A new evaluation procedure for the strut-and-tie models of the disturbed regions of reinforced concrete structures

Engineering Structures 148 (2017) 660–672

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A new evaluation procedure for the strut-and-tie models of the disturbed regions of reinforced concrete structures J.T. Zhong a,b,⇑, L. Wang a,b, P. Deng a,b, Man Zhou c a

College of Civil Engineering and Architecture, Shandong University of Science and Technology, Qingdao, China Shandong Provincial Key Laboratory of Civil Engineering Disaster Prevention and Mitigation, Qingdao, China c Department of Civil and Earth Resource Engineering, Kyoto University, Kyoto, Japan b

a r t i c l e

i n f o

Article history: Received 3 January 2017 Revised 4 July 2017 Accepted 5 July 2017

Keywords: Reinforced concrete Evaluation system Strut-and-tie Optimization Dapped beam Crack

a b s t r a c t For the same disturbed region (D-region) of concrete structures, different strut-and-tie models (STMs) may be verified by different researchers, and the reinforcement layouts thus vary significantly. To assess conveniently the performance of these D-regions designed using different STMs, an evaluation system is proposed in this paper. The numerical procedure of the evaluation system is developed based on the ANSYS parametric design language (APDL) and the computer-aided strut-and-tie (CAST) design tool. The evaluation process of this system is mainly divided into three parts: preliminary evaluation, further evaluation, and final evaluation. At the beginning, in the preliminary evaluation, proper STMs in which the distribution of ties agrees well with the tensile stress regions are initially selected according to a finite element analysis. Furthermore, a crack propagation simulation is qualitatively executed to further evaluate and determine the locations of the most significant ties, where primary cracks may arise. For a rational STM, there should exist corresponding ties at these crack locations. Finally, by a load carrying capacity simulation, the most efficient design can be quantitatively confirmed. A classical dapped beam with openings is adopted to demonstrate the efficiency and reliability of the evaluation system. The analysis results indicate that the evaluation system exhibits more robust characteristics than traditional experimental methods. Moreover, the use of this system in the design of concrete structures will bring about significant economic benefits (e.g., reductions in time and costs) and produce high-performance structures. Ó 2017 Elsevier Ltd. All rights reserved.

1. Introduction A strut-and-tie model (STM) is a visual representation of load transfer in the disturbed regions (D-regions) of concrete structures. An STM idealizes a complex force flow in the structures as a collection of compression members (struts), tension members (ties), and the intersection of such members (nodes). This idealized truss-like model can thus indicate the force transfer in the actual structures in a discretized way. It is for this reason that the strut-and-tie model method is widely used in the design of concrete structures, especially for D-regions where the Bernoulli hypothesis does not apply [1]. As the saying goes, ‘‘One can’t make bricks without straw.” To use the STM method in the design of D-regions, a reliable strutand-tie model is first needed. Therefore, it becomes apparent that

⇑ Corresponding author at: College of Civil Engineering and Architecture, Shandong University of Science and Technology, Qingdao, China. E-mail address: [email protected] (J.T. Zhong). http://dx.doi.org/10.1016/j.engstruct.2017.07.012 0141-0296/Ó 2017 Elsevier Ltd. All rights reserved.

developing a technique for strut-and-tie modeling of D-regions is of significant importance. In fact, the strut-and-tie modeling techniques have been extensively investigated since a comprehensive work was reported by Schlaich et al. [1]. Subsequently, many different types of techniques and algorithms have been proposed by dozens of researchers. Xie and Steven [2] addressed an evolutionary structural optimization (ESO) method, which represents a pioneering approach for generating strut-and-tie models by topology optimization. Based on the ESO method, Yang et al. [3] developed a bidirectional evolutionary optimization (BESO) method. Afterwards, Liang et al. [4,5] proposed a performance-based optimization (PBO) method for strut-and-tie modeling. In the following decade, the strutand-tie modeling technique experienced rapid developments, and different versions of algorithms were introduced by many researchers [6–14]. In recent six years, the research field on this technique has stepped into a new stage, and many new procedures have been performed for strut-and-tie modelings, such as the full homogenization (FH) optimization method [15], the smooth

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2. STMs of the dapped beam 2.1. Novak model A dapped beam with a rectangular opening was utilized by Novak and Sprenger [42] as a strong example for the application of strut-and-tie modeling of reinforced concrete structures, as shown in Fig. 1. The beam as a whole is considered a D-region due to geometric and force discontinuity. It behaves as a socalled a beam-on-beam, that is, an upper span is supported on a lower span. Based on the expected behavior, Novak and Sprenger proposed the strut-and-tie model of the dapped beam, as shown in Fig. 2. The solid lines represent ties and the dashed lines indicate struts.

2.2. Reineck model The above-described new problem was first proposed for which no experiments had been conducted and no analytical solution was available in textbooks. Therefore, it is not surprising that several engineers may come to various solutions. In other words, different strut-and-tie models may be recommended by different engineers, leading to different reinforcement arrangements and detailing.

4000

8000

2000

3500

2000

2500

2000

F

6000

evolutionary structural optimization (SESO) method [16], the different material properties method [17–20] and the hybrid technique combining different methods [21,22]. In particular, a conceptual optimization procedure was utilized to determine only the distribution of the ties in STMs [23,24]. Contrary to that, Bruggi [25] proposed a numerical method to generate strut-only models in search of optimal and consistent STMs of concrete structures. In addition, several practical computer-aided design tools have been developed to provide a graphical user interface (GUI) environment for the STM-based design process [26–28]. With the exploitation of the above-described various modeling techniques, a problem emerges inevitably that different researchers can, and probably will, propose very different STM patterns for the same design. This variety of models leads to a discussion regarding which model is better. For this problem, Schlaich et al. [1] simply proposed an evaluation criterion that the STM with the least and shortest ties is the best. Nevertheless, in the other literatures, it is experimental verifications that are usually employed to compare between different STMs. Since 2000, a series of experiments have been conducted to solve the problem, taking some classical D-regions as research subjects, such as deep beams [29– 31], deep beams with openings [32,33], dapped beams [34–36], dapped beams with openings [37–39]. In particular, Park et al. [28] addressed an integrated STM design and finite-element analysis validation environment for disturbed regions of concrete structures. Until very recently, the experimental evaluation means are still being utilized by many researchers [40,41]. However, as is known to all, these experiments would demand a great deal of time and money along with well-trained labourers. This situation does not appear to have been improved during the last two decades. Therefore, it is clearly that an efficient and reliable technique is urgently needed for assessing the performance of complex regions designed using different STMs. The evaluation system presented in this study is an attempt to improve this situation. The evaluation process of the system mainly consists of three parts: preliminary evaluation, further evaluation, and final evaluation. A classical dapped beam with openings is used to demonstrate the efficiency and reliability of the evaluation system. As the research object, the available strut-and-tie models of this dapped beam are reviewed and analyzed in the next section.

350

2000

4000

2250

350

3750

12700 Fig. 1. Dapped beam geometry (mm).

F Strut Tie

Fig. 2. Strut-and-tie model built by Novak and Sprenger.

After the Novak model, Reineck [43] proposed and investigated several different strut-and-tie models, as shown in Fig. 3. The model in Fig. 3(a) is a variant of the Novak model. In addition, the models in Figs. (b) and (c) are obtained through a frame analysis, in which the upper beams are symmetrically supported and the lower beams are the same with each other. Contrary to that, the model in Fig. 3(d) exhibits a completely different feature. The left upper part is similar with a corbel, but the right part is a simple beam. Moreover, no transverse ties are provided in the lower beam. 2.3. Ley model Since Reineck’s summarizing investigation and discussion, this classical example of D-region has attracted widespread attention from other researchers. Ley et al. [38] developed some other STMs that are different from above-described ones and conducted a series of experiments to verify the application of strut-and-tie modeling. In the experiments, six specimens (i.e., No. 1, 2, 3, 4, 4i and 5) were designed with different reinforcement layouts determined by different strut-and-tie models, as shown in Fig. 4. In particular, specimen 4i used the strut-and-tie model in Fig. 3(d). Note that these specimens were small-scale models, including 1:10.5 and 1:6 scaled versions. The test results indicated that each reinforcing pattern derived from its strut-and-tie model yielded a safe design. For reinforced specimens, the scaled results were not dependent on the specimen size.

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F

F Strut

Strut

Tie

Tie

(a)

(b)

F

F Strut

Strut

Tie

Tie

(c)

(d)

Fig. 3. Strut-and-tie models built by Reineck: (a) refined model for the model in Fig. 2; (b) model based on frame analysis; (c) model with two corbels in upper beam; (d) load path model.

F

F Strut

Strut

Tie

Tie

(a)

(b)

F

F

(c)

F

Strut

Strut

Strut

Tie

Tie

Tie

(d)

(e)

Fig. 4. Strut-and-tie models built by Ley et al. for: (a) specimen 1; (b) specimen 2; (c) specimen 3; (d) specimen 4i; (e) specimen 5.

2.4. FH model

2.5. Hybrid model

Herranz et al. [15] utilized the full homogenization (FH) optimization method to determine the strut-and-tie model of the dapped beam. Fig. 5(a) shows the optimal topology obtained by the FH optimization method, and its corresponding strut-and-tie model is depicted in Fig. 5(b). In the STM, several supplementary struts (e.g., oc, od, and oe) are added to satisfy the conditions of static equilibrium. This FH model was evaluated and compared with other models, such as the Novak model and the Reineck model, using a nonlinear finite element (NLFE) analysis. The analysis results indicated that the FH model was efficient and could produce good solutions.

Afterwards, Gaynor et al. [44] employed a hybrid topology optimization method to confirm the strut-and-tie model of the dapped beam. In this method, a hybrid truss-continuum ground structure is introduced to solve the problem, in which the truss ground structure represents reinforcing bars and the continuum solid elements represent concrete compression struts. Fig. 6 displays the optimal topology obtained by the hybrid method. 2.6. STM in this study In this study, a ground structure method (GSM) is used to generate the strut-and-tie model of the dapped beam. In this method,

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F

F

Strut Tie

d

o c

e

(a)

(b)

Fig. 5. FH model: (a) optimal topology; (b) strut-and-tie model.

G is the shear modulus; t is the thickness of the micro-truss element. The design domain is divided into 1204 micro-truss elements with dimensions of 20 mm  20 mm, as shown in Fig. 8. A point load of F = 23.6 kN is applied at the top center of the upper beam. Young’s modulus E = 28.0 GPa, Poisson’s ratio k = 0.2 and the thickness of the dapped beam t = 38 mm are assumed. The evolutionary structural optimization (ESO) method [2] is employed to generate the optimal topology of the structure. During the optimization process, an axial stress criterion is used to remove inefficient bar elements in load transfer, which can be denoted as

F Strut Tie

Rdj  rsi;maxc 6 rti;e 6 Rdj  rsi;maxt where

r

s i;maxc

and

r

s i;maxt

ð3Þ

are the maximum compressive and tensile

stresses in the structure at the ith iteration,

is the removal ratio of eleof the eth element at the ith iteration, ments at the jth steady state and can be determined by

Fig. 6. Hybrid model.

Rdj ¼ Rd0 þ ðj  1Þ  Rdi

a new micro-truss element with 8 nodes [45] is proposed to replace the solid element equivalently. The 22-bar micro-truss element is shown in Fig. 7, in which the cross-sectional areas of the bars can be determined by

Ai ¼

K b;i li E

ði ¼ 1; 2; 3 . . . 22Þ

where

ð2Þ

2

1

6

2

7

5

K1

7

3

4

8

3

6

5

4

8

5

is an incre-

Rdi

ð5Þ

6

l1

6

is the initial removal ratio of elements and

where C i0 , Ai0 , li0 , and n are the strain energy, cross-sectional area, length of the ith bar element, and the number of bar elements in the initial structure, respectively, and C ij , Aij , lij , and n0 are the strain energy, cross-sectional area, length of the ith bar element, and the number of bar elements at the jth iteration in the current structure, respectively. In this study, the performance index reaches its peak value (Fig. 9) at iteration 541, which indicates that the optimal topology of the dapped beam is obtained, as shown in Fig. 10. According to the optimal topology, the strut-and-tie model of the dapped beam can be easily constructed, as displayed in Fig. 11

l1 þl2

l2

ð4Þ Rdi

Pn C i0 Ai0 li0 PI ¼ Pi¼1 n0 i¼1 C ij Aij lij

where K 1 , K 2 ; K 3 are the stiffnesses of different bar elements, and l1 , l2 are the dimensions of the micro-truss element, as shown in Fig. 7;

1

ðj ¼ 1; 2; 3; 4 . . .Þ

¼ 0:5% and ¼ 0:1% are specified, and mental removal ratio. the maximum iteration number is set to 600 in this study. Furthermore, to evaluate the efficiency of the resulting topology in each iteration, a performance index (PI) is defined as

ð1Þ

8 ðEl1 Gl2 Þt > > K 1 ¼ 2l1 2 > > > > 2 < ðEl1 Gl 2 Þt ffiffiffiffiffiffiffi ffi K2 ¼ p 2 2 l þl 2 1 2 > >  pffiffiffiffiffiffiffiffi > 3=2 > > El Gl2 4l þ l21 þl22 t G l2 þl2 t > ffi : K 3 ¼ ð 14l 2l Þ  ð 1 2 Þ p1ffiffiffiffiffiffiffi 2 2 1 2

Rd0

Rd0

where li is the length of the ith bar element; E is the elastic modulus; K b;i is the stiffness of the ith bar element and can be derived from

16l1

rti;e is the axial stress

Rdj

K2

1

6

2

5

K3

7

4

8

3

K1/2 7

8 Fig. 7. Micro-truss element.

5

7

8

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381

216

20

178

318

20

178

534

178

F

216

356

203

368

1143 Fig. 8. Dimensions of the numerical model built with micro-truss element (mm).

3. Elastic stress field determination (preliminary evaluation) As described above, numerous strut-and-tie models of the dapped beam have been proposed by different researchers using different approaches. Undoubtedly, each model has its own advantages and disadvantages. Although all of the models may exhibit the full desired capacity if they are designed and detailed properly, it is possible that one of them behaves more reasonably and economically. To preliminarily evaluate these strut-and-tie models, a finiteelement analysis (FEA), which opens the way for powerful analytical studies, may be a good choice. A two-dimensional (2-D) FEA for the dapped beam is performed, and the elastic stress fields in the structure are established. Fig. 12 depicts the principal stress contours and the tensile stresses along the paths. It is seen from Fig. 12(a) that there exist principal tensile stresses in region X. This can be further confirmed by the relative value of tensile stresses on the paths (i.e., paths 2, 5, 6, 7, 8) in Fig. 12(c) and (d). The relative value on path 2 and path 5 are 0.9 and 0.5, respectively, which are slightly smaller than the standard tensile stress (i.e., relative value 1.0 on path 10) along the bottom of the lower beam. On the other hand, the tensile stresses in region X should not be ignored when building strut-and-tie models of the dapped beam. Similarly, in

F

Fig. 10. Optimal topology of the dapped beam based on the micro-truss element.

F f

g

e

Strut

a

d c

Tie Node

b

h i m

j k

l

Fig. 11. Strut-and-tie model in this study.

Fig. 9. Performance index history of the dapped beam.

other tensile stress regions, the corresponding ties (reinforcing bars) should also be assigned. In a word, there should exist corresponding ties at primary tensile stress regions, which is essential and indispensable to a rational strut-and-tie model. From this point of view, strut-andtie models in these figures (i.e., Figs. 4(d), (e) and 11) are more reasonable. Consider, for example, five specimens designed according to different Ley models in Fig. 4. A comparison of the failure load,

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F

F

F

0.8

0.8

F 2.3

0.9 2.3 0.9

3.0

1.0

1.0

3.0 1.0

Fig. 12. Two-dimensional elastic FEA: (a) principal tensile stress contour; (b) principal compressive stress contour; (c) tensile stresses (relative value) on path 1  path 5; (d) tensile stresses (relative value) on path 6  path 10.

cracking load and deflection at failure among these specimens is illustrated in Fig. 13. The figure shows that the failure loads of specimen 4i and 5 are maximum (55.2 kN) and second maximum (42.7 kN), respectively. The cracking loads of both the specimens are also in the top two levels (31.1 kN and 28.5 kN). This further demonstrates that the strut-and-tie models in Fig. 4(d) and (e) are more robust than the other Ley models. 4. Crack propagation simulation (further evaluation) After the above preliminary evaluation, the Ley model in Fig. 4 (d) is found to exhibit comprehensively a better structural behavior. However, to compare this Ley model with the authors’ model

(Fig. 11), a further evaluation must be performed using a 2-D crack propagation simulation procedure. 4.1. Numerical model based on the Weibull distribution In the simulation procedure, the first step is to build a numerical model. A suitable model plays a critical role in the finite element analysis. Concrete can be modeled as a homogeneous material when studying the whole response of a concrete structure. However, it is not sufficient for analyzing the distribution of concrete cracks caused by loading. In fact, concrete is typically a non-homogeneous material; its mechanical properties, such as elastic modulus and strength, ordinarily satisfy some statistical distribution (e.g., Gauss distribution and Weibull distribution). In this study, the Weibull distribution is assumed, which can be stated as [46]

 m1  r m  r m r f ðrÞ ¼ e 0 r0 r0

ð6Þ

where r is the mechanical parameter; r 0 is a parameter relevant to average value EðrÞ; m is the homogeneous degree of mechanical properties among different elements. Further research indicates that there are some relationships between the mechanical properties at two arbitrary points of concrete and the distance of the two points. Generally, the mechanical properties of the two points will have a higher correlation with the decrease of the distance. In this study, a factor n is introduced to describe this relationship. Consider, for example, a twodimensional concrete region. The mechanical properties of one element i in this region can be expressed by

qi ¼ Fig. 13. A comparison among different Ley models (data from Table 2 in the literature [38]).

n X qj gi;j j¼1

ð7Þ

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where qi and qj are the mechanical parameters of element i and j, respectively; gðdij Þ is a function of the normal distribution, which can be defined as d2

i j 1  gðdij Þ ¼ pffiffiffiffiffiffiffi e 2n2 n 2p

ð8Þ

where n is the neighboring correlation coefficient; di j is the distance between element i and j, which can be calculated by

di j ¼

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðyj  yi Þ2 þ ðxj  xi Þ2

ð9Þ

where xi , yi and xj , yj are the coordinates of the centroids of element i and j, respectively. In this study, the dapped beam is modeled with 12182 fournode plane stress elements. A point load P = 23.6 kN is applied at the top center of the upper beam. The average value of Young’s modulus E = 28.0 GPa and Poisson’s ratio k = 0.2 are specified. The homogeneous degree m ¼ 9 in Eq. (6) and the neighboring correlation coefficient n ¼ 1:0 in Eq. (8) are assumed. The finite element model of the dapped beam based on the Weibull distribution is illustrated in Fig. 14. 4.2. Constitutive relations

ð10Þ

~ is the elastic modulus of damaged concrete, which can be where E described as

~ ¼ E0 ð1  DÞ ð0 6 D 6 1Þ E

ð11Þ

where E0 is the initial Young’s modulus; D is a damage variable and can be calculated by

D¼

8 0 > > > > f t þf m > < 1  2E 0e > 1 > > > > : 1  Dk

fm 2E0 e E0

0 < e 6 et

et < e 6 em em < e 6 ef ef < e

the form of stiffness degradation (e.g., Eqs. (11) and (12)) when their resulting stresses reach the initial tensile strength f t as shown in Fig. 15. In this study, the initial tensile strength of concrete f t ¼ 1:6 MPa and residual tensile strength f m ¼ 0:6 MPa are specified referring to the literature [38].

4.3. Numerical results and model evaluation

The second step of the procedure is to input the constitutive relation of concrete. In this study, a modified constitutive relation is introduced based on a three broken line model, as shown in Fig. 15. According to this stress–strain relationship, the damage constitutive equations of concrete can be expressed by

r ¼ E~e

Fig. 15. Constitutive relation of concrete.

ð12Þ

where f t and f m are the initial tensile strength of concrete and residual tensile strength at point M, respectively; Dk is a minimum value (e.g., 1  105). During the finite element analysis, a principal tensile stress criterion is adopted. It is assumed that elements begin to damage in

Fig. 16 depicts the numerical results of crack propagation simulation, including distributions of cracks and directions of crack propagation. It can be seen from the figure that three primary cracks (i.e., Cr1, Cr2, Cr3) are generated under the load. The first crack Cr1 originates from the bottom center of the upper beam, propagating towards the load acting point. Afterwards, cracks Cr2 and Cr3 arise almost at the same time. Crack Cr3 begins from the left lower corner of the rectangular opening to the nearer support. Contrary to Cr1 and Cr3, crack Cr2 starts at the left outer boundary of the dapped beam and develops towards the inner opening. These cracks obtained from the numerical simulation can be well employed to evaluate strut-and-tie models verified by different researchers. The fundamental principle of this evaluation method is based on the fact that the places where cracks appear represent the weak parts of a structure, and should be reinforced adequately in the reinforcement design process. Direct identification of all the weak regions of a structure is quite difficult but important task. Therefore, ideally, a tool for predicting where cracks may occur is available for use. The strut-and-tie model is one such tool. A robust strut-and-tie model should fully indicate the weak regions of structures. In other words, the ties of STMs, which may be regarded as reinforcing bars, should be arranged properly to resist the growth of cracks.

F Cr1

Cr2

Cr3

Fig. 14. Numerical model based on the Weibull distribution.

Direction of crack propagation

Fig. 16. Numerical results of crack propagation.

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667

Fig. 17. Location relationship between cracks and ties in STMs: (a) one of Ley models; (b) the authors’ model.

In the Ley model, as shown in Fig. 17(a), it is obvious that ties 0 0 b c0 and g 0 h are used to resist the growth of crack Cr1 and Cr2, respectively. However, for crack Cr3 which originates from the lower left corner of the opening, there is a lack of relevant ties. In contrast, the authors’ model in Fig. 17(b) agrees well with the distribution of primary cracks. Three key ties bc, gh and ij are generated to resist the growth of cracks Cr1, Cr2, and Cr3, respectively. From this point of view, the authors’ model shows a better robustness. To compare the practicality of the two models, a final evaluation will be performed by a 3-D numerical study on the load carrying capacity of the dapped beam.

F Strut Tie

5. Load carrying capacity simulation (final evaluation) 5.1. Simplified strut-and-tie model Fig. 18. Simplified strut-and-tie model.

P=23.6 T1-3=2.5

3

1

2

5

C3

2. =2

C2

-5 =

22. 5

T4-5=7.4

.3

T3-6=2.9

=4

-4

C 1-4

5

C7

7 T7-11=2.9

C6-10=11.8

5.2

= -10

9=

T7-8=4.4

T11-12=17.8

13

(unit: kN)

.3

C8-9=7.1

8

9

C9

-13 =

2

22.5 C 2-5=

T10-11=7.4 11

10

5-

C5-8=4

17 -6 =

C4 T6-7=8.6

6

C

.0

.1

4

=2. -12

T9 12

18. 2

5.2. Reinforcement design 13

T12-13=15.8

According to this simplified model, a reinforcement design process is able to be conducted by the computer aided strut-and-tie (CAST) design tool [47]. The CAST design tool is developed by Kuchma and Tjhin at the University of Illinois at UrbanaChampaign. It is a graphical design tool that allows users to customize D-regions, draw an internal truss, and select member dimensions and tie reinforcement. The STM design procedure can be stated as follows:

Fig. 19. Member forces of the strut-and-tie model.

P=23.6 kN

1 4 1 2 43 8 (0.389) .0 .7 8 (0 .0/7 3 (0. /38.7 / ) 0 .9 .3 . 90 0 18 43 .90 0) ) (0 1 6 4 (0.529)

10

(0.446)

10-gauge

1 6 7 (0.615) .9 8 / 9.9 892) (0. 1 6 (0.525) 11

1 4 (0.676) 8.7 20.9/1 2) (0.89 2 6 (0.635)

Re (Sr)

25 .2 (0. /23 91 .0 0)

8

12

ws,pro/ws,req (Sr)

Tie:

5

10/1.7 (0.173)

32

25.0/20.3 (0.812)

6

Strut:

.9 7.8/6 1) (0.88

1 4 .5/ (0.451) (0. 29.3 90 3)

3

As indicated by the above evaluation, the authors’ model appears to perform better than the Ley model. However, before a final comparison conclusion is safely drawn, the load carrying capacity test should be conducted for the dapped beams designed according to both STMs. Consider, for example, the authors’ strut-and-tie model shown in Fig. 17(b). This idealized model is generated by a ground structure method without any modification. Although this model fully reflects the load transfer mechanism in the dapped beam, it is difficult to provide a direct guidance for the reinforcement design. Therefore, a simplified model based on the idealized model is developed as shown in Fig. 18.

13.9/12.2 (0.877) 4 1 48) (0.3

9

2 6 (0.563)

34. 6 (0. /31.3 903 )

Fig. 20. Evaluation for the tie reinforcement and the widths of struts.

13

(1) Define the dimensions and boundaries of the dapped beam according to Fig. 8. (2) Plot the internal truss model in accordance with Fig. 18 and then apply the load on the bearing plate whose length and width can be obtained referring to the literature [38]. Next, the truss is analyzed, with the results displayed along the truss members as shown in Fig. 19. (3) Select reinforcing steel to provide the necessary tension tie capacity and then evaluate the dimensions of struts and nodes, as illustrated in Fig. 20. The definitions of the data

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5.3. Load carrying capacity test

Fig. 21. Layout of reinforcing steel bars.

fy

o

y Fig. 22. Stress–strain curve of reinforcing bars.

Fig. 23. Failure mode of the plain concrete specimen.

along members are presented at the right top corner of the figure, in which ws;pro is the provided width of struts, ws;req is the required width of struts, Sr is the ratio of actual stress to allowable stress in the member and Re is the provided reinforcing bars. (4) Distribute the reinforcing steel bars according to Fig. 20. The layout of reinforcing steel bars is depicted in Fig. 21. The primary reinforcement for the dapped beam includes U4 and U6 bars. 10-gauge and 12-gauge wires, as additional steel reinforcement, are mainly used to provide confinement to the compression struts in the strut-and-tie model. Especially, at the load acting point where high stresses are anticipated, 12-gauge wires are provided to prevent a local failure of the concrete. Note that, gauge is the unit of thickness of a steel wire. Based on the Birmingham Wire Gauge (BWG), the diameters of 10-gauge and 12-gauge wires are 3.4 mm and 2.7 mm, respectively.

5.3.1. Constitutive relations of reinforcing bars The above-described reinforcing bars are distributed at the middepth of the cross-section of the dapped beam. After the reinforcement layout is determined, a three-dimensional numerical test for the load carrying capacity can be performed for a final evaluation. Moreover, the 2-D crack propagation simulation procedure in the previous section can be easily extended to a 3-D reinforced concrete structure only by introducing the stress–strain relationship of reinforcing bars. In this study, the reinforcing bars are assumed to be elastic-perfectly plastic material, which shows an elastic behavior until the yield stress is reached. Subsequently, the behavior is perfectly plastic with a constant stress, as shown in Fig. 22. 5.3.2. Validation of the 3-D numerical procedure To validate the above numerical procedure, two cases are performed for comparison with the existing experimental results in the literature [38]. In the first case, a plain concrete specimen is tested to examine its load carrying capacity. This specimen is modeled with 42952 tetrahedral elements. A gradually increased load is applied on the top center of the specimen with a load increment of 2 kN. Young’s modulus E = 28.0 GPa, Poisson’s ratio k = 0.2 are assumed. As stated in the previous section, the initial tensile strength of concrete f t ¼ 1:6 MPa and the residual tensile strength f m ¼ 0:6 MPa are specified. Fig. 23 shows the failure mode of the plain concrete specimen in the first case. Different from the experimental results in the literature, cracks in this case primarily originate from the bottom middle of the upper beam and the left bottom corner of the opening. The failure at different locations reveals the variable nature of the unreinforced concrete. During the failure process of the specimen, the load–deflection curve can be plotted. Fig. 24 presents a comparison of load deflection response between the numerical and experimental results. It can be seen from the figure that the numerical results show an ultimate load capacity of 21.2 kN, which is slightly larger than the experimental results of 18.1 kN. The idealized numerical simulation, which does not consider any defect in the actual structure, may lead to this difference. In the second case, a reinforced concrete specimen, as shown in Fig. 25, is numerically tested to determine its load carrying capacity. The concrete component is modeled with 54691 tetrahedral elements, and the reinforcing bar component is discretized into 4950 line elements. The two components are tied together by generating constraint equations that connect the selected nodes of the reinforcing bars to the selected elements of the concrete. The yield strength of the reinforcing bars f y ¼ 565 MPa is specified in reference to the literature [38]. The other relevant parameters are the same as those in the first case. The failure mode of the reinforced concrete specimen is presented in Fig. 26. It can be seen from the figure that cracks mainly arise at the middle of the lower beam where insufficient steel bars are provided (Fig. 25). Similar to the first case, a primary crack also originates from the bottom center of the upper beam, although the relevant reinforcing bars are distributed. During the simulation process, a load–deflection curve for the reinforced specimen is drawn in Fig. 27. The numerical result of this curve shows a load carrying capacity of 60.5 kN, which is slightly higher than the experimental result of 55.2 kN. From the above two cases, although the numerical results of the load carrying capacity do not completely agree with the experimental results, the agreement is sufficient to justify the 3-D numerical procedure. In other words, this procedure is able to be used to predict numerically the load carrying capacity of actual reinforced concrete structures.

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Fig. 24. Comparison of load–deflection curves of the plain concrete specimen.

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Fig. 27. Comparison of load–deflection curves of the reinforced concrete specimen 4i.

F

Fig. 28. Failure mode of the reinforced specimen designed by the authors. Fig. 25. Reinforcing layout of the specimen 4i in the literature [38]

Fig. 26. Failure mode of the reinforced specimen 4i.

5.3.3. Final comparison Using the above-verified procedure, a load carrying capacity test of the specimen designed by the authors’ model can be numerically conducted. The concrete component is modeled with 60329 tetrahedral elements, and the reinforcing bar component (Fig. 21) is discretized into 5377 line elements. The yield strength of steel bars U4 and U6 are f y1 ¼ 565 MPa and f y2 ¼ 517 MPa, respectively. The other relevant parameters are the same as those in the above cases. Note that the constraint equation method is adopted to deal with the interaction between steel bars and concrete.

Fig. 29. Load-deflection curves of the specimens designed using different STMs.

Fig. 28 displays the failure mode of the reinforced concrete specimen designed according to the authors’ model (Figs. 18–21). The fracture occurs at the bottom center of upper and lower beam. The load–deflection curve of the specimen during the failure process is plotted in Fig. 29. For the purpose of comparison, the curve of the specimen designed according to the Ley model is also illus-

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Table 1 Summary of results for the dapped beam designed using different STMs. Model

Failure load (kN)

D at failure (mm)

Requirement of steel (kg)

Efficiency (kN/kg)

Ley model Authors’ model

60.5 69.7

25.1 30.6

0.95 1.03

64 68

Fig. 30. Flow chart of the evaluation process.

trated in the figure. It can be seen obviously that the specimen reinforced referring to the authors’ model exhibits a larger load carrying capacity than that of the Ley model. A detailed summary of results for the specimen is tabulated as shown in Table 1. The table further presents that the authors’ model carries a failure load of 69.7 kN, which is 9.7 kN larger than that of the Ley model. In addition, the deflections of the proposed model and the Ley model are 30.6 mm and 25.1 mm, respectively. Although the authors’ model requires a few more steel bars (1.03 kg) than the Ley model (0.95 kg), the former shows comprehensively a higher efficiency. Therefore, a final conclusion can be safely made that the authors’ model is demonstrated to display a more robust behavior compared to other models according to the results of the evaluation system. 5.4. Frame of the evaluation system

(2) Evaluate preliminarily these strut-and-tie models by executing a two-dimensional finite element analysis and then remove the unreasonable models according to the principal stress contours and tensile stresses on paths. (3) Evaluate further the remaining STMs after the above step by performing a two-dimensional crack propagation simulation and then unselect those STMs whose location of ties does not agree well with the distribution of cracks. (4) Design the D-region referring to the remaining STMs by using the CAST design tool. (5) Evaluate finally the remaining STMs by conducting numerically a three-dimensional load carrying capacity test on the specimen, which is reinforced by different layouts of steel bars. (6) Choose the most robust strut-and-tie model and draw a conclusion. The entire process is illustrated in Fig. 30.

The evaluation process of the system is summarized as follows:

6. Conclusions

(1) Collect a variety of strut-and-tie models built by different researchers for one disturbed region and then perform an initial evaluation of the characteristics of these STMs.

This paper presented an evaluation system for judging varied strut-and-tie models that were verified by different researchers for D-regions of reinforced concrete structures. The evaluation pro-

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cess of this system involved three steps, i.e., preliminary evaluation, further evaluation, and final evaluation, after the STMs for comparison were prepared. The evaluation process was performed predominantly based on two analytical tools, the ANSYS software and the CAST (computer-aided strut-and-tie) design tool. During the process, the ANSYS software is mainly used to execute numerical simulations, such as crack propagation simulation and load carrying capacity test, while the CAST design tool is usually employed to determine the layout of reinforcing steel bars. Based on the work presented in this study, the following conclusions are drawn: (1) The evaluation system is a reliable and efficient tool that can be utilized to evaluate different strut-and-tie models of the same design domain of D-regions, especially for those concrete structures where no existing validation experiments are available. (2) The evaluation system can not only identify the best among many STMs but also examine an individual strut-and-tie model and predict its load carrying capacity numerically instead of experimentally. (3) Furthermore, it can be seen from the evaluation process that another function of the system is to simplify rationally the idealized models generated by some techniques. (4) Consider, for example, the dapped beam in this study. The evaluation system indicates that the authors’ model shows more robust characteristics in comparison with other STMs. In other words, this system is able to indirectly demonstrate the advantage and efficiency of a model generating method. (5) The evaluation system overcomes the limitations of experimental validation method in evaluating the strut-and-tie models of reinforced concrete structures. The use of this system in the design of concrete structures will result in significant reductions in time and costs and produce highperformance structures. (6) Although the evaluation system seems to be too theoretical and complicated, it is, in fact, a very convenient tool after a parametric program is compiled successfully. Parameterization means that the users only need to input relevant parameters; then, the system starts running. With the help of this system, professional engineers can design a complex concrete structure efficiently and reliably, and an effect of ‘‘half work with double results” can be expected.

Acknowledgements This study is supported by China Postdoctoral Science Fund (No. 2016M602167) and the Qingdao Postdoctoral Sustentation Fund of PR China (No. 2015201). The financial supports are gratefully acknowledged. We would like to extend our heartfelt gratitude to the anonymous reviewers for their helpful suggestions on the quality improvement of our present paper. References [1] Schlaich J, Schäfer K, Jennewein M. Toward a consistent design of structural concrete. PCI J 1987;32(3):74–150. [2] Xie YM, Steven GP. A simple evolutionary procedure for structural optimization. Comput Struct 1993;49(5):885–96. [3] Yang XY, Xie YM, Steven GP, et al. Bidirectional evolutionary method for stiffness optimization. AIAA J 1999;37(11):1483–8. [4] 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):322–32. [5] Liang QQ, Xie YM, Steven GP. Generating optimal strut-and-tie models in prestressed concrete beams by performance-based optimization. ACI Struct J 2001;98(2):226–32.

671

[6] Ali MA, White RN. Formulation of optimal strut-and-tie models in design of reinforced concrete structures. Spec Publ 2000;193:979–98. [7] Buhl T, Pedersen CBW, Sigmund O. Stiffness design of geometrically nonlinear structures using topology optimization. Struct Multidiscipl Optimiz 2000;19 (2):93–104. [8] Biondini F, Bontempi F, Malerba PG. Modelli Strut-and-Tie emergenti dall’ottimizzazione topologica evolutiva di strutture composite. In: Proc national workshop Tecniche di progettazione strut and tie di elementi in cemento armato; 2002. [9] Cai CS. Three-dimensional strut-and-tie analysis for footing rehabilitation. Practice Periodical Struct Des Constr 2002;7(1):14–25. [10] Leu LJ, Huang CW, Chen CS, et al. Strut-and-tie design methodology for threedimensional reinforced concrete structures. J struct Eng 2006;132(6):929–38. [11] Nagarajan P, Pillai TMM. Development of strut and tie models for simply supported deep beams using topology optimization. Sonklanakarin J Sci Technol 2008;30(5):641. [12] Bruggi M. Generating strut-and-tie patterns for reinforced concrete structures using topology optimization. Comput Struct 2009;87(23):1483–95. [13] Guest JK. Imposing maximum length scale in topology optimization. Struct Multidiscip Optimiz 2009;37(5):463–73. [14] Guest JK, Moen CD. Reinforced concrete design with topology optimization. In: Proceedings of the 19th analysis & computation specialty conference. Reston: American Society of Civil Engineers; 2010. p. 445–54. [15] Herranz JP, Santa María H, Gutiérrez S, et al. Optimal strut-and-tie models using full homogenization optimization method. ACI Struct J 2012;109 (5):605–14. [16] Almeida VS, Simonetti HL, Neto LO. Comparative analysis of strut-and-tie models using smooth evolutionary structural optimization. Eng Struct 2013;56:1665–75. [17] Liu S, Qiao H. Topology optimization of continuum structures with different tensile and compressive properties in bridge layout design. Struct Multidiscip Optimiz 2011;43(3):369–80. [18] Victoria M, Querin OM, Martí P. Generation of strut-and-tie models by topology design using different material properties in tension and compression. Struct Multidiscip Optimiz 2011;44(2):247–58. [19] Luo Y, Kang Z. Layout design of reinforced concrete structures using twomaterial topology optimization with Drucker-Prager yield constraints. Struct Multidiscip Optimiz 2013;47(1):95–110. [20] Kim BH, Chae HS, Yun YM. Refined 3-dimensional strut-tie models for analysis and design of reinforced concrete pile caps. J Korean Soc Civ Eng 2013;33 (1):115–30. [21] Palmisano F, Elia A. Shape optimization of strut-and-tie models in masonry buildings subjected to landslide-induced settlements. Eng Struct 2015;84:223–32. [22] Palmisano F, Alicino G, Vitone A. Nonlinear analysis of RC discontinuity regions by using the bi-directional evolutionary structural optimization method. In: Proc. of the OPT-I, an international conference on engineering and applied sciences optimization; 2014. p. 749–58. [23] Bogomolny M, Amir O. Conceptual design of reinforced concrete structures using topology optimization with elastoplastic material modeling. Int J Numer Meth Eng 2012;90(13):1578–97. [24] Amir O, Sigmund O. Reinforcement layout design for concrete structures based on continuum damage and truss topology optimization. Struct Multidiscip Optimiz 2013;47(2):157–74. [25] Bruggi M. A numerical method to generate optimal load paths in plain and reinforced concrete structures. Comput Struct 2016;170:26–36. [26] Tjhin TN, Kuchma DA. Computer-based tools for design by strut-and-tie method: advances and challenges. ACI Struct J 2002;99(5):586–94. [27] Yun YM, Kim BH. Two-dimensional grid strut-tie model approach for structural concrete. J Struct Eng 2008;134(7):1199–214. [28] Park JW, Yindeesuk S, Tjhin T, et al. Automated finite-element-based validation of structures designed by the strut-and-tie method. J Struct Eng 2010;136 (2):203–10. [29] Foster SJ, Malik AR. Evaluation of efficiency factor models used in strut-and-tie modeling of nonflexural members. J Struct Eng 2002;128(5):569–77. [30] Brown MD. Minimum transverse reinforcement for bottle-shaped struts. ACI Struct J 2006;103(6):813–21. [31] Tuchscherer RG, Birrcher DB, Williams CS, et al. Evaluation of existing strutand-tie methods and recommended improvements. ACI Struct J 2014;111 (6):1451–60. [32] Maxwell BS, Breen JE. Experimental evaluation of strut-and-tie model applied to deep beam with opening. ACI Struct J 2000;97(1):142–8. [33] Garber DB, Gallardo JM, Huaco GD, et al. Experimental evaluation of strut-andtie model of indeterminate deep beam. ACI Struct J 2014;111(4):873–80. [34] Nagrodzka-Godycka K, Piotrkowski P. Experimental study of dapped-end beams subjected to inclined load. ACI Struct J 2012;109(1):11–20. [35] Nagy-György T, Sas G, Da˘escu AC, et al. Experimental and numerical assessment of the effectiveness of FRP-based strengthening configurations for dapped-end RC beams. Eng Struct 2012;44:291–303. [36] Jatziri Y. Moreno-Martínez, Meli R. Experimental study on the structural behavior of concrete dapped-end beams. Eng Struct 2014;75:152–63. [37] Chen BS, Hagenberger MJ, Breen JE. Evaluation of strut-and-tie modeling applied to dapped beam with opening. ACI Struct J 2002;99(4):445–50. [38] Ley MT, Riding KA, Widianto, et al. Experimental verification of strut-and-tie model design method. ACI Struct J 2007;104(6):749–55.

672

J.T. Zhong et al. / Engineering Structures 148 (2017) 660–672

[39] Kuchma D, Yindeesuk S, Nagle T, et al. Experimental validation of strut-and-tie method for complex region. ACI Struct J 2008;105(5):578–89. [40] Oviedo R, Gutiérrez Sergio, Santa María Hernán. Experimental evaluation of optimized strut-and-tie models for a dapped beam. Struct Concr 2016;17 (3):469–80. [41] Shuraim AB, El-Sayed AK. Experimental verification of strut and tie model for HSC deep beams without shear reinforcement. Eng Struct 2016;117:71–85. [42] Novak LC, Sprenger H. Example 4: deep beam with opening. in: examples for the design of structural concrete with strut-and-tie models. In: ACI Fall Convention; 2002. [43] Reineck KH. Modeling structural concrete with strut-and-tie modelssummarizing discussion of the examples as per Appendix A of ACI 318-02. Spec Publ 2002;208:225–42.

[44] Gaynor AT, Guest JK, Moen CD. Reinforced concrete force visualization and design using bilinear truss-continuum topology optimization. J Struct Eng 2013;139(4):607–18. [45] Zhong J, Wang L, Li Y, et al. A practical approach for generating the strut-andtie models of anchorage zones. J Bridge Eng 2016. http://dx.doi.org/10.1061/ (ASCE)BE.1943-5592.0001013. [46] Mier JGMV, Vliet MRAV, Tai KW. Fracture mechanisms in particle composites: statistical aspects in lattice type analysis. Mech Mater 2002;34(11):705–24. [47] Kuchma DA, Tjhin TN. CAST (Computer Aided Strut-and-Tie) design tool. In: Structures congress; 2001. p. 1–7.