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Lateral Load effects on buckling of cracked plates under tensile loading
Thin-Walled Structures 72 (2013) 37–47
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Lateral Load effects on buckling of cracked plates under tensile loading Rahman Seifi n, Ali Reza Kabiri Faculty of Engineering, Bu-Ali Sina University, Hamedan, Iran
art ic l e i nf o
a b s t r a c t
Article history: Received 2 March 2013 Received in revised form 20 June 2013 Accepted 20 June 2013
Fractured thin plates may be buckled locally due to applying the tensile loads in certain conditions. Sometimes the buckling mode can occur before fracturing or collapsing. Lateral loads can affect the buckling loads and modes. In thin cracked plates, compressive stress region near the crack or defect can be changed due to the lateral loads, so critical local buckling loads can be increased or decreased. Experimental and numerical study of the buckling load of plates with central crack under axial tensile loading is done. Effects of lateral load and constraint, relative crack length and its direction are investigated. & 2013 Elsevier Ltd. All rights reserved.
Keywords: Local buckling Tensile loading Lateral load Cracked plate
1. Introduction Sheets or plates are components that are used in a wide range of structures. Ships, offshore structures and airplanes are examples of complex structures made by thin-walled plates under different load combinations. A thin plate can contain different types of defects such as cracks produced by different causes like corrosion or fatigue loadings. It is important to study the behavior of these plates with or without faults in order to get the safety assessment of the structures. Buckling under compressive loads is known to happen for thin plates especially when some imperfections as holes or cracks exist, buckling under tension loadings is also important and can heavily affect the structural strength. Many researchers in the past decade studied the buckling of cracked plates under compression, tension or other types of loads for different structures. Most of these researches were done on perfect plates and some of them were about the imperfect plates. Numerical analysis, such as finite element method, was used helpfully for studying the buckling. This method is widely used for the study of the behavior of imperfect shells and plates. Brighenti [1,2] studied the buckling of cracked plates under tension and compression, by considering the effects of geometrical, mechanical and boundary conditions. He showed numerically that the boundary conditions affect the compression buckling but it almost has not effect on the buckling under tension. Crack length and orientation also have a significant influence in reducing the buckling loads. Also he proposed an approximate analytical method to evaluate the critical buckling load of cracked tensioned plates [2]. In plates under tension, based on the values of some
n
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parameters such as fracture toughness, yield strength and critical buckling stress, local buckling can occur before fracture or plastic failures [3]. Riks et al. [4] proposed a numerical method for the analysis of buckling and post buckling of cracked tensioned plates. The results show that the buckling deformations cause a significant increase in the stress intensity factor of the crack tip. This effect is due to the change of stresses distributions in the plate which increases by increasing the crack length. Zielsdorff and Carlson [5] investigated the local buckling of the cracked plate under tensile loading by using experimental method. Effects of the crack tip curvature and crack length were also studied. Influence of the buckling process on plane stress fracture toughness was obtained from center cracked tensioned plates. They obtained some formulas for the determination of the critical buckling stress of cracked plates under tensile loading perpendicular to the length of crack. Seifi and Khoda-yari [6] studied the buckling of the fractured thin plates under uniform compression, experimentally and numerically. Effects of crack length and orientation and thickness of plates, different types of supports and loadings, such as partial edge acting forces and supports, were considered. Khedmati et al. [7] used the finite element method to study the buckling of plates with cracks. Effects of location, length and orientation of crack and plate aspect ratio on the critical loads were studied. They showed that the behavior of the cracked plate is different when the crack is either on the edge or inside the plate. Wrinkling of stretched thin sheets is a commonly occurrence in local buckling, leading to complex deflection patterns, particularly in regions close to the cracks. Sih and Lee [8] performed some predictions of the buckled displacement modes for cracked plates which was made on the basis of variational methods. Markstrom and Storakers [9] analyzed the buckling characteristics of cracked plates subjected to uniaxial tensile loads by the aid of the finite element method based on linear bifurcation
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theory. Interaction of local buckling and fracture behaviors of stretched cracked plate were also discussed. Shaw and Huang [10] studied the effect of bi-axial tension on the critical load. It is found that lateral load stabilizes the plate and buckling load increases as this load increases. This is due to the lateral tension that reduces the magnitude and area of the compression stress around the crack. Alinia et al. [11] used the finite element method for buckling analysis of cracked shear panels containing central or edge cracks. They showed that the long cracks have significant influence on the buckling capacity of shear panels, especially if the cracks exist in the tension region of the plates. Paik et al. [12] carried out an experimental and numerical study on the ultimate strength of cracked steel plates under compressive or tensile axial loadings. They also developed some relations, based on their results, for predicting the ultimate strength of cracked plates. They concluded that the ultimate strength of a cracked plate subjected to tensile loads can be determined on the basis of the reduced cross-sectional area due to crack. Vafai and Estekanchi [13] investigated the general behavior of cracked sheets and shells by using the finite element method. Effects of various parameters such as the mesh size in the near crack tip region, boundary conditions, Poisson’s ratio, crack length and curvature of the membrane have been studied. Dyshel [14] experimentally studied the stability of cracked plates under biaxial tensile loading. He showed that the critical stress is proportional to the elastic modulus and square of the ratio of thickness to crack length. Also it was shown that local loss of stability depends strongly on the magnitude of the lateral load. As can be seen, almost all researches about the buckling of cracked plates are performed numerically and for axial loading. In this study, experimental and numerical determination of buckling loads on the cracked plates under uniform tensile load is discussed. Effects of tensile or compressive lateral loads were studied. Parameters such as the length and orientation of cracks, aspect ratio of plates, variation of lateral loads, can affected the buckling load of cracked plate with a central crack; such effects were evaluated numerically and experimentally.
perturbation loads, we can extract the tensile buckling load. The following method is used in eigenvalue extraction for linear buckling calculation: we suppose an arbitrarily achieved base configuration with initial stresses sE , in equilibrium state under some surface E tractions t E , and body forces b . We consider an elastic deformation with very ‘small’ deformation gradients, under additional surface tractions Δt, body forces Δb and boundary displacements Δu. Applying these incremental external loads causes arising of the additional stresses, Δs. Since the problem is linear, then by applying λ times of these loads, the stress response will be λΔs. Each distinct value of λ corresponds with a linear perturbation of the initial equilibrium state. Among these perturbed states we can find special values of λ that allow for the existence of nontrivial incremental deformations with arbitrary magnitudes as valid solutions to the problem. Such nontrivial incremental displacements are referred to as buckling modes, and λ as buckling eigenvalue. Buckling load is the summation of initial load and λ times the load is applied in the perturbation step. For structures with nonlinear behavior, it is necessary to obtain nonlinear static equilibrium solutions. In numerical methods, a load-deflection method (such as Riks [15] and modified Riks method [16]) may be performed to investigate the nonlinear buckling. These methods use the load magnitude as an additional unknown and solved simultaneously for loads and displacements. The development of the solution requires that we traverse the path of equilibrium in a space defined by the loading parameter (λ) and some variables, such as displacements. The basic algorithm is the Newton–Raphson method (Finding the root of equation as FðxÞ ¼ 0 by iteration from xnþ1 ¼ xn −Fðxn Þ=F′ðxn Þ) as depicted in Fig. 1a; therefore, at any time there will be a finite radius of convergence with proper increment size i.e. the length of the tangent line, Δl. The increment size is chosen in a way that the ratio of the norm of error to norm of known variables vector norm is smaller than the prescribed tolerance (ε); in the other words, if Ku ¼ f , then ‖Ku−f ‖=‖f ‖ o ε where K is the global stiffness matrix, u is the unknown displacement vector and f is the known load vector. In the modified Riks method, the increment size is limited by moving a given distance along the tangent line (Δl) to the current
2. Numerical study of tensile buckling of cracked plates For thin plates under tensile loading, the compressive stress zone is induced near the crack or fault and the local buckling could occur due to these stresses. Some numerical procedures can be used to evaluate the tensile buckling load. For structures with linear behavior before buckling, it is proper to estimate the elastic buckling load by eigenvalue extraction. In this method, the buckling load is calculated as a multiplier of the pattern of perturbation loads (in buckling step), which is added to a set of base state loads (in static loading step). Based on the state of the
Fig. 2. Definition of variables for tensile buckling of cracked plate.
Fig. 1. (a) The Newton–Raphson method. (b) Modified Riks algorithm.
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solution point and then searching for equilibrium in the plane that passes through the point that is obtained and is orthogonal to the same tangent line (Fig. 1b).
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The Riks method for determination of tensile buckling load of cracked plate is described in Ref. [4]. Based on the works of Riks et al. [4,15] and results of the numerical methods, it is observed
Fig. 3. Dimensions and configurations of samples.
Table 1 Numbering of samples with different length and angle. Number
1
2
3
4
5
6
7
8
9
a/W Angle(1)
0.5 0.0
0.6 0.0
0.7 0.0
0.5 15
0.6 15
0.7 15
0.5 30
0.6 30
0.7 30
Fig. 4. Mesh of the model for case ‘e3’.
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that the crack parameter values such as energy release rate (G or J-integral), crack opening displacement and stress intensity factors as a function of the load, will bifurcate (with finite slope) from relation that belong to the (linear) pre-buckling state. In pre-buckling stage, there are not any out of plane displacements in the plate while in buckling and post-buckling stages, these displacements are introduced and also increased by increasing the load. These deformations cause variations in the crack parameters such as J-integral or G values. In this paper, we compared the variations of J-integral versus load for linear and buckling modes. In linear analysis, out of plane displacement was restricted, while in buckling mode, this restriction was removed. Separation point was assumed for 5% deviation in J-integral values. The effects of lateral load on the tensile buckling of cracked plates were studied in this paper. Schematic view of the cracked plate with lateral load is depicted in Fig. 2. As is shown, plate has 2L 2W t dimensions with a center crack with length 2a. Lateral load (P) was applied at two opposite points in x-direction, while θ is the orientation angle of crack with respect to x-axis (normal to loading direction). The edge placed at y¼−L line is clamped, while the tensile load N is applied to the edge y¼ +L (Fig. 2). The parameters of the problem were assumed as described below: two different plate sizes were adopted: 0.24 m 0.24 m and
0.24 m 0.18 m both with t¼1 mm. Crack lengths were equal to a/W¼ 0.5,0.6,0.7 while their orientation angles are θ¼ 0, 15, 301. Lateral loads are assumed as P¼−400, −200, 0, 200, 400 N. These values are smaller than one-third of the buckling load for cracked plates under lateral load only. All the studied cases are shown in Fig. 3. For each case, crack has three different length and orientation angle. We identify the samples in the following manner: a letter for the condition of sample (as in Fig. 3) followed by a number according to Table 1 for describing the crack length and its angle. For example, ‘j4’ is for case ‘j’ in Fig. 3 with a/W¼0.5 and θ¼ 151. Elastic properties of aluminum 1200 A were used in the numerical calculations. All cases in Fig. 3 were modeled with proper mesh pattern, density and elements (Fig. 4). For instance, finite element model of sample ‘e3’ in abaqus software is showed in Fig. 4, characterized by 6830 elements type S8R (quadratic eight node reduced integration points plane stress element). As can be seen, there are small elements in the positions of concentrated lateral loads and around the crack tips. Calculation of the linear tensile buckling gives the first eigenvalue equal to λ ¼25,523 for an initial load equal to N ¼1.0 N/m and so Ncr ¼25,523 N/m. Nonlinear buckling of this sample is calculated by using modified Riks algorithm and obtaining the variation of J-integral as depicted in Fig. 5. For deviation of 5%, the buckling load was
Fig. 5. Varation of load versus J-integral.
Fig. 8. distribution of compressive stress sx along y-axis in the Fig. 1.
Fig. 6. Buckling load versus lateral load for 2 W ¼0.24 m and (a) θ ¼01 (b) θ ¼151 (c) θ¼ 301.
Fig. 7. Buckling load versus lateral load for 2 W ¼0.18 m and (a) θ ¼01 (b) θ ¼151 (c) θ¼ 301.
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Fig. 9. Upper fixture.
Fig. 10. Lower fixture.
Fig. 11. Assembled fixtures and sample (a) without, and (b) with lateral load.
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estimated as Ncr ¼25,000 N/m. As can be seen, there is 2% difference between the linear and nonlinear results. The results for other models were calculated by using the mentioned method and are shown in Fig. 6 for plates with width 2W¼ 0.24 m and in Fig. 7 for 2W¼ 0.18 m. The total Buckling load is obviously the product of the distributed buckling load Ncr and width of plate 2W. According to these figures, applying lateral constraints and loads increases significantly the buckling load in comparison with uniaxial tensile buckling. In all cases, buckling load decreases by increasing the crack length but increases with increasing the lateral loads. It must be noted that the effect of constraint without any lateral load is also considerable in comparison with changing the lateral load. As can be seen, tensile lateral loads induce a decrease but compressive loads induce an increase of the buckling load. This is because of changing the compressive stresses around the crack line. For instance, distribution of the stresses parallel to crack line along the vertical line near the midpoint of crack (sx along y-axis in Fig. 2) is depicted in Fig. 8 for plates with W¼ 120 mm, a/W¼0.7, θ ¼ 0∘ under axial tensile loads as N¼ 28,400 N/m (buckling load for free edge case) and different lateral loads. This figure shows that the lateral
load alternates the compressive stress distribution and so buckling load. It must be noted that the lateral load changes the compressive stress values but its region around the crack line remains almost constant hence the buckling mode does not change. An important aspect for cracked plates under tensile load is the possibility of crack growing before buckling. For determination of collapse mode, it is necessary to compute the fracture parameters, such as stress intensity factor or J-integral, due to buckling load and comparing with fracture toughness. In this paper, because of angled cracks, a mixed mode fracture arises, so we calculated the J-integral values. These values are compared with strength JC. For series 1000 of aluminum alloys the plane strain fracture toughness (JIC) is about 14.0 KN/m. The J C −K C relationship can be computed in the following manner: K IC ¼
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 EJ IC =ð1−v2 Þ; K C ¼ K IC ð1 þ Bk e−ðAk t=t 0 Þ Þ; J C ¼ K 2C =E
ð1Þ
where t0 ¼ 2.5KIC/sy; for aluminum alloys we have elastic modulus equal to E¼70.0 GPa, Poisson’s ratio as v¼ 1/3, material constants equal to Ak ¼Bk ¼ 1.0 and yield stress equal to sy ¼137 MPa. After some calculations, JC is derived to be 56.0 KN/m. J-integral values due to the buckling load for all cases of Fig. 3 were derived and compared with J C . The maximum value of calculated J-integrals is about 12,230 N/m for case ‘d7’. Thus for all cases, buckling occurs before crack growth. It must be noted that for crack angles, usually larger than 301, in some cases the calculated J-integral is larger than JC so crack grows before buckling.
Table 3 Mechanical properties of 1200 A Al alloy [6]. Yield stress
Ultimate strength
Elongation Hardness Elastic modulus
Poisson’s Ratio
137 MPa
183 MPa
2%
0.33
54.4 HB
70 GPa
Fig. 13. Force-extension curve for case ‘l3’.
Fig. 12. Measuring the out of plane displacement by extensometer.
Table 2 Chemical composition of samples and comparing with standard [6]. Sample
Si 0.36
Standard ASTM E 1251-07 Min Max
Fe
Cu
Mn
Mg
Cr
Zn
Al
0.62
0.01
0.01
Trace
Trace
0.01
Base
Fe+Si
Cu
Mn
Mg
Cr
Zn
Other each
Other total
Al
* 1.00
* 0.10
* 0.30
* 0.3
* 0.1
* 0.1
* 0.05
* 0.15
99.00 *
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3. Experimental Investigation of tensile buckling of cracked plates 3.1. Design and fabrication of fixtures All tests were done by using the SANTAM tensile machine. For testing the samples two proper fixtures were designed and fabricated. Upper fixture is a channel with set screws on its edges. Two suitable slabs were mounted to clamp the samples. Details of this fixture is depicted in Fig. 9. Lower Fixture is similar to upper one with additional devices for applying and measuring the lateral loads as shown in Fig. 10.
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Assembled fixtures on the machine and mounted sample were shown in Fig. 11. First figure shows the sample with free edges and the other is a sample with lateral load. A proper extensometer was used for measuring the out of plane displacement on the crack edges (Fig. 12).
3.2. Material and samples Samples were made from aluminum 1200 A with 1 mm thickness. Chemical compositions of material are depicted in Table 2 and compared with ASTM E 1251-07 standard. Mechanical properties of this alloy are given in Table 3. Samples sizes were 0.24 m 0.24 m and 0.18 m 0.24 m, obtained from a 1.0 m 2.0 m aluminum sheet (Fig. 3). Cracks were Table 5 The factor of the effects of different parameters on the buckling load of 0.24 m 0.24 m samples. Levels
L1 L2 L3 L4 L5 L6 ΔL
Fig. 14. Load-out of plane displacement curve for case ‘j4’.
Factors Lateral load factor
Crack angle factor
Crack length factor
5523 9856 11,000 11,846 9877 8793 6324
7775 8567 12,105 – – – 4330
12,629 8908 6910 – – – 5719
Fig. 15. Determination of slope (m) for Eq. (2).
Table 4 L18(61 32) Taguchi design factors and levels for tensile buckling load. Test
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18
Factor Lateral load (N)
Crack angle (1)
Crack length (a/W)
Free Free Free 0 0 0 200 200 200 400 400 400 −200 −200 −200 −400 −400 −400
0 15 30 0 15 30 0 15 30 0 15 30 0 15 30 0 15 30
0.5 0.6 0.7 0.5 0.6 0.7 0.6 0.7 0.5 0.7 0.5 0.6 0.6 0.7 0.5 0.7 0.5 0.6
Experimental results (N) for 0.24 m 0.24 m
Experimental results (N) for 0.18 m 0.24 m
6361.025 4614.65 5591.97 11,859.01 8262.64 9445.2 8470.47 7541.2 16,987.88 7670.6 14,393.68 13,474.42 7401.8 6325.5 15,904.03 4884.24 10,267.3 11,226.2
8208.38 6207.88 5700.34 13,207.34 9730.6 10,954.33 10,237.25 8512.25 19,603.43 8991.72 15,371.63 14,444.67 9488.5 6787.26 17,953.38 6258.91 11,913.3 11,539.38
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created in samples by wire electrical discharge machining method with 0.1 mm thickness. Created cracks have about 0.2 mm radius at their tips. Based on our numerical calculations, the small curvature on crack tip does not have considerable effect on the buckling load. For example, for case ‘e3’ with sharp crack and with 0.2 mm radius at crack tip, the buckling loads are 25,523 and 25,510 N/m (0.05% difference). Thirty-six specimens were made with different conditions based on the results of Taguchi method [17]. All samples were tested and values of applied force versus in-plane extensions and out of plane displacements at middle point of crack edges were recorded and saved.
3.3. Determination of buckling load The cracked plates under tensile load may buckle locally in the regions with compressive stress around the crack’s edges. This local buckling may not change the variation of applied force versus in-plane displacement so we cannot use the load-extension curve for determination of the buckling load. For example, the forceextension curve for case ‘l3’ was depicted in Fig. 13. As can be seen there are no considerable changes in trend of the curve due to the local buckling. Instead of in-plane extension we can use the out of plane displacement in the middle point of crack’s edge. For instance, the load-displacement curve for case ‘j4’ is depicted in Fig. 14. As can be seen, there is nonlinear tendency for load versus out of plane displacement but it is still difficult to obtain the buckling load value. It is clear that the curve has a knee and before it, the Table 6 The factor of the effects of different parameters on the buckling load of 0.18 m 0.24 msamples. Levels
L1 L2 L3 L4 L5 L6 ΔL
Factors Lateral load factor
Crack angle factor
Crack length factor
6706 11,297 12,784 13,366 11,410 9904 6661
9399 9817 13,518 – – – 4120
14,439 10,427 7867 – – – 6571
curve has linear trend. The lower limit was defined as the stage when the out of plane displacement, in the vicinity of the crack edge, begins to increase rapidly due to the applied load. This linear limit of the curve will be used as an approximate value of pre-load for calculation of tensile buckling loads. Based on Southwell method [18,19] and modified Southwell method [5,20], we can derive the buckling load from the following relation: P cr ¼ P 0 þ 1=m
ð2Þ
where m is the slope of linear part of the curve of ðw−w0 Þ=ðP−P 0 Þ versus ðw−w0 Þ. P 0 and w0 are the pre-load (usually load at the end of linear part of the force-out of plane extension curve) and out of plane extension due to P 0 , respectively. For example, the slope value for sample ‘c7’ was derived. ðP−P 0 Þ and ðw−w0 Þ=ðP−P 0 Þ curves versus ðw−w0 Þ were determined as depicted in Fig. 15. For P0 ¼ 5020 N and 1/m ¼1/0.0842 KN according to Fig. 15, we have Pcr ¼16.897 KN for sample ‘c7’. This method was used for other samples. As shown in the Fig. 3, there are 12 cases with nine different conditions for each case (three crack lengths and three crack angles) so 108 samples must be made. 3.4. Designing the experiments by Taguchi method For reducing the number of tests, we designed them based on the Taguchi method. There are three signal to noise (S/N) ratios of common interest for optimization of problems in Taguchi method. Smaller the better: This is usually the chosen S/N ratio for all undesirable characteristics like defects for which the ideal value is zero. Also, when an ideal value is finite and its maximum or minimum value is defined then the difference between measured data and ideal value is expected to be as small as possible. Larger the better: This case has been converted to ‘smaller the better’ by taking the reciprocals of measured data and then taking the S/N ratio as in the ‘smaller the better’ case. Nominal the best: This case arises when a specified value is most desired, meaning that neither a smaller nor a larger value is desirable. We have one factor (lateral load) with six levels and two factors (crack length and orientation) with three levels, hence the array L18(61 32)must be used. For this purpose, we did 18 tests for each plate sets 0.24 m 0.24 m and 0.18 m 0.24 m as depicted in Table 4. The results of Taguchi method were obtained by using Minitab software and are depicted in Tables 5 and 6.
Table 7 Comparison of experimental and numerical tensile buckling load. 0.24 m 0.24 m
0.18 m 0.24 m
Sample
Numerical (N)
Experiment (N)
Difference (%)
Sample
Numerical (N)
Experiment (N)
Difference (%)
a1 a5 a9 b1 b5 b9 c2 c6 c7 d3 d4 d8 e2 e6 e7 f3 f4 f8
6820.32 5204.64 5765.28 11,531.52 9163.68 10,093.68 9170.88 7917.84 18,414.72 8065.68 13,843.92 14,489.52 7866.96 6634.32 16799.04 5426.16 10985.76 11729.76
6361.025 4614.65 5591.97 11,859.01 8262.64 9445.2 8470.47 7541.2 16,896.5 7670.6 14,393.68 13,474.42 7401.8 6325.5 15904.03 4884.24 10267.3 11226.2
6.73 11.33 3 −2.83 9.83 6.42 7.63 4.75 8.4 4.89 −3.97 7 5.91 4.65 5.32 9.98 6.53 4.29
g1 g5 g9 h1 h5 h9 i2 i6 i7 j3 j4 j8 k2 k6 k7 13 14 18
7998.66 5897.34 6220.8 13,984.56 10,508.04 11,430.54 10,814.58 8952.12 20,582.82 9298.08 16,534.8 16,176.24 9131.94 7252.38 18619.2 6036.48 12925.98 12450.96
8208.38 6207.88 5700.34 13,207.34 9730.6 10,954.33 10,237.25 8512.25 19,603.43 8991.72 15,371.63 14,444.67 9488.5 6787.26 17953.38 6258.91 11913.3 11539.38
−2.62 −5.26 8.36 5.55 7.39 4.16 5.33 4.91 4.75 3.29 7.03 10.70 −3.90 6.41 3.57 −3.68 7.83 7.32
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Tables 5 and 6 show that the effects of lateral load and constraints are greater than the crack length and angle. Crack angle has a low effects on the tensile buckling load. These results agree with the numerical results in Figs. 5 and 6. Based on ‘larger the better’ S/N ratio,
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the fourth level of first factor, third level of second factor and first level of third factor give the maximum tensile buckling load while minimum value based on ‘smaller the better’ S/N ratio, is obtained for first level of first and second factors and third level of third factor.
Fig. 16. Some samples in buckling step.
Fig. 17. Effects of lateral loads for different crack length in 0.24 m 0.24 m plates.
Fig. 18. Buckled form of plates with a/W ¼0.7.
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Fig. 19. Effects of crack length on buckling load for 0.24 m 0.24 m.
Fig. 20. Effects of crack length on buckling load for 0.24 m 0.24 m.
4. Results and discussion Based on the results of the Taguchi method, all of the 36 tests were done and their results were compared with numerical calculations. Differences of results for two methods were depicted in Table 7. Some samples in buckling step were shown in Fig. 16. Observing this table shows that the data for experimental and numerical methods are in good agreement.
4.1. Effects of lateral load on the critical condition Under uniaxial loading, tensile buckling load increases by increasing the crack angle. Critical condition (minimum buckling load) occurs for crack with zero angle. Applying the lateral load can change this situation. Effects of lateral load versus crack length for different crack lengths were depicted in Fig. 17.
As can be seen, applying small lateral loads does not change the condition of the minimum buckling load for all the considered crack lengths. The angle of crack for minimum load, increases by increasing the crack length and lateral load. The resultant of axial and lateral loads changes due to increasing the lateral load so that minimum buckling load can be achieved for a position where crack is almost normal to the resultant force. Difference between the buckling loads decreases by increasing the crack length and orientation. This is due to the fact that the crack orientation is close to normal direction with respect to the resultant force, while for zero angle crack, the lateral load is parallel to crack line. For instance, buckled form for plates with a/W¼0.7, crack angles 01 and 301 and lateral loads equal to 200 N and 9600 N were depicted in Fig. 18. As can be seen, for small values of lateral loads the eigenvalue for mode 1 is larger for θ¼301 while for large lateral loads the condition is reversed.
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4.2. Effects of crack length For uniaxial loading, the buckling load decreases by increasing the crack length for all conditions. This behavior is also true for bi-axial loading as depicted in Fig. 19 for 0.24 m 0.24 m and in Fig 20 for 0.12 m 0.24 m. Reduction of loads is larger for large inclinations. For both series of plates, constraints (case P ¼0) considerably increase the buckling load versus free edge case for all lengths and inclinations of cracks. This is due to the creation of a resistance over the out of plane deformations of plates. Increasing the tensile in-plane lateral loads on the constraints causes increasing the buckling load while increasing the compressive lateral load decreases the critical load. Although, these variations are not considerable in comparison with cases with constraints. 4.3. Conclusion In this paper the tensile buckling behavior of cracked plate was studied. Effects of crack length and orientations, constraints and lateral load were considered. Based on the results, the following conclusions can be obtained: Tensile buckling load decreases with increasing crack length and orientation. Due to the presence of lateral supports, buckling load increases significantly in comparison with uniaxial buckling. Lateral tensile (compressive) load increases (decreases) the critical buckling load with respect to the case with only lateral support. Minimum tensile buckling load due to change of crack orientation can be altered by lateral load. Usually lateral load does not alter the first mode shape of buckling. References [1] Brighenti R. Numerical buckling analysis of compressed or tensioned cracked thin plates. Engineering Structures 2005;27:265–76. [2] Brighenti R. Buckling of cracked thin plates under tension or compression. Thin-Walled Structures 2005;43:209–24.
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[3] Brighenti R. Buckling sensitivity analysis of cracked thin plates under membrane tension or compression loading. Nuclear Engineering and Design 2009;239:965–80. [4] Riks E. The buckling behavior of a central crack in a plate under tension. Engineering Fracture Mechanics 1992;434:529–48. [5] Zielsdroff GF, Carlson RL. On the buckling of thin tensioned sheets with cracks and slots. Engineering Fracture Mechanics 1972;4:939–50. [6] Seifi R, Khoda-yari N. Experimental and numerical studies on buckling of cracked thin-plates under full and partial compression edge loading. ThinWalled Structures 2011;49:1504–16. [7] Khedmati MR, Edalat P, Javidruzi M. Sensitivity analysis of the elastic buckling of cracked plate elements under axial compression. Thin-Walled Structures 2009;47:522–36. [8] Sih GC, Lee YD. Tensile and compressive buckling of plates weakened by cracks. Theoretical and Applied Fracture Mechanics 1986;6:129–38. [9] Markstrom K, Storakers B. Buckling of cracked members under tension. International Journal of Solids and Structures 1980;16:217–29. [10] Shaw D, Huang YH. Buckling behavior of a central cracked thin plate under tension. Engineering Fracture Mechanics 1990;35:1019–27. [11] Alinia MM, Hosseinzadeh SAA, Habashi HR. Numerical modeling for buckling analysis of cracked shear panels. Thin-Walled Structures 2007;45:1058–67. [12] Paik JK, Kumar YV, Lee JM. Ultimate strength of cracked plate elements under axial compression or tension. Thin-Walled Structures 2005;43:237–72. [13] Vafai A, Estekanchi HE. A parametric finite element study of cracked plates and shells. Thin-Walled Structures 1999;33:211–29. [14] Dyshel MS. Stability of thin plates with cracks under biaxial tension. Soviet Applied Mechanics 1982;18:924–8. [15] Riks E. An incremental approach to the solution of snapping and buckling problems. International Journal of Solids and Structures 1979;15:529–51. [16] Crisfield MA. A fast incremental/iterative solution procedure that handles snap-through. Composite Structures 1981;13:55–62. [17] Rosa JL, Robin A, Silva MB, Baldan CA, Peres MP. Electrodeposition of copper on titanium wires: Taguchi experimental design approach. Journal of Materials Processing Technology 2009;209:1181–8. [18] Southwell RV. On the analysis of experimental observations in problems of elastic stability. Proceedings of the Royal Society London 1932;135:601–16. [19] Kalkan I. Application of Southwell method on the analysis of lateral torsional buckling tests on reinforced concrete beams. International Journal of Engineering Research and Development 2010;2:58–66. [20] E.E. Lundquist, Generalized analysis of experimental observations in problems of elastic stability, NACA TN-658 (1938).
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Lateral Load effects on buckling of cracked plates under tensile loading Rahman Seifi n, Ali Reza Kabiri Faculty of Engineering, Bu-Ali Sina University, Hamedan, Iran
art ic l e i nf o
a b s t r a c t
Article history: Received 2 March 2013 Received in revised form 20 June 2013 Accepted 20 June 2013
Fractured thin plates may be buckled locally due to applying the tensile loads in certain conditions. Sometimes the buckling mode can occur before fracturing or collapsing. Lateral loads can affect the buckling loads and modes. In thin cracked plates, compressive stress region near the crack or defect can be changed due to the lateral loads, so critical local buckling loads can be increased or decreased. Experimental and numerical study of the buckling load of plates with central crack under axial tensile loading is done. Effects of lateral load and constraint, relative crack length and its direction are investigated. & 2013 Elsevier Ltd. All rights reserved.
Keywords: Local buckling Tensile loading Lateral load Cracked plate
1. Introduction Sheets or plates are components that are used in a wide range of structures. Ships, offshore structures and airplanes are examples of complex structures made by thin-walled plates under different load combinations. A thin plate can contain different types of defects such as cracks produced by different causes like corrosion or fatigue loadings. It is important to study the behavior of these plates with or without faults in order to get the safety assessment of the structures. Buckling under compressive loads is known to happen for thin plates especially when some imperfections as holes or cracks exist, buckling under tension loadings is also important and can heavily affect the structural strength. Many researchers in the past decade studied the buckling of cracked plates under compression, tension or other types of loads for different structures. Most of these researches were done on perfect plates and some of them were about the imperfect plates. Numerical analysis, such as finite element method, was used helpfully for studying the buckling. This method is widely used for the study of the behavior of imperfect shells and plates. Brighenti [1,2] studied the buckling of cracked plates under tension and compression, by considering the effects of geometrical, mechanical and boundary conditions. He showed numerically that the boundary conditions affect the compression buckling but it almost has not effect on the buckling under tension. Crack length and orientation also have a significant influence in reducing the buckling loads. Also he proposed an approximate analytical method to evaluate the critical buckling load of cracked tensioned plates [2]. In plates under tension, based on the values of some
n
Corresponding author. Tel.: +988118292630; fax: +988118292631. E-mail address: rseifi@basu.ac.ir (R. Seifi).
0263-8231/$ - see front matter & 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.tws.2013.06.010
parameters such as fracture toughness, yield strength and critical buckling stress, local buckling can occur before fracture or plastic failures [3]. Riks et al. [4] proposed a numerical method for the analysis of buckling and post buckling of cracked tensioned plates. The results show that the buckling deformations cause a significant increase in the stress intensity factor of the crack tip. This effect is due to the change of stresses distributions in the plate which increases by increasing the crack length. Zielsdorff and Carlson [5] investigated the local buckling of the cracked plate under tensile loading by using experimental method. Effects of the crack tip curvature and crack length were also studied. Influence of the buckling process on plane stress fracture toughness was obtained from center cracked tensioned plates. They obtained some formulas for the determination of the critical buckling stress of cracked plates under tensile loading perpendicular to the length of crack. Seifi and Khoda-yari [6] studied the buckling of the fractured thin plates under uniform compression, experimentally and numerically. Effects of crack length and orientation and thickness of plates, different types of supports and loadings, such as partial edge acting forces and supports, were considered. Khedmati et al. [7] used the finite element method to study the buckling of plates with cracks. Effects of location, length and orientation of crack and plate aspect ratio on the critical loads were studied. They showed that the behavior of the cracked plate is different when the crack is either on the edge or inside the plate. Wrinkling of stretched thin sheets is a commonly occurrence in local buckling, leading to complex deflection patterns, particularly in regions close to the cracks. Sih and Lee [8] performed some predictions of the buckled displacement modes for cracked plates which was made on the basis of variational methods. Markstrom and Storakers [9] analyzed the buckling characteristics of cracked plates subjected to uniaxial tensile loads by the aid of the finite element method based on linear bifurcation
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theory. Interaction of local buckling and fracture behaviors of stretched cracked plate were also discussed. Shaw and Huang [10] studied the effect of bi-axial tension on the critical load. It is found that lateral load stabilizes the plate and buckling load increases as this load increases. This is due to the lateral tension that reduces the magnitude and area of the compression stress around the crack. Alinia et al. [11] used the finite element method for buckling analysis of cracked shear panels containing central or edge cracks. They showed that the long cracks have significant influence on the buckling capacity of shear panels, especially if the cracks exist in the tension region of the plates. Paik et al. [12] carried out an experimental and numerical study on the ultimate strength of cracked steel plates under compressive or tensile axial loadings. They also developed some relations, based on their results, for predicting the ultimate strength of cracked plates. They concluded that the ultimate strength of a cracked plate subjected to tensile loads can be determined on the basis of the reduced cross-sectional area due to crack. Vafai and Estekanchi [13] investigated the general behavior of cracked sheets and shells by using the finite element method. Effects of various parameters such as the mesh size in the near crack tip region, boundary conditions, Poisson’s ratio, crack length and curvature of the membrane have been studied. Dyshel [14] experimentally studied the stability of cracked plates under biaxial tensile loading. He showed that the critical stress is proportional to the elastic modulus and square of the ratio of thickness to crack length. Also it was shown that local loss of stability depends strongly on the magnitude of the lateral load. As can be seen, almost all researches about the buckling of cracked plates are performed numerically and for axial loading. In this study, experimental and numerical determination of buckling loads on the cracked plates under uniform tensile load is discussed. Effects of tensile or compressive lateral loads were studied. Parameters such as the length and orientation of cracks, aspect ratio of plates, variation of lateral loads, can affected the buckling load of cracked plate with a central crack; such effects were evaluated numerically and experimentally.
perturbation loads, we can extract the tensile buckling load. The following method is used in eigenvalue extraction for linear buckling calculation: we suppose an arbitrarily achieved base configuration with initial stresses sE , in equilibrium state under some surface E tractions t E , and body forces b . We consider an elastic deformation with very ‘small’ deformation gradients, under additional surface tractions Δt, body forces Δb and boundary displacements Δu. Applying these incremental external loads causes arising of the additional stresses, Δs. Since the problem is linear, then by applying λ times of these loads, the stress response will be λΔs. Each distinct value of λ corresponds with a linear perturbation of the initial equilibrium state. Among these perturbed states we can find special values of λ that allow for the existence of nontrivial incremental deformations with arbitrary magnitudes as valid solutions to the problem. Such nontrivial incremental displacements are referred to as buckling modes, and λ as buckling eigenvalue. Buckling load is the summation of initial load and λ times the load is applied in the perturbation step. For structures with nonlinear behavior, it is necessary to obtain nonlinear static equilibrium solutions. In numerical methods, a load-deflection method (such as Riks [15] and modified Riks method [16]) may be performed to investigate the nonlinear buckling. These methods use the load magnitude as an additional unknown and solved simultaneously for loads and displacements. The development of the solution requires that we traverse the path of equilibrium in a space defined by the loading parameter (λ) and some variables, such as displacements. The basic algorithm is the Newton–Raphson method (Finding the root of equation as FðxÞ ¼ 0 by iteration from xnþ1 ¼ xn −Fðxn Þ=F′ðxn Þ) as depicted in Fig. 1a; therefore, at any time there will be a finite radius of convergence with proper increment size i.e. the length of the tangent line, Δl. The increment size is chosen in a way that the ratio of the norm of error to norm of known variables vector norm is smaller than the prescribed tolerance (ε); in the other words, if Ku ¼ f , then ‖Ku−f ‖=‖f ‖ o ε where K is the global stiffness matrix, u is the unknown displacement vector and f is the known load vector. In the modified Riks method, the increment size is limited by moving a given distance along the tangent line (Δl) to the current
2. Numerical study of tensile buckling of cracked plates For thin plates under tensile loading, the compressive stress zone is induced near the crack or fault and the local buckling could occur due to these stresses. Some numerical procedures can be used to evaluate the tensile buckling load. For structures with linear behavior before buckling, it is proper to estimate the elastic buckling load by eigenvalue extraction. In this method, the buckling load is calculated as a multiplier of the pattern of perturbation loads (in buckling step), which is added to a set of base state loads (in static loading step). Based on the state of the
Fig. 2. Definition of variables for tensile buckling of cracked plate.
Fig. 1. (a) The Newton–Raphson method. (b) Modified Riks algorithm.
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solution point and then searching for equilibrium in the plane that passes through the point that is obtained and is orthogonal to the same tangent line (Fig. 1b).
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The Riks method for determination of tensile buckling load of cracked plate is described in Ref. [4]. Based on the works of Riks et al. [4,15] and results of the numerical methods, it is observed
Fig. 3. Dimensions and configurations of samples.
Table 1 Numbering of samples with different length and angle. Number
1
2
3
4
5
6
7
8
9
a/W Angle(1)
0.5 0.0
0.6 0.0
0.7 0.0
0.5 15
0.6 15
0.7 15
0.5 30
0.6 30
0.7 30
Fig. 4. Mesh of the model for case ‘e3’.
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that the crack parameter values such as energy release rate (G or J-integral), crack opening displacement and stress intensity factors as a function of the load, will bifurcate (with finite slope) from relation that belong to the (linear) pre-buckling state. In pre-buckling stage, there are not any out of plane displacements in the plate while in buckling and post-buckling stages, these displacements are introduced and also increased by increasing the load. These deformations cause variations in the crack parameters such as J-integral or G values. In this paper, we compared the variations of J-integral versus load for linear and buckling modes. In linear analysis, out of plane displacement was restricted, while in buckling mode, this restriction was removed. Separation point was assumed for 5% deviation in J-integral values. The effects of lateral load on the tensile buckling of cracked plates were studied in this paper. Schematic view of the cracked plate with lateral load is depicted in Fig. 2. As is shown, plate has 2L 2W t dimensions with a center crack with length 2a. Lateral load (P) was applied at two opposite points in x-direction, while θ is the orientation angle of crack with respect to x-axis (normal to loading direction). The edge placed at y¼−L line is clamped, while the tensile load N is applied to the edge y¼ +L (Fig. 2). The parameters of the problem were assumed as described below: two different plate sizes were adopted: 0.24 m 0.24 m and
0.24 m 0.18 m both with t¼1 mm. Crack lengths were equal to a/W¼ 0.5,0.6,0.7 while their orientation angles are θ¼ 0, 15, 301. Lateral loads are assumed as P¼−400, −200, 0, 200, 400 N. These values are smaller than one-third of the buckling load for cracked plates under lateral load only. All the studied cases are shown in Fig. 3. For each case, crack has three different length and orientation angle. We identify the samples in the following manner: a letter for the condition of sample (as in Fig. 3) followed by a number according to Table 1 for describing the crack length and its angle. For example, ‘j4’ is for case ‘j’ in Fig. 3 with a/W¼0.5 and θ¼ 151. Elastic properties of aluminum 1200 A were used in the numerical calculations. All cases in Fig. 3 were modeled with proper mesh pattern, density and elements (Fig. 4). For instance, finite element model of sample ‘e3’ in abaqus software is showed in Fig. 4, characterized by 6830 elements type S8R (quadratic eight node reduced integration points plane stress element). As can be seen, there are small elements in the positions of concentrated lateral loads and around the crack tips. Calculation of the linear tensile buckling gives the first eigenvalue equal to λ ¼25,523 for an initial load equal to N ¼1.0 N/m and so Ncr ¼25,523 N/m. Nonlinear buckling of this sample is calculated by using modified Riks algorithm and obtaining the variation of J-integral as depicted in Fig. 5. For deviation of 5%, the buckling load was
Fig. 5. Varation of load versus J-integral.
Fig. 8. distribution of compressive stress sx along y-axis in the Fig. 1.
Fig. 6. Buckling load versus lateral load for 2 W ¼0.24 m and (a) θ ¼01 (b) θ ¼151 (c) θ¼ 301.
Fig. 7. Buckling load versus lateral load for 2 W ¼0.18 m and (a) θ ¼01 (b) θ ¼151 (c) θ¼ 301.
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Fig. 9. Upper fixture.
Fig. 10. Lower fixture.
Fig. 11. Assembled fixtures and sample (a) without, and (b) with lateral load.
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estimated as Ncr ¼25,000 N/m. As can be seen, there is 2% difference between the linear and nonlinear results. The results for other models were calculated by using the mentioned method and are shown in Fig. 6 for plates with width 2W¼ 0.24 m and in Fig. 7 for 2W¼ 0.18 m. The total Buckling load is obviously the product of the distributed buckling load Ncr and width of plate 2W. According to these figures, applying lateral constraints and loads increases significantly the buckling load in comparison with uniaxial tensile buckling. In all cases, buckling load decreases by increasing the crack length but increases with increasing the lateral loads. It must be noted that the effect of constraint without any lateral load is also considerable in comparison with changing the lateral load. As can be seen, tensile lateral loads induce a decrease but compressive loads induce an increase of the buckling load. This is because of changing the compressive stresses around the crack line. For instance, distribution of the stresses parallel to crack line along the vertical line near the midpoint of crack (sx along y-axis in Fig. 2) is depicted in Fig. 8 for plates with W¼ 120 mm, a/W¼0.7, θ ¼ 0∘ under axial tensile loads as N¼ 28,400 N/m (buckling load for free edge case) and different lateral loads. This figure shows that the lateral
load alternates the compressive stress distribution and so buckling load. It must be noted that the lateral load changes the compressive stress values but its region around the crack line remains almost constant hence the buckling mode does not change. An important aspect for cracked plates under tensile load is the possibility of crack growing before buckling. For determination of collapse mode, it is necessary to compute the fracture parameters, such as stress intensity factor or J-integral, due to buckling load and comparing with fracture toughness. In this paper, because of angled cracks, a mixed mode fracture arises, so we calculated the J-integral values. These values are compared with strength JC. For series 1000 of aluminum alloys the plane strain fracture toughness (JIC) is about 14.0 KN/m. The J C −K C relationship can be computed in the following manner: K IC ¼
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 EJ IC =ð1−v2 Þ; K C ¼ K IC ð1 þ Bk e−ðAk t=t 0 Þ Þ; J C ¼ K 2C =E
ð1Þ
where t0 ¼ 2.5KIC/sy; for aluminum alloys we have elastic modulus equal to E¼70.0 GPa, Poisson’s ratio as v¼ 1/3, material constants equal to Ak ¼Bk ¼ 1.0 and yield stress equal to sy ¼137 MPa. After some calculations, JC is derived to be 56.0 KN/m. J-integral values due to the buckling load for all cases of Fig. 3 were derived and compared with J C . The maximum value of calculated J-integrals is about 12,230 N/m for case ‘d7’. Thus for all cases, buckling occurs before crack growth. It must be noted that for crack angles, usually larger than 301, in some cases the calculated J-integral is larger than JC so crack grows before buckling.
Table 3 Mechanical properties of 1200 A Al alloy [6]. Yield stress
Ultimate strength
Elongation Hardness Elastic modulus
Poisson’s Ratio
137 MPa
183 MPa
2%
0.33
54.4 HB
70 GPa
Fig. 13. Force-extension curve for case ‘l3’.
Fig. 12. Measuring the out of plane displacement by extensometer.
Table 2 Chemical composition of samples and comparing with standard [6]. Sample
Si 0.36
Standard ASTM E 1251-07 Min Max
Fe
Cu
Mn
Mg
Cr
Zn
Al
0.62
0.01
0.01
Trace
Trace
0.01
Base
Fe+Si
Cu
Mn
Mg
Cr
Zn
Other each
Other total
Al
* 1.00
* 0.10
* 0.30
* 0.3
* 0.1
* 0.1
* 0.05
* 0.15
99.00 *
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3. Experimental Investigation of tensile buckling of cracked plates 3.1. Design and fabrication of fixtures All tests were done by using the SANTAM tensile machine. For testing the samples two proper fixtures were designed and fabricated. Upper fixture is a channel with set screws on its edges. Two suitable slabs were mounted to clamp the samples. Details of this fixture is depicted in Fig. 9. Lower Fixture is similar to upper one with additional devices for applying and measuring the lateral loads as shown in Fig. 10.
43
Assembled fixtures on the machine and mounted sample were shown in Fig. 11. First figure shows the sample with free edges and the other is a sample with lateral load. A proper extensometer was used for measuring the out of plane displacement on the crack edges (Fig. 12).
3.2. Material and samples Samples were made from aluminum 1200 A with 1 mm thickness. Chemical compositions of material are depicted in Table 2 and compared with ASTM E 1251-07 standard. Mechanical properties of this alloy are given in Table 3. Samples sizes were 0.24 m 0.24 m and 0.18 m 0.24 m, obtained from a 1.0 m 2.0 m aluminum sheet (Fig. 3). Cracks were Table 5 The factor of the effects of different parameters on the buckling load of 0.24 m 0.24 m samples. Levels
L1 L2 L3 L4 L5 L6 ΔL
Fig. 14. Load-out of plane displacement curve for case ‘j4’.
Factors Lateral load factor
Crack angle factor
Crack length factor
5523 9856 11,000 11,846 9877 8793 6324
7775 8567 12,105 – – – 4330
12,629 8908 6910 – – – 5719
Fig. 15. Determination of slope (m) for Eq. (2).
Table 4 L18(61 32) Taguchi design factors and levels for tensile buckling load. Test
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18
Factor Lateral load (N)
Crack angle (1)
Crack length (a/W)
Free Free Free 0 0 0 200 200 200 400 400 400 −200 −200 −200 −400 −400 −400
0 15 30 0 15 30 0 15 30 0 15 30 0 15 30 0 15 30
0.5 0.6 0.7 0.5 0.6 0.7 0.6 0.7 0.5 0.7 0.5 0.6 0.6 0.7 0.5 0.7 0.5 0.6
Experimental results (N) for 0.24 m 0.24 m
Experimental results (N) for 0.18 m 0.24 m
6361.025 4614.65 5591.97 11,859.01 8262.64 9445.2 8470.47 7541.2 16,987.88 7670.6 14,393.68 13,474.42 7401.8 6325.5 15,904.03 4884.24 10,267.3 11,226.2
8208.38 6207.88 5700.34 13,207.34 9730.6 10,954.33 10,237.25 8512.25 19,603.43 8991.72 15,371.63 14,444.67 9488.5 6787.26 17,953.38 6258.91 11,913.3 11,539.38
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created in samples by wire electrical discharge machining method with 0.1 mm thickness. Created cracks have about 0.2 mm radius at their tips. Based on our numerical calculations, the small curvature on crack tip does not have considerable effect on the buckling load. For example, for case ‘e3’ with sharp crack and with 0.2 mm radius at crack tip, the buckling loads are 25,523 and 25,510 N/m (0.05% difference). Thirty-six specimens were made with different conditions based on the results of Taguchi method [17]. All samples were tested and values of applied force versus in-plane extensions and out of plane displacements at middle point of crack edges were recorded and saved.
3.3. Determination of buckling load The cracked plates under tensile load may buckle locally in the regions with compressive stress around the crack’s edges. This local buckling may not change the variation of applied force versus in-plane displacement so we cannot use the load-extension curve for determination of the buckling load. For example, the forceextension curve for case ‘l3’ was depicted in Fig. 13. As can be seen there are no considerable changes in trend of the curve due to the local buckling. Instead of in-plane extension we can use the out of plane displacement in the middle point of crack’s edge. For instance, the load-displacement curve for case ‘j4’ is depicted in Fig. 14. As can be seen, there is nonlinear tendency for load versus out of plane displacement but it is still difficult to obtain the buckling load value. It is clear that the curve has a knee and before it, the Table 6 The factor of the effects of different parameters on the buckling load of 0.18 m 0.24 msamples. Levels
L1 L2 L3 L4 L5 L6 ΔL
Factors Lateral load factor
Crack angle factor
Crack length factor
6706 11,297 12,784 13,366 11,410 9904 6661
9399 9817 13,518 – – – 4120
14,439 10,427 7867 – – – 6571
curve has linear trend. The lower limit was defined as the stage when the out of plane displacement, in the vicinity of the crack edge, begins to increase rapidly due to the applied load. This linear limit of the curve will be used as an approximate value of pre-load for calculation of tensile buckling loads. Based on Southwell method [18,19] and modified Southwell method [5,20], we can derive the buckling load from the following relation: P cr ¼ P 0 þ 1=m
ð2Þ
where m is the slope of linear part of the curve of ðw−w0 Þ=ðP−P 0 Þ versus ðw−w0 Þ. P 0 and w0 are the pre-load (usually load at the end of linear part of the force-out of plane extension curve) and out of plane extension due to P 0 , respectively. For example, the slope value for sample ‘c7’ was derived. ðP−P 0 Þ and ðw−w0 Þ=ðP−P 0 Þ curves versus ðw−w0 Þ were determined as depicted in Fig. 15. For P0 ¼ 5020 N and 1/m ¼1/0.0842 KN according to Fig. 15, we have Pcr ¼16.897 KN for sample ‘c7’. This method was used for other samples. As shown in the Fig. 3, there are 12 cases with nine different conditions for each case (three crack lengths and three crack angles) so 108 samples must be made. 3.4. Designing the experiments by Taguchi method For reducing the number of tests, we designed them based on the Taguchi method. There are three signal to noise (S/N) ratios of common interest for optimization of problems in Taguchi method. Smaller the better: This is usually the chosen S/N ratio for all undesirable characteristics like defects for which the ideal value is zero. Also, when an ideal value is finite and its maximum or minimum value is defined then the difference between measured data and ideal value is expected to be as small as possible. Larger the better: This case has been converted to ‘smaller the better’ by taking the reciprocals of measured data and then taking the S/N ratio as in the ‘smaller the better’ case. Nominal the best: This case arises when a specified value is most desired, meaning that neither a smaller nor a larger value is desirable. We have one factor (lateral load) with six levels and two factors (crack length and orientation) with three levels, hence the array L18(61 32)must be used. For this purpose, we did 18 tests for each plate sets 0.24 m 0.24 m and 0.18 m 0.24 m as depicted in Table 4. The results of Taguchi method were obtained by using Minitab software and are depicted in Tables 5 and 6.
Table 7 Comparison of experimental and numerical tensile buckling load. 0.24 m 0.24 m
0.18 m 0.24 m
Sample
Numerical (N)
Experiment (N)
Difference (%)
Sample
Numerical (N)
Experiment (N)
Difference (%)
a1 a5 a9 b1 b5 b9 c2 c6 c7 d3 d4 d8 e2 e6 e7 f3 f4 f8
6820.32 5204.64 5765.28 11,531.52 9163.68 10,093.68 9170.88 7917.84 18,414.72 8065.68 13,843.92 14,489.52 7866.96 6634.32 16799.04 5426.16 10985.76 11729.76
6361.025 4614.65 5591.97 11,859.01 8262.64 9445.2 8470.47 7541.2 16,896.5 7670.6 14,393.68 13,474.42 7401.8 6325.5 15904.03 4884.24 10267.3 11226.2
6.73 11.33 3 −2.83 9.83 6.42 7.63 4.75 8.4 4.89 −3.97 7 5.91 4.65 5.32 9.98 6.53 4.29
g1 g5 g9 h1 h5 h9 i2 i6 i7 j3 j4 j8 k2 k6 k7 13 14 18
7998.66 5897.34 6220.8 13,984.56 10,508.04 11,430.54 10,814.58 8952.12 20,582.82 9298.08 16,534.8 16,176.24 9131.94 7252.38 18619.2 6036.48 12925.98 12450.96
8208.38 6207.88 5700.34 13,207.34 9730.6 10,954.33 10,237.25 8512.25 19,603.43 8991.72 15,371.63 14,444.67 9488.5 6787.26 17953.38 6258.91 11913.3 11539.38
−2.62 −5.26 8.36 5.55 7.39 4.16 5.33 4.91 4.75 3.29 7.03 10.70 −3.90 6.41 3.57 −3.68 7.83 7.32
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Tables 5 and 6 show that the effects of lateral load and constraints are greater than the crack length and angle. Crack angle has a low effects on the tensile buckling load. These results agree with the numerical results in Figs. 5 and 6. Based on ‘larger the better’ S/N ratio,
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the fourth level of first factor, third level of second factor and first level of third factor give the maximum tensile buckling load while minimum value based on ‘smaller the better’ S/N ratio, is obtained for first level of first and second factors and third level of third factor.
Fig. 16. Some samples in buckling step.
Fig. 17. Effects of lateral loads for different crack length in 0.24 m 0.24 m plates.
Fig. 18. Buckled form of plates with a/W ¼0.7.
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Fig. 19. Effects of crack length on buckling load for 0.24 m 0.24 m.
Fig. 20. Effects of crack length on buckling load for 0.24 m 0.24 m.
4. Results and discussion Based on the results of the Taguchi method, all of the 36 tests were done and their results were compared with numerical calculations. Differences of results for two methods were depicted in Table 7. Some samples in buckling step were shown in Fig. 16. Observing this table shows that the data for experimental and numerical methods are in good agreement.
4.1. Effects of lateral load on the critical condition Under uniaxial loading, tensile buckling load increases by increasing the crack angle. Critical condition (minimum buckling load) occurs for crack with zero angle. Applying the lateral load can change this situation. Effects of lateral load versus crack length for different crack lengths were depicted in Fig. 17.
As can be seen, applying small lateral loads does not change the condition of the minimum buckling load for all the considered crack lengths. The angle of crack for minimum load, increases by increasing the crack length and lateral load. The resultant of axial and lateral loads changes due to increasing the lateral load so that minimum buckling load can be achieved for a position where crack is almost normal to the resultant force. Difference between the buckling loads decreases by increasing the crack length and orientation. This is due to the fact that the crack orientation is close to normal direction with respect to the resultant force, while for zero angle crack, the lateral load is parallel to crack line. For instance, buckled form for plates with a/W¼0.7, crack angles 01 and 301 and lateral loads equal to 200 N and 9600 N were depicted in Fig. 18. As can be seen, for small values of lateral loads the eigenvalue for mode 1 is larger for θ¼301 while for large lateral loads the condition is reversed.
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4.2. Effects of crack length For uniaxial loading, the buckling load decreases by increasing the crack length for all conditions. This behavior is also true for bi-axial loading as depicted in Fig. 19 for 0.24 m 0.24 m and in Fig 20 for 0.12 m 0.24 m. Reduction of loads is larger for large inclinations. For both series of plates, constraints (case P ¼0) considerably increase the buckling load versus free edge case for all lengths and inclinations of cracks. This is due to the creation of a resistance over the out of plane deformations of plates. Increasing the tensile in-plane lateral loads on the constraints causes increasing the buckling load while increasing the compressive lateral load decreases the critical load. Although, these variations are not considerable in comparison with cases with constraints. 4.3. Conclusion In this paper the tensile buckling behavior of cracked plate was studied. Effects of crack length and orientations, constraints and lateral load were considered. Based on the results, the following conclusions can be obtained: Tensile buckling load decreases with increasing crack length and orientation. Due to the presence of lateral supports, buckling load increases significantly in comparison with uniaxial buckling. Lateral tensile (compressive) load increases (decreases) the critical buckling load with respect to the case with only lateral support. Minimum tensile buckling load due to change of crack orientation can be altered by lateral load. Usually lateral load does not alter the first mode shape of buckling. References [1] Brighenti R. Numerical buckling analysis of compressed or tensioned cracked thin plates. Engineering Structures 2005;27:265–76. [2] Brighenti R. Buckling of cracked thin plates under tension or compression. Thin-Walled Structures 2005;43:209–24.
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