Strength Assessment of Imperfect Stiffened Panels Using Modified Stress Strain Curves

Transcription

1 Strength Assessment of Imperfect Stiffened Panels Using Modified Stress Strain Curves Mesut Tekgoz Center for Marine Technology and Engineering (CENTEC), Technical University of Lisbon, Instituto Superior Técnico, Lisboa, Portugal ABSTRACT: The objective of the present thesis is to analyze the ultimate strength of steel plates and stiffened panels accounting for residual stresses, welding sequences and corrosion degradation and to develop a fast finite element approach based on modified stress strain curves. The analysis is performed by finite element method employing commercial software. The heat source of welding is defined both as pre-stresses and a moving heat one, in order to simulate welding processes and consequently residual stresses. The modified stress strain curves are developed for three finite element models of structural components and accounting for different levels of residual stresses and boundary conditions. The modified material stress strain curves also account for the effect of the welding sequence, shakedown and corrosion degradation. The modified stress strain curves can be directly used for nonlinear finite element analyses of ultimate strength of steel plates and stiffened panels. This work also analyses the effect of shakedown on ultimate strength. A mathematical model is developed to account for a time dependent residual stress reduction which is used to evaluate ultimate strength. 1 Introduction The assessment of the structural response and its ultimate strength is rather complicated under the compressive loads because many factors influence its response in contrast to the tensile loads. Under the compressive loads, the structural behavior is affected by corrosion degradation, welding residual stresses, imperfection shape and magnitude, boundary conditions, bending, twisting, buckling forces and etc. The geometric and material nonlinearities of ship structures turn the problem definition into a more complicated stage. The ultimate strength of structural components and systems is a real measure in a strength assessment in a sense that the ultimate strength is the maximum load carrying capacity that they can withstand. No additional load can carry beyond the ultimate strength [1]. Achieving the ultimate strength implies that the resting capacity deteriorates due to plasticity, which appears since the axial rigidity is set to zero. The ultimate strength assessment involves a large amount of uncertainties and many factors may affect it. Garbatov, et al. [2] implemented a Monte Carlo simulation in order to find the most influential parameters on the ultimate strength. It has been found that slenderness ratio and plate thickness have the most significant effect on the ultimate strength of the stiffened panels. In addition to that, residual stresses and different loading conditions, imperfection magnitude, the cross-section of the structure and material properties and corrosion etc. Welding induced distortions and residual stresses lead to a structural strength reduction due to a nonuniform heat distribution and are a major concern in steel construction. Ship structures are subjected to cyclic loadings and because of that, the accumulated residual stresses are reduced through local plastic strains (shakedown) because residual stresses only reside in the rigid part of the structure. In addition to the shakedown effect, the structure is vulnerable to a corrosion degradation, which leads to a plate thickness reduction and in turn, the ultimate load carrying capacity is reduced. Guedes Soares and Kmiecik [3] have found that the residual stresses tend to be more important for thicker plates and are likely to contribute to geometrical nonlinearity due to the presence of buckling and plastic propagation. They also found that when the compressive forces are predominant, the transversal residual stresses effect is very small 1

2 until the ultimate load is reached, but they influence the post-collapse regime. Ueda and Yao [4] showed that, both welding residual stresses and initial geometrical imperfections reduce the compressive buckling and ultimate strength of plates, and this reduction achieves its maximum when the plate slenderness is about 1.8. Welding residual stresses reduce the compressive buckling and ultimate strength of stiffened panels when local buckling takes place, but increase them when overall buckling occurs. Ueda, et al. [5] concluded that, welding residual stresses reduce the buckling strength remarkably, but have a little effect on the ultimate strength when the plate is thin. On the other hand, when the plate is thick, welding residual stresses reduce the ultimate strength remarkably if there is an initial geometrical imperfection accompanied by local bending stresses. Paik, et al. [6] found that the longitudinal residual stresses were reduced by 36% and 21% in tension and compression respectively when the loading is applying in three cycles, which produces on the extreme fiber stresses equal to 88% of the yield stress. There are several types of heat source models developed in the literature. Pavelic, et al. [7] proposed the disc model, which is considered to be associated with the Gaussian distribution. Friedman [8], Krutz and Segerlind [9] suggested an alternative method to the Pavelic disc model, which is expressed with the moving coordinate system along with the heat source that has been adopted herein. Goldak, et al. [10] proposed a double ellipsoidal heat source model to simulate deep or shallow penetrated welding processes. Long, et al. [11] used the Goldak model to find out the welding induced distortions and residual stresses for a single plate. They found out that, the plate thickness and heat source speed have a significant influence on the welding induced stresses and distortions. Biswas, et al. [12] used the disc model in order to simulate a line-heating process for a single plate changing the plate thicknesses. They found that as the plate thickness is increased, the out-of plate deflection becomes smaller for the same heat and speed input. Gannon, et al. [13] used the disc model to simulate the welding influence on the ultimate strength for a stiffened panel. They concluded that the welding residual stresses may decrease the ultimate load carrying capacity by 16.5 %. Gannon, et al. [13] investigated the effect of shakedown on the reduction of welding induced stresses and on the improvement of ultimate strength. They concluded that the shake down might increase the ultimate load carrying capacity by 6% and it decreases the tensile and compressive residual stresses as much as 40%, when the applied load causes 50% yielding stress on the structure. Ships operate in corrosive environments and due to the interaction with ship structures; it leads to a gradual thickness reduction, which influences the ship durability. Several corrosion models have been developed to predict corrosion degradation in ship structures. Corrosion involves a big amount of uncertainty, which is based on many parameters. It is normally not straightforward to develop a corrosion wastage model solely based on theory, because corrosion is a function of many variables and uncertainties involved, such as the type of the corrosion protection system employed, type of cargo, temperature, humidity, etc. The corrosion models developed based on statistical analysis of operational data will usually be different according to the types of ships and cargoes or structural member locations and categories [14-16]. It has been shown on many occasions that the nonlinear corrosion wastage model is well accepted in representing realistic situations for steel plates in different areas of ship structures. A non-linear time dependent corrosion model developed by Guedes Soares and Garbatov [17] has been adopted herein. Faulkner [18] suggested a model to account for residual stresses by implementing a modification in the elastic perfectly plastic stress-strain curve. Through this type of stress strain curve, the structure is subjected to premature yielding and in turn reduces both stiffness and strength. The structural yielding stress point is reduced allowing the structure to absorb more strain and reduces the strength per strain. This implies that the structural linear behaviour is reduced. 2 Modified material stress-strain definition Residual stresses reduce the strength of the structural components. The approach presented here accounts for the presence of residual stresses and corrosion degradation and can be directly used during the design for estimate the ultimate strength of steel structures subjected to compressive load. The approach, deals with the residual stress modeling for an ultimate strength assessment by using modified stress strain elastic-plastic curve accounting for different effects. This approach employs finite element method to estimate not only 2

3 the ultimate strength, but also the pre, and post collapse regime behavior. Figure 1: Modified stress-strain relationships To find out the shape of the modified stress-strain curve an iterative procedure has been developed. In this iterative procedure, the material stress-strain curve is modified according to the best-fitted results of the pre and post collapse estimates. The entire procedure is divided into two stages. In the first one, the moving heat or pre-stresses, which cause residual stresses, is applied to the plate accounting for the thickness, heat speed and level and welding sequence applied over the structure. Then the corrosion degradation is applied accounting for the current time estimated of a plate thickness reduction. The structural normalized strength against strain is estimated. In the second stage, the iterative procedure is implemented to identify the best estimate of the parameters describing the material stress-strain curve used in the finite element analysis of ultimate strength to find out the best match of the structural response to that of the first stage. However, the pre and post-collapse regime assessment, based on the modified stress-strain curve, is influenced by the structural configuration, boundary conditions of the plate edges and the level of residual stresses and the corrosion degradation. The first step in defining the modified stress-strain curve is to define the proportional stress to capture the first yielding point. Once the first yielding is defined then the next point, in the stress-strain curve definition, is to define the changing of the slope as can be seen from Figure 1, up or by adding a new point as can be seen in Figure 1, down. The iterations are concluded when the two structural responses are found identical. In the case of corrosion degradation, the first stage of the linear structural behavior changes and in order to capture it the first tangent elasticity modulus and the material yielding point is reduced through a slope modification of the stress strain curve. 3 Thermal-structural analysis The ultimate strength of the structural components is analyzed based on finite element method using commercial software ANSYS [19]. The software enables modeling of elastic plastic material properties and large deformations. For the nonlinear static analysis, the kinematic assumption of finite element analysis calculations is large displacement and rotation, but small strain. The material yielding stress, y 350 MPa, the elastic modulus is E 2.05 E+11 Pa and the Poison coefficient is Stress stiffening effect, in order to avoid sharp post-collapse regime, has been accounted for. The applied load is uni-axial compression. As for the thermal analysis, The material properties have been taken from Gery, et al. [20] for the thermal analysis. The circular moving heat source proposed by Friedman [8], Krutz and Segerlind [9] has been adopted to simulate the welding heat sources (see Figure 2). The mathematical heat source model is expressed as: q z, 3Q 2 c x 2 c e (1) where Q is energy input rate and c is the characteristic radius of surface flux distribution, x is the transversal distance and is the moving longitudinal location which is expressed as: z v( t) (2) where v is the welding speed and t is time and is a lag factor needed to define the source at time 0. Figure 2: The moving heat source types A formula is developed to account for the welding induced stress reduction as a function of time to simulate the shakedown, which is expressed as: 3

4 t t 0exp SD, R T, R (3) where t is time, is the specified time at which TR, the welding residual stresses drastically degrease, 0is the welding residual stress at time zero which is equal to the material yielding stress in the tensile region. It is important to note that the reduction of the compressive and tensile residual stresses have been considered equal and SD, R represent the time up to where no welding residual stress reduction is present. As for the corrosion wastage model, the one developed by Guedes Soares and Garbatov [17] has been adopted here: elements have been implemented to represent the structural behavior and pre-stresses have been applied to simulate the welding induced residual stresses and to catch the same response, the material stress-strain curves have been developed. Similarly, in the second stage, a moving heat source has been introduced in a single and stiffened plate in order to assess the strength accounting for the residual stress and welding sequence and similarly the material stress-strain curves have been developed accounting for these effects. In the third stage, the timedependent shakedown and corrosion have been accounted for and to catch the same response the material stress-strain curves have been developed for the intact single plate. d n t t c d exp, t c t 0, t c (4) (5) StDev d t aln t b where dn t is the mean corrosion depth, c is the coating life, which is equal to the time interval between the painting of the surface and the time when its effectiveness is lost and T is the transition time under average conditions and d is the longterm corrosion wastage which is assumed here as 1.85 mm and a and b are the coefficient defined as and for the standard deviation of the corrosion degradation in ballast tanks [21]. Schematic representation of the effect of shakedown and corrosion deterioration is shown in Figure 3 and Figure 4 respectively. Figure 4: Corrosion application, plate 3 Ultimate strength assessment 3.1 Pre-stress application Figure 3: Residual stress and corrosion depth as a function of time In the finite element method, in the first stage shell In the pre-stress application, at first, the possible residual stress distribution is defined and it is applied as static pre-stresses before the analysis is conducted to find out the welding induced stress effect on the ultimate strength. In the second phase, as told in the material stress-strain definition, the material stress strain curves are established to find out the same response to that of the welding induced stresses. Different structural shapes have been employed to find out the parameters that affect the material stress-strain definition. Figure 5 to 8 show the stress-strain response and their respective material stress-strain definitions for Model 1. Figure 9 and 10 show the stress-strain response and their respective material stress-strain definitions for Model 2. 4

5 Figure 5: Stress-strain response, Model 1 (SSM- modified stress-strain, RS-residual stresses, WRS-welding residual stresses) Figure 8: Material stress strain relationship, Model 1 Figure 9: Stress-strain response, Model 2 Figure 6: Stress-strain response, Model 1 (SSM- modified stress-strain, RS-residual stresses, WRS-welding residual stresses) Figure 7: Material stress strain relationship, Model 1 Figure 10: Material stress strain relationship The material stress-strain definition is affected by the structural imperfections and edge boundary conditions. The ultimate load carrying capacity is well captured through this method (see Figure 6 to 10). 5

6 3.2 Moving heat source application In the moving heat source application, the welding process is simulated with the use of two different traveling heat sources as shown in Figure 2. The nodal temperature has been used as a nodal body load to find out the welding induced stresses in turn the ultimate load carrying capacity. Two different structural models namely, a single plate and stiffened panel have been employed to find out the thickness and the welding sequence effect on the material stress strain definition. In the second phase, as told in the material stress-strain definition, the material stress strain curves are established to find out the same response to that of the welding induced stresses. once the welding has been completed, the ultimate load carrying capacity gets larger. This is true between the plate thickness of 4 and 5 mm. To capture the same response, certain strain hardening has been implemented (see Figure 11 and 12). In the stiffened panel, the best welding sequence has been selected based on the ultimate load carrying capacity and distortions and the material stressstrain curve have been established (see Figure 13 and 14). Figure 13: Stress-strain, residual stresses and material stressstrain modification, Stiffened panel Figure 11: Stress-strain, residual stresses and material stressstrain modification Figure 14: Modified stress-strain curve, stiffened plate 3.3 Shakedown and corrosion degradation application Figure 12: Modified stress-strain curve, single plate As for the single plate, due to the deformed shape In the shakedown and corrosion degradation application, the welding process is simulated with the use of traveling heat source as shown in Figure 2. The nodal temperature has been used as a nodal 6

7 body load to find out the welding induced stresses. A new mathematical model has been developed to simulate the shakedown and the calculated stresses have been applied over the single plate to find out the ultimate load carrying capacity. As for the corrosion degradation, the corrosion wastage model given herein have been implemented (see Figure 4) In the second phase, as told in the material stressstrain definition, the material stress strain curves are established to find out the same response to that of the shakedown and corrosion degradation combination. cyclic loadings have been applied. As can be seen, the compressive stresses get larger through the plate thickness to satisfy the equilibrium. Figure 17 shows the applied cyclic loadings on the plate that have been used to analyze the shakedown effect on the tensile and compressive stresses. Figure 18 shows the cyclic loading effect on the welding residual stress. The cyclic loading-1 leads to 40% and 22.2% reduction on the tensile and compressive stresses respectively. The cyclic loading-2 results in 63.1% and 27.2% reduction in the tensile and compressive stresses respectively. However, it has been considered that the reduction is proportional for the tensile and compressive stress herein. Figure 15: Longitudinal residual stress distribution Figure 15 shows the welding induced stresses after the welding process has been completed. The welding zone is under tension while the rest is under compression stresses. Figure 17: Applied cyclic loadings Figure 18: Residual stress loading effect, third layer Figure 16: Longitudinal stress distribution Figure 16 shows the welding residual stress distribution through the plate thickness before the Figure 19 shows the shakedown effect on the ultimate load carrying capacity. As can be seen, the presence of the welding induced stresses cause the structure prematurely yield and in turn the ultimate load carrying capacity reduction. 7

8 Figure 19: Normalized stress-strain, shakedown effect Table 1: The data values for the shakedown and corrosion 1st year 2nd year 3rd year 6th year 9th year stress within a 2 years period and after then it contains both the partial residual stress and corrosion degradation and in the 9 th year, only contains the corrosion degradation. Figure 20 to 22 show the ultimate stress distribution accounting for both the time dependent shakedown and corrosion effects in the respective time. As can be seen in Figure 23, the response of the structure that has been modeled with the stressstrain modified curve is almost the same as for the one accounting for the effect of the shakedown and corrosion. It has to be noted that the roughness of the surface of the plate changes towards compressive loads. This implies that with the same strain absorption, in the corroded plate, there is less stress development present in relation to the uncorroded plate, which can be expressed as directional tangent elasticity. Figure 24 shows the material stress-strain definition defined for the shakedown and corrosion effect. Mean value N/A N/A St Dev. N/A N/A Residual stress Coating life Full RS %100 %100 %57 % years 2 years Figure 21: Longitudinal stress distribution of the ultimate state at 6th year, shakedown and corrosion deterioration effects Figure 20: Longitudinal stress distribution of the ultimate state at 3th year, shakedown and corrosion deterioration effects The time dependent corrosion degradation has been assumed after the welding process has been made. As given in Table 1, the structure has full residual Figure 22: Longitudinal stress distribution of the ultimate state at 9th year, shakedown and corrosion deterioration effects 8

9 Figure 23: Normalized modified stress-strain, shakedown and corrosion degradation effects Figure 26: Material yielding strain, mean corrosion depth Figure 25 and 26 show the material stress strain behavior as a function of time and the mean corrosion depth. It is important to point out that the first two years have not been incorporated due to the material stress-strain definition change. Once the corrosion becomes active, the elastic-perfectly plastic material has been implemented with different tangent modulus depending on the corrosion degradation. In contrast, the welding residual stress does change the linear behavior to some degree. Figure 24: Normalized modified stress-strain curves Figure 25: Material yielding stress, mean corrosion depth 4 Conclusions The work just presented here dealt with ultimate strength assessment of steel plates and stiffened panels. It has been found that the residual stresses decrease the first yielding point of the structure response in turn the ultimate strength. The shape effect of deformed plate results in larger ultimate strength in certain BT ratio. It has been found that the welding sequence is the most influential parameter that affects the lateral displacement of the stiffener due to a moving heat source leading to more ultimate load carrying capacity. The corrosion degradation has significant influence on the ultimate strength leading to different failure locations. The new developed stress-strain curves have been found as a good and fast solution to introduce the welding induced stresses and corrosion degradation in a non-linear finite element analysis of ultimate strength. 9

10 5 References [1] Paik, J., Branner, K., Choo, J., Czujko, J., Fujikubo, M., Gordo, J. M., Parmentier, G., Iaccarino, R., O Neil, S., Pasqualino, I., Wang, D., Wang, X. and Zhang, S. Committee III.1 Ultimate Strength. In: 17th International Ship and Offshore Structures Congress (ISSC2009), Vol. 1. C. Jang and S. Hong, editors. Seul, South Korea: University of Seoul, 2009, pp [2] Garbatov, Y., Tekgoz, M. and Guedes Soares, C. Uncertainty Assessment of the Ultimate Strength of a Stiffened Panel. In: Advances in Marine Structures. C. Guedes Soares and W. Fricke, editors. London, UK: Taylor & Francis Group, 2011, pp [3] Guedes Soares, C. and Kmiecik, M., 1993, Simulation of the Ultimate Compressive Strength of Unstiffened Rectangular Plates, Marine Structures, 6, pp [4] Ueda, Y. and Yao, T., 1991, Fundamental Behavior of Plates and Stiffened Plates with Welding Imperfections, Transactions of JWRI, 20, (2), pp [5] Ueda, Y., Yasukawa, W., Yao, T., Ikegami, H. and Ohminami, R., 1997, Effect of Welding Residual Stresses and Initial Deflection on Rigidity and Strength of Square Plates Subject- ed to Compression (Report II) Transactions of JWRI, 6, (1), pp [6] Paik, J. K., Hughes, O. F., Hess, P. E. and Renaud, C., 2005, Ultimate limit state design technology for aluminium multi-hull ship structures, Transactions of SNAME, 113, pp [7] Pavelic, V., Tanbakuchi, R., Uyehara, O. A. and Myers, P. S., 1969, Experimental and computed temperature histories in Gas Tungsten Arc Welding of thin plates, Welding Journal Research Supplement, 48, pp [8] Friedman, E., 1975, Thermomechanical analysis of the welding process using the finite element method, Journal of Pressure Vessel Technology, 97, pp [9] Krutz, G. W. and Segerlind, L. J., 1978, Finite element analysis of weld structures, Welding Journal Research Supplement, 57, pp [10] Goldak, J., Chakravarti, A. and Bibby, M., 1984, A new finite element model for welding heat sources, Met. Transaction, 15B, pp [11] Long, H., Gery, D., Carlier, A. and Maropoulus, P., 2009, Prediction of welding distortion in butt joint of thin plates, Journal Materials and Design, 30, pp [12] Biswas, P., Mandal, N. R. and Sha, O. P., 2006, Three dimensional finite element prediction of transient thermal history and residual deformation due to line heating, Proceedings of IMechE, Part M: J. Engineering for the Maritime Environment, Vol [13] Gannon, L., Liu, Y., Pegg, N. and Smith, M., 2010, Effect of welding sequence on residual stress and distortion in flat-bar stiffened plates, Journal Marine Structures, 23, pp [14] Guedes Soares, C., Garbatov, Y. and Zayed, A., 2011, Effect of environmental factors on steel plate corrosion under marine immersion conditions, Corrosion Engineering, Science and Technology, 46, (4), pp [15] Guedes Soares, C., Garbatov, Y., Zayed, A. and Wang, G., 2009, Influence of Environmental Factors on Corrosion of Ship Structures in Marine Atmosphere, Corrosion Science, 51, (9), pp [16] Guedes Soares, C., Garbatov, Y., Zayed, A. and Wang, G., 2008, Corrosion Wastage Model for Ship Crude Oil Tanks, Corrosion Science, 50, (11), pp [17] Guedes Soares, C. and Garbatov, Y., 1999, Reliability of Maintained, Corrosion Protected Plates Subjected to Non-Linear Corrosion and Compressive Loads, Marine Structures, pp [18] Faulkner, D. Compression tests on welded eccentrically stiffened plate panels. Steel plated structures. P. J. Dowling, editor. London: Crosby Lockwood Staples, 1977, pp [19] ANSYS, 2009, Online Manuals, Release 11. [20] Gery, D., Long, H. and Maropoulos, P., 2005, Effects of welding speed, energy input and heat source distribution on temperature variations in butt joint welding, Journal of Materials Processing Technology, 167, pp [21] Garbatov, Y., Guedes Soares, C. and Wang, G., 2007, Nonlinear time dependent corrosion wastage of deck plates of ballast and cargo tanks of tankers, Journal of Offshore Mechanics and Arctic Engineering, 129, (1), pp