**3. Proposed fracture criteria**

The yield and ultimate tensile strengths of structural steels tend to increase with a decrease in the temperature [49], 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 *Pu*<sup>2</sup> at cryogenic condition is greater than the ductile collapse-induced *Pu*<sup>1</sup> 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 *E*<sup>3</sup> at cryogenic condition can be smaller than *E*<sup>1</sup> or *E*<sup>2</sup> at room temperature or a temperature higher than the DBTT (ductile-to-brittle fracture transition temperature). For this purpose, the entire

**Figure 6.**

*Ultimate strength and post-ultimate strength behaviour at room temperature (or a temperature higher than the DBTT) versus cryogenic condition.*

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

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 *σYC* ¼ *σYT*, *εYc* ¼ *εYt* and *εfc* ¼ *εft*.

In the present model, it is hypothesised that brittle fracture occurs if the equivalent stress (*σeq*) 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:

$$
\sigma\_{eq} \ge \sigma\_{FT} \text{ with } \sigma\_{FT} = \chi\_t \sigma\_{YT} \tag{8}
$$

(2) In compression:

$$
\sigma\_{eq} \ge \sigma\_{\rm FC} \text{ with } \sigma\_{\rm FC} = \chi\_{\rm c} \sigma\_{\rm YC} \tag{9}
$$

where *σeq* is the equivalent stress, *σFT* and *σYT* 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), *σFC* and *σYC* are the fracture or yield stresses in compression at cryogenic condition or in the region of ductile-to-brittle fracture transition, *γ<sup>t</sup>* and *γ<sup>c</sup>* 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.

In Eqs. (6) and (7), *σeq* can be calculated as a function of principal stresses by the von Mises stress [51] as follows:

$$
\sigma\_{eq} = \frac{1}{\sqrt{2}} \sqrt{\left(\sigma\_1 - \sigma\_2\right)^2 + \left(\sigma\_2 - \sigma\_3\right)^2 + \left(\sigma\_3 - \sigma\_1\right)^2} \tag{10}
$$

For plane stress state, *σeq* can be simplified as follows:

$$
\sigma\_{eq} = \frac{1}{\sqrt{2}} \sqrt{\left(\sigma\_1 - \sigma\_2\right)^2 + \sigma\_2^2 + \sigma\_1\right^2} \tag{11}
$$

#### **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, *Ts*, is above the ductile-to-brittle fracture transition temperature (DBTT), i.e., *Ts* > DBTT, the material behaves according to the ductile region. In this case, the stress– strain relation in tension is expressed as follows:

$$\sigma = \begin{cases} \begin{array}{l} \text{E\\_for } \sigma < \sigma\_{YT} \\ \sigma\_{YT} \text{ for } \sigma = \sigma\_{YT} \text{ or } \varepsilon = \varepsilon\_{Y} \\ \mathbf{0} \text{ } for \, \varepsilon \ge \varepsilon\_{ft} \end{array} \tag{12}$$

*Low-Temperature Technologies and Applications*

where *σ* is the engineering stress, *ε* is the engineering strain, *E* is the elastic modulus, *εYt* is the yield strain in tension, and *εft* 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.

$$\sigma = \begin{cases} \begin{array}{l} \text{E\{for } \sigma < \sigma\_{\text{YC}} \\ \sigma\_{\text{YC}} \text{ for } \sigma = \sigma\_{\text{YC}} \text{ or } \varepsilon = \varepsilon\_{\text{Yc}} \\ \mathbf{0} \text{ for } \varepsilon \ge \varepsilon\_{\text{fc}} \end{array} \end{cases} \tag{13}$$

where *εYt* is the yield strain in tension and *εfc* is the fracture strain in compression.

If *Ts* ≤ 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:

$$
\sigma = \begin{cases}
\text{Ee} & \text{for } \sigma < \sigma\_{FT} \\
\text{0} & \text{for } \sigma \ge \sigma\_{FT}
\end{cases}
\tag{14}
$$

In compression, a similar equation to Eq. (14) is obtained as follows:

$$
\sigma = \begin{cases}
\text{E\,\sigma} & \text{for } \sigma < \sigma\_{\text{FC}} \\
\text{0} & \text{for } \sigma \ge \sigma\_{\text{FC}}
\end{cases}
\tag{15}
$$

**Figure 7.**

*Proposed model of the engineering stress-engineering stress relation with full or partial ductility at room temperature or sub-zero temperatures.*

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

In the region of entire brittle fracture, i.e., with *Ts* ≤ 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.
