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ScienceDirect Procedia Engineering 193 (2017) 273 – 280
International Conference on Analytical Models and New Concepts in Concrete and Masonry Structures AMCM’2017
The application of a probabilistic method to the reliability analysis of longitudinally reinforced concrete beams Marta Sáowika,*, Izabela Skrzypczakb, Renata Kotyniac, Monika Kaszubskac a
Lublin University of Technology, 40 Nadbystrzycka,Lublin 20-618, Poland Rzeszow University of Technology, 2 PoznaĔska, Rzeszów 35-084, Poland Lodz University of Technology, 6 Aleja Politechniki, àódĨ 90-924, Poland
b c
Abstract The failure in longitudinally reinforced concrete beams without transverse reinforcement is the most often observed in a support zone due to bending moment and shear force acting simultaneously in a cross section. This kind of failure is called shear failure and in case of slender beams it can lead to dangerous, brittle damage. Therefore the evaluation of reliability of these members is of the paramount importance, all the most because the design procedure given in Eurocode 2 for shear capacity is of an empirical character. In the paper, the reliability analysis of longitudinally reinforced concrete beams designed according to the rules given in Eurocode 2 was performed. Furthermore, the own experimental investigations were presented. The experiments focused on determining shear capacity and observing the failure process in longitudinally reinforced concrete beams without stirrup of a rectangular and a T-shape cross section. The obtained experimental results were used to calculate the difference of safety margins of the designed shear resistance calculated on the basis of the formula from Eurocode 2 according to reinforced concrete beams without transverse reinforcement of different cross sections. © Published by Elsevier Ltd. This © 2017 2017The TheAuthors. Authors. Published by Elsevier Ltd.is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/). Peer-review under responsibility of the scientific committee of the International Conference on Analytical Models and New Peer-review responsibility of the scientific committee of the International Conference on Analytical Models and New Concepts in Concrete and Masonry Structures. Concepts inunder
Concrete and Masonry Structures
Keywords: reliability; concrete beams; shear capacity
* Corresponding author. Tel.: +48-81-5384392; fax: +48-81-5384390. E-mail address:
[email protected]
1877-7058 © 2017 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license
(http://creativecommons.org/licenses/by-nc-nd/4.0/). Peer-review under responsibility of the scientific committee of the International Conference on Analytical Models and New Concepts in Concrete and Masonry Structures
doi:10.1016/j.proeng.2017.06.214
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1. Introduction The shear failure is the type of failure which can appear in reinforced concrete beams in a support zone and it is caused by bending moment and shear force acting simultaneously in a cross section. This type of failure can reduce flexural capacity and shear governs the resistance of a member. Shear capacity of reinforced concrete flexural members is affected by several parameters. The most important ones which are mentioned in the professional literature are: shear span-to-depth ratio (a/d where a is a distance between an applied load and a support and d is an effective depth of a cross section), concrete compressive strength (fc), longitudinal reinforcement ratio (ρl), the size of the member (d). In typical slender beams with a moderate reinforcement ratio the dangerous, brittle failure due to inclined cracking was usually observed during the experiments. Mostly the experimental investigations were performed on beams of the rectangular section and therefore the database of shear capacity for such members predominates. Much more smaller database for shear capacity can be found for beams of a T-shape cross section. The comparative experimental investigation on beams of rectangular and T-shape section which is described in [1] showed that shear forces at failure were higher in the beams of a T-shape cross section compared to ultimate shear forces in beams of rectangular section. Unfortunately, the experiment [1] was performed only on 3 rectangular beams and 3 T-shape cross section beams. The conclusion about the contribution of concrete in flanges in shear resistance should be verified on a big enough database of test results. Usually it is not possible to compare test results from different experiments exactly as the tests are not performed in the same or similar conditions. For evaluating the conclusions about the influence of the type of cross section on shear capacity the probabilistic analysis was performed. The errors between the design theoretical shear resistance evaluated from the model given in EN 1992-1-1 (Eurocode 2) [2] and the design experimental shear resistance were determined. The obtained errors allowed to conclude about the model uncertainty for shear resistance of beams both of a rectangular and a T-shape cross section. 2. Shear capacity of beams without stirrups The experimental investigation concerning reinforced concrete beams of rectangular cross section was performed in the Faculty of Civil Engineering and Architecture of Lublin University of Technology. Twenty five beams were tested. All beams had the rectangular cross section of the width b = 0.12 m, the total depth h = 0.25 m, and the effective depth d = 0.22 m. The beams had the different effective span leff, from 0.9 m to 1.8 m and therefore the shear span-to-depth ratio a/d varied from 1.8 to 4.1. The beams were made from concrete of the compressive strength 35 MPa, the tensile splitting strength 3.5 MPa, the Young’s modulus 41400 MPa. The reinforcement ratio was 0.9%, 1.3% or 1.8%. The steel bars in the beams of the symbol “S” were of 34GS category of the yield stress fy = 453 MPa and the tensile strength ft = 698 MPa. The bars in the beams of the symbol “O” and “P” were of RB500 category of the yield stress fy = 545 MPa and the tensile strength ft = 631 MPa. In the beams, as the first, flexural cracks formed within the mid-span. When the load reached approximately F = 80 kN the shear stress caused the appearance of diagonal cracks. The specimens exhibited two different modes of shear failure as it was described in depth in [3]. The beams of the shear span-to-depth ratio a/d 2.5 failed suddenly in shear, soon after the appearance of the diagonal crack. The shear transfer run along one major diagonal crack at one side of the beam in the mid-span of the support zone and the ultimate load was close to the diagonal cracking load. In the beams of the shear span-to-depth ratio a/d < 2.5 the failure process went in a different way. Two major diagonal cracks formed symmetrically at both opposite support zones of the beam and the ultimate load at failure was significantly higher than the diagonal cracking load. The change of the mode of failure at a/d = 2.5 was confirmed by the results of numerical analysis presented in [4]. According to shear span-to-depth ratio, the beams were divided on typical slender beams of a/d 2.5 and short beams of a/d < 2.5. As the character of failure and shear capacity of short beams differed considerably comparing to slender beams they were not included in the analysis performed in the paper. In the performed experimental investigation, the diagonal failure was mostly observed in slender beams except of one beam in which the full flexural capacity was attained (the beam of a/d = 4.1 and ρl = 0.9%). The obtained ultimate shear forces Vult and diagonal cracking forces Vcr for the beams of the symbol “S” are presented in Fig. 1.
Marta Słowik et al. / Procedia Engineering 193 (2017) 273 – 280
275
V [kN] 120
Vcr Vult
100 80 60 40 20
a/d
0 0
1
2
3
4
5
Fig. 1. Ultimate shear forces and diagonal cracking forces obtained in the experimental investigation [3]
Typical shear failure was also observed in beams of a T-shape cross section during the experimental investigation which was performed at the Laboratory of the Department of Concrete Structures in Lodz University of Technology. Seven real scale T-shape cross section reinforced concrete beams with the web width bw = 0.15 m, the height depth h = 0.40 m, the flange depth and width hf = 0.06 m and bf = 0.40 m, the effective span leff = 1.8 m. Two concrete cover thickness were assumed 15 mm and 35 mm, that caused differences in the effective depth in a range of d = 0.356 – 0.379 m, that means the shear span-to-depth ratio a/d varied from 2.9 to 3.1. The beams were made of C25/30 concrete mixture with the average cubic concrete compressive strength of 33.2 MPa (the mean cylinder compressive strength 26.6 MPa), the tensile splitting strength 2.9 MPa and the Young’s modulus 26600 MPa. The aim of the test was to investigate an effect of the longitudinal reinforcement ratio, bars diameter (12 mm, 16 mm, 18 mm), a number of bars and a number of reinforcement layers (one or two). Three values of longitudinal reinforcement ratio were assumed: 1.0%, 1.4% or 1.9%. The ordinary steel reinforcement was made of steel RB500 class with the yield stress fy = 529 MPa, 512 MPa and 541 MPa, respectively for bars of 12 mm, 16 mm and 18 mm diameter. Table 1. Test results of T-shape cross section beams No
Beam
ρl
Vult [N]
1
S-512-30-15
0.0099
55590
2
S-316-30-15
0.0107
52590
3
S-318-30-15
0.0135
56120
4
S-312/212-30-15
0.0102
50930
5
S-318/118-30-15
0.0185
61790
6
S-512-30-35
0.0105
45240
7
S-418-30-35
0.0191
52930
The question arises if the similar mode of failure in rectangular and T-shape cross section beams is connected with the same shear transfer mechanism and if the shear stress is carried only by the web of the cross section or the flanges have the contribution in shear transfer. It is not possible to confront the obtained shear capacity from the presented experimental investigations for rectangular and T-shape cross section beams as the test were realized in different condition. Conclusions about a higher shear resistance of T-shape cross section beams have been derived on the basis of a reliability analysis.
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3. The basis of the method used in the analysis The model uncertainty for shear resistance can be obtained after comparing experimental test data and model results. Both experimental results – re and theoretical results – rt should be treated as variables. In the first step the parameters and the type of distribution of the variables should be established. The model verification is based on a comparison of the design values of theoretical and experimental resistance. The design values are derived on the basis of the first order reliability method (FORM) from the formulas (1), (2) described in [5,6,7]. The procedure of calculating the model uncertainty has been also described in [8, 9]. Rtd = Rt + (−α R β )σ t
(1)
Red = Re + (−α R β )σ e
(2)
where: Rt – mean value of theoretical resistance, Re – mean value of experimental resistance,
σ t – standard deviation of theoretical resistance, σ e – standard deviation of experimental resistance, α R – sensitivity factor, β – reliability index. In order to measure the model uncertainty, the absolute error δa and the relative error δr can be calculated from the formulas (3) and (4). The graphical illustration of the resistance model verification is presented in Fig. 2.
δ a = Red − Rtd
(3)
Red − Rtd Red
(4)
δr =
2. Graphical illustration of a resistance model verification
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4. Probabilistic analysis of shear resistance 4.1. Model provided in Eurocode 2 For the probabilistic analysis the model resistance from the European Standard EN 1992-1-1:2004 Eurocode 2 [2] has been applied because Eurocode 2 is the basic code for designing concrete structures in several European countries and in Poland as well. The design value for the shear capacity in members not requiring design shear reinforcement is given by the Eq. (5). The formula was derived on the basis of test results and so it is of an empirical character. The experimental results performed mostly on rectangular beams gave the basis for the formulation the rules of checking shear capacity of reinforced concrete beams without stirrups in Eurocode 2 [2] and therefore the design formula does not include the type of cross section.
(
)
VRd ,c = CRd ,c k (100ρ l f ck ) 3 + k1σ cp bw d ≥ (ν min + k1σ cp )bw d 1
(5)
where: f ck – characteristic compressive strength of concrete in MPa, bw – smallest width of the cross section in the tension area in mm,
d – effective depth of the cross section in mm, ρ l – ratio of tensile reinforcement ρ l ≥ 0.02, 200 ≤2, d
k = 1+
σ cp =
N Ed ≤ 0.2 f cd – stress caused by axial force in MPa, Ac 3
1
ν min = 0.035k 2 f ck2 , C Rd ,c = 0.18 / γ c ,
γ c – safety coefficient for concrete. For probabilistic calculation the resistance model described by the Eq. (6) was assumed. In the model, the basis variables X are: fc, d, bw, ρl. 3 1 1 Rmod (X ) = max ª0.18k (100ρ l f c ) 3 bw d ;0.35k 2 f c2 bw d º «¬ »¼
(6)
4.2. Database The database for the calculations consisted of 84 tests of rectangular beams without shear reinforcement. The database covers beams with low and medium concrete compressive strength, moderately reinforced. The shear spanto-depth ratio in the beams was from 2.5 to 4.0 in order to exclude short beams in which arch action predominates and highly slender beams in which the full flexural capacity can be obtained. The database was collected on the basis of the test performed by Thamrin et al. [1] and the experiments described in the monograph [10], among them there were test results obtained by Sáowik [10], Kani [11], Moody et al. [12], Mphonde and Frantz [13], Desai [14]. The data base for T-shape cross section was smaller and consisted of 32 tests. It was built on the experimental investigation presented in the paper and on available test results from literature [15,16]. All considered experiments were performed, just as rectangular beams, on slender members of a/d from 2.5 to 4.0, made of low and medium concrete compressive strength and moderately reinforced. The scatter of variables included in the database is presented in Table 2.
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Marta Słowik et al. / Procedia Engineering 193 (2017) 273 – 280 Table 2. Scatter of variables included in the database Type of cross section
Variable
Distribution
Standard deviation
Mean
Min.
Max.
Rectangular
bw – smallest width of cross section [mm]
N
12
120
102
152
T-shape
d – effective depth [mm]
N
23
220
203
305
fc – concrete compressive strength [MPa]
LN
5.8
33.0
26.7
40.2
ρl – longitudinal reinforcement ratio
cut-off N
0.00449
0.0135
0.01
0.02
Vu – shear force at failure [N]
cut-off N
3780
42000
33000
52500
bw – smallest width of cross section [mm]
N
1.5
156
150
178
d – effective depth [mm]
N
38
370
279
379
fc – concrete compressive strength
cut-off LN
4.2
26.7
20.3
28.2
ρl – longitudinal reinforcement ratio
cut-off N
0.00403
0.0132
0.01
0.019
Vu – shear force at failure [N]
cut-off N
5976
53600
45250
101905
4.3. Results of the probabilistic analysis As the result of the probabilistic analysis the diagrams of frequency for shear resistance for the applied model – theoretical resistance Rt and for test results – experimental resistance Re were evaluated separately for rectangular and T-shape cross section beams, see Fig. 3 and Fig 4. The results of calculations are provided in Table 3 for rectangular beams and in Table 4 for T-shape cross section beams. In the calculation, the sensitivity factor was αR = 0.8 and the reliability index was β = 3.8, as they are recommended in EN 1990 [1] and in ISO 2394 [2] for design values of resistance. The obtained results for rectangular beams are similar to those presented in [8, 17].
Fig. 3. Diagrams of frequency for shear capacity of rectangular cross section beams: a) re – experimental, (b) rt – theoretical, c) re; rt – summary Table 3. Results of calculation for beams of rectangular cross section Resistance of rectangular cross section
Mean value [N]
Standard deviation [N]
Design value [N]
Theoretical resistance – rt
ܴത௧ ൌ 30959
ߪ௧ ൌ 1739.8
ܴ௧ௗ ൌ25670.0
Experimental resistance – re
ܴത ൌ 42044
ߪ ൌ 4245.7
ܴௗ ൌ29137.1
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Fig. 4. Diagrams of frequency for shear capacity of T-shape cross section beams: a) re – experimental, (b) rt – theoretical, c) re; rt – summary Table 4. Results of calculation for beams of T-shape cross section Resistance of T-shape cross section beams
Mean value [N]
Standard deviation [N]
Design value [N]
Theoretical resistance – rt
ܴത௧ ൌ 50792
Experimental resistance – re
ߪ௧ ൌ 5762.4
ܴ௧ௗ ൌ33274.3
ܴത ൌ 53221
ߪ ൌ 4535.2
ܴௗ ൌ39433.9
4. Conclusions On the basis of the performed calculations the model accuracy calculated as θd = Rtd/Red was determined: θd = 0.88 for rectangular beams and θd = 0.84 for T-shape cross section beams. The absolute error and the relative error calculated from the formulas (3) and (4) were: δa = 3476 N, δr = 0.119 for rectangular beams and δa = 6165 N, δr = 0.156 for T-shape cross section beams. After comparing the results of probabilistic analysis it appeared that higher relative error has been obtained for the beams of a T-shape cross section (see Fig. 5). The difference of relative errors reached 3.7%.
0.156
T-shape beams
Rectangular beams
0.119
δ 0.000
0.020
0.040
0.060
0.080
0.100
0.120
0.140
0.160
r
[-] 0.180
Fig. 5. Juxtaposition of relative errors calculated for rectangular beams and T-shape cross section beams
The level of relative errors allowed to conclude about the uncertainty of the analyzed model of shear resistance. The probabilistic calculations showed higher safety margin at designing shear capacity on the basis of the resistance
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model from Eurocode 2 [2] for T-shape cross section beams comparing to rectangular beams. This finding suggests that flanges work together with a web of concrete in compression zone in a shear stress transfer. This fact is not included in the design formula from Eurocode 2 [2] as it was evaluated on the basis of test results performed on rectangular beams. The performed analysis showed that the analyzed shear model does not take into account the type of cross section at determining the resistance of reinforced concrete members without stirrups. The conclusions which has been drown from the reliability analysis are supported by the experimental results presented in [1]. The researchers noticed that the shear capacity was higher averagely of 13% for T-section beams compared to the shear capacity for rectangular beams. Furthermore, the authors of the work [1] compared the experimental capacity with theoretical ones calculated on empirical equations, among them the equation from Eurocode 2 [2], and they obtained more conservative prediction of shear capacity for T-shape beams. The test results and the results of calculations have confirmed that the shear capacity depends on the shape of a beam’s cross section and the mechanism of shear stress transfer by concrete in the compression zone is more effective in T-shape beams because not only web but also flanges take part in carrying shear forces. The evaluation of reliability of reinforced concrete beams without stirrups is of the paramount importance and the description of the shear model uncertainty is a crucial problem. The performed probabilistic analysis gives the promising results and points that further research should be carried out to conclude about the influence of concrete in compression zone on shear resistance of reinforced concrete beams without transverse reinforcement, especially in beams of a T-shape cross section. Acknowledgements The work was financially supported by Lublin University of Technology (Grant No. S15/2016). References [1] R. Thamrin, J.Tanjung, R. Aryanti, O. Fitrah Nur, A. Devinus, Shear strength of reinforced concrete T-beams without stirrups, Journal of Engineering Science and Technology, 11 (2016) 548–562. [2] EN 1992-1-1:2004 Eurocode 2: Design of concrete structures – Part 1-1: General rules and rules for buildings. 2004 CEN. [3] M. Sáowik, Shear Failure Mechanism in Concrete Beams. Procedia Materials Science. 3 (2014), 1977–1982. [4] M. Sáowik, T. Nowicki, The Analysis of Diagonal Crack Propagation in Concrete Beams. Computational Materials Science. 52 (2012) 261– 267. [5] EN 1990:2002 Eurocode – Basis of structural design. 2002 CEN. [6] ISO 2394:2015 General principles on reliability for structures. 2015 ISO. [7] V. Cervenka, Global safety formats in fib Model Code 2010 for design of concrete structures. Proceedings of the 11th International Probabilistic Workshop, Brno 2013, 30–41. [8] M. Holicky, K. Jung, M. Sykora, Model uncertainty of shear resistance based on experimental data. 13th Bilateral Czech/German Symposium. Telc, Czech Republic (2012), 67–70. [9] Sz. WoliĔski, Ocena niepewnoĞci modeli noĞnoĞci elementów Īelbetowych, Budownictwo i Architektura. 12/1 (2013) 203–210. [10] M. Sáowik, Shear Capacity of Flexural Reinforced Concrete Members without Transverse Reinforcement. Monographs. Lublin University of Technology, Lublin, 2016. [11] G.N.J. Kani, Basic Facts Concerning Shaer Failure. Journal of the ACI. 63 (1966), 675–692. [12] K.G. Moody, I.M. Viest, R.C. Elstner, E. Hognestad, Shear Strength of Reinforced Concrete beams. Part 1 – Tests of Simple Beams. Journal of the ACI. 53 (1954) 317–333. [13] A.G. Mphonde, G.C. Frantz, Shear Test of High– and Low–Strength Concrete Beams Without Stirrups. ACI Journal. July–August (1984) 350–357. [14] S. Desai, Influence of Constituents of Concrete on Its Tensile Strength and Shear Strength. Magazine of Concrete Research. 55 (2003) 77–84. [15] R. Hakkenberg van Gaasbeek, The Shear Strength of Reinforced Concrete T-Beams. Thesis. Mc Gill University, Montreal, 1966. [16] E.W. Kenneth, Shear strength of reinforced concrete T-beams. Thesis. Purdue University, West Lafayette, 1967. [17] M. Sykora, M. Holicky, Assessment of model uncertainty for shear resistance of reinforced concrete beams without shear reinforcement, Technical University of Ostrava, Civil Engineering Series, 2 (2013) 371–380.
