Structural topology optimization with constraints on ...

Structural topology optimization with constraints on multi-fastener joint loads

Ji-Hong Zhu, Jie Hou, Wei-Hong Zhang & Yu Li

Structural and Multidisciplinary Optimization ISSN 1615-147X Volume 50 Number 4 Struct Multidisc Optim (2014) 50:561-571 DOI 10.1007/s00158-014-1071-5

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Author's personal copy Struct Multidisc Optim (2014) 50:561–571 DOI 10.1007/s00158-014-1071-5

RESEARCH PAPER

Structural topology optimization with constraints on multi-fastener joint loads Ji-Hong Zhu · Jie Hou · Wei-Hong Zhang · Yu Li

Received: 18 June 2013 / Revised: 27 January 2014 / Accepted: 14 February 2014 / Published online: 6 April 2014 © Springer-Verlag Berlin Heidelberg 2014

Abstract This paper addresses an important problem of design constraints on fastener joint loads that are well recognized in the design of assembled aircraft structures. To avoid the failure of fastener joints, standard topology optimization is extended not only to minimize the structural compliance but also to control shear loads intensities over fasteners. It is shown that the underlying design scheme is to ameliorate the stiffness distribution over the structure in accordance with the control of load distributions over fastener joints. Typical examples are studied by means of topology optimization with joint load constraints and the standard compliance design. The effects of joint load constraints are highlighted by comparing numerical optimization results obtained by both methods. Meanwhile, resin models of optimized designs are fabricated by rapid prototyping process for loading test experiments to make sure the effectiveness of the proposed method. Keywords Topology optimization · Multi-fastener joint · Joint load constraint · Loading test · Rapid prototyping

1 Introduction Sizing and shape optimizations are traditionally two basic techniques for stiffness and strength designs of engineering structures, especially in the least-weight design of aircraft J.-H. Zhu () · J. Hou · W.-H. Zhang () · Y. Li Engineering Simulation & Aerospace Computing, The Key Laboratory of Contemporary Design and Integrated Manufacturing Technology, Northwestern Polytechnical University, Xi’an, Shaanxi 710072, China e-mail: [email protected] e-mail: [email protected]

structures (Martinez et al. 2001; Vitali et al. 2002; Hansen and Horst 2008; Chintapalli et al. 2010). Since last two decades, topology optimization has been recognized as a promising technique in the layout design of structures (Bendsøe and Sigmund 2003; Allaire et al. 2004; Zhang and Sun 2006; Remouchamps et al. 2011). Literature surveys of Deaton and Grandhi (2013), Guo and Cheng (2010) have summarized recent advances and applications of topology optimization. These important achievements continue to motivate further studies about the applications of topology optimization and its capabilities in solving complex engineering design problems. In an assembled aircraft structure, bolts or rivets are widely used as multi-fastener joints that even act as critical elements in damage tolerance design. They are sometimes the weakest component of a structure due to the high intensity of joint load (Niu 1988; Barrett 1992; Wang 2000; Chintapalli et al. 2010). Earlier studies were focused on developing analytical and numerical methods for stress and failure predictions of multi-fastener joints. Typical models concerned panels joined by single or multiple joints in the analysis of bolted and riveted joint loads as well as stress distributions around pin holes etc. (Rowlands et al. 1982; Wang and Han 1988; Zhao et al. 1996; Camanho and Matthews 1997). In the works of Poon and Xiong (1995), Oh et al. (1997), key parameters such as fastener joint locations, ply angles and stacking sequences of laminates, fastener diameters and edge distances were optimized to avoid the failure of fasteners. Bianchi et al. (2007) developed an optimization procedure maximizing load carrying capability of the joint system to balance the number and size of bolts. Ekh and Sch¨on (2008) evaluated the effects of different parameters on the load distribution, such as the mismatch of member plates, length of the overlap region and fastener’s stiffness. Optimization was then carried out

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Fig. 3 I-shape cantilever beam connected with a thin sheet Fig. 1 Assembled structure with multi-fastener joints

to minimize the bearing stress. In the work of Oinonen et al. (2010), a “weakest link” method was proposed to optimize the layout of fasteners for the bracket-to-beam joints. The design objective was to ameliorate Von-Mises equivalent strain as well as shear loads in the joints. Attempts were made by Jiang and Chirehdast (1997), Buhl (2002), Zhu and Zhang (2006), Qiao and Liu (2009), Zhu and Zhang (2010) to introduce the concept of topology optimization into the layout design of fixations and fasteners. In the works of Chickermane and Gea (1997), Li et al. (2001) and Qian and Ananthasuresh (2004), structural patterns of different parts and the layout of fasteners were also sought by constraining these sub-domains with different volume fractions. Chickermane et al. (1999) also proposed a topology optimization method for the location optimization of fasteners in conjunction with fastener load constraints, while the connected components remain unchanged. Unfortunately, it was found in engineering practices that the adjustment of joint distributions, addition of more joints or enlargement of joint diameters might not lower the joint loads significantly due to the stiffness mismatch between the connected structure components. This is especially the

case for aircraft spar-skin structure with overloaded fastener joints. In fact, joint loads highly depend upon not only mechanical and geometrical attributes of joints themselves, but also stiffness distributions of the connected structures according to aircraft design manuals (Niu 1988; Wang 2000) and existing works (Poon and Xiong 1995; Oh et al. 1997; Ekh and Sch¨on 2008; Chintapalli et al. 2010). In this paper, standard topology optimization method is applied to deal with the above problem by introducing additional multi-fastener joint loads constraints for the first time. Unlike the existing works, dominant shear loads over joints are controlled by optimizing the stiffness distribution of connected structures. In the calculation of shear loads, short beam elements are used to model multi-fastener joints with the negligence of bolt-hole clearance, clampup and friction effects. The proposed optimization method can be considered as an effective way to limit the joint loads during the concept design procedure of assembled aircraft structures. Meanwhile, designers can also benefit from the optimized load carrying path for detailed structures design.

2 Basic formulation of topology optimization with joint loads constraints An assembled structure consisting of two components connected by large numbers of fasteners is schematically shown in Fig. 1. Topology optimization is to find such an optimized material distribution in the design domain that satisfies constraints on the structural volume and shear loads in the joints. The optimization problem can thus be expressed as: find : η= (η1 , η2 , ..., ηi , ..., ηn ) min : C = 12 uT Ku Ku = f s.t. Fj ≤ F0 , j = 1, 2, ..., m V ≤ V0

Fig. 2 Beam element and its coordinate system, tension force, moments and shear forces

(1)

where η is the design variables vector related to elements pseudo-densities describing material distribution in the design domain. C is the global elastic strain energy. f is the force vector and u is the corresponding nodal displacement vector. K is the global stiffness matrix. F j is the shear

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Fig. 4 The initial distribution of the shear loads in the joints

load value in the jth joint. F 0 is the allowable maximum shear load. V is the structural volume and V 0 is its upper bound. In the above topology optimization model, a precise calculation of shear loads in fasteners could involve deep studies of relative displacements of hole centers, fastener deformation, conforming contact between fasteners and member plates as well as effects of fastener clamp-up, friction and out of plane deformation etc. This will bring extra complexities and intractable computational difficulties. Here, multi-fastener joints are approximated by numbers of beam elements as used by Chickermane et al. (1999), Ekh and Sch¨on (2008). For a short beam of two nodes A and B shown in Fig. 2, the total shear load is calculated as

Fig. 5 Optimized design and joint load distribution without joint load constraint (Elastic strain energy 0.660 J, maximum joint load 30.13 N)

Fj =



2 Fj2x + Fjy

(2)

with Fj x =

12EI · uj Ax L3 (1+) 6EI + L2 (1+) · θj By Fjy = L312EI · uj Ay (1+) − L26EI · θj Bx (1+) γ  = 12EI GSL2

−

12EI L3 (1+)

· uj Bx +

6EI L2 (1+)

· θj Ay

−

12EI L3 (1+)

· uj By −

6EI L2 (1+)

· θj Ax

(3) where E and G are the elastic and shear moduli. I, S and L denote the moment of inertia, the cross section area and the beam length.  is the shear coefficient. γ = 10/9 is the

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Fig. 6 The iteration history of the elastic strain energy

shear factor of the cross section. uj Ax and θj Ax are the corresponding nodal displacement and rotation angle of node A in x direction. Similar definitions are used for uj Ay , θj Ay , uj Bx , θj Bx , uj By and θj By .

3 Sensitivity analysis of the joint loads Sensitivity analysis of elastic strain energy with respect to pseudo-densities is very popular. Here, we are focused on the sensitivity analysis of the joint load F j , whose components are noted by F j x and F jy . For example, F j x can be expressed as a linear function of nodal displacements Fig. 7 Optimized design and comparison of joint load distribution with joint load constraint (Elastic strain energy 0.667 J, maximum joint load 19.97 N)

Fj x =

12EI · uj Ax − L312EI L3 (1+) (1+) T u + L26EI · θ = λ j By jx (1+)

· uj Bx +

6EI L2 (1+)

· θj Ay (4)

where λj x is a constant vector with the same dimension as the displacement vector u. Items of λj x corresponding to the four degrees of freedom are defined by the coefficients in the above equation, while the rest are set to be zero. Accordingly, the sensitivity of F j x is calculated as   ∂ λTjx u ∂Fj x ∂u = = λTjx · ∂ηi ∂ηi ∂ηi

(5)

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Fig. 8 The iteration history of the elastic strain energy

Similar calculations of F jy can be done to obtain constant vector λjy . As a result, the derivative of the total joint load corresponds to ∂Fj ∂ηi

=

=

1 2Fj

∂

  2 Fj2x +Fjy ∂ηi

=

 2 2 Fj2x +Fjy

 · 2Fj x λTjx · =

1

∂u ∂ηi + Fj x λTjx +Fjy λTjy Fj ∂u = λTj · ∂η i

·

  2 ∂ Fj2x +Fjy

2Fjy λTjy · ·

∂ηi

∂u ∂ηi



∂u ∂ηi

(6)

Based on the finite element equilibrium equation, the above equation can further be written as λTj ·

     T  ∂f ∂f ∂u ∂K ∂K = λTj · −K −1 − u = − K −1 λj · − u ∂ηi ∂ηi ∂ηi ∂ηi ∂ηi (7)

The above expression can be simplified by the following notation K −1 λj = u∗

(8)

Notice that the stiffness matrix K is symmetric. u* can be interpreted as the displacement vector related to the adjoint

load vector λj applied on the structure. The substitution of (8) into (7) gives rise to   ∗ T ∂Fj ∂f ∂K =− u · − u (9) ∂ηi ∂ηi ∂ηi In the above equation, the derivative of the load vector is zero when only design-independent loads are applied to the structure. Derivatives of the stiffness matrix K with respect to the pseudo-densities are easily obtained by means of SIMP material interpolation model used in this paper. Take the ith element in the design domain for example, its stiffness matrix Ki can be expressed as p

K i = ηi K i0

(10)

where Ki0 is the stiffness matrix when the element is purely solid. p is the penalty factor set to be 3 as usual. The derivative of the stiffness matrix is then calcaulated as ∂K ∂K i p = = Ki (11) ∂ηi ∂ηi ηi In practice, sensitivity analyses of the joint loads will be computationally expensive due to the additional finite element analyses required for large numbers of fasteners and joint load constraints. To solve the problem, methods like KS function and P-norm function can be used to merge large numbers of design constraints into only one constraint. In addition, as shown in numerical examples, only a small

Table 1 Comparison of optimized results Items

Optimized design without joint load constraints

Optimized design with Fj ≤ 20N

Initial solid beam

Elastic strain energy (J) Maximum displacement (mm) Maximum shear loads (N) Volume fractions

0.660 8.887 30.13 0.3

0.667 8.988 19.97 0.3

0.335 4.53 26.3 1

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Fig. 9 Principal stresses of the optimized designs, a Fig. 5 and b Fig. 7

Fig. 10 Solid CAD model of the optimized beam and joints with engineering features

Fig. 11 Resin models fabricated according to the topology optimization result

Fig. 12 Resin model installed onto the static stretching machine

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Author's personal copy Topology optimization with constraints on fastener joint loads Table 2 Comparison of loading test results Minimum stretching forces (N)

Group 1 Group 2 Group 3

Model A

Model B

27.6 32.1 31.7

42.8 40.6 44.2

number of fasteners need to be constrained, because the rest joint loads are small enough to be neglected during optimization. In the following sections, two numerical examples are presented to illustrate the efficiency and validity of the proposed optimization model. One is further validated with loading test of fabricated resin models. We use the optimization algorithm GCMMA (Globally Convergent Method of Moving Asymptotes, Svanberg 1995) implemented in the general-purpose design platform Boss-Quattro (Radovcic and Remouchamps 2001). Besides, density filter technique (Bruns and Tortorelli 2001) is applied with the filter radius being three times of the average element size in both examples to avoid the checkerboards.

4 An I-shape cantilever beam connected with a thin sheet Here, we will design an I-shape cantilever beam joined with a thin sheet by two rows of evenly distributed fasteners. Dimensions and layout of joints are shown in Fig. 3. As the shear loads distributions are identical in both rows, fasteners are numbered only for one row from the beam root to the tip. The assembled structure is meshed with quadrangular shell elements of size 1 mm × 1 mm and thickness 1 mm. The design domain of the structure is the I-shape beam web. Elastic modulus of the solid material is 2.6 × 109 Pa and Poisson’s ratio is 0.3. A point force of 150 N is applied on the bottom right corner of the design domain. In Fig. 4, we present the shear load distribution over fastener joints after static analysis of the initial solid FE model. We can find the maximum shear

Fig. 13 Model B with broken joints

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load occurs at the beam root, which indicates the most vulnerable position. For the purpose of comparison, a standard compliance topology optimization is firstly carried out without joint load constraints. The only constraint is defined by limiting the volume fraction of the design domain to 30 %. Figure 5 presents the optimized design and related joint loads after 36 design iterations. The elastic strain energy is 0.660 J. The maximum tip displacement is 8.887 mm, while the maximum shear load rises to 30.13 N at the beam root. Figure 6 shows the iteration history of the objective function which converges stably. The optimized topology seems to be the classical topology solution of a pure cantilever beam. There are no materials distributed near the joint for the structure reinforcement even the maximum shear load occurs, which implies that the increase of the maximum joint load does not significantly affect the minimization of the elastic strain energy. Now, the same topology optimization problem is considered with the introduction of joint load constraints. The upper bound of the shear load is limited to 20 N. Due to the symmetry, we only constrain the shear loads in one row of fasteners. Figure 7 shows the optimized design and joint loads distribution after 30 design iterations. To make ease the comparison, load distribution in Fig. 5 is also plotted here. Figure 8 indicates that the iteration history of the objective function has also a stable convergence. Detailed results are listed and compared in Table 1. The elastic strain energy is now 0.667 J while the maximum tip displacement reaches 8.988 mm. Although the structural stiffness is slightly smaller than before, the maximum shear load in the joints is now 19.97 N with a great reduction by more than 10 N, i.e. 33.7 % of the previous design. The maximum joint load appears still at the beam root and all the loads now satisfy the design constraints. To have a deep insight into the design solution, the principal stresses near the beam root are plotted and compared in Fig. 9 for two optimized designs. Notice that the arrows indicate magnitudes and directions of principal stresses. In the optimized design of Fig. 7, the structural configuration near the beam root is redesigned significantly to provide a reasonable support near the joint where maximum shear loads occur. Compared with the optimized design of Fig. 5, an additional push is provided by the structure branch to offset the shear effect. Moreover, the maximum shear load given in Fig. 7 is even lower than its initial value related to the purely solid beam given in Fig. 4. This clarifies intuitive engineering misunderstanding that increasing beam stiffness can reduce joint load. To validate the optimized design, loading tests are carried out. We use rapid prototyping to fabricate resin models according to the above two topology optimization solutions.

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The resin material Somos 14120 is a standard ABS-like stereolithography resin. According to the data obtained from the supplier DSM Somos (www.dsmsomos.com, 14120 white documents) and some previous applications (see Norouzi et al. 2009; Gu et al. 2012), the stereolithography resin material can approximately maintain linear elastic properties within a small deformation. According to the optimized designs shown in Figs. 5 and 7, solid models of engineering features are built in CATIA as shown in Fig. 10a and then outputted as STL models for rapid prototyping. Clearly, structural configurations are globally maintained except some minor details. As thin-walled stereolithography resin parts have the problem of easy-buckling, some extra stiffeners are added on the structure. Because failures of practical bolt and rivet joints appearing around the joined holes on the member plates are not easily detectable, all the joints are directly fabricated as small vulnerable resin beams of identical size and cross section with approximately the same strength, as shown in Fig. 10b. These resin joints can thus be considered as simple breaking sensors of shear loads. Although the experiment cannot precisely determine the magnitude of the shear loads, a significant tendency should be persuasive due to the fact that the optimized design with joint load constraints reduces the shear loads remarkably. In Fig. 11, transparent outlines of the resin models are plotted over the material density distribution for comparison. Resin models marked as a and b in Fig. 11

Fig. 14 Wing-box structure with 4 rows of joints

Fig. 15 Initial distribution of the joint loads in the joints

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Fig. 16 The optimized design without joint load constraint (Elastic strain energy 1.033 J, maximum joint load 27.5 N)

are then fabricated with a SPS-350B stereolithography machine. Loading test is set up in the following way. Resin models are installed onto static stretching machine as cantilever beams with a fixture as shown in Fig. 12. A strong string bounds the lower right corner to the dynamometer. The experiment is to find the minimum stretching force breaking the joints in both models. Three groups of specimens are tested. By slowly increasing the stretching force, the force value is recorded from the dynamometer once the first joint is broken. All the experimental data are compared in Table 2. Model B with broken joints is shown in Fig. 13. Magnitudes of the stretching forces in Table 2 have clearly indicated that the resin joints related to model B can undertake larger stretching force. In other words, optimized design with joint load constraints can reduce the shear loads significantly. It should be noted that the displacements of the resin models are not evaluated in the experiment. This is because the maximum displacements are 0.621 mm and 0.624 mm respectively according to finite element analyses of both

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Fig. 17 The optimized design with joint load constraint (Elastic strain energy 1.082 J, maximum joint load 20 N)

models when a 20 N force is applied. The difference is only 0.003 mm, which cannot be precisely distinguished by our existing equipment in the experiment.

5 Wing-box structure with 4 rows of joints Consider now a wing box structure shown in Fig. 14. It consists of two tapered I-beams covered with two thin sheets. Both beams are clamped at one end. A distributed pressure of 5 × 104 Pa is applied on the lower skin. Due to the symmetry, only one of the two beams is considered.

Fig. 18 Joint load distributions for two optimized designs

Fig. 19 Principal stresses of the optimized designs, a Fig. 15 and b Fig. 16

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Suppose skins and webs of the I-beams have a thickness of 2 mm. The beams and skins are meshed with quadrangular shell elements of size 1 mm × 1 mm and 2 mm × 2 mm, respectively. Each beam is fastened to the thin sheets with four rows of joints marked as Rows A, B, C and D. For the whole structure, the elastic modulus is 2.6 × 109 Pa and the Poison’s ratio is 0.3. After the first FE analysis, the shear loads of the initial solid design are shown in Fig. 15. We can find that the maximum shear loads occur at the root of the I-beam in all the four rows. Likewise, topology optimization is to minimize the elastic strain energy. The constraint of 30 % volume fraction is imposed to the beam web. Similarly, two optimizations are performed for the purpose of comparison. One of them constrains the shear load in the joints to be less than 20 N. Figures 16 and 17 present the two optimized structural topologies. Corresponding joint loads are plotted and compared in Fig. 18. Figures 16–18 indicate that the maximum joint load appears in Row D in both cases. Compared with the initial joint loads presented in Fig. 15, the distributions of joint loads change considerably. The comparison between

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Figs. 16 and 17 indicates that materials are topologically distributed towards the beam root to offset the joint loads. Furthermore, the principal stresses near the beam roots in two optimized designs are plotted in Fig. 19. This confirms that the structure is optimized to provide a push for the upper plate and a pull for the lower one near the beam root.

6 Conclusion In this paper, we propose a structural topology optimization method with constraints on the shear loads of multi-fastener joints. The joints are modeled as short beam elements. Design sensitivities of shear loads with respect to pseudodensities are derived in terms of the derivatives of nodal displacements of beam elements. Two numerical examples are tested and optimization results are compared with standard topology optimization design. It is shown that the joint loads are strongly affected by the structural layout and the load carrying path. With an optimized distribution of structural stiffness, all the shear loads in the joints are perfectly controlled by the prescribed upper bound. To validate the proposed optimization method, loading tests are carried out with the help of stereolithography resin models. By comparing the minimum stretching force breaking the joints in the resin model, three groups of experiments have obviously verified the effect of the joint load constraints upon the optimized topology. Acknowledgments The authors would like to thank Prof Wang Boping from the University of Texas at Arlington and Dr Yang Jun from Chengdu Aircraft Design Institute of AVIC for the valuable discussions. This work is supported by National Natural Science Foundation of China (51275424, 11002113 and 11172236), 973 Program (2011CB610304), NPU Foundation for Fundamental Research (NPUFFR-JC20120229), and the 111 Project(B07050).

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