Elasto-plastic truss optimization under geometric nonlinearity using a genetic algorithm

Steel trusses are widely used in various structural applications owing to their numerous advantages, particularly their excellent strength-to-weight ratio and structural flexibility. The most common applications include bridges, roofs, and transmission towers, where proper design is essential to ensure safety and reliability. However, under unexpectedly excessive loads-such as extreme weather conditions [1] or seismic events [2]-trusses may experience significant stress and deformation near collapse, which can exceed the assumptions of conventional computational methods. Specifically, in such extraordinary situations, nonlinear effects—including yielding and large displacements—become significant and must be accounted for to achieve accurate predictions of structural behavior. This demonstrates the necessity for advanced design strategies capable of capturing complex nonlinear responses and providing safer and more efficient solutions beyond the limitations of traditional approaches.
In this context, plastic analysis offers a promising approach for enhancing structural design and mitigating the risk of catastrophic collapse. Depending on the structural type and boundary conditions, failure may be governed by the yielding of the members. Accordingly, plastic analysis focuses on the identification and assessment of inelastic deformations that accumulate and persist after unloading. Among the various theorems developed in recent decades [3-5], a notable solution method for elasto-plastic analysis is the work of Kaliszky [6], who introduced a nonlinear mathematical program for control of plastic performance. Building upon this work, Kaliszky and Lógó [7] proposed a design method for optimizing bar structures, which incorporates the complementary strain energy of residual forces to characterize plastic deformations. This approach was later extended to address the strengthening of elasto-plastic trusses [8]. In these studies, the developed technique was demonstrated using a geometrically perfect 9-bar structure, under the assumption of small displacements and solved using a gradient-based optimization algorithm.
Beyond the consideration of plastic behavior, the effect of large deformations is critical for accurately capturing the structural response [9], particularly in relatively slender trusses [10], where such deformations contribute significantly to geometric nonlinearity. Over the years, the finite element method (FEM) has proven to be a robust and versatile tool in this context [11-13], employing incremental-iterative solution strategies that systematically update the stiffness matrix to reflect the evolving stress-strain relationship during deformation. Based on this groundwork, several researchers have developed optimization frameworks for truss structures that explicitly incorporate geometric nonlinearity [14-16].

To ultimately improve the design of truss structures, increasing emphasis has recently been placed on integrating inelastic behavior and geometric nonlinearity within a unified optimization framework. Viet-Hung and Seung-Eock [17] proposed a reliability-based optimization approach aimed at minimizing structural weight, incorporating constraints related to strength, displacement, and failure probability. Subsequently, Viet-Hung et al. [18] introduced a machine learning-based method for size optimization that considers both serviceability and load-bearing requirements. Expanding on this, Viet-Hung et al. [19] further enhanced the optimization algorithm by integrating additional machine learning techniques. It is noteworthy that all three studies assumed initially perfect structural geometry and did not incorporate plastic analysis for the control of inelastic deformations. In contrast, Grubits and Movahedi [20] developed a novel optimization framework for elasto-plastic trusses, which integrates plastic behavior within a geometrically nonlinear context and allows for the inclusion of initial geometric imperfections. While earlier approaches-such as those proposed by Kaliszky and Lógó [6-8]-successfully addressed plastic behavior through complementary strain energy under the assumption of small displacements, the recent framework extends this capability to structures subjected to large deformations and imperfection-sensitive responses.
Building upon the earlier work of the Authors presented in [20], this research further advances the previously developed framework by incorporating serviceability constraints to limit displacements in accordance with code-based criteria, thereby enhancing the robustness and practical applicability of the proposed optimization approach. Consequently, an advanced design methodology is introduced for the elasto-plastic and elastic analysis of steel trusses. This is achieved by assessing and limiting plastic deformations through the complementary strain energy of residual forces, and by employing an incremental solution technique based on the finite element method to account for geometric nonlinearity. The proposed mathematical formulations enable a careful and rigorous design process in which, alongside the objective of minimizing structural weight, three penalty terms-related to load-bearing capacity, inelastic behavior, and global stability-ensure that the final solution is both materially efficient and structurally safe.
Ultimately, this approach allows for a precise characterization of structural response, with the capability to capture various failure modes such as elastic buckling, yielding, and inelastic post-buckling. By incorporating a newly introduced displacement constraint, serviceability requirements can also be directly addressed in the design process. This comprehensive and flexible framework offers a versatile design tool that accommodates a wide range of performance criteria and practical constraints. In this way, the methodology not only overcomes the limitations of conventional elastic design practices but also offers a practical computational framework that balances efficiency with reliability. By explicitly linking advanced nonlinear analysis with optimization, the study demonstrates how modern mathematical tools such as genetic algorithms can be effectively leveraged to deliver innovative solutions for the safe and sustainable design of steel trusses.
The effectiveness and capability of the proposed methodology are demonstrated through two widely recognized benchmark examples-the 37-bar planar truss and the 25-bar space truss. For the 37-bar truss, two primary design scenarios are investigated: (i) elastic design, where structural adequacy is ensured without any inelastic deformation while satisfying the required load-bearing capacity, and (ii) elasto-plastic design, in which plastic deformations are permitted but constrained within an acceptable limit under the applied load. In addition, a supplementary optimization case is conducted in the elastoplastic scenario by imposing displacement thresholds to assess the influence of serviceability requirements. In the 25-bar truss example, the analysis focuses exclusively on elasto-plastic design, where two cases are compared: one without displacement constraints and another where serviceability limitations are explicitly enforced. The obtained results confirm the efficiency of the developed framework, underlining the practical relevance of advanced optimization techniques in improving structural design precision and achieving safer, more materially efficient configurations.

This section outlines the fundamental theoretical components underlying the proposed framework. First, the complementary plastic work theory is introduced, which serves as the methodological basis by providing a physically meaningful measure of plastic deformations. Building on this foundation, the principles of elastic and elasto-plastic limit analysis are discussed, with particular emphasis on the role of complementary strain energy associated with residual forces. Finally, the core aspects of geometrically nonlinear analysis are presented, as incorporated into the structural design process to accurately reflect the behavior of structures under large deformations.

This section outlines the theoretical basis of the complementary plastic work theory. Consider a model composed of an elasto-plastic material, assumed to be independent of both time and temperature. The body occupies a volume $V$$V$VV$V$ and is bounded by a surface $S$$S$SS$S$, which is partitioned into two disjoint subsets: $S_u$$S_u$S_(u)S_{u}$S_u$, where zero surface displacement is prescribed,
and $S_q$$S_q$S_(q)S_{q}$S_q$, which is subjected to quasi-static surface tractions $q(t,x_i)$$q\left(t,x_i\right)$q(t,x_(i))q\left(t, x_{i}\right)$q(t,x_i)$, with $x_i$$x_i$x_(i)x_{i}$x_i$ denoting the position vector on the surface. Following the foundational work of Kaliszky [6], the field quantities at a given time $t$$t$tt$t$ are defined as follows:
${\sigma }_{ij}(t)=$${\sigma }_{ij}(t)=$sigma_(ij)(t)=\sigma_{i j}(t)=${\sigma }_{ij}(t)=$ actual stress components,
${\in }_{ij}(t)$${\in }_{ij}(t)$in_(ij)(t)\in_{i j}(t)${\in }_{ij}(t)$ and $u_i(t)=$$u_i(t)=$u_(i)(t)=u_{i}(t)=$u_i(t)=$ actual strain and displacement fields,
${\sigma }_{ij}^{el}(t)=$${\sigma }_{ij}^{el}(t)=$sigma_(ij)^(el)(t)=\sigma_{i j}^{e l}(t)=${\sigma }_{ij}^{el}(t)=$ fictitious stress components corresponding to an ideal purely elastic response,
${\epsilon }_{ij}^{el}(t)$${\epsilon }_{ij}^{el}(t)$epsi_(ij)^(el)(t)\varepsilon_{i j}^{e l}(t)${\epsilon }_{ij}^{el}(t)$ and $u_i^{el}(t)=$$u_i^{el}(t)=$u_(i)^(el)(t)=u_{i}^{e l}(t)=$u_i^{el}(t)=$ fictitious elastic strains and displacements associated with ${\sigma }_{ij}^{el}(t)$${\sigma }_{ij}^{el}(t)$sigma_(ij)^(el)(t)\sigma_{i j}^{e l}(t)${\sigma }_{ij}^{el}(t)$.
To further characterize the internal force state, two additional stress fields are introduced:
${\sigma }_{ij}^R(t)=$${\sigma }_{ij}^R(t)=$sigma_(ij)^(R)(t)=\sigma_{i j}^{R}(t)=${\sigma }_{ij}^R(t)=$ the actual residual stress distribution,
${\overline{\sigma }}_{ij}^R=$${\overline{\sigma }}_{ij}^R=$bar(sigma)_(ij)^(R)=\bar{\sigma}_{i j}^{R}=${\overline{\sigma }}_{ij}^R=$ any arbitrary, time-independent self-stress distribution.
According to the constitutive relation for time-independent elasto-plastic materials, the total strain ${\epsilon }_{ij}$${\epsilon }_{ij}$epsi_(ij)\varepsilon_{i j}${\epsilon }_{ij}$ can be additively decomposed into elastic and plastic components:

This relation is assumed to hold at every time step throughout the loading history. Consequently, the material behavior does not involve any intrinsic time dependence, and the evolution of inelastic strains can be fully described within a rateindependent framework.
The elastic strain is related to the actual stress tensor via the fourth-order elastic stiffness tensor $C_{ijkl}$$C_{ijkl}$C_(ijkl)C_{i j k l}$C_{ijkl}$, assuming linear elasticity:

Meanwhile, the evolution of plastic strain is governed by the associated flow rule [6]:

where $f$$f$ff$f$ denotes the yield function and $\dot{\lambda }$$\dot{\lambda }$lambda^(˙)\dot{\lambda}$\dot{\lambda }$ is the plastic multiplier, as schematically illustrated for the von Mises yield surface under plane stress in Fig. 1.

Considering again the body introduced earlier, the total stress field ${\sigma }_{ij}(t)$${\sigma }_{ij}(t)$sigma_(ij)(t)\sigma_{i j}(t)${\sigma }_{ij}(t)$ at time $t$$t$tt$t$ can be expressed, in accordance with the constitutive relations presented in Eqns. (1) and (2), as the sum of a fictitious elastic stress component ${\sigma }_{ij}^{el}(t)$${\sigma }_{ij}^{el}(t)$sigma_(ij)^(el)(t)\sigma_{i j}^{e l}(t)${\sigma }_{ij}^{el}(t)$, corresponding to the elastic part of the total strain, and a time-independent residual stress component ${\sigma }_{ij}^R$${\sigma }_{ij}^R$sigma_(ij)^(R)\sigma_{i j}^{R}${\sigma }_{ij}^R$, which accounts for the irreversible effects of plastic deformation:

The fictitious elastic strain field ${\epsilon }_{ij}^{el}(t)$${\epsilon }_{ij}^{el}(t)$epsi_(ij)^(el)(t)\varepsilon_{i j}^{e l}(t)${\epsilon }_{ij}^{el}(t)$, associated with the fictitious elastic stress ${\sigma }_{ij}^{el}(t)$${\sigma }_{ij}^{el}(t)$sigma_(ij)^(el)(t)\sigma_{i j}^{e l}(t)${\sigma }_{ij}^{el}(t)$, is related to the stress field through the linear elastic constitutive relation introduced in Eqn. (2):

It should be emphasized that Eqns. (2) and (5) refer to different stress fields: Eqn. (2) involves the actual stress ${\sigma }_{ij}(t)$${\sigma }_{ij}(t)$sigma_(ij)(t)\sigma_{i j}(t)${\sigma }_{ij}(t)$, while Eqn. (5) is defined in terms of the fictitious elastic stress ${\sigma }_{ij}^{el}(t)$${\sigma }_{ij}^{el}(t)$sigma_(ij)^(el)(t)\sigma_{i j}^{e l}(t)${\sigma }_{ij}^{el}(t)$. These are connected through the decomposition in Eqn. (4), where the residual stress ${\sigma }_{ij}^R$${\sigma }_{ij}^R$sigma_(ij)^(R)\sigma_{i j}^{R}${\sigma }_{ij}^R$ does not vanish.
This stress decomposition facilitates the evaluation of the complementary strain energy associated with the residual stress field, offering a scalar quantity to characterize inelastic behavior within the elasto-plastic framework. To further clarify this concept, consider the total complementary plastic work accumulated along a loading path from $t=0$$t=0$t=0t=0$t=0$ to $t=\tau$$t=\tau$t=taut=\tau$t=\tau$. This work serves as a meaningful measure for quantifying the extent of plastic deformation and evaluating the inelastic performance of the body under consideration.
If a time-independent, self-equilibrated stress field ${\overline{\sigma }}_{ij}^R$${\overline{\sigma }}_{ij}^R$bar(sigma)_(ij)^(R)\bar{\sigma}_{i j}^{R}${\overline{\sigma }}_{ij}^R$ can be identified such that the following yield condition is satisfied throughout the volume $V$$V$VV$V$ at any time $t\le \tau$$t\le \tau$t <= taut \leq \tau$t\le \tau$ :

then the total complementary plastic work $W_p(\tau )$$W_p(\tau )$W_(p)(tau)W_{p}(\tau)$W_p(\tau )$ is bounded from above by the expression:

To control the extent of plastic deformation, an upper limit $W_{p0}$$W_{p0}$W_(p0)W_{p 0}$W_{p0}$ is prescribed on the allowable complementary plastic work $W_p$$W_p$W_(p)W_{p}$W_p$. This constraint is enforced by relating the limit directly to the actual residual stress distribution. Specifically, it is assumed that:

This identification yields a conservative and practical estimate, enabling the formulation of a tractable constraint on allowable plastic work. The resulting condition that limits plastic dissipation takes the form:

Elastic and elasto-plastic limit analysis based on residual forces

To establish a load-dependent response, assume that the surface traction $\mathit{q}(t,x_i)$$\mathit{q}\left(t,x_i\right)$q(t,x_(i))\boldsymbol{q}\left(t, x_{i}\right)$\mathit{q}(t,x_i)$ applied to the body occupying volume $V$$V$VV$V$, is defined as:

where $m(t)$$m(t)$m(t)m(t)$m(t)$ is a time-dependent load multiplier that increases monotonically, and $q_0\left(x_i\right)$$q_0\left(x_i\right)$q_(0)(x_(i))q_{0}\left(x_{i}\right)$q_0\left(x_i\right)$ is a time-independent surface load distribution defined over the surface $S_q$$S_q$S_(q)S_{q}$S_q$.
In the case of truss structures with constant structural weight $G_s=\rho \sum _{i=1}^n{A}_i{l}_i$$G_s=\rho \sum _{i=1}^n A_i{l}_i$G_(s)=rhosum_(i=1)^(n)A_(i)l_(i)G_{s}=\rho \sum_{i=1}^{n} A_{i} l_{i}$G_s=\rho \sum _{i=1}^n{A}_i{l}_i$ where $\rho$$\rho$rho\rho$\rho$ is the material density, $A_i$$A_i$A_(i)A_{i}$A_i$ is the cross-sectional area, and $l_i$$l_i$l_(i)l_{i}$l_i$ is the length of the $i$$i$ii$i$-th bar, the loading expression can be reformulated in terms of the externally applied nodal load vector ${\mathit{P}}_0$${\mathit{P}}_0$P_(0)\boldsymbol{P}_{0}${\mathit{P}}_0$ and a prescribed reference load $\mathit{P}$$\mathit{P}$P\boldsymbol{P}$\mathit{P}$, such that:

Throughout the loading history, the maximum value of the load multiplier $m$$m$mm$m$ for which all bar elements remain within the elastic range is denoted by $m_{el}$$m_{el}$m_(el)m_{e l}$m_{el}$. Consequently, the corresponding elastic limit load ${\mathit{P}}^{el}$${\mathit{P}}^{el}$P^(el)\boldsymbol{P}^{e l}${\mathit{P}}^{el}$ is given by:

Up to this point, no plastic deformations occur in any of the individual members, and the structural response remains entirely elastic. Accordingly, the elastic internal force vector corresponding to the applied external load ${\mathit{P}}_0$${\mathit{P}}_0$P_(0)\boldsymbol{P}_{0}${\mathit{P}}_0$ can be expressed as:

where $\mathit{F}$$\mathit{F}$F\boldsymbol{F}$\mathit{F}$ denotes the flexibility matrix, $\mathit{G}$$\mathit{G}$G\boldsymbol{G}$\mathit{G}$ is the geometric matrix, and $\mathit{K}$$\mathit{K}$K\boldsymbol{K}$\mathit{K}$ is the global stiffness matrix. However, once the load exceeds ${\mathit{P}}^{el}$${\mathit{P}}^{el}$P^(el)\boldsymbol{P}^{e l}${\mathit{P}}^{el}$, plastic deformations begin to develop in certain members and continue to evolve until the plastic collapse multiplier $m_{pl}$$m_{pl}$m_(pl)m_{p l}$m_{pl}$ and its corresponding load level ${\mathit{P}}^{pl}$${\mathit{P}}^{pl}$P^(pl)\boldsymbol{P}^{p l}${\mathit{P}}^{pl}$ are reached:

At this point, excessive plastic deformations prevent the internal stresses from equilibrating the externally applied load ${\mathit{P}}_0$${\mathit{P}}_0$P_(0)\boldsymbol{P}_{0}${\mathit{P}}_0$. Consequently, for load multipliers in the range $m_{el}\le m\le m_{pl}$$m_{el}\le m\le m_{pl}$m_(el) <= m <= m_(pl)m_{e l} \leq m \leq m_{p l}$m_{el}\le m\le m_{pl}$ a residual internal force vector ${\mathit{N}}^R$${\mathit{N}}^R$N^(R)\boldsymbol{N}^{R}${\mathit{N}}^R$, representing the remaining forces after unloading, can be defined as the difference between the plastic internal force ${\mathit{N}}^{pl}$${\mathit{N}}^{pl}$N^(pl)\boldsymbol{N}^{p l}${\mathit{N}}^{pl}$ and the elastic component ${\mathit{N}}^{el}$${\mathit{N}}^{el}$N^(el)\boldsymbol{N}^{e l}${\mathit{N}}^{el}$ :

In this context, for a truss structure of structural weight $G_s$$G_s$G_(s)G_{s}$G_s$, the complementary strain energy associated with the residual forces can be expressed as:

where $E$$E$EE$E$ is the elasticity modulus of the material, $N_i^R$$N_i^R$N_(i)^(R)N_{i}^{R}$N_i^R$ denote the residual force of the $i$$i$ii$i$-th bar, and $W_{p0}$$W_{p0}$W_(p0)W_{p 0}$W_{p0}$ represents the allowable limit of plastic deformation in the structure, beyond which the internal stresses can no longer maintain equilibrium, leading to structural collapse.
During elastic limit analysis, the primary objective is to determine the elastic load multiplier $m_{el}$$m_{el}$m_(el)m_{e l}$m_{el}$ for a structure with a given volume $V$$V$VV$V$ and corresponding structural weight $G_s$$G_s$G_(s)G_{s}$G_s$, and to compute the associated elastic limit load ${\mathit{P}}^{el}$${\mathit{P}}^{el}$P^(el)\boldsymbol{P}^{e l}${\mathit{P}}^{el}$, provided that the condition $W_p=0$$W_p=0$W_(p)=0W_{p}=0$W_p=0$ remains satisfied. In contrast, the aim of elasto-plastic limit analysis is to determine the maximum load multiplier $m$$m$mm$m$ such that the complementary plastic work satisfies the inequality $W_p\le W_{p0}$$W_p\le W_{p0}$W_(p) <= W_(p0)W_{p} \leq W_{p 0}$W_p\le W_{p0}$. To improve safety and enable precise control over inelastic deformations, a more restrictive threshold $0<W_{p,max}\le W_{p0}$$0<W_{p,max}\le W_{p0}$0 < W_(p,max) <= W_(p0)0<W_{p, \max } \leq W_{p 0}$0<W_{p,max}\le W_{p0}$ may be specified, corresponding to the elasto-plastic limit load $P^{ep}$$P^{ep}$P^(ep)P^{e p}$P^{ep}$ and its associated multiplier $m_{ep}$$m_{ep}$m_(ep)m_{e p}$m_{ep}$ :

The response curves in Fig. 2 schematically illustrate the identification of the three characteristic loads over the loading history of a representative truss (Fig. 2(a)). In Fig. 2(b), the load-plastic deformation diagram starts at $W_p=0$$W_p=0$W_(p)=0W_{p}=0$W_p=0$; the elastic limit $P^{el}$$P^{el}$P^(el)P^{e l}$P^{el}$ is read at the last equilibrium state that still satisfies $W_p=0$$W_p=0$W_(p)=0W_{p}=0$W_p=0$. The elasto-plastic limit $P^{ep}$$P^{ep}$P^(ep)P^{e p}$P^{ep}$ is obtained at the prescribed threshold $W_{p,max}$$W_{p,max}$W_(p,max)W_{p, \max }$W_{p,max}$ (with $0<W_{p,max}\le W_{p0}$$0<W_{p,max}\le W_{p0}$0 < W_(p,max) <= W_(p0)0<W_{p, \max } \leq W_{p 0}$0<W_{p,max}\le W_{p0}$ ), such that $W_p=W_{p,max}$$W_p=W_{p,max}$W_(p)=W_(p,max)W_{p}=W_{p, \max }$W_p=W_{p,max}$ and $P^{ep}=m_{ep}P$$P^{ep}=m_{ep}P$P^(ep)=m_(ep)PP^{e p}=m_{e p} P$P^{ep}=m_{ep}P$. Finally, the plastic limit $P^{pl}$$P^{pl}$P^(pl)P^{p l}$P^{pl}$ corresponds to the maximum sustained load on the path. For clarity, the ordinate is marked at $\{P^{el},P^{ep},P^{pl}\}$$\left\{P^{el},P^{ep},P^{pl}\right\}${P^(el),P^(ep),P^(pl)}\left\{P^{e l}, P^{e p}, P^{p l}\right\}$\{P^{el},P^{ep},P^{pl}\}$, highlighting the progression from the purely elastic regime to controlled inelasticity and, ultimately, to the plastic limit. Fig. 2(c) reports the associated displacement history, which is consistent with the transitions observed on the load-plastic deformation curve and confirms that $P^{ep}$$P^{ep}$P^(ep)P^{e p}$P^{ep}$ is reached well before plastic collapse.
This implies that, according to structural design criteria, a limited and controlled level of inelastic deformation may be permitted for a structure of given weight $G_s$$G_s$G_(s)G_{s}$G_s$. Alternatively, the elastic and elasto-plastic limit analysis problem can be reformulated as an optimization task, where the primary objective is to attain a prescribed load multiplier-corresponding to the target load ${\mathit{P}}_0$${\mathit{P}}_0$P_(0)\boldsymbol{P}_{0}${\mathit{P}}_0$-while minimizing the structural weight $G_s$$G_s$G_(s)G_{s}$G_s$. Within this framework, the cross-sectional areas $A_i$$A_i$A_(i)A_{i}$A_i$ of the individual members serve as design variables, framing the task as a size optimization problem, which is one of the three principal categories of structural optimization [21].

Building upon the previously established linear-elastic and elasto-plastic framework, this section extends the analysis to account for geometric nonlinearities that arise in structures undergoing large deformations. In certain structural systems, such as slender truss configurations, large displacements and rotations may significantly affect the mechanical response. To accurately capture such behavior, the analysis must incorporate geometric nonlinearity within a total Lagrangian finite element framework. In this formulation, the Green-Lagrange strain tensor ${\epsilon }_{ij}$${\epsilon }_{ij}$epsi_(ij)\varepsilon_{i j}${\epsilon }_{ij}$ and the second Piola-Kirchhoff stress tensor
$s_{ij}$$s_{ij}$s_(ij)s_{i j}$s_{ij}$ are employed. These quantities form a work-conjugate pair suitable for the analysis of large deformations while remaining consistent with the underlying constitutive laws.
The Green-Lagrange strain tensor is defined as:

where $u$$u$uu$u$ denotes the displacement field, and indices $i,j,k$$i,j,k$i,j,ki, j, k$i,j,k$ refer to spatial directions.
In finite element implementations, the incremental form of the strain-displacement relationship is written as:

where $\mathit{B}(\mathit{U})$$\mathit{B}(\mathit{U})$B(U)\boldsymbol{B}(\boldsymbol{U})$\mathit{B}(\mathit{U})$ is the displacement-dependent strain-displacement matrix and $d\mathit{U}$$d\mathit{U}$dUd \boldsymbol{U}$d\mathit{U}$ is the vector of incremental nodal displacements. The matrix $\mathit{B}(\mathit{U})$$\mathit{B}(\mathit{U})$B(U)\boldsymbol{B}(\boldsymbol{U})$\mathit{B}(\mathit{U})$ evolves with the current configuration, capturing the geometric nonlinearity of the deformation process.
In consistency with the previously presented Eqn. (2), the constitutive relation in terms of the Green-Lagrange strain tensor and the second Piola-Kirchhoff stress tensor is expressed as:

To assess equilibrium in the deformed configuration, the global residual vector is defined as:

where ${\mathit{P}}_0$${\mathit{P}}_0$P_(0)\boldsymbol{P}_{0}${\mathit{P}}_0$ is the vector of externally applied loads, and $\mathit{s}$$\mathit{s}$s\boldsymbol{s}$\mathit{s}$ represents the current second Piola-Kirchhoff stress field. The residual vector $\mathit{R}(\mathit{U})$$\mathit{R}(\mathit{U})$R(U)\boldsymbol{R}(\boldsymbol{U})$\mathit{R}(\mathit{U})$ quantifies the imbalance between internal and external forces and vanishes when equilibrium is satisfied.
The nonlinear equilibrium equations are solved iteratively using the Newton-Raphson method, in which the tangent stiffness matrix is computed as:

This iterative scheme updates the displacement field until convergence is achieved, enabling the simulation of both elastic and elasto-plastic behavior under geometric nonlinearity. The resulting displacement vector $\mathit{U}$$\mathit{U}$U\boldsymbol{U}$\mathit{U}$, representing the total nodal displacements at equilibrium, becomes a key quantity in the assessment of deformation-based criteria, such as serviceability. As such, it may also serve as the basis for imposing constraints on maximum allowable displacements in subsequent optimization formulations.
Importantly, this geometrically nonlinear formulation remains fully consistent with the residual force-based framework and the evaluation of complementary plastic work introduced earlier. The use of Green-Lagrange strains and second PiolaKirchhoff stresses ensures a coherent treatment of both material and geometrical nonlinearities, enabling accurate assessment of residual internal forces and the corresponding inelastic energy contributions throughout the deformation history. Furthermore, by operating within a total Lagrangian framework, this approach inherently allows the incorporation of initial geometric imperfections, which can significantly influence the structural response-particularly in slender systems sensitive to stability effects.

Expanding on the foundational concepts discussed in the preceding sections, this part of the study presents the mathematical formulation of two optimization problems: one based on elastic limit analysis and the other on elastoplastic limit analysis. The proposed formulations are integrated into a unified computational framework, enabling flexible and accurate structural design under both material and geometrical nonlinearities.

In the context of elastic limit analysis, the principal objective is to determine the elastic limit load ${\mathit{P}}^{el}$${\mathit{P}}^{el}$P^(el)\boldsymbol{P}^{e l}${\mathit{P}}^{el}$, which marks the onset of yielding in a structural system characterized by a given configuration and associated structural weight $G_s$$G_s$G_(s)G_{s}$G_s$. Beyond this threshold, plastic deformations begin to develop. Based on this theoretical foundation, the elastic design problem is formulated as an optimization task, where the aim is to minimize the structural weight $G_s$$G_s$G_(s)G_{s}$G_s$ while ensuring that the structural response remains entirely elastic and that sufficient global stability is maintained. This approach enables a design strategy that is simultaneously material-efficient and structurally robust, expressed mathematically as follows:

where the objective function " fitness " quantifies the overall quality of a given design solution based on three components: the structural weight term $f\left(G_s\right)$$f\left(G_s\right)$f(G_(s))f\left(G_{s}\right)$f\left(G_s\right)$, and two penalty terms, $p_1(m\mathit{P})$$p_1(m\mathit{P})$p_(1)(mP)p_{1}(m \boldsymbol{P})$p_1(m\mathit{P})$ and $p_2(\overline{\lambda })$$p_2(\overline{\lambda })$p_(2)( bar(lambda))p_{2}(\bar{\lambda})$p_2(\overline{\lambda })$, which enforce the elastic limit and stability conditions, respectively. The function $f\left(G_s\right)$$f\left(G_s\right)$f(G_(s))f\left(G_{s}\right)$f\left(G_s\right)$ aims to minimize the total structural weight within prescribed design bounds, using a normalized expression that incorporates the minimum and maximum allowable cross-sectional areas $A_{\text{min}}$$A_{\text{min }}$A_("min ")A_{\text {min }}$A_{\text{min }}$ and $A_{\text{max}}$$A_{\text{max }}$A_("max ")A_{\text {max }}$A_{\text{max }}$ , which together define the design domain. Accordingly, $f\left(G_s\right)$$f\left(G_s\right)$f(G_(s))f\left(G_{s}\right)$f\left(G_s\right)$ is expressed as:

For further clarification, the term $p_1(m\mathit{P})$$p_1(m\mathit{P})$p_(1)(mP)p_{1}(m \boldsymbol{P})$p_1(m\mathit{P})$ is introduced to ensure that the optimized structure reaches the predefined load level ${\mathit{P}}_0$${\mathit{P}}_0$P_(0)\boldsymbol{P}_{0}${\mathit{P}}_0$, while remaining within the elastic regime and avoiding failure due to instability or yielding. This term evaluates the extent to which the structure satisfies the fundamental design requirement of attaining the desired load-carrying capacity. The formulation distinguishes between four physically meaningful scenarios that may arise during the load history:

Here, $m\mathit{P}$$m\mathit{P}$mPm \boldsymbol{P}$m\mathit{P}$ denotes the maximum load attained during the load history, as determined by the load multiplier $m$$m$mm$m$, while ${\mathit{P}}^{el}$${\mathit{P}}^{el}$P^(el)\boldsymbol{P}^{e l}${\mathit{P}}^{el}$ represents the elastic limit load marking the onset of yielding, in accordance with the principles of elastic limit analysis. The interpretation of each case is as follows:

where ${\lambda }_{\text{stab}}$${\lambda }_{\text{stab }}$lambda_("stab ")\lambda_{\text {stab }}${\lambda }_{\text{stab }}$ denotes a prescribed threshold value for the critical buckling load factor, which can be freely selected in accordance with specific design requirements. If chosen to be relatively low, the influence of the stability penalty term becomes negligible. This formulation ensures that the optimization penalizes configurations with insufficient resistance to global buckling, thereby promoting structurally stable solutions that meet the prescribed safety criteria.
As presented through Eqns. (23b)-(23d), the structural weight term $f\left(G_s\right)$$f\left(G_s\right)$f(G_(s))f\left(G_{s}\right)$f\left(G_s\right)$, along with the penalty terms $p_1(m\mathit{P})$$p_1(m\mathit{P})$p_(1)(mP)p_{1}(m \boldsymbol{P})$p_1(m\mathit{P})$ and $p_2(\overline{\lambda })$$p_2(\overline{\lambda })$p_(2)( bar(lambda))p_{2}(\bar{\lambda})$p_2(\overline{\lambda })$, are formulated using normalized expressions. As a result, their values always lie within the range $[0,1]$$[0,1]$[0,1][0,1]$[0,1]$. This solution ensures consistent scaling within the overall objective function and enables the coherent evaluation of competing design objectives-such as minimizing material usage, maintaining elastic behavior, and ensuring structural stability-within a unified optimization framework. Consequently, the proposed formulation facilitates an efficient and robust search for optimal solutions that effectively balance weight reduction with safety requirements.
In addition to the objective function and associated penalty terms, further constraints are imposed to enhance the effectiveness of the design process. These are defined as follows:

Subject to:

Here, Eqn. (23e) introduces a displacement constraint that plays a critical role in ensuring structural serviceability, particularly in scenarios where excessive deformations may compromise performance or user comfort. In this context, $U_k$$U_k$U_(k)U_{k}$U_k$ denotes the displacement component corresponding to the $k$$k$kk$k$-th degree of freedom in the global displacement vector $\mathit{U}$$\mathit{U}$U\boldsymbol{U}$\mathit{U}$, while $U_{\text{max}}$$U_{\text{max }}$U_("max ")U_{\text {max }}$U_{\text{max }}$ represents the prescribed maximum allowable displacement, defined in accordance with applicable serviceability criteria or design standards. This formulation offers the flexibility to impose displacement limits at any node and in any direction, thereby accommodating a wide range of functional design requirements. During the optimization process, this constraint may be omitted in cases where serviceability is not a governing design criterion or where the primary focus is structural strength and stability.
Furthermore, Eqn. (23f) expresses the elastic condition by requiring that the complementary plastic strain energy $W_p$$W_p$W_(p)W_{p}$W_p$ remains zero. Incorporating the calculation of $W_p$$W_p$W_(p)W_{p}$W_p$ during structural analysis enables the identification of the elastic limit
load ${\mathit{P}}^{el}$${\mathit{P}}^{el}$P^(el)\boldsymbol{P}^{e l}${\mathit{P}}^{el}$. As the load $m\mathit{P}$$m\mathit{P}$mPm \boldsymbol{P}$m\mathit{P}$ increases, the structure initially responds elastically, and the point at which $W_p$$W_p$W_(p)W_{p}$W_p$ first becomes nonzero marks the onset of plastic deformation. At this point, the corresponding load multiplier $m_{el}$$m_{el}$m_(el)m_{e l}$m_{el}$ is defined, such that ${\mathit{P}}^{el}=m_{el}\mathit{P}$${\mathit{P}}^{el}=m_{el}\mathit{P}$P^(el)=m_(el)P\boldsymbol{P}^{e l}=m_{e l} \boldsymbol{P}${\mathit{P}}^{el}=m_{el}\mathit{P}$. This provides a physically meaningful and computationally consistent criterion for detecting the elastic threshold within the proposed framework. Ultimately, this constraint serves as a formal guarantee of structural performance in applications where inelastic behavior is not permissible.
Finally, Eqn. (23g) introduces the representation of initial geometric imperfections within the design formulation. Here, the structural geometry $\mathit{X}$$\mathit{X}$X\boldsymbol{X}$\mathit{X}$ is expressed as the sum of the perfect configuration ${\mathit{X}}_0$${\mathit{X}}_0$X_(0)\boldsymbol{X}_{0}${\mathit{X}}_0$ and an imperfection vector $\mathrm{\Delta }\mathit{x}$$\mathrm{\Delta }\mathit{x}$Delta x\Delta \boldsymbol{x}$\mathrm{\Delta }\mathit{x}$, which captures perturbations in the nodal positions. The term $\mathrm{\Delta }\mathit{x}$$\mathrm{\Delta }\mathit{x}$Delta x\Delta \boldsymbol{x}$\mathrm{\Delta }\mathit{x}$ is constructed as a linear combination of eigenmodes ${\varphi }_j$${\varphi }_j$phi_(j)\phi_{j}${\varphi }_j$ derived from LBA, each scaled by a corresponding weighting factor ${\omega }_j$${\omega }_j$omega_(j)\omega_{j}${\omega }_j$ that governs its amplitude. When imperfections are not considered, the geometry reduces to the undeformed state, $\mathit{X}={\mathit{X}}_0$$\mathit{X}={\mathit{X}}_0$X=X_(0)\boldsymbol{X}=\boldsymbol{X}_{0}$\mathit{X}={\mathit{X}}_0$. Otherwise, the designer is provided with flexibility to incorporate selected buckling modes and their influence magnitudes, enabling a more realistic and stability-conscious design process.
The developed framework integrates all components presented in Eqns. (23a)-(23g), offering a comprehensive formulation for the elastic design scenario. In addition to its core features, the framework is structured to optionally incorporate the displacement constraint Eqn. (23e), and the representation of initial geometric imperfections Eqn. (23g). This modularity enables flexible adaptation to a wide range of structural requirements, allowing the designer to tailor the optimization process based on the specific demands of different structural systems and performance criteria.

Building upon the elastic optimization scenario and integrating the theoretical framework of elasto-plastic limit analysis, an extended optimization formulation is proposed. In this approach, plastic deformations are permitted during the loading process but are restricted by a predefined threshold value $W_{p,max}$$W_{p,max}$W_(p,max)W_{p, \max }$W_{p,max}$, representing the maximum allowable complementary strain energy for an acceptable design. This parameter must satisfy the condition $0<W_{p,max}\le W_{p0}$$0<W_{p,max}\le W_{p0}$0 < W_(p,max) <= W_(p0)0<W_{p, \max } \leq W_{p 0}$0<W_{p,max}\le W_{p0}$, where $W_{p0}$$W_{p0}$W_(p0)W_{p 0}$W_{p0}$ denotes the upper bound of plastic deformation beyond which internal stresses can no longer maintain equilibrium, ultimately resulting in structural collapse. The value of $W_{p,max}$$W_{p,max}$W_(p,max)W_{p, \max }$W_{p,max}$ can be freely specified in accordance with design codes, performance objectives, or regulatory requirements. While the primary objective remains the minimization of structural weight $G_s$$G_s$G_(s)G_{s}$G_s$, the extended formulation explicitly regulates the allowable inelastic response, thereby enabling more material-efficient designs compared to the purely elastic scenario. Accordingly, the complete optimization problem is reformulated as follows: