The cross-sectional topology optimization of a slender load carrying member such as a beam typi- cally begins with the choice of design space which is generally either a square or a rectangle. However, when subjected to bending and torsional loads, such design space may often lead to biased topolo- gies as the material distribution is directed towards the boundaries. As it may be di/uniFB03cult to avoid square/rectangular design space, this work introduces a constraint which may help in decreasing the dependency on the shape of the design space. The constraint is on polar /uniFB01rst moment of area and is di/uniFB00erent from the traditional weight constraint in a way that it simultaneously constrains the weight and enclosed volume. The outcomes from the new scheme are compared with those obtained through conventional compliance- and weight- based topology optimization formulation. /one.pnum Introduction Cross-section selection is one of the primary considerations in design of load carrying members such as beams, rods and shaft. Standard cross-section shapes such as I-beam, H-beam, box and circular sections are common choices of cross-sections which are easily available in di/uniFB00erent dimensions. However, /uniFB01nal- izing the dimensions for di/uniFB00erent loading and failure criteria is an important task. In current times, the task is typically performed through topology optimization (Zuberi et al. (/two.pnum/zero.pnum/one.pnum/zero.pnum); Yiru et al. (/two.pnum/zero.pnum/one.pnum/five.pnum); Vatanabe et al. (/two.pnum/zero.pnum/one.pnum/six.pnum); Zhou et al. (/two.pnum/zero.pnum/zero.pnum/two.pnum); Patel et al. (/two.pnum/zero.pnum/zero.pnum/nine.pnum); Kim and Kim (/two.pnum/zero.pnum/zero.pnum/zero.pnum)). Kim and Kim (/two.pnum/zero.pnum/zero.pnum/zero.pnum) used the technique for the /uniFB01rst time in optimization of a beam cross-section, where, bending and torsional rigidities of the section were maximized for a prescribed material constraint. However, it was the work of Zhou et al. (/two.pnum/zero.pnum/zero.pnum/two.pnum) which simpli/uniFB01ed the cross-section optimization process using a compliance-weight based formula- tion under realistic modelling and loading conditions. In fact, the work introduced extrusion manufacturing constraint which could also be used for the cross-section optimization. Most of the later works on beam optimization were inspired from the same extrusion constraint formulation (Patel et al. (/two.pnum/zero.pnum/zero.pnum/nine.pnum); Vatanabe et al. (/two.pnum/zero.pnum/one.pnum/six.pnum)). To e/uniFB00ectively demonstrate the e/uniFB03cacy of extrusion constraint, Zuberi et al. (/two.pnum/zero.pnum/one.pnum/zero.pnum) used di/uniFB00er- ent load combinations and design space geometries for yielding the optimal cross-sections. After this work, the focus on beam optimization algorithms relatively decreased until composite material beams became the topic of interest (Ren et al. (/two.pnum/zero.pnum/one.pnum/six.pnum); Yiru et al. (/two.pnum/zero.pnum/one.pnum/five.pnum); Blasques and Stolpe (/two.pnum/zero.pnum/one.pnum/two.pnum)). In such beams, apart from weight, the /uniFB01bre orientation is also optimized simultaneously. However, there are some limitations of the existing cross-section topology optimization formulations (i.e., compliance minimization against a weight constraint subject to an extrusion constraint). Firstly, the /uniFB01nal topologies are primarily governed by the direction of load which increases the moment of inertia about a particular axis whereas neglecting the other. This consideration is particularly important in cases such as selection of the web thickness in an I- section, as compliance-weight based topology optimization would result in a thinner web and thick /uniFB02anges for unidirectional vertical loading Patel et al. (/two.pnum/zero.pnum/zero.pnum/nine.pnum). Therefore, it is obvious that the thin web could either fail due to shear or buckling. In that case, cross-section optimization will have no meaning if web thick- ness would require a re-calculation. Secondly, the /uniFB01nal topologies depend upon the choice of initial design space. For example, for same value of weight constraint, beam subjected to vertical loads yield di/uniFB00erent /uniFB01nal topologies for square and rectangular design spaces Zuberi et al. (/two.pnum/zero.pnum/one.pnum/zero.pnum). Therefore, the current work proposes a topology optimization scheme which besides generating robust cross-sections (i.e. resistance to tension, compression, shear and warping), could also decrease the e/uniFB00ect of design space on the /uniFB01nal cross-sections. Thirdly, the weight constrain in the conventional topology optimization constraints the to- 1 tal weight and not the enclosed volume, the resulting design may not be space-e/uniFB03cient. To address the aforementioned issues, this work suggests an alternative to weight constraint where the weight constraint is replaced by a constraint on polar /uniFB01rst moment of area which simultaneously constrains both weight and space. As such, for a prescribed constraint value, the new scheme can generate di/uniFB00erent geometries for di/uniFB00erent weight. The scheme is tested for optimizing the cross-section of a prismatic cantilever beam un- der di/uniFB00erent loading conditions. Geometry of beam is represented by two-dimensional tiling of /one.pnum/six.pnum-noded hexahedral (brick) elements, where linear interpolation functions have been used along the cross-section (i.e. in transverse directions of the beam) and cubic interpolation functions have been used along the beam length (Fig. /one.pnum(a)). Such discretisation avoids the need for an extrusion constraint. The hybrid /uniFB01nite element model is /uniFB01rst tested against a fully discretised model in terms of stress distribution at /uniFB01xed end when subjected to a vertical end load. Subsequently, the newly developed topology optimization scheme is applied to optimize the cross-sections for di/uniFB00erent end loading conditions. The results are compared with those obtained using the conventional compliance-weight based topology optimization formulation. Further, the ability of the new scheme to generate design space independent topologies is tested which could carve circular and elliptical sections out of square design space when the beam is subjected to bi- axial bending loads. Organization of rest of this paper is as follows. Section /two.pnum describes the methodologies used for /uniFB01nite element modeling and topology optimization. The results have been presented in Section /three.pnum. Conclusions are given in Section /four.pnum, along with discussions and scope of future work. /two.pnum Methodologies /two.pnum./one.pnum Finite element model As mentioned in Section /one.pnum, the /uniFB01nite element model uses a hybrid /one.pnum/six.pnum-noded element which spans across the beam length. The idea for the element is derived from Euler-Bernoulli Beam Theory (Popov (/one.pnum/nine.pnum/six.pnum/eight.pnum)). For any load F applied on the free end of a cantilever beam: M=F:x=EId2uy dx2(/one.pnum) whereuyis the transverse de/uniFB02ection and M is the bending moment at a distance xfrom the free end, E is material sti/uniFB00ness and I is area moment of inertia. Hence, the transverse displacement uyshould at least be a cubic function of distance x. Accordingly, a cubic interpolation function is chosen along the length of the element. To keep the computational cost minimum, transverse interpolation functions are kept linear as accuracy in this direction can be taken care of by /uniFB01ne discretization. The shape functions for the element are accordingly derived and element sti/uniFB00ness matrix was computed by applying standard methods (Reddy (/one.pnum/nine.pnum/nine.pnum/three.pnum); Bathe (/two.pnum/zero.pnum/zero.pnum/six.pnum)). A standard beam model used by Liu et al. (/two.pnum/zero.pnum/zero.pnum/eight.pnum) is discretized using the /one.pnum/six.pnum-noded elements (as discussed in detail in Section /three.pnum). The computational work has been done using an in-house code written in MATLAB ®(Mathworks Inc., Boston). The /uniFB01nite element model developed here is then tested for a vertical end-loading case (Fig. /one.pnum(a)) against a fully discretised beam model developed and analysed in a commercial /uniFB01nite element analysis software (viz., ANSYS). The stress distribution at the /uniFB01xed ends in both beams was found to be similar and nearly equal in magnitude (error in maximum stress within /one.pnum/zero.pnum% , disregarding stress concentration at corners), as shown in Figs. /one.pnum(c) and (d). /two.pnum./two.pnum Optimization problems The conventional compliance-weight based topology optimization is /uniFB01rst discussed. Thereafter, the compliance- polar /uniFB01rst moment of area based topology optimization is introduced. /two.pnum./two.pnum./one.pnum Minimization of compliance subject to weight constraint The objective of this problem is to minimize structural compliance subject to a weight constraint, as in Sigmund (Sigmund /two.pnum/zero.pnum/zero.pnum/one.pnum). The method used for structural topology optimization has been adapted from Andreassen et al. (/two.pnum/zero.pnum/one.pnum/one.pnum). The basic optimization problem is given by: minc(x) =UTKU=NX e=1(xe)puT ek0ue (/two.pnum) /two.pnum Figure /one.pnum: Geometry representation of a beam using /one.pnum/six.pnum-noded brick elements. (a) Beam discretization (for illustration only, actual discretization is generally /one.pnum/five.pnum/zero.pnum /one.pnum/five.pnum/zero.pnum elements), (b) the /one.pnum/six.pnum-noded brick element; com- parison of normal stress distribution at the /uniFB01xed end of cantilever beam for a vertical load at the top edge of free end using a fully discretised (/two.pnum/zero.pnum /two.pnum/zero.pnum/two.pnum/five.pnum/one.pnum elements) /uniFB01nite element model analysed in ANSYS (c), and (d) the current modelling approach (/two.pnum/zero.pnum /two.pnum/zero.pnum/one.pnum elements) analysed using MATLAB. /three.pnum subject to: V(x) V0=f (/three.pnum) KU=F (/four.pnum) 0<xminxe1 (/five.pnum) where, c is total compliance of the structure. ‘x’ is a vector containing element densities. K is the global sti/uniFB00ness matrix, and U is the global displacement vector. Similarly, xe , ue and k/zero.pnum are element density, el- ement displacement vector and element sti/uniFB00ness matrix, respectively. N is the total number of elements and the penalty ‘p’ is a prescribed constant. V(x) is the material volume. V0is the total volume of de- sign space, f is the permissible volume fraction and F is the global force vector. The minimum density (xmin) for an element has been taken as /zero.pnum./zero.pnum/zero.pnum/one.pnum in accordance with Andreassen et al. (/two.pnum/zero.pnum/one.pnum/one.pnum). The Method of Moving Asymptotes (MMA) was used for optimization (Svanberg (/one.pnum/nine.pnum/eight.pnum/seven.pnum); Sigmund (/two.pnum/zero.pnum/zero.pnum/seven.pnum); Prasad and Diaz (/two.pnum/zero.pnum/zero.pnum/eight.pnum)). /two.pnum./two.pnum./two.pnum Minimization of compliance subject to constraint on polar moment of area This optimization problem is similar to the one in Section /two.pnum./two.pnuma, however, the weight constraint in (/three.pnum) is replaced by the constraint on polar moment of area. The constraint can be represented as: ar(x)ar0 (/six.pnum) wherear(x))is the sum of /uniFB01rst polar moment of area due to individual elements in the design space and 0ar00)is a user de/uniFB01ned value. Mathematical expression for ar(x))is as follows: ar(x) =NX e=1(xe:jrej:ae) (/seven.pnum) whereaerepresents area of an element which may be taken as unity and jrejis the magnitude of position vector of the element centroid with respect to centroid of the design space. /two.pnum./three.pnum Load application scheme To achieve design space-independency in /uniFB01nal results, it is important that loads are not applied at speci/uniFB01c points/nodes, as the /uniFB01nal design may be biased toward selected loading points. Accordingly, we used a modulus/density dependent loading scheme such that the load is distributed on complete design space rather than select points. The global force vector ’F’ in (/four.pnum) is assembled from individual element force vectorsf0 efollowing standard procedure. The distributed of total load (F/zero.pnum) on all elements in the design space is given by: Fe=xp eF0PN i=1(xp i)(/eight.pnum) The elemental force share (F(e))is then distributed to the corresponding element nodes. For a typical case of vertical load0F0 0applied only at free end of the beam (i.e. /uniFB01rst four nodes are loaded in 2nddegree of freedom direction), the element force vector is computed as follows: f0 e=f0nf0 0nf0 0nf0 0nf0 0 0 00 0g481 (/nine.pnum) where nf=xp e:F0 4:PN i=1(xp i)(/one.pnum/zero.pnum) A similar scheme was used for torsion application as well. For an externally applied torsional moment 0T0 0, the elemental force distribution should be such that it maintains constant angular displacement. The elemental share of force should, therefore, be multiplied with distance between element centroid and /four.pnum Section Load Constraint(Speci/uniFB01ed) /three.pnum./one.pnum Fvertical V0= 0:5,ar0= 1:09e+ 6 /three.pnum./two.pnum Torque V0= 0:5,ar0= 1:27e+ 6 /three.pnum./three.pnum Fhorizonal andFvertical V0= 0:5,ar0= 1:12e+ 6 /three.pnum./four.pnum Fvertical +Faxial V0= 0:5,ar0= 1:9e+ 5 *V0-Volume constraint, * ar0- speci/uniFB01ed value of polar /uniFB01rst moment of area Table /one.pnum: Problem-speci/uniFB01c data torsional axis. Accordingly, an all-new torsion application scheme is derived. The resultant elemental force is given by Pe=" T0PN e=1(xp e)re^ejrej# xp ejrej^e (/one.pnum/one.pnum) wherereis the position vector of element centroid with respect to torsional axis and (^e)is the unit vector at the centroid of the element in the direction of the force (i.e. inclined /nine.pnum/zero.pnum degrees with respect to re). For a torque0T0 0applied at the free end of a shaft, elemental force vector may be written as: f0 e=fnthntv0nthntv0nthntv0nthntv0nthntv00 0g481 (/one.pnum/two.pnum) wherenthandntvare the horizontal and vertical components of nodal force ntwhich is one fourth of Pe. /three.pnum Results The developed methodology is /uniFB01rst tested for /uniFB01nding optimal cross-sections in a 3000mm long beam having 240240mm2of square design space (Fig. /two.pnum(a-b)) (as in Liu and Jia (/two.pnum/zero.pnum/zero.pnum/eight.pnum) ). The material is assumed to be isotropic with Young’s modulus ‘ E’ of200GPa and Poisson’s ratio ‘ ’ of0:3. The design space is discretized using 150150(i.e., total /two.pnum/two.pnum/five.pnum/zero.pnum/zero.pnum) elongated brick elements. The associated data for problems to be discussed in Sections /three.pnum./one.pnum-/three.pnum./four.pnum are given in Table /one.pnum. /three.pnum./one.pnum A single vertical end load One end of the beam is /uniFB01xed and a vertical force of /one.pnum/zero.pnum kN is applied at the free end (Figs. /two.pnum(a,c)). As shown in Figs. /two.pnum(d-e), the standard I-section topologies are obtained from both optimization methods, which are also similar to those observed in the literature(Liu et al. (/two.pnum/zero.pnum/zero.pnum/eight.pnum); Patel et al. (/two.pnum/zero.pnum/zero.pnum/nine.pnum); Yiru et al. (/two.pnum/zero.pnum/one.pnum/five.pnum)). Despite the similarity, it is worth noticing that, for the same /uniFB01nal weight, the I-section obtained through the new method ( Fig. /two.pnum(e)) seems more robust in terms of thicker web, chamfered web-/uniFB02ange junction and tapered /uniFB02anges. In addition, the /uniFB01nal geometry does not utilize the complete breadth of design space. The aforementioned observations clearly indicate that new method can result in relatively stronger sections in such loading cases. /three.pnum./two.pnum Beam section under pure torsion The beam model used in previous sub-section is used. One end of the beam is /uniFB01xed whereas a twisting moment of 1:0kNm is applied on the free end as shown in Fig. /three.pnum(a, c). A hollow circular section similar to that observed in previous studies on shaft optimization is obtained from both the formulations discussed in Sections /two.pnum./two.pnuma and /two.pnum./two.pnumb (Figs. /three.pnum(d-e)) (Li et al. (/two.pnum/zero.pnum/zero.pnum/one.pnum); Kim and Kim (/two.pnum/zero.pnum/zero.pnum/zero.pnum); Ishii and Aomura (/two.pnum/zero.pnum/zero.pnum/four.pnum)). This indicates that the new method is also equally e/uniFB03cient for beam/shaft optimization under torsional loads. /three.pnum./three.pnum Bi-axial loading This is an example of two loading cases, i.e., a beam needs to be designed for two di/uniFB00erent loading cases similar to that in Ishii and Aomura (/two.pnum/zero.pnum/zero.pnum/four.pnum). Speci/uniFB01cally, the cantilever beam is subjected to independent /five.pnum Figure /two.pnum: Schematic representation of cantilever beam with a single end load (a), design space dimensions (/two.pnum/four.pnum/zero.pnummm x /two.pnum/four.pnum/zero.pnummm) for designing the beam’s cross-section (b), load distribution scheme over the free-end section (c), the optimal section obtained by minimizing compliance subject to the weight constrained to /five.pnum/zero.pnum% of the design space (d), and the optimal solution obtained by replacing the weight constraint with a constraint on polar /uniFB01rst moment of area (e). Figure /three.pnum: Beam subjected to pure torsion with square section as design space (a-b), load distribution scheme (c), section obtained after minimizing compliance against /five.pnum/zero.pnum% weight constraint (d) and that after minimization of weight subject to constraint on compliance (e). /six.pnum Figure /four.pnum: Bi-axial end loading of a cantilever beam (a), load distribution in vertical direction (b), load dis- tribution in horizontal direction (c), outcome of compliance minimization subject to /five.pnum/zero.pnum% weight constraint when both vertical and horizontal loads have equal magnitude (d), and solution of the corresponding prob- lem with polar moment of area constraint (e). end loads of 10kNin two mutually perpendicular directions (Figs. /four.pnum(a-c)). The compliance due to both the loading cases were added and minimized. The result is a square section with a nearly circular cavity, which is also one of the commonly used beam sections (Figs. /four.pnum(d-e)). The similarity of results in the two optimization methods indicate that both methods can be used interchangeably in such loading cases. /three.pnum./four.pnum Beam subjected to a multi-load case of simultaneous tensile and vertical loading at free end To test the e/uniFB03cacy of the method with respect to multi-loading conditions and changes in beam/element dimensions, we used a beam of size 100mm100mm1000mm which is discretised using the same /one.pnum/six.pnum- noded hybrid brick elements. The dimension of each element is 1mm1mm1000mm in this case. One end of the beam is /uniFB01xed while the other end is simultaneously subjected to a vertical load of magnitude /one.pnumkN and a tensile load of magnitude /one.pnum/zero.pnumkN, both distributed over the section. First, the compliance of the beam is minimized for a /five.pnum/zero.pnum% weight constraint (Fig. /five.pnum(a)) which is followed by a constraint on polar /uniFB01rst moment of area such that the /uniFB01nal weight remains /five.pnum/zero.pnum% of the design space (Fig. /five.pnum(b)). Interestingly, the /uniFB01nal topologies are considerably di/uniFB00erent as the area moment constraint tries to keep the material closer to the centre of the design space. The /uniFB01nal compliance values, however, were approximately the same (i.e. /three.pnum/six.pnum/four.pnum./seven.pnum/two.pnum J and /three.pnum/seven.pnum/three.pnum./four.pnum/nine.pnum/eight.pnum J for weight constraint and area moment constraint, respectively). This indicates that both methods are equally e/uniFB03cient in such cases, though the area moment based section seems more appropriate in terms of space utilization, more elegant and robust. /three.pnum./five.pnum Demonstrating the ability to generate design space independent beams sections From the comparison of /uniFB01nal topologies in Sections /three.pnum./one.pnum-/three.pnum./four.pnum, it may be said that the new scheme has a potential to be used as an alternative to the conventional weight constraint-based optimization scheme. However, it is important to notice that in all the cross-sections, most of the material distribution occurs at design space boundaries. This implies that in case a di/uniFB00erent shape of the design space such as a triangle is used, the /uniFB01nal topologies would have been very di/uniFB00erent from those obtained above. Therefore, in this section, the ability of the new scheme to generate topologies which could have a lower dependence on the shape of design space is demonstrated. This is done by using a considerably low value of constraint on /seven.pnum Figure /five.pnum: A cross-section topology obtained for a combination of tensile and vertical loads applied simulta- neously at free end of the cantilever beam with /five.pnum/zero.pnum% weight constraint (a), and the topology with a polar /uniFB01rst moment of area constraint such that /uniFB01nal weight is approximately /five.pnum/zero.pnum% of the maximum weight possi- ble for the design space. polar /uniFB01rst moment of area as in order to minimize the compliance, higher allowable values would obviously distribute the material at boundaries. Apart from this, dependence of /uniFB01nal shaft topologies on magnitude of the applied torsion is also described. /three.pnum./five.pnum./one.pnum Bending loads Single bending load on free end To demonstrate the design-space-independency feature, two di/uniFB00erent end-load cases are considered where in the /uniFB01rst case, load is applied vertically (similar to that in Section /three.pnum./one.pnum) and in the second case, load is ap- plied at an angle of 450to the horizontal. The beam models are the same as in Section /three.pnum./one.pnum with exception of /uniFB01nite element discretization ( 75751elements) and load magnitude (/seven.pnumN). Both the beams were /uniFB01rst optimized for an area moment constraint value of /five.pnum /one.pnum/zero.pnum/four.pnum mm/three.pnum (Figs. /six.pnum(a-b). The /uniFB01nal weight fractions were then used to optimize the same beam sections using a weight constraint-based formulation ((Figs. /six.pnum(c-d)). As anticipated, the area moment constraint based formulation resulted in approximate H-sections in both the cases whereas the weight constraint based formulation achieved two completely di/uniFB00erent topologies for change in loading direction (indirectly the shape of design space). Bi-axial bending All the beam model parameters are the same as in the bi-axial loading problem discussed in Section /three.pnum./three.pnum. Five di/uniFB00erent loading conditions are considered; where in the /uniFB01rst case, both vertical and horizontal load has the same magnitude of /five.pnum/zero.pnum/zero.pnumN. In the subsequent cases, the horizontal load is sequentially decreased to zero. All the /uniFB01ve cases are run for a /uniFB01xed value of area moment constraint (i.e., 105mm3). Accordingly, the horizontal load is /five.pnum/zero.pnum/zero.pnumN in the /uniFB01rst case, /four.pnum/zero.pnum/zero.pnumN in the second, /three.pnum/zero.pnum/zero.pnumN in the third, /one.pnum/zero.pnum/zero.pnumN in fourth, and zero in the last (single load case) (Figs. /seven.pnum(a-e)). To demonstrate the di/uniFB00erence between the outcomes of the two optimization schemes, the /uniFB01rst case (i.e., equal bi-axial load) is also solved using conventional weight constraint based formulation for a low constraint value viz. /two.pnum/zero.pnum% of the maximum possible (Fig. /seven.pnum(f)). Similar to the single vertical load case, innovative sectional geometries were obtained from the di/uniFB00erent loading conditions in bi-axial loading too. In particular, the circular and elliptical cross-sections (Figs. /seven.pnum(a- b)) are di/uniFB03cult to achieve through the conventional method when the beam is subjected to only bending loads. Interestingly, such sections are typically observed in biological tissues such as long bones (Goldman et al. (/two.pnum/zero.pnum/zero.pnum/nine.pnum)). Therefore, the introduction of the new scheme could enable the extension of the topology optimization to the area of anatomical research. /three.pnum./five.pnum./two.pnum Torsion loads Beam optimization under pure torsion is explored again. The model parameters in this case are adopted from Section /three.pnum./two.pnum. First, the beam/shaft is optimized for a low value of polar /uniFB01rst moment of area constraint (i.e., 105mm3) under two di/uniFB00erent torque magnitudes (i.e., 105Nmm and 5104Nmm, respectively) (Figs. /eight.pnum(a-b)). Both problems are then solved using a small weight constraint (i.e., /one.pnum/five.pnum% of design space) (Figs. /eight.pnum(c-d)). It can be observed from Figs. /eight.pnum(a-b) that the beam under higher torque uses more material than the one with lower torque while maintaining a constant constraint value. However, such di/uniFB00erence /eight.pnum Figure /six.pnum: Testing the design-space-independency feature of the new constraint on end-load problems; vertical end-load (a), load at an angle of 450to the horizontal (b), corresponding weight-constraint based topologies (c-d). Figure /seven.pnum: Exploring the design-space-independency feature for varying bi-axial bending loads under a small area moment constraint value; (a) /uniFB01nal cross-section for equal bi-axial loads of magnitude /five.pnum/zero.pnum/zero.pnumN, the opti- mal sections achieved by decreasing the horizontal load magnitude to /four.pnum/zero.pnum/zero.pnumN, /three.pnum/zero.pnum/zero.pnumN and /one.pnum/zero.pnum/zero.pnumN respectively (b-d), geometry with only vertical load applied (e), and the /uniFB01nal section obtained using a small weight constraint of /two.pnum/zero.pnum% with equal bi-axial loads of /five.pnum/zero.pnum/zero.pnumN (f). /nine.pnum Figure /eight.pnum: Test of design-space-independency under torsional moment of di/uniFB00erent magnitudes and small area moment constraint value ( 105mm3); (a-b) shaft cross-sections with torque magnitudes of 105Nmm and5104Nmm respectively, and corresponding cross-sections with weight constraints of /one.pnum/five.pnum% each (c-d). is not observed in the corresponding weight-constraint based results (Figs. /eight.pnum(c-d)) where both geometries are nearly identical. The results present the current scheme as an ideal choice for shaft design as it takes care of all the major design aspects i.e. shape, dimensions and material e/uniFB03ciency. /four.pnum Discussion and Conclusion A novel method for optimization of beam/shaft cross-sections is presented. In this method, the structural compliance is minimized against a constraint on polar /uniFB01rst moment of area (similar to polar moment of inertia) rather than a conventional weight constraint. The constraint is useful as in addition to constraining the weight of the beam, it also constrains the ‘enclosed area’ of the beam section and thus decreases the dependency on shape of the design space; as can be seen in Figs. /six.pnum-/eight.pnum. In addition, the method can suggest compact, realistic and robust I-beam sections having tapered /uniFB02anges which are typically used in the actual practice (e.g., rail pro/uniFB01le). Such compact designs are also expected to have better resistance to lateral (i.e. sectional) buckling (due to a lower lateral slenderness ratio) and twisting/warping (due to thicker/compact size), which are typical concerns during beam design for unforeseen eccentric loading (Liu et al. (/two.pnum/zero.pnum/zero.pnum/eight.pnum)). The inherent resistance for warping is an important advantage of this method as most of the existing beam optimization methods did not consider the e/uniFB00ect of twisting/warping and lateral buckling (Zuberi et al. (/two.pnum/zero.pnum/one.pnum/zero.pnum); Patel et al. (/two.pnum/zero.pnum/zero.pnum/nine.pnum); Ishii and Aomura (/two.pnum/zero.pnum/zero.pnum/four.pnum); Kim and Kim (/two.pnum/zero.pnum/zero.pnum/zero.pnum)). Further, the method also intro- duces and tests a design space-independent loading scheme where the total force or torque is prescribed, which is distributed on the section elements according to their elastic moduli. This scheme along with the polar moment of area constraint can lead to intuitive hollow circular and elliptical sections (Fig. /seven.pnum) for bi- axial bending cases which has never been achieved using traditional weight/volume/stress/area constraint. Apart from the design-space-independency, the complete formulation (i.e., the objective, constraint, and the loading scheme) also brings in a way to achieve material e/uniFB03ciency in shaft design as both torque and the constraint magnitude govern the /uniFB01nal weight of the section (Fig. /eight.pnum). Further, the dependency of force on elastic modulus of the elements may be particularly suitable for optimizing beam sections composed of multiple materials. The current optimization method may have some limitations as well. Due to its very formulation, the new method may have an inherent a/uniFB03nity towards the centre of design space. Hence, the /uniFB01nal topologies obtained through the new method may sometime be inferior to those obtained through /one.pnum/zero.pnum conventional weight constrained based method. Therefore, it is advised to use the method as a support rather than a replacement to the conventional method. Similarly, the optimization method may suggest di/uniFB00erent topologies from a change of parameters due to inherent limitation of gradient based optimization methods (such as the currently used MMA algorithm) to converge to local optima which may sometimes be redundant. Nevertheless, the limitation may be overcome by using global optimization methods (Wang et al. (/two.pnum/zero.pnum/zero.pnum/one.pnum)). Moreover, the new method presents an idea of constraints beyond weight and stress, and is a step towards making design independent of the design space. To summarize, the work is an attempt to overcome the e/uniFB00ect of the shape of the design space on topologically optimized beams/shaft sections. The goal is achieved through a new constraint accompanied by a /uniFB01tting force application method. In addi- tion to design-space-independency, the formulation has an added bene/uniFB01t of achieving material e/uniFB03ciency with respect to the cost of space/radial distance. Complete testing of the formulation was performed us- ing a simpli/uniFB01ed beam model which is also unique in itself, as it can easily incorporate material anisotropy and inhomogeneity through suitable alteration in element elastic sti/uniFB00ness matrix. Hence, this work has three major contributes to the /uniFB01eld of cross-section optimization : (i) the polar /uniFB01rst moment of area con- straint, (ii) the force and torsion application scheme, and (iii) the simpli/uniFB01ed model of prismatic beam that can take up realistic loads without requiring an extrusion constraint. Most importantly, all three formula- tions/contributions are simple enough to be integrated into most of the optimization platforms. Acknowledgements Authors acknowledge Department of Science and Technology (Grant No. SB/S/three.pnum/MMER//zero.pnum/zero.pnum/four.pnum/six.pnum//two.pnum/zero.pnum/one.pnum/three.pnum) for the computational facility. First author also acknowledges IIT Ropar for the institute fellowship.