As a case study, a steel tubular structure fabricated by CIMOLAI SPA is considered (Fig. 2). It is a 67 × 124-m double-layer steel space frame structure with hollow cross-section elements of different shapes and sizes. Due to the unusual shape of the structure, the members intersect in several ways. There is a consequent development of complex joint geometries with a high density of welded connections. The present analysis focuses on a typical joint with the scope of developing a general methodology, which can also be applied to other joints with different configurations.
Two conventional joint fabrication methods of this structure are schematized in Fig. 3. In the conventional solution 1, there are many welding and cutting points to connect the various elements and to add stiffeners inside the circular hollow steel section (HSS). The profiles had to be cut and welded to insert the internal stiffeners. In conventional solution 2, the connection is fabricated using thread pre-stressed bolts and milled surfaces welded to the HSS elements. In this case, a removable access panel was necessary to apply the bolts, weakening the members.
We propose an alternative to these highly complex and time/resource-consuming fabrication methods. Our design idea is to customize the complex node geometries at each joint with a free-form geometry optimized for high stiffness, resistance, and robustness. The result is a sort of “natural” stiffener inside the tubular joint. The optimized joints can be 3D printed and then welded (or bolted) to the rest of the structure made of conventional steel profiles, as described in Figs. 4 and 5.
We call this approach “Structural Design for Additive Manufacturing (SDfAM)”, and its advantages can be listed as follows:
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The optimized joint is “naturally” strengthened and stiffened. This eliminates most of the operations described above for the two conventional solutions.
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The joint can be welded perpendicularly to the tube profiles with a smoother transition in the welded region.
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The orthogonal welding removes the joint region from the node core to a less stressed area, reducing the stress concentrations and residual stresses.
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Lower stress concentrations may result in higher static and fatigue performance of the joint.
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Reduced stress concentrations and force eccentricities will result in a decreased weld size in the joint (with an extra possibility to decrease the tube wall thickness).
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The welded zone becomes more accessible for periodic inspection and maintenance during the service life of the structure.
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All these can result in less shop-welding time and consumption and hence, less environmental impact.
This paper focuses on the design and development of the “optimized joint” rather than the welded region. The latter will be the subject of a future paper. The final design is achieved after several steps:
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1st step: Computer-aided design (CAD) for the concept and geometric design.
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2nd step: Computer-aided engineering (CAE) to construct the finite element model (FEM) from CAD geometry. Mesh, loads, and boundary conditions are defined.
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3rd step: Topology optimization starting with a “solid” body for design space (DS). The objective function and constraints are defined. The minimum compliance design with a volume fraction and stress constraints are considered.
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4th step: Finite element analysis (FEA) is performed for all the cases using Optistruct-Hyperworks 2017 (Altair University 2017).
Definition of the design space
To build-up the 3D model of the joint, a parametric joint builder tool was developed (Fig. 6). The original structure consists of four different groups of joints with similar geometry (170 types of joint geometries in total). For this article, two representative versions are generated: a symmetric (S) and an asymmetric case (A) (Fig. 7a and b).
An unstiffened and a benchmark model have been simulated for each case (Fig. 8). The benchmark presents a conventionally cut-weld solution similar to the ones shown in Fig. 3. The design space is highlighted in Fig. 9. Boundary conditions (loads and support constraint) are linked to the non-design spaces that are constant volumes.
The new joint shapes are designed with the SDfAM approach. To analyze the new shapes, stainless steel 316 L (EN_6892–1 2009) has been chosen. Among the metal powder alloys available in the metal 3D printing market, this material has appropriate strength, durability, and weldability characteristics (Stratasys 2020) for the structure under investigation. The unstiffened and the benchmark shapes represent joints to be produced by traditional fabrication techniques. Therefore, for those shapes, a common steel material available in the market is selected (AISI, 1080), which has similar mechanical characteristics with SS 316 L. The yield strength of both materials is equal, while SS-316 L presents lower ultimate strength than the traditional steel, which keeps our results on the safe side. Figure 10 and Table 2 present the material properties used in the simulations.
Table 2 Material properties As boundary conditions, a simple support at the bottom and axial loads on top and four lateral faces have been applied. Four different loading conditions were considered to make sure the worst-case scenario is present in the stress analysis: full compression, full tension, and two asymmetric loadings (Fig. 11).
The finite element model consisted of a solid discretized in a tetra-mesh generated by solidThinking Inspire. Its algorithms use a combination of HyperMesh and Simlab for meshing. The tetra-mesh configuration created in solidThinking Inspire presents first-order elements, which were fair and accurate for the initial set of analysis performed in this study. More advanced analysis has been performed using Hyperworks (Altair Engineering 2017).
Topology optimization of the tubular joint design space
Figure 12 shows our SDfAM process using a truss-optimization analogy: the removal of material from a solid body where it is not needed, based on stress analysis. This approach is well-established in high-tech industries, such as automotive and aerospace. In our case, the inner material distribution is considered as a design variable. The solid isotropic material with penalization (SIMP) method (Bendsøe and Sigmund 2004) is used.
The design domain is discretized so that only the material densities equal to one remain and those equal to zero are cancelled. The topology optimization is performed with the Optistruct/Hyperworks software (Altair Engineering 2017) using the following inputs:
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Definitions of design and non-design spaces
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Minimum compliance, as objective
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Material density distribution, as variable
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Volume fraction and maximum stress, as constraints
Our objective of topology optimization is to minimize the compliance under volume and stress constraints (the joint stiffness is maximized when the compliance is minimum). For this reason, as constraints, we used different volume fractions of the design space (DS) that is described in Fig. 9. The optimized solution was found by the SIMP method when the objective function and the constraints reached the convergence. Material density is cancelled in the zones where the stresses are low and placed for high stress concentration zones according to the loading path. Thus, the full solid core of the joint is transformed into a sort of internal truss, resulting in “internal natural stiffeners”. These joints with new shapes are then validated through finite element analysis. Moreover, some authors have demonstrated the possibility to also improve the fracture resistance at an increasing volume fraction during the topology optimization process (Da et al. 2018) and to obtain a desired overall stiffness of the joints (Kang and Li 2017).
Table 3 shows the loading conditions considered for different shapes characterized by the degree of volume and stress constraints (σVon Mises < σyielding): full compression, full tension, asymmetric. “Robust”, “Medium-weight”, and “Light-weight” solutions are called respectively “DS-R”, “DS-M”, and “DS-L”, where the volume fractions of the design space (DS) are, respectively, 50%, 40%, and 30%.
Table 3 Topology optimization cases (The percentages refer to the volume fractions of the design space—DS) The entire structure is analyzed using SAP2000 (Computer and Structures 2017) to estimate the design loads for the joints (Fig. 13). Ultimate limit state (ULS) and service limit state (SLS) combinations are considered under code permanent and accidental loads for the structure considering Milan, Italy, as the location of the construction site. The joints were mainly loaded in different combinations of compression and tension actions. Multiple combinations of axial loads observed in this analysis are considered during topology optimization.
The weighted compliance CW approach was considered as the objective function. This global response is the weighted sum of the compliance of each subcase or load case (Altair University 2017), and it is defined by the following equation:
$$ {C}_W={\sum}_i{W}_i{C}_i=\frac{1}{2}{\sum}_i{W}_i\ {u}_i^T\ {f}_i $$
where Wi is scale factor per each load case,\( {u}_i^T \)is displacement vector per each load case, and fi: applied force per each load case.
The shapes obtained from the topology optimization process require a further post-processing. Before the 3D printing stage, it is necessary to smoothen and clean the part surfaces to achieve uniform shapes. The control of parameters, such as thickness and voids in the joints, is also essential to produce regular geometries. For this reason, a parametric post-processing tool is developed using Grasshopper (https://www.grasshopper3d.com/) (Fig. 14). The geometry was intersected with cutting planes to generate the required number of control points, and from these, the new and smoother surfaces were built up with the original thicknesses.
Finally, the stress values and distributions have been calculated through finite element analysis. An “envelope load combination” is created by linear superposition of results to find the maximum stress values. The different alternatives (DS-R, DS-M, DS-L, and a Benchmark) have been compared. The structural performance was monitored by a common reference point (RP) that is set at the intersection of structural profiles. This location showed the highest stress concentrations in the joint. The densities are plot as a scale of colours in Optistruct (Hyperworks solver) to interpret the results. The range of values is from 0 to 1 (Fig. 15) as an iso-contour plot. It represents the absence or presence of material, respectively. An intermediate value of 0.5 is selected because it shows defined shapes. Consequently, the values above 0.5 are displayed, and the other ones below 0.5 are filtered from the iso-contour plot.
Each one of the TO solutions reached the convergence after several iterations (in the range of 25 to 28 iterations) to obtain the minimum compliance and the lowest percentage of violated constraints. The optimization resulted in stiffening the hollow section joint. A higher percentage of DS results in more internal stiffeners. For the DS-R case, two sets of stiffeners are formed: two vertical and two horizontal ones, creating an internal void in the joint. For the DS-M case, just two horizontal stiffeners are generated. This also occurs to the DS-L case, differing only on the thickness, because the two resulting horizontal stiffeners are thinner compared to those of the DS-M case. The trend of results can be observed in both symmetric and asymmetric joint cases (Figs. 16, and 17). Although the new solutions (DS-R, DS-M) slightly weigh more than the benchmark, they are expected to have a significantly better structural performance with easier fabrication.
The first estimation of stresses with a lower computational time was obtained by linear analysis. For this initial part, two solutions were analyzed considering the weight: the robust case DS-R (50% DS) and the light case DS-L (30% DS) for the symmetric and asymmetric joints. Regarding the unstiffened symmetric joint cases (cross-section views are shown in Fig. 18), the most stressed regions correspond to the intersection zones where the structural profiles converge. In that case, the material reached yield stresses because of the high concentration of stresses. In current practice, the traditional (and costly) way of preventing this situation is to use welded stiffeners inside the geometry to decrease the stress concentrations in the critical zones. On the other hand, the new solutions provide much better performance with their “naturally” stiffened shapes thanks to the material distribution, which was optimized according to the different load paths.
The yielding stress was not exceeded in the proposed solutions. Figure 19 compares the structural performance in terms of maximum stresses and weight information among different topologies. The DS-R solution, with a slightly higher weight, significantly drops the maximum stresses in the joint. The light and benchmark joints perform similarly; however, the fabrication of the benchmark joint would require many cuts and welding operations.
In the asymmetric joint case (A), a similar trend of results is obtained (Fig. 20). The unstiffened joint presented the highest stress concentrations along the members’ intersection zone. As in the previous cases, the topology optimized joints with the “natural stiffener” formations represent a vast drop of the stress values and an improved distribution (Fig. 21).
A comparison of the linear analysis results of symmetric (S) and asymmetric (A) cases is presented in Table 4. Their optimized shapes presented a similar stress decrease against their respective unstiffened cases. Both DS-R solutions achieved a large percentage of reduction with respect to the corresponding unstiffened cases: 81% and 82% less, respectively, for the symmetric and asymmetric cases. Both DS-L cases reached an important stress decrease compared with the unstiffened cases: 76% and 78% less, respectively, for the symmetric and asymmetric. Despite the similarities in terms of shape and weight, the DS-L solutions presented an improved linear behaviour with respect to the benchmark joint: For example, the unstiffened joint stress reduction reaches 76%, with the symmetric symmetric (S) Vol = 30% DS case and 69% with the benchmark symmetric (S) case.
Table 4 Symmetric and asymmetric joints: stress reduction with respect to the unstiffened benchmark