#### Abstract

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.

#### 1. Introduction

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 [15]. An alternative for implementing limit analysis is the linear matching method (LMM) [6, 7]. The LMM is a class of programming methods based on elastic solutions with spatially distributed material stiffness. In the LMM, finite element (FE) methods are usually adopted to find solutions that satisfy equilibrium and compatibility under the given boundary conditions on the assumption of the linear-elastic material. The plastic behavior of the certain portion of the system is simulated by the sequential adjustment of the corresponding stiffness iteratively. For the system consisting of perfectly plastic materials without geometric nonlinearities, Ponter et al. [6] demonstrated that the displacement field generated from an elastic solution can be treated as a plastic displacement field in the limiting case. Therefore, the upper and lower bound solutions can be obtained simultaneously through the LMM, and the exact solution associated with the displacement field allowed by the FE formation is then guaranteed by the convergence of the upper and lower bound solutions with certain restrictions on the yield surface (e.g., an incompressible material under the von Mises criterion).

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. [8] from the material constitutive relation aspect and by Zienkiewicz and Taylor [9] from the numerical calculation aspect. For a specific class of proportional loading that loads can be expressed by a load multiplier (e.g., the pushover analysis), the softening branch in the global load-displacement curve can be simulated by a displacement-control approach, with the prescribed displacement in one degree [10, 11] or the prescribed incremental displacement norm [12, 13].

Recent development of structural numerical methods extended limit analysis to strain-softening materials by employing the mathematical programming with equilibrium constraint (MPEC) method [14, 15]. Barrera et al. [8] proposed a modified LMM to determine the peak load that can be applied to a structural system composed of strain-softening materials. However, the convergence of the solution is not guaranteed. Therefore, the motivation of this paper is to determine the peak load that a system with strain-softening materials can resist. As the first step in extending the LMM to strain-softening materials, this paper focuses on a classical problem, i.e., the static response of a beam column with nonlinear springs, to demonstrate the potential of the proposed 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 [1618]. The proposed method falls into the general group of determining the load multiplier in global displacement-softening problems in FE analysis. The novel features of the proposed method on the potential energy-based kinematic solution (mimic to the upper bound solution in limit analysis) and the yield criteria-based static solution (mimic to the lower bound solution in limit analysis) for bracketing the exact load-displacement response are discussed.

#### 2. Description of the Extended LMM Method

##### 2.1. Simplified Pile-Soil Interaction Model

As shown in Figure 1, a pile is assumed to satisfy the Bernoulli–Euler beam theory and then is discretized into a number of finite elements where two nodes are allocated at each end. Moreover, the interaction between pile-soil is simplified by a set of one-dimensional springs. To limit the scope of this study, only steel pipe is considered and the pile cross-sectional bending moment-thrust-curvature relation is taken following Chen and Han [19] (see Appendix). It is worthwhile to notice that the proposed method in the current study is also applicable to other pile materials as long as the bending moment-thrust-curvature curves can be obtained. The soil resistance is taken to follow the recommendations of API RP2GEO [20], i.e., the t-z (pile shaft) and q-z (pile end) curves acting along the pile vertical axis and p-y curves for providing lateral resistance to the pile. Such uncoupled behavior of soil axial and lateral resistance is consistent with conventional analysis procedures for offshore piles, where the majority of the axial capacity is contributed by the lower portion of the pile and the lateral capacity is developed within top 10–20 diameters of the pile below the mudline [21, 22].

##### 2.2. Generalized Stress and Strain

Prager’s [19] work-conjugate generalized stress and generalized strain are adopted in this study, where the subscript indicates the element in the discretized model. For the convenience of the notation, both pile elements and soil springs are denoted by the structural elements in the following context. Thus, the generalized stresses for a structural element are dependent on the pile cross-sectional axial force, the cross-sectional bending moment, and the axial and lateral soil resistances. The generalized strains are associated with pile axial deformation, bending curvature, and axial and lateral soil spring deformations.

##### 2.3. Procedure for Load-Displacement Response

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 can be chosen. The final solution is independent of since only provides an initial solution to be scaled and the analysis is controlled by the displacement criterion.

The proposed procedure retains the general characteristics of the LMM presented by Barrera et al. [8], with the extension of the LMM to satisfy the generalized stress-strain relation giving the full load-displacement response of a pile foundation, from which the peak capacity of the pile foundation can be evaluated from a displacement-control way. This full analysis loses the merit of the direct searching for the peak capacity, but gives the full load-displacement response as well as the peak capacity. Therefore, the proposed method provides an alternative for solving pile-soil interaction problems when compared to the conventional methods.

##### 2.4. Discussion on Kinematic and Static Solutions

Without geometric nonlinearities, the kinematic solution in equation (1) is obtained from the necessary condition for an exact solution: the first variation of the system potential energy is zero. At the kth iteration, when the scaling factor and the associated internal displacement have been determined, the potential energy of the system is given as follows:where is the generalized stress corresponding to the generalized strain from the generalized stress-strain relation, is the strain energy of the jth element at kth iteration, and is determined by setting as follows:where and are the arbitrary pair of external displacement and internal generalized strain, respectively since and are determined by the linear-elastic analysis and must be kinematically admissible. Thus, and can be related by a compatible matrix B such that . Hence, equation (5) can be manipulated to equation (6), from which equation (1) is obtained:

For the static solution, the linear scaling in equations (2a) and (2b) may not necessarily bring all the stress points below the constitutive curve for a concave-down constitutive curve in the prepeak zone. Besides, the constant load Pc is scaled to be , and the original problem has been changed unless ; however, the linear scaling for the static solution does not affect the convergence as it will be discussed below.

For rigid perfectly plastic materials (see Figure 2), is independent of . In this case, the left-hand side of equation (1) gives the input work to the pile-soil system, while the right-hand side gives the energy dissipation in the system. Thus, equation (1) gives the upper bound solution in classical limit analysis, and the solution is independent of . Nevertheless, in general, equations (2a) and (2b) does not constitute a lower bound solution to the rigid-perfectly-plastic system as would be expected since the constant load Pc is scaled to be as discussed before. Thus, the internal generalized scaled stresses are in equilibrium with the external load . Therefore, equations (2a) and (2b) give a lower bound load multiplier of only when or . The former case corresponds to zero constant gravity loads; the latter case implies that the lower bound solution coincides with the upper bound solution (see equations (2a) and (2b) with ), and the exact solution is obtained. Hence, the closeness of the upper bound solution and the static solution indicates the convergence of the solution for rigid perfectly plastic materials.

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., as can be seen from the scaling in equations (2a) and (2b). When , all the necessary conditions for an exact solution (compatibility, equilibrium, and constitutive laws) are satisfied. Hence, the solution will be exact for a specific FE formation (i.e., a specific mesh field). Therefore, the closeness of the kinematic and static solution also indicates the convergence of the solution for nonlinear or strain-softening materials.

##### 2.5. Geometric Nonlinearity

The geometric nonlinearity for pile foundation commonly refers to the effect that the axial load will cause secondary bending moment on the pile cross section for a laterally deformation pile. In this case, the axial load on the pile cross section needs to be determined in order to incorporate the effect, and equation (1) needs to be modified accordingly. A revised procedure is described as follows:Step 1:determine the axial load on the pile cross section without considering the effect using the proposed procedure as described in Section 2.3.Step 2: repeat the procedure as described in Section 2.3 to incorporate the effect with the following two modifications: (i) the elastic solution is obtained under the consideration of the effect, i.e., the element stiffness of a pile cross section needs to subtract the geometric stiffness that depends on the axial load on the pile cross section; (ii) to estimate the kinematic solution, the input energy from the secondary bending moment applied by the external axial load needs to be considered in equation (1) resulting in where is the secondary bending moment caused by the axial load at the jth pile cross section, and depends on the pile lateral deflection (i.e., ) and the applied axial load; is the rotation of the jth 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 effect can be incorporated and a convergent solution can be found through iterations.

#### 3. Convergence Discussion

From the linear matching process in equation (3), it can be seen that the updated stiffness is the secant stiffness at . Therefore, the proposed method bears the similarity to the incremental-secant (or quasi-Newton) method, and a superlinear asymptotic convergence rate is expected [9]. For rigid perfectly plastic materials, it has been proved that the upper bound solution obtained from the linear matching process in equation (3) decreases monotonically with iterations. However, such feature may not exist for strain-softening materials.

For the current beam-column problems without the effect, if the load multiplier remains constant during iterations (i.e., the analysis is load-control, and remains 1.0), the linear matching process reduces the system potential energy with each iteration. This statement remains valid both for strain-hardening and strain-softening pile materials and soil springs provided that the pile material and soil springs satisfy the second postulate of the classical plasticity theory [24] in the prepeak region in the generalized stress-strain curve, i.e., the constitutive function of the pile material and soil springs is concave-down with respect to the generalized strains nonincreasing with the increase of generalized strains. The proof is provided below.

As shown in Figure 3, the area of the trapezoid a-b-d-e is not less than the area of polygon a-b-e-c because the constitutive curve is concave-down in the prepeak zone. Therefore, the following inequality holds:

Using the definition in equations (3), equation (8) can be manipulated to equation (9) as follows (note that ):

The principle of minimum potential for the (k+1)th elastic solution gives

Manipulating equation (10) yields

Combining equations (9) and (11) gives

Inequality (12) is equivalent to equation (13) as follows:

The left-hand side of expression 13 is the potential energy of the system at the (k+1)th iteration, while the right-hand side is the potential energy of the system at the kth iteration. Hence, the matching process reduces the system potential energy monotonically for load-control analyses and ensures the convergence. The equality in equation (13) only occurs when the solution converges such that .

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 produces a change to the potential energy, and the effect violates the principles of the proof presented above. Therefore, the monotonic reduction of the potential energy with iterations has not been approved and may no longer remain valid. Nevertheless, the energy conservation-based kinematic solution (i.e., equations (1) and (7)) still preserves a strong convergence characteristic based on the actual implementation of the proposed method as will be presented in the case study in the next section.

#### 4. Case Study

##### 4.1. Platform Description

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 X) that failed in Hurricane Ike in the Gulf of Mexico in 2008. Both two-dimensional and three-dimensional upper and lower bound limit analysis techniques were used comprehensively to determine the pile system capacity of Platform X in the literature [16, 17, 25], and the system reliability technique was adopted to determine the survival/failure probability of the pile system of this platform [26]. The kinematic and static solutions derived from the current study extend the limit analysis and provide additional tools to determine the full load-displacement response of the pile system of Platform X. The extended LMM method proposed in the current study is implemented using the programming software MATLAB to analyse response of the pile foundation.

The description of Platform X is presented here as follows [17]. Platform X is a 3-leg jacket platform (tripod) located in the coast of Louisiana in about 110 m water depth. A plane view of the pile foundation is shown in Figure 4. Platform X failed in Hurricane Ike in 2008 due to the pull-out of the pile under tensile loading. The pile are steel pile, two of which (Piles B and C in Figure 4) are battered with the vertical-to-horizontal ratio of 5 : 1. Pile A is vertical. A well conductor presents just to the north of Pile A; however, this well conductor is neglected in the foundation analysis since the well conductor was not connected in the vertical direction with the platform. The yield strength of the pile is 248 MPa. The pile diameters, embedment lengths, and wall thickness schedules are shown in Table 1. The soil at the platform site is mainly clay. The design undrained shear strength and submerged unit weight profiles are shown in Figure 5. The pile cross-section bending moment-thrust-curvature relation is taken from Chen and Han [19], which consists of a linear-elastic portion, a perfectly plastic portion, and a nonlinear transition between the above two portions (see Appendix). The t-z and p-y curves for soil springs are taken following API RP2GEO [20].

##### 4.2. Single Pile Response

Take Pile A as an analysis example. Pile A is discretized into 60 equal length beam elements. The head of Pile A is assumed to be fixed against rotation and subjected to axial and lateral combined loading. The ratio between the axial and lateral loads is , which remains constant during the loading history. Hence, for small values of , a lateral failure mechanism will be expected, while for larges value of , an axial failure will occur.

###### 4.2.1. Single Ultimate Capacity

Figure 8 shows the result for . Both of the kinematic and the static solutions oscillate initially and tend to converge monotonically after 6 iterations. The difference between the two solutions is about 0.5% after 30 iterations and almost become zeros after 50 iterations. The oscillation of the kinematic solution is due to the adopted yield surface of the steel pile. The cross-sectional ultimate failure surface of the steel pile is shown in equation (A.1). This surface is adopted as the yield surface in the limit analysis. However, this yield surface is categorized as a general yield surface involving the interactions of axial load and bending moment. As the contour of constant complementary potential passing through a stress point on this yield surface does not enclose the whole yield surface, it does not meet the necessary condition for a monotonic decreasing upper bound solution in the limit analysis using the LMM [7]. Despite of this deficiency, the two solutions converge in all the cases studied in the current study.

Figure 10 shows the axial load-displacement response of a single pile for (tensile axial load). Similar to the conclusion from Figure 9, the axial load increases with the axial displacement if nonsoftening t-z curves are used, as shown by Curve 1, and the axial load is upper limited by the limit load shown by Curve 3. For softening t-z curves (the residual to the peak value in t-z curves is 0.8), the axial load exhibits a peak value and then decreases to a residual value. This effect is well known as the progressive failure of axial loaded offshore piles that the side shear of a pile is not mobilized simultaneously due to the flexibility of that pile. The effect has no effect in this case as the pile is governed by the axial failure mechanism.

##### 4.3. Pile System Response

Figure 11 shows the variations of the kinematic and the static horizontal loading multiplier (base load is 1 MN) with the number of iterations in the limiting case for nonsoftening soils. The kinematic horizontal loading multiplier decreases monotonically with iterations. The monotonically decreasing kinematic solution is consistent with the discussion in Sections 3 and 4.1 because none of the piles yields in this case, and the failure of the jacket is dominated by the pulling out of Pile C. Thus, the yield surface of pile steel cross-section does not play in this case, despite each pile is subjected to a combined loading pattern. The static horizontal loading multiplier oscillates initially and increases monotonically to the exact value after 8 iterations.

#### 5. Conclusions

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.

#### Appendix

Following Chen and Han [19], for elastic perfectly plastic steel, the steel pipe pile cross-sectional ultimate plastic moment resistance is given bywhere is the maximum plastic moment resistance and and are the cross-sectional axial load and axial yield resistance, respectively.

The normalized moment-thrust-curvature relation is given in Figure 13. The parameters , and depend on the axial load acting on that cross section. The values and related expressions are given in Chen and Han [19] and will not be repeated here. It is assumed that the axial strain is independent of the bending curvature. Thus, the axial strain can be calculated directly from the axial load.

#### Data Availability

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

#### Conflicts of Interest

The authors declare that they have no conflicts of interest.

#### Acknowledgments

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.