Shear resistance test analysis and calculation method of shear capacity of UHPC-NC beam without web reinforcement

Abstract

In order to study the failure phenomenon and shear resistance performance of UHPC-NC beams without web reinforcement, shear resistance tests were carried out on 15 UHPC-NC beams without web reinforcement. The variation parameters were shear span ratio, web thickness, steel fiber content, and compressive stress level. The load-deflection curve, failure phenomenon and crack development of the test beam are analyzed. The test results show that the joint between the top plate and the bottom plate of UHPC-NC I-beam without web reinforcement is the weakest, and the cracked web is easy to be “cut off” at the joint; The “banding crack area” of web is closely related to the shear span ratio, and the width of ribbon area increases with the increase of shear span ratio. The shear capacity of UHPC-NC I-beam without web reinforcement decreases with the increase of shear-span ratio. The shear capacity increases with the increase of web thickness and compressive stress level. The shear capacity has little relationship with steel fiber content, and the initial stiffness of the structure can be improved by applying prestress. Finally, considering the influence of the top slab compression-shear zone and the bottom slab tension-shear zone on the shear capacity of UHPC-NC prestressed composite beams without web reinforcement, the corresponding calculation method of shear capacity is given based on the modified compression field theory and the idea of sub-item superposition, and the applicability of the calculation method is verified by the test results.

Keywords

Ultra-high performance concrete shear resistance test modified compression field theory UHPC-NC (normal concrete) beam without web reinforcement crack

Introduction

Ultra-high performance concrete (UHPC) is a new type of super concrete with excellent properties such as ultra-high strength, high toughness and high corrosion resistance, it can improve the load carrying capacity of civil infrastructure significantly (Qi et al., 2016; Graybeal and Tanesi, 2007; Larrard and Sedran, 1994).

In recent studies, many experts and scholars have studied the influence of different parameters on its performance and its application in bridge structures. Among them, shear resistance studies are one of the main focuses of their research. Graybeal (2006) has carried out the shear performance test on the AASHTO I-shaped sections of 3 prestressed UHPC beams without web reinforcement, and it is found that they have different failure modes. Voo et al. (2010) and Wu and Han (2009) studied the influence of different factors on the shear resistance performance of UHPC beams, and based on the results, they put forward the calculation formulas of shearing capacity respectively. Baby et al. (2013, 2014a, 2014b) carried out shear resistance tests on 11 I-shaped UHPC beams, verified the shear capacity formula of French UHPC structural design guide (2015), improved the calculation method of shear cracking load, and then adopted the modified compression field theory method to predict the shear resistance performance of UHPC beams. Ferrier et al. (2015) studied the effect of FRP bars on the shear resistance performance of UHPC beams. Xu et al. (2014) carried out shear resistance tests of 9 T-shaped UHPC beams. The test parameters include shear span ratio, stirrup ratio and prestress level. Based on the results, the empirical formula for calculating shear bearing capacity of UHPC beams was fitted.

Voo et al. (2012, 2015) described the fabrication process, construction sequence, and design method of a combined highway bridge consisting of UHPC precast girders and cast-in-place NC deck slabs. Wangwang et al. (2020) found that the application of UHPC to the tensile zone of prestressed concrete beams can significantly improve the stiffness and ductility of the members. Existing studies have shown that the UHPC-NC composite structure has good mechanical properties and value for engineering applications.

At present, scholars have carried out a certain number of UHPC-NC beam shear tests, and some scholars have carried out some research on the calculation method of UHPC-NC shear bearing capacity, but they are mainly obtained by fitting and correcting their own test data, and the number of test samples is small, so its applicability needs further study. At the same time, the current calculation method ignores the contribution of top and bottom plates to the shear bearing capacity of UHPC beams. Meanwhile, UHPC-NC beam without web reinforcement as a completely new construction, has not widely promoted in engineering or formed calculation method of shear capacity. Therefore, a large number of tests are required for this structure, and a new calculation model of shear bearing capacity of UHPC-NC beams without web reinforcement needs to be established.

Test overview

Construction and parameters of test beam

Test girders are used for bridge structures and are all prefabricated. The test section adopts I-beam section, and the height of the test beam is 700 mm, in which the height of C50 (Concrete with cube compressive strength of 50 MPa) deck is 100 mm, and the height of UHPC I-beam is 600 mm; The width of the top plate is 950 mm, and the width of the bottom plate is 400 mm; The web thickness adopts 60 mm, 80 mm, and 100 mm, and the prestressed steel strand is 15.2 under-relaxation steel strand with tensile strength of 1860 MPa. 12 strands are arranged in the bottom plate and 5 strands are arranged in the top plate. See Figure 1 for the overall size layout.

Figure 1.

Dimensions of Test Beams (Unit: mm): (a) Web Thickness 60, (b) Web Thickness 80, (c) Web Thickness 100, (d) Experimental setup of Test Beams.

A total of 15 UHPC-NC beams were designed in this test. All beams were fabricated by pre-tensioning method with steam curing for 28 days, and the interface was fully bonded with steel anchors between UHPC and NC. The test parameters include: shear span ratio, steel fiber content, web thickness, and compressive stress level of the test beam. The test beam parameters are summarized in Table 1.

Table 1.

Main parameters of test beams.

Serial number	Test beam number	Shear span ratio λ	Steel fiber content ρ (%)	Compressive stress level S (MPa)	Web thickness t (mm)
1	B1-λ2.0-ρ2.0-S1-t60	2.0	2.0	1358	60
2	B2-λ3.0-ρ2.0-S1-t60	3.0	2.0	1358	60
3	B3-λ3.5-ρ2.0-S1-t60	3.5	2.0	1358	60
4	B4-λ4.0-ρ2.0-S1-t60	4.0	2.0	1358	60
5	B5-λ1.5-ρ1.8-S1-t60	1.5	1.8	1358	60
6	B6-λ1.5-ρ2.5-S1-t60	1.5	2.5	1358	60
7	B7-λ1.5-ρ2.0-S2-t60	1.5	2.0	1086	60
8	B8-λ1.5-ρ2.0-S1-t80	1.5	2.0	1358	80
9	B9-λ1.5-ρ2.0-S1-t100	1.5	2.0	1358	100
10	B10-λ3.0-ρ1.8-S1-t60	3.0	1.8	1358	60
11	B11-λ3.0-ρ2.5-S1-t60	3.0	2.5	1358	60
12	B12-λ3.0-ρ2.0-S2-t60	3.0	2.0	1086	60
13	B13-λ3.0-ρ2.0-S1-t80	3.0	2.0	1358	80
14	B14-λ3.0-ρ2.0-S1-t100	3.0	2.0	1358	100
15	B15-λ1.0-ρ2.0-S1-t60	1.0	2.0	1358	60

Note: The test beam is named B-λ-ρ-S-t, where B represents No. 1 to No. 15 test beams, λ represents shear span ratio of test beams, ρ represents steel fiber content of test beams, S represents compressive stress level (tension control stress of single prestressed tendon in the table), and t represents web thickness of test beams.

Test material

The mix ratio was designed in accordance with different steel fiber content, and the standard specimens were made of concrete materials in the same batch as the test beams. The standard curing was carried out in the test room in accordance with the “Standard for Test Methods of Concrete Physical and Mechanical Properties” (GB/T 50081-2019), and the basic mechanical properties of UHPC specimens were tested after 28 days of natural curing in a standard curing room with a temperature of 20°C ± 2°C and a relative humidity of more than 95 %. The test methods and dimensions are carried out in accordance with the specifications “Standard for Test Methods of Concrete Physical and Mechanical Properties” (GB/T 50081-2019), “Reactive Powder Concrete” (GB/T31387-2015) and the French norm “Concrete - Ultra-high performance fibre-reinforced concrete - Specifications, performance, production and conformity” (NF P18-470). The test indexes include compression strength of compression cube, splitting tensile strength, flexural strength and elasticity modulus. The mechanical properties of UHPC materials are summarized in Table 2.

Table 2.

Main parameters of UHPC.

Fiber content (%)	Mechanical property
Fiber content (%)	f_cu (MPa)	f_ts (MPa)	f_ct,fl (MPa)	f_c (MPa)	σ_f (MPa)	E_c (MPa)
1.8	163	15	31	120	7.21	40516
2.0	171	16	31	124	7.27	40983
2.5	175	18	32	127	7.50	42408

Note: f_cu is compression strength of compression cube, f_ts is splitting strength, f_ct,fl are flexural strength, f_c is prismatic axial compressive strength, σ_f is residual tensile strength after cracking, and E_c is elastic modulus.

Test equipment and test scheme

As shown in Figure 1(d), the test beam adopts mid-span and single-point loading mode and adopts simple supported beam. A 1000-ton long column compression testing machine is used for loading, and a force sensor is set at the ball joint. In accordance with the step loading system, each stage loading is 100 kN, and the measurement results of beam deflection and sensor are recorded in stages. After each stage load is stable, the development of cracks is observed and marked. The LVDTs and strain gauge arrangement is shown in Figure 2.

Figure 2.

LVDTs of test beams.

Test results and analysis

Main results of the test

The main test results are shown in Table 3. The failure types and crack distribution of 15 test beams are shown in Figure 3. The shear-deflection curve of the test beam is shown in Figure 4. The maximum deflection in the mid span of the test beam is the maximum deflection recorded during the test; When determining the crack load, the crack inclination angle and load value are recorded. There are significant differences in shear failure types between UHPC-NC beams without web reinforcement and ordinary concrete beams without web reinforcement, which are mainly reflected in the following aspects:

(1) Because of the connection of steel fiber, the diagonal cracks in the middle web of the part combined bending and shear of UHPC-NC beam without web reinforcement appear at first, which are close to the direction of principal stress, and then the cracks extend continuously, and new cracks appear continuously. A large number of parallel diagonal cracks appeared before the main cracks appeared, and then gradually developed into “banding crack areas”. In the whole process of beam failure, the diagonal cracks appear from thin to thick, the length of cracks from short to long, the number of cracks is numerous, the crack spacing is proportional, and the test beam presents very obvious ductility characteristics. This is in sharp contrast with the brittleness of conventional reinforced concrete beams in shear resistance failure.

(2) Different from the shear failure of traditional reinforced concrete beams, the shear capacity of UHPC beams without web reinforcement still has a long gentle descending stage after reaching the peak load. This further shows that the shear resistance failure of UHPC-NC prestressed composite beams without web reinforcement is no longer the brittle failure of ordinary reinforced concrete beams, and has good shear ductility.

(3) The main crack of UHPC-NC beam without web reinforcement is Z-shaped, which is different from that of ordinary reinforced concrete beam through top and bottom plate. The main crack of UHPC-NC beam without web reinforcement is Z-shaped at failure. When the main crack penetrates the web, it develops horizontally along the web and the chamfer of the top and bottom plates. The failure of the test beam is accompanied by the generation and development of horizontal cracks at the end of the beam. As the most important part of providing shear force for UHPC-NC prestressed composite beam without web reinforcement, the web is the weakest joint with the top and bottom plates of UHPC I-beam. At the same time, the top and bottom plates are equipped with prestressed reinforcement, and the cracked web is easy to be “cut off” at the joint during the shear process of the beam.

Table 3.

Main test results.

Serial number	Test beam number	Cracking load (kN)	Crack diagonal angle (°)	Ultimate load (kN)	Main fracture diagonal angle (°)	Ultimate load mid-span deflection (mm)	Maximum mid-span deflection (mm)
1	B1-λ2.0-ρ2.0-S1-t60	1340.0	46	2600.5	30	6.3	10.9
2	B2-λ3.0-ρ2.0-S1-t60	835.0	44	1894.3	41	14.5	46.3
3	B3-λ3.5-ρ2.0-S1-t60	1100.0	42	2769.6	35	32.2	47.6
4	B4-λ4.0-ρ2.0-S1-t60	758.0	25	1843.0	35	33.5	37.4
5	B5-λ1.5-ρ1.8-S1-t60	896.2	20	2877.2	20	6.3	17.2
6	B6-λ1.5-ρ2.5-S1-t60	859.7	39	3065.2	30	6.0	20.6
7	B7-λ1.5-ρ2.0-S2-t60	1051.3	39	2375.2	31	7.1	40.0
8	B8-λ1.5-ρ2.0-S1-t80	1135.8	37	3508.4	35	5.7	13.2
9	B9-λ1.5-ρ2.0-S1-t100	1881.4	44	3431.0	43	11.0	28.5
10	B10-λ3.0-ρ1.8-S1-t60	927.4	46	1778.5	-	11.8	40.0
11	B11-λ3.0-ρ2.5-S1-t60	916.4	41	1947.0	43	16.2	45.4
12	B12-λ3.0-ρ2.0-S2-t60	667.4	35	1720.9	-	15.9	20.6
13	B13-λ3.0-ρ2.0-S1-t80	1296.5	43	2072.3	41	17.0	46.0
14	B14-λ3.0-ρ2.0-S1-t100	1161.8	41	2450.6	40	27.1	45.5
15	B15-λ1.0-ρ2.0-S1-t60	2400.0	55	4500.0	/	4.6	7.9

Note: In the table, “-” means that the main cracks are inclined and horizontal cracks of web, and “/’ means that the main cracks are inclined and vertical cracks of web.

Figure 3.

Failure State of Test Beams: (a) Left side web, (b) Right side web.

Figure 4.

Shear Force-Displacement Curves: (a) Shear span ratio (Steel fiber content 1.8%), (b) Shear span ratio (Steel fiber content 2.0%), (c) Shear span ratio (Steel fiber content 2.5%), (d) Steel fiber content, (e) Web thickness, (f) Compressive stress level.

Failure type and crack distribution

As can be seen in Figure 3, when the test beams are loaded to a certain value (cracking load), fine diagonal cracks in the web begin to appear, and the strike is about 40°. With the increase of load, the fine cracks increase and gradually form “banding crack area”. It is found that the width of banding area is closely related to the shear span ratio, and with the increase of the ratio, the width of banding area also increases. As the load continues to increase, the number of web cracks no longer increases, and the concrete of web diagonal cracks begins to peel off and gradually connects into a main crack, but it is obvious that the direction of the main crack is inconsistent with the general direction of the previous diagonal crack. At the same time, for the test beam with thick web, the vertical crack of the bottom plate gradually widens during the concrete peeling off, and then runs through the main crack of the web plate to form a complete main crack, but due to the existence of prestressed reinforcement of the bottom plate, the bottom plate is not completely separated; In the test beams with thinner webs, horizontal or vertical main cracks at the web position occur simultaneously during the development of diagonal cracks. Because of the high compressive strength of C50 at failure, the concrete in the shear-compression area does not collapse in a large area, but the concrete at the upper end of the diagonal crack is delaminated, warped and fell off, which is significantly different from the shear failure of ordinary concrete beams. During the tests, delamination or crack at UHPC-NSC interface was not observed due to the weak web. It can be found that when the shear span ratio is small and the web is thin, the web is not full of shear diagonal cracks, and even horizontal shear failure occurs, that is, the shear resistance of the web cannot be fully exerted.

Analysis of main influencing factors

Figure 4 presents that the stress process of UHPC-NC beams without web reinforcement can be divided into three stages: elastic stage before bending-shear cracking; After bending and shear cracking and before steel fiber is pulled out; And after the steel fiber is pulled out. Before bending and shear cracking, the test beam is in elastic working state. After bending-shear cracking, the stiffness of the beam decreases. Because of the tensile action of steel fiber, the load-deflection curve is still approximately linear, but the slope decreases. After the longitudinal reinforcement yields or the fiber yields or is pulled out, the bearing capacity of the test beam decreases continuously until it is destroyed.

It can be seen from Figure 4 that the shear span ratio has a significant influence on the shear capacity of the test beam, and the ultimate shear capacity decreases with the increase of the shear span ratio. The content of steel fiber has a certain influence on the ultimate shear capacity of the test beams. The content of steel fiber increases from 1.8% to 2.0%, the ultimate shear capacity increases by 6.5%, the content of steel fiber increases from 2.0% to 2.5%, and the ultimate shear capacity increases by 2.8%; However, the content of steel fiber has little effect on the initial stiffness of the test beam. The web thickness has great influence on the ultimate shear capacity of the test beams, and the ultimate shear capacity of the test beams increases with the increase of web thickness. The ultimate shear capacity increases with the increase of compressive stress level. Prestressing the test beam can improve the ultimate shear capacity of the structure, and also can improve the initial stiffness and shear resistance strength of the test beam.

Calculation model of UHPC beam

At present, researchers in various countries have put forward many shear resistance calculation models, including limit equilibrium theory, truss model, compression field theory, arching theory and so on (Wang and Qi, 2013; Nielsen, 1998; Vecchio and Collins, 1986). The idea of sub-item superposition has been widely adopted by scholars, which is also reflected in the norms of various countries. Therefore, the shear resistance of UHPC beams in various countries consists of three parts: concrete contribution, stirrup contribution and fiber contribution, namely

V_{u} = V_{c} + V_{s} + V_{f}

(1)

Where:

V_{c}

is the shear force borne by the matrix.

V_{s}

is the shear force borne by stirrups.

V_{f}

is the shear force borne by fiber.

The calculation method of shear bearing capacity in French NF P 18-710 (2015) (hereinafter referred to as French norms) is based on classical frame model when calculating the contribution of UHPC matrix to shear capacity, and variable angle frame model when calculating the contribution of stirrup to shear capacity, and considering the contribution of fiber to shear bearing capacity. For prestressed section without reinforcement, the shear force calculation formula is as follows:

V_{c} = \frac{0.24}{γ_{c f} γ_{E}} k \sqrt{f_{c k}} b d

(2)

V_{s} = \frac{A_{s}}{s} z f_{y} \cot θ

(3)

V_{f} = A_{f v} σ_{t} \cot θ

(4)

Where:

γ_{1}

is the design safety factor for the shear resistance contribution of UHPC concrete is taken as 1.5 in analysis.

k_{1}

is the prestress influence coefficient

k = 1 + 3 \cdot σ_{c p} / f_{c k} = 1.20

σ_{c p}

is the axial force caused by prestress.

b

is the minimum width of the section.

d

is the effective height of the section.

A_{s}

is the cross-sectional area of stirrup.

s

is stirrup spacing.

z

is the distance between the active points of composite force of longitudinal reinforcement at the upper and lower edges of the member section.

θ

is the angle between concrete diagonal web member and beam axis in truss model, which is calculated by 45° in this paper.

A_{f v}

is the area of fiber action on diagonal crack.

f_{c k}

is the compressive strength of UHPC cylinder, and the test value in this paper is taken.

f_{y}

is Yield strength of stirrup.

σ_{t}

is the residual tensile strength of UHPC, and the test value in this paper is taken.

However, as shown in Figure 5(b), it can be seen from the test beam results that the bottom plate is not completely separated due to the existence of prestressed reinforcement in the bottom plate during ultimate failure. If only the web width is taken into account in the calculation, the calculation results must be conservative, and the angle of truss model is artificially assumed, which cannot correctly reflect the cracking inclination angle. Therefore, the modified compression field theory is introduced and combined with the idea of partial superposition, and the shear resistance calculation formula of UHPC beam is put forward as follows:

V_{u} = V_{t} + V_{w} + V_{b}

(5)

Where:

V_{t}

is the shear force borne by the top plate.

V_{w}

is shear force for web.

V_{b}

is shear force borne by the bottom plate.

Figure 5.

Calculation Model: (a) Fracture Distribution, (b) Vertical force, (c) Force area, (d) Initial strain state, (e) Shear resistance calculation model 1.

Constitutive relation of materials

The test in this paper is the shear test of UHPC-NC prestressed composite beams without web tendons, so the constitutive relations of prestressed reinforcements, C50 concrete and UHPC are introduced.

The compressive constitutive relation of C50 concrete used in calculation is as follows:

f_{c} (ε) = {\begin{cases} 35.5 [1 - {(1 - \frac{ε}{2000})}^{2}] & ε \leq 2000 \\ 35.5 & 2000 \leq ε \leq 3300 \end{cases}

(6)

Referring to the domestic design code of ultra-high performance concrete structure (CECS, 2020), the compressive constitutive of UHPC used in calculation is as follows:

f_{u} (ε) = {\begin{cases} f_{c k} \frac{ε}{2900} & ε \leq 2900 \\ f_{c k} & 2900 \leq ε \leq 3400 \end{cases}

(7)

Where:

f_{c k}

is Take the measured value of the test.

The constitutive relation of steel strand proposed by Naaman is as follows:

f_{p} (ε) = E_{p s} ε [Q + \frac{1 - Q}{{[1 + {(\frac{E_{p s} ε}{K f_{p y}})}^{N}]}^{1 / N}}]

(8)

Where:

E_{p s}

is 1.92 × 10⁵ MPa.

f_{p y}

is 1680.15 MPa. N, K, and Q are 5.05, 1.1155, 0.01699.

ε_{p u}

is 0.069.

f_{p u}

is 1918.2 MPa.

Equilibrium conditions of MCFT

The calculation model used in this paper is shown in Figure 5. It is assumed that the shear force of UHPC-NC beam section without web reinforcement is shared by the shear bearing capacity provided by concrete in the upper compression area, the aggregate bonding strength in the web tension-compression area and the tensile bearing capacity of steel fiber in the web tension-compression area, and the tension-shear area of bottom plates. As shown in Figure 5(a), the modified compression field theory (Collins and Mitchell, 1991; Bentz and Collins, 2006; Bentz et al., 2006) assumes that when there are fine cracks distributed in the region and it is regarded as a continuous region, for the UHPC-NC beam without web reinforcement, a micro-element of the web plate is taken for stress balance analysis, and the stress balance conditions can be established as follows:

f_{x} = f_{1} + ρ_{p} f_{p} - v \cot θ

(9)

f_{y} = f_{1} - v \tan θ = 0

(10)

f_{1} + f_{2} = v (\tan θ + \cot θ)

(11)

Where:

ρ_{p}

is reinforcement ratio of longitudinal prestressed reinforcement.

f_{1}

is the principal tensile stress perpendicular to the crack direction.

f_{2}

is the principal compressive stress parallel to the crack direction.

f_{x}

is Stress in the horizontal direction.

f_{y}

is Stress in the vertical direction.

θ

is the critical fracture diagonal angle.

Among them, in accordance with the strain compatibility conditions:

f_{1} = {\begin{cases} \frac{f_{c}}{2900} ε_{1} & ε_{1} \leq 170 \\ \frac{170 f_{c}}{2900} - f_{t u} \sqrt{\frac{ε_{1} - 170}{5000}} & ε_{1} > 170 \end{cases}

(12)

v_{c i} = \frac{0.18 \sqrt{f_{c}}}{0.31 + \frac{24 w}{d_{a} + 16}} \geq f_{1}

(13)

w = \frac{ε_{1}}{\frac{\sin θ}{s_{x}} + \frac{\cos θ}{s_{y}}}

(14)

Where:

v_{c i}

is the bonding strength between cracks.

d_{a}

is the aggregate particle size.

f_{c}

is UHPC uniaxial compressive strength.

f_{t u}

is UHPC residual tensile strength.

s_{x}

and

s_{y}

is the horizontal and vertical spacing of cracks.

ε_{1}

is the principal tensile strain perpendicular to the crack direction.

f_{2} = f_{2 \max} [2 (\frac{ε_{2}}{ε_{0}}) - {(\frac{ε_{2}}{ε_{0}})}^{2}]

(15)

f_{2 \max} = \frac{f_{c}}{0.8 - 0.34 ε_{1} / ε_{0}}

(16)

Where:

ε_{0}

is the peak compressive strain of UHPC under uniaxial compression is 0.0029.

ε_{2}

is the principal compressive strain parallel to the crack direction.

At this time, the average strain of the element conforms to the strain Mohr circle, that is:

ε_{x} = \frac{ε_{1} \tan^{2} θ + ε_{2}}{1 + \tan^{2} θ}

(17)

ε_{y} = \frac{ε_{1} + ε_{2} \tan^{2} θ}{1 + \tan^{2} θ}

(18)

Where:

ε_{x}

is horizontal average strain of concrete.

ε_{y}

is average vertical strain of concrete.

Contribution of compression-shear area

For UHPC-NC beam without web reinforcement, prestress produces initial upper edge compressive stress and lower edge compressive stress, as shown in Figure 5(d), which needs to be calculated by using converted section, and the initial strain of concrete at the characteristic height of the section is calculated as follows:

ε_{c 1} = \frac{f_{p 0} A_{s}}{E_{c} A_{0}} - \frac{f_{p 0} A_{s} (d - c)}{E_{c} I_{0}} c

(19)

ε_{n 1} = \frac{f_{p 0} A_{s}}{E_{c} A_{0}} - \frac{f_{p 0} A_{s} (d - c)}{E_{c} I_{0}} (c - x_{n})

(20)

ε_{0} = \frac{f_{p 0} A_{s}}{E_{c} A_{0}}

(21)

ε_{s 1} = \frac{f_{p 0} A_{s}}{E_{c} A_{0}} + \frac{f_{p 0} A_{s} (d - c)}{E_{c} I_{0}} (d - c)

(22)

Where:

f_{p 0}

is the effective prestress of prestressed reinforcement.

E_{c}

is the elastic modulus of UHPC.

A_{o}

is the converted sectional area.

A_{s}

is the area of prestressed reinforcement.

I_{o}

is the converted section moment of inertia.

ε_{c 1}

is the upper edge strain of compression-shear area.

ε_{n 1}

is the bottom edge strain of compression-shear area.

ε_{s 1}

is the strain of prestressed reinforcement. c is the height of neutral axis. x_n is the height of compression-shear zone. d is the effective height of the section.

Existing tests show that once the bending cracks in UHPC-NC beams occur, they will rapidly extend to the lower edge of the compression area and tend to be stable, and further increase the load, and the distribution range of cracks will increase, but the height changes little. Therefore, the height c of the compression area is still taken according to the linear elastic bending theory. Then for section concrete, the horizontal strain increment at different heights meets the following requirements:

\frac{ε}{x} = \frac{ε_{n 2}}{c - x_{n}} = \frac{ε_{c 2}}{c} = \frac{ε_{s 2}}{d - c}

(23)

Where:

ε_{n}

is the bottom edge strain of compression-shear area.

ε_{c}

is the upper edge strain of compression-shear area.

ε_{c}

is the strain of prestressed reinforcements. c is the height of neutral axis. x_n is the height of compression-shear area. d is the effective height of the section.

With the increase of shear span ratio $λ$ , the shear failure type of shallow beams without web reinforcement changes from shear-compression failure to diagonal tension failure. Therefore, the following simplified assumptions are made in reference (Zheng et al., 2017): when $λ \leq 1$ , the height coefficient $x_{n} / c = 1$ of shear-compression area, that is, the compression area is all compression-shear area; when $1 < λ \leq 5$ , with the increase of shear span ratio, the height of compression-shear area gradually decreases; when $λ > 5$ , $x / c$ approached 0. The relationship between $x_{n} / c$ and $λ$ is:

\frac{x_{n}}{c} = \frac{1}{λ}

(24)

At this time, the resultant force of compression-shear area is:

C = \int_{0}^{x_{n}} f_{c} (ε) \cdot β b (x) d x

(25)

Where: b (x) is the section width.

β

is the effective width reduction factor.

At the same time, in accordance with the constitutive relation of prestressed reinforcements, the effective prestress in the initial state is substituted into formula (8) to obtain the initial strain $ε_{s 0}$ , and then the strain of prestressed reinforcements in shear failure is as follows:

ε_{s} = ε_{s 0} + ε_{s 2}

(26)

Formula (26) is substituted into formula (8), and from the equilibrium condition of force, that is, the resultant force of compression-shear area, the horizontal component force of web is equal to the tensile prestressed reinforcement:

C = f_{p} A_{s} + f_{x} b (x) (d - x_{n})

(27)

Where:

f_{p}

is the final stress of prestressed reinforcement.

b

is the effective height of the section.

The test shows that when the UHPC-NC prestressed composite beam with C50 deck is at failure, the compressive stress and shear stress of concrete in shear-compression area are controlled by biaxial strength criterion, that is,

\frac{τ}{f_{c k}} = \sqrt{a + b (\frac{σ_{c}}{f_{c k}}) + c {(\frac{σ_{c}}{f_{c k}})}^{2}}

(28)

Where:,

σ_{c}

and

τ

are the normal stress and shear stress of concrete respectively.

f_{c k}

is the compressive strength of concrete. a, b, and c are the coefficient for the test certainty, take the suggested value given by Yoshikatsu Tsuboi (Zhou et al., 1989) as

a = 0.0089

b = 0.095

c = - 0.104

If the normal stress at different heights is substituted into formula (15) and its shear contribution is taken into account, the shear capacity of concrete in compression-shear area is as follows:

V_{t} = \int_{0}^{x_{n}} τ \cdot β b (x) d x

(29)

Contribution of tension-shear area

Based on the stress situation of the spacer, it can be found that the bottom plate located in the tension-shear area has stress $τ_{b}$ that cannot be ignored, as shown in Figure 5(e). Generally speaking, the shear resistance strength of concrete is generally 1∼2 times of the tensile strength (Huang, 1999), and the shear capacity of the tension-shear area can be obtained as follows:

V_{c b} = τ_{b} b (x) h_{b} \approx f_{t k} b (x) h_{b}

(30)

Where:

f_{t k}

is UHPC tensile strength, 7.2 MPa.

Web shear resistance contribution

Based on the modified compression field theory, the shear capacity of web plate is calculated, and the tension stress at cracks is included, which can better reflect the cracking angle of cracks and more accurately calculate the shear capacity of web plate without web reinforcement under bending and shear action. According to the hypothesis of modified compression field theory, the average strain of web concrete in horizontal direction is taken as the central strain of web:

\frac{ε_{x 0} - ε_{x}}{(d - x_{n}) / 2 - c} = \frac{ε_{s 2}}{d - c}

(31)

Where:

ε_{x}

is the final horizontal average strain of web.

ε_{x 0}

is the initial horizontal average strain of web.

For UHPC-NC beams without web reinforcement, assuming that the tension stress $f_{1}$ caused by the existence of steel fibers at the crack at failure is equal to the principal tension stress of the web, the shear capacity of the web is as follows:

V_{w} = v b (x) (h - x_{n} - h_{b})

(32)

Where:

v

is the average shear stress of web.

Calculation flow of shear capacity

In accordance with the material constitutive relationship, stress-strain coordination relation, and force balance, the unknown parameters in the calculation are $f_{1}$ and $θ$ , and in order to solve the unknown parameters, it is necessary to combine some assumptions in the modified compression field theory. The initial tension strain $ε_{1}$ and crack initiation angle $θ$ can be given first, and then the judgment can be made by loop iteration and finally by the equality of prestress of prestressed reinforcements. The judgment equation can be obtained by the following bending moment balance:

f_{p}^{'} = \frac{λ V d - f_{x} b_{w} (d - x_{n}) (d_{v} + x_{n} / 2 - d / 2)}{A_{s} d_{v}}

(33)

Where:

d_{v}

is the stress center of the concrete in the compression-shear area to the height of the prestressed reinforcement.

The specific calculation steps are shown in Figure 6.

Figure 6.

Shear strength calculation process.

Comparison of calculation formulas

Based on the test data of shear capacity of 15 test beams, the validity of the calculation method is verified. In this paper, PYTHON language is used to program and calculate the shear capacity and cracking initiation angle, which are compared with the test data and the calculation results of French norms. See Table 4 for specific data. Among them, V_TEST is the test value, V_CAL is the calculation value of this calculation method, and V_NFP is the result of French norms formula.

Table 4.

Comparison of calculation results.

Test beam no.	V_TEST/kN	$θ_{T E S T}$ /°	V_CAL/kN	$θ_{C A L}$ /°	V_NFP/kN	$\frac{V_{C A L}}{V_{T E S T}}$	$\frac{V_{N F P}}{V_{T E S T}}$
B1	1300.25	46	1213.2	43.0	1000.4	0.93	0.77
B2	947.15	44	951.7	39.9	1000.4	1.00	1.06
B3	1384.8	42	823.0	35.1	1000.4	0.59	0.72
B4	921.5	25	725.9	44.1	1000.4	0.79	1.09
B5	1438.6	20	1402.1	44.0	992.0	0.97	0.69
B6	1532.6	39	1402.4	43.8	1029.5	0.92	0.67
B7	1187.6	39	1205.9	43.9	996.5	1.02	0.84
B8	1754.2	37	1423.9	42.2	1079.2	0.81	0.62
B9	1715.5	44	1443.8	40.8	1157.7	0.84	0.67
B10	889.25	46	946.8	40.3	992.0	1.06	1.12
B11	973.5	41	954.7	38.8	1029.5	0.98	1.06
B12	860.45	35	856.6	43.6	996.5	1.00	1.16
B13	1036.15	43	981.3	38.2	1079.2	0.95	1.04
B14	1225.3	41	996.9	39.2	1157.7	0.81	0.94
B15	2250	55	2166.13	25.5	1000.4	0.96	0.44
Average value						0.909	0.859
Coefficient of variation						0.133	0.259

From Table 4, it can be seen that the calculation method in this article can calculate the cracking angle of the web plate, and the calculated cracking initiation angle is relatively consistent with the actual cracking initiation angle in the experiment. From Figure 7, it can be seen that the calculation results of the French code formulas are mostly lower than the experimental values. At the same time, the absolute values of the calculation results in the French code are not significantly different when the cross-sectional area of the beam does not change. The calculation results obtained by the method in this article can mostly be very close to the experimental values, and the discreteness is smaller than the French code Therefore, the calculation method based on the modified compression field in this article is more reasonable.

Figure 7.

Comparison of theoretical calculations and experimental values: (a) Calculation methods in this paper, (b) French norms.

In order to further analyze the applicability of the formula proposed in this paper, the influence laws of four factors, such as shear span ratio, steel fiber content, web thickness, and compressive stress level, are analyzed, and the variation laws of the ratios between the calculated values of this calculation method and the calculated values of French norms and the experimental values are listed, as shown in Figures 8∼11.

Figure 8.

Ratio of calculated values to experimental values changes with shear span ratio: (a) Calculation methods in this paper, (b) French norms.

Figure 9.

Ratio of calculated values to experimental values changes with steel fiber content: (a) Calculation methods in this paper, (b) French norms.

Figure 10.

Ratio of calculated values to experimental values changes with web thickness: (a) Calculation methods in this paper, (b) French norms.

Figure 11.

Ratio of calculated values to experimental values changes with compressive: (a) Calculation methods in this paper, (b) French norms.

As shown in Figure 8, with the increase of shear span ratio, the French norms has a large error when the shear span ratio is small, that is, the French norms does not consider the factor of shear span ratio enough; However, the results of this method in this paper are in good consistency, that is, the shear span ratio has little influence on the accuracy thereof. As shown in Figure 9, with the change of steel fiber content, there is no divergence between them, which shows that the steel fiber content has little influence on the calculation accuracy of both; However, the error of the calculation method in this paper is smaller than that of the French norms under all the steel fiber content conditions, that is, the calculation method in this paper is more accurate than that of the French norms. As shown in Figure 10, with the change of web thickness, the accuracy of the calculation method in this paper is better than that in French norms, but the results of the former are smaller than the experimental values when the web is thicker. Considering the web thickness has certain influence on the accuracy of the calculation formula in this paper. As shown in Figure 11, the accuracy of French norms varies greatly with the change of compressive stress level, while the calculation method in this paper is more accurate.

Conclusion

(1) The proportion of the platform section in the load deflection curve of the test beam is greater than that of the rise section, UHPC-NC prestressed composite beams without web reinforcements show ductility during the whole beam failure process; However, the web, as the main part of shear force for UHPC-NC prestressed composite beams without web reinforcement, is the weakest joint with the top and bottom plates of UHPC I-beams, and the cracked web is easy to be “cut off’ at the joint.

(2) The “banding crack area” of the web of UHPC-NC prestressed composite beams without web reinforcement is closely related to the shear-span ratio, and the width of the banding area increases with the increase of the shear-span ratio. When the shear span ratio is small and the web is thin, the web cannot be full of shear diagonal cracks, and even the horizontal shear failure of the web occurs, which cannot fully utilize the shear resistance of the web.

(3) The test shows that the shear span ratio has a significant influence on the shear capacity of the test beam, and the ultimate shear capacity decreases with the increase of the shear span ratio. The content of steel fiber has a certain influence on the ultimate shear capacity of the test beams. The content of steel fiber increases from 1.8% to 2.0%, the ultimate shear capacity increases by 6.5%, the content of steel fiber increases from 2.0% to 2.5%, and the ultimate shear capacity increases by 2.8%; However, the content of steel fiber has little effect on the initial stiffness of the test beam. The web thickness has great influence on the ultimate shear capacity of the test beams, and the ultimate shear capacity of the test beams increases with the increase of web thickness. The ultimate shear capacity increases with the increase of compressive stress level.

(4) In this paper, the influence of top slab compression-shear area and bottom slab tension-shear area on the shear capacity of UHPC-NC prestressed composite beams without web reinforcement is considered, and the corresponding calculation method is given based on the modified compression field theory and the idea of sub-term superposition. The coefficient of variation of the formula proposed in the paper based on the 15 beams is 0.133. The calculation flow established in this paper can accurately calculate the ultimate shear capacity of UHPC-NC prestressed composite beams without web reinforcement.

Footnotes

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work was supported by the Ministry of Industry and Information Technology of the People’s Republic of China (2022YFC3801100), Ministry of Transport of the People’s Republic of China (2021-MS1021).

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