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Ultimate strength of corroded steel plates with irregular surfaces under in-plane compression
Ocean Engineering 54 (2012) 261–269
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Ultimate strength of corroded steel plates with irregular surfaces under in-plane compression Ahmad Rahbar-Ranji n AmirKabir University of Technology, Department of Ocean Engineering, Hafez Avenue, Tehran 15914, Iran
a r t i c l e i n f o
abstract
Article history: Received 26 November 2011 Accepted 14 July 2012
Corrosion is one of the time dependent detrimental phenomena which reduces strength of structures and leads to catastrophic failures. All rules and regulations concerning strength of corroded plates are based on uniform thickness reduction. To estimate residual strength of corroded structures, typically a much higher level of accuracy is required, since, the actual corroded plate has irregular surfaces. There is little study on strength analysis of corroded plate with irregular surfaces especially as a function of corrosion parameters. It is the main aim of present work to study ultimate strength of corroded steel plates with irregular surfaces under in-plane compression. Nonlinear finite element method is employed to determine ultimate strength of corroded steel plates with irregular surfaces. Comparing the results with ultimate strength of corroded plates with uniform thickness, a reduction factor is introduced. Having done this, ultimate strength of corroded plates could be evaluated easily as a function of corrosion conditions. & 2012 Elsevier Ltd. All rights reserved.
Keywords: Corroded steel plate Ultimate strength FEM Irregular surface
1. Introduction It is believed that corrosion is one of the main reasons of failures in aged structures. There is an increased interest in implementing reliability approach for assessing strength of structures. The success of this approach depends on reliable estimates of time varying structural deterioration such as corrosion. For the structural safety assessment of corroded structures, it is necessary to determine how corrosion proceeds by time. Two main corrosion mechanisms, namely, general corrosion and pitting corrosion are recognized. Pitting is localized corrosion in the form of deep holes and general corrosion which occurs in relatively large area is due to coalescence of pits. Some research works are carried-out to investigate ultimate strength of corroded structures. Guedes Soares and Garbatov (1996, 1999) have studied the effect of corrosion on ultimate strength of ships and plates. Paik et al. (2003; 2004) have carried out a series of nonlinear finite element analyses (FEA) for steel plates with pit corrosion under axial compressive and shear loads. Wang et al. (2005) have given loss of buckling strength of 20 years aged deck plates. Teixeira and Guedes Soares (2008) have studied ultimate strength of plates with random fields of corrosion. Huang et al. (2010) have used nonlinear FEA approach for ultimate strength evaluation of corroded plate with pit corrosion. Jiang and Guedes Soares (2010, 2011) have studied ultimate strength of pitted plates. Silva et al. (2011) have studied the
effect of non-linear randomly distributed non-uniform corrosion on ultimate strength of plates. They have used Monte Carlo simulation to generate corroded surface of un-stiffened plates. Traditional engineering approach uses uniform thickness reduction for general corrosion of plates. To estimate the remaining safe life of an existing structure, typically a much higher level of accuracy is required. Limited research works are investigated time dependent surface geometries of plates due to corrosion. Rahbar-Ranji (2001) has proposed a spectrum for geometry of corroded surface based on surface measurements of a corroded plate. Rahbar-Ranji (2012) has used this spectrum to analyze plastic collapse load of one- and both-sided corroded steel plates. The main aim of present work is to study ultimate strength of corroded plate with irregular surfaces. Proposed power spectrum is used to generate irregular surfaces of corroded plates, and nonlinear large deflection finite element method (FEM) by using ANSYS code (version 5.6) is carried-out to determine ultimate strength of plates. All influential parameters and different corrosion conditions are investigated. 2. Geometry of corroded surface Steel plates that have been exposed to corrosive environments exhibit a characteristically non-uniform surface and one would expect that thickness of the plate varies from point-to-point as follows: þ
tðx1 ,x2 Þ ¼ t avr þ z ðx1 ,x2 Þ þ z ðx1 ,x2 Þ n
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ð1Þ
where z þ (x1, x2) and z (x1, x2) are distance of points on top and bottom of undulated surfaces from average thickness plane, respectively (Fig. 1).
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12 points per millimeters and was positioned at every half millimeter by hand. Therefore, the distances between measured points in this experiment were: 1 Dx1 ¼ 12 mm, Dx2 ¼ 12 mm
To cover all points, sensor was moved in both directions and on either side of the plate automatically. Two dimensional spectrums were calculated by using Fast Fourier Transform (FFT) method on top and bottom surfaces, and in longitudinal and transverse directions. Following features were observed: Fig. 1. Geometry of a plate with undulated surfaces.
1. Upper cutoff wave number in all cases was less than 1 Rad/mm, 2. Spectra have a well-defined main peak, and in some cases there was second smaller peak, 3. Spectra at k¼0, were zero then abruptly reach to its main peak. For wave numbers bigger than main peak, spectra have approached to zero with smaller slope. Based on these observations, an equation for spectrum of corroded surface was proposed in the following form (RahbarRanji, 2001): 8 3 > 11:88abs > Exp 23 2:97bsjkj Dtavr r2:97s > < k2 ð3Þ sðkÞ ¼ 3 > sÞ2 > > abðDtavr þ 2:97 Exp 23 Dt b jkj Dtavr Z2:97s 2 : Dtavr k
avr
where k is wave number, Dtavr and s are average and standard deviation of thickness diminution, respectively, and a and b are two fitting constants which are depend on corrosion conditions and lie in the following ranges:
a ¼ 0:010:15 Fig. 2. Record of data in an undulated plate.
b ¼ 0:020:15
Monte Carlo simulation methodology is used to generate z and z þ , since it is not feasible to measure ordinates of points in all corroded structures. Among the various methods that have been developed to generate such sample functions, the spectral representation method (Goda, 1970; Shinozuka, 1987) is one of the most widely used today. Power spectrum is another way of representing a series of sampling data, z(x1, x2), based on wave number, (k1, k2), which shows the contribution of different wave numbers in the series. If z(x1, x2) exhibits some approximate repetition on (k1, k2), the spectrum at vicinity of (k1, k2) would have a local peak. Direct Fourier transformations of original sampling points can be used to develop corresponding spectrum function. Consider a series of sampling points, which are recorded with increments Dx1, and Dx2 in x1 and x2 directions, respectively (Fig. 2). If the number of points in x1 and x2 directions denoted by NPT1 and NPT2, respectively, spectrum of this series is calculated as follows: Sðk1 ,k2 Þ ¼ 2
These two parameters are selected in such a way that statistical characteristic of simulated surface and target surface be the same. Though, Eq. (3) was deduced based on measurements on a plate, Melchers et al. (2010) have shown that smaller coupon size of corroded surface can provide adequate estimates of corrosion loss. Fig. 3 shows comparisons of some of the calculated spectrums against proposed spectrum by Eq. (3). An isotropic spectrum in two directions is expressed by Eq. (3), since, the stochastic characteristics of corroded surface in all directions are identical. Equivalent wave number is defined as follows: qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 keq ¼ k1 þ k2 ð4Þ
2 NPT 1 1 NPT2 1 X X zðx1n1 ,x2n2 Þeiðk1 x1n1 þ k2 x2n2 Þ ð2pNPT1 Þð2pNPT2 Þ n ¼ 0 n ¼ 0
Dx1 Dx2
1
2
ð2Þ where x1n1 ¼n1Dx1 and x2n2 ¼ n2Dx2. A corroded plate with dimensions 250 mm and 104 mm from bottom plate of a heavily corroded ship (Saito and Suzuki, 1998) was chosen and subjected to surface ordinate measurements. The geometry of both sides of the plate was scanned by a laser displacement sensor. The sensor was moved in one direction automatically with adjustable speed, and was positioned by hand in opposite direction. Speed of sensor was adjusted to measure
Fig. 3. Calculated and proposed spectrum of corroded surface.
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Three dimensional geometry of corroded surface could be generated as follows: zðx1 ,x2 Þ ¼
N1 X N 2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffi X 2 2Sðk1i ,k2j ÞDk1 Dk2 ½cosðk1i x1 þ k2j x2 þ j1ij Þ i¼1j¼1
þ Cosðk1i x1 k2j x2 þ j2ij Þ
ð5Þ
where N1 and N2 are number of discretization of spectrum in x1 and x2 directions, respectively, f1ij and f2ij are random phase angles uniformly distributed between 0 and 2p, Dk1 and Dk2 are wave number increments in x1 and x2 directions, respectively, and k1i ¼iDk1 and k2j ¼ jDk2. Corrosion in cargo space of ships shows a wide variation, dependent upon trade route, ship’s age, type of cargo carried, and frequency of ballasting, cleaning and repairing. One would expect to express spectrum of corroded surface as a function of abovementioned variables, which are called external environmental variables. It is not feasible to repeat above experiment for all possible environments. Therefore, spectrum of corroded surface is expressed as a function of geometrical parameters (average and standard deviation of thickness diminution), which are called internal parameters. Many research works have been carried out to estimate the average and standard deviation of thickness diminution in any environments. Guo et al. (2008) have given nonlinear equations for calculation of mean and standard deviation of corrosion wastages in deck plate of single hull tankers as a function of ships age based on measured data. They have summarized the existing corrosion wastage models and have calculated the mean and standard deviation of corrosion wastages. They have concluded that corrosion wastage measurements spread over a wide range, mean and standard deviation fluctuate with time and maximum corrosion wastage is much higher than average value. Therefore, according to their study, expressing corrosion wastage model only as a function of mean value or with constant standard deviation of thickness diminution is not realistic. Melchers (2003) has given mean and standard deviation of thickness diminution in web and flange of primary members in ships based on survey data. Melchers et al. (2010) have given average uniform corrosion and standard deviation of corroded plates as a function of exposure periods based on measured data on carefully controlled tests on plates which were put in a corrosive environment. Wang et al. (2003, 2005) have given mean and standard deviation of thickness diminution based on 110000 data measurements. Yamamoto and Ikegami (1998) have reported corrosion lost in bulk carriers based on data measurements. Guedes Soares et al.
Fig. 4. Simulated geometry of corroded surface after 5 years.
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(2008, 2009, 2011) have studied corrosion wastages in different ships and have proposed some models for corrosion lost estimation taking into account more parameters. Figs. 4 to 8 show simulated surfaces corresponding to mean and standard deviations of thickness diminution for bulkhead plate of ballast tank in a bulk carrier (Yamamoto and Ikegami, 1998) after 5, 10, 15, 20 and 25 years, respectively. Comparing Figs. 4 and 8, it reveals that fluctuation of surface in corroded plate (irregularities of surface) is increases by increasing age of structure. Table 1 shows the values of mean and standard deviation of thickness diminution of simulated and target surfaces. As can be seen, parameters a and b are chose in such a way that mean and standard deviations of thickness diminution for target and simulated surface be the same. 2.1. One-sided corroded plate For one-sided corroded plate all thickness reductions occur in one side of plate. After generating an irregular surface by using Eq. (5), this surface and a flat surface on opposite side are placed in such a way that the average residual thickness of simulated plate and target corroded plate be the same.
Fig. 5. Simulated geometry of corroded surface after 10 years.
Fig. 6. Simulated geometry of corroded surface after 15 years.
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on each surface. As can be seen, for both-sided corroded plates, more statistical information is needed. In this work, ultimate strength of both-sided corroded plates with the same irregular surfaces at each side are investigated.
3. Ultimate strength of plate under in-plane compression Plate elements are the main structural components in ship, offshore and bridge structures which are supported at their edges by stiffeners and girders. At certain level of axial compression loads, central part of plate would be buckled while edges remain straight. By increasing applied load, stresses at the edges of the plate would be increased while stresses at central part remain unchanged. When stresses at the edges of the plate reach to yield stress, plate would be failed. Generally, an approximate method by using uniform stress distribution assumption is used to determine ultimate load, since, exact ultimate load determination based on real stress distribution is not practical for day-today engineering practice. This uniform stress is distributed on a smaller width of plate, called effective width. The effective width concept was first introduced by Von-Karman (1932). Based on this assumption, ultimate strength of plate is found as follows:
Fig. 7. Simulated geometry of corroded surface after 20 years.
PUlt ¼ b t sm ¼ bef f t sY
ð6Þ
where b and t are plate width and thickness, respectively, sm is mean stress, beff is the effective width and sY is the material yield stress. Prediction of PUlt or sm was the subject of many studies in recent years. In these studies ultimate strength of plate was calculated for different loading conditions, aspect ratios of plate, initial imperfections and residual stresses. These studies mostly were done by applying non linear finite element method (FEM) and in some instances are supported by experiments. Based on these studies some approximate equations are proposed for ultimate strength calculation of plates in the following form: sm a ¼ f b, sred ,wp , ð7Þ b sY where sred represent the residual stress, wp is the initial imperfection of plate, a is the plate length, and b is the plate slenderness ratio and is defined as follows: rffiffiffiffiffiffi b sY b¼ ð8Þ t E
Fig. 8. Simulated geometry of corroded surface after 25 years.
Table 1 Average and standard deviations of thickness diminution of corroded surfaces of ballast tank in a bulk carrier (Yamamoto and Ikegami, 1998). Corroded surface
Target
Simulated
Corrosion parameters
Mean value (mm) Standard deviation (mm) Mean value (mm) Standard deviation (mm)
Age (years) 5
10
15
0.80 0.20
1.40 2.40 0.225 0.25
20
25
3.40 4.40 0.275 0.30
0.832 1.431 2.423 3.395 4.393 0.201 0.221 0.243 0.269 0.294
2.2. Two-sided corroded plate To simulate both-sided corroded plate, two irregular surfaces are generated and positioned in such a way that average residual thickness of generated plate and target corroded plate be the same. Some statistical studies are needed to determine how these thickness reductions are divided between two opposite sides, since, corrosion data are limited to thickness measurements. Another parameter which could have influence on ultimate strength of corroded plate is relative position of deepest point
where E is the Young’s modulus. Imperfections are always present in plate elements mainly due to distortions originated from welding and cutting. Though, shape of initial imperfections is irregular, however, sinusoidal wave with half wave equal to plate dimension is generally accepted. Measurements show that average value of maximum out-of-plane amplitude is equal to (Smith et al., 1975): 2
wp ¼ 0:1b t
ð9Þ
4. Numerical analysis and discussions Large deflection, non linear FEM by using ANSYS code (version 5.6) has been carried-out for numerical analysis. To simulate onesided corroded plate and both-sided corroded plate with different corrosion degrees at each side, solid element should be used. The computational time depends on number of elements, type of analysis, number of load steps, and computer configurations. Solid elements cannot be used since: 1. Non-linear analysis needs so much iteration, 2. Corroded surface model needs small mesh size, 3. Additional restrain is needed to force plane cross section remain plane (Kirchhoff hypothesis).
A. Rahbar-Ranji / Ocean Engineering 54 (2012) 261–269
Shell elements can be used to model both-sided corroded plate with the same irregular surfaces at each side. In other words, in this case mid-plane of plate is completely flat and by using shell elements, only thickness varies from node to node. In this study shell elements, SHELL181 with variable thickness at each node is used to model corroded plate. The element has four nodes with six degrees of freedom at each node and large deflection capabilities are included. Further, an automatic incremental-iterative solution procedure was performed until they reached to the ultimate load. Fig. 9 shows the FE model of a corroded plate. 4.1. Verification of finite element model accuracy Table 2 shows three empirical formulae (Wang et al., 2005) for ultimate strength of plates as a function of plate slenderness ratio. The differences between these three equations lie on ratio of ultimate strength over yield stress denoted by coefficient, C. According to IACS CSR, for values of b less than 1.58, this ratio is one. However, according IACS S11, this ratio is equal to one, only at b equal to zero. To check the accuracy of FE models some preliminary un-corroded plate are analyzed. Fig.10 shows comparison of ultimate strength of un-corroded plate with these three formulae. As can be seen good agreements between numerical method and IACS S11 is observed.
265
In all cases, four edges of plate are assumed simply supported. A uniformly distributed normal stress is applied over one end. The loaded edge and the edge perpendicular to it are kept straight and free to deform in the plane of plate while holding the other ends. The loading is continued until and after the ultimate strength is reached. The data measurements for applied loads and displacements are recorded in the computer as average stress and average strain. An automatic incremental iterative solution procedure was applied until the displacement exceeds the limit value. For each case, ultimate strength of corroded surface with irregular surface and uniform surface are calculated and a reduction factor is defined as follows: Rd ¼
ðsm =sY ÞFlush ðsm =sY ÞRough ðsm =sY ÞFlush
ð11Þ
Ultimate strength and reduction factors for a square corroded plate with different corrosion conditions are given in Table 4. As can be seen, reduction factor is increases as corrosion condition is harder. In some cases, reduction factor is negative, which differences between ultimate strength of plate with uniform thickness and irregular surface are very small (in fourth digit after decimal point). Therefore, it can be concluded that they are almost the same. This indicates that depending on corrosion degree, ultimate
4.2. Worked-out examples
mpx py wðx,yÞ ¼ 0:1b tsin sin a b 2
ð10Þ
1.1
Ultimate strength (Sm/SY)
Table 3 shows three different corrosion conditions considered for corroded plates. To study influence of plate aspect ratio, square plate (a/b¼1) and rectangular plate (a/b¼4) with thickness varies from 6 mm to 14 mm are considered. Based on studies of RahbarRanji and Zakeri (2010), yield stress, Young’s modulus and Poisson’s ratio are insensitive to corrosion. Therefore, material is considered as mild steel with E¼ 206 GPa, v¼0.3, and yield stress 245 MPa. Elastic perfectly plastic and Von-Mises yield criterion are assumed for material properties. Residual stress is ignored and initial imperfection is considered in the following form:
1 0.9 0.8 Analyzed
0.7
IACS S11 ABS SafeHull
0.6
IACS CSR
0.5 0.4 0.3 0.2 1.3
1
where m is the number of half waves in longitudinal direction and depends on plate aspect ratio.
1.6
1.9
2.2
2.5
2.8
Plate Slenderness ratio Fig. 10. Ultimate strength of corroded plate with uniform thickness reduction.
Table 3 Corrosion conditions considered in this work. Corrosion condition
Fig. 9. Finite element model of corroded plate generated with Shell element (displayed with eshape command).
Cor. 1 Cor. 2 Cor. 3
Corrosion parameter Mean value (mm)
Standard deviation (mm)
0.5 1.0 1.0
0.2 0.3 0.4
Table 2 Buckling and ultimate strength of plate panels (Wang et al., 2005).
Terminology Strength C
IACS S11 Critical buckling scr ¼ C sY
ABS SafeHull Effective width su ¼ C sY
IACS CSR Critical stresses scr ¼ C sY
8 3:6 < b2
( 2:25
forb Z 1:25
( 2:14
forb o 1:25
1
: 1
forb Z 2:68 b2 14:4
forb o 2:68
b
1
1:25 2 b
su: Ultimate strength; scr: critical compressive stress; sY: specified minimum yield stress of the material; b plate slenderness ratio.
b
089 2 b
forb Z 1:58 forb o 1:58
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Table 4 Ultimate strength and reduction factor of a square corroded plate under uni-axial compression. ðsm =sY ÞFlush
Plate slenderness ratio
1.20 1.30 1.40 1.60 1.70 1.90 2.20 2.50 2.90
Cor. 1
0.958 0.930 0.8901 0.8390 0.7787 0.7121 0.6435 0.5750 0.5036
Cor. 2
ðsm =sY ÞRough
Rd (%)
ðsm =sY ÞRough
Rd (%)
ðsm =sY ÞRough
Rd (%)
0.9558 0.9284 0.8900 0.8378 0.7771 0.7119 0.6432 0.5736 0.5023
0.23 0.17 0.01 0.15 0.21 0.03 0.04 0.24 0.25
0.957 0.9301 0.8902 0.8401 0.7790 0.7129 0.6442 0.5740 0.5024
0.10 0.01 0.01 0.13 0.04 0.11 0.11 0.17 0.24
0.9502 0.9215 0.8818 0.8302 0.7696 0.7026 0.6336 0.5629 0.4889
0.8 0.91 0.93 1.05 1.16 1.33 1.50 2.10 2.90
0.5
Sig/SigY
0.4
Cor.3
0.3
Cor.2 Cor.1
0.2
uniform thickness
0.1
0 0
0.5
1
1.5
2
Cor. 3
2.5
Eps/EpsY Fig. 11. Load shortening curves for corroded plate with different corrosion conditions.
strength of corroded plate with irregular surface and uniform thickness could be the same. Also from Table 4, it can be concluded that, reduction of ultimate strength of corroded plate is proportional to plate slenderness ratio. By increasing slenderness ratio of plate, which indicates that mode of collapse changes from yielding to buckling, reduction factor of ultimate strength increases. In another word, for slender plate which buckling controls collapse mode, reduction of ultimate strength due to corrosion is more prominent. Fig. 11 shows load-shortening curves for square corroded plates with slenderness ratio 2.90, for the case of uniform thickness and irregular surfaces with three different corrosion conditions. As can be seen, all plates behave the same till elastic buckling point. The differences between them are in post-buckling region, and ultimate strength value. Figs. 12 and 13 show distribution of deflection and Von-Mises stress in a square corroded plate with different corrosion conditions at ultimate stage for sturdy plate (b ¼1.20) and slender plate (b ¼2.90), respectively. As can be seen, for corroded plate with uniform thickness assumption, distribution of deflection and VonMises stress are symmetric for both sturdy plate and slender plate. However, for corroded plates with irregular surfaces deflection and Von-Mises stress distribution are asymmetric, which are more prominent in sturdy plates than slender plates. Examining these figures more closely, it reveals that maximum and minimum values of deflection and Von-Mises stresses do not vary other than by small amounts for plates with the same slenderness ratio and different corrosion conditions. For sturdy plates, maximum deflection is limited to 2 mm, while for slender plates it reaches to 8.5 mm. Minimum value of Von-Mises stress for sturdy plates is about 210 MPa, while for slender plates is about 40 MPa.
So it can be concluded that in plates with b ¼1.20, at ultimate stage, Von-Mises stress is almost evenly distributed and is equal to yield stress. However, for plates with b ¼ 2.90, differences between maximum and minimum values of Von-Mises stress at ultimate stage is significant. To study the influence of aspect ratio on reduction factor, a rectangular plate with aspect ratio four and corrosion condition, Cor. 3, under uni-axial longitudinal and transverse compression is analyzed (Table 5). As can be seen, generally, reduction factor of ultimate strength shows similar tendencies for both longitudinal and transverse loading. By increasing slenderness ratio of plate, reduction factor of ultimate strength in both cases increases. Maximum value of reduction factor of ultimate strength reaches to 3.5% for plate under transverse in-plane load and slenderness ratio of 2.90. Comparing Tables 4 and 5, it can be concluded that reduction of ultimate strength of corroded plate is unaffected by plate aspect ratio. To study the effect of material properties on reduction factor, three types of material are considered as follows: Mat. 1 Mat. 2 Mat. 3
Elastic perfectly plastic, sY ¼245 MPa and E¼206 GPa, Elastic with kinematics strain hardening, sY ¼245 MPa, sult ¼367.5 MPa, E¼206 GPa and H0 ¼E/100, Elastic perfectly plastic, sY ¼314 MPa, E¼206 GPa.
Table 6 shows reduction factor and ultimate strength of a square plate for corrosion condition, Cor. 1 and three different materials. As can be seen, ultimate strength and reduction factor for both Mat. 1 and Mat. 2 are almost the same. This implies that strain hardening rate of material has no influence on reduction of ultimate strength of plates due to corrosion. Comparing Mat. 3 and Mat. 1, it can be seen that for small values of slenderness ratio, both materials have almost the same reduction factor. When slenderness ratio increases, reduction factor for Mat. 3 is higher than for Mat. 1. This indicates that yield stress has influence on reduction factor when buckling controls collapse mode. In this case, by increasing yield stress, reduction factor also increases.
5. Conclusions There is little study on strength of corroded plate with irregular surface, especially as a function of corrosion parameters. Nonlinear, large deflection FEM is used for ultimate strength calculation of corroded steel plate with both-sided irregular surfaces. A reduction factor is introduced as a ratio of reduction of ultimate strength of corroded plate with non-uniform surfaces over ultimate strength of corroded plate with uniform thickness. Influential parameters are studied and it is found that the reduction of ultimate strength of plate due to corrosion is very
A. Rahbar-Ranji / Ocean Engineering 54 (2012) 261–269
267
Fig. 12. Deflection and Von-Mises stress distribution at ultimate stage in a square corroded plate (b ¼ 1.20). (a) Uniform thickness assumption (b) irregular surface (Cor. 1) (c) irregular surface (Cor. 2) (d) irregular Surface (Cor. 3).
sensitive to plate slenderness ratio, but less sensitive to plate aspect ratio, strain hardening rate of steel, and direction of inplane loading. Yield stress of steel has influence on reduction factor for slender plates. By increasing yield stress, reduction factor increases.
Results show that the potential for decrease in ultimate strength of plate as a consequence of corrosion is found to depend on the dominating collapse mode. By increasing slenderness ratio of plate, which indicates that mode of collapse changes from yielding to buckling, reduction factor of ultimate strength
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Fig. 13. Deflection and Von-Mises stress distribution at ultimate stage in a square corroded plate (b ¼ 2.90). (a) Uniform thickness assumption (b) irregular surface (Cor. 1) (c) irregular surface (Cor. 2) (d) irregular surface (Cor. 3).
increases. Residual ultimate strength is reduced by as much as 3.5% in rectangular plates under transverse compression and up to 3% in square plates when buckling is dominant collapse mode. Though,
these are small quantities, it shows that corroded plate with irregular surfaces has lower ultimate strength which in ultimate strength evaluation of ship hull girder could be significant.
A. Rahbar-Ranji / Ocean Engineering 54 (2012) 261–269
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Table 5 Ultimate strength and reduction factor of a rectangular corroded plate (a/b¼ 4), corrosion condition, Cor. 3. Plate slenderness ratio
1.23 1.33 1.44 1.56 1.70 1.90 2.20 2.50 2.90
Longitudinal compression
Transverse compression
ðsm =sY ÞFlush
ðsm =sY ÞRough
Rd (%)
ðsm =sY ÞFlush
ðsm =sY ÞRough
Rd (%)
1.00 1.00 1.00 1.00 0.9999 0.9920 0.9408 0.7099 0.5714
1.00 1.00 1.00 1.00 0.9885 0.9823 0.9287 0.6937 0.5603
0 0 0 0 1.14 0.98 1.37 2.29 1.94
0.4405 0.3921 0.3450 0.2995 0.2565 0.2170 0.1833 0.1556 0.1328
0.4378 0.3890 0.3420 0.2939 0.2536 0.2147 0.1807 0.1522 0.1282
0.62 0.79 0.87 1.04 1.13 1.08 1.40 2.18 3.47
Table 6 Ultimate strength and reduction factor of a square corroded plate, uniaxial loading, corrosion condition, Cor. 1. Plate slenderness ratio
1.70 1.90 2.20 2.50 2.90
Mat. 1
Mat. 2
Mat. 3
ðsm =sY Þ Uniform thickness
ðsm =sY Þ Irregular thickness
Rd(%) ðsm =sY Þ Uniform thickness
ðsm =sY Þ Irregular thickness
Rd(%) ðsm =sY Þ Uniform thickness
ðsm =sY Þ Irregular thickness
Rd (%)
0.7787 0.7121 0.6435 0.5750 0.5036
0.7696 0.7026 0.6336 0.5629 0.4889
1.16 1.33 1.50 2.10 2.90
0.7716 0.70.48 0.6357 0.5649 0.4906
1.06 1.34 1.47 2.19 2.90
0.7119 0.6559 0.5803 0.5128 0.4436
1.19 0 1.79 2.42 3.32
0.7799 0.7144 0.6452 0.5776 0.5053
References Goda, Y., 1970. Numerical experiments on wave statistics with spectral simulation. Rep. Port Harbor Res. Inst. 9 (3), 3–57. Guedes Soares, C., Garbatov, Y., 1996. Reliability of maintained ship hull subjected to corrosion. J. Ship Res. 40 (3), 225–243. Guedes Soares, C., Garbatov, Y., 1999. Reliability of maintained, corrosion protected plates subjected to non-linear corrosion and compressive loads. Mar. Struct. 12 (6), 425–446. Guedes Soares, C., Garbatov, Y., Zayed, A., Wang, G., 2008. Corrosion wastage model for the inside of ship crude oil tanks. Corros. Sci. 50 (11), 3095–3106. Guedes Soares, C., Garbatov, Y., Zayed, A., Wang, G., 2009. Influence of environmental factors on corrosion of ship structures in marine atmosphere. Corros. Sci. 51 (9), 2014–2026. Guedes Soares, C., Garbatov, Y., Zayed, A., 2011. Effect of environmental factors on steel plate corrosion under marine immersion conditions. Corros. Eng. Sci. Technol. 46 (4), 524–541. Guo, J., Wang, G., Ivanov, L., Perakis, A.N., 2008. Time varying ultimate strength of aging tanker deck plate considering corrosion effect. Mar. Struct. 21, 402–419. Huang, Y., Zhang, Y., Liu, G., Zhang, Q., 2010. Ultimate strength assessment of hull structural plate with pitting corrosion damnification under biaxial compression. Ocean Eng. 37, 1503–1512. Jiang, X. and Guedes-Soares, C., 2010. Ultimate compressive capacity of rectangular plates with partial depth pits. Proceecings of 29th Intenational Conference Offshore Mechanical Arctic Engineering (OMAE 2010); Shanghai, China. New York, USA: ASME; OMAE 2010-21050. Jiang, X. and Guedes-Soares, C., 2011. Ultimate capacity behaviour of pitted steel plates under biaxial compression. Proceecings of 30th Intenational Conference Offshore Mechanical Arctic Engineering (OMAE 2011); Rotterdam, The Netherland. New York, USA: ASME; OMAE 2011-49980. Melchers, R.E., 2003. Probabilistic models for corrosion in structural reliability assessment, Part 1: Empirical models. Trans. ASME 125, 264–271. Melchers, R.E., Ahammed, M., Jeffery, R., Simundic, G., 2010. Statistical characterization of surfaces of corroded steel plates. Mar. Struct. 23, 274–287.
0.7205 0.6559 0.5909 0.5255 0.4588
Paik, J.K., Lee, J.M., Ko, M.J., 2003. Ultimate compressive strength of plate elements with pit corrosion wastage. J. Eng. Mar. Environ. 217 (4), 185–200. Paik, J.K., Lee, J.M., Ko, M.J., 2004. Ultimate shear strength of plate elements with pit corrosion wastage. Thin Walled Struct. 42 (8), 1161–1176. Rahbar-Ranji, A., 2001. Stress analysis of a randomly undulated plate due to corrosion in marine structures. Ph.D. Thesis. Yokohama National University, Department of Naval Architecture, Japan. Rahbar-Ranji, A., Zakeri, A.H., 2010. Mechanical properties and corrosion resistance of normal strength and high strength steels in chloride solution. J. Naval Arch. Mar. Eng. 7, 93–100. Rahbar-Ranji, A., 2012. Plastic collapse load of corroded steel plates. SadhanaAcademy Proceedings of Engineering Science 37 (3), 341–349. Saito, M., Suzuki, S., 1998. Statistical analysis of corrosion on corroded plate. Graduate thesis, Yokohama National University, Department of Naval Architecture, Japan. Shinozuka, M., 1987. Stochastic Fields and their digital simulation. Stochastics Mech. I, 1–43. Silva, J.E., Garbatov, Y., Guedes Soares, C., 2011. Ultimate strength assessment of ageing steel plates subjected to random non-uniform corrosion wastage. In: Guedes-Soares, C., Fricke, W. (Eds.), Advances in Marine Structures. Taylor & Francis Group, London, UK, pp. 213–220. Smith, C.S., Davidson, P.C., Chapman, J.C., Dowling, P.J., 1975. Strength and stiffness of ship’s plating under in-plane compression and tension. J. Ship Res. 9 (1), 1–17. Teixeira, A.P., Guedes Soares, C., 2008. Ultimate strength of plates with random fields of corrosion. Struct. Infrastruct. Eng. 4 (5), 363–370. Von-Karman, T., Sechler, E.E., Donnell, L.H., 1932. The strength of thin plates in compression. Trans. ASME 54, 53–57. Wang, G., Spencer, J. and Elsayed T., 2003. Estimation of corrosion rates of structural members in oil tankers. ASME, 22nd International Conference on Offshore Mechanics and Arctic Engineering (OMAE2003) Cancun, Mexico. Wang, G., Spencer, J., Sun, H., 2005. Assessment of corrosion risks to aging ships using an experience database. J. Offshore Mech. Arctic Eng. 127 (2), 167–174. Yamamoto, N., Ikegami, K., 1998. A study on the degradation of coating and corrosion of ship’s hull based on the probabilistic approach. J. Offshore Mech. Arctic Eng. 120, 121–128.
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Ultimate strength of corroded steel plates with irregular surfaces under in-plane compression Ahmad Rahbar-Ranji n AmirKabir University of Technology, Department of Ocean Engineering, Hafez Avenue, Tehran 15914, Iran
a r t i c l e i n f o
abstract
Article history: Received 26 November 2011 Accepted 14 July 2012
Corrosion is one of the time dependent detrimental phenomena which reduces strength of structures and leads to catastrophic failures. All rules and regulations concerning strength of corroded plates are based on uniform thickness reduction. To estimate residual strength of corroded structures, typically a much higher level of accuracy is required, since, the actual corroded plate has irregular surfaces. There is little study on strength analysis of corroded plate with irregular surfaces especially as a function of corrosion parameters. It is the main aim of present work to study ultimate strength of corroded steel plates with irregular surfaces under in-plane compression. Nonlinear finite element method is employed to determine ultimate strength of corroded steel plates with irregular surfaces. Comparing the results with ultimate strength of corroded plates with uniform thickness, a reduction factor is introduced. Having done this, ultimate strength of corroded plates could be evaluated easily as a function of corrosion conditions. & 2012 Elsevier Ltd. All rights reserved.
Keywords: Corroded steel plate Ultimate strength FEM Irregular surface
1. Introduction It is believed that corrosion is one of the main reasons of failures in aged structures. There is an increased interest in implementing reliability approach for assessing strength of structures. The success of this approach depends on reliable estimates of time varying structural deterioration such as corrosion. For the structural safety assessment of corroded structures, it is necessary to determine how corrosion proceeds by time. Two main corrosion mechanisms, namely, general corrosion and pitting corrosion are recognized. Pitting is localized corrosion in the form of deep holes and general corrosion which occurs in relatively large area is due to coalescence of pits. Some research works are carried-out to investigate ultimate strength of corroded structures. Guedes Soares and Garbatov (1996, 1999) have studied the effect of corrosion on ultimate strength of ships and plates. Paik et al. (2003; 2004) have carried out a series of nonlinear finite element analyses (FEA) for steel plates with pit corrosion under axial compressive and shear loads. Wang et al. (2005) have given loss of buckling strength of 20 years aged deck plates. Teixeira and Guedes Soares (2008) have studied ultimate strength of plates with random fields of corrosion. Huang et al. (2010) have used nonlinear FEA approach for ultimate strength evaluation of corroded plate with pit corrosion. Jiang and Guedes Soares (2010, 2011) have studied ultimate strength of pitted plates. Silva et al. (2011) have studied the
effect of non-linear randomly distributed non-uniform corrosion on ultimate strength of plates. They have used Monte Carlo simulation to generate corroded surface of un-stiffened plates. Traditional engineering approach uses uniform thickness reduction for general corrosion of plates. To estimate the remaining safe life of an existing structure, typically a much higher level of accuracy is required. Limited research works are investigated time dependent surface geometries of plates due to corrosion. Rahbar-Ranji (2001) has proposed a spectrum for geometry of corroded surface based on surface measurements of a corroded plate. Rahbar-Ranji (2012) has used this spectrum to analyze plastic collapse load of one- and both-sided corroded steel plates. The main aim of present work is to study ultimate strength of corroded plate with irregular surfaces. Proposed power spectrum is used to generate irregular surfaces of corroded plates, and nonlinear large deflection finite element method (FEM) by using ANSYS code (version 5.6) is carried-out to determine ultimate strength of plates. All influential parameters and different corrosion conditions are investigated. 2. Geometry of corroded surface Steel plates that have been exposed to corrosive environments exhibit a characteristically non-uniform surface and one would expect that thickness of the plate varies from point-to-point as follows: þ
tðx1 ,x2 Þ ¼ t avr þ z ðx1 ,x2 Þ þ z ðx1 ,x2 Þ n
Tel.: þ98 21 64543114; fax: þ 98 21 66412495. E-mail address: [email protected]
0029-8018/$ - see front matter & 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.oceaneng.2012.07.030
ð1Þ
where z þ (x1, x2) and z (x1, x2) are distance of points on top and bottom of undulated surfaces from average thickness plane, respectively (Fig. 1).
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A. Rahbar-Ranji / Ocean Engineering 54 (2012) 261–269
12 points per millimeters and was positioned at every half millimeter by hand. Therefore, the distances between measured points in this experiment were: 1 Dx1 ¼ 12 mm, Dx2 ¼ 12 mm
To cover all points, sensor was moved in both directions and on either side of the plate automatically. Two dimensional spectrums were calculated by using Fast Fourier Transform (FFT) method on top and bottom surfaces, and in longitudinal and transverse directions. Following features were observed: Fig. 1. Geometry of a plate with undulated surfaces.
1. Upper cutoff wave number in all cases was less than 1 Rad/mm, 2. Spectra have a well-defined main peak, and in some cases there was second smaller peak, 3. Spectra at k¼0, were zero then abruptly reach to its main peak. For wave numbers bigger than main peak, spectra have approached to zero with smaller slope. Based on these observations, an equation for spectrum of corroded surface was proposed in the following form (RahbarRanji, 2001): 8 3 > 11:88abs > Exp 23 2:97bsjkj Dtavr r2:97s > < k2 ð3Þ sðkÞ ¼ 3 > sÞ2 > > abðDtavr þ 2:97 Exp 23 Dt b jkj Dtavr Z2:97s 2 : Dtavr k
avr
where k is wave number, Dtavr and s are average and standard deviation of thickness diminution, respectively, and a and b are two fitting constants which are depend on corrosion conditions and lie in the following ranges:
a ¼ 0:010:15 Fig. 2. Record of data in an undulated plate.
b ¼ 0:020:15
Monte Carlo simulation methodology is used to generate z and z þ , since it is not feasible to measure ordinates of points in all corroded structures. Among the various methods that have been developed to generate such sample functions, the spectral representation method (Goda, 1970; Shinozuka, 1987) is one of the most widely used today. Power spectrum is another way of representing a series of sampling data, z(x1, x2), based on wave number, (k1, k2), which shows the contribution of different wave numbers in the series. If z(x1, x2) exhibits some approximate repetition on (k1, k2), the spectrum at vicinity of (k1, k2) would have a local peak. Direct Fourier transformations of original sampling points can be used to develop corresponding spectrum function. Consider a series of sampling points, which are recorded with increments Dx1, and Dx2 in x1 and x2 directions, respectively (Fig. 2). If the number of points in x1 and x2 directions denoted by NPT1 and NPT2, respectively, spectrum of this series is calculated as follows: Sðk1 ,k2 Þ ¼ 2
These two parameters are selected in such a way that statistical characteristic of simulated surface and target surface be the same. Though, Eq. (3) was deduced based on measurements on a plate, Melchers et al. (2010) have shown that smaller coupon size of corroded surface can provide adequate estimates of corrosion loss. Fig. 3 shows comparisons of some of the calculated spectrums against proposed spectrum by Eq. (3). An isotropic spectrum in two directions is expressed by Eq. (3), since, the stochastic characteristics of corroded surface in all directions are identical. Equivalent wave number is defined as follows: qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 keq ¼ k1 þ k2 ð4Þ
2 NPT 1 1 NPT2 1 X X zðx1n1 ,x2n2 Þeiðk1 x1n1 þ k2 x2n2 Þ ð2pNPT1 Þð2pNPT2 Þ n ¼ 0 n ¼ 0
Dx1 Dx2
1
2
ð2Þ where x1n1 ¼n1Dx1 and x2n2 ¼ n2Dx2. A corroded plate with dimensions 250 mm and 104 mm from bottom plate of a heavily corroded ship (Saito and Suzuki, 1998) was chosen and subjected to surface ordinate measurements. The geometry of both sides of the plate was scanned by a laser displacement sensor. The sensor was moved in one direction automatically with adjustable speed, and was positioned by hand in opposite direction. Speed of sensor was adjusted to measure
Fig. 3. Calculated and proposed spectrum of corroded surface.
A. Rahbar-Ranji / Ocean Engineering 54 (2012) 261–269
Three dimensional geometry of corroded surface could be generated as follows: zðx1 ,x2 Þ ¼
N1 X N 2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffi X 2 2Sðk1i ,k2j ÞDk1 Dk2 ½cosðk1i x1 þ k2j x2 þ j1ij Þ i¼1j¼1
þ Cosðk1i x1 k2j x2 þ j2ij Þ
ð5Þ
where N1 and N2 are number of discretization of spectrum in x1 and x2 directions, respectively, f1ij and f2ij are random phase angles uniformly distributed between 0 and 2p, Dk1 and Dk2 are wave number increments in x1 and x2 directions, respectively, and k1i ¼iDk1 and k2j ¼ jDk2. Corrosion in cargo space of ships shows a wide variation, dependent upon trade route, ship’s age, type of cargo carried, and frequency of ballasting, cleaning and repairing. One would expect to express spectrum of corroded surface as a function of abovementioned variables, which are called external environmental variables. It is not feasible to repeat above experiment for all possible environments. Therefore, spectrum of corroded surface is expressed as a function of geometrical parameters (average and standard deviation of thickness diminution), which are called internal parameters. Many research works have been carried out to estimate the average and standard deviation of thickness diminution in any environments. Guo et al. (2008) have given nonlinear equations for calculation of mean and standard deviation of corrosion wastages in deck plate of single hull tankers as a function of ships age based on measured data. They have summarized the existing corrosion wastage models and have calculated the mean and standard deviation of corrosion wastages. They have concluded that corrosion wastage measurements spread over a wide range, mean and standard deviation fluctuate with time and maximum corrosion wastage is much higher than average value. Therefore, according to their study, expressing corrosion wastage model only as a function of mean value or with constant standard deviation of thickness diminution is not realistic. Melchers (2003) has given mean and standard deviation of thickness diminution in web and flange of primary members in ships based on survey data. Melchers et al. (2010) have given average uniform corrosion and standard deviation of corroded plates as a function of exposure periods based on measured data on carefully controlled tests on plates which were put in a corrosive environment. Wang et al. (2003, 2005) have given mean and standard deviation of thickness diminution based on 110000 data measurements. Yamamoto and Ikegami (1998) have reported corrosion lost in bulk carriers based on data measurements. Guedes Soares et al.
Fig. 4. Simulated geometry of corroded surface after 5 years.
263
(2008, 2009, 2011) have studied corrosion wastages in different ships and have proposed some models for corrosion lost estimation taking into account more parameters. Figs. 4 to 8 show simulated surfaces corresponding to mean and standard deviations of thickness diminution for bulkhead plate of ballast tank in a bulk carrier (Yamamoto and Ikegami, 1998) after 5, 10, 15, 20 and 25 years, respectively. Comparing Figs. 4 and 8, it reveals that fluctuation of surface in corroded plate (irregularities of surface) is increases by increasing age of structure. Table 1 shows the values of mean and standard deviation of thickness diminution of simulated and target surfaces. As can be seen, parameters a and b are chose in such a way that mean and standard deviations of thickness diminution for target and simulated surface be the same. 2.1. One-sided corroded plate For one-sided corroded plate all thickness reductions occur in one side of plate. After generating an irregular surface by using Eq. (5), this surface and a flat surface on opposite side are placed in such a way that the average residual thickness of simulated plate and target corroded plate be the same.
Fig. 5. Simulated geometry of corroded surface after 10 years.
Fig. 6. Simulated geometry of corroded surface after 15 years.
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on each surface. As can be seen, for both-sided corroded plates, more statistical information is needed. In this work, ultimate strength of both-sided corroded plates with the same irregular surfaces at each side are investigated.
3. Ultimate strength of plate under in-plane compression Plate elements are the main structural components in ship, offshore and bridge structures which are supported at their edges by stiffeners and girders. At certain level of axial compression loads, central part of plate would be buckled while edges remain straight. By increasing applied load, stresses at the edges of the plate would be increased while stresses at central part remain unchanged. When stresses at the edges of the plate reach to yield stress, plate would be failed. Generally, an approximate method by using uniform stress distribution assumption is used to determine ultimate load, since, exact ultimate load determination based on real stress distribution is not practical for day-today engineering practice. This uniform stress is distributed on a smaller width of plate, called effective width. The effective width concept was first introduced by Von-Karman (1932). Based on this assumption, ultimate strength of plate is found as follows:
Fig. 7. Simulated geometry of corroded surface after 20 years.
PUlt ¼ b t sm ¼ bef f t sY
ð6Þ
where b and t are plate width and thickness, respectively, sm is mean stress, beff is the effective width and sY is the material yield stress. Prediction of PUlt or sm was the subject of many studies in recent years. In these studies ultimate strength of plate was calculated for different loading conditions, aspect ratios of plate, initial imperfections and residual stresses. These studies mostly were done by applying non linear finite element method (FEM) and in some instances are supported by experiments. Based on these studies some approximate equations are proposed for ultimate strength calculation of plates in the following form: sm a ¼ f b, sred ,wp , ð7Þ b sY where sred represent the residual stress, wp is the initial imperfection of plate, a is the plate length, and b is the plate slenderness ratio and is defined as follows: rffiffiffiffiffiffi b sY b¼ ð8Þ t E
Fig. 8. Simulated geometry of corroded surface after 25 years.
Table 1 Average and standard deviations of thickness diminution of corroded surfaces of ballast tank in a bulk carrier (Yamamoto and Ikegami, 1998). Corroded surface
Target
Simulated
Corrosion parameters
Mean value (mm) Standard deviation (mm) Mean value (mm) Standard deviation (mm)
Age (years) 5
10
15
0.80 0.20
1.40 2.40 0.225 0.25
20
25
3.40 4.40 0.275 0.30
0.832 1.431 2.423 3.395 4.393 0.201 0.221 0.243 0.269 0.294
2.2. Two-sided corroded plate To simulate both-sided corroded plate, two irregular surfaces are generated and positioned in such a way that average residual thickness of generated plate and target corroded plate be the same. Some statistical studies are needed to determine how these thickness reductions are divided between two opposite sides, since, corrosion data are limited to thickness measurements. Another parameter which could have influence on ultimate strength of corroded plate is relative position of deepest point
where E is the Young’s modulus. Imperfections are always present in plate elements mainly due to distortions originated from welding and cutting. Though, shape of initial imperfections is irregular, however, sinusoidal wave with half wave equal to plate dimension is generally accepted. Measurements show that average value of maximum out-of-plane amplitude is equal to (Smith et al., 1975): 2
wp ¼ 0:1b t
ð9Þ
4. Numerical analysis and discussions Large deflection, non linear FEM by using ANSYS code (version 5.6) has been carried-out for numerical analysis. To simulate onesided corroded plate and both-sided corroded plate with different corrosion degrees at each side, solid element should be used. The computational time depends on number of elements, type of analysis, number of load steps, and computer configurations. Solid elements cannot be used since: 1. Non-linear analysis needs so much iteration, 2. Corroded surface model needs small mesh size, 3. Additional restrain is needed to force plane cross section remain plane (Kirchhoff hypothesis).
A. Rahbar-Ranji / Ocean Engineering 54 (2012) 261–269
Shell elements can be used to model both-sided corroded plate with the same irregular surfaces at each side. In other words, in this case mid-plane of plate is completely flat and by using shell elements, only thickness varies from node to node. In this study shell elements, SHELL181 with variable thickness at each node is used to model corroded plate. The element has four nodes with six degrees of freedom at each node and large deflection capabilities are included. Further, an automatic incremental-iterative solution procedure was performed until they reached to the ultimate load. Fig. 9 shows the FE model of a corroded plate. 4.1. Verification of finite element model accuracy Table 2 shows three empirical formulae (Wang et al., 2005) for ultimate strength of plates as a function of plate slenderness ratio. The differences between these three equations lie on ratio of ultimate strength over yield stress denoted by coefficient, C. According to IACS CSR, for values of b less than 1.58, this ratio is one. However, according IACS S11, this ratio is equal to one, only at b equal to zero. To check the accuracy of FE models some preliminary un-corroded plate are analyzed. Fig.10 shows comparison of ultimate strength of un-corroded plate with these three formulae. As can be seen good agreements between numerical method and IACS S11 is observed.
265
In all cases, four edges of plate are assumed simply supported. A uniformly distributed normal stress is applied over one end. The loaded edge and the edge perpendicular to it are kept straight and free to deform in the plane of plate while holding the other ends. The loading is continued until and after the ultimate strength is reached. The data measurements for applied loads and displacements are recorded in the computer as average stress and average strain. An automatic incremental iterative solution procedure was applied until the displacement exceeds the limit value. For each case, ultimate strength of corroded surface with irregular surface and uniform surface are calculated and a reduction factor is defined as follows: Rd ¼
ðsm =sY ÞFlush ðsm =sY ÞRough ðsm =sY ÞFlush
ð11Þ
Ultimate strength and reduction factors for a square corroded plate with different corrosion conditions are given in Table 4. As can be seen, reduction factor is increases as corrosion condition is harder. In some cases, reduction factor is negative, which differences between ultimate strength of plate with uniform thickness and irregular surface are very small (in fourth digit after decimal point). Therefore, it can be concluded that they are almost the same. This indicates that depending on corrosion degree, ultimate
4.2. Worked-out examples
mpx py wðx,yÞ ¼ 0:1b tsin sin a b 2
ð10Þ
1.1
Ultimate strength (Sm/SY)
Table 3 shows three different corrosion conditions considered for corroded plates. To study influence of plate aspect ratio, square plate (a/b¼1) and rectangular plate (a/b¼4) with thickness varies from 6 mm to 14 mm are considered. Based on studies of RahbarRanji and Zakeri (2010), yield stress, Young’s modulus and Poisson’s ratio are insensitive to corrosion. Therefore, material is considered as mild steel with E¼ 206 GPa, v¼0.3, and yield stress 245 MPa. Elastic perfectly plastic and Von-Mises yield criterion are assumed for material properties. Residual stress is ignored and initial imperfection is considered in the following form:
1 0.9 0.8 Analyzed
0.7
IACS S11 ABS SafeHull
0.6
IACS CSR
0.5 0.4 0.3 0.2 1.3
1
where m is the number of half waves in longitudinal direction and depends on plate aspect ratio.
1.6
1.9
2.2
2.5
2.8
Plate Slenderness ratio Fig. 10. Ultimate strength of corroded plate with uniform thickness reduction.
Table 3 Corrosion conditions considered in this work. Corrosion condition
Fig. 9. Finite element model of corroded plate generated with Shell element (displayed with eshape command).
Cor. 1 Cor. 2 Cor. 3
Corrosion parameter Mean value (mm)
Standard deviation (mm)
0.5 1.0 1.0
0.2 0.3 0.4
Table 2 Buckling and ultimate strength of plate panels (Wang et al., 2005).
Terminology Strength C
IACS S11 Critical buckling scr ¼ C sY
ABS SafeHull Effective width su ¼ C sY
IACS CSR Critical stresses scr ¼ C sY
8 3:6 < b2
( 2:25
forb Z 1:25
( 2:14
forb o 1:25
1
: 1
forb Z 2:68 b2 14:4
forb o 2:68
b
1
1:25 2 b
su: Ultimate strength; scr: critical compressive stress; sY: specified minimum yield stress of the material; b plate slenderness ratio.
b
089 2 b
forb Z 1:58 forb o 1:58
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A. Rahbar-Ranji / Ocean Engineering 54 (2012) 261–269
Table 4 Ultimate strength and reduction factor of a square corroded plate under uni-axial compression. ðsm =sY ÞFlush
Plate slenderness ratio
1.20 1.30 1.40 1.60 1.70 1.90 2.20 2.50 2.90
Cor. 1
0.958 0.930 0.8901 0.8390 0.7787 0.7121 0.6435 0.5750 0.5036
Cor. 2
ðsm =sY ÞRough
Rd (%)
ðsm =sY ÞRough
Rd (%)
ðsm =sY ÞRough
Rd (%)
0.9558 0.9284 0.8900 0.8378 0.7771 0.7119 0.6432 0.5736 0.5023
0.23 0.17 0.01 0.15 0.21 0.03 0.04 0.24 0.25
0.957 0.9301 0.8902 0.8401 0.7790 0.7129 0.6442 0.5740 0.5024
0.10 0.01 0.01 0.13 0.04 0.11 0.11 0.17 0.24
0.9502 0.9215 0.8818 0.8302 0.7696 0.7026 0.6336 0.5629 0.4889
0.8 0.91 0.93 1.05 1.16 1.33 1.50 2.10 2.90
0.5
Sig/SigY
0.4
Cor.3
0.3
Cor.2 Cor.1
0.2
uniform thickness
0.1
0 0
0.5
1
1.5
2
Cor. 3
2.5
Eps/EpsY Fig. 11. Load shortening curves for corroded plate with different corrosion conditions.
strength of corroded plate with irregular surface and uniform thickness could be the same. Also from Table 4, it can be concluded that, reduction of ultimate strength of corroded plate is proportional to plate slenderness ratio. By increasing slenderness ratio of plate, which indicates that mode of collapse changes from yielding to buckling, reduction factor of ultimate strength increases. In another word, for slender plate which buckling controls collapse mode, reduction of ultimate strength due to corrosion is more prominent. Fig. 11 shows load-shortening curves for square corroded plates with slenderness ratio 2.90, for the case of uniform thickness and irregular surfaces with three different corrosion conditions. As can be seen, all plates behave the same till elastic buckling point. The differences between them are in post-buckling region, and ultimate strength value. Figs. 12 and 13 show distribution of deflection and Von-Mises stress in a square corroded plate with different corrosion conditions at ultimate stage for sturdy plate (b ¼1.20) and slender plate (b ¼2.90), respectively. As can be seen, for corroded plate with uniform thickness assumption, distribution of deflection and VonMises stress are symmetric for both sturdy plate and slender plate. However, for corroded plates with irregular surfaces deflection and Von-Mises stress distribution are asymmetric, which are more prominent in sturdy plates than slender plates. Examining these figures more closely, it reveals that maximum and minimum values of deflection and Von-Mises stresses do not vary other than by small amounts for plates with the same slenderness ratio and different corrosion conditions. For sturdy plates, maximum deflection is limited to 2 mm, while for slender plates it reaches to 8.5 mm. Minimum value of Von-Mises stress for sturdy plates is about 210 MPa, while for slender plates is about 40 MPa.
So it can be concluded that in plates with b ¼1.20, at ultimate stage, Von-Mises stress is almost evenly distributed and is equal to yield stress. However, for plates with b ¼ 2.90, differences between maximum and minimum values of Von-Mises stress at ultimate stage is significant. To study the influence of aspect ratio on reduction factor, a rectangular plate with aspect ratio four and corrosion condition, Cor. 3, under uni-axial longitudinal and transverse compression is analyzed (Table 5). As can be seen, generally, reduction factor of ultimate strength shows similar tendencies for both longitudinal and transverse loading. By increasing slenderness ratio of plate, reduction factor of ultimate strength in both cases increases. Maximum value of reduction factor of ultimate strength reaches to 3.5% for plate under transverse in-plane load and slenderness ratio of 2.90. Comparing Tables 4 and 5, it can be concluded that reduction of ultimate strength of corroded plate is unaffected by plate aspect ratio. To study the effect of material properties on reduction factor, three types of material are considered as follows: Mat. 1 Mat. 2 Mat. 3
Elastic perfectly plastic, sY ¼245 MPa and E¼206 GPa, Elastic with kinematics strain hardening, sY ¼245 MPa, sult ¼367.5 MPa, E¼206 GPa and H0 ¼E/100, Elastic perfectly plastic, sY ¼314 MPa, E¼206 GPa.
Table 6 shows reduction factor and ultimate strength of a square plate for corrosion condition, Cor. 1 and three different materials. As can be seen, ultimate strength and reduction factor for both Mat. 1 and Mat. 2 are almost the same. This implies that strain hardening rate of material has no influence on reduction of ultimate strength of plates due to corrosion. Comparing Mat. 3 and Mat. 1, it can be seen that for small values of slenderness ratio, both materials have almost the same reduction factor. When slenderness ratio increases, reduction factor for Mat. 3 is higher than for Mat. 1. This indicates that yield stress has influence on reduction factor when buckling controls collapse mode. In this case, by increasing yield stress, reduction factor also increases.
5. Conclusions There is little study on strength of corroded plate with irregular surface, especially as a function of corrosion parameters. Nonlinear, large deflection FEM is used for ultimate strength calculation of corroded steel plate with both-sided irregular surfaces. A reduction factor is introduced as a ratio of reduction of ultimate strength of corroded plate with non-uniform surfaces over ultimate strength of corroded plate with uniform thickness. Influential parameters are studied and it is found that the reduction of ultimate strength of plate due to corrosion is very
A. Rahbar-Ranji / Ocean Engineering 54 (2012) 261–269
267
Fig. 12. Deflection and Von-Mises stress distribution at ultimate stage in a square corroded plate (b ¼ 1.20). (a) Uniform thickness assumption (b) irregular surface (Cor. 1) (c) irregular surface (Cor. 2) (d) irregular Surface (Cor. 3).
sensitive to plate slenderness ratio, but less sensitive to plate aspect ratio, strain hardening rate of steel, and direction of inplane loading. Yield stress of steel has influence on reduction factor for slender plates. By increasing yield stress, reduction factor increases.
Results show that the potential for decrease in ultimate strength of plate as a consequence of corrosion is found to depend on the dominating collapse mode. By increasing slenderness ratio of plate, which indicates that mode of collapse changes from yielding to buckling, reduction factor of ultimate strength
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Fig. 13. Deflection and Von-Mises stress distribution at ultimate stage in a square corroded plate (b ¼ 2.90). (a) Uniform thickness assumption (b) irregular surface (Cor. 1) (c) irregular surface (Cor. 2) (d) irregular surface (Cor. 3).
increases. Residual ultimate strength is reduced by as much as 3.5% in rectangular plates under transverse compression and up to 3% in square plates when buckling is dominant collapse mode. Though,
these are small quantities, it shows that corroded plate with irregular surfaces has lower ultimate strength which in ultimate strength evaluation of ship hull girder could be significant.
A. Rahbar-Ranji / Ocean Engineering 54 (2012) 261–269
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Table 5 Ultimate strength and reduction factor of a rectangular corroded plate (a/b¼ 4), corrosion condition, Cor. 3. Plate slenderness ratio
1.23 1.33 1.44 1.56 1.70 1.90 2.20 2.50 2.90
Longitudinal compression
Transverse compression
ðsm =sY ÞFlush
ðsm =sY ÞRough
Rd (%)
ðsm =sY ÞFlush
ðsm =sY ÞRough
Rd (%)
1.00 1.00 1.00 1.00 0.9999 0.9920 0.9408 0.7099 0.5714
1.00 1.00 1.00 1.00 0.9885 0.9823 0.9287 0.6937 0.5603
0 0 0 0 1.14 0.98 1.37 2.29 1.94
0.4405 0.3921 0.3450 0.2995 0.2565 0.2170 0.1833 0.1556 0.1328
0.4378 0.3890 0.3420 0.2939 0.2536 0.2147 0.1807 0.1522 0.1282
0.62 0.79 0.87 1.04 1.13 1.08 1.40 2.18 3.47
Table 6 Ultimate strength and reduction factor of a square corroded plate, uniaxial loading, corrosion condition, Cor. 1. Plate slenderness ratio
1.70 1.90 2.20 2.50 2.90
Mat. 1
Mat. 2
Mat. 3
ðsm =sY Þ Uniform thickness
ðsm =sY Þ Irregular thickness
Rd(%) ðsm =sY Þ Uniform thickness
ðsm =sY Þ Irregular thickness
Rd(%) ðsm =sY Þ Uniform thickness
ðsm =sY Þ Irregular thickness
Rd (%)
0.7787 0.7121 0.6435 0.5750 0.5036
0.7696 0.7026 0.6336 0.5629 0.4889
1.16 1.33 1.50 2.10 2.90
0.7716 0.70.48 0.6357 0.5649 0.4906
1.06 1.34 1.47 2.19 2.90
0.7119 0.6559 0.5803 0.5128 0.4436
1.19 0 1.79 2.42 3.32
0.7799 0.7144 0.6452 0.5776 0.5053
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