Dimensions of the tested structure.
This chapter presents a practical method to investigate the effects of brittle fracture on the ultimate compressive strength of steel stiffened-plate structures under cryogenic conditions. Computational models are developed to analyse the ultimate compressive strength of steel stiffened-plate structures, triggered by brittle fracture, under cryogenic condition. A phenomenological form of the material model for the high-strength steel at cryogenic condition is proposed, that takes into account the Bauschinger effect, and implemented into a nonlinear finite element solver (LS-DYNA). Comparison between computational predictions and experimental measurements is made for the ultimate compressive strength response of a full-scale steel stiffened-plate structure, showing a good agreement between them.
- Bauschinger effect
- brittle fracture
- compressive strength
- cryogenic condition
- nonlinear finite element method
- steel stiffened plate structure
- ultimate strength
Steel stiffened panels are used in naval, offshore, mechanical and civil engineering structures as their primary strength sub-structures. Occasionally, they are exposed to cryogenic conditions, e.g., due to the unwanted release of liquefied gas such as LNG (liquefied natural gas) or liquefied hydrogen as discussed in Paik et al. .
The ultimate strength is a primary criterion for the design of steel stiffened-plate structures [2, 3, 4, 5, 6], and it is essential to characterise the effects of cryogenic condition on the ultimate strength of such structures. The authors of this chapter have previously conducted collapse tests on full-scale steel stiffened-plate structures under axial-compressive loading at room temperature , at cryogenic condition  and at -80°C . This chapter is part of a sequel to investigate the brittle fracture of steel stiffened-plate structures under cryogenic conditions. The tested structures were designed from a reference plate panel in an as-built containership carrying 1,900 TEU (twenty-foot equivalent units). They were fabricated at a shipyard using exactly the same welding technology as used in today’s shipbuilding industry .
Structural fracture modes are classified into three groups [3, 4]: rupture, ductile fracture and brittle fracture. Material rupture occurs when failure occurs by cracking associated with necking localisation during large plastic flow. If the strain at which a material fractures is small, with very little ductility, it is a brittle fracture. An intermediate fracture mode between rupture and brittle fracture is called ductile fracture with partial ductility. Fracture behaviour of ductile materials such as carbon steels is quite different from that of inherently brittle materials. Ductile materials generally exhibit slow stable crack growth during crack extension, but they can show a similar behaviour to brittle materials at specific environments such as very low temperatures or lower than the ductile-to-brittle fracture transition temperatures (DBTT) and/or impact loading. It is also recognised that the Bauschinger effect of materials cannot be neglected at sub-zero temperatures and cryogenic condition as the material behaviour in compression is distinct from that in tension [3, 10, 11]. Figure 1 illustrates the transition of the ductile-to-brittle behaviour for structural steel.
To compute the failure behaviour of structures at cold (sub-zero) temperatures (or higher than the ductile-to-brittle fracture transition temperatures), constitutive equations of materials have been proposed in the literature [12, 13, 14, 15]. It is recognised that most of previous studies are associated with predominantly ductile behaviour or at least with partial ductility, but studies applicable to entirely brittle fracture at cryogenic condition are lacking. It is also recognised that the approaches using the constitutive equations are not always practical for the ultimate strength analysis of supersized structures because they are too complex to apply for the problem.
This chapter presents a method for computing the ultimate compressive strength of steel stiffened-plate structures by nonlinear finite element method (NLFEM) using the multi-physics software package LS-DYNA implicit code. Mechanical properties of high-strength steel with grade AH32 used for fabricating the tested structure were obtained from tension and compression tests at low temperatures and cryogenic condition [7, 8], and a phenomenological relation of engineering stress versus engineering strain of the material was formulated. The material model is implemented into the LS-DYNA implicit code. To demonstrate the validity of the computational model, the NLFEM is compared to experimental results from a full-scale physical test.
2. Literature survey on structural behaviour at cold (sub-zero) temperatures
A number of studies in modelling of material behaviour for structural steels at cold (sub-zero) temperatures are available in the literature. Most of the studies dealt with predominantly ductile behaviour of materials with the focus on how crack initiates in association with ductile fracture. Ehlers and Varsta  and Ehlers  derived the true stress versus true strain relation of ordinary steel. The effects of stress triaxiality on ductile fracture have been one of research topics [18, 19, 20, 21, 22, 23, 24]. The works of the Choung group have provided useful insights for ductile fracture behaviour of structural steels [14, 25, 26, 27, 28, 29, 30, 31, 32].
It is recognised that structural steel behaviour is predominantly ductile at temperatures higher than the temperature of the ductile-to-brittle fracture transition, as shown in Figure 1. As the temperature decreases approaching cryogenic condition, the material behaves predominantly in a brittle manner with partial or no ductility [33, 34, 35, 36, 37, 38, 39, 40]. Majzoobi et al.  observed that the ductile-to-brittle fracture transition of carbon steel occurs at about -80°C, and the material behaviour becomes entirely brittle at -196°C.
Although there are considerable uncertainties associated with the ductile-to-brittle fracture transition temperature (DBTT), a number of evidences for brittle fracture behaviour of steel structures at cryogenic condition have been seen in the literature, depending on the type of materials and loading conditions (e.g., quasi-static or impact), among other factors. Crushing testing of steel tubes under quasi-static loads at -60°C [42, 43] showed ductile fracture, as shown in Figure 2. Dropped-object impact testing of steel stiffened plate panels at -60°C  showed brittle fracture, as shown in Figure 3. Full-scale collapse testing of a steel stiffened-plate structure under axial-compressive loading showed that the ultimate strength was reached by a trigger of brittle fracture , as shown in Figures 4 and 5. At room temperature, the structures reached the ultimate strength by flexural-torsional buckling , but brittle fracture triggered the global failure at cryogenic conditions .
Here, an attempt is made to develop new fracture criteria based on the hypothesis that crack initiates if an equivalent stress exceeds a critical value, to model the fracture phenomenon of high-strength steel (AH32) under cryogenic conditions. Existing material models for the fracture analysis is first reviewed.
2.1 Maximum principal stress based fracture criterion
The maximum principal stress-based fracture criterion is the simplest among all fracture criteria. It is useful to predict fracture behaviour of brittle materials under predominantly tensile loads. In this criterion, brittle fracture is expected to occur when the largest principal normal stress reaches the ultimate tensile strength () of the material, which is usually obtained from tension tests of coupon specimens. The maximum principal stress-based fracture criterion is expressed as follows:
where , and are the principal stress components.
2.2 Coulomb-Mohr fracture criterion
The Coulomb-Mohr fracture criterion gives reasonably accurate predictions of fracture in brittle materials for which the compressive strength far exceeds the tensile strength, e.g., concrete or cast iron . It is presumed that fracture occurs in a certain stress plane of material when a critical combination of normal stress and shear stress acts on the plane. The linear relation of the combination of critical stresses is given by:
where is the shear stress, is the normal stress, and are constants for a given material.
2.3 Johnson-Holmquist fracture criterion
The Johnson-Holmquist fracture criterion  is useful for modelling brittle materials, e.g., ceramic and glass, over a range of strain rates. It is one of the most widely used models in dealing with the ballistic impact on ceramics, which is expressible as follows:
where is the uniaxial failure strength of intact material, see Eq. (4), is the uniaxial failure strength of completely fractured material, see Eq. (5), and is a damage accumulation variable, see Eq. (6).
where , , , , are material constants, is time, is the inelastic strain, and is the plastic strain to fracture. The asterisk indicates a normalised quantity, where the quantities of each variable are defined as follows:
where and are the stresses normalised by the stress at the Hugoniot elastic limit, and is the tensile hydrostatic pressure normalised by the pressure at the Hugoniot elastic limit.
The Johnson-Holmquist damage model was modified by Deshpande and Evans  and Bhat et al. , where it is considered that the propagation of an initial crack is a function of the stress state, the fracture toughness and the flaw characteristics.
3. Proposed fracture criteria
The yield and ultimate tensile strengths of structural steels tend to increase with a decrease in the temperature , and subsequently the maximum load-carrying capacity (ultimate strength) of steel structures at cold temperatures is greater than that at room temperature [43, 50]. Figure 6 shows a schematic of ductile and brittle failure behaviour, where the brittle fracture-induced ultimate strength at cryogenic condition is greater than the ductile collapse-induced at room temperature. However, the post-ultimate strength behaviour becomes very unstable if brittle fracture triggers the structural collapse at cryogenic condition. In this case, the strain energy absorption capability of structures can be more useful than the ultimate strength itself in terms of the structural safety assessment as it is obtained by integrating the area below the load–displacement curve until or after the ultimate strength is reached. The absorbed energy at cryogenic condition can be smaller than or at room temperature or a temperature higher than the DBTT (ductile-to-brittle fracture transition temperature). For this purpose, the entire behaviour of structural collapse involving brittle fracture at cryogenic condition must be quantified efficiently and accurately.
3.1 Fracture criteria
A practical model is proposed for carbon steels which can be used for the ultimate strength analysis triggered by brittle fracture at cryogenic condition or in the region of ductile-to-brittle fracture transition. An elastic-perfectly plastic material model without the strain-hardening effect is used similar to a typical application at room temperature (20°C). However, the material behaviour in compression is different at low (sub-zero) temperatures or cryogenic condition from that in tension as the Bauschinger effect plays a role. However, the Bauschinger effect is usually neglected at room temperature with , and .
In the present model, it is hypothesised that brittle fracture occurs if the equivalent stress () reaches a fracture stress which is defined as the yield strength of material at the corresponding temperature, which can be expressed as follows:
(1) In tension:
(2) In compression:
where is the equivalent stress, and are the fracture or yield stresses in tension at cryogenic condition or in the region of ductile-to-brittle fracture transition (which depends on types of materials), and are the fracture or yield stresses in compression at cryogenic condition or in the region of ductile-to-brittle fracture transition, and are test constants for a given steel in tension or compression, which may depend on various sources of parameters including chemical composition (grade), temperature and strain rate.
For plane stress state, can be simplified as follows:
3.2 Formulation of the engineering stress-engineering strain relations
The relations of the engineering stress versus engineering strain can be formulated following the fracture criterion defined in Section 3.1. If the steel temperature, , is above the ductile-to-brittle fracture transition temperature (DBTT), i.e., DBTT, the material behaves according to the ductile region. In this case, the stress–strain relation in tension is expressed as follows:
where is the engineering stress, is the engineering strain, is the elastic modulus, is the yield strain in tension, and is the fracture strain in compression. Figure 7 shows a schematic view of the proposed model of the engineering stress-engineering stress relation with full or partial ductility at room temperature or low temperatures.
In compression, the following equation similar to Eq. (13) is obtained.
where is the yield strain in tension and is the fracture strain in compression.
If DBTT or the steel temperature is lower than DBTT, the material behaves according to the brittle region. In this case, the stress–strain relation in tension is expressed as follows:
In compression, a similar equation to Eq. (14) is obtained as follows:
In the region of entire brittle fracture, i.e., with DBTT, Figure 7 can be redrawn as shown in Figure 8 when elongation can be neglected after reaching the yield stress in tension or compression as the material exhibits brittle fracture immediately.
4. Nonlinear finite element method modelling
The proposed material model is implemented into the LS-DYNA implicit solver with 4-node shell elements (element formulation 16) for the ultimate compressive strength analysis of a structure tested at cryogenic condition as shown in Figures 4 and 5 . Table 1 with Figure 9 shows the dimensions of the tested structure. The nonlinear ultimate compressive strength analysis is simulated in a quasi-static analysis using the LS-DYNA implicit solver. The 4-node shell elements are used to model plating, stiffeners and transverse frames. In order to ensure sufficient resolution in the mesh size, a convergence study was performed by varying the element size following a typical approach as described in Paik . The resulting convergence study provided the element size of 40 mm 40 mm which was chosen to obtain sufficiently accurate results while minimising the computational cost. The thermal shrinkage effects for steel at low temperatures were not considered. Figure 10 shows the FE model of the tested structure. Figure 11 shows the loading and boundary conditions which were modelled as much as close to the tested structure, where unloaded edges were kept straight and loaded edges were entirely fixed.
|Material||AH32 high-strength steel|
|Spacing between transverse frames ()||3150 mm|
|Spacing between longitudinal stiffeners ()||720 mm|
|Plating thickness ()||10 mm|
|Longitudinal stiffener (middle bay)||290 90 10/10 (T) (mm)|
|Longitudinal stiffener (side bays)||290 90 20/10 (T) (mm)|
|Transverse frame||665 150 10/10 (T) (mm)|
Only the middle bay of the tested structure was exposed to the cryogenic condition as shown in Figure 12. Table 2 summarises the measured data of steel temperatures during the collapse testing. For details of Table 2, see Paik et al. .
|Part||Highest temperature (°C)||Lowest temperature (°C)||Average temperature (°C)|
|Web of longitudinal stiffener||−72.1||−167.2||−128.8|
|Flange of longitudinal stiffener||−58.8||−99.1||−79.3|
In room temperature, the mechanical properties of steel in compression are typically defined in the same manner as in tension without considering. However, the Bauschinger effect cannot be neglected at low (sub-zero) temperatures and cryogenic condition [3, 10, 11]. To define the mechanical properties of AH32 steel at different temperatures (20°C, −80°C, −130°C and − 160°C), material tests in tension and compression were conducted. Details of these test data are presented in separate papers [7, 8]. Tables 3 and 4 summarize the test data for the mechanical properties of the AH32 steel. It is found that the yield stress of steel in tension or compression increases as the temperature decreases, while the fracture strain in tension decreases with decrease in the temperature. The elastic modulus of steel remains unchanged regardless of sub-zero temperatures. This chapter focuses on the ultimate strength of steel structures under monotonically applied compressive loads, but fatigue crack resistance at sub-zero temperatures must be associated with microstructural characteristics which are closely related to low-temperature impact toughness of steel [52, 53].
|Parameter||At 20°C||At −80°C||At −130°C||At −160°C|
|Elastic modulus, (GPa)||205.8||205.8||205.8||205.8|
|Yield stress, (MPa)||358.0||433.4||546.7||672.9|
|Fracture strain, (−)||0.376||0.430||0.409||0.336|
|Poisson’s ratio (−)||0.3||0.3||0.3||0.3|
|Parameter||At 20°C||At −80°C||At −130°C||At −160°C|
|Elastic modulus, (GPa)||205.8||205.8||205.8||205.8|
|Yield stress, (MPa)||359.6||382.0||387.2||411.5|
|Poisson’s ratio (−)||0.3||0.3||0.3||0.3|
In the present case study, an elastic-perfectly plastic material model was applied without considering the strain-hardening effect. To implement the material model, *MAT_PLASTICITY_COMPRESSION_TENSION, MAT124 in LS-DYNA was used as it is an isotropic elastic–plastic material which can distinguish material behaviour in tension and compression. The von Mises yield criterion was applied using MAT124. Tension or compression was determined by the sign of the mean stress (hydrostatic stress). A positive sign which means a negative pressure is indicative of tension, or a negative sign is indicative of compression. The mean stress, can be expressed as follows:
where , and are the stress components in the , and directions, respectively.
Majzoobi et al.  observed that the ductile-brittle transition of steel occurs at −80°C, and the material behaviour of steel is completely brittle at −196°C. With Figure 12 and Table 2, the average steel temperatures of plating and web of stiffeners in the middle bay of the tested structure were −160°C and −130°C, respectively. Therefore, the plating and web of stiffeners in the middle bay of the tested structure were modelled using the engineering stress-engineering stress relation of Eqs. (12) and (13). The rest of structural members in ductile region (above −80°C) were modelled using the engineering stress-engineering stress relation of Eqs. (12) and (13). See Paik et al.  for details.
Three types of fabrication-related initial deformations are considered as shown in Figure 13. The measurement data of welding-induced initial deformations for the tested structure  as shown in Figure 14 was directly applied to the FE model.
The initial deformations of the tested structure were formulated so as to make easier implementation into the FE model as shown in Figure 15.
Initial deflection of plating:
Column-type initial deformation of stiffener:
Sideways initial deformation of stiffener:
where is the coordinate in the direction of stiffener web height, and is the stiffener web height.
Biaxial residual stresses developed in the plating of the tested structure between the support members because the welding was conducted in both the longitudinal and the transverse directions to attach the longitudinal stiffeners and the transverse frames. Measurement data of the fabrication-induced residual stresses in the tested structure  was also directly applied to the FE model although the biaxial residual stress distributions were idealised as shown in Figure 16 with the measurement data indicated in Table 5.
|Transverse direction||Longitudinal direction|
|Simplified model||Smith average level model||Simplified model||Smith average level model|
|39.61 mm||56.80 mm||51.47 mm||64.70 mm|
Stress concentration in structural details or fillet weld toe locations happens due to geometrical discontinuity, and it is a critical factor that must be considered for fatigue limit state analysis [3, 56]. Figure 17 shows an example of the effective plastic strain distribution which was obtained from the FE analysis of the ultimate compressive strength of the tested structure. It is obvious from Figure 17 that the effective plastic strain is comparatively large along the weld lines between plating and stiffeners. For the ultimate strength analysis in ductile region, e.g., at room temperature, the stress concentration at the fillet weld toes is usually ignored.
For brittle fracture analysis at sub-zero or cryogenic condition, however, the effects of stress concentration cannot be neglected [4, 44]. This is because the weld toes can reach the yield condition earlier, leading to local brittle fracture which can trigger the ultimate limit states at cryogenic condition. Therefore, the nonlinearity at weld toes along the fillet weld lines needs to take into account in the FE modelling.
One of approaches is to model the weld toes directly in the FE model using shell elements with specific properties of weld metal. Figure 18 shows a schematic of modelling the weld toes using shell elements along the plate-stiffener junction. A similar approach was used to model weld toes by Kim et al.  and Nam et al. . The tested structure was fabricated using flux-cored arc welding (FCAW) method and the consumable was CSF-71S, and the mechanical properties of weld metal with the CSF-71S are presented in Table 6. As such, the weld metal was modelled using the engineering stress-engineering strain relation of Eqs. (12) and (13). It is assumed that the yield strength of the weld metal at −160°C increases linearly in the same proportion as the steel (Table 6).
|At 20°C||At −160°C (tension)||At −160°C (compression)|
|Elastic modulus, (GPa)||268.0||268.0||268.0|
|Yield strength, (MPa)||513.0||929.3||571.1|
|Poisson’s ratio (−)||0.3||0.3||0.3|
5. Computed results and discussion
Figure 19, Tables 7 and 8 present the comparison results between the test and the FE analysis. The difference of ultimate strength between them is 16.6% by the ductile material model, but it becomes at most 2.3% by the brittle material model. When only the ductile material model was applied for all structure members without considering brittle fracture, the FE analysis overestimates the ultimate strength significantly. As the yield strength of the material at cold temperature is greater than that at room temperature, the ultimate strength becomes much larger as far as brittle fracture is not allowed to happen. On the other hand, the ultimate strength obtained from the FE analysis with the brittle material model is in good agreement with the test results. Figure 20 compares the ultimate strength behaviour with or without the weld elements along the plate-stiffener junctions. It is seen from Figure 20 that the weld metal model increased the ultimate strength by 4.7%. This is due to the mechanical properties of weld metal which are larger than those of base metal (Table 9).
|Parameter||Test ||FE analysis with ductile material model||Difference|
|Ultimate strength (ton)||1149.06||1340.14||+ 16.6%|
|Stiffness (ton/mm)||72.38||80.80||+ 11.6%|
|Axial shortening up to collapse (mm)||16.90||19.24||+ 13.8%|
|Strain energy up to collapse (ton·mm)||10179.96||14541.84||+ 42.8%|
|Parameter||Test ||FE analysis with brittle material model||Difference|
|Ultimate strength (ton)||1149.06||1176.01||+ 2.3%|
|Stiffness (ton/mm)||72.38||80.82||+ 11.7%|
|Axial shortening up to collapse (mm)||16.90||14.75||- 12.7%|
|Strain energy up to collapse (ton·mm)||10179.96||8740.46||- 14.1%|
|Parameter||FEA with consideration of weld metal||FEA without consideration of weld metal||Difference|
|Ultimate strength (ton)||1176.01||1121.15||- 4.7%|
|Axial shortening up to collapse (mm)||14.75||14.05||- 4.7%|
|Strain energy up to collapse (ton·mm)||8740.46||7908.81||- 9.5%|
Figure 21 shows the deformed shape at the ultimate limit state of the tested structure obtained from the FE analysis without brittle fracture model (with only ductile material model). It is seen from Figure 21 that the tested structure reached the ultimate limit state by tripping mode of stiffeners (without brittle fracture) which is similar to the collapse mode at room temperature . However, the brittle fracture model represents brittle fracture behaviour which triggered the ultimate strength as shown in Figure 22, where deformed and fracture shapes of the test structures are compared between physical testing and FE analysis.
6. Concluding remarks
This chapter presents a practical method to compute the ultimate compressive strength of steel stiffened-plate structures at cryogenic condition which is triggered by brittle fracture. Case studies were carried out using the method, and the following conclusions were obtained together with modelling recommendations for NLFEM simulations.
A useful material model was formulated to analyse the brittle fracture behaviour of structural steels at cryogenic condition, where the Bauschinger effect was taken into account as the material properties in compression are distinct from those in tension
An elastic-perfectly plastic material model was applied without considering strain-hardening effect.
The developed material model was implemented into the LS-DYNA implicit code with *MAT_PLASTICITY_COMPRESSION_TENSION, MAT124.
Weld elements which are the same type of shell elements but with specific properties of material were introduced to model weld metal (weld toes) at the plate-stiffener junctions where stress concentrations develop. As the mechanical properties of weld metal are typically greater than those of base metal, the ultimate strength usually becomes larger with weld elements at weld toes.
Comparisons were made between the test results and the FE computations on a full-scale steel stiffened-plate structure at cryogenic condition. It is confirmed that the nonlinear FE analysis with the proposed material model, and the weld element model, gives a reasonably good solution of the ultimate compressive strength behaviour for steel stiffened plate structures at cryogenic condition.
This study was undertaken in the International Centre for Advanced Safety Studies, www.icass.center (the Korea Ship and Offshore Research Institute) at Pusan National University which has been a Lloyd’s Register Foundation Research Centre of Excellence since 2008. Part of the work was supported by the Swedish Research Council by the project “Fundamental research on the ultimate compressive strength of ship stiffened-plate structures at Arctic and cryogenic temperatures”, contract no. 2018-06864.