Behaviour and analysis of ultra high performance fibre reinforced concrete (UHPFRC) skew slabs

Engineering Structures 199 (2019) 109588

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Behaviour and analysis of ultra high performance fibre reinforced concrete (UHPFRC) skew slabs

T

⁎

Tianyu Xie , M.S. Mohamed Ali, Phillip Visintin School of Civil, Environmental and Mining Engineering, The University of Adelaide, South Australia 5005, Australia

A R T I C LE I N FO

A B S T R A C T

Keywords: Ultra-high performance fibre reinforced concrete Skew slab Closed-form equations Novel yield-line analysis Generic methods

This study presents an experimental and theoretical investigation of the performance of ultra-high performance fibre reinforced concrete (UHPFRC) skew bridge decks. Five simply-supported UHPFRC slabs with different skew angles were tested under monotonically increasing concentrated load. It was observed that an increase in the skewness of a slab led to a more unsymmetrical vertical deflection profile, and also resulted in a decrease in the magnitude of reaction forces at the acute corners and a corresponding increase in the reaction forces at the obtuse corners. The test results also show that, although some reduction in ductility was seen with increasing skewness of a slab, the specimen with over 25° skew angle still maintained a desirable load carrying capacity and ductility compared to those of a conventional RC straight slab. In addition to the experimental investigations, a fundamental mechanics based closed-form model was developed based on strain energy theorem to predict the performance of a UHPFRC skew slab under linear elastic material condition. At its ultimate limit state, a novel yield-line analysis using a numerically-generated mechanics based moment-rotation relationship was adopted to predict the ultimate load carrying capacity of a skew UHPFRC slab. Both the generic analytical procedures were compared and against test results obtained from this study, as well as those reported in literature, and the results show that the models developed can be applied to UHPFRC skew slabs.

1. Introduction Transport infrastructure is critical to the productivity of cities as it connects businesses, ideas, goods and services. Physical space constraints in urban areas mean it is often necessary to design bridges with non-orthogonal decks. When skew angles are small (< 20°), a skew slab performs similarly to an orthogonal slab in terms of the moment distribution, support reaction and shear force profiles [1,2], and in this scenario the distribution of shear and moment in the skew slab with a smaller skew angle can be easily defined from simple mechanics. However, for slabs with a large skew angle (20°), the reactions and distributions of shear and moment are significantly different form orthogonal slabs and as such more advanced analysis procedures are required [3–5]. For example, the findings of existing studies on skew plate elements reveal the decreases in the longitudinal moment and vertical deflection of a slab with a larger skew angle offsetting by the increasing transverse-moment [6–8]. The majority of existing codes and design guidelines for skew slabs adopt an approach in which design is first conducted assuming the slab is orthogonal and then empirical factors are applied to allow for the influence of skew angle. However, due to the empirical nature of the code approaches, none of them fully characterize the underlying ⁎

mechanisms of the actions in a skew slab and hence only provide conservative design. For example, AASHTO-2007 [9] suggests using a reduction factor for longitudinal bending moments while no magnification factors applied for shear forces. Canadian Standards Association, CSA-16 [10] imposes a limit for using an equivalent-beam method to design skewed bridge decks, but no additional guidance beyond this limit. Most of the existing theoretical work on skews bridge decks adopt finite element analysis (FEA) [1,2,11] or grillage analogy method [7,8,12,13], while yielding accurate results, these approaches can be complex and time consuming and do not provide mathematical expressions that can easily be generalised for guidelines or codes of practice. Ultra-high performance fibre reinforced concrete (UHPFRC) is characterized by its high compressive strength, superior ductility and non-negligible post-cracking tensile strength [14–16]. The fibres incorporated in UHPFRC not only prevent wider cracks [15,17,18] but also effectively act as additional reinforcements that resist stresses arising from the shear and torsion forces present in skew and curved elements [14,19]. These characteristics make UHPFRC a promising material for constructing structural elements such as beams [20,21], columns [22–24], frames [25], and slabs [26,27], with longer spans, reduced member sizes and increased design lives. Despite the benefits

Corresponding author.

https://doi.org/10.1016/j.engstruct.2019.109588 Received 10 September 2018; Received in revised form 31 May 2019; Accepted 23 August 2019 0141-0296/ © 2019 Elsevier Ltd. All rights reserved.

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m

Nomenclature

moment of resistance of slab per length crossing the yieldline in (mm/mm) P applied load RA reaction force at the centre of the supports A RB reaction force at the centre of the supports B R1,R2 ,R3,R 4 the reaction forces at the corners 1, 2, 3 and 4 T torsional moment TP torsional moment due to the applied load P only torsional moment due to the redundant action TB TTB T¯ torsional moment when TB = 1 t thickness of each slab V shear force VP shear force due to the applied load P only shear force due to the redundant action TB VTB v Poisson’s ratio of the UHPFRC w length the shorter edge of a slab α skew angle of a slab Δ vertical displacement of the load in the yield line analysis ΔB deflection caused by torsional moment (TB ) δb virtual deflection when TB = 1 θ rotation of slab due to the applied load λ factor to represent the location of the concentrated load with reference to central line of the slab reinforcement ratio in the longitudinal direction ρl ρt reinforcement ratio in the transverse direction

List of notations

B0 Ec EI EI / GJ Fz f c' fr fst GJ L l y − max l y − max l y − min M MP MTB Mx My ¯ M

projection of supported edge of slab on the line that is perpendicular to the central-line of slab elastic modulus of the UHPFRC bending stiffness bend-to-twist ratio force at z direction axial compressive strength of the UHPFRC modulus of rupture of the UHPFRC splitting tensile strength of the UHPFRC torsional stiffness span length of a slab or the length of the longer edge of a slab length of yield-line upper bound of yield-line length lower bound of yield-line length bending moment bending moment due to the applied load P only bending moment due to the redundant action TB moment in x direction moment in y direction bending moment when TB = 1

understand the effect of slab skewness on the performance of UHPFRC slabs. (2) Yield information on the effect of skew angle on the structural response of simply-supported UHPFRC skew slabs at both the serviceability- and ultimate- limit states. (3) Establish generic approaches with strong physical significance for analysing the behaviour of UHPFRC skew slabs and also conventional RC skew slabs at both the serviceability- and ultimate- limit states. (4) Assist in developing design guideline for UHPFRC skew slabs that avoids the need to conduct multiple large-scale tests. This outcome arises due to the generic nature of the analytical work including both the closed-form solutions and the yield-line analysis incorporating the displacement based segmental analytical approach.

of UHPFRC application, it is yet to receive attention in the application to skew slabs. To address the research gaps identified above, in this research work, experimental and analytical investigations are undertaken to: (1) Experimentally study the behaviour of simply-supported UHPFRC slabs with different skew angles at both their serviceability- and ultimate- limit states. (2) Derive a closed-form solution based on the fundamental mechanics for quantifying the behaviour of skew decks and to model the structural response of a skew slab under their linear elastic material conditions. The advantage of this closed-form solution is that it is less demanding on computational effort and therefore may have greater potential to be incorporated into practical design guidelines. (3) Carry out a novel yield-line analysis for determining the ultimate load carrying capacity of slabs with different skewness at their ultimate limit state, where the peak moment of each yield-line section in this analysis is generated using a generic partial-interaction segmental analytical approach [28].

2. Test program 2.1. Materials 2.1.1. UHPFRc All skew slabs and material test specimens were manufactured using a UHPFRC mix developed at the University of Adelaide as a part of a project aimed at producing UHPFRC using only conventional concrete mixing ingredients, equipment and curing techniques [29]. The mix proportions are given in Table 1 in which copper coated, straight steel fibres, with 13 mm nominal length and 0.2 mm nominal diameter were added at a volume fraction of 2.25% to ensure the homogenous dispersion [30,31]. The manufacturer-reported elastic modulus and ultimate tensile strength of the steel fibres are 210 GPa and 2850 MPa, respectively.

To achieve these objectives, five UHPFRC slabs with a skew angle of 0, 15, 25, 35 or 45° are tested under monotonically increasing concentrated load applied at various in-plane positions. Deflection profiles and reactions at the supports of each slab are experimentally recorded. The experimental results are then used to validate the predictions using the closed-form equation and the novel yield-line method. It is expected that the outcomes of this work will: (1) Initiate experimental investigations of UHPFRC skew slabs and Table 1 Mix proportion of UHPFRC.

UHPFRC

Sulfate Resistant Cement (kg/m3)

Silica fume (kg/m3)

Sand (kg/m3)

Water (kg/m3)

SP* (kg/m3)

Fibre** (kg/m3)

834.6

222.0

1056.6

126.5

64.1

246.2

* Containing 70% water. ** At 2.25% volume fraction. 2

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determined according to ASTM C496-04 [34] and ASTM C78-18 [35], respectively. Fig. 5 shows the typical failure modes of the UHPFRC specimens for indirect tensile and flexural tests. The constitutive relationships of the UHPFRC are shown in Fig. 6, in which Fig. 6(a) shows the axial compressive stress-strain relationship, and Fig. 6 (b) shows the direct tensile stress-strain relationship prior to cracking and the stress-crack width relationship post-cracking.

In preparation of the UHPFRC, two binders were utilized, sulphate resisting cement and silica fume both sourced from Adelaide Brighton Cement Ltd. The physical and mechanical properties of these two binder materials can be found in references [16,29]. The only aggregate in the mix was natural river sand with a nominal particle size of 4 mm. To ensure sufficient flow of the concrete a third generation superplasticizer (Sika viscocrete 10) was used. To produce the UHPFRC all the dry materials (sand and binders) were mixed in a rotating pan mixer with 80 L capacity for approximately 5 min. The water and superplasticizer were then gradually poured into the mixer and mixing continued until the mortar exhibited the desired flow. Finally, the discrete steel fibres were added and mixed for further ten minutes to form the UHPFRC.

3. Experimental results and discussion 3.1. Deflection profile for slab at the linear elastic stage The effect of skew angle on the deflection profile of the skew slabs loaded at point A, C and E are shown in Figs. 7–9. These deflection profiles were selected for comparison because they corresponded to the following cases: (1) close to an acute corner (at point A); (2) close to an obtuse corner (at point C) and (3) at the centre of the slab (at point E). For load applied at points A and C, it is evident from Figs. 7 and 8 that even for the straight slab, the deflection was unsymmetrical. This behaviour was observed because of the out-plane moment induced by the load eccentricity and the vertical deflection at any of the in-plane points of the five slabs decreased with an increase in the distance between the point and the loading position. As can also be seen from Figs. 7 and 8, the difference in the deflections among all companion reference points increased with increasing skew angle. Specifically, with an increase in the skewness of the slab, for both of the loading cases (at points A and C), the vertical deflections measured adjacent to the acute corners (points (x1, y1) and (x4, y4)) increased, on the contrary, the vertical deflections measured adjacent to the obtuse corners (i.e. points (x1, y1) and (x4, y4) decreased. For the centrically-loaded slabs (load applied at point E), it can be seen from Fig. 9 that the degree of non-symmetricity of the deflection profile increased with increasing skew angle. Also shown in Fig. 9, it can be seen that regardless of the change in the skewness of the slabs, the maximum deflection of each centrically-loaded slab is still located at the centre of the slab where the maximum bending moment occurred. Similar to those observed in the tests of the slabs loaded at the points A and C, an increase in skewness of the slab also led to larger deflections at locations adjacent to the obtuse corners of the slabs (i.e. points (x1, y4) and (x4, y1)), whereas the deflections measured at the locations adjacent to the acute corners of the slabs (i.e. points (x1, y1) and (x4, y4)) decreased with increasing skew angle. These observations can be attributed to the fact that the out-plane moment (anti-clockwise twist along each supporting edge) increases with increasing skewness of the slab, that is, the regions near each of the obtuse corners were pushed down whereas the regions close to the acute corners were lifted up [1,8]. This in turn caused the more significantly unsymmetrical deflection profiles of the slab with a lager skewness. The above observations also indicate that the deflection caused by bending moment

2.1.2. Welded steel wire mesh Welded steel wire mesh with a 4 mm diameter and a horizontal and vertical spacing of 50 mm was used as reinforcement in the slabs. The tensile stress-strain relationship of the steel used in the mesh is shown in Fig. 1 and Fig. 2(a)–(e) illustrates the arrangement for the mesh in each slab. 2.2. Test specimens, set up, instrumentation and test procedure 2.2.1. Slab test Five simply-supported slabs with a skew angle of 0, 15, 25, 35 or 45° were tested under monotonically increasing concentrated load, where each specimen has the thickness of 60 mm, the free edge of 1420 mm and the supported edge of 720 mm. To establish the deflection profiles of each slab test, a total of 17 LVDTs were used; 16 were used to record the vertical deflection profile of the slab (Fig. 3a and c) and the final LVDT was located on the loading ram to measure the ram displacement. A load cell was placed under all four corners of the slabs to measure the vertical reaction forces. During testing, each slab was initially loaded in the elastic region, to a maximum load of 20 kN at a load rate of 5 kN/min. The maximum elastic loading limit of 20 kN was adopted to avoid the yielding of reinforcement and the magnitude of the load was found based on a lower bound prediction using the PI-based yield line analysis for the most extreme case (i.e. centrically-loaded S45), in which yielding of the reinforcement was found to initiate at a load of 28.7 kN. The elastic response of the slab was recorded five times by shifting the loading points to five different in-plane points in Fig. 3(b). Finally, the slabs were tested to failure by applying the point load at the centre of the slabs (point E as illustrated in Fig. 3(b)). When loading to failure, the load was continuously applied at a rate of 5 kN/min until the peak load was attained and thereafter the load was applied under displacement control at a displacement rate of 0.1 mm/min until failure to obtain the complete load-deflection relationship. Note that the maximum vertical deflection of each slab test was directly measured by the LVDT attached to the loading ram.

400

Tensile stress (MPa)

2.2.2. Material tests for UHPFRC The axial compressive strength (f′c), elastic modulus (Ec) and Poisson’s ratio (v) of the concrete were established according to ASTM standard C39/C39M-05 [32] and ASTM standard C469/C469m-14 [33], respectively. For this testing, cylindrical specimens of 100 mm in diameter and 200 mm in height were used and tests were conducted using a 3000 kN universal testing machine at a loading rate of 0.3 MPa per second. Total deformation of the cylinder was measured with two LVDTs mounted on opposite faces of the specimen. The tensile behaviour of the UHPFRC, including the stress-strain relationship prior to cracking, and the post-cracking stress-crack-width behaviour was determined using a dog-bone shaped specimens shown schematically in Fig. 4. The details of the dog-bone specimens and test set-up can be found in Singh et al. [31]. The splitting tensile strength (fst) and the modulus of rupture (fr) reported in Table 2 were

300 200 100 0 0

6

Strain (%)

12

Fig. 1. Stress-strain relationship of steel mesh. 3

18

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Fig. 2. Slab geometries and steel mesh arrangement.

forces measured at diagonally opposite corners were nearly identical, which indicates the symmetrically distributed flexure, torsion and shear on the slab. Similar to the those observed in the eccentrically-loaded cases, the increase in the skewness of the slab also resulted in decreases in the reaction forces measured at the acute corners (i.e. corners 2 and 4) and increases in the reaction forces measured at the obtuse corners. Note that for the centrically-loaded S0 slab, there were slight differences among the measured reaction forces. This unexpected observation might be attributed to the unintended load eccentricity generated on loading ram or due to slight slippage between the load cells and the supports. It is worth noting that the offset is a minor one (less than 5% of the total span length). It may also be noted that this minor error occurred only in the case E of specimen S0; there were no such discrepancies observed in all the other tests.

predominates the vertical deflection profile of a skew slab, whereas the effect of torsional is only minor. It is also worth mentioning that for all the loading cases, the slabs with the skew angle smaller than 25° exhibited comparable deflection profiles to that of their straight slab counterpart, and this observation agrees well with those reported in previous studies on conventional RC skew slabs [2,6,36].

3.2. Reaction force at the linear elastic region The reaction forces at the four corners of each slab within the elastic region were experimentally-recorded using the load cells. Figs. 10–12 show the relationship between the load applied and the corner reaction forces measured, and it also displays the sum of the four corner reaction forces for the cases of loads applied at point A, C and E. It can be seen that the sum of the four corner reaction forces is equal to the applied load and this indicates the load cells used in the present study can provide valid and reliable measurements of the corner reaction forces. Figs. 10 and 11 show that when loaded at points A and C, due to the torsional moment induced by the load eccentricity, the reaction forces measured at four corners were all different and the larger reaction forces were measured at the corners close to the loading point. The differences among the reaction forces measured at each corner changed with the change in the skewness of the slab, where an increase in the skew angle of the slab led to decreases in the reaction forces measured at the acute corners (i.e. corners 2 and 4) and increases in the reaction forces measured at the obtuse corners (i.e. corners 1 and 3). As discussed in Section 3.1, this is due to the increased torsional moment generated in the plane of the slabs as the consequence of the increased skew angle which pushed the obtuse corners down (resulting in an increase in the reaction force) and lifted the acute corners up (resulting in a reduction in the reaction force). For the cases of the slabs loaded at its centre (at point E), it can be seen from Fig. 12 that the reaction

3.3. Yield-line pattern of skew slabs In addition to the tests of the slabs under linear elastic material conditions, all the slabs were also loaded at their centre (i.e. point E) until failure. Fig. 13 illustrates the final failure patterns of the slabs where it is evident that the yield-lines originated from the centre of the slabs and propagated along central-line that was parallel to the supported edges. This indicates that for a one-way slab, flexural failure dominates even for skew slabs. Based on the failure patterns shown in Fig. 13, the upper bound of the length of the yield-line is the length of the supported edge of the slab and its lower boundary is the projection of the short edge of slab on the line that is perpendicular to the centralline of the slab. The lower and upper values of the length of the yieldline are used later for the yield-line analysis.

4

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Fig. 3. Slab test set up: (a) set up; (b) loading positions; (c) LVDTs locations.

observed when compared those measured from S0 and S45 series of specimens. No significant change in load carrying capacity was observed with increasing skewness of the slabs. Note that, although the ductility of the slab reduced slightly with increasing skewness of the slabs, all the specimens exhibited desirable ductility as that their ductility factors (Δfailure/Δyield) are all over 2.5. Note that some experimental variations were also observed, for instance the slightly higher load carrying capacity of the S35 series of slabs compared to those of its companion specimens. This was mostly caused by the presence of the

3.4. Complete load-deflection relationship of centrically-loaded skew slabs The complete load-deflection relationships of all the slabs tested to failure are illustrated in Fig. 14. The load-deflection relationship of all the slabs exhibited nearly identical trend up to the peak load. However, the skewness of the slab significantly affected the post-peak load-deflection response, in which the increase in the skewness resulted in a more rapid descending branch on its load-deflection curve. An up to around 115% decrease in the maximum mid-span deflection was

Fig. 4. Dogbone shaped specimen for direct tension test: (a) specimen details; (b) test setup. 5

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Table 2 Material properties of UHPFRC. Compressive strength (MPa)

Elastic modulus (GPa)

Direct tensile strength (MPa)

Splitting tensile strength (MPa)

Modulus of rupture (MPa)

162.1

42.3

6.3

12.1

17.8

Fig. 5. Test specimens for tensile properties of UHPFRC: (a) indirect tensile test; (b) flexural test.

Compressive stress (MPa)

180 150 120 90 60 30 0 0

1

2

Strain (%)

3

4

(a) 8

Tensile stress (MPa)

Tensile stress (MPa)

8 6 4 2 0

6 4 2 0

0

0.05

0.1

0.15

Strain (%)

0.2

0.25

0

1

2

3

Crack width (mm)

4

5

(b) Fig. 6. Materials properties: (a) compressive stress-strain relationship of UHPFRC; (b) direct tensile characterizations of UHPFRC under direct tension.

randomly distributed fibres in the specimens whose orientations and dispersions were hard to control during the casting process of the specimens. To show the improvements of the performance of the skew slab by using the UHPFRC materials, the result of a similar test on a conventional simply-supported reinforced concrete (RC) skew slab (series RC slab S2-502) subjected to a point load as reported in Kabir et al. [37] is adopted to compare with the results of the current study. Moreover, this test result is also used later in this paper together with the current test results to validate the proposed analytical models. The

material properties and geometry of this RC slab (S2-502) are reported in Table 3 and the load- mid-span deflection relationship of the slab is shown in Fig. 14. As can be seen from Table. 3 and Fig. 14, the S2-502 slab reported in [37] with around 67% larger cross-sectional area, approximately 5 times higher tensile reinforcement ratio in the longitudinal direction and a only slightly shorter span length (i.e. 1200 mm vs. 1400 mm), exhibited less than 6% improvement in load carrying capacity and up to 70% higher mid-span deflection at the peak load compared to those of the two slabs S35 and S45 with similar skew 6

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2 0

x1

x2 x3 Span length

Deflection, (mm)

6

6

Skew angle = 15º @ point A

2 0

x4

x1

x2 x3 Span length

6

y1 y2 y3 y4

4

4

6

y1 y2 y3 y4

Skew angle = 35º @ point A

2 0

x1

x3 x2 Span length

4

y1 y2 y3 y4

Skew angle = 25º @ point A

2 0

x4

y1 y2 y3 y4

4

x1

x2 x3 Span length

x4

Skew angle = 45º @ point A

2 0

x4

Deflection, (mm)

Skew angle = 0º @ point A

Deflection, (mm)

4

y1 y2 y3 y4

Deflection, (mm)

Deflection, (mm)

6

284 x1

0

x2 x3 Span length

x4

Fig. 7. Deflection profiles of the skew slabs loaded at point A: (a) S0; (b) S15; (c) S25; (d) S35; (e) S45.

2 0

x1

x2 x3 Span length

Deflection, (mm)

6 4

x4

6 4

6

y1 y2 y3 y4

Skew angle = 15º @ point C

2 0

x1

x2 x3 Span length

6

y1 y2 y3 y4

Skew angle = 35º @ point C

2 0

x1

x3 x2 Span length

y1 y2 y3 y4

4

4

y1 y2 y3 y4

Skew angle = 25º @ point C

2 0

x1

x2 x3 Span length

x4

Skew angle = 45º @ point C

2 0

x4

x4

Deflection, (mm)

Skew angle = 0º @ point C

Deflection, (mm)

4

y1 y2 y3 y4

Deflection, (mm)

Deflection, (mm)

6

x1

x2 x3 Span length

x4

Fig. 8. Deflection profiles of the skew slabs loaded at point C: (a) S0; (b) S15; (c) S25; (d) S35; (e) S45.

problem using the sign convention and the coordinate system shown in Fig. 15. Based on the equilibrium of moments and forces: ∑ Mx = 0 , ∑ My = 0 , and ∑ Fz = 0 , by applying the virtual work and force methods considering the torsional moment on one of the supports to be redundant (TB ), the reaction force at the centre of the supports A (RA) and B (RB) is:

angles in the presented study. These comparisons indicate that, due to its superior material properties, the application of UHPFRC materials in a skew slab can significantly reduce the size of the specimen as well as the ratio of reinforcements required to achieve the similar performance to that of conventional concrete structures. 4. Analytical study 4.1. Development of closed-form mathematic solutions for a skew UHPFRC bending element

RA = (1 − λ )·P

(1)

RB = λ·P

(2)

where λ represent the location of the concentrated load with reference to central line of the slab and is expressed as:

Even under simply-support boundary conditions, a skew slab is statically indeterminate as a result of the associated torsional moments that develop. In this section, a closed-form solution is developed using virtual work and force methods to solve the indeterminate skew slab

λ=

7

x L

(3)

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2 0

x2 x3 Span length

x1

Deflection, (mm)

4

0

x2 x3 Span length

x1

Skew angle = 35º @ point E

2 0

x3 x2 Span length

x1

0

x1

x2 x3 Span length

x4

Skew angle = 45º @ point E

2 0

x4

Skew angle = 25º @ point E

2

y1 y2 y3 y4

4

y1 y2 y3 y4

4

x4

6

y1 y2 y3 y4

4

Skew angle = 15º @ point E

2

x4

6

6

y1 y2 y3 y4

Deflection, (mm)

4

6

Skew angle = 0º @ point E

Deflection, (mm)

y1 y2 y3 y4

Deflection, (mm)

Deflection, (mm)

6

x2 x3 Span length

x1

x4

Fig. 9. Deflection profiles of the skew slabs loaded at point E: (a) S0; (b) S15; (c) S25; (d) S35; (e) S45.

redundant torsional moment (TB ) are:

The shear force, bending and torsional moments due to only the applied load P when 0 ≤ x ≤ λ are:

MP = x·(1 − λ )·PL

(4)

TP = 0

(5)

VP = (1 − λ )·P

(6)

MTB = −TB sinα

(10)

TTB = −TB cosα

(11)

VTB = 0

(12)

Based on the virtual work method:

and for λ ≤ x ≤ 1 (7)

TP = 0

(8)

VP = −λ·P

(9)

where δb is the virtual deflection when TB = 1; and ΔB is the deflection caused by torsional moment (TB ), and

δB =

The shear force, bending and torsional moments caused by the 2

3

4

Ram

25

SUM

Reaction force (kN)

1

20 Skew angle=0° @ point A

15 10 5

1

2

3

4

∫L

0

¯2 M dx + EI

Ram

∫L

0

Skew angle=15° @ point A

15

T¯ 2 dx GJ

10 5

10

15

Applied load (kN)

20

25

2

3

5

10

15

Applied load (kN)

(a)

1

2

3

4

Ram

Skew angle=35° @ point A

10 5 0

5

20

25

0

5

10

1

2

3

4

Ram

SUM

20 Skew angle=45° @ point A

15

5

10

15

Applied load (kN)

20

25

10 5 0

(d)

5

10

15

Applied load (kN)

20

25

(e)

Fig. 10. Reaction forces of the skew slabs loaded at point A: (a) S0; (b) S15; (c) S25; (d) S35; (e) S45. 8

15

Applied load (kN)

0 0

SUM

10

(c) 25

SUM

20 15

Ram

Skew angle=25° @ point A

15

(b) 25

4

0 0

Reaction force (kN)

5

1

20

0 0

(14)

25

SUM

20

0

Reaction force (kN)

Reaction force (kN)

25

(13)

Reaction force (kN)

MP = λ·(1 − x )·PL

δB TB + ΔB = 0

20

25

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2

3

4

Ram

25

SUM

20 Skew angle=0° @ point C

15 10 5

1

2

3

4

Ram

25

SUM

Reaction force (kN)

1

Reaction force (kN)

Reaction force (kN)

25

20 Skew angle=15° @ point C

15 10 5

0 10

15

20

Applied load (kN)

Reaction force (kN)

25

25

1

2

3

4

5

Ram

10

15

Applied load (kN)

25

SUM

20 Skew angle=35° @ point C

15 10 5 0 0

5

10

3

4

Ram

SUM

10 5 0

0

Reaction force (kN)

5

2

Skew angle=25° @ point C

15

0 0

1

20

15

Applied load (kN)

20

20

25

1

2

0

3

4

5

Ram

10

15

Applied load (kN)

20

25

SUM

20 Skew angle=45° @ point C

15 10 5 0

25

0

5

10

15

Applied load (kN)

20

25

Fig. 11. Reaction forces of the skew slab loaded at point C: (a) S0; (b) S15; (c) S25; (d) S35; (e) S45.

ΔB =

∫L

0

¯ MM dx + EI

∫L

0

¯ TT dx GJ

TB = −TA = −

(15)

¯ and T¯ are the bending and torsional moment when TB = 1. where M By substituting Eqs. (10)–(12) into Eqs. (14) and (15) respectively, δB and ΔB are: EI L δB = sin2 α·⎛1 + cot2 α ⎞· GJ ⎝ ⎠ EI ΔB =

∫L

0

¯ MM dx + EI

∫L

0

(

)

(18)

The reaction force at the corners 1 (R1), 2 (R2), 3 (R3), and 4 (R4) can be calculated using TB and the reaction force (i.e. RA and RB ) at the centre of each support:

(16)

RA T cosα L−x ⎡ x cotα ·P·⎢1 − − B = EI 2 B0 2L ⎢ B0 1 + GJ cot2 α ⎣

R1 =

¯ TT PL2 ·(1 − λ )·λ dx = −sinα· GJ 2EI

ΔB λ (1 − λ )·PL = EI δB 2·sinα· 1 + GJ cot2 α

(

(17)

)

⎤ ⎥ ⎥ ⎦

(19)

The torsional moment at support A (TA) and B (TB) is therefore: 3

4

Ram

25

SUM

20 Skew angle=0° @ point E

15 10 5

2

3

4

Ram

25

SUM

20 Skew angle=15° @ point E

15 10 5 0

0

5

10

15

Applied load (kN)

25

20

1

25

2

3

4

5

Ram

SUM

10

15

Applied load (kN)

25

20 Skew angle=35° @ point E

15

1

2

3

10 5 0 5

10

15

Ram

SUM

Skew angle=25° @ point E

15 10 5

20

1

25

2

3

4

0

5

Ram

SUM

10

20 Skew angle=45° @ point E

15

Applied load (kN)

20

25

10 5 0

5

10

15

Applied load (kN)

20

25

Fig. 12. Reaction forces of the skew slab loaded at point E (a) S0; (b) S15; (c) S25; (d) S35; (e) S45. 9

15

Applied load (kN)

0 0

4

20

0 0

Reaction force (kN)

0

1

Reaction force (kN)

2

Reaction force (kN)

1

Reaction force (kN)

Reaction force (kN)

25

20

25

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Fig. 13. Yield-line patterns for the skew slabs.

50

Load (kN)

40

S0 S15 S25 S35 S45 S2-502

30 20 10 0 0

10

20

30

40

50

Deflection (mm)

60

Fig. 14. Complete load-deflection relationship of the skew slabs loaded at the centre (i.e. point E).

R T cosα L−x ⎡ x cotα ·P·⎢1 + R2 = A + B = EI 2 B0 2L ⎢ B0 1 + GJ cot2 α ⎣

(

R3 =

⎡ RB T cosα x (L − x )cotα ·P·⎢1 − + B = EI 2 B0 2L ⎢ B0 1 + GJ cot2 α ⎣

(

)

⎡ R T cosα x (L − x )cotα ·P·⎢1 + R4 = B − B = EI 2 B0 2L ⎢ B0 1 + GJ cot2 α ⎣

(

)

Fig. 15. Sign convention and coordinate system of simply-supported skew slab loaded at its central line.

⎤ ⎥ ⎥ ⎦

(20)

⎤ ⎥ ⎥ ⎦

)

Table 4 Materials properties for the validation of the analytical model.

UHPFRC

λ

(

2 1+

EI cot2 α GJ

)

L (mm)

w (mm)

t (mm)

EI/GJ

0.233

1420

720

60

0.611

(21)

T=−

⎤ ⎥ ⎥ ⎦

λ (1 − λ )·P·L·cotα

(

2 1+ (22)

EI cot2 α GJ

)

(24)

V = (1 − λ )·P

where B0 is the projection of supported edge of slab on the line that is perpendicular to the central-line of slab, as shown in Fig. 15. The generic expressions of bending moment (M), torsional moment (T) and shear force (V) along the central line of the slab when For 0 ≤ x ≤ λ is

M = (1 − λ )(x −

ν

(25)

and for λ ≤ x ≤ 1

⎡⎛ 1 M = ⎢ ⎜1 − EI ⎢⎜ 2 1 + GJ cot2 α ⎣⎝

(

)·P·L

T=−

(

)

⎞⎤ ⎟ ⎥·λ·P·L ⎟⎥ ⎠⎦

(26)

λ (1 − λ )·P·L·cotα

(

2 1+

(23)

)

⎞ ⎛ λ ⎟ − ⎜x − EI ⎟ ⎜ 2 1 + GJ cot2 α ⎠ ⎝

EI cot2 α GJ

)

(27)

Table 3 Details of S2-502 series of specimen reported in [41]. Material properties

S2-502

ν

L (mm)

w (mm)

t (mm)

EI/GJ

α

f'c (MPa)

ft (MPa)

ρl (%)

ρt (%)

0.2

1200

1000

75

0.6

40°

30.5

2.9

0.86

0.48

10

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Table 5 Comparison between experimental and analytical results for the slabs loaded at point D. Corner

Reaction force

Deflection

Exp (kN)

Mod (kN)

Mean

SD

Exp (mm)

Mod (mm)

Mean

SD

S0

1 2 3 4

6.82 8.23 1.94 2.71

7.46 7.46 2.54 2.54

1.09 0.91 1.31 0.94

0.46 0.54 0.43 0.12

2.21

2.08

0.94

0.09

S15

1 2 3 4

8.87 6.11 4.19 1.25

8.96 5.96 4.04 1.04

1.01 0.98 0.96 0.83

0.06 0.11 0.11 0.15

2.12

1.94

0.92

0.12

S25

1 2 3 4

9.63 5.55 3.88 0.51

9.75 5.18 4.82 0.25

1.01 0.93 1.24 0.50

0.09 0.26 0.67 0.18

1.62

1.80

1.11

0.13

S35

1 2 3 4

9.25 6.20 4.29 0.04

10.31 5.39 4.61 0.03

1.11 0.87 1.08 0.75

0.75 0.58 0.23 0.01

1.47

1.29

0.87

0.13

S45

1 2 3 4

9.63 5.75 4.55 −0.20

10.67 5.75 4.25 −0.67

1.11 1.00 0.93 3.32

0.74 0.00 0.21 0.33

1.15

0.99

0.86

0.11

Table 6 Comparison between experimental and analytical results for the slabs loaded at point E. Corner

Reaction force

Deflection

Exp (kN)

Mod (kN)

Mean

SD

Exp (mm)

Mod (mm)

Mean

SD

S0

1 2 3 4

4.39 5.53 4.30 5.43

5.00 5.00 5.00 5.00

1.14 0.90 1.16 0.92

0.43 0.38 0.49 0.31

2.71

2.73

1.00

0.01

S15

1 2 3 4

7.03 2.81 7.00 2.67

6.81 2.95 6.81 2.95

0.97 1.05 0.97 1.10

0.16 0.10 0.14 0.20

2.76

2.56

0.93

0.14

S25

1 2 3 4

8.38 1.65 7.90 1.51

8.02 1.98 8.02 1.98

0.96 1.21 1.01 1.31

0.26 0.24 0.08 0.33

2.53

2.33

0.92

0.14

S35

1 2 3 4

8.59 1.27 8.83 0.74

8.76 1.24 8.76 1.24

1.02 0.98 0.99 1.67

0.12 0.02 0.05 0.35

2.35

2.24

0.95

0.08

S45

1 2 3 4

9.82 0.48 9.18 0.65

9.24 0.76 9.24 0.76

0.94 1.58 1.01 1.18

0.42 0.20 0.04 0.08

2.22

2.03

0.91

0.14

Table 7 Validation of the proposed model using the result from [41]

θ=

Mid-span deflection at 16.04 kN

S2-502

Exp (mm)

Mod (mm)

Mean

SD

1.74

1.72

0.99

0.014

(

)

⎤ ⎥ ⎥ ⎦

(29)

and for λ ≤ x ≤ 1

θ=

V = −λ·P

⎡ Pl3 3·λ (1 − 2·x ) (1 − λ )·⎢ (2λ − λ2 − 3·λ2) − EI 6EI ⎢ 2 1 + GJ cot2 α ⎣

(28)

Pl3 ⎡ 3·(1 − λ )(1 − 2·x ) ⎤ ⎥ λ·⎢ (2 − 6·x + 3·x 2 + λ2) − EI 6EI ⎢ 2 1 + GJ cot2 α ⎥ ⎣ ⎦

(

)

(30)

Similarly, the generic expression of the deflection (Δ ) along the central line of the slab when 0 ≤ x ≤ λ is:

The generic expression for the rotation (θ ) along the central line of the slab when For 0 ≤ x ≤ λ is:

Δ=

11

⎡ 3·λ (1 − x ) PL3 (1 − λ )·x·⎢ (2λ − λ2 − x 2) − EI 6EI ⎢ 2 1 + GJ cot2 α ⎣

(

)

⎤ ⎥ ⎥ ⎦

(31)

Engineering Structures 199 (2019) 109588

T. Xie, et al.

0.8

0.4

P=1 @ centre; Skew angle=45°; L=1.42 m; W=0.72 m; t=0.06 m

EI/GJ=0.2

0.4

Moment (kNm)

Moment (kNm)

P=1 @ centre; L=1.42 m; W=0.72 m; t=0.06 m; EI/GJ=0.6

0 -0.4

Simply supported (0 degree)

Skew angle=30 degree

Skew angle=60 degree

Skew angle=45 degree

Fixed-end (90 degree)

-0.8

0.284

EI/GJ=0.6

EI/GJ=0.8

0.2

0

-0.2

0.568

0.852

1.136

Span length (m)

Moment (kNm)

0

EI/GJ=0.4

0.5 0.4 0.3 0.2 0.1 0 -0.1 -0.2 -0.3

1.42

0

0.284

0.568

0.852

Span length (m)

1.136

1.42

P=1; EI/GJ=0.6; Skew angle=45 degree; L=1.42 m; W=0.72 m; t=0.06 m

x/L=0.8

0

0.284

x/L=0.6

0.568

x/L=0.4

0.852

Span length (m)

x/L=0.2

1.136

1.42

0

-0.1

Torsional moment (kNm)

Torsional moment (kNm)

Fig. 16. Influence diagram for bending moment by: (a) varying the skew angle; (b) varying the bend-to-twist ratio (EI/GJ); (c) varying the loading position at the central line.

P=1 kN @ centre L=1.42 m W=0.72 m t=0.06 m

-0.2 EI/GJ=0.2 EI/GJ=0.8

-0.3 0

10

20

EI/GJ=0.4 EI/GJ=1

30

40

50

EI/GJ=0.6

60

Skew angle (degree)

70

80

0 -0.05 P=1 kN L=1.42 m W=0.72 m t=0.06 m EI/GJ=0.6

-0.1 -0.15 x/L=0.1 x/L=0.4

-0.2

90

0

10

20

x/L=0.2 x/L=0.5 30

40

50

x/L=0.3 60

Skew angle (degree)

70

80

90

Shear force (kN)

1.5

Torsional moment (kNm)

Fig. 17. Influence diagram for torsional moment by: (a) varying the bend-to-twist ratio (EI/GJ); (b) varying the loading position at the central line.

P=1; EI/GJ=0.6; Skew angle=45º; L=1.42 m; W=0.72 m; t=0.06 m

1 0.5 0 -0.5 -1

x/L=0.5 x/L=0.2

-1.5 0

0.284

x/L=0.4 x/L=0.1

0.568

0.852

Span length (m)

x/L=0.3

1.136

1.42

0

-0.1

P=1 kN @ centre L=1.42 m W=0.72 m t=0.06 m

-0.2 EI/GJ=0.2 EI/GJ=0.8

-0.3 0

Fig. 18. Influence diagram for shear force by varying the loading position at the central line.

10

20

EI/GJ=0.4 EI/GJ=1

30

40

50

EI/GJ=0.6

60

Skew angle (degree)

70

80

90

Fig. 19. Influence diagram for deflection by: (a) varying the skew angle; (b) varying the span length.

12

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Table 8 Summary of the results from yield–line analysis. Pexp (kN)

S0 S15 S25 S35 S45

Upper bonds

34.0 34.2 32.9 36.2 34.2

Lower bonds

ly-max (mm)

My-max (kN·m)

Pmod (kN)

Mean

SD

ly-min (mm)

My-min (kN·M)

Pmod (kN)

Mean

SD

720 720 720 720 720

11.4 11.1 10.8 10.5 10.5

33.0 32.2 31.4 30.5 30.5

1.0 0.9 1.0 0.8 0.9

0.7 1.4 1.1 4.0 2.6

720 695 653 590 509

11.4 10.9 10.2 9.3 7.9

33.0 31.5 29.4 26.8 22.8

0.97 0.92 0.89 0.74 0.67

0.69 1.91 2.45 6.64 8.03

such as for R4 of S25 loaded at point D, R4 of S45 loaded at point D, R4 of S35 loaded at point E and R2 of S45 loaded at point E. As discussed in Sections 3.1 and 3.2, unlike the experimental variation seen in the statically determinate case of the straight slab S0 in which the reaction forces at the supports are only related to the geometry of the specimen, the reaction forces at the supports of the statically indeterminate skew slabs system are related not only to the geometry but also the material properties. Therefore, the unexpected errors seen between the experimental and theoretical results of the skew slabs could be mostly attributed to the fact that the fibres distributions and orientations in the specimens were rather random, leading to the inhomogeneous material properties, in which the theoretical models assumed the homogenous material properties So as to establish its generic nature and reliability, the analytical models were also used to simulate the response of a skew slab tested by [37]. The details of this experiment are shown in Section 3.4 and the load-mid-span deflection curve and the marital properties used for the model validations are shown in Fig. 14 and reported in Table 3, respectively. Table 7 presents the comparison between the experimentally-obtained and the predicted mid-span deflections within the linear elastic region and the statistics presented in this table indicates that the predictions correlate well with the test results.

Fig. 20. Yield-line analysis.

and for λ ≤ x ≤ 1

4.2. Generation of influence line using the analytical method

⎡ 3·x (1 − λ ) (1 − x )·λ·⎢ (2x − x 2 − λ2) − Δ= EI 6EI ⎢ 2 1 + GJ cot2 α ⎣ PL3

(

)

⎤ ⎥ ⎥ ⎦

Using the generic expressions developed [Eqs. (11) and (15) to (18)] for calculating bending moment, torsional moment and shear force, the values for the M, T and V are calculated for simply-supported skew slabs with different skew angles α (i.e. 0–180°) and different dimensionless bend-to-twist ratios EI (i.e. 0–10) which were subjected to a unit load GJ (P) moving along the central-line of the skew slabs. The calculated values are plotted where the unit load is applied and subsequently, the obtained value of M, T, V and Δ for each point along the central-line of the skew slabs is used to generate the influence line diagrams as shown in Fig. 16(a)–(c). As can be seen from Fig. 16(a), the slab with a skew angle of 90° performs identically to a fixed-end straight slab with infinite rigidity at the supports, in which the maximum moment and deflection of the centrically -loaded slab with a skew angle of 90° are pL

(32)

Table 4 summarizes the key material properties and specimen geometries of the slab used for model validation, including Poisson’s ratio (ν), span length (L) and width (w). Tables 4 and 5 report the experimentally- and analytically-obtained maximum deflection and reactions that were measured at four corners of each slab loaded at point E and D respectively. The statistics shown in Tables 5 and 6 indicate that the analytical procedure developed in this study reproduces the test results with a good degree of accuracy. Although the model simulates well the structural response of the skew slabs for the most of the cases (see Tables 5 and 6), some predictions exhibited over 50% errors,

8EI

15

Upper bounds

12 9

Moment (kN.m)

Moment (kN.m)

15

S0 S15 S25 S35 S45

6 3 0 0

0.01

0.02

0.03

0.04

0.05

Lower bounds

12 9

S0 S15 S25 S35 S45

6 3 0

0.06

Rotation (Radius)

0

0.01

0.02

0.03

0.04

0.05

Rotation (Radius)

Fig. 21. Moment-rotation relationship of the skew slab sections: (a) Upper bounds; (b) Lower bonds. 13

0.06

Engineering Structures 199 (2019) 109588

T. Xie, et al.

and

pL3 , 192EI

analytical approach and the calculated the upper and lower bounds of the ultimate load carrying capacity of each slab. The comparisons shown in Table 8 suggest that the yield-line analysis using the partialinteraction segmental analytical approach is capable of estimating the ultimate bending capacity of skew slabs.

respectively. Fig. 16(b) shows that for a given skew angle of

the slab, the increase in bend-to-twist ratio EI leads to an increase in the GJ bending moment and Fig. 16(c) shows the influence of loading position on the bending moment of the skew slab, where the largest bending moment occurs when the load is applied at the centre of the slab. Fig. 17(a) and (b) illustrates that for a slab with a given skew angle, the absolute value of the in-plane torsional moment (T) increases with a decrease in EI , and for a given EI , the increase in T with increasing skew GJ GJ angle only is up to α of 45°; further increase in skew angle leads to a decrease in T. For the influence lines generated for shear force, shown in Fig. 18, the shear force profile for a skew slab strongly depends on the loading position and this is the same as that of a simply-supported straight slab. Fig. 19 presents that the deflection of a skew slab decreases with increasing skew angle and the deflection is also seen to increase with an increase in EI .

5. Conclusions In this research, experimental program and analytical studies have been conducted to investigate the performance of UHPFRC skew slabs/ bridge decks under applied concentrated loads. Based on the results obtained the following inferences can be made: 1. The skewness of a slab significantly affects the deflection profile and reaction at the supports; an increase in the skewness of the slab leads to a more unsymmetrical vertical deflection distribution and also results in a decrease in the magnitude of reaction at the acute corners of and a corresponding increase in the reaction at the obtuse corner. The effect of slab skewness on the performance of a slab is particularly significant when the skew angle exceeds 25°. 2. It is observed that compared to the minor effect caused by torsional moment, the maximum vertical deflection of a centrically-loaded UHPFRC skew slab predominates by the flexure bending. However, reactions forces measured at the supports of the slabs significantly affected by the torsion, in which under a significantly higher outplane moment, reaction forces measured at the acute corner even change from compression to tension. 3. Increasing the skew angle of a centrically-loaded slab has a minor effect on the load-deflection relationship prior to attaining the ultimate load, whereas a significant reduction in ductility with an increase (i.e. up to 115% reduction in the maximum mid-span deflection) in the skew angle is observed at the post-peak stage of the load-deflection curve of the slab. 4. The application of UHPFRC material in a skew slab can achieve comparable performance to that of a conventional normal strength concrete with a similar skew angle, larger cross-sectional area and 5 times higher longitudinal reinforcement ratio. 5. Closed-form analytical solutions are derived for indeterminate, simply supported skew slab/bridge decks based on the virtual work method to generate influence lines for bending moment, shear force, reaction force and torsional moment. 6. Under the linear elastic material conditions, the predictions using the developed mechanic-based closed-form equations correlate well with experimentally established results. The experimental variations and some errors seen in the model predictions are mostly caused by the inhomogeneous material properties of the slabs as the consequences of the randomly distributed fibres in the specimens. 7. Yield-line analysis using moment-curvature relationship from the partial-interaction segmental analytical approach can properly determine the upper and lower bounds of the ultimate load carrying capacity of a skew slab.

GJ

4.3. Yield-line analysis In this section, the ultimate bending capacity of the skew slabs is determined using yield-line analysis. The upper (ly-max) and lower (ly-min) bounds of the yield-line length as explained in Section 3.4 and can be defined as:

l y − max = w

(33)

l y − min = w·cosα

(34)

The calculated upper and lower bounds of the yield-line length of each slab are summarized in Table 8. To conduct yield-line analysis, it is required that at failure the potential energy expended by loads displacing the slab must equal the energy dissipated (or work done) rotating yield lines, which is depicted in Fig. 20 and can be defined as:

∑ P × Δ = ∑ m × ly × θ

(35)

in which P is the peak load in (kN), Δ is the vertical displacement of the load in (m), m is the moment of resistance of slab per length crossing the yield-line in (mm/mm), ly is the total length of yield-line in (mm), and θ is the rotation of slab. For the case as shown in Fig. 20, Eq. (35) can be simplified to:

P × Δ = 2 × m × ly × θ

(36)

The moment capacity (M = m × l y ) of each section where the yieldline appeared can be obtained using a partial-interaction segmental analytical approach developed in [28] as shown in Fig. 21 and the details of this modelling approach can be found in references [38–43]. The approach is a direct utilization of the Euler-Bernoulli theorem of plane section remain plane without the corollary of a linear strain profile. Hence a similar solution technique to that which is followed when undertaking a moment-curvature analysis is applied, with the major difference being the starting point is the imposition of deformation rather than strain profile. This change is significant as it allows for the direct application of partial interaction theory to simulate localized behaviours such as crack formation, widening and tension stiffening [42] in the tension region and concrete softening using shear friction theory in the compression region [44]. In Eq. (36) the rotation of the slab can be determined using the vertical displacement of the load (Δ) and half of the span length of the slab (L/2), as the expression given in Eq. (37)

Δ L/2

θ=

Acknowledgements The authors would like to extend their gratitude to Messrs. Cecchin, Chesini, Di Girolamo, and Mazzone, who performed the experimental program presented in this paper. We also wish to acknowledge the financial support of the Australian Government Research Training Program Scholarship awarded to the first author.

(37)

References

By substituting Eq. (37) in to Eq. (36)

P=

4M L

[1] Menassa C, Mabsout M, Tarhini K, Frederick G. Influence of skew angle on reinforced concrete slab bridges. J Bridge Eng 2007;12:205–14. [2] Dhar A, Mazumder M, Karmakar MCS. Effect of skew angle on longitudinal girder (support shear, moment, torsion) and deck slab of an IRC skew bridge. IndIan

(38)

Table 8 also summaries the peak moment of the yield-line sections of each slab established using the partial-interaction segmental 14

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