The Prediction of Cracking Load and Peak Load of UHPC-NC Composite Structure

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Introduction
Ultra-high-performance concrete (UHPC) is a new construction material developed by Richard and Cheyrezy in 1995 [1]. UHPC has been widely used in protecting structures and giant structures with slim designs. Te UHPC has excellent mechanical properties, which include high strength in compression and tension [2][3][4], strain hardening [5] low permeability [3], high energy absorption, and durability [6]. Due to its superior performance, many researchers have studied the structural response of UHPC structures. Compared with the traditional reinforced concrete (RC) elements, the overall performance of the UHPC structure, including ultimate strength, stifness, ductility behavior, and strain hardening, has been signifcantly improved. And it can efectively control crack width and ductility [7,8].
UHPC also has excellent impact resistance and energyabsorbing capacity. Wei Fan et.al examined impact performances of UHPC columns by using the drop-hammer the impact test system, the results show the crashworthiness of the axially-loaded UHPC column was confrmed to be considerably superior to that of the conventional RC column [9]. Doo-Yeol Yoo et al. investigated the impact and blast resistances of UHPC and found that UHPC can dissipate much higher energy by impact than ordinary concrete [10]. To analyze the low-speed impact resistance of UHPC, Wei Guo et al. established the modifed CSCM model, which was used to calculate the low-speed impact resistance of UHPC, and it was in good agreement with the experimental results [11]. Based on excellent properties, UHPC is used in structures that need to resist impact loads. It is used to strengthen traditional concrete columns. Tree reinforcement forms have been completed by Wei Fan et.al, strengthened columns with two-end UHPC jackets are the better strengthening method [9]. Te reinforcement method is used to improve the crashworthiness of bridge piers, and the impact of various parameters on the crashworthiness of bridge piers is analyzed by the response surface model [12,13]. Te combination of UHPC and steel structure has also been proven to have good impact resistance. Diferent types of core structures were experimentally investigated by scholars, and various models are simulated by the fnite

Specimen Test Collection and Classifcation.
Tis paper collects some experimental samples of the UHPC-NC composite structure. Seven slab specimens were tested by AI-Osta et al. Te UHPC strength RC beams from the bottom edge, longitudinal sides, and jacket, respectively [18]. Lampropoulos and Paschalis also conducted experiments with the top, bottom edge, and jacket, respectively [15]. Yin Hor et al. have tested two types of specimens. Te frst tested UHPC as a patch material for repairing deteriorated concrete structures; the second tested the UHPC as an overlay series used to retroft softs of RC beams [23]. Paschalis and Lampropoulos have tested full-scale RC beams strengthened with UHPC layers. Additional UHPU layers with and without steel reinforcing bars have been tested [24]. Safdar and Matsumoto have strengthened the RC beam in tension and compression zones, with UHPC of varying thicknesses [22]. LI wang-wang et al. have tested seven I-beams composed of UHPC and NC, and also considered the infuence of varying steel reinforcing bars and prestressed reinforcement [25].

Material Properties of Specimens.
Typically, the compressive strength of UHPC is much greater than that of normal concrete. All of the compressive strengths of UHPC in this paper exceeded 100 MPa, as shown in Figure 1; the tensile strength of UHPC is 11 MPa-18 MPa, as shown in Figure 2; the compressive strength of normal concrete is 30 MPa-60 MPa, as shown in Figure 3; and the strength of ordinary steel bars is 500 MPa-600 MPa. LI wang-wang's samples are added with high-strength prestress tendons in UHPC, and the design value of tensile strength is 1860 MPa [15,18,[22][23][24][25][26].
All the samples consider the damage of UHPC in the tension zone; some use two-stage stress-strain curves, and some use multistage curves, as shown in Figure 4. Generally, the mechanical calculation model uses a two-stage formula, while the fnite element model uses a two-stage or multistage formula.
Te two-stage formula is adopted in the samples of Al-O and Safdar, and the slope of the second stage is assumed to be negative to consider the damage to the tensile zone of UHPC [13,20]. Tis type of stress-strain curve is also used in the fnite element model calculation in reference [21]. According to the literature conclusion, the results of fnite element calculation are in complete accord with the experimental results [27]. In Kazutaka Shirai's sample, only the bottom edge of UHPC is strengthened, and it is assumed that UHPC reaches the design value of tensile strength. Tis assumption is also considered a two-stage type, and the bottom edge UHPC is in the second stage [28].
In the fnite element calculation of Spyridon A. Paschalis's sample, a multistage stress-strain curve is adopted. Te stress in the ascending and descending sections is expressed as a function of strain [24]. In Giovanni Martinola's sample, a multistage mode stress-strain curve is also adopted, with the stress in the ascending part as a function of strain and the stress in the descending part as a function of crack width, as shown in Figure 4(b) [24]. Te two modes are essentially the same, except that the frst mode converts the crack width into cross-section strain.
Due to the super-high compressive performance of UHPC, the concrete in the compression zone rarely reaches the ultimate compressive strength, so UHPC in the compression zone can be assumed to be elastic.
Te tensile strength of UHPC was obtained by a splitting strength test [29] or a fexural test [30], while the compressive strength of UHPC was obtained by cylinder specimens [31] or cubic specimens [32].
Tis paper mainly studies the quick calculation method of cracking load and bearing capacity, so a two-stage stressstrain curve is adopted, and the second stage is assumed to be horizontal. Te UHPC in the compression zone is also considered to be in the elastic stage, and the stress-strain relationship conforms to Hooke's law.
All the tensile steel reinforcing bars used in the calculation models adopt two-section stress-strain curves, and the second section of most samples is a horizontal straight line, as shown in Figure 5. Tis stress-strain model is also adopted in this paper.

Combined Section of Specimens.
UHPC is mainly used to improve the ability and durability of reinforced concrete structures, with various combinations. All specimens can be classifed into 5 types.
Te bottom edge of the reinforced concrete beam is combined with UHPC, with or without reinforcement in the UHPC layer, as shown in Figure 6(a). Tis type is the most commonly used combination of UHPC and reinforced concrete, which can signifcantly improve the bearing capacity and durability of reinforced concrete beams, and the combined efect is obvious. In the case of the UHPC        Te top of the reinforced concrete beam is combined with UHPC, generally used to reinforce the original beam when the top edge is cracked or waterproof performance of the beam, as shown in Figure 6(b). As this combination has little infuence on the bearing capacity, it will not be discussed in this paper.
Reinforced concrete beams are combined with UHPC on two vertical sides, as shown in Figure 6(c). Tis form has a good efect on improving shear capacity but a poor efect on improving bending capacity. Te combination form can restrain the development of web cracks, is convenient for construction, and can be used to reinforce the webs of box structures.
In jacket combination, reinforced concrete beams are combined with UHPC from the bottom edge and both vertical sides, as shown in Figure 6(d). According to the experimental data, this strengthening method is efective [15,18].
Full-enclosed reinforcement, UHPC wraps all the original concrete structure, like a hoop, as shown in Figure 6(e). Tis combination form is less used.

Te Existing Design Model of Flexural Members.
Te experiments in the literature prove that the junction between UHPC and NC is well connected and can be regarded as the same cross-section [18][19][20][21]. So, all calculation models follow the assumption of the plane section even at the junction of UHPC and NC.

Te Calculation Mode of Reinforced Concrete.
According to design code ACI318 (2008) for RC structures, the simplifed rectangular stress diagram is adopted for the compressive stress of ordinary reinforced concrete members, and the tensile stress of concrete is ignored. Tis calculation model is also adopted in this paper, in which the stress and coefcients are recommended by ACI318 [32].

Te Calculation Mode of UHPC in the Tension Zone.
In the peak load calculation model, the UHPC stress in the tension zone is distributed as a curve along with the beam height, so the calculation is inconvenient. Scholars simplify the stress curve. Te simplifed stress distribution along the beam height can be divided into one-stage and two-stage, as shown in Figure 7.
(1) One-stage type Tis model assumes that the stress diagram in the tension zone is rectangular, and the UHPC stress value in the tension zone is βσ t , rectangular height is αh ut . Where β is the stress reduction factor and the height of the tensile zone gives α a reduction factor [33]. Tis method is easy to calculate, and the acting point of the resultant force is clear. However, the situation that UHPC in the tension zone is still in the elastic stage cannot be considered. Te method is  suitable for bottom edge reinforcement or combination, and the UHPC layer is thin. (2) Two-stage type two-stage calculation mode divides the UHPC layer into elastic and elastic-plastic (with cracks), and considers the constraint of steel reinforcing bars on the cracking of UHPC, without considering the reduction in height and width of the stress diagram. Te stress diagram of this method is more accurate, but the calculation process is slightly complicated. Diferent damage calculation methods will have diferent stress distributions.

Prediction Model of the Flexural Capacity of UHPC-Concrete Composite Members.
Based on the analysis and research of the above data, to quickly predict the cracking load and peak load of UHPC-NC composite beams, the following calculation model is established. Since the UHPC on the top surface is usually used for waterproofng, the case of UHPC full-enclosure is seldom used in practical projects. So the two types will not be considered in this paper. Cracking load and peak load are calculated by plane section assumption and equilibrium internal force, as shown in Figure 8. When the cracking load is calculated, UHPC in the tension zone and compression zone is in the elastic stage, and the ordinary concrete in the compression zone is also in the elastic stage. Te UHPC at the edge of the tensile zone reaches the tensile strength. Te stress of steel reinforcing bars and prestress tendons is calculated by Hooke's law. When the peak load is calculated, the ordinary concrete in compression was represented using the Whitney stress block by the ACI Code [33]; the UHCP in compression was also represented by the rectangular stress block [34]. Te UHPC in tension zone adopts a two-stage model, but the plastic UHPC considers two reduction factors. β is the stress reduction factor, and the height of the tensile zone gives α a reduction factor, the values of these two parameters are obtained by trial calculation.
When calculating the cracking load, through the balance of force and moment, we can get the following equations : where C s ′ is resultant force of top reinforcement. A sn ′ is the area of top edge reinforcement, E sn ′ is the modulus of top edge reinforcement, ε sn ′ is the strain of top edge reinforcement; C u is UHPC resultant force in compression zone. b u is width of UHPC one side in compression zone, x is distance from neutral axis to top edge, σ uc ′ is UHPC stress on the top; C c is UHPC resultant force in the compression zone. b n is width of NC in compression zone, σ c ′ is NC stress on the top; T sn is the resultant force of steel bars in the bottom NC; it is tensile force. A sn is the area of reinforcement in the bottom NC, E sn is the modulus of reinforcement in the bottom NC, ε sn is the strain of reinforcement in the bottom NC; T u1 and T u2 is resultant force of UHPC in bottom and sides. h u height of UHPC in bottom edge, f ut is the cracking stress of UHPC, h is the height of section; T y is resultant force of prestressed tendon in UHPC, A y is the area of prestressed tendon in UHPC, E y is the modulus of prestressed tendon in UHPC, ε y is the strain of prestressed tendon in UHPC; T su is resultant force of steel bars in bottom UHPC, it's tensile force. A su is the area of reinforcement in the bottom UHPC, E su is the modulus of reinforcement in the bottom UHPC, ε su is the strain of reinforcement in the bottom UHPC; when calculating the peak load, through the balance of force and moment, we can get the following equations: T u1 Figure 7: Te calculation mode of UHPC in the tension zone (a) One-stage type (b) Two-stage type.
Where α u is the stress reduction coefcient of UHPC, β u is a factor relating the depth of the equivalent rectangular compressive stress block to the depth of the neutral axis of UHPC. Trough data trial calculation, α u � 0.75, β u � 0.565. α c is stress reduction coefcient of NC in compression zone, β c is factor relating depth of equivalent rectangular compressive stress block to the depth of neutral axis of NC in the compression zone. According to the ACI code [32], α c � 1, β c � 0.85 − 0.05f c ′ − 28/7f c ′ > 28MPa. y is the height of UHPC in elastic. Whether cracking load or peak load, the infuence of UHPC in the elastic state on axial force and bending moment is considered. In the calculation of cracking load, the upper and lower edge reinforcement and prestress tendons are calculated by Hooke's law; in the calculation of peak load, it is considered that all steel reinforcement bars in the tensile zone have reached the failure stress. However, if the UHPC has poor deformation capacity, the steel bar will not reach the yield stress. So when equations (3a) and 3b are used, the ultimate deformation of UHPC needs to be large enough and it is recommended to exceed 0.005.
Tis calculation model applies to all reinforcement forms, no matter whether the section of the ordinary concrete is a rectangle, I-section, or box. Tis model is also suited for reinforced concrete structures and prestressed reinforced concrete structures.
Te cracking load of type 3 combination is determined by ordinary concrete, and the tensile performance of ordinary concrete is poor so this paper does not calculate the cracking load of type 3, but only the peak load.
It can be seen from Table 1 that no matter which combination form is used, the calculation method in this paper is close to the experimental data.
It can be seen from Figure 9 that the diference between the experimental cracking load and the cracking load calculated in this paper is very small. In references [13,20,21], the maximum COV of the cracking load is 11.49. So the calculation of cracking load in this paper is accurate. However, the calculation method in this paper of cracking load cannot distinguish the diference between several connection types of UHPC and NC, such as anchoring, epoxy resin bonding, sandblasting, and pouring. According to the original experimental data, after the accidental factors are excluded, the diference between diferent connection types is small, which can be calculated by the same calculation formula [20]. In reference [12], there is no steel bar in the UHPC layer. In reference [18], steel bars are added to ordinary reinforced concrete and UHPC. In reference [19], the UHPC layer has prestressed tendons. Te calculation shows that the cracking load calculated in this paper is very close to the experimental cracking load in references [12,18,19], so the model in this paper is suitable for UHPC with and without reinforcement [ [13] is P and the unit is kN, but the peak load is M and the unit is kN·m. To avoid misunderstanding, P is converted into M according to the experimental structure, and the units are kN·m.
It can be seen from Figure 10 that all the peak loads calculated in this paper are close to the experimental values, the maximum COV of the peak load is 14.2. All of the other COVs of the peak load are less than 10; most of the COV are less than 5. Some of the calculation results are closer to the experimental data than the original calculation models. Especially in the calculation of reference [19], there is a big gap between the original calculation results and the experimental values, but the calculation results in this paper are very close to the experimental values. Terefore, this model can calculate the peak load more accurately than other calculation models. Figure 11 plots the ratio of analysis results to experimental results, including cracking load and peak load, most of which are vertically concentrated between 0.9 and 1.2. Figure 12 plots the measured cracking load versus the corresponding literature cracking load and the current calculated cracking load. Te average of M C (calculation)/ M C (experiment) in this paper is 0.98, with a standard deviation of 0.122. Figure 13 plots the measured peak loads versus the originally calculated peak load and the current calculated peak load. Te average of M p (calculation)/ M p (experiment) in this paper is 1.04, with a standard deviation of 0.08.

Analyze the Correlation between Model and Sample.
Trough the comparison of experimental data and calculation results, the calculation results of this model are related to the cracking load and peak load results of all test specimens. Te COV of the maximum cracking load is 11.64%, and the maximum creaking load is 14.2; all of them are in the frst kind of reinforcement samples in reference [22]. Te maximum COV of the cracking load is the BL-20 specimen, in which the UHPC layer is 20 cm; and the maximum COV of peak load is BL-60, in which the UHPC layer is 60 cm.
Tere is little diference between the calculated results of this model and the experimental results of many projects, and the coefcient of variation is also small. Tis shows that the calculation model and experimental data adopted in this paper have good applicability. From the data analysis, except for some data, the model adopted in this paper is closer to the experimental value than the original calculation model, especially the peak load in reference [25].

Analysis and Discussion
In this study, it can be seen from the abovementioned calculation that the cracking load and peak load of the UHPC-NC beam are afected by many factors. For example, UHPC tensile and compression properties, UHPC ultimate strain, reinforcement strength, and reinforcement area, prestress tendons, and so on.

UHPC Tensile and Compression
Properties. Generally, the tensile strength and compressive strength of UHPC increase together, and the tensile strength is about 1/10 of the compressive strength. It can be seen from equations. (1a) and 1b that the cracking loads are afected by the performance of UHPC. In combination with types 1 and 4, the cracking load was controlled by UHPC tensile stress. Terefore, the cracking load varies with the performance of UHPC. In combination with type 3, the cracking load is controlled by the NC tensile stress, so the crack resistance is no diferent from ordinary reinforced concrete. If it is structural reinforcement, the cracking signs of type 1 and type 4 reinforcements will be changed from ordinary reinforced concrete cracking to UHPC cracking, and the cracking load will be greatly increased. However, type 3 reinforcement is still marked by the cracking of ordinary reinforced concrete, and the cracking load does not increase too much.
It can be seen from equations (3a) and 3b that UHPC in the tension zone can provide considerable tensile stress, the increase in tensile and compressive strength of UHPC in all reinforcement forms can increase the peak load of the structure.
Based on the experimental sample in reference 12, the compressive strength of UHPC was adjusted, and the tensile strength was assumed to be 1/10 of the compressive strength, and. Figure 14 was obtained. Figure 14 shows the peak load change along with the performance of UHPC in the form of 1.3.4. Te peak load of jacket reinforcement increases obviously with UHPC compressive strength, and the bottom edge reinforcement increases most slowly, but the infuence of bottom edge reinforcement thickness should also be considered. If the UHPC layer thickens, it can be inferred from equations (3a) and 3b that the slope of type 1 in Figure 14 will become larger.

Reinforcement Strength and Reinforcement Area.
In the design of the UHPC-NC composite beam, if the bottom UHPC layer is thick, steel reinforcing bars can be added; reinforcement cannot be added if the UHPC layer is thin. Te strength and area of steel reinforcing bars have a direct impact on the bending performance, whether in the UHPC layer or NC layer. Normally, the steel reinforcing bars in the UHPC layer are far from the neutral axis and deform together with the UHPC. Te strength and area have a great infuence on the bending performance of the structure. Te reinforcement in the UHPC layer is best made of highstrength reinforcement, which matches the good deformation capacity of UHPC. In reference 18, it can be seen that the combination form of adding steel reinforcing bars is 89.6% higher than the original peak load of ordinary reinforced concrete beams, and the combination form of the UHPC layer without steel reinforcing bars is 1.3% higher than that of ordinary reinforced concrete beams. Terefore, if the purpose of combination or reinforcement is to improve peak load, the UHPC layer needs sufcient thickness  Advances in Civil Engineering and high-strength steel reinforcing bars. It is necessary to prevent the super-reinforced beam from crushing the concrete at the top edge when adding steel reinforcing bars and prestressing tendons.

Prestress Tendons.
If UHPC is used to reinforce ordinary concrete, and the prestress tendons are located in ordinary concrete, it is necessary to comply with the specifcation of ordinary prestressed concrete; if it is the UHPC-NC composite beam, prestress tendons are a better choice in the UHPC layer, which ensures the deformation coordination between UHPC and high-strength prestress tendons and increases cracking load and peak load. So this paper only considers the prestress efect of the UHPC layer.
Te width of structural cracks and the structural damage can be reduced by applying prestress, and the bearing capacity of the structure can also be improved. Using a sample in reference [25], adjust the calculation of tension stress controlled by prestressing and get Figure 15. It shows the change of structural cracking load when a diferent percentage of controlled tensile stress is applied. With the increase of control tension stress of prestress tendons, the cracking load on the structure increases linearly.

Conclusions
In this study, a mechanical calculation model for predicting the performance of UHPC fexural members is established and verifed. Te experimental data of 31 specimens proves that the model is suitable for predicting the cracking load and peak load of UHPC-NC composite beams. Te results show that the fexural performance of composite beams can be improved by increasing the strength of UHPC, the area and strength of steel reinforcing bars, and the area and strength of prestress tendons.
(1) In the calculation of cracking load, UHPC is assumed to be elastic, and the stress is distributed in a triangle. In calculating peak load, UHPC is divided into plasticity and elasticity. Te elastic part is calculated according to the actual stress distribution; for the plastic part, the calculated height reduction factor β and plastic stress reduction factor α are introduced, and their values are 0.565 and 0.75, respectively, through trial calculation. Te calculated results are in good agreement with the data of references. (2) In the calculation of cracking load, both ordinary steel bars and prestressed tendons are elastic, and the plane section assumption and Hooke's law are used to calculate the stress. In the calculation of peak load, both ordinary steel bars and prestressed tendons are plastic, and the failure load of steel bars is substituted instead of the yield load.   combination. If it's necessary, it can also be extended to top combination and all-inclusive combination calculations. In this paper, ordinary reinforcement in the NC layer, ordinary reinforcement in the UHPC layer, and prestress tendons in the UHPC layer are also considered. Te parameters are considered in detail and have strong applicability. Terefore, the calculation model in this paper has strong applicability and can be applied to various combinations of UHPC-NC beams. (4) Te parameters afecting cracking load and peak load are studied in the paper. Te efects of UHPC performance on the bottom edge combination, two vertical combinations, and jacket combination are analyzed. Te infuence of strength and area of steel bars and prestress tendons is analyzed and calculated. Based on the sample in Ref. [19], the infuence of prestress tendons on controlling tensile stress is analyzed. With the increase in the controlled tensile stress of prestress tendons, the cracking load will increase. Te model does not consider the diferent connection modes between UHPC and ordinary concrete. In future research tasks, the infuence of these factors will be considered.

Data Availability
Te data are available from the corresponding author upon reasonable request.