**2.9 Comprehensive assessment of risk, strength, in-service life, reliability and safety**

Characterization of initial strength, in-service life, risk and safety of the bearing elements of the sea oil and gas production platform in terms of impact of a complex of loads (including such specific service conditions as collisions with the drifting ice floes, impact of storm and gale-force winds, existence of the corrosive environment, low-temperature embrittlement effects, etc.) is the comprehensive problem considering occurrence of the cyclic dynamic loads corresponding to these conditions and, consequently, nonlinear change in time of the kinetic fields of stresses and deformations in these elements of SP under the impact of irregular loads [1–4]. In this regard in zones of design concentration, the local stresses and deformations have the increased values and the processes of material damage run more intensively leading to appearance of local destructions zones (cracks) eventually developing into macrodestructions (loss of bearing capacity). In such conditions, depending on the nature of loading and the operational environment, various mechanisms of accumulation of damages and destruction are implemented.

For the analysis of operational load of SP (as well as on other objects of energy, transport, oil and gas chemistry) at all stages of the life cycle, curves of the parameters dependent on calculated or real force impact on the bearing elements of the oil and gas production platform (set in the specification or measured during operation) are plotted. Among these parameters are number of loading cycles *N*, time *τ*, temperature *t* as well as service forcing *P*, stress *σ* and deformation *e*. The curves of parameters *P*, *t, σ* and *e* as function of time (**Figure 12**) are plotted for all stages and operational phases.

**Figure 12.** *Diagram of operational loads and their basic parameters.*

When determining safety of the most important objects, the following types of ultimate limit states have to be considered: plastic deformation and forming; shortduration elastic failure; delayed or fast brittle failure; long-term static fracture; cyclic (low- and multi-cycle) destruction; creep strain accumulation; cyclic strain accumulation; buckling; dangerous vibrations occurrence; coupled units wear; single loading cracks initiation and propagation; cyclic cracks initiation and propagation; corrosion, corrosion and mechanical, cavitation and erosive damages; leakages; and change of structures and a condition of the bearing components.

The ultimate limit states listed above identify methods, structure and criteria of safety analysis by integrated approaches of mechanics, physics and chemistry of

In the process of design of structure, its components and, at the bottom, the following groups of the ultimate limit states are taken into consideration. The first group with unacceptable plastic strain and damages includes ultimate limit states surpassing of which will cause total unusability of the structure or total (or partial) loss of supporting capacity of the platform substructure. The second group with damages accumulation and development includes the ultimate limit states where surpassing makes impossible the normal operation of the platform

It should be noted that the above-listed ultimate limit states were taken into account at design of the reinforced concrete support substructure of gravity type for offshore stationary platforms on the sites of the Sakhalin-II project for Piltun-

The design elements of the platform substructure can be split into criticality

**High criticality design elements**—these are elements whose destruction can cause fatalities, serious damages to constructions and environment contamination. **Low criticality design elements**—these are elements whose destruction will not cause fatalities, serious damages to constructions and environment contamination. Between high criticality design elements, the following ones shall be listed:

• column walls in areas of their connection with the bottom and top plates of

• design elements of supporters of the critical and safety equipment including

• structures which damage and destruction will cause dramatic environment

categories depending on the external impacts taken into account:

• design elements of skirt and skirt interface with caisson bottom;

• parts of walls and columns overlapping subject to significant loads

Astokhsky (PA-B) and Lunsky (LUN-A) fields.

overlapping of a caisson;

• internal waterproof walls;

contamination including risers.

riser holders; and

**94**

• design elements contacting with ice;

• connection of deck with the column;

• outer walls, floor slabs and caisson bottom;

concentration;

disasters.

*Probability, Combinatorics and Control*

substructure.

These dependences are initial for the analysis of strength, in-service life, risk and safety of elements of engineering designs both for their initial states and for the damaged states. Values P, *t* and *τ*, are, as a rule, set by the modes of operation and can be registered by instrumentation and control diagnostic systems or by monitoring equipment.

At the same time, *σ* and *e* parameters of the general and local stress-deformed states can be obtained with the help of calculation based on the values of parameters *P, t* and *τ* or purposely measured by means of full-scale strain gauging and thermometry.

In **Figure 12** where a block of external and internal technological operational loadings are presented, the following standard modes of loading of the SP elements are highlighted: assembling (*AS*), tests (*TS*), start-up (*SU*), stationary the mode with maintenance of set operating parameters (*SO*), basic parameters adjustment (*PA*), accident occurrence (*AO*) (including those after of earthquakes), protection systems actuation (*PS*) and shut-down after planned or fault situation (*SD*).

When analyzing the initial and residual strength, service-life, survivability, risk and safety of the oil and gas production platform, the key phase is decomposition of SP and selection and identification of its potentially dangerous critical components, defining the greatest risks of accidents and disasters occurrence. The critical zones of SP components and critical points in them are identified on the basis of experimental and computational studies of stress-deformed and ultimate limit states. In such zones and points, as a rule, processes of local destructions are initiated followed by tramline destructions. At the same time for further experimental and computational evaluations of initial and residual strength, service-life, survivability, risk and safety, the following characteristics of history of loading (**Figure 12**) are accepted:

where *σа*, *еа* are strength and deformation amplitudes; *Е* is the elasticity modulus; *N* is the number of cycles prior to destruction; *r<sup>σ</sup>* is the stress ratio; *σт*, *σ<sup>в</sup>* are

> , ½ �¼ *<sup>N</sup> <sup>N</sup> nN*

When making stress assessment, the characteristics *σ<sup>В</sup>* and *σ<sup>Т</sup>* have to be set with taking into account service conditions—impact of loading cycling, temperatures

**2.10 Criteria of strength, in-service life, safety and protection level (security)**

As it was noted above, the solution of fundamental problems of provision of safety, risks and security of critically and strategically important infrastructure facilities is based on the analysis and development of fundamental scientific approaches to issues relevant to strength and in-service life, development of engineering methods of calculations and tests, creation of norms and rules regulating design and fabrication of objects of offshore technosphere, ensuring their functioning within identified limits of the design and beyond-design modes and parameters. Nowadays, the analysis and development of all components of the criterial sequence "Strength ! rigidity ! consistency ! in-service life ! reliability ! survivability ! safety ! risk ! protection level (security)" became the basic ones, step by step raising requirements imposed on their routine (normal) functioning and ensuring

*:* (20)

On **Figure 13**: *σai*, (*σaв*)*<sup>i</sup>* is the amplitude of basic and vibratory stresses for *i*-mode; *ni*, (*nв*)*<sup>i</sup>* is the number of cycles for basic and vibratory loads; *Ni, N<sup>в</sup><sup>i</sup>* is the number of destructive cycles; *σт*, *σ<sup>в</sup>* are yield and strength limits; [*σa\**], [*N*] are acceptable tensions amplitudes [*σа*] and endurance capability [*N*] are defined on the basis of traditional calculations with consideration of ultimate factor of safety *n<sup>σ</sup>*

*Diagram that is used for identification of static, cyclic and long-term initial strength and in-service life*

½ �¼ *<sup>σ</sup> <sup>σ</sup> nσ*

yield and ultimate stress limits of the structural material.

*parameters: I—AS, SU,TS, PS, SD; II—SO; III—PA; and IV—V (vibration Δσ*v�*).*

*Hybrid Modeling of Offshore Platforms' Stress-Deformed and Limit States…*

*DOI: http://dx.doi.org/10.5772/intechopen.88894*

realization of design parameters at all stages of life cycle.

and marginal life *nN*

**97**

**Figure 13.**

and operating environment.


With the help of this history of loading set are additional design parameters:


From the analysis of all *i* modes according to standard calculation, the most adverse combinations of *P* and *t* are identified: ð Þ *Pmax*, *tmax* —for heavy loadings and areas of increased temperatures impacts, and ð Þ *Pmin*, *tmin* —for heavy loadings and low temperatures (including cryogenic). A set of such combinations is defined by taking into account the number and geometrical shape of the designed details or elements and number of critically dangerous zones, sections and points in them.

For quantitative evaluation of static and cyclic strength, as well as in-service life [1, 2, 5], experimental and computational diagram in coordinates of "*σa*-*N*" (**Figure 13**) is used.

$$
\sigma\_{\mathfrak{a}} = \mathfrak{e}\_{\mathfrak{a}} \cdot \mathbf{E} = F\{N, r\_{\sigma}, \sigma\_{\mathfrak{r}}, \sigma\_{\mathfrak{a}}\}, \tag{19}
$$

*Hybrid Modeling of Offshore Platforms' Stress-Deformed and Limit States… DOI: http://dx.doi.org/10.5772/intechopen.88894*

#### **Figure 13.**

These dependences are initial for the analysis of strength, in-service life, risk and safety of elements of engineering designs both for their initial states and for the damaged states. Values P, *t* and *τ*, are, as a rule, set by the modes of operation and can be registered by instrumentation and control diagnostic systems or by moni-

At the same time, *σ* and *e* parameters of the general and local stress-deformed states can be obtained with the help of calculation based on the values of parameters *P, t* and *τ* or purposely measured by means of full-scale strain gauging and ther-

In **Figure 12** where a block of external and internal technological operational loadings are presented, the following standard modes of loading of the SP elements are highlighted: assembling (*AS*), tests (*TS*), start-up (*SU*), stationary the mode with maintenance of set operating parameters (*SO*), basic parameters adjustment (*PA*), accident occurrence (*AO*) (including those after of earthquakes), protection systems actuation (*PS*) and shut-down after planned or fault situation (*SD*).

When analyzing the initial and residual strength, service-life, survivability, risk and safety of the oil and gas production platform, the key phase is decomposition of SP and selection and identification of its potentially dangerous critical components, defining the greatest risks of accidents and disasters occurrence. The critical zones of SP components and critical points in them are identified on the basis of experimental and computational studies of stress-deformed and ultimate limit states. In such zones and points, as a rule, processes of local destructions are initiated followed by tramline destructions. At the same time for further experimental and computational evaluations of initial and residual strength, service-life, survivability, risk and safety, the

• time of standard load conditions *τ<sup>i</sup>* and total time pf all modes and blocks of

With the help of this history of loading set are additional design parameters:

• peak-to-peak range of vibration loads *ΔP<sup>В</sup>* (dual- or multi-frequency) loads.

From the analysis of all *i* modes according to standard calculation, the most adverse combinations of *P* and *t* are identified: ð Þ *Pmax*, *tmax* —for heavy loadings and areas of increased temperatures impacts, and ð Þ *Pmin*, *tmin* —for heavy loadings and low temperatures (including cryogenic). A set of such combinations is defined by taking into account the number and geometrical shape of the designed details or elements and number of critically dangerous zones, sections and points in them. For quantitative evaluation of static and cyclic strength, as well as in-service life

[1, 2, 5], experimental and computational diagram in coordinates of "*σa*-*N*"

*σ*<sup>а</sup> ¼ еа � Е ¼ *F N*f g ,*rσ*, *σ*т, *σ*<sup>в</sup> , (19)

following characteristics of history of loading (**Figure 12**) are accepted:

• maximum (minimum) rated temperature *t*max(*t*min);

• peak-to-peak range of forcing Δ*P* and forcing amplitude;

• peak-to-peak range of temperature variations Δ*t*; and

• maximum rated load *P*max*;*

modes *τ*<sup>Σ</sup> (life capacity).

(**Figure 13**) is used.

**96**

toring equipment.

*Probability, Combinatorics and Control*

mometry.

*Diagram that is used for identification of static, cyclic and long-term initial strength and in-service life parameters: I—AS, SU,TS, PS, SD; II—SO; III—PA; and IV—V (vibration Δσ*v�*).*

where *σа*, *еа* are strength and deformation amplitudes; *Е* is the elasticity modulus; *N* is the number of cycles prior to destruction; *r<sup>σ</sup>* is the stress ratio; *σт*, *σ<sup>в</sup>* are yield and ultimate stress limits of the structural material.

On **Figure 13**: *σai*, (*σaв*)*<sup>i</sup>* is the amplitude of basic and vibratory stresses for *i*-mode; *ni*, (*nв*)*<sup>i</sup>* is the number of cycles for basic and vibratory loads; *Ni, N<sup>в</sup><sup>i</sup>* is the number of destructive cycles; *σт*, *σ<sup>в</sup>* are yield and strength limits; [*σa\**], [*N*] are acceptable tensions amplitudes [*σа*] and endurance capability [*N*] are defined on the basis of traditional calculations with consideration of ultimate factor of safety *n<sup>σ</sup>* and marginal life *nN*

$$\left[\sigma\right] = \frac{\sigma}{n\_{\sigma}}, \left[N\right] = \frac{N}{n\_N}.\tag{20}$$

When making stress assessment, the characteristics *σ<sup>В</sup>* and *σ<sup>Т</sup>* have to be set with taking into account service conditions—impact of loading cycling, temperatures and operating environment.

#### **2.10 Criteria of strength, in-service life, safety and protection level (security)**

As it was noted above, the solution of fundamental problems of provision of safety, risks and security of critically and strategically important infrastructure facilities is based on the analysis and development of fundamental scientific approaches to issues relevant to strength and in-service life, development of engineering methods of calculations and tests, creation of norms and rules regulating design and fabrication of objects of offshore technosphere, ensuring their functioning within identified limits of the design and beyond-design modes and parameters. Nowadays, the analysis and development of all components of the criterial sequence "Strength ! rigidity ! consistency ! in-service life ! reliability ! survivability ! safety ! risk ! protection level (security)" became the basic ones, step by step raising requirements imposed on their routine (normal) functioning and ensuring realization of design parameters at all stages of life cycle.

The specified requirements implemented in this knowledge area are imposed on operability of critical structures and expressed by means of the corresponding characteristic parameters of criteria dependences for the above sequence.

destructions or instability; *Lld* is the survivability determined by ability of an object to perform limited functions at damages d and dimensions of defects *l* that are inadmissible according to norms; *PPR* is the reliability determined by ability of an object to perform specified functions in the known or defected state at specified loadings *P* or service-life *RNτ*; and *S* is the safety determined by the ability of an object not to pass into a catastrophic state causing significant damages to the

*Hybrid Modeling of Offshore Platforms' Stress-Deformed and Limit States…*

As it was already mentioned, operational conditions of loads of SP are characterized by a significant amount of various factors and parameters; among them are loading conditions and levels of static and dynamic mechanical loads (**Figure 15a**) and impact of corrosive environment, of external factors, etc. These factors taken together and each one individually can cause significant change of nature of behavior of material, its mechanical properties, ability to resist cyclic deformation in comparison with standard design loading specifications (stationary application of cyclic load, room temperature, etc.) at which standard experiments are usually conducted to define the corresponding characteristics. They also may contribute changes in the corresponding patterns of damages accumulation in the material of the equipment components experiencing their influence when in operation.

Cyclic loading waveform of random operating modes as a rule has more sophisticated nature than widely used in experimental practice sinusoidal or triangular

In some cases, it is obviously possible to schematize and replace actual conditions of loadings by more simple, single-frequency modes. However, generally, the patterns of change of the loadings influencing the structural elements have random

The actual loading modes are schematized (**Figure 15c,e**) in the process of the loading history tracking (**Figure 15e**). Approximation of simulated loading conditions of the equipment as accurate, in respect to reality, as possible for each factor occurring during equipment operation and taking into consideration of impact of these factor on parameters of the characteristic equations and equations describing damages accumulation process is an effective step for adjustment of applied methods for calculations of strength, endurance capability and reliability of the oil and gas production platform components' and hence to identification of really

Cyclic strength *σ* and endurance capacity *N* are defined by the use of the stress-

þ

1 ð Þ <sup>4</sup>*<sup>N</sup> me* � *SK*

*mp* <sup>¼</sup> 0, 36 <sup>þ</sup> <sup>2</sup> � <sup>10</sup>�<sup>4</sup>*σB*, *me* <sup>¼</sup> 0, 132 � *lg S*ð Þ *<sup>K</sup>=σ*�<sup>1</sup> , (23)

*<sup>E</sup>* <sup>1</sup> <sup>þ</sup> <sup>1</sup>þ<sup>2</sup> 1�2

, (22)

100 � *ψ<sup>K</sup>*

where *ψ<sup>K</sup>* is the limit plastic yield of contraction, *SK* is the rupture strength of contraction and *E* is the elasticity modulus defined in the process of standard tests of static tension. Value of index of plastic *mp* and elastic *me* components of deformation *еа* in the absence of direct data on their values can be determined with the help of material yield stress and ultimate stress values, which are as follows

where value of fatigue limit *σ*�<sup>1</sup> can be defined as *σ*�<sup>1</sup> ffi 0, 45*σB*, and rupture strength of contraction *SK*, dependent on ultimate stress limit *σ<sup>B</sup>* and relative narrowing of contraction *ψK*, correspondingly comes from relation *SK* ¼ *σB*ð Þ 1 þ 1, 4 � *ψ<sup>K</sup> :* Parameters *σв*, *ψK*, *SK* in general case are dependent on time *τ*, operational temperature and full size cross-sections of SP bearing elements.

person, the technosphere and the environment.

*DOI: http://dx.doi.org/10.5772/intechopen.88894*

grounded and justified their safe in-service life.

*ea* <sup>¼</sup> <sup>1</sup>

<sup>2</sup> � ð Þ <sup>4</sup>*<sup>N</sup> mp ln* <sup>100</sup>

cycle relationship and the equation

waveforms of cyclic loadings.

nature (**Figure 15b**).

**99**

A "pyramid" of provision of technosphere objects' operability according to the main criteria (**Figure 14**) was constructed based on requirements and parameters providing safe operation conditions of these objects.

From **Figure 14**, it is clear that every element located above the other one is supported by the lower elements, i.e., it is laid on it as on foundation. It eventually means that the solution of the task of security, risk and safety provision has to rest upon the solution of problems of "survivability ! reliability ! in-service life ! rigidity ! consistency ! strength" with passing through traditional stages of their interaction I ! VIII. Fundamental results of identification and provision of strength (stage I) were obtained in the beginning of the nineteenth century and it took a long time, while complete analysis of rigidity and resistance (consistency) (stage II) came to the end by the end of this century. In the twentieth century, the theory and practice of provision of "in-service life ! reliability ! survivability" (stages III, IV, V) were formed. At the end of the last century, the fundamental problem of the analysis and safety and risk provision (stage VI) was formulated for all potentially hazardous civilian and defense objects with transition to management (stage VII) of safety and security according to risks criteria. At these stages, safety and security requirements were formulated like governing, and this provoked development of the new line where consequence "VII ! I" becomes the basis for the future technosphere development. At the beginning of this century, the new task (stage VIII) was formulated and this is provision of safety and security of crucial objects based on anti-accidents and anti-disasters of technogenic, natural and anthropogenic character performance.

According to abovementioned and expressions (1)–(9) and **Figure 14**, the proofness of SP is the function of function (functional) Fz of the basic change in time τ parameters

$$Z\_{\kappa}(\tau) = F\_{\pi} \left\{ R\_{\sigma} \left( \tau \right), R\_{N\tau}(\tau), L\_{ld}(\tau), P\_{PR}(\tau), S(\tau) \right\}, \tag{21}$$

where *Zк* is the proofness determined by the ability of an object to resist to accidents occurrence and development of adverse situations in normal and abnormal conditions; *Rσ* is the strength determined by resistance of the bearing object elements to destruction under normal and emergency impacts; *RN<sup>τ</sup>* is the in-service life (endurance capability) determined by time τ or cycles number *N* prior to

**Figure 14.** *General structure of provision of technosphere objects operability.*

### *Hybrid Modeling of Offshore Platforms' Stress-Deformed and Limit States… DOI: http://dx.doi.org/10.5772/intechopen.88894*

destructions or instability; *Lld* is the survivability determined by ability of an object to perform limited functions at damages d and dimensions of defects *l* that are inadmissible according to norms; *PPR* is the reliability determined by ability of an object to perform specified functions in the known or defected state at specified loadings *P* or service-life *RNτ*; and *S* is the safety determined by the ability of an object not to pass into a catastrophic state causing significant damages to the person, the technosphere and the environment.

As it was already mentioned, operational conditions of loads of SP are characterized by a significant amount of various factors and parameters; among them are loading conditions and levels of static and dynamic mechanical loads (**Figure 15a**) and impact of corrosive environment, of external factors, etc. These factors taken together and each one individually can cause significant change of nature of behavior of material, its mechanical properties, ability to resist cyclic deformation in comparison with standard design loading specifications (stationary application of cyclic load, room temperature, etc.) at which standard experiments are usually conducted to define the corresponding characteristics. They also may contribute changes in the corresponding patterns of damages accumulation in the material of the equipment components experiencing their influence when in operation.

Cyclic loading waveform of random operating modes as a rule has more sophisticated nature than widely used in experimental practice sinusoidal or triangular waveforms of cyclic loadings.

In some cases, it is obviously possible to schematize and replace actual conditions of loadings by more simple, single-frequency modes. However, generally, the patterns of change of the loadings influencing the structural elements have random nature (**Figure 15b**).

The actual loading modes are schematized (**Figure 15c,e**) in the process of the loading history tracking (**Figure 15e**). Approximation of simulated loading conditions of the equipment as accurate, in respect to reality, as possible for each factor occurring during equipment operation and taking into consideration of impact of these factor on parameters of the characteristic equations and equations describing damages accumulation process is an effective step for adjustment of applied methods for calculations of strength, endurance capability and reliability of the oil and gas production platform components' and hence to identification of really grounded and justified their safe in-service life.

Cyclic strength *σ* and endurance capacity *N* are defined by the use of the stresscycle relationship and the equation

$$e\_d = \frac{1}{2 \cdot (4N)^{m\_p}} \ln \frac{100}{100 - \psi\_K} + \frac{1}{(4N)^{m\_e}} \cdot \frac{S\_K}{E\left(1 + \frac{1+2}{1-2}\right)},\tag{22}$$

where *ψ<sup>K</sup>* is the limit plastic yield of contraction, *SK* is the rupture strength of contraction and *E* is the elasticity modulus defined in the process of standard tests of static tension. Value of index of plastic *mp* and elastic *me* components of deformation *еа* in the absence of direct data on their values can be determined with the help of material yield stress and ultimate stress values, which are as follows

$$m\_p = 0, \mathfrak{H} + 2 \cdot 10^{-4} \sigma\_{\mathcal{B}}, m\_\epsilon = 0, 132 \cdot \lg(\mathbb{S}\_K/\sigma\_{-1}),\tag{23}$$

where value of fatigue limit *σ*�<sup>1</sup> can be defined as *σ*�<sup>1</sup> ffi 0, 45*σB*, and rupture strength of contraction *SK*, dependent on ultimate stress limit *σ<sup>B</sup>* and relative narrowing of contraction *ψK*, correspondingly comes from relation *SK* ¼ *σB*ð Þ 1 þ 1, 4 � *ψ<sup>K</sup> :* Parameters *σв*, *ψK*, *SK* in general case are dependent on time *τ*, operational temperature and full size cross-sections of SP bearing elements.

The specified requirements implemented in this knowledge area are imposed on

A "pyramid" of provision of technosphere objects' operability according to the main criteria (**Figure 14**) was constructed based on requirements and parameters

From **Figure 14**, it is clear that every element located above the other one is supported by the lower elements, i.e., it is laid on it as on foundation. It eventually means that the solution of the task of security, risk and safety provision has to rest upon the solution of problems of "survivability ! reliability ! in-service life ! rigidity ! consistency ! strength" with passing through traditional stages of their interaction I ! VIII. Fundamental results of identification and provision of strength (stage I) were obtained in the beginning of the nineteenth century and it took a long time, while complete analysis of rigidity and resistance (consistency) (stage II) came to the end by the end of this century. In the twentieth century, the theory and practice of provision of "in-service life ! reliability ! survivability" (stages III, IV, V) were formed. At the end of the last century, the fundamental problem of the analysis and safety and risk provision (stage VI) was formulated for all potentially hazardous civilian and defense objects with transition to management (stage VII) of safety and security according to risks criteria. At these stages, safety and security requirements were formulated like governing, and this provoked development of the new line where consequence "VII ! I" becomes the basis for the future technosphere development. At the beginning of this century, the new task (stage VIII) was formulated and this is provision of safety and security of crucial objects based on anti-accidents and anti-disasters of technogenic, natural and anthropogenic character performance. According to abovementioned and expressions (1)–(9) and **Figure 14**, the proofness of SP is the function of function (functional) Fz of the basic change in

*Zк*ð Þ¼ *τ Fz* f g *R<sup>σ</sup>* ð Þ*τ* , *RNτ*ð Þ*τ* , *Lld*ð Þ*τ* , *PPR*ð Þ*τ* , *S*ð Þ*τ* , (21)

where *Zк* is the proofness determined by the ability of an object to resist to accidents occurrence and development of adverse situations in normal and abnormal conditions; *Rσ* is the strength determined by resistance of the bearing object elements to destruction under normal and emergency impacts; *RN<sup>τ</sup>* is the in-service life (endurance capability) determined by time τ or cycles number *N* prior to

operability of critical structures and expressed by means of the corresponding characteristic parameters of criteria dependences for the above sequence.

providing safe operation conditions of these objects.

*Probability, Combinatorics and Control*

time τ parameters

**Figure 14.**

**98**

*General structure of provision of technosphere objects operability.*

is written as *g*ð Þ *x* , where *x* ¼ ð Þ *x*1, *x*2, ⋯, *xn* is the vector of variables describing the element state. Then the element failure can be generally described as follows (**Figure 16**)

*Probability density functions for bearing capacity and loading. Probability curves for design parameters at*

Conditional probability of failure in case when the element is under load *L* ¼ *l* is defined by function *FR*ð Þ *x* (this is due to the fact that *FR*ð Þ¼ *x P R*ð Þ < *l* ). Then, using the theorem of total probability, it is possible to write expression for the probability

*P*ð Þ¼ *F*

*Hybrid Modeling of Offshore Platforms' Stress-Deformed and Limit States…*

*DOI: http://dx.doi.org/10.5772/intechopen.88894*

where *R* is the bearing capacity and *x* ¼ *l* is the load.

*σ<sup>M</sup>* ¼

expectation and rms deviation of values *R* и *L*:

ð<sup>∞</sup> �∞

Let us consider the random variable of margin of safety, in-service life and proofness (safety) *M* ¼ *R* � *L* equal to excess of bearing capacity over load. As *R* and *L* are random variables, *M* is also the random variable with mathematical expectation *μ<sup>M</sup>* and rms deviation *σM*. They can be calculated from mathematical

μ<sup>M</sup> ¼ μ<sup>R</sup> � μ<sup>L</sup>

The probability of system failure which is equal to the probability of value *M* be

where *β* is the proofness (safety) index (this variable sometimes is called reliability index) of the element analyzed upon its ultimate limit state *g*ð Þ *x* . Value *β* characterizes the distance of the ultimate limit state surface and can be treated as safety (proofness or security) characteristics of element relative to analyzed failure

If the destruction mechanism relative to excess of maximum permissible load is

considered, then equation of the surface of ultimate limit states takes the form

*σ*2 *<sup>R</sup>* <sup>þ</sup> *<sup>σ</sup>*<sup>2</sup>

q

*<sup>P</sup>*ð Þ¼ *<sup>F</sup> P M*ð Þ¼ <sup>≤</sup><sup>0</sup> *<sup>Φ</sup>* � *<sup>μ</sup><sup>M</sup>*

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

*<sup>L</sup>* þ 2*ρRLσRσ<sup>L</sup>*

*σ<sup>M</sup>* � �

of element failure as follows:

*assessment of chances of failure.*

**Figure 16.**

less or equal to 0.

mechanism.

**101**

*F* ¼ f g *x*j*g*ð Þ *x* ≤0 (25)

*FR*ð Þ *x f <sup>L</sup>*ð Þ *x dx*, (26)

¼ *Φ*ð Þ �*β* , (28)

(27)

#### **Figure 15.**

*Methods of schematization of operational modes of loads. a) Sign-variable and sign-constant service loading mode. b) Random and routine loading modes. c) Service loading modes schematization. d) Modeling mode of random loading with equally probable change of stress amplitude in set range. e) Rainflow technique.*

### **2.11 Probabilistic analysis of strength, in-service life and risks**

Because SP is functioning in the conditions of the high level of uncertainty concerning external impacts during operation period and bearing capacity level changing due to structures degradation, the criteria in expressions (21)–(23) have to be probabilistic [2–6].

Let function of ultimate limit states for the considered platform element is defined by a ratio of bearing capacity and loading *l*. Generally, function of ultimate limit states

$$\mathbf{g}(r,l) = r - l \tag{24}$$

*Hybrid Modeling of Offshore Platforms' Stress-Deformed and Limit States… DOI: http://dx.doi.org/10.5772/intechopen.88894*

**Figure 16.**

*Probability density functions for bearing capacity and loading. Probability curves for design parameters at assessment of chances of failure.*

is written as *g*ð Þ *x* , where *x* ¼ ð Þ *x*1, *x*2, ⋯, *xn* is the vector of variables describing the element state. Then the element failure can be generally described as follows (**Figure 16**)

$$F = \{ \mathfrak{x} | \mathfrak{g}(\mathfrak{x}) \le 0 \}\tag{25}$$

Conditional probability of failure in case when the element is under load *L* ¼ *l* is defined by function *FR*ð Þ *x* (this is due to the fact that *FR*ð Þ¼ *x P R*ð Þ < *l* ). Then, using the theorem of total probability, it is possible to write expression for the probability of element failure as follows:

$$P(\mathbf{F}) = \int\_{-\infty}^{\infty} F\_R(\mathbf{x}) f\_L(\mathbf{x}) d\mathbf{x},\tag{26}$$

where *R* is the bearing capacity and *x* ¼ *l* is the load.

Let us consider the random variable of margin of safety, in-service life and proofness (safety) *M* ¼ *R* � *L* equal to excess of bearing capacity over load. As *R* and *L* are random variables, *M* is also the random variable with mathematical expectation *μ<sup>M</sup>* and rms deviation *σM*. They can be calculated from mathematical expectation and rms deviation of values *R* и *L*:

$$
\mu\_{\rm M} = \mu\_{\rm R} - \mu\_{\rm L}
$$

$$
\sigma\_{\rm M} = \sqrt{\sigma\_{\rm R}^2 + \sigma\_{\rm L}^2 + 2\rho\_{\rm RL}\sigma\_{\rm R}\sigma\_{\rm L}}\tag{27}
$$

The probability of system failure which is equal to the probability of value *M* be less or equal to 0.

$$P(F) = P(M \le 0) = \Phi\left(-\frac{\mu\_M}{\sigma\_M}\right) = \Phi(-\beta),\tag{28}$$

where *β* is the proofness (safety) index (this variable sometimes is called reliability index) of the element analyzed upon its ultimate limit state *g*ð Þ *x* . Value *β* characterizes the distance of the ultimate limit state surface and can be treated as safety (proofness or security) characteristics of element relative to analyzed failure mechanism.

If the destruction mechanism relative to excess of maximum permissible load is considered, then equation of the surface of ultimate limit states takes the form

**2.11 Probabilistic analysis of strength, in-service life and risks**

be probabilistic [2–6].

*Probability, Combinatorics and Control*

limit states

**100**

**Figure 15.**

Because SP is functioning in the conditions of the high level of uncertainty concerning external impacts during operation period and bearing capacity level changing due to structures degradation, the criteria in expressions (21)–(23) have to

*Methods of schematization of operational modes of loads. a) Sign-variable and sign-constant service loading mode. b) Random and routine loading modes. c) Service loading modes schematization. d) Modeling mode of random loading with equally probable change of stress amplitude in set range. e) Rainflow technique.*

Let function of ultimate limit states for the considered platform element is defined by a ratio of bearing capacity and loading *l*. Generally, function of ultimate

*g r*ð Þ¼ , *l r* � *l* (24)

*Probability, Combinatorics and Control*

$$\mathbf{g}\_U(\mathbf{x}) = \mathbf{R} - L,\tag{29}$$

In this case, the probability of failure becomes the function of time:

*Hybrid Modeling of Offshore Platforms' Stress-Deformed and Limit States…*

The probability of system failure is

*DOI: http://dx.doi.org/10.5772/intechopen.88894*

achieve extreme (critical) values *σf*, *ef*.

"*e*–*l*";

**103**

the construction materials, the following are plotted:

stresses "*σ*–*t*" and deformations "*e*–*t*"; and

*Pf*ðÞ¼ *t*

realization or permission to operate running offshore objects.

ð

**2.12 Engineering justification of strength, in-service life and safety**

*g*ð Þ *X*ð Þ*t* ≤0

The identification of time moment *t* <sup>∗</sup> when loading *L t*ð Þ for the first time will exceed the bearing capacity of an element *R t*ð Þ is an important task (**Figure 17**).

As it was noted above, continuously raising requirements to regular (normal) and abnormal functioning are imposed on modern SP. In modern conditions of the analysis and provision of safe operation of technosphere objects, the new task about identification and safety and security provision upon criteria of actual *R*ð Þ*τ* and acceptable [*R*ð Þ*τ* ] bearing capacities are used in expressions (7) and (9). Within that narrative [1–4, 7], only characteristics of safety with the set levels of risks give justification to acceptance (or rejection) of decisions on permission of new projects

Operational impacts on the SP elements in general (periodically arising ice loadings, service, wind and seismic loads) are characterized by the following parameters, in particular numbers of loading cycles N, time of loading *τ* and ambient temperature t. At the same time, *N* and *τ* define in-service life of the examined object, while *t* defines its cold brittleness. The imperfection of bearing structures is defined by the sizes of cracks of *l*, their shape and location. Sizes *l* are initial for determination of objects survivability. Characteristic of flexibility, rigidity, stability *λ* of the bearing component of the analyzed element depends on a shape and dimensions of cross section, length and type of supporting. It defines his stability. External routine and abnormal impacts (including accidents and catastrophic) generate in the analyzed element design stress level *σ* and deformations *e*; they depend on the applied loads (mechanical, temperature, aero hydrodynamic, seismic, etc.), a way of their application, the sizes and shapes of cross-sections. If these impacts increase, then at some point in the bearing elements, the ultimate state limits (critical) are achieved, and these elements are destructed, losing stability and getting inadmissible deformations. Stresses and deformations at this moment

Values of characteristics *σf*, *ef* according to **Figures 12–14** depend on values *N*, *τ*, *l*, *t*, *λ*. Based on these dependences and upon experimental and laboratory studies of

• fatigue curves (live curve) for stresses "*σ*–*N*" and deformations "*e*–*N*";

• crack resistance curve (survivability) for stresses "*σ*–*l*" and deformations

• temperature resistance curve (cold- and heat resistance) in coordinates of

• stress rupture curves for stresses "*σ*–*τ*" and deformations "*e*–*τ*";

where *gXt* ð Þ¼ ð Þ *M t*ð Þ, depending on time proofness margins as per bearing capacity.

*Pf*ðÞ¼ *t PRt* f g ð Þ ≤*L t*ð Þ ¼ *PgXt* f g ð Þ ð Þ ≤ 0 , (33)

*<sup>f</sup> <sup>X</sup>*ð Þ*<sup>t</sup> <sup>x</sup>*ð Þ*<sup>t</sup> <sup>d</sup>x*ð Þ*<sup>t</sup> :* (34)

where *R* is the strength (bearing capacity) of the element and *L* is the maximal load during the analyzed period.

Safety (proofness or security) upon the criterion of exceed of maximal permissible load will be presented by the expression:

$$\beta\_U = \frac{\mu\_R - \mu\_L}{\sqrt{\sigma\_R^2 + \sigma\_L^2 + 2\rho\_{RL}\sigma\_R\sigma\_L}}. \tag{30}$$

If to talk about the fatigue mechanism of element destruction, then equation of the surface of ultimate limit states takes the form *gF*ð Þ¼ *х N* � *n* where N is the number of cycles prior to destruction at the set level of stresses range and n is the number of cycles to which the element is exposed during use. Then the proofness of element upon criterion of fatigue failure will look as follows:

$$\beta\_F = \frac{\mu\_N - \mu\_K}{\sqrt{\sigma\_N^2 + \sigma\_K^2 - 2\rho\_{NK}\sigma\_N\sigma\_K}} \,. \tag{31}$$

Because of hostile environment influence on the OGPF elements and relevant degradation processes in them, the function of element ultimate limit states has to depend on time. In the considered statement, the proofness (safety or security) reserve of a critical element is estimated in the form of *M = R-L,* where *R* is the bearing capacity in critical cross-section and *L* is the loading in the same crosssection. If to consider that both random variables of *R* and *L* in real systems can depend on time, then the bearing capacity can change because of degradation of material properties (corrosion, fatigue, etc.); loading, in its turn, can change due to change of service conditions, of external environment, etc. At that their mathematical expectations *μR*ð Þ*t μL*ð Þ*t* and rms deviations *σR*ð Þ*t σL*ð Þ*t* will change as well. Then the margins of bearing capacity can be written as follows:

$$M(t) = R(t) - L(t). \tag{32}$$

**Figure 17.** *Load and bearing capacity changing in time.*

*Hybrid Modeling of Offshore Platforms' Stress-Deformed and Limit States… DOI: http://dx.doi.org/10.5772/intechopen.88894*

In this case, the probability of failure becomes the function of time:

$$P\_f(t) = P\{R(t) \le L(t)\} = P\{\mathbf{g}(X(t)) \le \mathbf{0}\},\tag{33}$$

where *gXt* ð Þ¼ ð Þ *M t*ð Þ, depending on time proofness margins as per bearing capacity. The probability of system failure is

$$P\_f(t) = \int\_{\mathbf{g}(\mathbf{X}(t))} f\_{\mathbf{X}(t)} \mathbf{x}(t) d\mathbf{x}(t). \tag{34}$$

The identification of time moment *t* <sup>∗</sup> when loading *L t*ð Þ for the first time will exceed the bearing capacity of an element *R t*ð Þ is an important task (**Figure 17**).

#### **2.12 Engineering justification of strength, in-service life and safety**

As it was noted above, continuously raising requirements to regular (normal) and abnormal functioning are imposed on modern SP. In modern conditions of the analysis and provision of safe operation of technosphere objects, the new task about identification and safety and security provision upon criteria of actual *R*ð Þ*τ* and acceptable [*R*ð Þ*τ* ] bearing capacities are used in expressions (7) and (9). Within that narrative [1–4, 7], only characteristics of safety with the set levels of risks give justification to acceptance (or rejection) of decisions on permission of new projects realization or permission to operate running offshore objects.

Operational impacts on the SP elements in general (periodically arising ice loadings, service, wind and seismic loads) are characterized by the following parameters, in particular numbers of loading cycles N, time of loading *τ* and ambient temperature t. At the same time, *N* and *τ* define in-service life of the examined object, while *t* defines its cold brittleness. The imperfection of bearing structures is defined by the sizes of cracks of *l*, their shape and location. Sizes *l* are initial for determination of objects survivability. Characteristic of flexibility, rigidity, stability *λ* of the bearing component of the analyzed element depends on a shape and dimensions of cross section, length and type of supporting. It defines his stability.

External routine and abnormal impacts (including accidents and catastrophic) generate in the analyzed element design stress level *σ* and deformations *e*; they depend on the applied loads (mechanical, temperature, aero hydrodynamic, seismic, etc.), a way of their application, the sizes and shapes of cross-sections. If these impacts increase, then at some point in the bearing elements, the ultimate state limits (critical) are achieved, and these elements are destructed, losing stability and getting inadmissible deformations. Stresses and deformations at this moment achieve extreme (critical) values *σf*, *ef*.

Values of characteristics *σf*, *ef* according to **Figures 12–14** depend on values *N*, *τ*, *l*, *t*, *λ*. Based on these dependences and upon experimental and laboratory studies of the construction materials, the following are plotted:


*gU*ð Þ¼ *x R* � *L*, (29)

<sup>p</sup> *:* (30)

<sup>p</sup> *:* (31)

*M t*ðÞ¼ *R t*ðÞ� *L t*ð Þ*:* (32)

where *R* is the strength (bearing capacity) of the element and *L* is the maximal

Safety (proofness or security) upon the criterion of exceed of maximal permis-

*<sup>β</sup><sup>U</sup>* <sup>¼</sup> *<sup>μ</sup><sup>R</sup>* � *<sup>μ</sup><sup>L</sup>* ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

*<sup>β</sup><sup>F</sup>* <sup>¼</sup> *<sup>μ</sup><sup>N</sup>* � *<sup>μ</sup><sup>K</sup>* ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

Because of hostile environment influence on the OGPF elements and relevant degradation processes in them, the function of element ultimate limit states has to depend on time. In the considered statement, the proofness (safety or security) reserve of a critical element is estimated in the form of *M = R-L,* where *R* is the bearing capacity in critical cross-section and *L* is the loading in the same crosssection. If to consider that both random variables of *R* and *L* in real systems can depend on time, then the bearing capacity can change because of degradation of material properties (corrosion, fatigue, etc.); loading, in its turn, can change due to change of service conditions, of external environment, etc. At that their mathematical expectations *μR*ð Þ*t μL*ð Þ*t* and rms deviations *σR*ð Þ*t σL*ð Þ*t* will change as well. Then

If to talk about the fatigue mechanism of element destruction, then equation of the surface of ultimate limit states takes the form *gF*ð Þ¼ *х N* � *n* where N is the number of cycles prior to destruction at the set level of stresses range and n is the number of cycles to which the element is exposed during use. Then the proofness of

*<sup>L</sup>* þ 2*ρRLσRσ<sup>L</sup>*

*<sup>K</sup>* � 2*ρNKσNσ<sup>K</sup>*

*σ*2 *<sup>R</sup>* <sup>þ</sup> *<sup>σ</sup>*<sup>2</sup>

*σ*2 *<sup>N</sup>* þ *σ*<sup>2</sup>

element upon criterion of fatigue failure will look as follows:

the margins of bearing capacity can be written as follows:

**Figure 17.**

**102**

*Load and bearing capacity changing in time.*

load during the analyzed period.

*Probability, Combinatorics and Control*

sible load will be presented by the expression:

• stability curves (general or local) in coordinates of stresses "*σ*–*λ*" and deformations "*e*–*λ*."

At relatively low levels of external routine impacts when occurring deformations are elastic, the calculations relevant to stresses and deformations have identical results. At the increased abnormal and stress impacts when occurred are general and local plastic deformations, the calculations made with respect to stresses *σ* and deformations *e* are divergent—the values of stresses *σ<sup>f</sup>* happen to be insensitive to *N*, *τ*, *l*, *λ* variation. This fact predetermines the importance of transition from the traditional determined calculations in terms of stresses *σ<sup>f</sup>* to probabilistic calculations in terms of deformations *ef* [2, 5, 7–9].

In case of the integral analysis of strength, in-service life and safety, the deformation curve in true coordinates (the true stress *σ* and true deformations *e*) is presented as follows

$$
\sigma = \sigma\_T \left( e/e\_T \right)^m,\tag{35}
$$

The entire system of experimentally defined (*E*,*σT*, *σB, ψк*) and designed (*m*, *Sк*,

The real bearing SP components have various zones of concentration and

ical decisions (of Neuber type), it is possible to receive correlation of tension concentration factor *K<sup>σ</sup>* and deformations *Ke* in elastoplastic domain with theoretical concentration factor *ασ* in elastic domain, taking into account the relative level of

For existing offshore structures 1 ≤ *ασ* ≤ 5, 1 ≤ *K<sup>σ</sup>* ≤ *ασ*, *ασ < Ke* ≤ *ασ*

For experimental evaluation of size facto impact (sizes *F* of transverse crosssection) on mechanical properties of large-size SP components a set of polynomial

<sup>В</sup> <sup>¼</sup> *<sup>σ</sup>*Вð Þ *<sup>F</sup>*0*=<sup>F</sup> <sup>m</sup>*В*<sup>F</sup>* , *<sup>ψ</sup><sup>F</sup>*

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

where *σ<sup>с</sup>* is the failure stress for the sample with limitation *σ<sup>с</sup>* ≤ (0.9–1.0)*σT*. At the same time, by numerous experiments, it was shown that at change of *l*, *F*, *Q*, and temperature *t*, time *τ*, deformation velocity and stress voluminosity *Iσ*, *De* the basic characteristic *KIc* changes (in the same manner as change other basic design

As the first assumption in technical practice use is made of minimal values of *KIc* depending on the temperature *t* as this not always is counted as safety factor. The most acceptable in comprehensive assessment of strength, in-service life and safety of the SP components is the use of the minimum values defined on cylindrical samples with a circular crack with further calculation of *KIc* value as per basic characteristics *σT*, *m*, *e<sup>к</sup>* with regard to changes caused by variation of parameters *l*, *F*, *Q*, *t*, *τ*, *Iσ*, *De*. In more general case when conditions of linear mechanics of destruction are not satisfied and there are considerable deformations of plasticity and creep, instead of the standard characteristics *KIc* (or critical integral *Jc* and critical cracks opening *δс*), the deformation criterion of *KIec* is developed and implemented, where *KIec* is the critical factor of deformations intensity [5–7]. Factually this factor plays the same role as deformations concentration factor *Ke* in (41) upon condition of similarity of *ασ* and *KI*. At the same time, the modified

*<sup>π</sup><sup>l</sup>* � *F l* f g , *<sup>F</sup>*, *<sup>Q</sup>* <sup>p</sup> , (42)

where *mTF*, *mBF*, *mψ<sup>F</sup>* –characteristics not separate steels, but their groups (as per the stress level and doping level (*mTF* ≈ *mBF* = 0.013, *mψ<sup>F</sup>* = 0.024–0.04). For assessment of survivability characteristics based on crack resistance criteria in presence in the SP bearing structures of cracks like defects, the standard, unified and special tests with variation of cracks sizes *l,* cross-sections *F* and loads technique *Q* shall be conducted. The critical value of the stress intensity factor within the frameworks of the linear fracture mechanics is generally viewed as fundamental

assessment of sensitivity to a factor of tension concentration (in elastic and inelastic areas) and size factor represents essential methodical difficulties and is time-

For big group of constructive metal materials due to use of the modified analyt-

various sizes of cross-sections. Performance of the mechanical tests for

*Hybrid Modeling of Offshore Platforms' Stress-Deformed and Limit States…*

*<sup>T</sup>*) characteristics is identified with regard to results of

f g *Kσ*, *Ke* ¼ *Fk*f g *ασ*, *σ=σT*, *m :* (40)

*2 .*

*<sup>k</sup>* <sup>¼</sup> *<sup>ψ</sup>k*ð Þ *<sup>F</sup>*0*=<sup>F</sup> <sup>m</sup>ψ<sup>F</sup>* , (41)

*mp*, *me*, *mB*, *mψ*, *βT*, *βB*, *ne*\_

equations is recommended:

*σF*

characteristics *σT*, *σB*, *ψк*.).

**105**

consuming.

*<sup>B</sup>*, *ne*\_

the effective stress *σ*/*σ<sup>T</sup>* and work-hardening index *m*

*<sup>T</sup>* <sup>¼</sup> *<sup>σ</sup>T*ð Þ *<sup>F</sup>*0*=<sup>F</sup> mTF* , *<sup>σ</sup><sup>F</sup>*

characteristic of crack resistance at cyclic loading

*KIc* ¼ *σ<sup>c</sup>*

analytical solution with regard to (4.14) type gives dependence

mechanical tests of smooth standard samples.

*DOI: http://dx.doi.org/10.5772/intechopen.88894*

$$m = \lg(\mathbb{S}\_k/\sigma\_T) / \lg(e\_k/e\_T),\tag{36}$$

where *σ*<sup>Т</sup> is the yield stress; *m* is the work hardening exponent (0 ≤ m ≤ 0.3); *Sk* is the tension strength; *eT = σТ/E*; *ek = Sk/E*; and *E* is the elasticity modulus.

The strength-duration curves *σ<sup>t</sup> <sup>B</sup><sup>τ</sup>* and ductility property *ψ<sup>t</sup> <sup>k</sup><sup>τ</sup>* for time *τ* are the basic ones in case of long-term loading at increased temperature

$$
\sigma\_{B\tau}^t = \sigma\_B^t \left(\tau\_o/\tau\right)^m{}\_B, \psi\_{k\tau}^t = \psi\_k^t \left(\tau\_o/\tau\right)^{m\_\psi} \tag{37}
$$

where *τ*<sup>0</sup> is the time of short-time tests (*τ*<sup>0</sup> ≈ 0.05 h); and *mB*, *m<sup>ψ</sup>* are the material characteristics depending on temperature *t* and yield stress *σ<sup>t</sup> <sup>T</sup>* (0 ≤ *m<sup>в</sup>* ≤ 0.08, 0 ≤ *m<sup>ψ</sup>* ≤ 0.15). Then, it is possible to obtain the cyclic stress curve "*σ* <sup>∗</sup> - *N*" as per parameter *τ*.

In estimating the effect of temperatures *t*, different from room temperature *t*<sup>0</sup> = 20°С (both in the range of low climatic temperatures 20°С ≥ *t* ≥ �60°С, including cryogenic range -60°С ≥ *t* ≥ �270°С and elevated 20°С ≤ *t* ≤ 350°С and high temperatures 350°С ≤ *t* ≤ 1000°С), standard tests are carried out in thermocryocameras. In the absence of such tests' results, the estimated dependences of mechanical properties on temperature of *t* °С or *T* °К (*T* = *t* + 273) T are plotted

$$\left\{\sigma\_{m}^{l},\sigma\_{\mathfrak{n}}^{l}\right\} = \left\{\sigma\_{m},\sigma\_{\mathfrak{n}}\right\} \cdot \left\{\beta\_{m},\beta\_{\mathfrak{n}}\right\} \left(\frac{\mathbf{1}}{\mathbf{T}} - \frac{\mathbf{1}}{\mathbf{T}\_{\mathfrak{o}}}\right) \tag{38}$$

where *Т* is the temperature in Kelvin degrees (*Т = t<sup>o</sup>* + 273); and *β<sup>T</sup>* and *β<sup>B</sup>* are designed material characteristics dependent on *σT*. Limiting yielding is calculated via *ψк*, *σ<sup>T</sup>* and *σ<sup>B</sup>* at room temperature.

For dynamically loaded components of the SP, the values of *β<sup>T</sup>* decrease from 120 to 50 with *σ<sup>T</sup>* changing from 300 to 700 MPa, and at increased deformation velocities *<sup>e</sup>*\_ <sup>¼</sup> *de=d<sup>τ</sup>* (10<sup>0</sup> <sup>с</sup> �<sup>1</sup> ≤ *e*\_ ≤ 10<sup>3</sup> с�<sup>1</sup> ), there is increase of yield stress and ultimate stress limit defined experimentally or calculated with the help of polynomial equation

$$
\sigma\_\mathbf{a}^\circ = \sigma\_\mathbf{a} (\dot{\mathbf{e}} / \dot{\mathbf{e}}\_0)^{m\_m}, \sigma\_\mathbf{r}^\circ = \sigma\_\mathbf{r} (\dot{\mathbf{e}} / \dot{\mathbf{e}}\_0)^{m\_T} \tag{39}
$$

Dynamic plasticity performance calculation is done via *ψк*, *σ<sup>T</sup>* and *σB,* with the help of the same relations that are used for temperature effects description. Eqs. (37)–(39) provide possibility to calculate work-hardening index *m* in Eq. (36). *Hybrid Modeling of Offshore Platforms' Stress-Deformed and Limit States… DOI: http://dx.doi.org/10.5772/intechopen.88894*

The entire system of experimentally defined (*E*,*σT*, *σB, ψк*) and designed (*m*, *Sк*, *mp*, *me*, *mB*, *mψ*, *βT*, *βB*, *ne*\_ *<sup>B</sup>*, *ne*\_ *<sup>T</sup>*) characteristics is identified with regard to results of mechanical tests of smooth standard samples.

The real bearing SP components have various zones of concentration and various sizes of cross-sections. Performance of the mechanical tests for assessment of sensitivity to a factor of tension concentration (in elastic and inelastic areas) and size factor represents essential methodical difficulties and is timeconsuming.

For big group of constructive metal materials due to use of the modified analytical decisions (of Neuber type), it is possible to receive correlation of tension concentration factor *K<sup>σ</sup>* and deformations *Ke* in elastoplastic domain with theoretical concentration factor *ασ* in elastic domain, taking into account the relative level of the effective stress *σ*/*σ<sup>T</sup>* and work-hardening index *m*

$$\{K\_{\sigma}, K\_{\epsilon}\} = F\_k\{a\_{\sigma}, \sigma/\sigma\_T, m\}. \tag{40}$$

For existing offshore structures 1 ≤ *ασ* ≤ 5, 1 ≤ *K<sup>σ</sup>* ≤ *ασ*, *ασ < Ke* ≤ *ασ 2 .*

For experimental evaluation of size facto impact (sizes *F* of transverse crosssection) on mechanical properties of large-size SP components a set of polynomial equations is recommended:

$$
\sigma\_T^F = \sigma\_T(F\_0/F)^{m\_{T\overline{\sigma}}},\\\sigma\_\mathbf{B}^F = \sigma\_\mathbf{B}(F\_0/F)^{m\_{\overline{\sigma}}},\\\psi\_k^F = \psi\_k(F\_0/F)^{m\_{\overline{\sigma}}},\tag{41}
$$

where *mTF*, *mBF*, *mψ<sup>F</sup>* –characteristics not separate steels, but their groups (as per the stress level and doping level (*mTF* ≈ *mBF* = 0.013, *mψ<sup>F</sup>* = 0.024–0.04).

For assessment of survivability characteristics based on crack resistance criteria in presence in the SP bearing structures of cracks like defects, the standard, unified and special tests with variation of cracks sizes *l,* cross-sections *F* and loads technique *Q* shall be conducted. The critical value of the stress intensity factor within the frameworks of the linear fracture mechanics is generally viewed as fundamental characteristic of crack resistance at cyclic loading

$$K\_{lc} = \sigma\_c \sqrt{\pi l \cdot F\{l, F, Q\}},\tag{42}$$

where *σ<sup>с</sup>* is the failure stress for the sample with limitation *σ<sup>с</sup>* ≤ (0.9–1.0)*σT*.

At the same time, by numerous experiments, it was shown that at change of *l*, *F*, *Q*, and temperature *t*, time *τ*, deformation velocity and stress voluminosity *Iσ*, *De* the basic characteristic *KIc* changes (in the same manner as change other basic design characteristics *σT*, *σB*, *ψк*.).

As the first assumption in technical practice use is made of minimal values of *KIc* depending on the temperature *t* as this not always is counted as safety factor. The most acceptable in comprehensive assessment of strength, in-service life and safety of the SP components is the use of the minimum values defined on cylindrical samples with a circular crack with further calculation of *KIc* value as per basic characteristics *σT*, *m*, *e<sup>к</sup>* with regard to changes caused by variation of parameters *l*, *F*, *Q*, *t*, *τ*, *Iσ*, *De*. In more general case when conditions of linear mechanics of destruction are not satisfied and there are considerable deformations of plasticity and creep, instead of the standard characteristics *KIc* (or critical integral *Jc* and critical cracks opening *δс*), the deformation criterion of *KIec* is developed and implemented, where *KIec* is the critical factor of deformations intensity [5–7]. Factually this factor plays the same role as deformations concentration factor *Ke* in (41) upon condition of similarity of *ασ* and *KI*. At the same time, the modified analytical solution with regard to (4.14) type gives dependence

• stability curves (general or local) in coordinates of stresses "*σ*–*λ*" and

are elastic, the calculations relevant to stresses and deformations have identical results. At the increased abnormal and stress impacts when occurred are general and local plastic deformations, the calculations made with respect to stresses *σ* and deformations *e* are divergent—the values of stresses *σ<sup>f</sup>* happen to be insensitive to *N*, *τ*, *l*, *λ* variation. This fact predetermines the importance of transition from the traditional determined calculations in terms of stresses *σ<sup>f</sup>* to probabilistic

At relatively low levels of external routine impacts when occurring deformations

In case of the integral analysis of strength, in-service life and safety, the deformation curve in true coordinates (the true stress *σ* and true deformations *e*) is

where *σ*<sup>Т</sup> is the yield stress; *m* is the work hardening exponent (0 ≤ m ≤ 0.3); *Sk*

*<sup>В</sup>*, *ψ<sup>t</sup>*

where *τ*<sup>0</sup> is the time of short-time tests (*τ*<sup>0</sup> ≈ 0.05 h); and *mB*, *m<sup>ψ</sup>* are the material

0 ≤ *m<sup>ψ</sup>* ≤ 0.15). Then, it is possible to obtain the cyclic stress curve "*σ* <sup>∗</sup> - *N*" as per

In estimating the effect of temperatures *t*, different from room temperature *t*<sup>0</sup> = 20°С (both in the range of low climatic temperatures 20°С ≥ *t* ≥ �60°С, including cryogenic range -60°С ≥ *t* ≥ �270°С and elevated 20°С ≤ *t* ≤ 350°С and

*<sup>B</sup><sup>τ</sup>* and ductility property *ψ<sup>t</sup>*

*<sup>k</sup><sup>τ</sup>* <sup>¼</sup> *<sup>ψ</sup><sup>t</sup>*

is the tension strength; *eT = σТ/E*; *ek = Sk/E*; and *E* is the elasticity modulus.

*<sup>B</sup>* ð Þ *<sup>τ</sup>o=<sup>τ</sup> <sup>m</sup>*

high temperatures 350°С ≤ *t* ≤ 1000°С), standard tests are carried out in thermocryocameras. In the absence of such tests' results, the estimated dependences of mechanical properties on temperature of *t* °С or *T* °К (*T* = *t* + 273) T are

<sup>¼</sup> f g *<sup>σ</sup>т*, *<sup>σ</sup>*<sup>в</sup> � f g *<sup>β</sup>m*, *<sup>β</sup>*<sup>в</sup>

�<sup>1</sup> ≤ *e*\_ ≤ 10<sup>3</sup> с�<sup>1</sup>

<sup>в</sup> <sup>¼</sup> *<sup>σ</sup>*вð Þ <sup>е</sup>\_*=*е\_<sup>0</sup> *<sup>m</sup>*ев , *<sup>σ</sup>*<sup>е</sup>\_

help of the same relations that are used for temperature effects description.

where *Т* is the temperature in Kelvin degrees (*Т = t<sup>o</sup>* + 273); and *β<sup>T</sup>* and *β<sup>B</sup>* are designed material characteristics dependent on *σT*. Limiting yielding is calculated

For dynamically loaded components of the SP, the values of *β<sup>T</sup>* decrease from 120 to 50 with *σ<sup>T</sup>* changing from 300 to 700 MPa, and at increased deformation

ultimate stress limit defined experimentally or calculated with the help of polyno-

Dynamic plasticity performance calculation is done via *ψк*, *σ<sup>T</sup>* and *σB,* with the

Eqs. (37)–(39) provide possibility to calculate work-hardening index *m* in Eq. (36).

basic ones in case of long-term loading at increased temperature

characteristics depending on temperature *t* and yield stress *σ<sup>t</sup>*

*<sup>σ</sup>* <sup>¼</sup> *<sup>σ</sup><sup>T</sup>* ð Þ *<sup>е</sup>=еТ <sup>m</sup>*, (35)

*<sup>k</sup><sup>τ</sup>* for time *τ* are the

*<sup>T</sup>* (0 ≤ *m<sup>в</sup>* ≤ 0.08,

(38)

*<sup>k</sup>*ð Þ *<sup>τ</sup>o=<sup>τ</sup> <sup>m</sup><sup>ψ</sup>* (37)

*m* ¼ *lg S*ð Þ *<sup>k</sup>=σ<sup>T</sup> =lg e*ð Þ *<sup>k</sup>=eT* , (36)

1 <sup>Т</sup> � <sup>1</sup> То 

), there is increase of yield stress and

<sup>т</sup> <sup>¼</sup> *<sup>σ</sup>*тð Þ <sup>е</sup>\_*=*е\_<sup>0</sup> *mT* (39)

deformations "*e*–*λ*."

*Probability, Combinatorics and Control*

presented as follows

parameter *τ*.

plotted

calculations in terms of deformations *ef* [2, 5, 7–9].

The strength-duration curves *σ<sup>t</sup>*

*σt <sup>B</sup><sup>τ</sup>* <sup>¼</sup> *<sup>σ</sup><sup>t</sup>*

*σt <sup>m</sup>*, *σ<sup>t</sup>* в

*σ*е\_

via *ψк*, *σ<sup>T</sup>* and *σ<sup>B</sup>* at room temperature.

velocities *<sup>e</sup>*\_ <sup>¼</sup> *de=d<sup>τ</sup>* (10<sup>0</sup> <sup>с</sup>

mial equation

**104**

$$
\overline{K}\_{le} = \overline{K}\_{I}^{p\_{le}},
\tag{43}
$$

Theoretical and practical solutions of the considered problems of strength, inservice life, reliability, crack resistance were already performed for such high-risk objects as nuclear reactors, hydraulic and thermal power stations, aircraft, main

The ground for the analysis and risk management directed to quantitative evaluation of critical and acceptable risks is based on the matrix of risks. Qualitative and quantitative risk assessment is based on the standard matrices of criticality determined by probabilities of adverse events occurrence (destructions, failures, etc.) and consequences of these events. However, within risk matrixes, the mechanisms of material and the bearing SP components degradation relevant to the erosion and

The listed above approaches, methods, criteria, design schemes and calculation dependences give the chance to carry out assessment of SP technical condition and

**3. Development of methods of calculations and justification of strength,**

**3.1 Techniques of provision and enhancement of strength, in-service life and**

Taking into account a possibility of reaching in time of the ultimate limit states in the wide range of loading parameters, further it is required to define the following groups of situations occurring during SP functioning as presented in **Table 2**. Each class of situations corresponds to diminution of safety level of the analyzed objects while diminution of safety level can be estimated on expressions (1)–(9) as

According to **Table 2**, the last three abovementioned groups of the situations (T5, T4, T3) occurring during objects functioning can be referred to a kind of the risks which are monotonously increasing up to critical values. Such risks, mainly, are caused by the controlled processes of damages and degradations of physicalmechanical properties of material relevant to its aging. The first two groups (T2,

extreme impact parameters (earthquakes, tsunami, acts of terrorism and military

modeling, diagnostics, monitoring and protection. In this case, classic methods of a material consumption justification, constructability and efficiency are insufficient.

In case of use of foreign and domestic safety standards for risk analysis, the

T1) correspond to the occurrence of the most dangerous situations with

actions). These cases require use of the most difficult calculations, tests,

In such statement, the approaches presented in clauses 2.9–2.12 have to be

approaches given in [1, 2, 10, 11] can be rather efficient:

*<sup>i</sup>*ð Þ*t* of objects operation on a specified time interval of opera-

*<sup>i</sup>*ð Þ*t* . At the same time, the condition of safety provision

*<sup>t</sup>*ð Þ*t* is designed risk for the moment of operation *t* for mode *i*

*<sup>i</sup>*ð Þ*t* are calculated as product of the probability of

*<sup>t</sup>*ð Þ*t* where *Rc*ð Þ*t* is critical (inadmissible,

*<sup>i</sup>*ð Þ*t* by economic losses values as

pipelines and unique engineering constructions.

*DOI: http://dx.doi.org/10.5772/intechopen.88894*

*Hybrid Modeling of Offshore Platforms' Stress-Deformed and Limit States…*

corrosion processes are considered.

**in-service life and safety**

tion. Quantitative values of risks *R<sup>э</sup>*

takes the following form *nR* <sup>¼</sup> *Rc*ð Þ*<sup>t</sup> <sup>=</sup>R<sup>э</sup>*

and *nR* is the safety margin as per risks.

occurrence of each of the specified situation *i* - *Р<sup>э</sup>*

risks monitoring.

**safety**

per values of risks *R<sup>э</sup>*

per analyzed situation *U<sup>э</sup>*

unacceptable risk), *R<sup>э</sup>*

implemented.

**107**

**3.2 Risk-based inspections**

where *KI* <sup>¼</sup> *KI <sup>σ</sup><sup>т</sup>* ; *<sup>σ</sup><sup>n</sup>* <sup>¼</sup> *<sup>σ</sup>n=σ*т; *<sup>σ</sup><sup>т</sup>* is the yield stress; *Pke* <sup>¼</sup> <sup>2</sup>�*n*ð Þ <sup>1</sup>�*<sup>m</sup>* ð Þ <sup>1</sup>�*σ<sup>n</sup>* <sup>1</sup>þ*<sup>m</sup>* is the generalized parameter depending on work-hardening index *m* and relative level of rated stresses; *m* is the work-hardening index for deformation curve; and *n* is the characteristic of structural material type *n* ≈ 0.5.

The value of stress intensity factor in terms of operation at stress *σ<sup>n</sup>* with regard t0 (4.14) equals to

$$
\overline{K}\_l = \overline{\sigma}\_\pi \sqrt{\pi l \cdot F\{l, F, Q\}}.\tag{44}
$$

Expressions (41) and (44) make it possible to get conditions of local destruction—crack formation (41) and its development according to (43).

In presence of cracks and use of local criterion obtained is expression to plot the fracture diagram connecting increment of the crack length Δ*l* with the rated stress value and designed parameters of mechanical properties

$$
\Delta l = \frac{1}{2\pi} \left( \frac{\overline{K}\_{le}}{\overline{e}\_f} \right). \tag{45}
$$

where *ef* <sup>¼</sup> <sup>1</sup> *e*т ð Þ е*<sup>k</sup>* .

If loading process is cyclic, the value Δ*l* is equivalent to crack increment in preplanned cycle *<sup>Δ</sup><sup>l</sup>* <sup>¼</sup> *dl dN*, and the main parameter of loading appears to be the peak-to-peak range of deformations intensity factor *ΔK*ð Þ *<sup>N</sup> Ie* in this very loading cycle *N* with a variable work-hardening index *m* = *m*(*N*). The value *m* = *m*(*N*) depends on cyclic properties of materials that can be as follows:


$$\frac{dl}{dN} = \frac{1}{2\pi} \left(\frac{\Delta \overline{K}\_{le}^{(k)}}{\overline{e}\_f}\right)^2 = \frac{1}{2\pi \overline{e}\_f^2} \left(\Delta \overline{K}\_{le}^{(k)}\right)^2 = \mathbf{C}\_\epsilon \left(\Delta \overline{K}\_I\right)^{m'}.\tag{46}$$

Expression (46) with regard to expressions (43), (44) is similar to known Paris-Erdogan equation when *С* and *mk* are material constants; however, in expression (46), the values *C* and *m* are variables and are calculated. Mechanical tests for identification of *KIe*, *KIec*, *dl=dN* within the frames of nonlinear destruction mechanics are more comprehensive than those in linear destruction mechanics when identified are values of *KIc* and *dl/dN*. In non-routine events, emergency and catastrophic situations in nonlinear setting of the problem analyzed are the following essential effects of redistribution of the local plastic deformations and creep deformations depending on *m*, *t*, *τ*, *N*, *F*, *Iσ*, *De* in case of probabilistic approach. Noted complexity is overcome within deformation destruction criteria at setting of the general problems of strength, in-service life, reliability, survivability, risks, safety and SP equipment protection.

*Hybrid Modeling of Offshore Platforms' Stress-Deformed and Limit States… DOI: http://dx.doi.org/10.5772/intechopen.88894*

Theoretical and practical solutions of the considered problems of strength, inservice life, reliability, crack resistance were already performed for such high-risk objects as nuclear reactors, hydraulic and thermal power stations, aircraft, main pipelines and unique engineering constructions.

The ground for the analysis and risk management directed to quantitative evaluation of critical and acceptable risks is based on the matrix of risks. Qualitative and quantitative risk assessment is based on the standard matrices of criticality determined by probabilities of adverse events occurrence (destructions, failures, etc.) and consequences of these events. However, within risk matrixes, the mechanisms of material and the bearing SP components degradation relevant to the erosion and corrosion processes are considered.

The listed above approaches, methods, criteria, design schemes and calculation dependences give the chance to carry out assessment of SP technical condition and risks monitoring.
