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Comparative analysis of strut-and-tie models using Smooth Evolutionary Structural Optimization
Engineering Structures 56 (2013) 1665–1675
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Engineering Structures journal homepage: www.elsevier.com/locate/engstruct
Comparative analysis of strut-and-tie models using Smooth Evolutionary Structural Optimization Valério S. Almeida a, Hélio Luiz Simonetti b,⇑, Luttgardes Oliveira Neto c a
Department of Geotechnical and Structural Engineering, School of Engineering of the University of São Paulo (EPUSP), Brazil Department of Civil Engineering, Federal University of Ouro Preto (UFOP), Brazil c Department of Civil Engineering, Faculty of Engineering (FE), UNESP, Brazil b
a r t i c l e
i n f o
Article history: Received 29 May 2012 Revised 15 February 2013 Accepted 4 July 2013
Keywords: Strut-and-tie models Topology optimization Reinforced concrete structures Evolutionary Structural Optimization
a b s t r a c t The strut-and-tie models are widely used in certain types of structural elements in reinforced concrete and in regions with complexity of the stress state, called regions ‘‘D’’, where the distribution of deformations in the cross section is not linear. This paper introduces a numerical technique to determine the strut-and-tie models using a variant of the classical Evolutionary Structural Optimization, which is called Smooth Evolutionary Structural Optimization. The basic idea of this technique is to identify the numerical flow of stresses generated in the structure, setting out in more technical and rational members of strutand-tie, and to quantify their value for future structural design. This paper presents an index performance based on the evolutionary topology optimization method for automatically generating optimal strut-andtie models in reinforced concrete structures with stress constraints. In the proposed approach, the element with the lowest Von Mises stress is calculated for element removal, while a performance index is used to monitor the evolutionary optimization process. Thus, a comparative analysis of the strutand-tie models for beams is proposed with the presentation of examples from the literature that demonstrates the efficiency of this formulation. Ó 2013 Elsevier Ltd. All rights reserved.
1. Introduction With the growing worldwide competition demanding lower costs, the application of topological optimization has become a profit-oriented research tool for the industry, especially the steel industry, because it facilitates material molding. This application has the computational ability to solve large problems using lowcost microprocessors, therefore creating an immense opportunity for research in applied mechanics. As such, topological evaluation has been used in the field of mechanics to optimize structural components or elements in one, two, or three dimensions. To achieve this, stress and displacement analysis techniques, for example, by finite element (FEM) or boundary element methods (BEM), together with optimization criteria, such as Von Mises stress constraints, natural frequency or buckling, are used in search of solutions for numerical–mathematical problems. Topology Optimization (TO) is a recent topic in the structural optimization field. However, the basic concepts that support the theoretical method have been established for over a century, as described by Rozvany et al. [1]. The great advantage of TO compared ⇑ Corresponding author. E-mail addresses: [email protected] (V.S. Almeida), heliosimonet [email protected] (H.L. Simonetti), [email protected] (L.O. Neto). 0141-0296/$ - see front matter Ó 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.engstruct.2013.07.007
to traditional optimization methods, such as shape or parametric optimization, is that these methods are not able to change the layout of the original structure, failing to help the project conceptual framework for designing adequate flow stress. In topological analysis, two methodologies are important: the micro and macro approaches. The micro approach considers the existence of a microporous structure that depends on its geometry and the volumetric density of a unit cell representative of the material properties and its constitutive relations. An example for this group is the SIMP (simple isotropic material with penalization) method [1–3]. In the macro approach, the topology of the structure is modified by the insertion of holes in the field. As an example of this group of TO, ESO (Evolutionary Structural Optimization) can be mentioned, which is based on the solution of the objective function when an element is removed from the finite element mesh and TSA (Topological Sensitivity Analysis), based on a scalar function, called derived topology, which provides each set point in the problem domain the sensitivity of the cost function when a small hole is created [4–5]. Strut-and-tie models are used to idealize the load transfer mechanism in a cracked structural concrete member at the ultimate limit states. The design task is mainly to identify the load transfer mechanism in a structural concrete member and reinforce the member such that this load path will safely transfer the applied
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loads to the supports. Obviously, some regions of a structural concrete member are not as effective in carrying loads as others. By eliminating these underutilized portions from a structural concrete member, the actual load transfer mechanism in the cracked concrete member can be found. The SESO technique is able to identify the underutilized portions of a structure and to remove them from the structure, to improve its performance. Developing an appropriate strut-and-tie model in a structural concrete member can be transformed into a topology optimization problem of a continuum structure. The optimal topology of a plane stress continuum structure, using the SESO technique, tends towards a model of truss structures. Therefore, it is appropriate to apply this technology to the automatic generation of strut and tie models in concrete structures. In order to propose an effective tool for developing strut-and-tie models, this work uses the technique in topology optimization SESO (Smooth-ESO), which is a variant of ESO, founded on verifying whether the element is not really necessary to the structure. Hence, the contribution to the ESO technique is the reduction of stiffness until it no longer has influence. That is, their removal is conducted smoothly, reducing the values of the constitutive element of the array, as if it is in the process of damage and capable of generating ideal members of struts-and-ties models. Thus, in the SESO technique used herein, the tensile (steel) and compressive (concrete) regions are incorporated with topology optimization to generate more efficient strut-and-tie models. To evaluate elasticity problems numerically, finite element analysis is applied, but instead using a plane–stress triangular finite element implemented with high-order modes for solving complex geometric problems. Three examples are presented to show this effect: (1) single short corbel; (2) corbel jointed with column and (3) deep beam with hole. 2. Smooth Evolutionary Structural Optimization (SESO) Xie and Steven [6] developed a very simple way to impose modifications on the topology of a structure, using a heuristic gradual removal of the mesh of the finite elements, corresponding to the regions that do not effectively contribute to a better performance of the structure. Therefore, in this paper, the parameters of interest for the optimization problem are evaluated in an iterative process in order to decrease the weight by employing the maximum stress criterion of the structure. An initial finite element mesh is defined circumscribing the entire structure, or an extended domain of the design to include the boundary conditions (forces, displacements, cavities and other initial conditions). Thus, the elastic problem is solved via FEM iteratively; the principal stress is evaluated at each element, the highest stress values are taken as reference and the finite element which presents stress values above the defined condition as described in inequality (1a): vm Þ rev m ¼ RRk ðri;max
ð1aÞ
RRkþ1 ¼ RRk þ RE
ð1bÞ
v m are, respectively, the principal Von Mises where rev m and ri;max stress of element in the ith iteration and the maximum stress effective structure in the iteration ‘‘i’’; RRk is the rejection ratio at the kth steady state (0.0 6 RR 6 1.0), which is an input datum that is updated using the evolutionary rate (RE). The evolutionary process is defined by adding the rate of evolution (RE) to the RR factor, Eq. (1b), which is applied to control the removal process of the structure. In the subsequent iteration, the elastic problem is solved again, without the finite elements removed from the design body. The
elimination of the rejected elements is observed to adopt Young’s modulus values or thickness in the order of initial values times 1e8. This procedure avoids a re-meshing approach of the body and a subsequent interference in the relative field of responses. The same cycle of removing the elements used by inequality (1a) is repeated until there are no more elements that satisfy inequality (1a); when this occurs, or when the design volume is obtained, the steady state is reached. This procedure is known as hard kill and can be interpreted as follows:
DðjÞ ¼
D0
if j 2 C
0
if j 2 C
ð2Þ
where D(j) is the constitutive matrix of element j e X, D0 is the initial constitutive matrix, X ¼ C þ C is the structure’s domain, v m ðXÞÞ P RRg is the amount of elements that will not C ¼ fðre =rmax be removed from the structure (solid), and C ¼ X C ¼ v m ðXÞÞ < RRg is the set of elements that are removed fX=ðrev m =rmax from the structure (empty parts), in the ith iteration, due to meeting inequality (1a). Tanskanen [7] proved mathematically that the ESO heuristic removal approach of these elements is associated to regions with low level of strain energy and it is a non-explicit minimization procedure in relation to their weight if compared to a deterministic method, such as the sequential linear programming (SLP) of the optimization algorithm technique. However, the removal of an element may unduly affect the optimum; one way to correct this deviation would be the possibility of reinserting the element in the structure. In this sense, a variant of the ESO stands out, the BESO – Bidirectional Evolutionary Structural Optimization [8]. The SESO, Smooth ESO, comes from this mathematically consistent philosophy, weighting the Young’s modulus (E) and making the strain energy of the element increases tending to the strain energy of the structure; then the gradient tends to zero and the direction of minimum is restored. This is a smooth heuristic of the withdrawal of finite elements of the ESO method. Thus, the elements that meet the inequality (1a) are divided into groups and organized in order of increasing stress. Then the p% of those groups are excluded (the elements with the least stress, CLS domain) and (1 p%) is weighted by a regulatory function ð0 6 gðjÞ 6 1Þ and returned to the structure (CGS domain). This removal and devolution procedure of elements is performed by a v m within the C hyperbolic function, that weights the rate rve m =rmax domain; that is, it permits the high-stress elements (closest to m rvmax but fulfilling the ESO constraint in the CGS domain) providing a better path for flow of tensions in each iteration. The elements near the maximum limit stress are maintained in the structure, defining the procedure as a non-‘‘hard-kill’’ withdrawal, but a smooth process in the heuristic of elements removal.
Fig. 1. Smoothing of the volume of the elements removed in iteration i.
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The soft kill procedure used in the SESO technique can be interpreted as follows:
8 9 if j 2 Ci > > < D0 = Di ðjÞ ¼ D0 gj ðCÞ if j 2 CGS > > : ; 0 if j 2 CLS
ð3Þ
where C ¼ CLS þ CGS , for 0 6 gj ðCÞ 6 1 is the regulating function v m within the C domain, that weights the value of the rate rev m =rmax and this procedure can eliminate the checkerboard problem. The proposed smoothing can be, for example, performed by gðCÞ using a linear function of the gðCÞ ¼ aj þ b type or a trigonometric function of the gðCÞ ¼ senðajÞ type. Because these two functions are continuous, they can be differentiated over all of the C domain, and they have an image varying from 0 to 1, Fig. 1. It is noteworthy that for smaller values of the element removal ratio used by the SESO optimization criterion, the more accurate is the final design, the more expensive with larger computational time.
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3. Performance index for the SESO formulation The heuristic of the withdrawal of the undesired elements of the ESO/SESO methods is equivalent to the optimization procedure of the weight function of structure (W) to find a minimum stationary point, as demonstrated by Tanskanen [7]. It is known that the design criteria are defined in technical standards for acceptable limits of stresses or strains and the problem consists of the minimization of the objective function in terms of weight, subject to an allowable stress constraint (rproject) which can be defined as:
minimize W ¼
NE X we ðt e Þ e¼1
subject to
ð4Þ
v m rproject 6 0 rj;max
where W is the total weight of the structure, we is the weight of the eth element, te is the thickness of the eth element that is also treated v m is the maximum Von Mises stress of each as a design variable, re;max element in the structure and NE the number of elements.
Fig. 2. Flowchart for the Smooth ESO procedure.
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In [9] is proposed that, instead of solving the optimization problem directly as previously indicated by Eq. (4), a monitoring parameter of the history of its performance, called performance index (PI), can be implemented for continuous structures with stress constraints, and this index is defined as:
PI ¼
W s0 W si
ð5Þ
with:
W s0 ¼
W si ¼
vm r
0;max project
r
W0
vm ri;max Wi project
r
ð6aÞ
ð6bÞ
where W0 and Wi are the weight of the initial and current design at vm v m are the initial and ith-iteration the ith iteration, r0;max and ri;max maximum Von Mises stresses. Replacing Eqs. (6a) and (6b) in (5) one obtains:
!
!
!
vm vm m r0;max r0;max rv0;max W0 q V0 V0 0 PI ¼ ¼ ¼ v m v m v m ri;max W i ri;max qi V i ri;max Vi
ð7Þ
where V0 and Vi are the initial and ith-iteration volumes, q0 and qi are the initial and ith-iteration densities, which are equal for an incompressible material. The smoothing generated due to Eq. (3) in terms of the constitutive matrix can be written in terms of thickness due to the direct linear relation between them. In this context, the performance index in formula (7), which takes into account expression (3) in terms of each thickness and the regulating function from the SESO procedure, is written as:
!
!
vm vm r0;max r0;max A0 t A0 t 0 PI ¼ ¼ PNE PNE vm vm ri;max r i;max j¼1 Aj t j j¼1 Aj t j gðjÞ
ð8Þ
where t0 is the initial thickness and tj is the thickness of the jth element at the ith iteration. The optimal control is exerted by this performance index, because it is a ‘‘monitoring factor’’ in the region optimal design. The control of maximizing this parameter refers to the minimiza-
Fig. 3. Examples of D and B-regions in shaded and blanked areas.
Fig. 4. (a) Design domain of a beam structure and (b–d) optimal topologies of the beam structure obtained by Kwar and Noh [20].
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tion of the control volume, since the ratio between the stresses does not change greatly, ranging around the unit value. Thus, the curve PI is characterized by the ratio of the volumes and maximizing it means that its minimum volume has been found. However, if the index falls sharply, it is a strong indication that it underwent a local optimum or stationary configuration. There is also no guarantee that this is a global optimum, but an optimal configuration for engineering design. Finally, the algorithm of the evolutionary process can be described as follows or as in the flowchart indicated in Fig. 2: 1. Discretize the domain for analysis with a refined mesh of finite elements. 2. Solve the linear elastic problem, applying displacement boundary conditions and external or body forces. 3. Determine the Von Mises stress distribution for each element and maximum at each iteration.
4. Divide the elements that violate Eq. (1a), into n groups, where p% of n groups are deleted, and (1 p%) of the n groups are returned to the structure, setting each element in domains C, CLS or CGS; 5. Remove the elements that are inside domain CLS, and modify the constitutive matrix of the elements inside CGS with the prescribed regulatory function; 6. If C ¼ 0, then the steady state is reached at the ith iteration; then, update RR = RR + ER. If C–0, do not update RR. 7. While PIi > PIi1 or Vi > Vfinal, repeat steps 2 to 6; if PIi PIi1, equilibrium has been reached, plot final topology. 4. Problems analyzed In structural engineering, most concrete linear elements are designed by a simplified theory, using Bernoulli’s hypothesis. For a real physical analysis of the behavior of these bending elements, the strut-and-tie model, a generalization of the classical analogy of the truss beam model, is usually employed. This analogy is shown by Ritter and Morsch at the beginning of the twentieth century, associated with a reinforced concrete beam in an equivalent truss structure. The discrete elements (bars) represent the fields of tensile stress (rods) and compression (compressed struts) that emerge inside the structural element in bending effect. This analogy has been improved and is still used by the technical standards in the design of reinforced concrete beams in flexural and shear force and laying down various criteria for determining safe limits in its procedures. However, the application of this hypothesis for any structural element can lead to over or under sizing of certain parts of the structure. This hypothesis is valid for parts of the frame in which there is no interference from hard regions, such as sections near the columns, the cavities or other areas where the influence of strain due to shear is not negligible. However, there are structural elements or regions where Bernoulli’s assumptions do not adequately represent the bending structural behavior and the stress distribution. Structural elements such as beams, walls, footings and foundation blocks, and special areas such as beam–column connection, openings in beams and geometric discontinuities, are examples, as shown in Fig. 3. These regions, denominated ‘‘discontinuity regions D’’, are limited to dis-
Fig. 5. Optimal topologies of beam structure obtained by the present model: (a) L/ D = 2, (b) L/D = 3 and (c) L/D = 4.
Fig. 6. Comparison of computational time for beam structure (L/D = 4).
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Fig. 7. Domain design of corbel jointed with column [11].
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Fig. 8. Optimization history of strut-and-tie model in corbel using in-plane elements [11].
Fig. 9. Optimization history of strut-and-tie model in corbel using truss elements [20].
Fig. 10. (a) Initial configuration and (b–d) topologies of the corbel structure obtained by the present model (SESO), 5664-element mesh.
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Fig. 11. (a) Initial configuration and (b–d) topologies of the corbel structure obtained by the present model (SESO), 22,656-element mesh.
tances of the dimension order of structural adjacent elements (Saint Vernant’s Principle), in which shear stresses are applicable and the distribution of deformations in the cross section is not linear. The pioneering work by Schlaich et al. [10] describes the strut-and-tie model more generally, covering the equivalent truss models and including these regions and special structural elements. Numerical analysis has provided these tools for years, with faster processing, new theories and formulations. Together with these tools, to techniques have been employed via strut-and-tie models in reinforced concrete structures as shown in [11–20]. Based on the formulation described in the previous sections, a computer system was developed applying the SESO in conjunction with the finite element method, using a linear-elastic formulation for plane stress state analysis arising from free formulation [21]. An automated mesh generation routine was developed based on Delaunay triangulation and a set of Fortran subroutines was used for solving sparse symmetric sets of linear equations, as indicated in [22]. The Von Mises stresses are calculated in the centroid of each element, at each iteration process, using the nodal stress values. Thus, some numerical examples are presented for evaluating and comparing the configurations obtained by the classical models of strut-and-tie. The optimization parameters RR and RE, if not mentioned, are equal to 1% and defined as regulatory function g(C) = 104.
Fig. 12. Computational time of the corbel jointed with column.
Fig. 13. Design domain [15].
4.1. Beam structures In this example, SESO was applied to find the best topology for optimal structure, the strut-and-tie models using triangular finite elements of high order, by comparing with the results presented in [11] and [20]; the authors respectively used quadrilateral finite elements and truss elements. The design domain and the boundary conditions are shown in Fig. 4a; Fig. 4b–d shows the optimal settings obtained by Kwar and Noh [20], using truss elements. The Young’s modulus of the material is 28,5567 MPa, Poisson’s ratio of 0.15 and thickness of 250 mm. The concentrated load of 1200 kN is applied at the coordinate point (L/2, D), with the regulatory function g(C) = 102. Fig. 5a–c shows the optimal topologies obtained by the present formulation, SESO, using refined meshes, respectively equal to 40 20, 60 20 and 80 20. The optimal topologies shown in Fig. 5a–c was obtained with final volumes equal to 21.75%, 45,75% and 41.65%; the regions in red1 and blue respectively indicate the compressed regions, struts, and tensioned, ties. For the optimal settings shown in Fig. 5a and b, the same optimization parameters (RR, rejection ratio and RE, evolutionary ratio) were used. The optimal configuration shown in Fig. 5c was obtained with parameters RR equal to 1% and RE equal to 3% by volume at
1 For interpretation of color in Figs. 5, 14 and 16 the reader is referred to the web version of this article.
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Fig. 14. (a) Optimal topology obtained by present model; (b) optimal setting by Liang et al. [11] and (c) Strut-and-tie model proposed by Schlaich et al. [10].
Table 1 Dimensioning of reinforcement ties (MN, cm2). Ties
Effort
AS,nec
Effort at ties
Reinforcement
Number of bars
T1 T2 T4 aux T4
1.50 2.24
34.60 51.53
2.20
50.70
1.50 0.966 T2 0.01 0.707 T4 0.707 T4
As1 = 34.60 As2 = 49.75 As4 As3 = 35.85 As3 = 35.85
2 5 £ 20 2 7 £ 20 2 2 £ 20 2 5 £ 20 2 5 £ 20
Fig. 15. Performance index versus number of iterations by the SESO procedure.
Fig. 17. Schematic representation of the reinforcement bars and strut-and-tie model using SESO.
Fig. 16. Proposed strut-and-tie model.
constant removed iteration equal 1.75%. Note that the proposed algorithm is sensitive to the variation of these parameters, boundary conditions and the geometry of the element. Fig. 6 shows the computational time spent for a final volume equal to 50% of the initial design, using a PC with a Intel Core 2 Quad 2.83 GHz processor, 4 GB memory, Windows 7. The present formulation, SESO, is compared with a SIMP formulation presented
in [23], using a penalization power equal 3 (p = 3). Good results are obtained by using the SESO formulation even for poor mesh over the entire structure simulated, without the symmetry resource used by the referenced work authors. The computational time spent by the SESO formulation (developed in Fortran90s language) grows with the mesh refining and shows higher efficiency if compared with the Matlab language, known for its low level computational language. 4.2. Corbel jointed with column The design domain and the boundary conditions of the corbel jointed with column are shown in Fig. 7, designed to support a
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concentrated load of 500 kN. The column is fixed at both ends. An initial thickness of 300 mm is assumed for corbel. The Young’s modulus is 28,5567 MPa and the Poisson ratio is 0.15. This structure is modeled using refined mesh with 88 216 triangular finite elements and the optimization procedure used parameters RE = 1% and RR% = 1. The optimal topology obtained and the strut-and-tie model are presented by Liang et al. [11], which used a method called PBO – Performance-Based Optimization, with 125-mm square, four-node, plane stress elements. Fig 8a–c shows the history of the optimal topology obtained by Liang et al. [11] and Fig. 8d shows the optimal topology and the proposed strut-and-tie model. Fig 9a–d shows the history of optimal topology obtained by Kwar and Noh [20] using truss elements. Figs. 10a–d and 11a–d show the history of optimal topology for the strut-and-tie model obtained by the present formulation, SESO, using refined meshes 44 108 and 88 216, respectively totaling 5664 and 22,756 triangular finite elements. The optimal topologies (Figs. 10d and 11d) show several similarities with the configurations of the strut-and-tie models presented by both referenced works. Fig. 12 shows the computational time of the corbel jointed with column using refined meshes, respectively equal to 22 54 (1416 elements), 44 108 (5664 elements), 44 216 (11,328 elements) and 88 216 (22,656 elements) and final volume equal to 40% of the initial design. The respective processing times of the final volumes are equal to 240, 916, 3036 and 8879 s.
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respectively indicate the compressed regions, strut, and tensioned regions, tie. Fig. 14b shows the optimal design presented by the author [11] who uses the ESO optimization method to obtain a final volume equal to 33%. Fig. 14c shows the optimal configuration for the strut-and-tie design proposed by Liang et al. [11], which uses the strut-and-tie model. The graph of Fig. 15 shows the monitoring made during the performance of the presented formulation to determine the optimal topology. The growth of the PI (performance index) values is plotted for each iteration path, and the point at which the PI drops sharply indicates that the previous iteration is thus the area of optimal design. A proposed strengthening for this structure, as presented by Liang et al. [11], was obtained by using a strut-and-tie model arising from the combination of the finite element method with a procedure to obtain the flow strength, using the ‘‘load path’’ method. Fig. 19 shows the disposition of the reinforcement proposed by the authors to strengthen the beam cavity. From the optimal topology obtained by the present formulation, a proposal for a strut-and-tie model can be directly made. Note the proposed sketch shown in Fig. 16, where the semi-straights in red (dotted) and blue (continuous) respectively indicate compressed members (C), struts, and tensioned members (T), ties. The efforts at the members where the flow stress stands out can be calculated by multiplying the average stress values of each member and their respective area, given by the product of beam thickness to the width of the average flow region.
4.3. Deep beam with hole This example was reported by Schlaich et al. [10]. It is a simply supported deep beam with a large hole, the geometry and load (P) of which are presented in Fig. 13, and which is used as an extended domain for the optimization process. The Young’s modulus is E = 20,820 MPa, the Poisson’s ratio is m = 0.15 and the thickness is 0.4 m. The design strengths of reinforced concrete are taken with values fyd = 434 MPa and fcd = 25 MPa. This structure has a ‘‘D’’ region due to the geometric discontinuity corresponding to the cavity imposed by the design. In this case, this region should be evaluated using the strut-and-tie model. For modeling with the present formulation, 13,200 triangular elements were used (mesh 150 47). Fig. 14a shows the optimal design obtained by the SESO formulation and compared with the work developed by Liang [15], as depicted in Fig. 14b, and Schlaich et al. [10], Fig. 14c. The optimum topology design shown in Fig. 14a was obtained with a final volume equal to 30.3%; the red and blue regions
Fig. 18. Disposition of the reinforcement proposed for the present model.
Fig. 19. Disposition of the reinforcement proposed, beam with hole [10].
Fig. 20. Computational time of the deep beam with hole and topologies.
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Table 2 Verification of compression stresses acting on the struts (MN, MPa). Ties
Effort
r
0.8 fcd = 20
C3 C5 C6 C7 C8
3.3 1.4 1.7 5.5 2.6
28 20 18.5 18.5 22
Strengthen Ok Ok Ok Strengthen
It is thus possible to calculate the requirement reinforcement areas in the tie region and to evaluate the concrete strength in each strut. Table 1 shows the values of the average efforts obtained at the ties. The T2 and T4 ties are inclined at 15° and 45°, respectively, from the horizontal line. The longitudinal bars which represents tie T2 are calculated from the decomposed horizontal portion of their effort, thus obtaining the required area of reinforcement As2. Tie T4 has its representation in the orthogonal mesh, As3, which covers the stretch along the edge of the cavity and the in-angle encounters struts C3 and C8 on the left end. An additional reinforcement As4, at 45°, is proposed, covering the in-angle encounters struts C2, C5 and C8. Conforming to the calculation procedures to obtain the representative reinforcement of ties (Table 1 and Fig. 17, the details of these reinforcement are shown in Fig. 18, where we can see the proposal extention of reinforcement As3 to cover the struts encounter areas. Fig. 20 shows the computational times and the optimal settings of deep beam with hole using refined meshes with 3690 (mesh 45 47), 6600 (mesh 75 47), 7918 (mesh 90 47) and 13,200 (mesh 150 47) triangular elements and final volume corresponding to 30% of the initial design. The processing times related to final volumes of the topology shown in Fig. 20 are, respectively, equal to 1.001, 4.322, 4.638 and 6.527 s. Table 2 presents the verification of the compression stresses acting at strut members where the non-attendance of the ultimate state of compression can be observed in the concrete at struts C3 and C8. 5. Conclusions Aiming to present a numerical formulation for the design of reinforced concrete structures under the focus of the strut-andtie model, a variant of topology optimization procedure, called Smooth Evolutionary Structural Optimization – SESO, was employed. The evolutionary procedure proposed, which smoothly removes the elements of the discretized body, is coupled to a FEM formulation in plane stress state analysis. A priori, an extended initial domain is defined and, iteratively, the method seeks an optimal topology configuration in which natural members are set indicated by struts and ties. Thus, efforts in the members may be evaluated to enable the design of the reinforcements needed for each section. In contrast to the ESO method [15], this formulation – SESO, in conjunction with a high order triangular finite element, presents robustness and efficiency to obtain optimal configurations. Three numerical examples demonstrated: (a) good accuracy with strutand-tie model configurations and the related effort values reported by other authors. The quantification and the disposition of reinforcements for a classic example described in the international specific literature was also proposed; (b) the computational time obtained by the present formulation is smaller than that presented in the literature, even for finer mesh. It is worth pointing out that, for the beam structure example, the computational time spent by the present formulation grows with the mesh refining and shows higher efficiency if compared with [20], and the other two examples were not compared in time processing, because Ref. [20] does not permit cavities or complex geometry for the initial design; (c)
another major improvement obtained by the SESO formulation is the smoothing in classic TO problems, such as ‘‘checkerboard’’ in the topology configurations of structures presented by the cited authors in this article; (d) due to the intrinsic withdrawal/replace heuristic approach, in which some finite elements return to the structure body (neighbor to restriction value), the SESO procedure shows less sensitivity to the development of fine ramifications which can occur during the definition of the main stress path. A best path to main stress flow is defined without oscillations, once the calculation of the effort at each member of the model can be done as a weighted average of values taken inside a circumference of radius R; (e) finally, we emphasize that SESO is an evolutionary numerical technique which can be used in future researches in TO problems that include more complex phenomena such as the nonlinearities and/or dynamic analysis problems mainly due to the quality achieved, to the optimal settings and also to the low computational cost obtained in the problems presented that require intrinsically high computational cost. Acknowledgements The authors thank the Department of Structural and Foundation Engineering, School of Engineering of the University of São Paulo (EPUSP), the University of Ouro Preto (UFOP) and the São Paulo State University (UNESP) for their financial support to this research. References [1] Rozvany GIN, Bendsøe MP, Kirsch U. Layout optimization of structures. Appl Mech Rev 1995;48:41–119. [2] Bendsøe MP. Optimal shape design as a material distribution problem. Struct Optim 1989;1:193–202. [3] Rozvany GIN, Zhou M, Birker T. Generalized shape optimization without homogenization. Struct Optim 1992;4:250–2. [4] Sokolowski J, Zochowski A. On topological derivative in shape optimization. SIAM J Control Optim 1999;37(4):1251–72. [5] Novotny AA, Feijóo RA, Padra C, Taraco E. Topological sensitivity analysis. Comput Meth Appl Mech Eng 2003;192:803–29. [6] Xie YM, Steven GP. A simple evolutionary procedure for structural optimization. Comput Struct 1993;49(5):885–96. [7] Tanskanen P. The evolutionary structural optimization method: theoretical aspects. Comput Meth Appl Mech Eng 2002;191:5485–98. [8] Querin OM. Evolutionary structural optimization stress based formulation and implementation. PhD thesis. University of Sydney; 1997. [9] Liang QQ, Xie YM, Steven GP. Optimal topology selection of continuum structures with stress and displacement constraints. In: The seventh East AsiaPacific conference on structural engineering and construction, Kochi, Japan; 1999. [10] Schlaich J, Schafer K, Jennewein M. Toward a consistent design of structural concrete. PCI J 1987;32(3):74–150. [11] Liang QQ, Xie YM, Steven GP. Topology optimization of strut-and-tie models in reinforced concrete structures using an evolutionary procedure. ACI Struct J 2000;97(2):322–30. [12] Ali M. Automatic generation of truss model for the optimal design of reinforced concrete structures. Dissertation. New York, United States of America, Cornell University, Ithaca; 1997. [13] Liang QQ, Steven GP. A performance-based optimization method for topology design of continuum structures with mean compliance constraints. Comput Meth Appl Mech Eng 2002;191(13–14):1471–89. [14] Liang QQ, Uy B, Steven GP. Performance-based optimization for strut–tie modeling of structural concrete. J Struct Eng 2002;190:815–23. [15] Liang QQ. Performance-based optimization of structures: theory and applications. London: Spon Press; 2005. [16] Reineck K-H. Shear and concrete tensile strength in the design concept of strut-and-tie models. Ibracon Struct J 2007;3(1):1–18. [17] Bruggi M. Generating strut-and-tie patterns for reinforced concrete structures using topology optimization. Comput Struct 2009;87:1483–95. [18] Bruggi M. On the automatic generation of strut and tie patterns under multiple load cases with application to the aseismic design of concrete structures. Adv Struct Eng 2010;13(6):1167–81. [19] Victoria M, Querin OM, Martí P. Generation of strut-and-tie models by topology optimization using different material properties in tension and compression. Struct Multidiscip Optim 2011;44:247–58. [20] Kwar HG, Noh SH. Determination of strut-and-tie models using evolutionary structural optimization. Eng Struct 2006;28:1440–9.
V.S. Almeida et al. / Engineering Structures 56 (2013) 1665–1675 [21] Bergan PG, Felippa CA. A triangular membrane element with rotational degrees of freedom. Comput Meth Appl Mech Eng 1985;50:25–69. [22] Duff IS, Reid JK. A set of Fortran subroutines for solving sparse symmetric sets of linear equations. Report R10533, AERE Harwell; 1982.
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[23] Andreassen E, Clausen A, Schevenels M, Lazarov BS, Sigmund O. Efficient topology optimization in MATLAB using 88 lines of code. Struct Multidiscip Optim 2011;43(1):1–16.
Contents lists available at SciVerse ScienceDirect
Engineering Structures journal homepage: www.elsevier.com/locate/engstruct
Comparative analysis of strut-and-tie models using Smooth Evolutionary Structural Optimization Valério S. Almeida a, Hélio Luiz Simonetti b,⇑, Luttgardes Oliveira Neto c a
Department of Geotechnical and Structural Engineering, School of Engineering of the University of São Paulo (EPUSP), Brazil Department of Civil Engineering, Federal University of Ouro Preto (UFOP), Brazil c Department of Civil Engineering, Faculty of Engineering (FE), UNESP, Brazil b
a r t i c l e
i n f o
Article history: Received 29 May 2012 Revised 15 February 2013 Accepted 4 July 2013
Keywords: Strut-and-tie models Topology optimization Reinforced concrete structures Evolutionary Structural Optimization
a b s t r a c t The strut-and-tie models are widely used in certain types of structural elements in reinforced concrete and in regions with complexity of the stress state, called regions ‘‘D’’, where the distribution of deformations in the cross section is not linear. This paper introduces a numerical technique to determine the strut-and-tie models using a variant of the classical Evolutionary Structural Optimization, which is called Smooth Evolutionary Structural Optimization. The basic idea of this technique is to identify the numerical flow of stresses generated in the structure, setting out in more technical and rational members of strutand-tie, and to quantify their value for future structural design. This paper presents an index performance based on the evolutionary topology optimization method for automatically generating optimal strut-andtie models in reinforced concrete structures with stress constraints. In the proposed approach, the element with the lowest Von Mises stress is calculated for element removal, while a performance index is used to monitor the evolutionary optimization process. Thus, a comparative analysis of the strutand-tie models for beams is proposed with the presentation of examples from the literature that demonstrates the efficiency of this formulation. Ó 2013 Elsevier Ltd. All rights reserved.
1. Introduction With the growing worldwide competition demanding lower costs, the application of topological optimization has become a profit-oriented research tool for the industry, especially the steel industry, because it facilitates material molding. This application has the computational ability to solve large problems using lowcost microprocessors, therefore creating an immense opportunity for research in applied mechanics. As such, topological evaluation has been used in the field of mechanics to optimize structural components or elements in one, two, or three dimensions. To achieve this, stress and displacement analysis techniques, for example, by finite element (FEM) or boundary element methods (BEM), together with optimization criteria, such as Von Mises stress constraints, natural frequency or buckling, are used in search of solutions for numerical–mathematical problems. Topology Optimization (TO) is a recent topic in the structural optimization field. However, the basic concepts that support the theoretical method have been established for over a century, as described by Rozvany et al. [1]. The great advantage of TO compared ⇑ Corresponding author. E-mail addresses: [email protected] (V.S. Almeida), heliosimonet [email protected] (H.L. Simonetti), [email protected] (L.O. Neto). 0141-0296/$ - see front matter Ó 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.engstruct.2013.07.007
to traditional optimization methods, such as shape or parametric optimization, is that these methods are not able to change the layout of the original structure, failing to help the project conceptual framework for designing adequate flow stress. In topological analysis, two methodologies are important: the micro and macro approaches. The micro approach considers the existence of a microporous structure that depends on its geometry and the volumetric density of a unit cell representative of the material properties and its constitutive relations. An example for this group is the SIMP (simple isotropic material with penalization) method [1–3]. In the macro approach, the topology of the structure is modified by the insertion of holes in the field. As an example of this group of TO, ESO (Evolutionary Structural Optimization) can be mentioned, which is based on the solution of the objective function when an element is removed from the finite element mesh and TSA (Topological Sensitivity Analysis), based on a scalar function, called derived topology, which provides each set point in the problem domain the sensitivity of the cost function when a small hole is created [4–5]. Strut-and-tie models are used to idealize the load transfer mechanism in a cracked structural concrete member at the ultimate limit states. The design task is mainly to identify the load transfer mechanism in a structural concrete member and reinforce the member such that this load path will safely transfer the applied
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loads to the supports. Obviously, some regions of a structural concrete member are not as effective in carrying loads as others. By eliminating these underutilized portions from a structural concrete member, the actual load transfer mechanism in the cracked concrete member can be found. The SESO technique is able to identify the underutilized portions of a structure and to remove them from the structure, to improve its performance. Developing an appropriate strut-and-tie model in a structural concrete member can be transformed into a topology optimization problem of a continuum structure. The optimal topology of a plane stress continuum structure, using the SESO technique, tends towards a model of truss structures. Therefore, it is appropriate to apply this technology to the automatic generation of strut and tie models in concrete structures. In order to propose an effective tool for developing strut-and-tie models, this work uses the technique in topology optimization SESO (Smooth-ESO), which is a variant of ESO, founded on verifying whether the element is not really necessary to the structure. Hence, the contribution to the ESO technique is the reduction of stiffness until it no longer has influence. That is, their removal is conducted smoothly, reducing the values of the constitutive element of the array, as if it is in the process of damage and capable of generating ideal members of struts-and-ties models. Thus, in the SESO technique used herein, the tensile (steel) and compressive (concrete) regions are incorporated with topology optimization to generate more efficient strut-and-tie models. To evaluate elasticity problems numerically, finite element analysis is applied, but instead using a plane–stress triangular finite element implemented with high-order modes for solving complex geometric problems. Three examples are presented to show this effect: (1) single short corbel; (2) corbel jointed with column and (3) deep beam with hole. 2. Smooth Evolutionary Structural Optimization (SESO) Xie and Steven [6] developed a very simple way to impose modifications on the topology of a structure, using a heuristic gradual removal of the mesh of the finite elements, corresponding to the regions that do not effectively contribute to a better performance of the structure. Therefore, in this paper, the parameters of interest for the optimization problem are evaluated in an iterative process in order to decrease the weight by employing the maximum stress criterion of the structure. An initial finite element mesh is defined circumscribing the entire structure, or an extended domain of the design to include the boundary conditions (forces, displacements, cavities and other initial conditions). Thus, the elastic problem is solved via FEM iteratively; the principal stress is evaluated at each element, the highest stress values are taken as reference and the finite element which presents stress values above the defined condition as described in inequality (1a): vm Þ rev m ¼ RRk ðri;max
ð1aÞ
RRkþ1 ¼ RRk þ RE
ð1bÞ
v m are, respectively, the principal Von Mises where rev m and ri;max stress of element in the ith iteration and the maximum stress effective structure in the iteration ‘‘i’’; RRk is the rejection ratio at the kth steady state (0.0 6 RR 6 1.0), which is an input datum that is updated using the evolutionary rate (RE). The evolutionary process is defined by adding the rate of evolution (RE) to the RR factor, Eq. (1b), which is applied to control the removal process of the structure. In the subsequent iteration, the elastic problem is solved again, without the finite elements removed from the design body. The
elimination of the rejected elements is observed to adopt Young’s modulus values or thickness in the order of initial values times 1e8. This procedure avoids a re-meshing approach of the body and a subsequent interference in the relative field of responses. The same cycle of removing the elements used by inequality (1a) is repeated until there are no more elements that satisfy inequality (1a); when this occurs, or when the design volume is obtained, the steady state is reached. This procedure is known as hard kill and can be interpreted as follows:
DðjÞ ¼
D0
if j 2 C
0
if j 2 C
ð2Þ
where D(j) is the constitutive matrix of element j e X, D0 is the initial constitutive matrix, X ¼ C þ C is the structure’s domain, v m ðXÞÞ P RRg is the amount of elements that will not C ¼ fðre =rmax be removed from the structure (solid), and C ¼ X C ¼ v m ðXÞÞ < RRg is the set of elements that are removed fX=ðrev m =rmax from the structure (empty parts), in the ith iteration, due to meeting inequality (1a). Tanskanen [7] proved mathematically that the ESO heuristic removal approach of these elements is associated to regions with low level of strain energy and it is a non-explicit minimization procedure in relation to their weight if compared to a deterministic method, such as the sequential linear programming (SLP) of the optimization algorithm technique. However, the removal of an element may unduly affect the optimum; one way to correct this deviation would be the possibility of reinserting the element in the structure. In this sense, a variant of the ESO stands out, the BESO – Bidirectional Evolutionary Structural Optimization [8]. The SESO, Smooth ESO, comes from this mathematically consistent philosophy, weighting the Young’s modulus (E) and making the strain energy of the element increases tending to the strain energy of the structure; then the gradient tends to zero and the direction of minimum is restored. This is a smooth heuristic of the withdrawal of finite elements of the ESO method. Thus, the elements that meet the inequality (1a) are divided into groups and organized in order of increasing stress. Then the p% of those groups are excluded (the elements with the least stress, CLS domain) and (1 p%) is weighted by a regulatory function ð0 6 gðjÞ 6 1Þ and returned to the structure (CGS domain). This removal and devolution procedure of elements is performed by a v m within the C hyperbolic function, that weights the rate rve m =rmax domain; that is, it permits the high-stress elements (closest to m rvmax but fulfilling the ESO constraint in the CGS domain) providing a better path for flow of tensions in each iteration. The elements near the maximum limit stress are maintained in the structure, defining the procedure as a non-‘‘hard-kill’’ withdrawal, but a smooth process in the heuristic of elements removal.
Fig. 1. Smoothing of the volume of the elements removed in iteration i.
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The soft kill procedure used in the SESO technique can be interpreted as follows:
8 9 if j 2 Ci > > < D0 = Di ðjÞ ¼ D0 gj ðCÞ if j 2 CGS > > : ; 0 if j 2 CLS
ð3Þ
where C ¼ CLS þ CGS , for 0 6 gj ðCÞ 6 1 is the regulating function v m within the C domain, that weights the value of the rate rev m =rmax and this procedure can eliminate the checkerboard problem. The proposed smoothing can be, for example, performed by gðCÞ using a linear function of the gðCÞ ¼ aj þ b type or a trigonometric function of the gðCÞ ¼ senðajÞ type. Because these two functions are continuous, they can be differentiated over all of the C domain, and they have an image varying from 0 to 1, Fig. 1. It is noteworthy that for smaller values of the element removal ratio used by the SESO optimization criterion, the more accurate is the final design, the more expensive with larger computational time.
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3. Performance index for the SESO formulation The heuristic of the withdrawal of the undesired elements of the ESO/SESO methods is equivalent to the optimization procedure of the weight function of structure (W) to find a minimum stationary point, as demonstrated by Tanskanen [7]. It is known that the design criteria are defined in technical standards for acceptable limits of stresses or strains and the problem consists of the minimization of the objective function in terms of weight, subject to an allowable stress constraint (rproject) which can be defined as:
minimize W ¼
NE X we ðt e Þ e¼1
subject to
ð4Þ
v m rproject 6 0 rj;max
where W is the total weight of the structure, we is the weight of the eth element, te is the thickness of the eth element that is also treated v m is the maximum Von Mises stress of each as a design variable, re;max element in the structure and NE the number of elements.
Fig. 2. Flowchart for the Smooth ESO procedure.
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In [9] is proposed that, instead of solving the optimization problem directly as previously indicated by Eq. (4), a monitoring parameter of the history of its performance, called performance index (PI), can be implemented for continuous structures with stress constraints, and this index is defined as:
PI ¼
W s0 W si
ð5Þ
with:
W s0 ¼
W si ¼
vm r
0;max project
r
W0
vm ri;max Wi project
r
ð6aÞ
ð6bÞ
where W0 and Wi are the weight of the initial and current design at vm v m are the initial and ith-iteration the ith iteration, r0;max and ri;max maximum Von Mises stresses. Replacing Eqs. (6a) and (6b) in (5) one obtains:
!
!
!
vm vm m r0;max r0;max rv0;max W0 q V0 V0 0 PI ¼ ¼ ¼ v m v m v m ri;max W i ri;max qi V i ri;max Vi
ð7Þ
where V0 and Vi are the initial and ith-iteration volumes, q0 and qi are the initial and ith-iteration densities, which are equal for an incompressible material. The smoothing generated due to Eq. (3) in terms of the constitutive matrix can be written in terms of thickness due to the direct linear relation between them. In this context, the performance index in formula (7), which takes into account expression (3) in terms of each thickness and the regulating function from the SESO procedure, is written as:
!
!
vm vm r0;max r0;max A0 t A0 t 0 PI ¼ ¼ PNE PNE vm vm ri;max r i;max j¼1 Aj t j j¼1 Aj t j gðjÞ
ð8Þ
where t0 is the initial thickness and tj is the thickness of the jth element at the ith iteration. The optimal control is exerted by this performance index, because it is a ‘‘monitoring factor’’ in the region optimal design. The control of maximizing this parameter refers to the minimiza-
Fig. 3. Examples of D and B-regions in shaded and blanked areas.
Fig. 4. (a) Design domain of a beam structure and (b–d) optimal topologies of the beam structure obtained by Kwar and Noh [20].
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tion of the control volume, since the ratio between the stresses does not change greatly, ranging around the unit value. Thus, the curve PI is characterized by the ratio of the volumes and maximizing it means that its minimum volume has been found. However, if the index falls sharply, it is a strong indication that it underwent a local optimum or stationary configuration. There is also no guarantee that this is a global optimum, but an optimal configuration for engineering design. Finally, the algorithm of the evolutionary process can be described as follows or as in the flowchart indicated in Fig. 2: 1. Discretize the domain for analysis with a refined mesh of finite elements. 2. Solve the linear elastic problem, applying displacement boundary conditions and external or body forces. 3. Determine the Von Mises stress distribution for each element and maximum at each iteration.
4. Divide the elements that violate Eq. (1a), into n groups, where p% of n groups are deleted, and (1 p%) of the n groups are returned to the structure, setting each element in domains C, CLS or CGS; 5. Remove the elements that are inside domain CLS, and modify the constitutive matrix of the elements inside CGS with the prescribed regulatory function; 6. If C ¼ 0, then the steady state is reached at the ith iteration; then, update RR = RR + ER. If C–0, do not update RR. 7. While PIi > PIi1 or Vi > Vfinal, repeat steps 2 to 6; if PIi PIi1, equilibrium has been reached, plot final topology. 4. Problems analyzed In structural engineering, most concrete linear elements are designed by a simplified theory, using Bernoulli’s hypothesis. For a real physical analysis of the behavior of these bending elements, the strut-and-tie model, a generalization of the classical analogy of the truss beam model, is usually employed. This analogy is shown by Ritter and Morsch at the beginning of the twentieth century, associated with a reinforced concrete beam in an equivalent truss structure. The discrete elements (bars) represent the fields of tensile stress (rods) and compression (compressed struts) that emerge inside the structural element in bending effect. This analogy has been improved and is still used by the technical standards in the design of reinforced concrete beams in flexural and shear force and laying down various criteria for determining safe limits in its procedures. However, the application of this hypothesis for any structural element can lead to over or under sizing of certain parts of the structure. This hypothesis is valid for parts of the frame in which there is no interference from hard regions, such as sections near the columns, the cavities or other areas where the influence of strain due to shear is not negligible. However, there are structural elements or regions where Bernoulli’s assumptions do not adequately represent the bending structural behavior and the stress distribution. Structural elements such as beams, walls, footings and foundation blocks, and special areas such as beam–column connection, openings in beams and geometric discontinuities, are examples, as shown in Fig. 3. These regions, denominated ‘‘discontinuity regions D’’, are limited to dis-
Fig. 5. Optimal topologies of beam structure obtained by the present model: (a) L/ D = 2, (b) L/D = 3 and (c) L/D = 4.
Fig. 6. Comparison of computational time for beam structure (L/D = 4).
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Fig. 7. Domain design of corbel jointed with column [11].
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Fig. 8. Optimization history of strut-and-tie model in corbel using in-plane elements [11].
Fig. 9. Optimization history of strut-and-tie model in corbel using truss elements [20].
Fig. 10. (a) Initial configuration and (b–d) topologies of the corbel structure obtained by the present model (SESO), 5664-element mesh.
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Fig. 11. (a) Initial configuration and (b–d) topologies of the corbel structure obtained by the present model (SESO), 22,656-element mesh.
tances of the dimension order of structural adjacent elements (Saint Vernant’s Principle), in which shear stresses are applicable and the distribution of deformations in the cross section is not linear. The pioneering work by Schlaich et al. [10] describes the strut-and-tie model more generally, covering the equivalent truss models and including these regions and special structural elements. Numerical analysis has provided these tools for years, with faster processing, new theories and formulations. Together with these tools, to techniques have been employed via strut-and-tie models in reinforced concrete structures as shown in [11–20]. Based on the formulation described in the previous sections, a computer system was developed applying the SESO in conjunction with the finite element method, using a linear-elastic formulation for plane stress state analysis arising from free formulation [21]. An automated mesh generation routine was developed based on Delaunay triangulation and a set of Fortran subroutines was used for solving sparse symmetric sets of linear equations, as indicated in [22]. The Von Mises stresses are calculated in the centroid of each element, at each iteration process, using the nodal stress values. Thus, some numerical examples are presented for evaluating and comparing the configurations obtained by the classical models of strut-and-tie. The optimization parameters RR and RE, if not mentioned, are equal to 1% and defined as regulatory function g(C) = 104.
Fig. 12. Computational time of the corbel jointed with column.
Fig. 13. Design domain [15].
4.1. Beam structures In this example, SESO was applied to find the best topology for optimal structure, the strut-and-tie models using triangular finite elements of high order, by comparing with the results presented in [11] and [20]; the authors respectively used quadrilateral finite elements and truss elements. The design domain and the boundary conditions are shown in Fig. 4a; Fig. 4b–d shows the optimal settings obtained by Kwar and Noh [20], using truss elements. The Young’s modulus of the material is 28,5567 MPa, Poisson’s ratio of 0.15 and thickness of 250 mm. The concentrated load of 1200 kN is applied at the coordinate point (L/2, D), with the regulatory function g(C) = 102. Fig. 5a–c shows the optimal topologies obtained by the present formulation, SESO, using refined meshes, respectively equal to 40 20, 60 20 and 80 20. The optimal topologies shown in Fig. 5a–c was obtained with final volumes equal to 21.75%, 45,75% and 41.65%; the regions in red1 and blue respectively indicate the compressed regions, struts, and tensioned, ties. For the optimal settings shown in Fig. 5a and b, the same optimization parameters (RR, rejection ratio and RE, evolutionary ratio) were used. The optimal configuration shown in Fig. 5c was obtained with parameters RR equal to 1% and RE equal to 3% by volume at
1 For interpretation of color in Figs. 5, 14 and 16 the reader is referred to the web version of this article.
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Fig. 14. (a) Optimal topology obtained by present model; (b) optimal setting by Liang et al. [11] and (c) Strut-and-tie model proposed by Schlaich et al. [10].
Table 1 Dimensioning of reinforcement ties (MN, cm2). Ties
Effort
AS,nec
Effort at ties
Reinforcement
Number of bars
T1 T2 T4 aux T4
1.50 2.24
34.60 51.53
2.20
50.70
1.50 0.966 T2 0.01 0.707 T4 0.707 T4
As1 = 34.60 As2 = 49.75 As4 As3 = 35.85 As3 = 35.85
2 5 £ 20 2 7 £ 20 2 2 £ 20 2 5 £ 20 2 5 £ 20
Fig. 15. Performance index versus number of iterations by the SESO procedure.
Fig. 17. Schematic representation of the reinforcement bars and strut-and-tie model using SESO.
Fig. 16. Proposed strut-and-tie model.
constant removed iteration equal 1.75%. Note that the proposed algorithm is sensitive to the variation of these parameters, boundary conditions and the geometry of the element. Fig. 6 shows the computational time spent for a final volume equal to 50% of the initial design, using a PC with a Intel Core 2 Quad 2.83 GHz processor, 4 GB memory, Windows 7. The present formulation, SESO, is compared with a SIMP formulation presented
in [23], using a penalization power equal 3 (p = 3). Good results are obtained by using the SESO formulation even for poor mesh over the entire structure simulated, without the symmetry resource used by the referenced work authors. The computational time spent by the SESO formulation (developed in Fortran90s language) grows with the mesh refining and shows higher efficiency if compared with the Matlab language, known for its low level computational language. 4.2. Corbel jointed with column The design domain and the boundary conditions of the corbel jointed with column are shown in Fig. 7, designed to support a
V.S. Almeida et al. / Engineering Structures 56 (2013) 1665–1675
concentrated load of 500 kN. The column is fixed at both ends. An initial thickness of 300 mm is assumed for corbel. The Young’s modulus is 28,5567 MPa and the Poisson ratio is 0.15. This structure is modeled using refined mesh with 88 216 triangular finite elements and the optimization procedure used parameters RE = 1% and RR% = 1. The optimal topology obtained and the strut-and-tie model are presented by Liang et al. [11], which used a method called PBO – Performance-Based Optimization, with 125-mm square, four-node, plane stress elements. Fig 8a–c shows the history of the optimal topology obtained by Liang et al. [11] and Fig. 8d shows the optimal topology and the proposed strut-and-tie model. Fig 9a–d shows the history of optimal topology obtained by Kwar and Noh [20] using truss elements. Figs. 10a–d and 11a–d show the history of optimal topology for the strut-and-tie model obtained by the present formulation, SESO, using refined meshes 44 108 and 88 216, respectively totaling 5664 and 22,756 triangular finite elements. The optimal topologies (Figs. 10d and 11d) show several similarities with the configurations of the strut-and-tie models presented by both referenced works. Fig. 12 shows the computational time of the corbel jointed with column using refined meshes, respectively equal to 22 54 (1416 elements), 44 108 (5664 elements), 44 216 (11,328 elements) and 88 216 (22,656 elements) and final volume equal to 40% of the initial design. The respective processing times of the final volumes are equal to 240, 916, 3036 and 8879 s.
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respectively indicate the compressed regions, strut, and tensioned regions, tie. Fig. 14b shows the optimal design presented by the author [11] who uses the ESO optimization method to obtain a final volume equal to 33%. Fig. 14c shows the optimal configuration for the strut-and-tie design proposed by Liang et al. [11], which uses the strut-and-tie model. The graph of Fig. 15 shows the monitoring made during the performance of the presented formulation to determine the optimal topology. The growth of the PI (performance index) values is plotted for each iteration path, and the point at which the PI drops sharply indicates that the previous iteration is thus the area of optimal design. A proposed strengthening for this structure, as presented by Liang et al. [11], was obtained by using a strut-and-tie model arising from the combination of the finite element method with a procedure to obtain the flow strength, using the ‘‘load path’’ method. Fig. 19 shows the disposition of the reinforcement proposed by the authors to strengthen the beam cavity. From the optimal topology obtained by the present formulation, a proposal for a strut-and-tie model can be directly made. Note the proposed sketch shown in Fig. 16, where the semi-straights in red (dotted) and blue (continuous) respectively indicate compressed members (C), struts, and tensioned members (T), ties. The efforts at the members where the flow stress stands out can be calculated by multiplying the average stress values of each member and their respective area, given by the product of beam thickness to the width of the average flow region.
4.3. Deep beam with hole This example was reported by Schlaich et al. [10]. It is a simply supported deep beam with a large hole, the geometry and load (P) of which are presented in Fig. 13, and which is used as an extended domain for the optimization process. The Young’s modulus is E = 20,820 MPa, the Poisson’s ratio is m = 0.15 and the thickness is 0.4 m. The design strengths of reinforced concrete are taken with values fyd = 434 MPa and fcd = 25 MPa. This structure has a ‘‘D’’ region due to the geometric discontinuity corresponding to the cavity imposed by the design. In this case, this region should be evaluated using the strut-and-tie model. For modeling with the present formulation, 13,200 triangular elements were used (mesh 150 47). Fig. 14a shows the optimal design obtained by the SESO formulation and compared with the work developed by Liang [15], as depicted in Fig. 14b, and Schlaich et al. [10], Fig. 14c. The optimum topology design shown in Fig. 14a was obtained with a final volume equal to 30.3%; the red and blue regions
Fig. 18. Disposition of the reinforcement proposed for the present model.
Fig. 19. Disposition of the reinforcement proposed, beam with hole [10].
Fig. 20. Computational time of the deep beam with hole and topologies.
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Table 2 Verification of compression stresses acting on the struts (MN, MPa). Ties
Effort
r
0.8 fcd = 20
C3 C5 C6 C7 C8
3.3 1.4 1.7 5.5 2.6
28 20 18.5 18.5 22
Strengthen Ok Ok Ok Strengthen
It is thus possible to calculate the requirement reinforcement areas in the tie region and to evaluate the concrete strength in each strut. Table 1 shows the values of the average efforts obtained at the ties. The T2 and T4 ties are inclined at 15° and 45°, respectively, from the horizontal line. The longitudinal bars which represents tie T2 are calculated from the decomposed horizontal portion of their effort, thus obtaining the required area of reinforcement As2. Tie T4 has its representation in the orthogonal mesh, As3, which covers the stretch along the edge of the cavity and the in-angle encounters struts C3 and C8 on the left end. An additional reinforcement As4, at 45°, is proposed, covering the in-angle encounters struts C2, C5 and C8. Conforming to the calculation procedures to obtain the representative reinforcement of ties (Table 1 and Fig. 17, the details of these reinforcement are shown in Fig. 18, where we can see the proposal extention of reinforcement As3 to cover the struts encounter areas. Fig. 20 shows the computational times and the optimal settings of deep beam with hole using refined meshes with 3690 (mesh 45 47), 6600 (mesh 75 47), 7918 (mesh 90 47) and 13,200 (mesh 150 47) triangular elements and final volume corresponding to 30% of the initial design. The processing times related to final volumes of the topology shown in Fig. 20 are, respectively, equal to 1.001, 4.322, 4.638 and 6.527 s. Table 2 presents the verification of the compression stresses acting at strut members where the non-attendance of the ultimate state of compression can be observed in the concrete at struts C3 and C8. 5. Conclusions Aiming to present a numerical formulation for the design of reinforced concrete structures under the focus of the strut-andtie model, a variant of topology optimization procedure, called Smooth Evolutionary Structural Optimization – SESO, was employed. The evolutionary procedure proposed, which smoothly removes the elements of the discretized body, is coupled to a FEM formulation in plane stress state analysis. A priori, an extended initial domain is defined and, iteratively, the method seeks an optimal topology configuration in which natural members are set indicated by struts and ties. Thus, efforts in the members may be evaluated to enable the design of the reinforcements needed for each section. In contrast to the ESO method [15], this formulation – SESO, in conjunction with a high order triangular finite element, presents robustness and efficiency to obtain optimal configurations. Three numerical examples demonstrated: (a) good accuracy with strutand-tie model configurations and the related effort values reported by other authors. The quantification and the disposition of reinforcements for a classic example described in the international specific literature was also proposed; (b) the computational time obtained by the present formulation is smaller than that presented in the literature, even for finer mesh. It is worth pointing out that, for the beam structure example, the computational time spent by the present formulation grows with the mesh refining and shows higher efficiency if compared with [20], and the other two examples were not compared in time processing, because Ref. [20] does not permit cavities or complex geometry for the initial design; (c)
another major improvement obtained by the SESO formulation is the smoothing in classic TO problems, such as ‘‘checkerboard’’ in the topology configurations of structures presented by the cited authors in this article; (d) due to the intrinsic withdrawal/replace heuristic approach, in which some finite elements return to the structure body (neighbor to restriction value), the SESO procedure shows less sensitivity to the development of fine ramifications which can occur during the definition of the main stress path. A best path to main stress flow is defined without oscillations, once the calculation of the effort at each member of the model can be done as a weighted average of values taken inside a circumference of radius R; (e) finally, we emphasize that SESO is an evolutionary numerical technique which can be used in future researches in TO problems that include more complex phenomena such as the nonlinearities and/or dynamic analysis problems mainly due to the quality achieved, to the optimal settings and also to the low computational cost obtained in the problems presented that require intrinsically high computational cost. Acknowledgements The authors thank the Department of Structural and Foundation Engineering, School of Engineering of the University of São Paulo (EPUSP), the University of Ouro Preto (UFOP) and the São Paulo State University (UNESP) for their financial support to this research. References [1] Rozvany GIN, Bendsøe MP, Kirsch U. Layout optimization of structures. Appl Mech Rev 1995;48:41–119. [2] Bendsøe MP. Optimal shape design as a material distribution problem. Struct Optim 1989;1:193–202. [3] Rozvany GIN, Zhou M, Birker T. Generalized shape optimization without homogenization. Struct Optim 1992;4:250–2. [4] Sokolowski J, Zochowski A. On topological derivative in shape optimization. SIAM J Control Optim 1999;37(4):1251–72. [5] Novotny AA, Feijóo RA, Padra C, Taraco E. Topological sensitivity analysis. Comput Meth Appl Mech Eng 2003;192:803–29. [6] Xie YM, Steven GP. A simple evolutionary procedure for structural optimization. Comput Struct 1993;49(5):885–96. [7] Tanskanen P. The evolutionary structural optimization method: theoretical aspects. Comput Meth Appl Mech Eng 2002;191:5485–98. [8] Querin OM. Evolutionary structural optimization stress based formulation and implementation. PhD thesis. University of Sydney; 1997. [9] Liang QQ, Xie YM, Steven GP. Optimal topology selection of continuum structures with stress and displacement constraints. In: The seventh East AsiaPacific conference on structural engineering and construction, Kochi, Japan; 1999. [10] Schlaich J, Schafer K, Jennewein M. Toward a consistent design of structural concrete. PCI J 1987;32(3):74–150. [11] Liang QQ, Xie YM, Steven GP. Topology optimization of strut-and-tie models in reinforced concrete structures using an evolutionary procedure. ACI Struct J 2000;97(2):322–30. [12] Ali M. Automatic generation of truss model for the optimal design of reinforced concrete structures. Dissertation. New York, United States of America, Cornell University, Ithaca; 1997. [13] Liang QQ, Steven GP. A performance-based optimization method for topology design of continuum structures with mean compliance constraints. Comput Meth Appl Mech Eng 2002;191(13–14):1471–89. [14] Liang QQ, Uy B, Steven GP. Performance-based optimization for strut–tie modeling of structural concrete. J Struct Eng 2002;190:815–23. [15] Liang QQ. Performance-based optimization of structures: theory and applications. London: Spon Press; 2005. [16] Reineck K-H. Shear and concrete tensile strength in the design concept of strut-and-tie models. Ibracon Struct J 2007;3(1):1–18. [17] Bruggi M. Generating strut-and-tie patterns for reinforced concrete structures using topology optimization. Comput Struct 2009;87:1483–95. [18] Bruggi M. On the automatic generation of strut and tie patterns under multiple load cases with application to the aseismic design of concrete structures. Adv Struct Eng 2010;13(6):1167–81. [19] Victoria M, Querin OM, Martí P. Generation of strut-and-tie models by topology optimization using different material properties in tension and compression. Struct Multidiscip Optim 2011;44:247–58. [20] Kwar HG, Noh SH. Determination of strut-and-tie models using evolutionary structural optimization. Eng Struct 2006;28:1440–9.
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