In this paper, the kinematic and static solutions for solving the static response of the beam column with nonlinear springs are presented by adopting the extended linear matching method (LMM). The extended LMM can be used to predict the displacement response of the beam-column system consisting of perfectly plastic and strain-softening materials. It is found that the kinematic solution generated by the extended LMM demonstrates a monotonic decrease for perfect plastic materials with certain restrictions on the yield surface. The potential energy of the system is proved to decrease with iterations for both perfect plastic and strain-softening materials if the loading multiplier remains constant. The extended LMM method is then applied to analyse the response of the pile system in a 3-leg offshore platform. An incremental procedure is recommended to determine the peak load for the soil exhibiting strain-softening. A displacement-control approach is used with the loading multiplier obtained from the variation of the potential energy. Good convergence of the method is obtained.

Classical limit analysis has been conducted extensively in the engineering due to its sound theoretical fundamental and practicability since 1950s. The upper and lower bound theorems provide the basis for bracketing the plastic limit load of a structural system that consists of the perfectly plastic materials subjected to small strain [

For materials exhibiting strain-softening, the fundamental assumptions of limit analysis are violated. Hence, the analysis in general has to be conducted in an incremental way in order to determine the peak load since the plastic response of a material depends both on the accumulated plastic strains and the strain history. The computational issues associated with strain-softening materials are briefly discussed by Barrera et al. [

Recent development of structural numerical methods extended limit analysis to strain-softening materials by employing the mathematical programming with equilibrium constraint (MPEC) method [

The objective of this paper is to derive FE-based kinematic and static solutions for bounding the load-displacement responses of pile foundations under monotonic loading by adopting the extended linear matching method (LMM). The general procedure of the extended LMM is presented first, and then, a kinematic solution from the variation of potential energy and a static solution by satisfying the material constitutive law are derived. The convergence of the extended LMM method for strain-softening materials is discussed thereafter. A case study is conducted on an actual offshore platform failed in pile foundation overturning in Hurricane Ike [

As shown in Figure

Pile-soil interaction model.

Only a monotonic loading is considered in the current study, and the total loading exerted on the pile can be expressed as

Prager’s [

A displacement-control loading is adopted to determine the full load-displacement response of the pile with strain-softening

Step 1: during the

Step 2: the elastic solution is scaled to satisfy the specified displacement criterion, i.e., the analysis is conducted through a displacement-control way that one component of

where the subscript “kin” indicates that the solution is kinematic based on the variation of the potential energy,

Step 3: by imposing the constitutive laws that all the stress points corresponding to the exact solution can never pass beyond the constitutive curves, a simple approximation is used to scale the elastic solution to obtain a static load multiplier as follows:

where the subscript “

Step 4: the stiffness of each element is updated based on the generalized strain

Let

Schematic representation of the linear matching process.

Once a converged solution is obtained at the prescribed displacement, the prescribed displacement is increased and the iteration is then repeated to find a converged solution for the new prescribed displacement. The convergence is obtained when the kinematic solution equals the static solutions yielding the exact solution. The kinematic solution approaches the exact solution from the upper side, while the static solution approaches the exact solution from the lower side. Thus, the load-displacement response of the system can be determined accordingly. To start the iteration at a specific prescribed displacement, an arbitrary load multiplier

The proposed procedure retains the general characteristics of the LMM presented by Barrera et al. [

Without geometric nonlinearities, the kinematic solution in equation (^{th} iteration, when the scaling factor ^{th} element at ^{th} iteration, and

For the static solution, the linear scaling in equations (_{c} is scaled to be

For rigid perfectly plastic materials (see Figure _{c} is scaled to be

For the nonlinear or strain-softening materials, the kinematic and static solutions are not strictly the upper and lower bound solutions. Nevertheless, the determination of the kinematic solution has a sound theoretical basis that the first variation of the potential energy needs to be zero for an exact solution. Besides, the static solution in general ensures the constitutive laws and will not be larger than the previous kinematic solution, i.e.,

The geometric nonlinearity for pile foundation commonly refers to the

Step 1:determine the axial load on the pile cross section without considering the

Step 2: repeat the procedure as described in Section

where ^{th} pile cross section, and ^{th} pile cross section that can be taken as the difference of the rotation at the two nodes of a cross section. With the above modifications, the

From the linear matching process in equation (

For the current beam-column problems without the

As shown in Figure

Schematic for area calculation.

Using the definition in equations (

The principle of minimum potential for the (^{th} elastic solution gives

Manipulating equation (

Combining equations (

Inequality (

The left-hand side of expression 13 is the potential energy of the system at the (^{th} iteration, while the right-hand side is the potential energy of the system at the ^{th} iteration. Hence, the matching process reduces the system potential energy monotonically for load-control analyses and ensures the convergence. The equality in equation (

The shortcoming of the load-control analysis is that it is unable to determine the postpeak part of the global load-displacement response of a pile foundation. Instead, the displacement-control analysis is preferred to trace the postpeak response. However, the scaling of the linear-elastic solution in the displacement-control analysis to

The proposed kinematic and static solutions are used to determine the global load-displacement response of the pile foundation system of an actual 3-leg offshore platform (Platform

The description of Platform

Pile foundation plan view of Platform

Pile diameter and wall thickness schedule.

Segment no. | Pile | Pile | ||
---|---|---|---|---|

Penetration below mudline (m) | Wall thickness (mm) | Penetration below mudline (m) | Wall thickness (mm) | |

5 | 0 to 15.2 | 44.1 | 0 to 16.8 | 37.8 |

4 | 15.2 to 27.4 | 37.8 | 16.8 to 22.9 | 31.5 |

3 | 27.4 to 30.5 | 31.5 | 22.9 to 65.5 | 25.2 |

2 | 30.5 to 79.2 | 25.2 | 65.5 to 67.1 | 31.5 |

1 | 79.2 to 80.8 | 31.5 | N/A |

Design undrained shear strength and submerged unit weight profiles.

Take Pile

If the

Variations of the kinematic and the static loading multipliers for

Variation of the convergence rate.

Figure

Variations of the kinematic and the static loading multipliers for

For the soil with strain-softening behavior, a detailed load-displacement analysis using the displacement-control approach is adopted. The displacement in one degree of freedom is chosen as an independent variable. The proposed procedure in Section

Lateral load-displacement response of a single pile.

Figure

Axial load-displacement response of a single pile.

Figure

Variations of the kinematic and the static loading multipliers for the pile system.

Figure

Pile system horizontal load-displacement curves.

This paper presents kinematic and static solutions for solving the static response of the beam column with nonlinear springs using the linear matching method (LMM). For perfectly plastic materials, the convergence will be guaranteed if the kinematic solution decreases monotonically to the least upper bound solution allowed by the FE formation. For strain-softening materials, although the convergence is not proved directly, the difference between the kinematic and static solutions in each step of iteration can be used as an indicator of the convergence.

The application of the method in offshore pile foundations reveals that a good convergence can be obtained. For the steel pile and soil with perfectly plastic materials, the ultimate loads for a single pile and a pile foundation system can be derived in one step by iterations. For soils exhibiting strain-softening, it is convenient to conduct an incremental analysis to determine the peak load. For a single pile, the pile resistance is upper-bounded by the pile limiting load using the peak strength and lower-bounded by the pile limiting load using the residual strength. However, for a pile foundation system, the system load may drop below the pile system limiting load using the residual strength as the capacities of piles are not mobilized simultaneously. The current analysis indicates that the extended LMM proposed in this study has a higher convergence, and further applications in offshore pile foundation analysis can be expected.

Following Chen and Han [

The normalized moment-thrust-curvature relation is given in Figure

Representation of moment-thrust-curvature relation.

The data used to support the findings of the study are available from the corresponding author upon request.

The authors declare that they have no conflicts of interest.

The authors would like to express the special acknowledgement to Dr. Jinbo Chen, formerly with The University of Texas at Austin, for discussing the case study platform and for sharing his knowledge on the offshore piles and the linear matching method. Without his help, this study would not be completed. Grateful acknowledgment is made to National Natural Science Foundation of China (no. 52078288) to support this research.