Crack Propagation Modeling of Strengthening of RC Deep Beams With CFRP Plates

Fracture analysis of reinforced concrete deep beam strengthened with carbon fiber-reinforced polymer (CFRP) plates was carried out. The present research aims to find out whether the crack propagation in a strengthened deep beam follows linear elastic fracture mechanics (LEFM) theory or nonlinear fracture mechanics theory. To do so, a new energy release rate based on nonlinear fracture mechanics theory was formulated on the finite element method and the discrete cohesive zone model (DCZM) was developed in deep beams. To validate and compare with numerical models, three deep beams with rectangular cross-sections were tested. The code results based on nonlinear fracture mechanics models were compared with experimental results and ABAQUS results carried out based on LEFM. The predicted values of initial stiffness, yielding point and failure load, energy absorption, and compressive strain in the concrete obtained by the proposed model were very close to the experimental results. However, the ABAQUS software results have greater differences with the experimental results. For example, the predicted failure load for the shear-strengthened deep beam using the proposed model has only 6.3% differences compared to the experimental result. However, the predicted failure load using ABAQUS software based on LEFM has greater differences (25.1%) compared to the experimental result.


Introduction
Reinforced concrete deep beams play an essential role in bridges, buildings, offshore structures, foundations, and military structures [1,2,3]. Pile caps in Foundations, coupling beams in buildings, transfer beams, load-bearing walls, and bunker walls behave as deep beams [4]. In the deep beam, the span to effective depth ratio is less than or equals to 2.0 [5] and the transverse plane sections before bending do not remain plane after bending [6].
In general, cracks in a concrete structure, such as deep beams, start in the tension zone due to increasing stresses or the presence of initial cracks [7]. Therefore, these cracks must be studied properly. Two approaches are existing to study crack propagation in concrete structures [8]. They can be divided into two general categories such as Linear Elastic Fracture Mechanics (LEFM) and nonlinear fracture mechanics theories. LEFM theory, for the first time, was applied to analyze fracture mechanics on ships used in World War II [9]. In the LEFM theory, to calculate the real stress, a factor is multiplied on the stress near ahead of crack which the researchers called it stress intensity factor (SIF). If the principal stress in crack-tip reaches the fracture toughness, the crack propagates. The SIF changes with the size of the crack, load, geometry of the structure, and the material properties. One of the most characteristics of LEFM theory is stress singularity at the crack tip. Kaplan's [10]research showed that LEFM theory is not acceptable for normal-sized concrete structures and can be used to investigate the propagation of cracks in mass structures such as concrete dams. Later, researchers found a zone in front of cracks in a normal-sized concrete structure and called it the fracture process zone (FPZ). The behavior of this zone is non-linear and a lot of energy is accumulated in this area that if it exceeds a certain level (fracture energy) the behavior will cause softening [11] [12,13,14]. Modeling of this area is important for concrete structural members, such as beams, joints, and deep beams, and it has been studied in recent years. The FPZ is not used in the LEFM theory [15,16]. One of the most successful and accurate methods in modeling this area is the discrete cohesive zone model (DCZM) that was developed in the present study [17,18,19].
On the other hand, by using different kinds of strengthening, the crack propagation of concrete changes. In recent years, Carbon Fiber Reinforced Polymer (CFRP) plates have been applied in concrete structures. Advantages of this type of retrofitting are low weight, corrosion resistant, easy installation, and high tensile strength [20, 21,22]. Nowadays, strengthening using CFRP is of interest from an economic point of view [23]. The use of CFRP composites is now identified as a successful, suitable, and efficient technique to strengthen structures [24].
Because CFRP plates affect the initiation and propagation of cracks in concrete members, including deep beams, it is necessary to model, test, and study these members. Due to shear crack, the strengthening of deep beams in the shear region is significant. shear cracks propagate more in deep beams compare with flexural cracks.
Although, it was expected in advance that flexural strengthening of a deep beam in the flexural-zone (soffit strengthen of the deep beam) could not enhance their load-displacement behavior. To verify this issue, however, such strengthening was performed in this study. Indeed, capacity of deep beams is rather affected by shear-strengthening (side face of the deep beam) than flexural-strengthening.
There are many experimental works on deep beams and CFRP shear-strengthened deep beams [25,26]. However, there is limited knowledge about crack propagation in a concrete deep beam strengthened with CFRP plates in flexural and shear regions can be found in the literature. It is vital to carry out experimental tests to better understand the crack propagation pattern of concrete deep beams strengthened with CFRP plates.
The present research aims to find out whether the crack propagation in a strengthened deep beam follows LEFM theory or nonlinear fracture mechanics theory. Therefore, a new energy release rate based on nonlinear fracture mechanics theory was formulated using finite element method. In this study, a numerical model was proposed to simulate the FPZ. To validate and compare with the numerical models, three deep beam specimens with rectangular cross-sections were tested. One of the deep beams was considered as a control specimen. Another deep beam was strengthened in flexural with CFRP plats at the bottom and the last one was strengthened in shear with CFRP plates in both sides of the shear span of the beam. The three beams were loaded with two-point monotonic loads until failure. The results of the FEAPpv® program code based on the nonlinear fracture mechanics were compared with experimental results with the results of the FEAPpv® program code based on the nonlinear fracture mechanics model and ABAQUS software based on LEFM..

Materials and Methods
2.1 Numerical model 2.1.1. Proposed numerical model based on nonlinear fracture mechanic The Finite Element Analysis Programs personal version (FEAPpv) are processor evaluation methods considered for utilizing in an instructional package to show the presence of the platform of elements and simulation. FEAPpv has only command language and designed for research by Taylor [27]. In this program, the solution algorithm is written by the operator. Therefore, each operator can describe a solution plan that meets specific needs. The system includes enough commands that can be applied for use in mechanics, structural, heat transfer, fluid, and other areas by differential equations. Several numerical models have been accomplished by FEAPpv [28,29,30,31,32,33].
To estimate shear crack propagation for concrete deep beam, a new formulation of energy release rate based on the finite element method is introduced. To compute the strain energy release, the virtual crack closure technique (VCCT), which is the most popular and powerful tool on DCZM, is used [17]. A small concrete deep beam element in the shear span is shown in Figure 1. A truss element as the interface is set between interfacial node pairs, nodes 1 and 2. Initially, nodes '1' and '2' have the same coordinates. The stiffness of the interface element that selected as truss element, in the elastic zone (no crack propagates) based on VCCT is given by [18]: where , , , ∆ and ℎ are Young's modulus of concrete, shear modulus, the width of the deep beam, mesh size, and height of the deep beam, respectively.
, and are the stiffness of the interface element in x and y directions, respectively. The cracking stress proposed by Thomasa et al. [34] is adopted to model the initial shear crack as: where σ start and f c ' are cracking stress and concrete compressive strength, MPa, respectively. If the principal stress in Node '1' or '2' was equal to Eq. (3), the stiffness's in Eqs.
(1) and (2) were zero and cracks were created. The normal stress of crack propagation by Walraven [35] is adopted to find strain energy release rates in the deep beam.
where σ N , ρ, K, and C are normal stress, longitudinal reinforcement ratio, size factor, and a coefficient (about 0.12), respectively. The K is given by: where d is the effective depth in mm. It was assumed that cross section of the truss element is equal to A=B×∆. The strain energy release rates of Mode I, G I is expressed as follows [36]: where F N , x 1 and x 2 are nodal force, displacement of node '1'and '2' in x direction, respectively. The shear stress of crack in the deep bem is adopted by zhang et al. [37] as: where A and B are the shear cohesive and frictional that are: Therefore, the strain energy release rates for shear, G II , is based on work by Xie et al [18] as follows: On the other hand, the G F is the critical fracture energy of concrete deep beam which is equal to [38]: where f t is the tensile strength of concrete. Therefore, when G I + G II < G F , the stiffnesses are equal to Eqs. (1) and (2). When G I + G II > G F , the crack is propagated, the interface element is removed and Eqs. (1) and (2) becomes zero. In addition, to find failure stress, ultimate shear stress by Thomas et al. [39] is used.
Furthermore, a numerical method is developed to model crack propagation based on the mentioned new strain energy release rates in the reinforced concrete deep beam which is flexural-strengthened with CFRP [40,41]. Also, in this study, a computer code was developed specifically to model shear-strengthening of the reinforced concrete deep beam by modifying the crack propagation criteria [42].

Numerical model based on LEFM
To compare with the experimental results and FEAPpv based on nonlinear fracture mechanics, the ABAQUS software is used to model crack propagation by conventional cohesive elements. ABAQUS software can only model the crack propagation based on linear fracture mechanics and it cannot capable of simulating nonlinear fracture mechanics. Concrete parameters used for deep beam based on LEFM in ABAQUS are given in Table 1. Experimental tests were conducted under static monotonic load to validate the simulation results. Results from both models were compared with the experimental results of three deep beams. Figure 2 illustrates the deep beam dimensions which are 500 mm in depth, 150 mm in width, and 900 mm in length. The geometry, supports, and details of the deep beams are also shown in Figure 2. The deep beams were reinforced longitudinally with two 16 mm diameter steel rebars at bottom and two 12 mm diameter steel rebars at top with 30 mm clear concrete cover. The deep beams were not reinforced transversally in order to guarantee the shear failures. Only two stirrups were used out of the shear zone to hold the longitudinal reinforcement in place as shown in Figure 2. The yield strength of the bars was 400 MPa based on the manufacturer report. Average concrete compressive strength was 28 MPa (at 28 days) based on testing three cylinders with diameter of 150 mm and height of 300 mm. Table 2 shows mix design for the concrete used to cast the deep beams. The mechanical properties of the CFRP plates were found according to ASTM D7565 [43]. The properties of the CFRP plates and steel reinforcement are given in Table 3. B-0, B-1 and B-2 are used to designate the control deep beam, the flexural strengthened and the shear strengthened deep beams, respectively. All the deep beams have the same steel reinforcement as shown in Figure 2.   Table 3. Properties of the CFRP plates and steel reinforcement.

Specimens
Three plywood formwork (Figure 3(a)) are built to cast the deep beams. Figure 3(b) shows the reinforcement cage layout inside the formwork. After casting (Figure 4), the beams were tested after 28 days in the. The deep beams were simply supported and

Instrumentation
A hydraulic jack with 800 kN capacity was used to the deep beam. The mid-span displacements ( Figure 5) were recorded by Linear Variable Displacement Transformers (LVDTs). Two strain gauges were attached directly under the CFRP plates and concrete below the load to observe strain. The supports were placed at the bottom of the concrete beams. The applied loads were measured by a load cell. All the measurements were automatically recorded using a data logger.  Before strengthening the specimens with CFRP plates, the concrete surface was cleaned to remove any surface grease. The surfaces of the specimens were further cleaned by grinding and brushing. An epoxy adhesive was uniformly applied in a thin layer on the bonding surfaces as shown in Figure 6. The CFRP was placed over it and pressed firmly by plastic rollers. External strengthening with single-layer CFRP plates has a thickness of 0.8 mm and an elastic modulus of 250 GPa. The ultimate tensile stress of the CFRP plates was 210 GMPa.  Figure 7 illustrates the test setup details for the control beam (B-0), the beam with flexure strengthened by CFRP (B-1), and the beam with shear strengthened by CFRP plates (B-2). A strain gauge was mounted on the surface of concrete to record strain in the concrete. A four-point bending test setup was used to test the beams. One LVDT was set to monitor the displacements under deep beam. A load cell was utilized to measure the applied load. The test was conducted under a gradually increasing monotonic load (at a loading rate of 0.4 mm/min) until beam failure was reached.

Results
This section presents the validation and comparison of crack propagation obtained by the proposed model based on the nonlinear fracture mechanics theory, the experimental results, and ABAQUS software results based on LEFM.

Crack propagation in the control beam (B-0)
The control beam was studed first to compare the proposed model with experimental results. A comparison of the proposed model based on nonlinear fracture mechanics theory with the experimental results is shown in Figure 8. A good agreement between the experimental results and proposed model results based on nonlinear fracture mechanics theory was found out for the load-deflection curves in the mid-span, as shown in Figure  8. The load-deflection curves showed nonlinear behavior. The initial stiffness obtained by the nonlinear fracture mechanics model coincides with the stiffness of the beam at the elastic zone, as predicted by the experimental results. The initial stiffness obtained by the linear elastic fracture mechanics model by ABAQUS was over-estimated and had significantly higher stiffness in the mid-span.   Figure 9 shows the crack paths of the control beam, which were modeled using the nonlinear fracture mechanics model. Only half of the beam was modeled by considering symmetry. Figure 10 illustrates the crack path through the experimental test. This crack pattern may be compared with the predicted result shown in Figure 9. The nonlinear fracture mechanics model predicted three cracks within the shear span compared with the three cracks observed in the experimental test. Shear span is the distance between a support and the nearest load point. The nonlinear fracture mechanics model observed one big and one small flexural crack within the flexural zone whereasonly one crack was observed in the experimental result. Flexural span is the distance between the point loads. The agreement between the crack paths obtained in the nonlinear fracture mechanics model and the experimental test is sufficient to justify the validity of the nonlinear fracture mechanics model. The crack patterns obtained by the nonlinear fracture mechanics model and experimental results, were propagated to the last quarter of the section height. Figure 11 shows the crack paths of the control beam simulated using the ABAQUS software based on LEFM. The shear crack near the support cannot be modeled by conventional ABAQUS software. Therefore, the nonlinear fracture mechanics model more objectively predicts crack propagation compared with the ABAQUS software. As shown in figures 9, 10, and 11 the model of control deep beam works better with FEAPpv based on nonlinear fracture mechanics than ABAQUS based on LEFM.   Figure 12 compares the load versus the mid-span deflection of the deep beam obtained by the proposed model based on nonlinear fracture mechanics theory, the results of the experimental test, and the ABAQUS software. The results of the proposed model are consistent with the results of the nonlinear fracture michanics. Stiffness of the deep beam predicted based on the LEFM had smaller differences compared with the experimental results. However, the stiffness of the deep beam predicted based on ABAQUS is more than the experimental results stiffness. The steel yield load predicted by LEFM has about 4% difference compared to the experimental result. However, the steel yield load by ABAQUS software has greater error (31%) compared to the experimental result, which indicates that it is less capable of estimating the steel yield load. Compared with that in the experimental results, load failure in the poroposed model was predicted within a difference range of 5.6% to 8%. The load-deflection curve of the nonlinear fracture mechanics model was similar to the experimental results. However, the load-deflection curve by ABAQUS software was higher than the experimental curve. The ABAQUS software results have a greater amount of difference (29.9-39.4%) compared with the experimental results. Remarkably, the proposed model based on the nonlinear fracture mechanics model could be used to perform analysis of reinforced deep beam flexural-strengthened by CFRP plates.  Figure 13 shows the comparison of crack patterns between the proposed model, the experiment, and the ABAQUS software in the deep beam with flexure strengthened by CFRP plates. The predicted cracking pattern of the proposed model is generally consistent with the experimental observations. In both cases, the flexural cracks were closely spaced near the beam mid-span, as shown in Figures 13(a) and 13(b). Within the shear span, the model predicted two shear-flexural cracks, whereas only one crack was observed in the experiment (left shear span) in half of the beam. The model predicted one shear crack near the support, whereas no crack was observed in the test. The shear cracks may be too small to be noticed in the experimental test. As shown in Figure 13(c), the ABAQUS software cannot simulate this crack pattern because neither the accurate stiffness of the FPZ nor the crack propagation criterion is considered in the software. Flexural crack was observed by the nonlinear fracture mechanics model in the control beam near the mid-span at a load level of 280 kN. Whereas, crack of the beam with strengthened in flexure occurred at a relatively higher load level (300 kN) than that for the control beam. Observed flexural crack load for the flexural strengthened deep beam was about 300 kN based on the nonlinear fracture mechanics and the experimental results. The deep beam that strengthened in flexure showed approximately 2% shorter flexural crack length compared to the control beam. As expected, the behavior of the flexural strengthened deep beam with CFRP plates was slightly changed compared to the control deep beam. Finally, the beam failed because of shear failure with a large shear crack.

Crack propagation in the shear-strengthened deep beam (B-2)
The load-deflection curves obtained from the experimental test, the nonlinear fracture mechanics model, and the ABAQUS software for the deep beam with shear strengthened by CFRP plates are shown in Figure 14. The results of the proposed model are close to that of the experimental one. This finding indicates that the nonlinear fracture mechanics model is validated by the test results. The yield point of the load-deflection curve in the nonlinear fracture mechanics model is similar to that in the experimental test result (about 3% difference). However, this point obtained in the LEFM simulations by ABAQUS was higher than that in the experimental test result (about 21% difference). The accuracy of the proposed model is also confirmed by the close value of the failure load obtained from the proposed model and the test (about 2% difference at 4 mm deflection). Also, the stiffness of the beam with shear strengthened by CFRP as analyzed by the ABAQUS software is over-estinated.    Figure 16 shows the effect of shear strengthened by CFRP on the crack propagation. Only one small crack formed under the CFRP plate, as shown in Figure 16. CFRP plates delay and control cracking in the beam and confine the crack propagation of concrete. Shear strengthened by CFRP plate reduces the width of cracks in concrete. Figure 16 shows that crack was not observed in the shear span because of the presence of CFRP plates. The crack length was restrained as well. In Table 4, some factors such as accuracy of predicted initial stiffness, yielding point (based on the same load), plastic zone, and failure point for each model were compared with the experimental results. The predicted values of initial stiffness, yielding point, and failure point obtained by non-linear fracture mechanics were very close to the experimental results. However, the ABAQUS software results based on LEFM have a greater discrepancy. For example, the steel yielding point in the control deep beam was predicted by non-linear fracture mechanics within a difference of 1.2%. However, the steel yield load by ABAQUS software based on LEFM has a higher difference (13%) compared to the control beam, which indicates that it is less capable of estimating the steel yield load. The failure loads based on the non-linear fracture mechanics and the experimental results for the shear strengthened deep beam, were about 485 and 498 kN corresponding to mid-span displacements of 4.1 and 4.5 mm, respectively. However, these values for the model based on LEFM were equal to 510 kN with mid-span deflections of 3.7 mm. For the shear strengthened deep beam (B-2), delamination of the CFRP plates was evident based on the non-linear fracture mechanics model and the experimental results. However, the model based on LEFM showed that failure occurs because of the rupture of the CFRP plates on the shear span.

Discussion
In this study, a numerical model was developed to predict crack propagation that occurs in RC deep beams strengthened by CFRP under static monotonic load. The results were compared to experiment results, and it was shown that the present numerical results were close to the experimental test results. The following topics are suggested for future work: a) In the present study a nonlinear material interface element was assumed based on small displacement. It would be interesting to reformulate the based proposed model on large deformation.
b) It is recommended to study creep action on deep beam strengthened by FRP and to consider the effect of environmental condition. c) To study the fracture of beams in shear by internally reinforced with FRP bars.

d)
Similar experimental studies using high performance concrete and light-weight concrete can be considered for future study.

Conclusions
In this study, two models for reinforced concrete deep beam strengthened with CFRP plates were proposed to predict the fracture behavior of a deep beam. One of the models is based on nonlinear fracture mechanics by FEAPpv. Also, a new energy release rate based on nonlinear fracture mechanics theory was formulated. The DCZM was developed in deep beams and the crack propagation criteria were proposed. Another model is based on linear elastic fracture mechanics by ABAQUS software. Experimental testing on reinforced concrete deep beams strengthened with CFRP plates was carried out to compare with the models.
The proposed model can reasonably predict the experimental results in terms of stiffness, yielding load, plastic zone, and failure load. While, the ABAQUS results in the mentioned parts were far from the experimental results. For example, the failure load of the shear-strengthened deep beam was predicted by proposed model with 6.3% difference compared to the experimental result. However, the failure load predicted by ABAQUS software based on LEFM has a greater difference (25.1%) with the experimental result. The proposed model captured the crack patterns of the strengthened reinforced concrete deep beam with CFRP plates quite satisfactorily compared to the experimental observations. For the shear strengthened deep beams, the energy absorption obtained by LEFM was much greater than those of the experimental results and the proposed model. Furthermore, the shear strengthened deep beam simulated by the proposed model showed an increase in the compressive strain of 1.2% compared to the experimental results, while this increase was 10.4% for the deep beam modeled by LEFM. Therefore, the nonlinear fracture mechanics model is a reasonable choice for simulating the fracture mechanics of the reinforced concrete deep beam strengthened with CFRP plates. Funding: The authors wish to gratefully acknowledge the Alexander von Humboldt Foundation for the funding support of this study under project number AvH0001AB32.