the application of differential quadrature method. The multiple delaminations could be of various configurations such as enveloped, overlapped and separated, etc. In general, the previous buckling analyses for homogeneous delaminated beam cannot be applied to the present tri-layer delaminated beam. The analytical analysis of asymmetrically located enveloped delaminations of trilayer beams made of three materials is cumbersome and complex, which is not yet studied. The large numbers of material and geometric parameters associated with multiple delaminated composite beams of present configuration make a generic analytical study of the buckling behavior difficult. Young’s modulus and thickness ratios of the individual layers of the tri-layer delaminated beam do not give a clear perceptive of the buckling behavior. Thus, in order to get a clear understanding of the buckling behavior for a wide range of materials of tri-layer delaminated beams, nondimensionalized axial and bending stiffnesses and effective-slenderness ratio (ESR) have been introduced in the present analysis. To predict the buckling load precisely, the Euler – Bernoulli beam theory is used combined with appropriate kinematical continuity conditions, equilibrium equations and boundary conditions. A closed form expression of the characteristic equation of tri-layer beam with enveloped delaminations is derived. The results of the present analysis are compared with literature for homogeneous delaminated beam cases and are extended to the present case. The newly introduced nondimensionalized parameters strongly influence the critical buckling load.

2. Formulation 2.1. Problem definition The geometry of the delaminated tri-layer beam employed in the present study is shown in Fig. 1a.

Fig. 1. (a) Tri-layer beam with multiple asymmetric enveloped delaminations, (b) equivalent beam model.

The beam is of length L; thickness H; and unit width. It has two pre-existing asymmetrical enveloped delaminations of length a1 and a2 at offset distances of d1 and d2 ; respectively (a1 , a2 and X2 , X3 , X4 , X5 ). Both ends of the beam are clamped and an external load P is applied along the neutral axis of the beam. The beam consists of three distinctive layers, with Young’s moduli E3 ; E4 ; and E5 and thicknesses H3 ; H4 and H5 ; respectively. The two enveloped delamination divides the tri-layer beam into seven interconnected beams in which virgin beams are 1 and 7, and the remaining are sub-beams 2, 3, 4, 5 and 6, shown in the Fig. 1a, which have thicknesses of Hi and beam lengths of Li ði ¼ 1 – 7Þ; respectively. The equivalent beam model is shown in Fig. 1b. Points A and F denote the support ends and points B; C; D and E denote the delamination ends. The coordinate axis for the sub-beams is shown in the Fig. 1a and X is measured from the left end of the beam. Wi ðXÞ denotes the small elastic deflection of segment i ( ¼ 1 2 7). The delamination closest to the top surface is denoted as delamination I and the second delamination, further inside as delamination II. The sizes and locations of the sub-beams are defined by their length Li and depth Hi ; which are given by: L a a a 2 2 2 d2 ; L2 ¼ 2 2 1 2 2 2 2 a2 L3 ¼ a2 ; L4 ¼ a1 ¼ L5 ; L6 ¼ 2 2 L a L7 ¼ 2 2 þ d2 ; H1 ¼ H7 ¼ H; 2 2

L1 ¼

þ d1 þ d2 ; a1 2 d1 2 d2 ; 2 H2 ¼ H6 ¼ H3 þ H4 :

Following assumptions are made to simplify the present analysis: Each layer of the beam is homogeneous, isotropic and linearly elastic in nature, the compressive load is uniform and uniaxial, delaminations exist before the application of the load, interference of the layers is neglected, ‘free mode

P.M.S. Rao et al. / Composites: Part B 36 (2005) 33–39

analysis’ of Shu [22] is incorporated and unit width is considered. 2.2. Basic equations

ð1Þ

The flexural rigidities of the sub-beams 3, 4 and 5 are given by ðEIÞ3 ¼ E3 H33 =12; ðEIÞ4 ¼ E4 H43 =12; and ðEIÞ5 ¼ E5 H53 =12; respectively. Sub-beams 2 and 6 are bi-material beams whereas the virgin beams 1 and 7 are tri-layer beams. Qi0 and Mi0 are the shear force and bending moment at one end of the beam segment under consideration, respectively. Pi is the axial force acting on the ith beam segment. The Qi0 and Mi0 terms vanish by differentiating Eq. (1) twice. Therefore, Eq. (1) is differentiated twice to yield the usual differential equation, governing the buckled configuration for ith segment as: ðEIÞi W 00i ðXÞ þ Pi W 00i ðXÞ ¼ 0 ði ¼ 1 2 7Þ

L a 2 1 þ d1 : W2 ðX3 Þ ¼ W4 ðX3 Þ; 2 2 W2 ðX3 Þ ¼ W5 ðX3 Þ; W 02 ðX3 Þ ¼ W 04 ðX3 Þ; W 02 ðX3 Þ ¼ W 05 ðX3 Þ;

II: At X ¼ X3 ¼

Q2 ðX3 Þ ¼ Q40 þ Q50 ; M2 ðX3 Þ ¼ M40 þ M50

The bending moment distribution along the beam segment Mi ðXÞ can be expressed as 2Ei Ii W 00i ðXÞ (Timoshenko and Gere, [32]). For a beam segment from one end to an internal cross-section at X; the moment equilibrium yields: ðEIÞi W 00i ðXÞ þ Pi Wi ðXÞ þ Qi0 X þ Mi0 ¼ 0 ði ¼ 1 2 7Þ

35

ð2Þ

The general solution for Eq. (2) is: Wi ðXÞ ¼ Ai cosðli XÞ þ Bi sinðli XÞ þ Ci X þ Di ði ¼ 1 2 7Þ ð3Þ pffiffiffiffiffiffiffiffiffiffi with li ¼ Pi =ðEIÞi ði ¼ 1 2 7Þ where Ai ; Bi ; Ci and Di are unknown coefficients. The necessary equations, including kinematic relations, constitutive relations, relations between loads and moments, and equilibrium equations are given as follows. 2.2.1. Boundary conditions The boundary conditions for clamped end conditions are given by: At X ¼ X1 ¼ 0 : W1 ðX1 Þ ¼ 0 and W 01 ðX1 Þ ¼ 0: At X6 ¼ L : W7 ðX6 Þ ¼ 0 and W 07 ðX6 Þ ¼ 0: ð4Þ 2.2.2. Kinematic, shear force and moment continuity conditions L a I: At X ¼ X2 ¼ 2 2 2 d2 : W1 ðX2 Þ ¼ W2 ðX2 Þ; 2 2 W1 ðX2 Þ ¼ W3 ðX2 Þ; W 01 ðX2 Þ ¼ W 02 ðX2 Þ; W 01 ðX2 Þ ¼ W 03 ðX2 Þ;

2 DP5 ðlY2 2 Y5 lÞ þ DP4 ðlY2 2 Y4 lÞ:

ð6Þ

L a þ 1 þ d1 : W6 ðX4 Þ ¼ W4 ðX4 Þ; 2 2 W6 ðX4 Þ ¼ W5 ðX4 Þ; W 06 ðX4 Þ ¼ W 04 ðX4 Þ; W 06 ðX4 Þ ¼ W 05 ðX4 Þ;

III: At X ¼ X4 ¼

Q6 ðX4 Þ ¼ Q40 þ Q50 ; M6 ðX4 Þ ¼ M40 þ M50 2 DP5 ðlY6 2 Y5 lÞ þ DP4 ðlY6 2 Y4 lÞ:

ð7Þ

L a þ 2 2 d2 : W7 ðX5 Þ ¼ W6 ðX5 Þ; 2 2 W7 ðX5 Þ ¼ W3 ; ðX5 Þ W 07 ðX5 Þ ¼ W 06 ðX5 Þ; W 07 ðX5 Þ ¼ W 03 ðX5 Þ;

IV: At X ¼ X5 ¼

Q7 ðX5 Þ ¼ Q30 þ Q60 ; M7 ðX5 Þ ¼ M30 þ M60 2 DP3 ðlY3 2 Y7 lÞ þ DP6 ðlY7 2 Y6 lÞ:

ð8Þ

where Y is the neutral axis position of the tri-layer beam, Yi ði ¼ 2 – 6Þ are the neutral axis positions of the ith beam, DPi ði ¼ 2 – 6Þ are unknown incremental axial forces acting in the sub-beams 2, 3, 4, 5 and 6 arising from the axial extension/compression, Mi ðXÞ ¼ 2Ei Ii W 00i ðXÞ and Qi ðXÞ ¼ 2Ei Ii Wi 00 ðXÞ: 2.2.3. Additional conditions The systems of linear, homogeneous equations consist of 33 unknowns (28 unknown coefficients Ai ; Bi ; Ci ; Di i ¼ 1 – 7; and five unknown axial forces DP2 ; DP3 ; DP4 ; DP5 and DP6 ) and there exist 28 boundary and continuity conditions. Therefore, five additional equations are required to solve the systematic equations. The axial force balances at X ¼ X2 ; X ¼ X3 and X ¼ X4 are given by DP2 þ DP3 ¼ 0; and DP4 þ DP5 ¼ DP6 ; respectively. The compatibility between the stretching/shortening of the sub-beams 2, 3, 4, 5, and 6 is given by: DP3 L3 þ lY 2 Y3 lðu1 2 u4 Þ ¼ 0 ðEHÞ3 and DP2 L2 þ DP6 L6 DP5 L5 þ þ u1 lY 2 Y2 l 2 u2 lY2 2 Y5 l ðEHÞ2 þ ðEHÞ6 ðEHÞ5 þ u3 lY5 2 Y6 l ¼ 0 ð9Þ where

shear force continuity,

u1 ¼ W 01 ðX ¼ X2 Þ; u2 ¼ W 04 ðX ¼ X3 Þ;

Q1 ðX2 Þ ¼ Q20 þ Q30 ;

u3 ¼ W 04 ðX ¼ X4 Þ and u4 ¼ W 03 ðX ¼ X5 Þ:

bending moment continuity, M1 ðX2 Þ ¼ M20 þ M30 þ DP2 ðlY 2 Y2 lÞ 2 DP3 ðlY 2 Y3 lÞ ð5Þ

2.2.4. Non-dimensionalization Delamination lengths (a1 and a2 ; Fig. 1a) and delamination locations along the spanwise position (d1 and d2 ; Fig. 1a)

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are nondimensionalized with respect to beam length as, a 1 ¼

a1 a d d ; a 2 ¼ 2 ; d 1 ¼ 1 ; d 2 ¼ 2 : L L L L

ð10Þ

The following nondimensionalized parameters are introduced, sffiffiffiffiffiffiffiffi pffiffi Pi L2    ði ¼ 2 2 6Þ: l1 ¼ l1 L ¼ 2p P ¼ l7 ; li ¼ li L ¼ ðEIÞi ð11Þ P ¼ P=Pcr where Pcr is the critical buckling load of the undelaminated beam and for clamped boundary conditions, Pcr ¼

4p2 ðEIÞ1 : L2

The unknown coefficients (Ai ; Bi ; Ci and Di ) are nondimensionalized as:

where K¼

ð1 2 NAS3 2 NAS4Þ NAS4 ;T¼ ;Z¼ NAS3 NAS3

rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi NAS3NBS4 ; NAS4NBS3

and T1 ¼ NBS3½1 þ Z 2 T þ 3NAS32 {ðZ 2 þ 1 þ 2ZÞT 2 þ K 2 þ TK 2 Z 2 þ Tð1 þ Z 2 þ 2ZÞ þ KT 2 Z 2 þ K þ 2KTð1 þ Z þ Z 2 Þ} 2 1 After H5 =H3 is calculated, H4 =H3 ; E5 =E3 ; and E4 =E3 are determined from the definitions of the nondimensionalized axial and bending stiffness, which will be substituted in the system of linear homogeneous equations to solve for the critical buckling load as the equations are functions of Yang’s moduli and thickness ratios. The derivations are omitted for brevity.

A B CL  Di A i ¼ i ; B i ¼ i ; C i ¼ i ; D ði ¼ 1 2 7Þ: i ¼ H3 H3 H3 H3 ð12Þ

2.2.5. Solutions for nondimensionalized equations The system of linear, homogeneous algebraic equations, which govern buckling of the delaminated tri-layer beam, is written in matrix form as

The unknown incremental axial forces are nondimensionalized as:

½P{A} ¼ 0;

DP2 DP3 DP4 DP5 ; DP 3 ¼ ; DP 4 ¼ ; DP 5 ¼ ; Pcr Pcr Pcr Pcr DP6 DP 6 ¼ Pcr ð13Þ

DP 2 ¼

New nondimensionalized parameters, nondimensionalized axial and bending stiffnesses, have been introduced in the present analysis instead of conventional Young’s moduli ratio and thickness ratios of the tri-layer delaminated beam to analyze the buckling. Nondimensionalized axial stiffness (NAS) of the subbeam 3 is defined as: NAS3 ¼ ðEAÞ3 =ðEAÞ1 : Similarly for sub-beam 4, NAS4 ¼ ðEAÞ4 =ðEAÞ1 : Nondimensionalized bending stiffness (NBS) of the subbeam 3 is defined as NBS3 ¼ ðEIÞ3 =ðEIÞ1 : Similarly for subbeam 4, NBS4 ¼ ðEIÞ4 =ðEIÞ1 : For given nondimensionalized axial and bending stiffnesses (NAS3, NAS4, NBS3 and NBS4), the thickness ratio and Young’s moduli ratio of the individual sub-beams will be calculated and substituted in the system of homogeneous equations (Eqs. (5) – (9)). H5 =H3 is calculated from the following equation:

ð14Þ

where ½P is a 33 £ 33 matrix and it’s elements contain geometrical and material properties in addition to load parameter, li that are expressed in terms of newly introduced nondimensionalized axial and bending stiffnesses, and {A} is a 33 £ 1 (column) matrix which is given by  1 ; A 2 ; B 2 ; C 2 ; D  2 ; A 3 ; B 3 ; C 3 ; D  3 ; A 4 ; {A} ¼{A 1 ; B 1 ; C 1 ; D  4 ; A 5 ; B 5 ; C 5 ; D  5 ; A 6 ; B 6 ; C 6 ; D  6 ; A 7 ; B 4 ; C 4 ; D B7 ; C 7 ; D  7 ; DP 2 ; DP 3 ; DP 4 ; DP 5 ; DP 6 }: The characteristic equation is given by lPij l ¼ 0; ði; j ¼ 1 2 33Þ

ð15Þ

and the lowest eigenvalue is a measure of the critical buckling load. A computer program has been written for finding the critical buckling load, which uses LDU decomposition method to find out the roots of the characteristic equation, Eq. (15). 3. Results

þ 3TK 2 þ 12T þ 3KT 2 þ 3K þ 18KTÞ

This section presents results obtained using the analytical model described above to study the buckling behavior of tri-layer beam with enveloped double delaminations. The present analysis is first compared with the literature for single and double delaminations of a homogeneous beam cases. Selected results for double delaminations of tri-layer beam follow this.

þ ðH5 =H3 ÞNAS32 NBS3ð12T 2 ð1 þ ZÞ þ 6K 2

3.1. Homogeneous beam

ðH5 =H3 Þ2 NBS3ðK þ NAS32 ½12T 2 þ 3K 2

2

2

þ 6TK Z þ 12Tð1 þ ZÞ þ 6KT Z þ 6K þ 18KTð1 þ ZÞÞ þ T1 ¼ 0

The accuracy of the method employed in this study is verified by comparing the results of the homogeneous

P.M.S. Rao et al. / Composites: Part B 36 (2005) 33–39 Table 1 Comparison of critical buckling loads with Simitses et al. [4] for a homogeneous beam having single delamination with clamped boundary conditions a=L

0.2 0.4 0.6 0.8 1.0

H3 ¼ 0:1H

H3 ¼ 0:3H

H3 ¼ 0:5H

Present

Ref. [4]

Present

Ref. [4]

Present

Ref. [4]

0.2495 0.0624 0.0278 0.0156 0.01

0.2495 0.0624 0.0278 0.0156 0.01

0.9924 0.5314 0.2435 0.1390 0.09

0.9924 0.5314 0.2435 0.1390 0.09

0.9956 0.8481 0.5411 0.3514 0.25

0.9956 0.8481 0.5411 0.3514 0.25

delaminated beam for clamped boundary conditions with Simitses et al. [4]. The delamination I is shrunk to a length of 0.001 L to model it as a single delamination case, always ensuring that the developed system of homogeneous equations are valid in the entire analysis. There is an excellent agreement between the results is observed as shown in Table 1. Further, to check the validity of the present model, Table 2 shows a comparison between the present analysis and Haiying and Kardomateas [29], for the case of enveloped delaminations of a homogeneous beam (H3 ¼ 0:75H; H4 ¼ 0:125H). A good consensus is obtained. 3.2. Tri-layer beam After successful verification of the present analysis for homogeneous delaminated beam cases, it is extended to the tri-layer beam having enveloped double delaminations. Attention here is focused on the ESR and non-dimensionalized axial and bending stiffnesses on the critical buckling load, as it is rather difficult to consider all the possible combinations of geometric and material properties. 3.2.1. Effective-slenderness ratio (ESR) vs critical buckling load ðP=Pcr Þ The ESR chosen to represent the range of applicability of the thin-film buckling analysis. For clamped beam, ESR5 ¼ 1:0 represents the case that sub-beam 5 and the whole beam are geometrically similar. For sub-beam 5,

37

ESR is defined as qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ESR5 ¼ a 21 NAS5=ðNBS5Þ: Similarly ESR3, ESR4 are defined for sub-beams 3 and 4. Fig. 2 shows the strong effect of ESR5 on the P=Pcr for various delamination lengths of delamination I. The length of second delamination a 2 is 0.6. The change in P=Pcr with respect to the ESR5 is observed in three regions. In the region I (ESR5 , 2.3), the global buckling prevail and the influence of the delamination fades. In the region II (2.3 , ESR5 , 3.35), the buckling load decreases rapidly, where the mixed mode buckling is observed. The sub-beams 5 and 3 buckle together. In the region III (ESR5 . 3.35), the decrease of the buckling load is less significant and local buckling occurs in the sub-beam 5 as its ESR is higher than that of the sub-beams 3 and 4. Hence the ESR is the measure of global, mixed and local buckling phenomenon. In Fig. 2, all the curves are merging to a single line, indicating the effect of delamination I is minimal when compared to delamination II as delamination II is greater than delamination I in all the cases. The global, mixed and local mode shapes are observed at different ESR5s (2.1, 3.1 and 6.35, respectively) that are shown at the bottom of Fig. 2. 3.2.2. Nondimensionalized bending stiffness (NBS) vs critical buckling load ðP=Pcr Þ Fig. 3 shows the effect of NBS on the P=Pcr for various NASs. The delamination lengths are a 1 ¼ 0:4 and a 2 ¼ 0:6: For sub-beam 4, the nondimensionalized stiffnesses are NAS4 ¼ 0:15 and NBS4 ¼ 0:05: It is worth noting that P=Pcr does not vary monotonically with the NBS3. The behavior could be divided into three regions. In the region I, P=Pcr increases as the NBS3 increases, where ESR3 . ESR5 . ESR4: Local buckling mode is observed at lower NBSs. Region II is called transition region in which the ESR of sub-beam 3 reduces and that of sub-beam 5 increases, in turn it affects the buckling behavior. In the region III, ESR5 . ESR3 . ESR4; and the buckling behavior changes from global to local buckling mode. As the NBS3 increases further, ESR5 increases rapidly and the sub-beam 5 buckles locally. This is contrary to a beam with single delamination, where the corresponding variation is always monotonic [33].

Table 2 Comparison of critical buckling loads with Haiying and Kardomateas [29] for homogenous beams having enveloped delaminations (H3 ¼ 0:75H; H4 ¼ 0:125H) with clamped boundary conditions a2 =L

0.2 0.4 0.6 0.8 1.0

a1 =L ¼ 0:1

a1 =L ¼ 0:3

a1 =L ¼ 0:5

Present

Ref. [29]

Present

Ref. [29]

Present

Ref. [29]

0.9627 0.3754 0.1709 0.0969 0.0625

0.9628 0.3754 0.1709 0.0970 0.0625

0.1730 0.1508 0.1191 0.0852 0.0602

0.1731 0.1508 0.1191 0.0867 0.0602

0.0624 0.0624 0.0571 0.0497 0.0431

0.0624 0.0624 0.0571 0.0497 0.0431

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Fig. 4. Effect of normalized axial stiffness on the critical buckling load; a1 ¼ 0:4L; a2 ¼ 0:6L; NAS4 ¼ 0:05; NBS4 ¼ 0:05: Fig. 2. Effect of effective-slenderness ratio on the critical buckling load; a2 ¼ 0:6L; NAS3 ¼ 0:5; NBS3 ¼ 0:15; NAS5 ¼ 0:15; mode shapes shown at different effective-slenderness ratios (ESR5 ¼ 2:1; 3.1, 6.35, respectively).

3.2.3. Nondimensionalized axial stiffness (NAS) vs critical buckling load ðP=Pcr Þ Fig. 4 shows the influence of NAS on the P=Pcr for various NBSs. The delamination lengths are a 1 ¼ 0:4 and a 2 ¼ 0:6: For sub-beam 4, the nondimensionalized stiffnesses are NAS4 ¼ 0:05 and NBS4 ¼ 0:05: As the NAS increases the buckling load increases, for given NBS. P=Pcr varies monotonically with the NAS. This is due to the decrease of ESR3 and increase of ESR5. In the present case, ESR5 dominates the buckling behavior. 3.2.4. Delamination length vs critical buckling load Fig. 5 shows the variation of the P=Pcr with the delamination length a2 =L for various NBSs. P=Pcr decreases as the delamination length increases. The delamination length a1 is kept constant at 0:15L and the delamination length a2 is varied from 0:2L to 1:0L: Three regions of buckling behavior are observed. In the region I, the buckling load is insensitive to the larger NBSs as the delamination I is shorter than delamination II. At shorter enveloped delaminations, P=Pcr is not sensitive to delamination length a1 ;

Fig. 3. Effect of normalized bending stiffness on the critical buckling load; a1 ¼ 0:4L; a2 ¼ 0:6L; NAS4 ¼ 0:15; NBS4 ¼ 0:05:

and a2 ; up to the threshold value a2 =L ¼ 0:3:P=Pcr decreases thereafter, (in the region II) and the decrease is slower for longer delaminations up to a2 =L . 0:75: The local buckling mode is observed as a2 =L increases further.

4. Conclusions For the first time, the critical buckling load ðP=Pcr Þ of an enveloped delaminated tri-layer beam was accurately solved for clamped boundary conditions. This solution provides an attractive alternative to the usual lengthy FEM solution for delaminations in layer composite laminates and debonding in coated materials. The accurate solution can serve as a useful reference for other numerical schemes. Nondimensionalized axial and bending stiffnesses were introduced in the discussion instead of the moduli and thicknesses of the individual sub-beams. Consequently, P=Pcr varies monotonically with the normalized axial stiffness whereas it is worth noting that P=Pcr does not vary monotonically with the normalized bending stiffness and a transition region was observed. This is contrary to single delaminated beam case, where the corresponding variation is always monotonic. The stiffnesses have strong

Fig. 5. Effect of delamination length II on the critical buckling load; NAS3 ¼ 0:2; NAS4 ¼ 0:40; NBS4 ¼ 0:1; a1 ¼ 0:15L:

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effect on the buckling load. New parameter, ESR was introduced. P=Pcr strongly depends on the ESR. At low ESRs (, 2.3), the global buckling prevails and the influence of the delamination fades. At higher ESRs (. 3.35) local buckling occurs in the sub-beam having higher ESR. Hence the ESR is the measure of global, mixed and local buckling phenomenon. The critical buckling load is not sensitive to shorter enveloped delaminations whereas it is strongly influenced by longer enveloped delaminations.

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