Optimization of an additively manufactured self-supporting lattice-filled clevis component under combined loading

Abstract

Additively manufactured lattice structures are highly desirable in various fields because of their effectiveness in lightweight design and superior mechanical properties. Even with increasing popularity in different manufacturing fields, the requirement for post-processing and the limited usage of enclosed volume remains challenging. Therefore, the study investigates strut-based self-supported lattice design variables on mechanical properties via experimental validation and optimization. The clevis bracket, which is infilled with 3-, 4-, and 6-fold self-supporting lattice, is subjected to combined loading to simulate real case scenarios. Compression-shear and tension-shear loads were applied to the clevis while changing its variables: strut diameter, height, and overhang angle. The design is manufactured using the fused deposition modeling (FDM) with polylactic acid (PLA) material. The finite element model is validated with experimental results of manufactured specimens. Bayesian optimization (BO) algorithm minimizes stress and maximizes weight-saving value by alternating lattice type and its variables. The compression-shear performance of each specimen is much better than the tension-shear performance. While a 3-fold lattice gives the best result and broad design flexibility, a 6-fold lattice has the lowest performance. By changing variables, nearly the same result can be achieved for 3- and 4-fold strut lattices. The results show that significant weight saving is possible by using a self-supported lattice design.

Introduction

Thin-shell design is widely used in many industries, especially in aerospace and automotive. 80% of space vehicles and 50% of aerospace and automotive vehicles are designed and manufactured from sheets¹. It offers good mechanical properties with weight advantages compared to a machined design. Designing a part with complex shapes, low rigidity, and high quality is possible. By bending sheet material, it is possible to create various shapes that could be generated according to the imposed load². Due to its efficiency, design flexibility, and exceptional mechanical properties compared to its weight, it is a necessary design option in various applications.

Although it offers various benefits in the industry, it cannot be used in hard point connections like lug, fitting, and clevis designs. These designs create enclosed volumes inside the outer surface and cannot be filled with conventional manufacturing techniques. However, recent developments in 3D printing have made it possible to generate any enclosed shape and customized infill design. Lattice infill strategies have attracted significant attention due to their lightweight design and tailored mechanical properties^3,4. These techniques are highly desirable in various engineering applications such as aerospace, automotive, biomedicine, architecture, and heat transfer⁵. Even numerous studies have investigated infilled lattice designs; these investigations mostly focused on idealized lattice coupons. Despite its practical relevance, mechanical behavior and optimization of lattice filled structures are remain insufficiently explored. The most of the lattice optimization studies have disregarded manufacturability constraints especially in enclosed volumes.

The enclosed structures can be filled with various shapes according to their usage. Many alternative unit lattice cells have novel mechanical and physical properties in the literature⁶. The most used lattice designs are strut-based, sheet-based, and triply periodic minimal surface (TPMS) unit lattices. Strut lattices have high mechanical properties, and sheet-based lattices have superior mechanical properties and ease of manufacturing. TPMS lattices have a very low weight-over-volume ratio and superior heat transfer properties⁷.

Even though lattice has diverse alternatives, a critical design constant exists in enclosed structures, which is that post-processing is not possible after manufacturing. Since post-processing is not possible, the infilled lattice has to be self-supporting. In general, 45° is accepted as the threshold value to manufacture a self-supporting design in multiple 3D printing techniques⁸. This property eliminates most of the lattice options⁹. Self-supporting strut lattices are highly used because of their shape alternatives and design flexibility¹⁰.

Optimization is a well-established method for comparing design variables and finding optimum solutions for specific circumstances in specific domains. Several optimization algorithms have been proposed in the literature over the last decades¹¹. This paper uses the BO algorithm to investigate the effect of design variables and find the optimum solution. BO is highly suitable for black-box optimization problems with no objective function. A scaling method is needed to compare various lattice analysis results and loading. The equal-weight scaling method is preferred because of its simplicity¹².

To address these gaps in literature, the present study proposes a geometric based optimization approach for self-supporting lattice-filled clevis subjected a realistic combined loading condition. The 3-, 4-, and 6-fold strut-based self-supporting lattices are investigated due to their compatibility of self-supporting design with various design alternatives and manufacturability. Both compression-shear and tension-shear loading cases were experimentally tested and numerically modeled. Neuber’s energy-based correction technique is used to correlate experimental result with a linear finite element model to enable optimization without computationally expensive simulation.

Novelty of this study lies in four contributions. A self-supporting lattice-filled structural part under combined tension-shear and compression shear were investigated. The manufacturability constraints like self-supporting capability is taken into consideration in lattice optimization space. A realistic structural part is validated with experimental data rather than idealized lattice test coupons. The comparative optimization of three different strut lattice is revealed the critical role of nodal topology and joint density effect under combined loading. These findings provide new insight into lattice selection for additively manufactured load bearing joints and practical design concepts for self-supporting lattice-integrated designs.

Material and method

Material

The test specimen is a clevis model. The model is filled with three types of lattice design, which are 3-, 4-, and 6-fold strut lattices. The lattice types are given in Fig. 1 (a). Only strut types lattice is studied in order to guarantee self-supporting structures. Alternative lattice categories such as TPMS excluded because of local overhang violations¹³. Continuous surfaces make them unsuitable for enclosed clevis geometries.

Fig. 1.

(a) Lattice types, (b) Loading direction of clevis experiment specimen, (c) Optimization design variables.

Lattice diameter is taken as 1.6 m. 45°, 50°, 55°, and 60° strut overhang angles are selected for each type of strut lattice. The strut height is 12.8 mm for all specimens. Specimens were manufactured with PLA using FDM. Creality CR-5 Pro H 3D printer was used in manufacturing. All specimens were manufactured with a 0.2 mm thickness layer, 0.4 mm diameter nozzle, and 100% infill ratio. In the experiment, two sets of 12 specimens were tested. Each of the 12 test specimens has unique physical properties compared to the others, and 12 specimens were loaded in compression and shear direction. The remaining half of the specimens were loaded in tension and shear direction. Identical test specimens were tested in both loading directions.

Mechanical properties of the PLA are taken as 7200 MPa of elastic modulus and 36 MPa yield stress value were used as mechanical properties for samples¹⁴. However, these properties cannot be directly used in a voided structure. Solid material’s mechanical properties result in higher stress and lower strain. Mechanical properties of cellular material should be predicted numerically. Gibson-Ashby suggests that a cellular solid’s mechanical properties depend on the structure’s density compared to the full structure’s density¹⁵. If solid mechanical properties are used in finite element analysis (FEA), the results are stiffer and stronger than the test result, as expected. That’s why the Gibson-Ashby model should theoretically be used to predict cellular specimens’ mechanical properties. In analysis, stretch-dominated constants and modulus are used in tension-shear loading, and bending-dominated constants and modulus are used in compression-shear loading. These constants are denoting as CE in calculations. Table 1 gives constants used in a formulation.

Table 1.

Specimen mechanical properties.

Type	Weight (g)	Density ratio	CE (Stretch-dom.)	CE (Bending-dom.)
3F45	95	0.22	0.87	4.47
3F50	104	0.24	0.80	4.08
3F55	115	0.27	1.18	4
3F60	123	0.29	1.35	4
4F45	94	0.22	0.44	3.42
4F50	100	0.23	0.65	3.28
4F55	107	0.25	0.83	2.97
4F60	116	0.27	0.51	2.44
6F45	91	0.21	0.65	3.96
6F50	94	0.22	0.57	3.99
6F55	97	0.23	0.52	3.48
6F60	111	0.26	0.69	2.25

Loading

The experiment aims to find the capacity and effect of strut lattice type and its variables on mechanical properties under combined loading. In industry, parts undergo loading in various directions. However, in literature, most studies only simulate single-part loading, which does not simulate real problems. To simulate a real case scenario, the clevis is loaded in multiple axes instead of loading the specimen in a single direction. The loading direction creates a challenge because multiple actuators must load in various directions. To decrease complexity, new experiment setups were designed and manufactured. The specimen fixation is rotated 45°, and the experiment setup is loaded in one direction. By doing this, the specimen remains loaded in two directions, which are axial and shear. Specimens were loaded by using the unidirectional testing machine. All specimens were subjected to loading using an Instron 8801 testing system. The force capacity of the system is 100 kN. The experiment was conducted at room temperature with a constant 1 mm/min speed. Force-displacement data is collected from the experiment.

Numerical model

Experiments were simulated on nTopology software, version 5.13.2. The clevis model FEA details are given in Table 2.

Table 2.

FEA solver details of Clevis model.

Solver	nTopology 5.13.2
Analysis type	Linear Static analysis
Material modeling	Elastic
Element type	Tetrahedral
Contact	No
Boundary Condition	Constrained at the holes (1, 2, 3, 4, 5, 6)
Loading	Displacement

Even though multiple lattice design and analysis software are available, a limited number of them can allow a customized lattice to fill into the desired geometry. nTopology allows infill geometry with customized lattice and change design variables. However, nonlinear material modeling and analysis of clevis loading are not possible in software. Thus, simulation analysis can only be conducted with linear material and analysis. To bridge the gap between linear numerical analysis and nonlinear experimental results, Neuber’s energy-based correction method was employed. Moreover, the distortion energy theory was also used. Neuber’s method was used for this transition in the literature. These methods equalize energy between linear and nonlinear analysis¹⁶. By enforcing this energy equivalence, nonlinear stress-strain representation can be derived from linear FEA results^16,17. This method transformed the FEA results into a nonlinear curve. The method is mainly used for crack tip stress analysis to find correlations between linear and nonlinear material properties. Although Neuber’s method is mainly applied in notch and crack-tip analysis, recent studies have demonstrated its applicability to linear-nonlinear correlation^18,19. Neuber’s method provides a computationally efficient alternative to simulate fully nonlinear simulations while preserving analysis accuracy^19,20. In this approach, since all specimens have the exact dimensions, the area is accepted as the same for all specimens. As a result, correlation methods can be used in the clevis experiment FEA validation.

The clevis specimen is modeled using nTopology software. Outer shell thickness is defined as 2 mm from outer to inner. The lattices are infilled inside the voided volume. Linear static analysis is conducted because of the software’s incapability. Since large deformations are not the case, linear analysis is accepted. Fastener hole locations are constrained in the displacement direction. The total force is applied to the clevis from two holes equally. FEA model, boundary conditions, and load application holes are given in Fig. 1 (b). As a result of the analysis, force-displacement data and delta weight from the full specimen were collected. Force displacement data is used in experiment validation and stress optimization. Delta weight is evaluated while finding the optimum solution.

Optimization

In the optimization process, a clevis with a 2 mm wall thickness is filled with three types of self-supporting strut lattice and various design variables. Design variables are strut diameter, height, and overhang angle, as shown in Fig. 1 (c). Six different optimization analyses were performed because of different loading directions and lattice types. The algorithm evaluates the analysis result and repeats the cycle until the required analyses are completed.

The optimization objective is defined as minimization of peak von Mises stress and component weight simultaneously as stated in Eq. 1:

1

where is the FEA maximum von Mises stress result. W is the component weight. and are equal weighting factors applied in equal-weight scaling approach.

The design variables are,

• Strut diameter d [0.5, 3] mm,

• Strut height h ^10,30 mm,

• Strut overhang angle [45°, 60°] mm.

Outer shell thickness is fixed as 2 mm. Manufacturability constraints are the lowest diameter and overhang angle. Strut height higher than 30 mm cannot be applied to experiment specimen because of its dimensions. Remaining boundaries are selected to create voided structure and prevent stiff concentrations.

In optimization, the Bayesian optimization algorithm (BOA) is used. The optimization aims to find a lattice-filled clevis’s minimum stress level and minimum weight under multiple-direction loading. A sequential optimization technique is used to find the global minimum of a black-box function. BOA has four steps: promising solution selection, promising solution fitting, new solution generation by sampling, and incorporating the new solution into the original population¹². The population update and optimization steps are continued until the termination criteria are met. Termination criteria are accepted as a number of iterations. A convergence study is conducted to determine the required number of iterations. The iterations are increased from 25 to 500 iterations for selected cases. All objective function evaluations were performed using FEM simulations. No surrogate model was used to replace the finite element solver. 4-Fold lattice model is used for this convergence study. After 300 iterations, the error in the optimum solution decreases below 1% and stabilized, as given in Table 3. The cumulative moving average (CMA) formula was used in the convergence study. It is a statistical tool used to calculate the average of a data set. It smooths out short-term fluctuations and finds longer term trends. The CMA is used as a stopping criterion. The aim is to find certain number of iterations that change in CMA drop 1%. The CMA of each step is calculated, and the error value is calculated by comparing the CMA result of the former value. As a result of this, 300 iterations are accepted as the termination criteria.

Table 3.

Convergence result of optimization by using CMA method.

Iterations	Stress (MPa)	CMA Stress (MPa)	CMA stress change (%)
25	17.69	17.69	-
50	17.03	17.36	1.87
100	16.47	17.06	1.73
200	15.51	16.67	2.29
300	16.01	16.54	0.78
400	15.88	16.43	0.67
500	15.75	16.33	0.61

Clevis’s FEA model is parametrically modeled in nTopology, and the FEA model and outputs are self-generated with changed design variables. Self-supporting property and manufacturability are taken into consideration while determining design space. While optimizing the clevis, lattice types and variables were evaluated using collected results. The results were compared by using the equal-weight scaling method. It is a way of approach to allocate value equally across set of items. Scaling approach is required when the aim is to optimize multiple optimization objective same time. The goal is no longer to find best solution rather than a set of Pareto optimal solutions. In Pareto optimal solution, it cannot be improved one objective without making another objective worse. The outputs were ranked separately from minimum to maximum for the Von-Mises stress value and the weight value. Each result has an index value for stress and weight saving. The index is calculated as stated in Eq. 2. The stress and weight-saving indexes are added to include and evaluate two effects. n is the number of total solutions. The solution with the minimum index gives the best solution.

2

Results and discussion

Experimental validation

The numerical model is compared with experimental results. This work’s primary focus is optimizing infilled lattice parameters and understanding the effect of lattice type and variables on the strength and weight of the infilled specimens. Although optimization is the main focus, AM-related imperfections, mechanical properties of specimens, and loading need to be validated with experimental results. Different lattice types and overhang angle specimens were loaded up to failure in tension-shear and compression-shear directions in experiments. As a result of an experiment, the force-displacement curves are derived and compared with FEA outputs. The FEA result of the self-supporting strut lattice-filled clevis and the experiment results were compared. 3-, 4-, and 6-fold strut lattices with 45°, 50°, 55°, and 60° overhang angle specimens were loaded in compression-shear and tension-shear directions. The strut diameter is 1.6, and the strut height is 12.8 mm. The results were compared for the same lattice type with different overhang angles and the same overhang angle with varying lattice types. The experiment and FEA comparison of compression-shear and tension-shear loaded specimens are given in Fig. 2.

Fig. 2.

Experiment and FEA comparison of 3-, 4-, and 6-fold lattice filled clevis under compression-shear (a, b, c) and tension-shear (d, e, f) loading.

As a result of the comparison, FEA results correlate well with the experiment’s force-displacement result. Imperfections in the manufacturing cause acceptable discrepancies. In manufacturing, it is possible to encounter strut waviness, thickness variation, and strut oversizing²¹. This variable radically affects the carrying capability and displacement of the specimen. Most of the FEA results show a very good correlation with experimental results. Only limited experiment results show variation with the results. This supports the idea that some of the specimens have imperfections. The other reason that can create a difference between FEA and experiment is the solver type and analysis method. The linear analysis is conducted using FEA analysis. The local failures and more accurate results can be found with progressive failure analysis. However, this is impossible because of the high computational need and solver capability²². The differences between FEA and experiment generally resulted from these two effects.

As seen in Fig. 2, tensile-shear loading and compression-shear loading show discrete load-carrying behavior. The comparison of compression-shear and tension-shear loading FEA results is given in Fig. 3 (a). It can carry less load compared to compression-shear loading. Its failure is more brittle than compression-shear loading. Since compression-shear loaded specimens undergo bending-dominated loading, they deform longer than tension-shear loaded specimens²³. It has short deformations compared to compression-shear loading and hardly shows plastic behavior. This is mainly related to the effect of shear loading. The literature investigates the impact of shear load on the strut lattice. Shear loading creates tensile stress on struts and creates early failure²⁴. The applied shear load creates deformation of overall structure and struts are deformed. This deformation mostly involves bending. This bending enhances tensile loading on specimens. It generates a crack in the tensile stress part. Tensile-shear loading superposes this effect and fails earlier than compression-shear loading²⁵.

Fig. 3.

(a) FEA stress result, (b) Specimen failure types.

As seen from Fig. 3 (a), the compression and outer shell can carry a high load, so the stress result is highly red in those areas. This difference resulted from the load-carrying mechanisms of specimens. Struts can take a higher load in compression-shear loading and support the outer shell. However, struts cannot withstand a tensile load and fail first when tension-shear loading is performed. The support on the outer shell is collapsed. The results are mostly related to the overhang angle. This is consistent with experimental and FEA results.

When the failure of specimens was investigated, two different types of failure were observed for both types of loading. It is either lattice failure from the joint or intersection location between the shell and lattice, or the lattice can satisfy the loading, and the clevis hole region fails from bearing, as shown in Fig. 3 (b). The failure mode is directly related to the specimen’s load-carrying capacity. The lattice effect on hole area is limited because it has a very limited impact on area calculation. This outcome is consistent with Fig. 2. The specimens with the same failure modes have close load-carrying capacity. Even with different lattice types and overhang angles, the failure forces are very close to each other under the same failure mode. These results are also seen in Figs. 4 and 5.

Fig. 4.

Compression-shear loading experimental result comparison for same overhang angle.

Fig. 5.

Tension-shear loading experimental result comparison for same overhang angle.

The results were also compared for constant overhang angle for three different lattices in Figs. 4 and 5. The lower overhang angle results in lower failure force and deformation in compression. In the literature, similar results were found in different studies. Abdulhadi and Mian have investigated the effect of strut length and orientation. Elastic modulus and mechanical properties are improved with increasing overhang angle under axial compression²⁶. Under compression, the nodes undergo higher stress²⁷. Since a high overhang angle contains more joints than a low one, it can resist a higher load. Huang et al. found that a higher overhang angle contributes to buckling resistance. The deformation mode changes from uniform bending to folding²⁸. Since bending-dominated deformations are longer, it is consistent with lower overhang angles elongating larger than in specimens with higher overhang angles. The twinning-induced plasticity also explains this. It leads to a decrease in the effective dislocation glide distance. It acts as an obstacle to dislocation motion. This is called the Hall-Petch effect. The strain-hardening rate increases at high strength and creates uniform elongation^29,30. Since higher overhang angles have higher cooling surfaces, they solidify faster than low overhang angles, which lead to fine grains. They increase the elongation according to the Hall-Petch effect³¹.

In experimental results, the clevis can withstand a nearly equal force for the same lattice-type with a few exceptions. Specimens with the same failure force have the same failure mode. The specimens with different failure forces fail in other locations than the remaining specimens. Stronger lattices satisfy enough support in the most critical area and prolong failure. As a result, some other critical regions fail first. McClintock et al. found different failure modes with changing lattice types³². Lee et al. showed that while the BCC lattice performs worst under compressive loading, it is the best design option under shear loading³³. The microscopic failure highly depends on the loading type³⁴. That’s why some of the lattice combinations fail earlier than other lattice designs.

Optimization

The effect of parameters was compared and summarized in Figs. 6 and 7. The stress results are plotted for lattice variables. One of the objectives is to minimize the stress value by changing variables. The impact of radius, strut height, and overhang angle on stress can be seen in Figs. 6 and 7. Minimize stress and weight objects are defined in BOA. Since optimization algorithm models are different, each analysis result needs an index while combining results. The effectiveness of different optimization results must be compared. Since multiple independent runs exist, indexing is needed to eliminate randomness and obtain statistically significant results. Indexed results are more appropriate for comparing each other. As a result of optimization, the optimum design configuration for compression-shear and tension-shear loading is given in Table 4.

Fig. 6.

The effect of variables on optimization indices under compression-shear loading (Orange = 3-Fold, Yellow = 4-Fold, Green = 6-Fold).

Fig. 7.

The effect of variables on optimization indices under tension-shear loading (Orange = 3-Fold, Yellow = 4-Fold, Green = 6-Fold).

Table 4.

Optimum configuration of lattice under combined loading.

Loading	Lattice type	Radius (mm)	Height (mm)	Overhang angle (°)
Compression-shear	3-Fold	2.80	10	51
Tension-shear	3-Fold	2.42	18.60	59.6

Figures 6 and 7 illustrate the influence of lattice design variables on the optimization index, where lower optimization index values indicate improved stress-weight performance. While distributed results emphasize the increased sensitivity, clustering of index values highlights variable’s dominant influence for each loading condition. In Fig. 6, strut radius is the dominant parameter whereas strut height and overhang angle exhibit secondary influence. In Fig. 7, under tension-shear loading, the optimization response shows increased sensitivity to lattice geometry, particularly strut height and overhang angle.

In optimization, 3-fold lattices filled clevis show superior and weight-effective results. 6-fold lattice shows the worst possible design. This result can be attributed to a result of joint regions. In the same volume, a 3-fold lattice design has more joint locations than a 4-fold lattice. A 4-fold lattice also has more joint locations than a 6-fold lattice. Although strut junction points do not create weight and density, they create high stiffness^35,36. Many studies in literature investigate the joint effects on mechanical performance. Smith et al. used joints to increase specimens’ stiffness³⁷. Labeas and Sunaric increased joint end thickness to improve stiffness³⁸. Luxner et al., increased the stiffness of the beam 1000 times better by modifying nodal regions instead of the strut³⁹.

Figures 6 and 7 show that strut radius is a dominant design variable for both loading conditions. The larger strut radius improves net section area, load-carrying capacity and stiffness. For compression-shear loading, this behavior is consistent with previous studies on strut lattice structures. Researches show strut diameter strongly governs bending stiffness and stress distribution under compression loading. Increased radius enhances local bending resistance and results in lower stress concentration^40–42. For tension-shear loading, increased enhanced mechanical properties but lower efficiency. Although larger radii generally improve performance, wider scatter of optimization plot indicates that strut radii are insufficient to satisfy optimal behavior when tensile loading is dominant. This result is consistent with tensile dominant loading is more susceptible to individual stress localization rather than global stiffness increase⁴⁰.

The influence of lattice height has a limited effect on both loading conditions. As it can be seen from Figs. 6 and 7, higher optimization index values tend to group around intermediate height values. This group is mostly resulted from 6-fold for compression-shear loading and 4-fold tension-shear loading. While low strut height limit deformation capacity and increased specimen weight, high strut height increases strut slenderness, reduce bending resistance and stress level⁴³. Both lower and higher heights tend to decrease optimization performance. These results indicate that medium height range balances structural effective load transfer⁴⁴. Since 3-fold lattice has more voided design, it offers design alternatives for different lattice heights.

The effect of overhang angle differs between compression-shear and tension-shear loading. Although overhang angle change shows poor optimization index sensitivity under compression-shear loading, the change exhibits considerable influence on optimization performance under tensile-shear loading especially for 6-fold strut lattice. While overhang angle increasing, stress development and densification is increased^45,46.

As a result of optimization, it is observed that the effectiveness of lattice design is highly dependent on the applied loading mode⁴⁵. Strut radius dominates optimization behavior of compression-shear loaded specimens while lattice height and overhang angle are more influential on tension-shear loaded specimen. Thus, lattice optimization based on a single loading conditions may fail to find optimum design under realistic loading conditions.

Conclusion

This study investigates the effect of self-supporting strut lattice design variables on mechanical properties. The strut diameter, height, and overhang angle are set as design variables of 3-, 4-, and 6-fold self-supporting strut lattices. In industrial applications, structures undergo tension, compression, shear, and combination of these loads. However, in literature, most of the studies focus on single-direction loading scenarios, which is not possible in industrial problems. The study aims to find an answer to whether it is possible to use additively manufactured 3-, 4- and 6-fold self-supporting strut lattice structures were loaded with both compression-shear and tension-shear.

In conclusion, this study demonstrates that strut-based self-supporting lattice structures, when optimized appropriately, offer significant potential for weight reduction and structural efficiency under realistic loading conditions, where loaded with both compression-shear and tension-shear. Among the lattice configurations, 3-strut lattices provide the most balanced performance in terms of stress mitigation and lightweight design. The experimental validation confirms the predictive accuracy of the numerical models, and the design space defined here can serve as a baseline for future lattice-integrated structural components. These findings highlight the relevance of geometry-aware optimization in AM and pave the way for implementation in high-performance engineering applications.

Author contributions

M.O.T. conducted experiments and analysis. M.O.T. wrote the manuscript. Z.E. reviewed and edited the manuscript.

Data availability

Data is provided within the supplementary material.

Declarations

Competing interests

The authors declare no competing interests.