**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** [7]. **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 [4]. 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.

#### **Figure 8.**

*Proposed model of the engineering stress-engineering strain relation without ductility or entire brittle behaviour below the ductile-to-brittle transition temperature or at cryogenic condition.*


#### **Table 1.**

*Dimensions of the tested structure.*

**Figure 9.** *Nomenclature of the scantlings for the tested structure.*

**Figure 10.** *Finite element mesh model of the tested structure.*

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. [7].

In room temperature, the mechanical properties of steel in compression are typically defined in the same manner as in tension without considering. However, *Ultimate Compressive Strength of Steel Stiffened-Plate Structures Triggered by Brittle… DOI: http://dx.doi.org/10.5772/intechopen.97155*

**Figure 11.**

*The loading and boundary conditions applied to the FE model.*


#### **Figure 12.**

*Middle bay of the structure exposed to cryogenic condition.*


#### **Table 2.**

*Measured steel temperatures of the tested structure [7].*

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


#### **Table 3.**

*Mechanical properties of AH32 steel at room and low temperatures in tension [7, 8].*


**Table 4.**

*Mechanical properties of AH32 steel at room and low temperatures in compression [7, 8].*

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].

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, *σmean* can be expressed as follows:

$$
\sigma\_{mean} = \frac{\sigma\_{\chi} + \sigma\_{\chi} + \sigma\_{x}}{3} \tag{16}
$$

where *σx*, *σ<sup>y</sup>* and *σ<sup>z</sup>* are the stress components in the *x*, *y* and *z* directions, respectively.

Majzoobi et al. [41] 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. [7] 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 [54] 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**.

**Figure 13.** *Three types of welding-induced initial deformations in a stiffened plate structure.*

*Ultimate Compressive Strength of Steel Stiffened-Plate Structures Triggered by Brittle… DOI: http://dx.doi.org/10.5772/intechopen.97155*

**Figure 14.** *Measured and idealised deformations of the tested structure due to fabrication by welding.*

**Figure 15.**

*Welding-induced initial deformations applied to the FE model (with an amplification factor of 100 for plating and column-type, and 20 for sideways initial deformations).*

Initial deflection of plating:

$$
\omega\_{opl} = \omega\_{opl}^\* + \omega\_{opl}^{\*\;\*} \tag{17}
$$

$$\begin{aligned} w\_{opl}^{\*} &= w\_{o\max} \sin\left(\frac{m\varpi}{a}\right) \Big|\sin\left(\frac{n\varpi p}{b}\right)\Big|\text{ with } m = 1, \ n = 1, \\ w\_{o\max} &= \begin{cases} 3.5 & \text{for } 0 \le \varpi \le a, 2a \le \varpi \le 3a \\\\ 1.5 & \text{for } a \le \varpi \le 2a \end{cases} \end{aligned} \tag{18}$$

$$w\_{qpl}^{\*,\*} = w\_{o\max} \sin\left(\frac{m\pi\chi}{a}\right) \left| \sin\left(\frac{n\pi\chi}{b}\right) \right| \text{ with } w\_{o\max} = 1.0, \ m = 3, \ n = 1 \tag{19}$$

Column-type initial deformation of stiffener:

$$w\_{oc} = w\_{oc \text{ max}} \sin\left(\frac{m\pi\text{x}}{A}\right) \sin\left(\frac{n\pi y}{B}\right) \text{ with } w\_{oc \text{ max}} = 1.0, \ m = 1, \ n = 1 \tag{20}$$

Sideways initial deformation of stiffener:

$$w\_{os} = w\_{o\,\max} \left(\frac{z}{h\_w}\right) \sin\left(\frac{\pi\infty}{a}\right) \text{with } w\_{os\,\max} = 0.000133a \tag{21}$$

where *z* is the coordinate in the direction of stiffener web height, and *hw* 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 [55] 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**.

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

**Figure 16.** *Idealised distribution of biaxial residual stresses in plating of the tested structure.*

*Ultimate Compressive Strength of Steel Stiffened-Plate Structures Triggered by Brittle… DOI: http://dx.doi.org/10.5772/intechopen.97155*


**Table 5.**

*Measured data of the biaxial residual stresses in the plating of the tested structure.*

**Figure 17.**

*Effective plastic strain () distribution in FE analysis on ultimate compressive strength of stiffened plate structure.*

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

**Figure 18.** *Weld elements at the plate-stiffener junction.*


**Table 6.**

*Mechanical properties of the weld metal with the CSF-71S at room temperature and assumed mechanical properties at 160°C.*

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. [44] and Nam et al. [35]. 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**).
