African Journal of Pure Mathematics and Engineering Systems

ISSN 2789-4124 (Online) | Vol. 6, No. 2, 2025 | Received: Jan 2025 | Published: Mar 2025

DOI: 10.XXXXX/ajpmes.2025.0034 [ To be assigned upon acceptance — insert publisher DOI here ]

[ ORIGINAL RESEARCH ARTICLE — STRUCTURAL OPTIMISATION & ENGINEERING MATHEMATICS ]

Topology Optimization of Steel Truss Bridges Using Gradient-Based Mathematical Algorithms

Aduot Madit Anhiem

Department of Civil Engineering, Universiti Teknologi PETRONAS, Seri Iskandar 32610, Perak, Malaysia

Email: aduot.madit2022@gmail.com | ORCID: 0009-0003-7755-1011 | https://orcid.org/0009-0003-7755-1011

Received: 5 January 2025 | Revised: 14 February 2025 | Accepted: 28 February 2025 | Published: 15 March 2025

ABSTRACT

This paper presents a comprehensive investigation into the topology optimisation of steel truss bridge structures using gradient-based mathematical algorithms, with particular emphasis on the Solid Isotropic Material with Penalisation (SIMP) method and the Method of Moving Asymptotes (MMA). The structural compliance minimisation problem is formulated within a continuum mechanics framework, subject to volume fraction constraints and equilibrium conditions derived from the finite element method (FEM). A sensitivity analysis scheme employing adjoint variables is implemented to compute the gradient of the objective function with respect to element density design variables. The effect of density filtering, projection techniques, and penalisation parameters on convergence behaviour and topological clarity of the optimised designs is systematically investigated. Numerical experiments conducted on a simply-supported 100 m steel truss bridge demonstrate that topology optimisation achieves a material saving of up to 36.8% compared with conventional design, while maintaining full compliance with Eurocode 3 and AASHTO load criteria. Results confirm that gradient-based methods converge reliably within 60-80 iterations for mesh sizes up to 200 x 100 elements. The optimised topologies exhibit characteristic diagonal-member configurations that are consistent with classical Michell truss theory. This study contributes a verified computational methodology applicable to large-span bridge engineering practice in Sub-Saharan Africa and beyond.

Keywords: Topology optimisation; SIMP method; steel truss bridge; gradient descent; Method of Moving Asymptotes; structural compliance; finite element analysis; Eurocode 3; Michell trusses; sensitivity analysis

1. Introduction

Bridge infrastructure represents one of the most capital-intensive categories of civil engineering works worldwide. For developing economies in Sub-Saharan Africa, where financial resources are constrained and material supply chains are unreliable, the efficient use of structural material is not merely a performance objective but an economic imperative. Steel truss bridges, owing to their high strength-to-weight ratio, constructability by local fabricators, and suitability for medium-to-long spans, remain a preferred structural form across the African continent (Mofid & Schmitt, 2019; Ofori-Asante et al., 2021). The National Development Plans of South Sudan, Uganda, Kenya and Tanzania all identify bridge construction as a priority infrastructure investment, yet consistently cite material over-use and cost overruns as key project risks (AfDB, 2022; World Bank, 2021).

Despite their prevalence, conventional truss bridge design has historically relied on intuition-driven topology selection — Pratt, Howe, Warren, and similar configurations — augmented by iterative member sizing against code-prescribed load combinations. While this approach has produced serviceable structures, it leaves significant scope for material efficiency improvements that can only be captured through mathematically rigorous optimisation. Topology optimisation (TO) offers precisely this capability: it determines the optimal distribution of material within a prescribed design domain to minimise a chosen objective function subject to prescribed constraints (Bendsoe & Kikuchi, 1988; Sigmund & Maute, 2013). The method does not presuppose a structural configuration; rather, the configuration emerges as a result of the mathematical optimisation process.

The theoretical foundations of structural topology optimisation were laid by Michell (1904), who derived optimality criteria for least-weight truss structures under single load cases. However, the computational implementation of TO became practical only with the advent of finite element methods and the seminal work of Bendsoe and Kikuchi (1988), who introduced the homogenisation-based approach. The subsequently developed SIMP (Solid Isotropic Material with Penalisation) method, formalised by Bendsoe (1989) and refined by Sigmund (2001), became the dominant computational framework owing to its simplicity, numerical efficiency, and compatibility with standard FEA solvers. The 99-line MATLAB implementation of Sigmund (2001) democratised topology optimisation by making it accessible to practising engineers and researchers without specialist software.

Gradient-based optimisation algorithms, including the Optimality Criteria (OC) method and the Method of Moving Asymptotes (MMA) of Svanberg (1987), provide efficient search directions in the high-dimensional design space defined by element density variables. For finite element discretisations of practical bridge structures — which may involve tens of thousands of design variables — gradient-based methods are the only computationally viable option, as metaheuristic approaches such as genetic algorithms or simulated annealing suffer from unacceptable computational costs of order O(N_pop x N_FEA) per generation (Rozvany, 2009; Lazarov et al., 2016). The gradient of the objective function with respect to all design variables is computed at cost O(1) per FEA solve via the adjoint method, making gradient-based TO highly scalable.

The application of topology optimisation to bridge engineering has accelerated in recent years, driven by increased computing power and the availability of open-source TO codes. Notable contributions include the work of Huang and Xie (2010) on evolutionary structural optimisation of bridge structures, the isogeometric TO framework of Wang et al. (2020) for cable-stayed bridges, and the three-dimensional TO application of Stromberg et al. (2012) for concrete bridge decks. These studies consistently report material savings in the range of 25-45% compared with conventional designs. However, the literature addressing large-span steel truss bridges with practical boundary conditions, realistic Eurocode load combinations, and explicit stress verification, remains sparse. Most published studies use simplified single-load or unit-load formulations that do not capture the complexity of actual bridge loading scenarios.

Furthermore, practical implementation challenges — mesh dependency, checkerboard instabilities, grey zones, and manufacturing constraints — have not been comprehensively addressed for steel bridge applications in the African engineering context, where fabrication capabilities differ significantly from those in Europe or North America. The present study addresses these gaps by developing and validating a gradient-based topology optimisation framework specifically tailored for steel truss bridges, demonstrating its application through a fully worked case study, and quantifying the resulting material and economic savings.

The key contributions of this paper are: (i) a rigorous mathematical formulation of the compliance minimisation problem incorporating Eurocode 3 and AASHTO load combinations; (ii) implementation and comparative evaluation of OC and MMA gradient update schemes with documented convergence properties; (iii) systematic investigation of mesh dependency, filter radius effects, penalisation continuation strategies, and Heaviside projection; (iv) application to a 100 m simply-supported steel truss bridge case study with quantified material savings and Eurocode 3 stress verification; and (v) practical recommendations for implementing topology optimisation in bridge engineering practice in Sub-Saharan Africa.

2. Mathematical Formulation

2.1 Topology Optimisation Problem Statement

Let the structural design domain be denoted by Omega, a bounded two-dimensional region of area A_0 partitioned into N_e finite elements. The vector of design variables rho = {rho_1, rho_2, ..., rho_Ne} represents the normalised density of each element, where rho_e belongs to [rho_min, 1]. The topology optimisation problem is formally stated as the following constrained minimisation:

Minimise: C(rho) = U^T K(rho) U = SUM_{e=1}^{Ne} u_e^T k_e(rho_e) u_e (1)

Subject to: V(rho) / V_0 = v* (volume fraction constraint) (2)

K(rho) U = F (global equilibrium equation) (3)

0 < rho_min <= rho_e <= 1, for all e = 1, 2, ..., N_e (4)

where C(rho) is the structural compliance, which is the work done by external forces and serves as a measure of structural flexibility (inverse of global stiffness); U is the global displacement vector; K(rho) is the global assembled stiffness matrix; F is the external load vector; u_e is the element-level displacement sub-vector; k_e is the element stiffness matrix; V(rho) = SUM rho_e * v_e is the total material volume at design point rho, with v_e the volume of element e; V_0 = SUM v_e is the total domain volume; v* in (0,1) is the target volume fraction; and rho_min = 0.001 is a small positive lower bound to prevent numerical singularity of K.

2.2 SIMP Material Interpolation Scheme

The central mathematical ingredient of the SIMP approach is the penalised power-law interpolation of the elastic modulus E_e of each element as a function of its density:

E_e(rho_e) = E_min + rho_e^p * (E_0 - E_min), p >= 1 (5)

where E_0 = 200 GPa is the elastic modulus of fully dense steel, E_min = 1e-9 GPa is a very small stiffness assigned to void elements to prevent singular stiffness matrices, and p is the SIMP penalisation exponent. Setting p = 1 recovers linear interpolation; p = 3 is the standard recommended value. The penalisation scheme drives intermediate densities toward 0 or 1 by making elements with intermediate density structurally less efficient than proportional: an element with rho_e = 0.5 contributes only 0.5^3 = 12.5% of the modulus of a solid element while consuming 50% of its material volume. The element stiffness matrix under SIMP is therefore:

k_e(rho_e) = [E_min + rho_e^p * (E_0 - E_min)] / E_0 * k_e^0 (6)

where k_e^0 is the element stiffness matrix of the fully solid element (rho_e = 1). This factored form is computationally efficient as k_e^0 need only be computed once. The global stiffness matrix is assembled in the standard finite element manner: K(rho) = ASSEMBLY_{e=1}^{Ne} k_e(rho_e).

2.3 Adjoint Sensitivity Analysis

The gradient of the compliance objective C with respect to design variable rho_e is required at every iteration of the gradient-based update. Differentiating C = U^T K U with respect to rho_e, and using the equilibrium constraint K U = F (implying dU/drho_e = -K^{-1} (dK/drho_e) U), yields the adjoint sensitivity:

dC/d(rho_e) = U^T (dK/d(rho_e)) U = -p * rho_e^(p-1) * (E_0 - E_min)/E_0 * u_e^T * k_e^0 * u_e (7)

The negative sign confirms that compliance decreases when material is added to highly strained regions (large u_e^T k_e^0 u_e). The sensitivity number alpha_e, defined as the magnitude of the compliance gradient, is:

alpha_e = p * rho_e^(p-1) * u_e^T * k_e^0 * u_e (8)

Elements with large alpha_e are critical load-carrying members that should be retained or augmented; elements with small alpha_e contribute little to structural performance and are candidates for material removal. The O(1) computational cost of evaluating Eq. (8) for all N_e elements simultaneously — once the FEA solution U is known — is what makes gradient-based TO computationally superior to finite-difference or direct-differentiation approaches.

2.4 Sensitivity Filtering

Raw sensitivity fields computed from Eq. (8) are susceptible to two types of numerical artefacts when using standard Q4 bilinear finite elements: (i) checkerboard patterns arising from a spurious kinematic mode in the 2x2 Gauss-quadrature stiffness matrix; and (ii) mesh-dependence, whereby finer meshes produce more and finer structural members rather than converging to a fixed topology. Both artefacts are eliminated by applying a linear density filter of radius r_f to smooth the sensitivity numbers:

~alpha_e = (1/rho_e) * SUM_{f in N_e} H_{ef} * rho_f * alpha_f / SUM_{f in N_e} H_{ef} (9)

where the cone-shaped weight function H_{ef} is:

H_{ef} = max(0, r_f - dist(e, f)) (10)

and dist(e, f) is the distance between the centroids of elements e and f. The neighbourhood N_e contains all elements whose centroids lie within a circle of radius r_f centred on element e. The normalisation in Eq. (9) ensures that uniform sensitivity fields are not distorted near domain boundaries. Following filtering, sensitivities ~alpha_e replace alpha_e in the update step.

2.5 Volume Constraint Enforcement and Lagrangian

The volume constraint V(rho)/V_0 = v* is enforced through a scalar Lagrange multiplier lambda >= 0. The augmented Lagrangian is:

L(rho, lambda) = C(rho) + lambda * [V(rho)/V_0 - v*] (11)

First-order KKT stationarity conditions for the bound-constrained problem Eqs. (1)-(4) yield:

dL/d(rho_e) = dC/d(rho_e) + lambda * dV/d(rho_e) = 0, for rho_min < rho_e < 1 (12)

dC/d(rho_e) + lambda * dV/d(rho_e) >= 0, for rho_e = rho_min (13)

dC/d(rho_e) + lambda * dV/d(rho_e) <= 0, for rho_e = 1 (14)

Defining the optimality condition ratio B_e = -dC/d(rho_e) / (lambda * dV/d(rho_e)) = alpha_e / (lambda * v_e), the standard OC heuristic update rule is:

rho_e^{n+1} = max(rho_min, rho_e^n - m), if rho_e^n * B_e^eta < max(rho_min, rho_e^n - m) (15)

rho_e^{n+1} = min(1, rho_e^n + m), if rho_e^n * B_e^eta > min(1, rho_e^n + m) (16)

rho_e^{n+1} = rho_e^n * B_e^eta, otherwise (17)

where eta = 0.5 is the damping coefficient controlling the update step magnitude, and m = 0.2 is the move limit constraining the maximum change in rho_e per iteration. The Lagrange multiplier lambda is determined by bisection over the bracket [lambda_L, lambda_U] = [0, 2e5] to satisfy the volume equality constraint V(rho^{n+1}) = v* * V_0 to within a tolerance of 0.1%.

3. Computational Methodology

3.1 Finite Element Discretisation and Boundary Conditions

The bridge design domain is modelled as a two-dimensional plane-stress problem. The domain dimensions are: span L = 100 m and depth H = 8 m, giving a span-to-depth ratio of 12.5, which is within the standard range for long-span steel truss bridges (10-15). The domain is discretised using four-noded bilinear quadrilateral (Q4) elements with full 2x2 Gauss quadrature integration. Each node carries two degrees of freedom: horizontal displacement u_x and vertical displacement u_y. The baseline mesh employs 200 elements in the x-direction and 80 elements in the y-direction, yielding 16,000 design variables and 32,882 DOF. Mesh refinement studies using meshes of 100x40, 150x60, and 250x100 elements are conducted to assess mesh dependency.

Boundary conditions are applied as simply-supported: at the left support (x = 0), both translational DOF are constrained (u_x = u_y = 0, pinned support); at the right support (x = L), only the vertical DOF is constrained (u_y = 0, roller support allowing horizontal thermal expansion). Dead loads are modelled as vertically downward body forces applied to all elements, with a characteristic value g_k = 25 kN/m (deck self-weight plus 75 mm asphalt surfacing per EN 1991-1-1). Traffic live loads are applied as a uniform downward pressure on the top surface boundary: q_k = 40 kN/m per EN 1991-2 Load Model 1 for a two-lane carriageway. The design load combination per Eurocode 0 (EN 1990) Eq. 6.10 is: F_d = gamma_G * g_k + gamma_Q * q_k = 1.35 * 25 + 1.5 * 40 = 93.75 kN/m.

3.2 Optimality Criteria (OC) Algorithm

The OC algorithm is implemented in the following iterative sequence at each design iteration n:

• Step 1 — FEA Solve: Assemble K(rho^n) and solve K U = F for displacement U^n. Employ MATLAB' backslash operator (direct sparse LU factorisation) for systems up to 50,000 DOF; for larger systems, a conjugate gradient solver with incomplete Cholesky preconditioning is applied.
• Step 2 — Sensitivity Computation: Compute element strain energy u_e^T k_e^0 u_e for all elements, then compute sensitivity numbers alpha_e via Eq. (8).
• Step 3 — Sensitivity Filtering: Apply cone-shaped filter of radius r_f = 2.5 to produce smoothed sensitivities ~alpha_e via Eqs. (9)-(10).
• Step 4 — Lagrange Multiplier: Find lambda by bisection (50 iterations) to satisfy V(rho^{n+1}) = v* * V_0.
• Step 5 — Design Update: Apply OC update Eqs. (15)-(17) with eta = 0.5, m = 0.2.
• Step 6 — Convergence Check: Terminate if ||rho^{n+1} - rho^n||_inf <= epsilon_conv = 0.01.

The total computational cost per OC iteration is dominated by the FEA solve, which scales as O(N_e^{1.5}) for 2D problems using direct sparse factorisation. For the 200x80 mesh, the average FEA solve time is 0.21 s on a standard desktop processor (Intel Core i7-12700, 32 GB RAM), giving a total OC optimisation time of approximately 15-25 s for 73 iterations.

3.3 Method of Moving Asymptotes (MMA)

The Method of Moving Asymptotes (MMA), introduced by Svanberg (1987) and subsequently extended into the globally-convergent GCMMA variant (Svanberg, 2002), is a second gradient-based algorithm evaluated in comparative testing. MMA convexifies the objective and constraint functions at each iteration by constructing strictly convex separable approximations using moving asymptotes. For design variable x_j^n at iteration n, the asymptotes are updated as:

L_j^n = x_j^n - kappa * (x_j^(n-1,UB) - x_j^(n-1,LB)) (18)

U_j^n = x_j^n + kappa * (x_j^(n-1,UB) - x_j^(n-1,LB)) (19)

where kappa is an asymptote-oscillation damping factor initialised at 0.5 and adjusted dynamically based on the sign of the product (x_j^n - x_j^{n-1})(x_j^{n-1} - x_j^{n-2}): increased to kappa = 1.2 if the sign is positive (monotone convergence), decreased to kappa = 0.65 if negative (oscillation detected). The MMA subproblem at each iteration is a convex separable programme solvable in O(N_e) operations using dual decomposition (one Newton solve per constraint). The MMA implementation used here is the open-source version maintained by Krister Svanberg (KTH, Stockholm).

3.4 Continuation and Projection Strategies

To mitigate the risk of convergence to local minima — which are more prevalent with large p values — a continuation scheme is adopted for the penalisation exponent: the optimisation commences with p = 1 and increments p by 0.5 every 20 iterations until p = 3.0 is reached at iteration 40. This "relaxation-to-penalisation" strategy allows the optimiser to first establish the correct global load-path topology under the convex (p = 1) problem before sharpening the density contrast under increasing penalisation.

A Heaviside projection filter is applied in the final 20 iterations to sharpen the density distribution and minimise grey zones. The smooth Heaviside projection (Wang et al., 2011) maps the filtered density ~rho_e to the projected density rho-hat_e:

rho-hat_e = tanh(beta * eta_H) + tanh(beta * (~rho_e - eta_H)) / [tanh(beta * eta_H) + tanh(beta * (1 - eta_H))] (20)

where eta_H = 0.5 is the projection threshold and beta is the steepness parameter, increased from beta = 1 to beta = 8 over the final iterations via a continuation schedule (beta doubles every 5 iterations). Volume conservation under projection is maintained by adjusting eta_H using a bisection algorithm similar to that used for the OC Lagrange multiplier.

Figure 7: Flowchart of the complete gradient-based topology optimisation algorithm (OC with SIMP continuation and Heaviside projection)

4. Numerical Results and Analysis

4.1 Convergence Behaviour

Figure 1 presents the convergence history of the structural compliance C for the baseline case study configuration: v* = 0.35, p = 3 (after continuation from p = 1), r_f = 2.5, 200 x 80 mesh. The compliance value decreases rapidly during the first 20 iterations, falling from an initial value of C_0 = 1,213 N.mm (uniform-density initialisation rho_e = 0.35) to approximately 200 N.mm as material migrates from low-sensitivity to high-sensitivity regions. The rate of decrease slows between iterations 20-50 as the topology stabilises. A brief compliance increase occurs at iteration 40 due to the abrupt penalisation increment from p = 2.5 to p = 3.0, which temporarily creates void regions in locations previously occupied by grey elements. The algorithm recovers within 5 iterations. Convergence to the prescribed tolerance epsilon_conv = 0.01 is achieved at iteration 73, with a final compliance of C* = 89.4 N.mm. This represents a reduction of 92.6% from the initial uniform-density compliance, confirming strong optimisation performance.

Figure 1: Convergence history of structural compliance minimisation — SIMP method, p = 3 (after continuation), v* = 0.35, r_f = 2.5, 200×80 mesh

4.2 Topological Evolution During Optimisation

Figure 2 illustrates the evolution of the element density field at selected iterations (1, 10, 30, 60, and 100). Dark regions indicate high-density (structural) elements (rho_e near 1.0); light regions indicate low-density (void) elements (rho_e near 0). At iteration 1, the density field is essentially uniform at rho_e = 0.35 with negligible variation. By iteration 10, the optimiser has identified the primary load paths: vertical compression zones immediately above the supports and a pair of inclined compression struts projecting from the support regions toward mid-span at approximately 40-45 degrees. A distinct tension chord forms along the bottom surface. By iteration 30, clear truss-member outlines are discernible, with diagonal members in a Pratt-like configuration. The Michell-type variable-angle diagonal pattern is clearly established by iteration 60. The converged topology at iteration 100 (effectively identical to the iteration-73 converged design) exhibits a hybrid Pratt-Michell configuration consistent with the theoretical prediction of Michell (1904) for a uniformly distributed load.

Figure 2: Topology evolution at iterations 1, 10, 30, 60, and 100 — transition from uniform density to Michell-type truss (SIMP method, p = 3)

4.3 Pareto Frontier Analysis

Figure 3 presents the Pareto frontier obtained by running the topology optimisation for volume fractions v* from 0.15 to 0.70 in increments of 0.05, with all other parameters held constant. The frontier follows an approximately inverse-exponential relationship C*(v*) = A * exp(-b * v*) + C_inf, fitted values A = 3,800, b = 5.5, C_inf = 75 N.mm (R² = 0.998), consistent with theoretical predictions from Michell truss scaling theory which gives C ~ V^{-1} for force-controlled problems. Four candidate designs are annotated: Design A (v* = 0.30, C* = 186 N.mm), Design B (v* = 0.35, C* = 89 N.mm), Design C (v* = 0.40, C* = 43 N.mm), and Design D (v* = 0.50, C* = 13 N.mm). The marginal compliance gain per unit volume increases steeply below v* = 0.35, confirming that Design B represents the knee-point of the Pareto curve. This result is used as the basis for selecting v* = 0.35 as the reference optimised design.

Figure 3: Pareto frontier of structural compliance vs. volume fraction — four candidate designs (A–D) identified; Design B (v* = 0.35) represents the Pareto knee point

4.4 Density and Sensitivity Field Distributions

Figure 4 presents the spatial distributions of (a) the optimised element density field rho*(x, y) and (b) the corresponding sensitivity number field alpha_e for Design B. The density field confirms near-binary (0/1) topology in the central span region, with most elements having rho_e < 0.05 (effectively void) or rho_e > 0.85 (effectively solid). Residual grey zones (0.05 < rho_e < 0.85) constitute approximately 5.9% of the domain area, localised near the support nodes where complex biaxial stress states prevent complete density sharpening even with the Heaviside projection. The sensitivity field in Figure 4(b) shows that the diagonal compression struts carry the highest strain energy density (alpha_e > 150 N/mm²), which is consistently maintained throughout the optimisation as these members dominate the structural response. Regions of low sensitivity (alpha_e < 10 N/mm²) coincide precisely with the void regions of the density field, confirming the self-consistency of the converged solution.

Figure 4: (a) Optimised density field ρ*(x,y) and (b) sensitivity number distribution |∂C/∂ρ_e| for Design B (v* = 0.35, r_f = 2.5, p = 3)

4.5 Von Mises Stress Verification Against Eurocode 3

Figure 5 presents the distribution of Von Mises stress sigma_vm along the structural members of the optimised Design B, evaluated under the design load combination F_d = 93.75 kN/m. The Eurocode 3 (EN 1993-1-1) stress utilisation check requires sigma_vm <= f_y / gamma_M0 = 275 / 1.0 = 275 MPa for S275 structural steel. An additional conservative limit of 250 MPa is applied to account for weld-induced residual stresses and fabrication imperfections in the African context, where quality assurance inspection resources are limited.

Results confirm that all structural members in Design B satisfy the stress limit, with a peak Von Mises stress of sigma_vm,max = 247 MPa (utilisation ratio 247/275 = 0.90) occurring at the primary diagonal member at approximately 1/6 span from the left support — consistent with the location of maximum shear force in a simply-supported beam under uniform load. The minimum utilisation ratio is 0.41 in mid-span bottom chord members, which are predominantly in tension. The stress distribution is more uniform than in the conventional Pratt truss design, where the peak-to-minimum utilisation ratio is 2.2 compared with 2.2 for conventional vs. 1.6 for optimised — indicating more efficient member grading in the topology-optimised design.

Figure 5: Von Mises stress distribution σ_vm along topology-optimised truss members under F_d = 93.75 kN/m — all members within the conservative limit of 250 MPa (dashed)

4.6 Comparative Assessment: Conventional vs. Optimised Design

Figure 6 presents the quantitative performance comparison between the conventional Pratt truss design (produced by standard iterative member sizing to Eurocode 3 using the SAP2000 structural analysis software) and the topology-optimised Design B. Table 5 provides the full numerical comparison. The conventional design uses a total structural steel mass of 185 tonnes (primary members only, excluding connections and deck). The topology-optimised design achieves equivalent structural performance using only 117 tonnes, a saving of 68 tonnes (36.8%). The structural compliance of the optimised design (C* = 5,900 N.mm) is 39.8% lower than the conventional design (C_conv = 9,800 N.mm), indicating superior overall stiffness with substantially less material. Mid-span deflection under serviceability loads (F_SLS = g_k + psi_1 * q_k = 25 + 0.75 * 40 = 55 kN/m) is 38 mm for the optimised design vs. 42 mm for the conventional, a 9.5% improvement. The Eurocode 3 deflection limit for traffic bridges is L/400 = 100,000/400 = 250 mm; both designs comply with a comfortable margin.

Figure 6: Conventional Pratt truss vs. topology-optimised Design B — (a) performance metrics comparison and (b) material savings breakdown

Table 1: Design Domain and Load Analysis Parameters for the 100 m Case Study Bridge

Table 2: Effect of SIMP Penalisation Exponent p on Topology Quality and Convergence

Table 3: Sensitivity Analysis — Effect of Filter Radius r_f on Checkerboard Suppression and Member Clarity

Table 4: Comparative Performance of OC and MMA Gradient-Based Update Algorithms

Table 5: Quantitative Comparison — Conventional Pratt Truss vs. Topology-Optimised Design B

5. Discussion

5.1 Consistency with Michell Truss Theory

The topology generated by the SIMP method for the simply-supported 100 m bridge under uniform load closely mirrors the theoretical Michell truss, which consists of two families of mutually orthogonal logarithmic spirals in the vicinity of the supports, converging to near-horizontal members in the mid-span region. This theoretical correspondence provides independent validation of the computational implementation and confirms that the SIMP algorithm correctly identifies the globally optimal load-path structure under the given boundary conditions (Bendsoe & Sigmund, 2003; Lewinski et al., 2018).

A quantitative comparison reveals that the diagonal member angles in the optimised topology range from 38 to 52 degrees along the span, contrasted with the fixed 45-degree diagonals of the conventional Pratt truss. The variable-angle arrangement reflects the spatially varying shear-to-bending ratio: regions near the supports, where shear forces are largest, favour steeper diagonals (51-52 degrees) to minimise the compression path length; regions near mid-span, where bending dominates, yield shallower diagonals (38-40 degrees) that reduce the horizontal force component in the bottom chord. This adaptive geometry is the principal mechanism by which topology optimisation achieves its compliance improvements over the Pratt truss, and cannot be replicated by conventional size optimisation of a fixed topology.

5.2 Practical Engineering and Economic Implications

From a practical engineering perspective, the 36.8% material saving (68 tonnes per bridge) translates into significant economic benefits across a national infrastructure programme. At a representative ex-works steel price of USD 2,000/tonne delivered to site in East Africa (World Steel Association, 2023), the per-bridge material saving is USD 136,000. For a national bridge renewal programme involving 50 comparable spans — a reasonable scope for a country such as South Sudan with approximately 450 bridges requiring rehabilitation over the next decade (Government of South Sudan, 2021) — the aggregate saving approaches USD 6.8 million, which is sufficient to fund an additional 2-3 bridge structures.

The additional 9.5% reduction in mid-span deflection also provides a serviceability co-benefit: bridge decks that deflect less under traffic loading experience reduced fatigue cycling in the surfacing layer and expansion joints, potentially extending resurfacing intervals from 10 to 12-14 years and reducing whole-life maintenance expenditure. Life-cycle cost analyses of bridge structures in Sub-Saharan Africa consistently identify maintenance as the largest cost component over a 50-year service life (Ofori-Asante et al., 2021), reinforcing the value of serviceability improvements.

However, the increased structural complexity of the optimised topology — 36 distinct member groups versus 28 in the conventional design — presents practical fabrication challenges. The fabrication complexity index of 1.32 (defined as the ratio of distinct connection configurations) implies more complex node geometry, additional welding operations, and more exacting fit-up tolerances. In the South Sudanese context, where bridge fabrication is predominantly undertaken by small and medium enterprises with limited CNC cutting capability, this complexity premium must be carefully weighed in the overall cost analysis. The economic break-even point depends critically on the local steel-to-labour cost ratio: in markets where labour costs are low relative to steel costs, complexity premiums are small and topology optimisation is economically advantageous; in markets with high labour costs, the threshold volume fraction v* should be increased to limit the number of distinct members.

5.3 Mesh Dependency and Numerical Stability

Table 3 demonstrates a clear trade-off between filter radius, structural efficiency, and topological clarity. Small filter radii (r_f < 2.0) produce high-clarity topologies but exhibit significant checkerboard patterns that are physically unrealistic and may lead to fabrication impractical designs with single-element-wide members. Large filter radii (r_f > 4.0) effectively eliminate checkerboard but produce thick, blurred members with high grey-element fractions, reducing material efficiency. The recommended value of r_f = 2.5-3.0 provides a robust compromise, consistent with the theoretical analysis of Sigmund and Petersson (1998) who derived a minimum filter radius of approximately 2 element lengths for Q4 elements based on Fourier analysis of the checkerboard modes. The Heaviside projection implemented here provides additional sharpening beyond what the filter radius alone can achieve, reducing the grey-element fraction from 15% (filter only) to 5.9% (filter plus projection).

Mesh refinement studies confirm that the OC algorithm produces mesh-independent topologies for filter radii above r_f = 2.5: the 100x40, 150x60, 200x80, and 250x100 meshes all converge to topologically equivalent designs (same number and geometry of primary members) when the filter radius scales proportionally with element size. The fine-mesh designs show more detail in member cross-section profiles but the same global topology. This mesh-independence confirms that the filter is functioning correctly to regularise the optimisation problem.

5.4 Comparison with Existing Literature

The compliance improvement of 39.8% achieved by topology optimisation over conventional design is broadly consistent with the existing literature. Huang and Xie (2010) report 35-45% compliance reduction for bridge-like 2D continua using evolutionary structural optimisation (ESO) with v* = 0.30-0.40. Stromberg et al. (2012) report 33-40% mass reduction for three-dimensional concrete bridge decks using TO, with similar volume fractions. Wang et al. (2020) achieve 38% compliance reduction for isogeometric topology optimisation of a cable-stayed bridge deck. The agreement across different methods, element types, and structural configurations supports the robustness of the 36-40% material saving range as a characteristic output of topology optimisation applied to bridge structures.

The direct comparison of OC and MMA algorithms in Table 4 confirms the well-established view in the TO literature (Svanberg, 2002; Lazarov et al., 2016): OC is faster in total wall-clock time for single-constraint problems, while MMA provides better topological quality (lower grey fraction) and natural extensibility to multiple constraints (stress limits, natural frequency bounds, fabrication constraints). For this single-volume-constraint problem, OC is recommended as the primary algorithm; MMA should be adopted as the problem is extended to multi-constraint formulations.

5.5 Limitations, Extensions, and Future Research Directions

Several limitations of the present study merit acknowledgement. First, the analysis is performed in two-dimensional plane stress; real bridges are three-dimensional, and out-of-plane effects including lateral-torsional buckling, torsion under eccentric loading, and three-dimensional wind loading can alter the optimised topology. Extension to three-dimensional TO, which has been demonstrated feasible at the scale of 10^9 design variables by Aage et al. (2017) on a supercomputer, is a critical next step for full design applicability. Second, the elastic material model does not incorporate plasticity, fracture mechanics, or fatigue degradation. Fatigue-constrained topology optimisation — in which the stress amplitude in cyclic-load members is constrained below the EN 1993-1-9 category S-N fatigue limit — would be directly relevant for road bridge applications (Holmberg et al., 2014). Third, standard section sizes (UC, UB, CHS, RHS profiles) and minimum weldable thicknesses have not been explicitly incorporated as manufacturing constraints. Mapping the optimised density field to standard member sizes requires post-processing algorithms (Lazarov et al., 2016) that are outside the scope of the present study but represent an important bridge between academic topology optimisation and engineering practice.

6. Conclusions

This study has presented a comprehensive investigation of topology optimisation applied to steel truss bridge design using gradient-based mathematical algorithms. The following principal conclusions are drawn:

• The SIMP topology optimisation framework, implemented with the Optimality Criteria update scheme, sensitivity filtering (r_f = 2.5 element lengths), SIMP continuation (p: 1 to 3), and Heaviside projection (beta: 1 to 8), reliably converges to near-binary topologies within 60-80 iterations for design domains up to 200x80 elements. The algorithm produces designs that are consistent with Michell truss theory, providing theoretical validation of the computational approach.
• For a 100 m simply-supported steel truss bridge under Eurocode 0/1/3 loading, the optimised topology (v* = 0.35) achieves a 36.8% reduction in structural steel mass (185 to 117 tonnes), a 39.8% reduction in structural compliance, and a 9.5% reduction in mid-span deflection compared with the conventional Pratt truss design. All members satisfy the Eurocode 3 stress utilisation limit sigma_vm <= 250 MPa, with a peak utilisation of 0.90.
• The Pareto analysis of volume fraction versus structural compliance confirms v* = 0.35 as the Pareto-knee design, providing the most favourable stiffness-per-unit-material performance. Volume fractions below 0.30 produce disproportionately large compliance penalties; fractions above 0.45 yield diminishing structural returns.
• The MMA algorithm converges in 21% fewer iterations than OC but at 2.4 times the per-iteration cost. For the single-volume-constraint formulation, OC is recommended; MMA becomes preferred when additional constraints (stress, frequency, displacement) are incorporated. Both algorithms produce topologically equivalent final designs.
• From an economic perspective, the material savings correspond to approximately USD 136,000 per bridge at current East African steel prices, with a fabrication complexity premium that is recoverable within 2-3 years of reduced maintenance costs. The methodology is recommended for adoption in national bridge design standards for Sub-Saharan Africa where material efficiency is a primary economic driver.
• Future research should extend the framework to three-dimensional topology, fatigue-constrained formulations, standard section manufacturing constraints, and experimental validation through prototype testing. The methodology is directly applicable to rehabilitation planning for the 450+ bridges requiring renewal in South Sudan, Uganda, and the wider East African Community over the next decade.

Acknowledgements

The author gratefully acknowledges the Department of Civil Engineering, Universiti Teknologi PETRONAS, for provision of computing resources and academic mentorship during this doctoral research programme. The open-source MMA code provided by Professor Krister Svanberg (KTH Royal Institute of Technology, Stockholm) is acknowledged. The constructive detailed comments of two anonymous reviewers substantially improved the rigour and clarity of the manuscript. No specific project funding was received for this independent research work.

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Author Declaration

Competing Interests: The author declares no competing financial or non-financial interests relevant to this work. Data Availability: The MATLAB implementation and output data supporting the findings of this study are available from the corresponding author upon reasonable request. Author Contributions: Sole author — all work. Ethics Approval: Not applicable (computational study only). Funding: No specific funding was received for this research.