Conic Programming Approach to Limit Analysis of Plane Rigid-Plastic Problems

Abstract

This paper presents the application of conic programming methods to the limit analysis of plane rigid-plastic problems in structural and geotechnical engineering. The approach is based on the formulation of yield criteria as second-order cone constraints and on the dual optimization problem, which directly provides collapse mechanisms and limit loads. Two benchmark examples are investigated. The first concerns a deep beam under uniform top pressure, analyzed with linear and quadratic finite elements. The results confirm the ability of the method to reproduce realistic collapse mechanisms and demonstrate the effect of mesh refinement and element type on convergence. The second example addresses the ultimate bearing capacity of a strip footing on cohesive-frictional soil. The numerical implementation was carried out in MATLAB using CVX with MOSEK as the solver, which ensures practical applicability and efficient computations. Different soil models are considered, including Mohr–Coulomb and two Drucker–Prager variants, and the results are compared with the classical Terzaghi solution. Additional elastoplastic FEM simulations carried out in a commercial program are also presented. The comparison highlights the differences between rigid-plastic optimization and incremental elastoplastic analyses, showing that both conservative and liberal estimates of bearing capacity can be obtained. The study shows that conic programming is an efficient and flexible framework for limit analysis of plane rigid-plastic problems, providing engineers with complementary tools for assessing ultimate loads, while also ensuring good computational efficiency.

1. Introduction

Classic analytical methods are widely used to determine the ultimate load capacity of structures. These methods typically rely on the simplified assumption of ideal rigid-plastic behavior. Ideal rigid-plastic behavior means that the material does not experience elastic deformation. Instead, it remains rigid until it reaches its yield limit. After reaching the yield limit, the structure deforms plastically without any further increase in load. This simplification helps in understanding structural behavior at maximum loads.

One of the most commonly used approaches for beams and frames is the plastic hinge method [1]. A plastic hinge forms at the point of a structural member where the bending moment reaches the plastic moment capacity. At that location, the cross-section can rotate freely, like a mechanical hinge. As the applied load increases, additional plastic hinges gradually develop throughout the structure. Eventually, enough hinges appear to transform the structure into a collapse mechanism. Once this mechanism forms, it is possible to determine the ultimate load capacity analytically. Important classic works that describe the plastic hinge analysis include [2,3], both providing practical examples of this method applied to steel beams and frames. Practical design aspects of plastic methods are summarized, for example, by [4]. Recent developments also include worst-case formulations for frame structures [5].

Two fundamental analytical approaches are used to estimate the ultimate load based on plastic hinges. The first is called the upper-bound, or kinematic, method [2,3,6]. In this approach, a collapse mechanism that satisfies kinematic compatibility conditions is assumed. The collapse load is then calculated by equating the external work performed by loads to the internal plastic work performed via rotations at plastic hinges. The second method, known as the lower-bound or static method, involves proposing a statically admissible bending moment distribution. This moment distribution cannot exceed the plastic moment capacity anywhere in the structure. The lower-bound method always provides a safe, conservative estimate of the actual ultimate load. The true collapse load lies between the values obtained from these upper-bound and lower-bound calculations.

For plates and slabs, the classical analytical technique is called the yield-line method [7]. This method was initially introduced by [8] and further developed by [9]. The yield-line approach is based on the observation that slabs collapse along straight lines, referred to as yield lines. These lines divide the slab into rigid segments that rotate relative to each other at collapse. By assuming various possible yield-line patterns, one can calculate the collapse load corresponding to each of these patterns. The actual collapse mechanism is the one that results in the lowest calculated load. Later research, such as that conducted by [10], introduced linear programming techniques to find optimal yield-line patterns more effectively. Other early applications can be found in [11]. Classical approaches to shells are presented, for example, by [12].

Another important class of analytical solutions for determining ultimate load relates to two-dimensional problems, specifically under plane stress and plane strain conditions. For such problems, analytical methods using simplified yield criteria, such as the Mohr–Coulomb or Tresca criteria, are often employed. Classic textbooks and standard references on limit analysis, such as [6,13], present analytical solutions for these plane problems. Such solutions typically involve identifying statically admissible stress fields or kinematically admissible collapse mechanisms in two-dimensional domains.

Classical analytical methods remain essential in structural analysis and design due to their clarity and simplicity. Despite recent developments in advanced numerical techniques, such as finite element methods and conic optimization, classical approaches remain popular. These traditional methods allow quick and straightforward verification of more complex numerical results. They also help develop an intuitive understanding of structural collapse mechanisms, making them a key part of structural analysis education.

A significant challenge for the finite element method (FEM) in estimating ultimate load capacity is that the rigid-plastic material model cannot be directly applied. Real materials exhibit elastic behavior before reaching plastic yielding. Therefore, FEM analyses must include elasticity to properly simulate structural behavior. This requires the use of elastic–plastic constitutive models instead of the simpler rigid-plastic assumption [6]. In practice, this means performing analyses in multiple incremental load steps. At each increment, the equilibrium equations must be solved numerically, resulting in increased computational effort [2]. Furthermore, precisely identifying the load level at which structural collapse occurs is challenging. The transition from elastic to plastic deformation and then to structural instability is gradual. There is no clear numerical indication marking the exact moment of collapse. As a result, accurately determining the ultimate load based on the elastic–plastic FEM becomes complicated. It should be emphasized that the rigid-plastic model idealizes material behavior by omitting the elastic domain and the gradual transition to yielding. This simplification is justified in limit analysis, which is concerned solely with the ultimate load at collapse, where plastic strains dominate and elastic effects are of secondary importance. The comparison with the incremental elastoplastic FEM later in this paper illustrates how the two approaches complement each other. A detailed discussion of computational plasticity can be found in standard textbooks [14,15,16].

An important advantage of conic programming is its capability to perform rigid-plastic analyses. This makes conic programming methods directly comparable to classical analytical solutions, such as those obtained using the plastic hinge method [2,3]. Classical methods rely explicitly on rigid-plastic assumptions and provide clear mechanisms of structural collapse. By using conic programming, engineers can thus validate numerical results against well-established classical solutions. Additionally, rigid-plastic conic programming clearly identifies collapse mechanisms, making structural failure easier to understand and visualize. This combination of numerical efficiency, direct comparability, and intuitive results highlights the advantages of conic programming over elastic–plastic FEMs in limit load analysis [17,18].

The aim of this study is to explore the potential of conic programming as a framework for rigid-plastic limit analysis of plane problems in structural and geotechnical engineering. The scope of the work covers both theoretical formulation and practical verification. The formulation includes the representation of classical yield criteria such as Huber–Mises–Hencky, Mohr–Coulomb, and Drucker–Prager in the form of second-order cone constraints and the use of dual optimization to identify collapse mechanisms. Further developments have shown that more advanced porous plasticity models can also be treated within this framework [19]. The practical part of the study is based on two benchmark problems: the deep beam under uniform top pressure and the strip footing on cohesive-frictional soil. The results of conic programming are compared with analytical solutions and with incremental elastoplastic FEM simulations. In this way the paper demonstrates both the strengths and the limitations of the method, showing how it can complement traditional numerical approaches in engineering practice.

2. Conic Programming Approach

The following section is presented in a detailed step-by-step manner to ensure clarity of the formulation and reproducibility of the results.

Conic programming (or conic optimization) is a generalization of linear optimization. It solves optimization problems where the feasible region is defined as the intersection of an affine subspace and a convex cone. Such optimization problems have the following general form [17],

$$\begin{array}{cc}\underset{\mathbf{x}}{\mathrm{minimize}}& {\mathbf{c}}^{\mathrm{T}}\mathbf{x}\\ \mathrm{subject}\mathrm{to}:& \mathbf{A}\mathbf{x}=\mathbf{b}\\ .& \mathbf{x}\in \mathcal{K}\end{array}$$

(1)

where $\mathbf{x}\in {\mathbb{R}}^{\mathrm{n}}$ is the vector of optimization variables, $\mathbf{c}\in {\mathbb{R}}^{\mathrm{n}}$ is the objective vector, $\mathbf{A}\in {\mathbb{R}}^{\mathrm{m}\times \mathrm{n}}$ and $\mathbf{b}\in {\mathbb{R}}^{\mathrm{m}}$ represent linear constraints, and $\mathcal{K}\subseteq {\mathbb{R}}^{\mathrm{n}}$ is a convex cone. A convex cone is a set of points closed under non-negative linear combinations [18]. Conic programs commonly use three types of convex cones [20]. The linear (non-negative orthant) cone is defined as

$${\mathcal{K}}_{\mathrm{L}\mathrm{P}}={\mathbb{R}}_{+}^{\mathrm{n}}=\left\{\mathbf{x}\in {\mathbb{R}}^{\mathrm{n}}: {\mathrm{x}}_{\mathrm{i}}\ge 0, \mathrm{i}=1,2,\dots ,\mathrm{n}\right\}$$

(2)

Using this cone, the conic program simplifies to the well-known linear program (LP) [17]. Second-order cone programming (SOCP) involves the second-order (or Lorentz) cone, defined as

$${\mathcal{K}}_{\mathrm{S}\mathrm{O}\mathrm{C}}=\left\{\left(\mathbf{x},\mathrm{t}\right)\in {\mathbb{R}}^{\mathrm{n}+1}: {‖\mathbf{x}‖}_2\le \mathrm{t}\right\}$$

(3)

Second-order cones naturally represent constraints involving Euclidean norms, appearing often in engineering and finance [21]. Semidefinite programming (SDP) uses positive semidefinite cones

$${\mathcal{K}}_{\mathrm{S}\mathrm{D}\mathrm{P}}=\left\{\mathbf{X}\in {\mathbb{S}}^{\mathrm{n}}: \mathbf{X}\succcurlyeq 0\right\}$$

(4)

where ${\mathbb{S}}^{\mathrm{n}}$ is the set of symmetric matrices and $\mathbf{X}\succcurlyeq 0$ means matrix $\mathbf{X}$ is a positive semidefinite [22]. SDP problems appear frequently in control theory, structural optimization, and machine learning [23,24].

2.1. Duality in Conic Programming

Every conic program has a dual problem. The dual form provides important theoretical insights and computational benefits [25]. The dual problem of a conic optimization problem can be expressed as [18]

$$\begin{array}{cc}\underset{\mathbf{y},\mathbf{s}}{\mathrm{maximize}}& {\mathbf{b}}^{\mathrm{T}}\mathbf{y}\\ \mathrm{subject}\mathrm{to}:& {\mathbf{A}}^{\mathrm{T}}\mathbf{y}+\mathbf{s}=\mathbf{c}\\ .& \mathbf{s}\in {\mathcal{K}}^{\mathrm{*}}\end{array}$$

(5)

where ${\mathcal{K}}^{*}$ is the dual cone of $\mathcal{K}$, defined as

$${\mathcal{K}}^{*}=\left\{\mathbf{y}\in {\mathbb{R}}^{\mathrm{n}}: {\mathbf{x}}^{\mathrm{T}}\mathbf{y}\ge 0 \forall \mathbf{x}\in \mathcal{K}\right\}$$

(6)

Strong duality, meaning the equality of the primal and dual optimal values, typically requires additional conditions, such as Slater’s constraint qualification [17,20].

2.2. Computational Tools and Applications

Conic optimization problems are typically addressed using specialized algorithms, primarily interior-point methods, which are efficient, robust, and particularly well-suited for large-scale problems [22,26]. Modern computational tools such as MOSEK, SeDuMi, YALMIP, and CVX significantly facilitate formulating and solving conic programs [17,27,28,29]. In the present study all computations were performed with CVX using MOSEK, chosen for its robustness and efficiency in large-scale conic optimization. In general, interior-point algorithms require a limited number of iterations almost independent of mesh size, while the cost per iteration grows with the number of unknowns, leading to a predictable polynomial increase in CPU time with problem size.

Second-order cone programming has also been applied to elastoplastic problems with warm start strategies that improve computational performance [30]. Recent advancements also involve first-order methods designed to manage extremely large optimization tasks, a capability highly valuable in machine learning and signal processing applications [31,32]

Due to its inherent versatility, conic programming is widely applied across numerous scientific and engineering disciplines. In structural mechanics, it is effectively used in the optimal design of structural components, particularly in formulating and solving free material design problems, which involve optimal mass distribution and selection of material properties to achieve the best structural performance [18,33,34]. Conic optimization also plays a crucial role in control theory, enabling rigorous stability analyses and robustness verification of control systems [24]. Within financial engineering, these methods support sophisticated portfolio optimization and quantitative risk assessment [21]. Moreover, in the field of signal processing, conic optimization contributes significantly to the design of optimal filters and advanced compressed sensing techniques [32].

Overall, the broad applicability and computational effectiveness of conic programming have established it as an essential optimization framework in engineering, economics, and data science. All computations were carried out in MATLAB R2024b using CVX 2.2 (Build 1148) with MOSEK 9.1.9 as the solver. In all optimization runs CVX was set to the default high accuracy mode (“cvx_precision high”), which guarantees reliable convergence of the MOSEK solver. For the presented benchmark problems the MOSEK solver required between 15 and 25 iterations to reach convergence under the specified accuracy settings.

3. Formulation of Plane Problems in Limit Analysis

In structural mechanics, limit analysis is used to determine the maximum load that a structure can carry. The method relies on the rigid-plastic assumption. This means that materials behave rigidly until the yield limit is reached, and then deform plastically without any further increase in load. Plane limit analysis is often used to simplify calculations for two-dimensional problems, such as walls, slabs, or footings. There are two main types of plane problems: plane stress and plane strain. In-plane stress, the stress perpendicular to the plane is zero. In plane strain, deformation perpendicular to the plane is zero.

In limit analysis, problems can be expressed in two ways: the primal formulation, based on stresses, and the dual formulation, based on velocities or displacement rates. The dual formulation is important because it provides upper bounds to the ultimate load. Solving both primal and dual formulations allows verification of the numerical results, as the solutions converge from above and below.

The limit analysis primal problem can be written as an optimization problem, where the goal is to find the largest possible load factor $\mathsf{\lambda }$. A standard form of such a problem in terms of stress $\mathsf{\sigma }$ is

$$\begin{array}{cc}\mathrm{maximize}& \mathsf{\lambda }\\ \mathrm{subject}\mathrm{to}:& {\mathbf{B}}^{\mathrm{T}}\mathsf{\sigma }=\mathsf{\lambda }\mathbf{p}+{\mathbf{p}}_0\\ .& \mathsf{\sigma }\in \mathcal{K}\end{array}$$

(7)

where $\mathsf{\lambda }$ is the load multiplier, ${\mathbf{B}}^{\mathrm{T}}$ is the equilibrium matrix (relating stresses to external loads), $\mathbf{p}$ and ${\mathbf{p}}_0$ are vectors defining variable and constant loads, respectively, and $\mathcal{K}$ is a convex cone representing yield constraints of the material.

3.1. Specific Yield Constraints

3.1.1. Huber–Mises–Hencky Yield Criterion

For plane stress conditions, the Huber–Mises–Hencky (HMH) yield criterion simplifies as follows:

$${\mathsf{\sigma }}_{\mathrm{x}\mathrm{x}}^2-{\mathsf{\sigma }}_{\mathrm{x}\mathrm{x}}{\mathsf{\sigma }}_{\mathrm{y}\mathrm{y}}+{\mathsf{\sigma }}_{\mathrm{y}\mathrm{y}}^2+3{\mathsf{\sigma }}_{\mathrm{x}\mathrm{y}}^2\le {\mathrm{f}}_{\mathrm{y}}^2$$

(8)

where ${\mathsf{\sigma }}_{\mathrm{x}\mathrm{x}}$, ${\mathsf{\sigma }}_{\mathrm{y}\mathrm{y}}$ are the normal stresses components, ${\mathsf{\sigma }}_{\mathrm{x}\mathrm{y}}$ is the shear stress component, and ${\mathrm{f}}_{\mathrm{y}}$ denotes the material yield strength (uniaxial tension/compression test).

The above criterion can be represented as a second-order cone constraint, convenient for numerical implementation. To achieve this, define the auxiliary vector ${\mathbf{q}}^{\mathrm{H}\mathrm{M}\mathrm{H}}$.

$${\mathbf{q}}^{\mathrm{H}\mathrm{M}\mathrm{H}}=\left[\begin{array}{c}{\mathrm{f}}_{\mathrm{y}}\\ {\mathsf{\sigma }}_{\mathrm{x}\mathrm{x}}-{\displaystyle \frac{{\mathsf{\sigma }}_{\mathrm{y}\mathrm{y}}}{2}}\\ {\displaystyle \frac{\sqrt{3}}{2}}{\mathsf{\sigma }}_{\mathrm{y}\mathrm{y}}\\ \sqrt{3}{\mathsf{\sigma }}_{\mathrm{x}\mathrm{y}}\end{array}\right]$$

(9)

Then the HMH yield criterion for plane stress can be expressed as the second-order cone constraint:

$${\mathbf{q}}^{\mathrm{H}\mathrm{M}\mathrm{H}}\in {\mathcal{K}}^{\mathrm{H}\mathrm{M}\mathrm{H}},\hspace{1em}{\mathcal{K}}^{\mathrm{H}\mathrm{M}\mathrm{H}}:=\left\{{\mathbf{q}}^{\mathrm{H}\mathrm{M}\mathrm{H}}\in {\mathbb{R}}^4: {\mathrm{q}}_1^{\mathrm{H}\mathrm{M}\mathrm{H}}\ge \sqrt{{\left({\mathrm{q}}_2^{\mathrm{H}\mathrm{M}\mathrm{H}}\right)}^2+{\left({\mathrm{q}}_3^{\mathrm{H}\mathrm{M}\mathrm{H}}\right)}^2+{\left({\mathrm{q}}_4^{\mathrm{H}\mathrm{M}\mathrm{H}}\right)}^2}\right\}$$

(10)

It should be noted that equivalent cone definitions can be used, for example, by rescaling or permuting the components of the auxiliary vector. Such variants lead to the same yield surface and identical results in the optimization, with differences only in numerical conditioning [14,16]. This conic formulation is directly suitable for second-order cone programming using solvers such as MOSEK or YALMIP. The advantage of expressing yield conditions as second-order cone constraints is that they can be handled very efficiently by modern interior-point solvers. Such constraints represent Euclidean norms, which ensures numerical stability and predictable convergence even for large finite element meshes. This makes the conic formulation attractive compared with ad hoc linearizations of nonlinear yield surfaces.

3.1.2. Mohr–Coulomb Yield Criterion

The plane strain Mohr–Coulomb (MC) yield criterion is given as [35]

$$\sqrt{{\left({\mathsf{\sigma }}_{\mathrm{x}\mathrm{x}}-{\mathsf{\sigma }}_{\mathrm{y}\mathrm{y}}\right)}^2+4{\mathsf{\sigma }}_{\mathrm{x}\mathrm{y}}^2}+\left({\mathsf{\sigma }}_{\mathrm{x}\mathrm{x}}+{\mathsf{\sigma }}_{\mathrm{y}\mathrm{y}}\right)\sin \mathsf{\varphi }-2\mathrm{c}\cos \mathsf{\varphi }\le 0$$

(11)

where c is the cohesion and ϕ is the internal friction angle.

To formulate this criterion as a second-order cone constraint, define an auxiliary vector ${\mathbf{q}}^{\mathrm{M}\mathrm{C}}$:

$${\mathbf{q}}^{\mathrm{M}\mathrm{C}}=\mathbf{D}\mathsf{\sigma }+\mathbf{d}$$

(12)

where the stress vector $\mathsf{\sigma }$ and matrices $\mathbf{D}$ and $\mathbf{d}$ are defined as

$$\mathsf{\sigma }=\left[\begin{array}{c}{\mathsf{\sigma }}_{\mathrm{x}\mathrm{x}}\\ {\mathsf{\sigma }}_{\mathrm{y}\mathrm{y}}\\ {\mathsf{\sigma }}_{\mathrm{x}\mathrm{y}}\end{array}\right],\mathbf{D}=\left[\begin{array}{ccc}-\sin \mathsf{\varphi }& -\sin \mathsf{\varphi }& 0\\ 1& -1& 0\\ 0& 0& 2\end{array}\right],\mathbf{d}=\left[\begin{array}{c}2\mathrm{c}\cos \mathsf{\varphi }\\ 0\\ 0\end{array}\right]$$

(13)

The MC yield criterion can thus be expressed through the second-order cone constraint

$${\mathbf{q}}^{\mathrm{M}\mathrm{C}}\in {\mathcal{K}}^{\mathrm{M}\mathrm{C}},\hspace{1em}{\mathcal{K}}^{\mathrm{M}\mathrm{C}}:=\left\{{\mathbf{q}}^{\mathrm{M}\mathrm{C}}\in {\mathbb{R}}^3: {\mathrm{q}}_1^{\mathrm{M}\mathrm{C}}\ge \sqrt{{\left({\mathrm{q}}_2^{\mathrm{M}\mathrm{C}}\right)}^2+{\left({\mathrm{q}}_3^{\mathrm{M}\mathrm{C}}\right)}^2}\right\}$$

(14)

This formulation is directly suitable for second-order cone programming (SOCP) and can be efficiently solved using optimization software such as MOSEK or YALMIP.

3.1.3. Drucker–Prager Yield Criterion

The Drucker–Prager (DP) criterion is commonly used to describe yielding of geomaterials like soils and concrete. It generalizes the HMH criterion by accounting for pressure-dependent yield behavior. For plane strain conditions, the DP yield criterion is expressed as follows:

$$\sqrt{{\mathsf{\sigma }}_{\mathrm{x}\mathrm{x}}^2-{\mathsf{\sigma }}_{\mathrm{x}\mathrm{x}}{\mathsf{\sigma }}_{\mathrm{y}\mathrm{y}}+{\mathsf{\sigma }}_{\mathrm{y}\mathrm{y}}^2+3{\mathsf{\sigma }}_{\mathrm{x}\mathrm{y}}^2}+\mathsf{\alpha }\left({\mathsf{\sigma }}_{\mathrm{x}\mathrm{x}}+{\mathsf{\sigma }}_{\mathrm{y}\mathrm{y}}\right)-\mathrm{k}\le 0$$

(15)

Here, parameters α and k depend on material cohesion c and internal friction angle ϕ. In this study the inscribed form of the Drucker–Prager criterion is used, providing a conservative approximation of the Mohr–Coulomb yield surface. The corresponding material parameters are defined as

$$\mathsf{\alpha }={\displaystyle \frac{2\sin \mathsf{\varphi }}{\sqrt{3}\left(3+\sin \mathsf{\varphi }\right)}},\hspace{1em}\mathrm{k}={\displaystyle \frac{6\mathrm{c}\cos \mathsf{\varphi }}{\sqrt{3}\left(3+\sin \mathsf{\varphi }\right)}}$$

(16)

To conveniently represent this criterion as a second-order cone constraint for numerical computations, introduce an auxiliary vector ${\mathbf{q}}^{\mathrm{D}\mathrm{P}}$

$${\mathbf{q}}^{\mathrm{D}\mathrm{P}}=\left[\begin{array}{c}\mathrm{k}-\mathsf{\alpha }\left({\mathsf{\sigma }}_{\mathrm{x}\mathrm{x}}+{\mathsf{\sigma }}_{\mathrm{y}\mathrm{y}}\right)\\ {\mathsf{\sigma }}_{\mathrm{x}\mathrm{x}}-{\displaystyle \frac{1}{2}}{\mathsf{\sigma }}_{\mathrm{y}\mathrm{y}}\\ {\displaystyle \frac{\sqrt{3}}{2}}{\mathsf{\sigma }}_{\mathrm{y}\mathrm{y}}\\ \sqrt{3}{\mathsf{\sigma }}_{\mathrm{x}\mathrm{y}}\end{array}\right]$$

(17)

The corresponding convex cone ${\mathcal{K}}^{\mathrm{D}\mathrm{P}}$, associated explicitly with the DP yield criterion, is thus clearly defined as

$${\mathbf{q}}^{\mathrm{D}\mathrm{P}}\in {\mathcal{K}}^{\mathrm{D}\mathrm{P}},\hspace{1em}{\mathcal{K}}^{\mathrm{D}\mathrm{P}}:=\left\{{\mathbf{q}}^{\mathrm{D}\mathrm{P}}\in {\mathbb{R}}^4: {\mathrm{q}}_1^{\mathrm{D}\mathrm{P}}\ge \sqrt{{\left({\mathrm{q}}_2^{\mathrm{D}\mathrm{P}}\right)}^2+{\left({\mathrm{q}}_3^{\mathrm{D}\mathrm{P}}\right)}^2+{\left({\mathrm{q}}_4^{\mathrm{D}\mathrm{P}}\right)}^2}\right\}$$

(18)

3.2. Dual Formulation of the Problem

The corresponding dual formulation can be written as follows

$$\begin{array}{cc}\mathrm{minimize}& -{\mathbf{p}}_0^{\mathrm{T}}\mathbf{u}\\ \mathrm{subject}\mathrm{to}:& \mathbf{B}\mathbf{u}+\mathbf{e}=0\\ .& {\mathbf{p}}^{\mathrm{T}}\mathbf{u}=1\\ .& \mathbf{e}\in {\mathcal{K}}^{\mathrm{*}}\end{array}$$

(19)

Here $\mathbf{u}$ represents displacement rates, $\mathbf{e}$ represents strain rates associated with plastic deformation, $\mathbf{B}$ is the compatibility matrix linking displacements to strains, and ${\mathcal{K}}^{\mathrm{*}}$ is the dual cone, describing the flow rule related to the yield surface.

The dual cone ${\mathcal{K}}^{\mathrm{*}}$ defines admissible plastic strain rates $\mathbf{e}$. Its definition is directly linked to the yield condition cone $\mathcal{K}$ through:

$${\mathcal{K}}^{\mathrm{*}}=\left\{\mathbf{e}\in {\mathbb{R}}^{\mathrm{n}}: {\mathsf{\sigma }}^{\mathrm{T}}\mathbf{e}\ge 0 \forall \mathsf{\sigma }\in \mathrm{K}\right\}$$

(20)

This means plastic deformation $\mathbf{e}$ must not result in negative plastic work. If $\mathcal{K}$ describes the Mohr–Coulomb yield condition, the dual cone ${\mathcal{K}}^{\mathrm{*}}$ describes an associated flow rule, which implies normality between the yield surface and the flow direction. For example, the dual cone for a second-order cone (used for von Mises yield criterion or Mohr–Coulomb criterion in numerical form) is itself a second-order cone.

The present formulation is restricted to associated plasticity, where the flow rule is derived directly from the yield cone. Extension to non-associative flow rules would require additional modifications, such as relaxation techniques or auxiliary constraints, to ensure positive plastic dissipation. This important but a more complex case that lies outside the scope of the current study.

The dual formulation describes a collapse mechanism. The displacements $\mathbf{u}$ represent velocities describing the rigid-body motion of elements, and the strain rates $\mathbf{e}$ represent plastic deformation occurring at collapse. The solution of the dual formulation thus provides a clear view of the collapse mechanism, allowing engineers to understand how the structure fails under the limit load. The dual cone condition ensures that these strain rates are consistent with the yield criterion and that plastic dissipation remains non-negative, which gives the computed velocity field a clear physical interpretation as an admissible collapse mechanism. It should be noted that while the computed ultimate load is unique, the corresponding collapse mechanism may not be. The boundary conditions restrict the admissible velocity fields and may enforce uniqueness in some cases, but in other cases multiple mechanisms can coexist, all consistent with the same limit load.

In practice, both primal and dual formulations can be solved using conic optimization software, such as MOSEK. Solving both formulations simultaneously provides confidence in the computed ultimate load, as the primal formulation provides lower bounds, and the dual formulation gives upper bounds. The convergence of these two values confirms the accuracy of numerical results.

In the presented numerical implementation, the kinematic matrix $\mathbf{B}$ is computed for linear triangular elements. For each element, a local matrix ${\mathbf{B}}^{\mathrm{e}}$ is derived based on its nodal coordinates. This matrix relates nodal displacements to element strains. The global matrix $\mathbf{B}$ is then assembled from these element matrices by appropriately combining contributions from all elements according to the global node numbering. The assembly ensures that equilibrium conditions are correctly represented throughout the structure. The described approach is straightforward and effectively supports numerical computations in the limit analysis framework. Element integrals are evaluated with the classical three-point Strang rule, exact for quadratic polynomials [36]. For linear elements this is sufficient, although other point sets could also be applied.

The numerical implementation described in this paper was performed using MATLAB software, with the PDE Toolbox utilized to generate the geometry and finite element mesh. Specifically, the PDE Toolbox allowed convenient creation and discretization of the model geometry into linear triangular (CST) elements. The equilibrium conditions and yield criterion were formulated and solved using CVX. The Mosek solver was chosen within CVX for efficiently solving the resulting second-order cone programming (SOCP) problem, ensuring reliable and rapid convergence to the optimal solution.

4. Limit Load Analysis of a Deep Beam Under Uniform Top Pressure

To demonstrate the practical application of the presented method, consider a deep beam modeled under plane stress conditions. The beam has a length of $\mathrm{L}=0.2 \mathrm{m}$ and a height of $\mathrm{H}=0.05 \mathrm{m}$. It is loaded by a uniform pressure $\mathrm{q}\left[\mathrm{N}/{\mathrm{m}}^2\right]$ applied along its top edge, acting vertically downward. The bottom edges of the beam are free, while the vertical edges at $\mathrm{x}=0$ and $\mathrm{x}=\mathrm{L}$ are fully restrained, allowing no horizontal or vertical displacements.

4.1. CST Element Model

The beam was discretized using linear triangular finite elements (CST). A maximum element size of 0.001 m was assumed initially leading to 2654 elements (see Figure 1). The discretization was generated using the MATLAB PDE Toolbox, which provides only triangular elements. As a result, the mesh options are restricted and alternative structured configurations such as chevron or unidirectional arrangements cannot be tested in this environment.

The material is assumed to follow the rigid-plastic von Mises criterion, with a yield stress of ${\mathrm{f}}_{\mathrm{y}}=400 \mathrm{MPa}$. The equilibrium conditions are expressed using the kinematic matrix $\mathbf{B}$, assembled from individual triangular elements as described earlier. A conic programming optimization problem is formulated and solved numerically to determine the maximum collapse load factor $\mathsf{\lambda }$.

The resulting collapse mechanism is visualized by plotting the corresponding displacement field obtained from the dual solution of the optimization problem. Additionally, the distribution of the von Mises equivalent stresses within the beam at collapse is computed and presented. This provides a clear view of the plastic zones within the beam at the limit state.

The computed collapse load factor $\mathsf{\lambda }$ obtained numerically can be compared with classical analytical solutions available for validation, confirming the effectiveness of the numerical method described in this paper.

The numerical solution obtained for the presented plane stress problem indicates a collapse pressure of approximately ${\mathrm{q}}_{\mathrm{l}\mathrm{i}\mathrm{m}}=\mathsf{\lambda }·1 \mathrm{P}\mathrm{a}=92.78 \mathrm{M}\mathrm{P}\mathrm{a}$. For comparison, an analytical beam-theory solution was computed by using plastic hinge method [1], providing a collapse load of ${\mathrm{q}}_{\mathrm{l}\mathrm{i}\mathrm{m}}^{\mathrm{b}\mathrm{e}\mathrm{a}\mathrm{m}}={\displaystyle \frac{4{\mathrm{H}}^2}{{\mathrm{L}}^2}}{\mathrm{f}}_{\mathrm{y}}=100.00\mathrm{MPa}$. It is important to note, however, that this analytical solution is based on classical beam theory, which assumes slender structural elements. The structure analyzed here has an aspect ratio $\mathrm{L}/\mathrm{H}=4$, indicating a relatively deep beam geometry. Classical beam theory typically requires an aspect ratio of at least $\mathrm{L}/\mathrm{H}=8$ to be fully valid. Therefore, the beam-theory solution may not accurately represent the actual limit load of this particular deep-beam structure. The discrepancy between numerical and analytical results is thus expected, reflecting limitations inherent in classical beam theory when applied to deep structural members. The discrepancy between the numerical and analytical results also reflects the influence of geometric nonlinearity in deep beams. The 2D rigid-plastic continuum formulation accounts for shear and stress redistribution, whereas the classical beam solution neglects these effects and is valid only for slender members.

The graphical results presented in consequent figures show rigid-plastic deformation pattern (Figure 2) and equivalent stress distribution at the limit state for the analyzed structure (Figure 3).

The deformation plot visualized in Figure 2 clearly reveals the collapse mechanism of the deep beam under uniform top pressure. The beam forms a distinct three-hinge mechanism, typical for bending failure. Plastic hinges are evident at the supports (both ends) and at the midpoint of the beam. These three plastic hinges define a clear mechanism leading to collapse, consistent with classical plastic collapse theory. The equivalent von Mises stress distribution plot (Figure 3) further confirms the plastic hinge formation, showing zones of maximum stress concentration clearly corresponding to the identified plastic hinge locations. The stress distribution pattern exhibits symmetry, aligning with the geometry and boundary conditions of the problem.

4.2. Quadratic Element Model

After the initial CST study, the model was extended to second-order triangular elements with six nodes per element. The geometry and mesh were generated in MATLAB. For the quadratic variant, the formulation did not rely on PDE Toolbox element routines. The strain displacement matrix B and the numerical integration were coded directly. In the quadratic case the equilibrium matrix was built from the derivatives of quadratic shape functions evaluated at three Gauss points. The local matrices were then assembled into the global equilibrium system in the same way as for linear elements. This approach is consistent with formulations described in the literature [37] but here it was implemented directly in MATLAB with CVX and MOSEK.

In practice, three barycentric Gauss points per element were used and the global equilibrium matrix was assembled from the element B matrices and the Gauss weights. The same conic programming setup with CVX and MOSEK was used.

The quadratic mesh increases the number of nodes and displacement unknowns compared to CST for a similar mesh size. This raises memory use and solution time. At the same time it improves the smoothness of the kinematic fields and gives a faster and more regular convergence of the limit load. The convergence study for both meshes is shown in Figure 4. The CST sequence shows a slower approach with small oscillations. The quadratic sequence is smoother and reaches the same limit pressure close to 90 MPa with fewer refinement steps. This confirms the benefit of quadratic elements in rigid plastic limit analysis, with a clear balance between accuracy and computational cost. This sequence of refinements illustrates a mesh convergence study, demonstrating that quadratic elements achieve a stable limit pressure with fewer refinement steps compared to linear CST elements.

In addition, Figure 5 presents the CPU time for the same mesh sequences. The results show a monotonic increase in computational cost with the number of elements. Quadratic triangular elements require more memory and time than CST elements, which is consistent with their faster convergence of limit pressure shown in Figure 4.

5. Bearing Capacity of a Strip Footing

5.1. Coulomb–Mohr Soil Model

The second benchmark problem concerns the ultimate bearing capacity of a strip footing on a cohesive-frictional soil. The analysis was carried out under plane strain conditions using the rigid-plastic Coulomb–Mohr yield criterion with cohesion $\mathrm{c}=10\mathrm{kPa}$ and friction angle $\mathsf{\varphi }=25^{\circ }$.

For reference, the classical Terzaghi solution for a rigid strip footing was computed, which gives a limit pressure of about ${\mathrm{q}}_{\mathrm{l}\mathrm{i}\mathrm{m}}^{\mathrm{T}}\approx 207\mathrm{k}\mathrm{P}\mathrm{a}$.

In the present formulation, the footing is modeled as flexible, with the load represented by a uniform pressure applied along the base. This choice follows directly from the dual optimization setup, where it is not possible to impose kinematic constraints that would enforce a rigid footing displacement. As a consequence, the numerical model corresponds to a flexible footing rather than the rigid footing assumed in Terzaghi’s theory.

The geometry of the soil block and loading line is shown in Figure 6. A locally refined mesh was applied below the footing to capture the failure zone. The final model contained 5227 nodes and 10,302 triangular elements.

The rigid-plastic deformation pattern (Figure 7) reveals a clear settlement mechanism under the footing, accompanied by two symmetric shear bands spreading into the soil. The equivalent stress distribution (Figure 8) shows concentration zones forming a typical fan-shaped pattern beneath the loaded area, consistent with the expected failure mechanism.

The computed collapse pressure reached ${\mathrm{q}}_{\mathrm{num}}\approx 232\mathrm{kPa}$. Although the analytical and numerical models are not directly comparable because of the difference between rigid and flexible footing assumptions, the obtained values are of the same order of magnitude. The numerical result is slightly higher, which is logical, since a flexible footing can redistribute pressure and develop a somewhat larger bearing capacity than a rigid footing. This observation highlights that the two formulations are not strictly comparable. The rigid footing assumption enforces uniform displacement, while the flexible model allows redistribution of stresses and a different collapse mechanism. Real foundations usually lie between these two extremes, which explains why numerical and analytical predictions differ while still being of the same order of magnitude. This confirms that the proposed conic programming approach provides realistic estimates of bearing capacity and reproduces characteristic collapse mechanisms of soil foundations. Other geotechnical stability problems have also been addressed with SOCP formulations, for example, reinforced retaining structures [38]. Previous numerical studies on the bearing capacity of pile foundations carried out by the authors also confirmed the sensitivity of the results to the adopted constitutive assumptions [39].

5.2. Drucker–Prager Soil Model

The Drucker–Prager yield criterion provides a smooth pressure dependent approximation of the Mohr–Coulomb model. It is particularly convenient for conic optimization because it can be expressed as a second-order cone. However, different variants of the criterion are possible and the choice influences the numerical results.

In the first variant we used the inscribed mapping of Mohr–Coulomb in a full 3D plane strain form including ${\mathsf{\sigma }}_{\mathrm{z}\mathrm{z}}$. The computed limit pressure was ${\mathrm{q}}_{\mathrm{l}\mathrm{i}\mathrm{m}}\approx 144 \mathrm{k}\mathrm{P}\mathrm{a}$. This value is smaller than the Mohr–Coulomb result because the inscribed Drucker–Prager surface lies inside the Mohr–Coulomb hexagon in the deviatoric plane. This makes the model more conservative.

In the second variant we applied a projected plane strain formulation where the out-of-plane stress ${\mathsf{\sigma }}_{\mathrm{z}\mathrm{z}}$ is omitted (see Equation (15)). This approach is widely used in conic programming because it leads to a simpler two-dimensional cone constraint and avoids numerical difficulties related to the mean stress term. The computed limit pressure was ${\mathrm{q}}_{\mathrm{l}\mathrm{i}\mathrm{m}}\approx 97.8 \mathrm{k}\mathrm{P}\mathrm{a}$. The result is more restrictive than the 3D variant and provides a safe lower bound.

It is important to note that the same simplification was adopted in our Mohr–Coulomb implementation. In both cases we formulated the criterion only in terms of in-plane stress components ${\mathsf{\sigma }}_{\mathrm{x}\mathrm{x}}$, ${\mathsf{\sigma }}_{\mathrm{y}\mathrm{y}}$, and ${\mathsf{\sigma }}_{\mathrm{x}\mathrm{y}}$ (see Equation (11)). A similar approach can be found in the literature. Krabbenhøft [35] present the plane strain Mohr–Coulomb criterion explicitly without ${\mathsf{\sigma }}_{\mathrm{z}\mathrm{z}}$. Although the problem is called “plane strain”, the yield condition is written in a purely two-dimensional form. This confirms that the projected formulation is consistent with established conic programming treatments of soil plasticity.

5.3. FEM Elastoplastic Solution

For comparison, the same strip footing problem was solved using the commercial finite element program ZSoil. The analysis was performed in plane strain conditions with an elastoplastic material model based on the Mohr–Coulomb criterion. The numerical mesh contained 75,642 enhanced assumed strain (EAS) elements. In the central zone under the footing a refined mesh with 5 cm elements was applied, while outside this zone the mesh was gradually coarsened to 15 cm (see Figure 9). This discretization is free from volumetric locking because of the EAS formulation.

In commercial FEM software ZSoil 25.04 a rigid-plastic formulation is not available. Instead, the soil must always be modeled with an elastic–plastic constitutive law. The analysis is carried out incrementally. The external load is increased step by step and equilibrium is solved at each step until the numerical procedure no longer converges. The load corresponding to the last convergent step is taken as the bearing capacity of the soil. This incremental strategy is fundamentally different from the optimization approach used in rigid-plastic limit analysis.

The ZSoil results are shown in Figure 10 and Figure 11 as the deformed mesh and the developed plastic zones. The obtained bearing capacity was ${\mathrm{q}}_{\mathrm{l}\mathrm{i}\mathrm{m}}\approx 173 \mathrm{k}\mathrm{P}\mathrm{a}$. This value is lower than the theoretical Terzaghi solution for a rigid footing (207 kPa) and also below the rigid-plastic Mohr–Coulomb result obtained with the optimization method (232 kPa). It is closer to the Drucker–Prager predictions, which gave 144 kPa for the 3D inscribed variant and 98 kPa for the projected variant. The differences follow from the modeling assumptions. The FEM elastoplastic solution accounts for both elastic deformations and gradual development of plasticity, while the rigid-plastic optimization models assume an idealized instantaneous collapse state. Despite these differences, the results are consistent in terms of order of magnitude and the observed shear failure mechanism matches the expected classical pattern. From a computational viewpoint, the optimization problem is solved in a single run of the interior-point method, which usually requires only tens of iterations irrespective of mesh refinement. By contrast, the elastoplastic FEM involves many incremental load steps and repeated iterations in each step. This structural difference in numerical strategy explains the higher computational efficiency of the conic optimization framework, even though exact runtimes were not reported. Comparable FEM strategies were applied by the authors in earlier studies of diaphragm walls in clay formations, confirming the practical usefulness of commercial FEM software for geotechnical applications [40,41].

5.4. Comparison of Bearing Capacity Results

The comparison in

Table 1. Comparison of bearing capacity results.

Method/Model	Footing Type	Limit Pressure [kPa]	Relative to Terzaghi
Terzaghi analytical	Rigid	207	100%
Mohr–Coulomb rigid-plastic (optimization)	Flexible	232	112%
Drucker–Prager 3D inscribed (optimization)	Flexible	144	70%
Drucker–Prager 2D projected (optimization)	Flexible	98	47%
ZSoil elastoplastic (incremental FEM)	Flexible	173	84%

shows that the computed bearing capacity values range from 98 kPa for the projected Drucker–Prager variant up to 232 kPa for the Mohr–Coulomb rigid-plastic model. The ZSoil elastoplastic result of 173 kPa lies between these limits and is closer to the analytical Terzaghi solution. This confirms that the different approaches, despite their varying assumptions, provide results of the same order of magnitude and together frame a realistic range for the bearing capacity of the analyzed footing.

Large discrepancies between analytical bearing capacity solutions and finite element simulations have also been reported in the literature. For example, ref. [42] shows that analytical estimates and FEM predictions may differ substantially because they rely on very different assumptions about the collapse mechanism. This confirms that the differences observed in Table 1 are not an exception but rather a typical feature of geotechnical analysis. An important advantage of our study is that several modeling approaches are presented side by side. By comparing Mohr–Coulomb, Drucker–Prager (inscribed and projected), and incremental FEM results, the engineer can select between more conservative or more liberal estimates of the bearing capacity. This flexibility allows the designer to balance safety and efficiency depending on the context of the problem.

It is worth noting that design codes such as Eurocode 7 also rely on simplified bearing capacity formulas combined with partial safety factors. The range of results obtained here by different rigid-plastic and elastoplastic models can therefore be seen as consistent with the philosophy of code-based design, where both conservative and less conservative estimates are used depending on safety requirements.

6. Conclusions

This work has shown that conic programming can be successfully applied to rigid-plastic limit analysis of structural and geotechnical problems. The method allows direct identification of collapse mechanisms without the step-by-step procedures required in elastoplastic FEM.

In the first example of a deep beam under uniform pressure the numerical solution reproduced a realistic collapse mechanism with plastic hinges at both supports and at mid-span. The comparison of linear and quadratic finite elements confirmed that higher order elements provide smoother displacement fields and lead to faster convergence. This demonstrates that the proposed approach is able to capture the expected mechanism of failure and that the accuracy depends on the quality of the discretization.

In the second example of a strip footing the method was applied with different soil models. The Mohr–Coulomb formulation produced higher values of bearing capacity, while the Drucker–Prager variants gave more conservative estimates. This shows that the framework can include alternative yield criteria and deliver a range of possible solutions. Additional FEM calculations carried out in a commercial code gave results of the same order of magnitude, although lower than the rigid-plastic optimization. This difference reflects the incremental elastoplastic procedure used in standard FEM software, where rigid-plastic analysis is not available and elasticity must always be included.

The comparison of both examples confirms that discrepancies between analytical, optimization-based, and FEM results are natural and well known in the literature. They should be treated not as inconsistencies but as complementary views of the same problem. An important outcome of this study is that by combining rigid-plastic optimization and the elastoplastic FEM it is possible to move between more conservative and more liberal estimates of the ultimate load. In practical terms, conic programming is best suited for rapid limit load assessment and mechanism identification, while the elastoplastic FEM should be used when elastic deformations or step-by-step development of plasticity are important for design. This provides the engineer with a practical tool to balance safety and efficiency in design.

The scope of examples in this study was intentionally limited to two plane problems analyzed in detail, but the framework can be extended to more complex cases such as slope stability or multi-span frames, which remain directions for future research. Recent hybrid formulations for tunnels in anisotropic rock masses [43] indicate that the framework can be extended to more complex geometries and materials. Future research could extend the present framework in several directions. Three-dimensional formulations would allow the method to cover a broader range of structural and geotechnical problems. Dynamic loading conditions represent another relevant field, as they are essential for assessing seismic or impact resistance. Finally, coupling conic optimization with probabilistic models could provide a robust way to address uncertainties in material properties and loading, further increasing the practical relevance of the approach.