- Email: [email protected]
Collapse load for a crack in a plate with a mismatched welded joint
Engineering Failure Analysis 13 (2006) 1065–1075 www.elsevier.com/locate/engfailanal
Collapse load for a crack in a plate with a mismatched welded joint Andrei Kotousov *, Mohd Fairuz Mohmed Jaffar School of Mechanical Engineering, University of Adelaide, SA 5005, Australia Received 20 April 2005; accepted 15 July 2005 Available online 8 September 2005
Abstract The accuracy of flaw assessment techniques for welded joints is directly related to the accurate estimate of the yield collapse load. A careful finite element limit analysis can provide a very good and reliable assessment of the collapse load. Unfortunately, such an analysis is normally time-consuming and requires a substantial effort in order to validate the finite element calculations. In this paper we apply the classical upper bound theorem of plasticity to develop simplified solutions for a through crack in a tensile plate with a mismatched welded joint. The weld configuration is idealised as a simple sandwich model and the collapse load is derived analytically from a solution of an extreme-value problem. The theoretical solutions are then verified by independent finite element calculations. The proposed solutions seem to be very effective and accurate and can be easily generalized for many other weld geometries and loading conditions. Ó 2005 Elsevier Ltd. All rights reserved. Keywords: Plastic collapse; Elastic–plastic material; Welds; Fracture in plates
1. Introduction Situations when the weld metal and base material have different values of yield stress are very common for many engineering structures such as bridges, ships, piping and pressure vessels. The accurate estimation of collapse loads is essential for assessing the integrity of cracked structures with welds. A wide range of techniques has been used in the past to provide sufficiently accurate values for the limit (or collapse) load. Based on slip line field (SLF) analysis, Hao et al. [1] obtained a yield load solution for mismatched middle *
Corresponding author. Tel.: +61 8 83035439; fax: +61 8 83034367. E-mail address: [email protected] (A. Kotousov).
1350-6307/$ - see front matter Ó 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.engfailanal.2005.07.007
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Nomenclature a B e F FB FM H M W e rY rYB rYW W v sYW sYB
half length of a centre crack plate thickness crack eccentricity applied force net section collapse load collapse load for mismatch configuration half width of weld metal strip mismatch factor defined for yield strength (M = rYW/rYB) (half) width of plate normalized crack eccentricity e/H yield strength, general yield strength for base material yield strength for weld material normalised remaining ligament (W a)/H geometry factor of the weld joint v = a/W yield stress of the weld material in shear yield stress of the base material in shear
crack tension M(T) plates. Based on the assumed deformation fields proposed by Joch et al. [2], Hornet and Eripret [3] proposed an approximate solution for mismatched M(T) and single edge cracked bend plates. Kim and Schwalbe [4] conducted a comprehensive review of these techniques and concluded that although time and cost consuming, the finite element (FE) analysis based on elastic-perfectly plastic material model is currently the most effective and reliable approach to the calculation of the limit load in defective mismatched structures. The main objective of the present paper is to develop an effective yet simple analytical approach for the assessment of the collapse load for plates with cracks in mismatched welds using the classical upper bound theorem of plasticity. Hornet and Eripret [3] have used this theorem before for mismatched welded structures by invoking simple assumed deformation fields. Unfortunately, the previous application of the upper bound theorem of plasticity led to unacceptably higher values than the actual collapse load for highly under-matched plates [4]. The current approach is different from what has been done before and reduces the problem under consideration to an extreme-value problem for which the limit load is derived analytically. As it will be shown later in this paper it provides a very effective way to analyse cracked structures with mismatched weldments and with some physical insight, the approach can be used to get a very accurate assessment of the collapse load with very little effort. It will be demonstrated that the proposed method overcomes problems with highly under-matches plates and agrees well with independent FE studies. It can be easily generalized for many other practically important cases of mismatched welded structures.
2. Deformation patterns An actual weld joint is very complicated, both on micro- and macro-levels. In practical situations, the welding process affects material properties in the heat-affected zone (HAZ). The presence of residual stresses makes the situation further complicated, rendering analytical solutions nearly impossible [5]. For this
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reason simplifications typical of those made when analysing this problem, will be made in the following analysis [4]. The effects of heat affected zones, residual stresses or other kinds of non-heterogeneity will be omitted, and the real welded joint will be modelled as a sandwich-like bi-material system shown in Fig. 1. The yield stress in the weld material can be lower or higher than that of the base material and this strength difference is referred as mismatch. The mismatch in the yield strength between the weld metal, rYW, and the base plate, rYB, is normally quantified by the mismatch factor M: M¼
rYM ; rYB
ð1Þ
with M < 1 referring to under-matching and M > 1 referring to over-matching. The slenderness of the weld and the crack eccentricity ratio are defined as W¼
W a ; H
ð2Þ
and e ¼ e=H ;
ð3Þ
where 2a is the crack length, 2W is the plate width, 2H is the weld width and e is the distance to the crack from the mid-line of the weld joint (see Fig. 1). Under fully plastic conditions, failures are controlled by the formation of slip planes in the case of tensile loading and by the formation of plastic hinges in bending [6]. Possible patterns of plastic deformation for mismatched plates with a crack extracted from FE limit analysis are shown in Fig. 2. In each figure, relevant information on M, W and v = a/W is given. In the first pattern the plastic deformation penetrates through the base plate. This case corresponds to the gross section yielding failure mechanism when almost all plastic deformation is concentrated in the gross section of the base plate. It is typical for highly over-matched welds. The decreasing of the mismatch factor M leads to the penetrating yielding characterized by the plastic deformation in the cracked ligament of both the weld and base plate. Important observation here is that the lower mismatch factor leads to a higher proportion of the plastic deformation in the weld. The limiting case is a confined yielding when the
Fig. 1. Model geometry of the problem.
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Fig. 2. Possible plastic deformation patterns for mismatched plate with centre crack extracted from the FE limit analysis.
base material is shielded from plastic deformation by the weld, which is likely to occur for small M and W. We will use these observations in the next section when we build a simple analytical model to calculate the collapse load based on the upper bound theorem of plasticity [7].
3. Applications of the upper bound theorem of plasticity The upper bound theorem of plasticity provides a simple way to calculate the collapse load for an elasticperfectly plastic solid including a rigid perfectly plastic solid as a limiting case. It will not always predict the exact value, but with some physical insight, the theorem can be used to get a very good estimate for collapse load with very little effort. The proof of the upper bound theorem of plasticity for bi-material solids, such as welded structures is presented in the Appendix A and using the same approach can be easily generalized for multi-material solids. An informal introduction to the classical upper bound theorem of plasticity can be given as follows. Consider a plastically deforming solid, which is subjected to a distribution of tractions on its boundary. The upper bound theorem of plasticity states that the solid will collapse if the work done by the tractions through any mechanism of collapse exceeds the internal plastic dissipation. Collapse mechanism can be described through a kinematically admissible displacement field i.e. displacement field which is continuous everywhere within the solid, differentiable everywhere and also satisfies the displacement boundary conditions if any exists. Now, we apply the upper bound theorem to the problem under consideration. We guess and consider different collapse mechanisms as shown in Fig. 3. The top, bottom half and side parts of the plate slide past each other as rigid blocks, as shown in Fig. 3. Here we consider four different situations as: fully confined
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Fig. 3. Guess collapse mechanism (a1 and a2 are unknown angles to be determined from further analysis).
yielding (i), confined-penetrating yielding (ii), fully penetrating yielding (iii) and yielding in the base material yielding (iv). In the beginning we derive the estimate of the collapse load FB for an all-base plate (assuming that the plate is wholly made of the base material, i.e. sYW = sYB and, because of the symmetry, a1 = a2). From the upper bound theorem of plasticity we have F B tan ad ¼ 2BðW aÞsY dsec2 a;
ð4Þ
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where sY is the yield stress of the weld and base material in shear pffiffiffi(recall that TrescaÕs yield criterion predicts sY = rY/2, while Von Mises yield criterion predicts sY ¼ rY = 3). The left part of the above equations represents the work done by the applied force FM through the specified (guessed) mechanism of collapse and, in accordance with the upper bound theorem of plasticity, at collapse conditions (FM) must be equal to the internal plastic dissipation, which is written in the right part of these equations. Eq. (4) produces a range of values for the loads leading to the plastic collapse as a function of a: F B ðaÞ ¼ 2B
W a sYB . sin a cos a
ð5Þ
The real value of the collapse load in accordance with the upper bound theorem will always be less than FB. Obviously, that the lowest value of FB = minFB(a) will represent the best estimate of the collapse load. The right hand side of the above Eq. (5) reaches the minimum at a = p/4 and the minimum value is F B ¼ 4BðW aÞsY ;
ð6Þ
which is exactly the same equation as obtained from the SLF analysis [1]. It is pretty astonishing that the proposed simple approach has produced the same answer, which was obtained using rather complicated considerations. Using formula (6), the equations for corresponding collapse mechanisms of the mismatched welded plate shown in Fig. 3 can be written as for a1 6 a1 and a2 6 a2 (fully confined yielding) FM M 1 1 ¼ þ ; ð7Þ F B 2ðtan a1 þ tan a2 Þ cos2 a1 cos2 a2 for a1 > a1 and a2 6 a2 (confined-penetrating yielding) FM M 1e 1 1e 1 ¼ þ þ ; F B 2ðtan a1 þ tan a2 Þ W cos a1 sin a1 M cos2 a1 MW cos a1 sin a1 cos2 a2 for a1 6 a1 and a2 > a2 (confined-penetrating yielding) FM M 1 1þe 1 1þe ¼ þ þ ; F B 2ðtan a1 þ tan a2 Þ cos2 a1 W cos a2 sin a2 M cos2 a2 MW cos a2 sin a2 for a1 > a1 and a2 > a2 (fully penetrating yielding) FM M 1e 1 1e 1þe ¼ þ þ F B 2ðtan a1 þ tan a2 Þ W cos a1 sin a1 M cos2 a1 MW cos a1 sin a1 W cos a2 sin a2 1 1þe þ ; M cos2 a2 MW cos a2 sin a1 where sYW and sYB are the yield stress of the weld and base material in shear. The critical angles, which determine the specific mechanism of plastic collapse, are 1 1 e a1 ¼ tan ; W and a2
¼ tan
1
1þe . W
ð8Þ
ð9Þ
ð10Þ
ð11Þ
ð12Þ
The forth mechanism (base material yielding) can be analysed similar to the confined yielding mechanism by replacing the yield stress in the weld by the yield stress of the base material.
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Unfortunately, the transcendental Eqs. (7)–(10) cannot be solved analytically to provide a closed form solution for the collapse load. However, it can be easily solved using a pocket calculator for any combination of M and W and should not significantly devaluate the proposed approach. The upper estimate of the collapse load can be found as a minimum value of FM (or FFMB ) calculated for different mechanisms of the collapse failure by varying a1 and a2 in the range corresponding to the particular collapse mechanisms (i)–(iv). The one giving the minimum value gives the best estimate of the collapse load. The collapse load is also limited by the gross sectional yielding mechanism for which it is easy to show that FM 1 . ¼ FB 1 v
ð13Þ
Finally, the best estimate of the collapse load is FM ¼ minfEqs. (7)–(13)g. FB
ð14Þ
4. Results Results of the calculations using the current approach (solid lines) for different values of the strength mismatch levels, ligament-to-weld width ratios, and crack lengths are presented in Figs. 4 and 5 together with FE results [4] (circles). From the comparison it is clear seen that the proposed method has a very good accuracy and agrees well with independent FE studies. As it can be seen from Figs. 4 and 5 in some cases the FE results predict a slightly higher value for the collapse load than those obtained from the application
Fig. 4. Comparison o the results obtained using the proposed method with FE results [5], (a) a/W = 0.5 and e = 0; (b) a/W = 0.1 and e = 0.
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Fig. 5. Comparison of the results obtained using the proposed method with the FE results [4], a/W = 0.5 and e = 1.
of the upper bound theorem of plasticity. This difference could give some indication of the accuracy of the FE results, which are always needed to be treated with care. The general tendencies in the collapse load for mismatched joints agrees well with those obtained from finite element limit analysis and can be formulated as follows: (1) The collapse load is close to that of an all base plate (the plate is wholly made of the base material) for large values of normalized ligament ratios, say W > 10. (2) The collapse load is close to that of all weld material for under-matched welded joints with small value of normalized ligament ratios. (3) The collapse load is close to that of the gross sectional yielding for highly overmatched welded joints with small value of normalized ligament ratios. (4) The higher crack eccentricity ratio leads to the decrease of the collapse load for overmatched and to the increase of the collapse load for under matched welded joints. It is believed that the similar tendencies take place for other geometries and loading conditions in welded structures. 5. Conclusion The proposed approach can be easily generalised for other important geometries and loading conditions, which may include edge crack, longitudinal or radial through crack in a pipe or other structures. Because the upper bound theorem of plasticity provides the upper estimate of the collapse load it means that this approach is not conservative. It should also be treated with caution and be always validated by independent studies, such as FE. One of the important results of the current paper is that in the conjunction with FE the current approach can replace the necessity in the use of fitting formulas for the assessment of the collapse loads in mismatched structures and can significantly reduce the volume of required FE calculations. The proposed approach can also serve as an independent validation of FE results and can give some estimate of its accuracy. Acknowledgement The work reported herein was supported by the Cooperative Research Centre for Welded Structures. The Cooperative Research Centre for Welded Structures was established and is supported under the Australian GovernmentÕs Cooperative Research Centres Program. The authors wish to thank Prof. Valerie Linton for reading this paper and critical comments.
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Appendix A. Prove the upper bound theorem of plasticity for bi-materials Assume that the solid consist of two different parts (1) and (2) with different materials properties separated by a simple surface A1,2. We assume that each of these materials can be idealized as an elastic-perfectly plastic solid. Let d^ ui be a kinematically admissible displacement field (i.e. continuous everywhere within the solid, differentiable everywhere and also satisfies boundary conditions). Then the strain tensor associated with d^ ui can be calculated as 1 od^ ui od^ uk P d^eik ¼ þ . ðA1Þ 2 oxk oxi ^ik be the stress associated with this kinematically admissible displacement field, i.e. the stress that Let r satisfies d^ePik ¼ dk
3 ^sik ; 2 rY
ðA2Þ
^ik . ^sik is the deviator part of the stress tensor r ^ik is not necessarily the actual stress field induced in the solid by the applied tractions ti, unless Note that r ^ik is the stress field that would cause d^ ui happens to be the true mechanism of collapse in the solid. Instead, r ^ik differs from the true distribution of stress in the solid at the plastic strain increment d^ePik . The stress field r collapse in that it need not necessarily be an equilibrium stress field. The upper bound theorem of plasticity states that, if any displacement field can be found for which Z Z P ^ik d^eik dV 6 ti dui dA; ðA3Þ r V
A
then the solid will collapse under the applied loads ti. In words, the solid will collapse if the work done by the external loads through any mechanism of collapse exceeds the internal plastic dissipation. To prove it we recall the Principle of Maximum Plastic Resistance, which can be written for both materials as ðrij rij Þ
dePij P 0; dt
ðA4Þ deP
where rij is a stress state which causes plastic deformation, and dtij is the resulting strain rate, while Now, rij is any other stress that can be imposed on the specimen that either does not reach yield, or else just satisfies the yield criterion. deP For our purposes, let dePij ¼ dtij dt be the strain increment associated with the kinematically admissible ^ij be the stress associated with this collapse mechanism, and let rij be the actual displacement field dui, let r state of stress induced in the solid by a safe distribution of tractions. Observe that the actual stress state must satisfy: orij ¼ 0 inside the volume V ¼ V ð1Þ þ V ð2Þ oxi ðiiÞYield criterion f ð1;2Þ ðrij Þ 6 0 ðiÞEquilibrium
ðiiiÞBoundary conditions rij ni ¼ ti on A ¼ A
ð1Þ
þA
ð2Þ
where upper superscript (1) relates to the first material and (2) to the second.
ðA5:1Þ ðA5:2Þ ðA5:3Þ
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Start by re-writing the principle of maximum plastic resistance in terms of our stress and strain fields Z Z dePij ^ij dePij dV P P0) r rij dePij dV ; ðA6Þ ðrij rij Þ ð1;2Þ ð1;2Þ dt V V Symbol (1,2) means that this equation is true for both materials and volumes (1) and (2). Now substitute for the strain increment in terms of the displacements: Z Z 1 odui oduj ^ij dePij dV P rij þ r dV ; 2 oxj oxi V ð1;2Þ V ð1;2Þ from symmetry of the stress tensor: Z Z 1 odui oduj odui rij þ rij dV ; dV ¼ 2 oxj oxi oxj V ð1;2Þ V ð1;2Þ Z Z Z odui oðrij dui Þ orij o rij dV ¼ dui dV ¼ ðrij dui Þ dV . ð1;2Þ ð1;2Þ ð1;2Þ ox ox ox ox i j j j V V V
ðA7Þ
ðA8Þ ðA9Þ
In the above we used the symmetry of the stress tensor and the equation of stress equilibrium (A5.1). Finally, apply the divergence theorem and use the boundary conditions Z Z ^ij dePij dV P rij nj dui dA; ðA10:1Þ r Að1Þ
V ð1Þ
and
Z V ð2Þ
^ij dePij dV P r
Z Að2Þ
ðA10:2Þ
rij nj dui dA.
Summing these two Eqs. (A10.1) and (A10.2) we can write Z Z Z P ^ij deij dV P ti dui dA þ ti dui dA. r V
Að1Þ
ðA11Þ
Að2Þ
ð1Þ
ð2Þ
Surfaces A(1) and A(2) contain a common part A(12) on which ti ¼ ti over that surface will be cancelled and finally we have Z Z ^ij dePij dV P ti dui dA. r V
and consequently the integration
ðA12Þ
A
If the traction distribution ti is safe, this condition must be satisfied for all possible collapse mechanisms dui. Therefore, if we can find any collapse mechanism for which Z Z ^ij dePij dV 6 ti dui dA; r V
A
the loading applied to the solid will definitely lead to plastic collapse.
References [1] Hao S, Cornec A, Schwalbe K-H. Plastic stress and strain fields and limit loads of a plane strain centre cracked tensile panel with a mis-matched welded joint. Int J Solids Struct 1997;34:297–326. [2] Joch J, Ainsworth RA, Hyde TH. Limit load and J estimates for idealised problems of deeply cracked welded joints in plane strain bending and tension. Fatigue Fract Engng Mater Struct 1993;16:1061–79.
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[3] Hornet P, Eripret C. Experimental J evaluation from a load–displacement curve for homogeneous and over-matched SENB or M (T) specimens. Fatigue Fract Engng Mater Struct 1995;26:679–92. [4] Kim YJ, Schwalbe K-H. Mismatch effect on plastic yield loads in idealised weldments I. Weld centre cracks. Mismatch effect on plastic yield loads in idealised weldments II. Heat affected zone cracks. Engng Fract Mech 2001;68:163–82. and 183–199. [5] Khan IA, Bhasin V, Vaze KK, Kushwaha HS, Srivastava R, Gupta SR. An estimation procedure to evaluate limit loads of bimetallic specimens. Part – 1, Int J Press Vessels 2004;81:451–62. [6] Kachanov LM. Fundamentals of the theory of plasticity. Moscow: Mir Publishers; 1974. [7] Collins IF. The upper bound theorem for rigid/plastic solids generalized to include coulomb friction. J Mech Phys Solids 1969;17:323–38.
Collapse load for a crack in a plate with a mismatched welded joint Andrei Kotousov *, Mohd Fairuz Mohmed Jaffar School of Mechanical Engineering, University of Adelaide, SA 5005, Australia Received 20 April 2005; accepted 15 July 2005 Available online 8 September 2005
Abstract The accuracy of flaw assessment techniques for welded joints is directly related to the accurate estimate of the yield collapse load. A careful finite element limit analysis can provide a very good and reliable assessment of the collapse load. Unfortunately, such an analysis is normally time-consuming and requires a substantial effort in order to validate the finite element calculations. In this paper we apply the classical upper bound theorem of plasticity to develop simplified solutions for a through crack in a tensile plate with a mismatched welded joint. The weld configuration is idealised as a simple sandwich model and the collapse load is derived analytically from a solution of an extreme-value problem. The theoretical solutions are then verified by independent finite element calculations. The proposed solutions seem to be very effective and accurate and can be easily generalized for many other weld geometries and loading conditions. Ó 2005 Elsevier Ltd. All rights reserved. Keywords: Plastic collapse; Elastic–plastic material; Welds; Fracture in plates
1. Introduction Situations when the weld metal and base material have different values of yield stress are very common for many engineering structures such as bridges, ships, piping and pressure vessels. The accurate estimation of collapse loads is essential for assessing the integrity of cracked structures with welds. A wide range of techniques has been used in the past to provide sufficiently accurate values for the limit (or collapse) load. Based on slip line field (SLF) analysis, Hao et al. [1] obtained a yield load solution for mismatched middle *
Corresponding author. Tel.: +61 8 83035439; fax: +61 8 83034367. E-mail address: [email protected] (A. Kotousov).
1350-6307/$ - see front matter Ó 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.engfailanal.2005.07.007
1066
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Nomenclature a B e F FB FM H M W e rY rYB rYW W v sYW sYB
half length of a centre crack plate thickness crack eccentricity applied force net section collapse load collapse load for mismatch configuration half width of weld metal strip mismatch factor defined for yield strength (M = rYW/rYB) (half) width of plate normalized crack eccentricity e/H yield strength, general yield strength for base material yield strength for weld material normalised remaining ligament (W a)/H geometry factor of the weld joint v = a/W yield stress of the weld material in shear yield stress of the base material in shear
crack tension M(T) plates. Based on the assumed deformation fields proposed by Joch et al. [2], Hornet and Eripret [3] proposed an approximate solution for mismatched M(T) and single edge cracked bend plates. Kim and Schwalbe [4] conducted a comprehensive review of these techniques and concluded that although time and cost consuming, the finite element (FE) analysis based on elastic-perfectly plastic material model is currently the most effective and reliable approach to the calculation of the limit load in defective mismatched structures. The main objective of the present paper is to develop an effective yet simple analytical approach for the assessment of the collapse load for plates with cracks in mismatched welds using the classical upper bound theorem of plasticity. Hornet and Eripret [3] have used this theorem before for mismatched welded structures by invoking simple assumed deformation fields. Unfortunately, the previous application of the upper bound theorem of plasticity led to unacceptably higher values than the actual collapse load for highly under-matched plates [4]. The current approach is different from what has been done before and reduces the problem under consideration to an extreme-value problem for which the limit load is derived analytically. As it will be shown later in this paper it provides a very effective way to analyse cracked structures with mismatched weldments and with some physical insight, the approach can be used to get a very accurate assessment of the collapse load with very little effort. It will be demonstrated that the proposed method overcomes problems with highly under-matches plates and agrees well with independent FE studies. It can be easily generalized for many other practically important cases of mismatched welded structures.
2. Deformation patterns An actual weld joint is very complicated, both on micro- and macro-levels. In practical situations, the welding process affects material properties in the heat-affected zone (HAZ). The presence of residual stresses makes the situation further complicated, rendering analytical solutions nearly impossible [5]. For this
A. Kotousov, M.F. Mohmed Jaffar / Engineering Failure Analysis 13 (2006) 1065–1075
1067
reason simplifications typical of those made when analysing this problem, will be made in the following analysis [4]. The effects of heat affected zones, residual stresses or other kinds of non-heterogeneity will be omitted, and the real welded joint will be modelled as a sandwich-like bi-material system shown in Fig. 1. The yield stress in the weld material can be lower or higher than that of the base material and this strength difference is referred as mismatch. The mismatch in the yield strength between the weld metal, rYW, and the base plate, rYB, is normally quantified by the mismatch factor M: M¼
rYM ; rYB
ð1Þ
with M < 1 referring to under-matching and M > 1 referring to over-matching. The slenderness of the weld and the crack eccentricity ratio are defined as W¼
W a ; H
ð2Þ
and e ¼ e=H ;
ð3Þ
where 2a is the crack length, 2W is the plate width, 2H is the weld width and e is the distance to the crack from the mid-line of the weld joint (see Fig. 1). Under fully plastic conditions, failures are controlled by the formation of slip planes in the case of tensile loading and by the formation of plastic hinges in bending [6]. Possible patterns of plastic deformation for mismatched plates with a crack extracted from FE limit analysis are shown in Fig. 2. In each figure, relevant information on M, W and v = a/W is given. In the first pattern the plastic deformation penetrates through the base plate. This case corresponds to the gross section yielding failure mechanism when almost all plastic deformation is concentrated in the gross section of the base plate. It is typical for highly over-matched welds. The decreasing of the mismatch factor M leads to the penetrating yielding characterized by the plastic deformation in the cracked ligament of both the weld and base plate. Important observation here is that the lower mismatch factor leads to a higher proportion of the plastic deformation in the weld. The limiting case is a confined yielding when the
Fig. 1. Model geometry of the problem.
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Fig. 2. Possible plastic deformation patterns for mismatched plate with centre crack extracted from the FE limit analysis.
base material is shielded from plastic deformation by the weld, which is likely to occur for small M and W. We will use these observations in the next section when we build a simple analytical model to calculate the collapse load based on the upper bound theorem of plasticity [7].
3. Applications of the upper bound theorem of plasticity The upper bound theorem of plasticity provides a simple way to calculate the collapse load for an elasticperfectly plastic solid including a rigid perfectly plastic solid as a limiting case. It will not always predict the exact value, but with some physical insight, the theorem can be used to get a very good estimate for collapse load with very little effort. The proof of the upper bound theorem of plasticity for bi-material solids, such as welded structures is presented in the Appendix A and using the same approach can be easily generalized for multi-material solids. An informal introduction to the classical upper bound theorem of plasticity can be given as follows. Consider a plastically deforming solid, which is subjected to a distribution of tractions on its boundary. The upper bound theorem of plasticity states that the solid will collapse if the work done by the tractions through any mechanism of collapse exceeds the internal plastic dissipation. Collapse mechanism can be described through a kinematically admissible displacement field i.e. displacement field which is continuous everywhere within the solid, differentiable everywhere and also satisfies the displacement boundary conditions if any exists. Now, we apply the upper bound theorem to the problem under consideration. We guess and consider different collapse mechanisms as shown in Fig. 3. The top, bottom half and side parts of the plate slide past each other as rigid blocks, as shown in Fig. 3. Here we consider four different situations as: fully confined
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Fig. 3. Guess collapse mechanism (a1 and a2 are unknown angles to be determined from further analysis).
yielding (i), confined-penetrating yielding (ii), fully penetrating yielding (iii) and yielding in the base material yielding (iv). In the beginning we derive the estimate of the collapse load FB for an all-base plate (assuming that the plate is wholly made of the base material, i.e. sYW = sYB and, because of the symmetry, a1 = a2). From the upper bound theorem of plasticity we have F B tan ad ¼ 2BðW aÞsY dsec2 a;
ð4Þ
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where sY is the yield stress of the weld and base material in shear pffiffiffi(recall that TrescaÕs yield criterion predicts sY = rY/2, while Von Mises yield criterion predicts sY ¼ rY = 3). The left part of the above equations represents the work done by the applied force FM through the specified (guessed) mechanism of collapse and, in accordance with the upper bound theorem of plasticity, at collapse conditions (FM) must be equal to the internal plastic dissipation, which is written in the right part of these equations. Eq. (4) produces a range of values for the loads leading to the plastic collapse as a function of a: F B ðaÞ ¼ 2B
W a sYB . sin a cos a
ð5Þ
The real value of the collapse load in accordance with the upper bound theorem will always be less than FB. Obviously, that the lowest value of FB = minFB(a) will represent the best estimate of the collapse load. The right hand side of the above Eq. (5) reaches the minimum at a = p/4 and the minimum value is F B ¼ 4BðW aÞsY ;
ð6Þ
which is exactly the same equation as obtained from the SLF analysis [1]. It is pretty astonishing that the proposed simple approach has produced the same answer, which was obtained using rather complicated considerations. Using formula (6), the equations for corresponding collapse mechanisms of the mismatched welded plate shown in Fig. 3 can be written as for a1 6 a1 and a2 6 a2 (fully confined yielding) FM M 1 1 ¼ þ ; ð7Þ F B 2ðtan a1 þ tan a2 Þ cos2 a1 cos2 a2 for a1 > a1 and a2 6 a2 (confined-penetrating yielding) FM M 1e 1 1e 1 ¼ þ þ ; F B 2ðtan a1 þ tan a2 Þ W cos a1 sin a1 M cos2 a1 MW cos a1 sin a1 cos2 a2 for a1 6 a1 and a2 > a2 (confined-penetrating yielding) FM M 1 1þe 1 1þe ¼ þ þ ; F B 2ðtan a1 þ tan a2 Þ cos2 a1 W cos a2 sin a2 M cos2 a2 MW cos a2 sin a2 for a1 > a1 and a2 > a2 (fully penetrating yielding) FM M 1e 1 1e 1þe ¼ þ þ F B 2ðtan a1 þ tan a2 Þ W cos a1 sin a1 M cos2 a1 MW cos a1 sin a1 W cos a2 sin a2 1 1þe þ ; M cos2 a2 MW cos a2 sin a1 where sYW and sYB are the yield stress of the weld and base material in shear. The critical angles, which determine the specific mechanism of plastic collapse, are 1 1 e a1 ¼ tan ; W and a2
¼ tan
1
1þe . W
ð8Þ
ð9Þ
ð10Þ
ð11Þ
ð12Þ
The forth mechanism (base material yielding) can be analysed similar to the confined yielding mechanism by replacing the yield stress in the weld by the yield stress of the base material.
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Unfortunately, the transcendental Eqs. (7)–(10) cannot be solved analytically to provide a closed form solution for the collapse load. However, it can be easily solved using a pocket calculator for any combination of M and W and should not significantly devaluate the proposed approach. The upper estimate of the collapse load can be found as a minimum value of FM (or FFMB ) calculated for different mechanisms of the collapse failure by varying a1 and a2 in the range corresponding to the particular collapse mechanisms (i)–(iv). The one giving the minimum value gives the best estimate of the collapse load. The collapse load is also limited by the gross sectional yielding mechanism for which it is easy to show that FM 1 . ¼ FB 1 v
ð13Þ
Finally, the best estimate of the collapse load is FM ¼ minfEqs. (7)–(13)g. FB
ð14Þ
4. Results Results of the calculations using the current approach (solid lines) for different values of the strength mismatch levels, ligament-to-weld width ratios, and crack lengths are presented in Figs. 4 and 5 together with FE results [4] (circles). From the comparison it is clear seen that the proposed method has a very good accuracy and agrees well with independent FE studies. As it can be seen from Figs. 4 and 5 in some cases the FE results predict a slightly higher value for the collapse load than those obtained from the application
Fig. 4. Comparison o the results obtained using the proposed method with FE results [5], (a) a/W = 0.5 and e = 0; (b) a/W = 0.1 and e = 0.
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Fig. 5. Comparison of the results obtained using the proposed method with the FE results [4], a/W = 0.5 and e = 1.
of the upper bound theorem of plasticity. This difference could give some indication of the accuracy of the FE results, which are always needed to be treated with care. The general tendencies in the collapse load for mismatched joints agrees well with those obtained from finite element limit analysis and can be formulated as follows: (1) The collapse load is close to that of an all base plate (the plate is wholly made of the base material) for large values of normalized ligament ratios, say W > 10. (2) The collapse load is close to that of all weld material for under-matched welded joints with small value of normalized ligament ratios. (3) The collapse load is close to that of the gross sectional yielding for highly overmatched welded joints with small value of normalized ligament ratios. (4) The higher crack eccentricity ratio leads to the decrease of the collapse load for overmatched and to the increase of the collapse load for under matched welded joints. It is believed that the similar tendencies take place for other geometries and loading conditions in welded structures. 5. Conclusion The proposed approach can be easily generalised for other important geometries and loading conditions, which may include edge crack, longitudinal or radial through crack in a pipe or other structures. Because the upper bound theorem of plasticity provides the upper estimate of the collapse load it means that this approach is not conservative. It should also be treated with caution and be always validated by independent studies, such as FE. One of the important results of the current paper is that in the conjunction with FE the current approach can replace the necessity in the use of fitting formulas for the assessment of the collapse loads in mismatched structures and can significantly reduce the volume of required FE calculations. The proposed approach can also serve as an independent validation of FE results and can give some estimate of its accuracy. Acknowledgement The work reported herein was supported by the Cooperative Research Centre for Welded Structures. The Cooperative Research Centre for Welded Structures was established and is supported under the Australian GovernmentÕs Cooperative Research Centres Program. The authors wish to thank Prof. Valerie Linton for reading this paper and critical comments.
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Appendix A. Prove the upper bound theorem of plasticity for bi-materials Assume that the solid consist of two different parts (1) and (2) with different materials properties separated by a simple surface A1,2. We assume that each of these materials can be idealized as an elastic-perfectly plastic solid. Let d^ ui be a kinematically admissible displacement field (i.e. continuous everywhere within the solid, differentiable everywhere and also satisfies boundary conditions). Then the strain tensor associated with d^ ui can be calculated as 1 od^ ui od^ uk P d^eik ¼ þ . ðA1Þ 2 oxk oxi ^ik be the stress associated with this kinematically admissible displacement field, i.e. the stress that Let r satisfies d^ePik ¼ dk
3 ^sik ; 2 rY
ðA2Þ
^ik . ^sik is the deviator part of the stress tensor r ^ik is not necessarily the actual stress field induced in the solid by the applied tractions ti, unless Note that r ^ik is the stress field that would cause d^ ui happens to be the true mechanism of collapse in the solid. Instead, r ^ik differs from the true distribution of stress in the solid at the plastic strain increment d^ePik . The stress field r collapse in that it need not necessarily be an equilibrium stress field. The upper bound theorem of plasticity states that, if any displacement field can be found for which Z Z P ^ik d^eik dV 6 ti dui dA; ðA3Þ r V
A
then the solid will collapse under the applied loads ti. In words, the solid will collapse if the work done by the external loads through any mechanism of collapse exceeds the internal plastic dissipation. To prove it we recall the Principle of Maximum Plastic Resistance, which can be written for both materials as ðrij rij Þ
dePij P 0; dt
ðA4Þ deP
where rij is a stress state which causes plastic deformation, and dtij is the resulting strain rate, while Now, rij is any other stress that can be imposed on the specimen that either does not reach yield, or else just satisfies the yield criterion. deP For our purposes, let dePij ¼ dtij dt be the strain increment associated with the kinematically admissible ^ij be the stress associated with this collapse mechanism, and let rij be the actual displacement field dui, let r state of stress induced in the solid by a safe distribution of tractions. Observe that the actual stress state must satisfy: orij ¼ 0 inside the volume V ¼ V ð1Þ þ V ð2Þ oxi ðiiÞYield criterion f ð1;2Þ ðrij Þ 6 0 ðiÞEquilibrium
ðiiiÞBoundary conditions rij ni ¼ ti on A ¼ A
ð1Þ
þA
ð2Þ
where upper superscript (1) relates to the first material and (2) to the second.
ðA5:1Þ ðA5:2Þ ðA5:3Þ
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Start by re-writing the principle of maximum plastic resistance in terms of our stress and strain fields Z Z dePij ^ij dePij dV P P0) r rij dePij dV ; ðA6Þ ðrij rij Þ ð1;2Þ ð1;2Þ dt V V Symbol (1,2) means that this equation is true for both materials and volumes (1) and (2). Now substitute for the strain increment in terms of the displacements: Z Z 1 odui oduj ^ij dePij dV P rij þ r dV ; 2 oxj oxi V ð1;2Þ V ð1;2Þ from symmetry of the stress tensor: Z Z 1 odui oduj odui rij þ rij dV ; dV ¼ 2 oxj oxi oxj V ð1;2Þ V ð1;2Þ Z Z Z odui oðrij dui Þ orij o rij dV ¼ dui dV ¼ ðrij dui Þ dV . ð1;2Þ ð1;2Þ ð1;2Þ ox ox ox ox i j j j V V V
ðA7Þ
ðA8Þ ðA9Þ
In the above we used the symmetry of the stress tensor and the equation of stress equilibrium (A5.1). Finally, apply the divergence theorem and use the boundary conditions Z Z ^ij dePij dV P rij nj dui dA; ðA10:1Þ r Að1Þ
V ð1Þ
and
Z V ð2Þ
^ij dePij dV P r
Z Að2Þ
ðA10:2Þ
rij nj dui dA.
Summing these two Eqs. (A10.1) and (A10.2) we can write Z Z Z P ^ij deij dV P ti dui dA þ ti dui dA. r V
Að1Þ
ðA11Þ
Að2Þ
ð1Þ
ð2Þ
Surfaces A(1) and A(2) contain a common part A(12) on which ti ¼ ti over that surface will be cancelled and finally we have Z Z ^ij dePij dV P ti dui dA. r V
and consequently the integration
ðA12Þ
A
If the traction distribution ti is safe, this condition must be satisfied for all possible collapse mechanisms dui. Therefore, if we can find any collapse mechanism for which Z Z ^ij dePij dV 6 ti dui dA; r V
A
the loading applied to the solid will definitely lead to plastic collapse.
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