Determination of rectangular stress block parameters for high performance concrete

Engineering Structures 25 (2003) 371–376 www.elsevier.com/locate/engstruct

Determination of rectangular stress block parameters for high performance concrete Ertekin Oztekin, Selim Pul ∗, Metin Husem Black Sea Technical, University Dept. of Civil Engineering, 61080 Trabzon, Turkey Received 10 July 2002; received in revised form 8 October 2002; accepted 8 October 2002

Abstract Despite so much research on high performance concrete, the properties of this concrete are not known as well as those of ordinary concrete. There have been a lot of equations, rules and suggestions in the codes which are used in the design of reinforced concrete and prestressed concrete structures. They are obtained from experimental studies made on concrete that have compressive strength of less than about 40 MPa. It is not exactly known whether they could be used in the design of structures constructed by using high performance concrete. Therefore in this study, stress-strain and equivalent block parameters were obtained from experimental stress-strain diagrams for calculation of high performance reinforced concrete beams in flexure. The conclusions obtained from this study showed that determined rectangular stress block parameters can be used in the design of high performance reinforced concrete members in flexure.  2002 Elsevier Science Ltd. All rights reserved. Keywords: High performance concrete; Ordinary concrete; Stress-strain model; Hognestad’s model; Modified Hognestad’s model; Compressive strength; Rectangular stress block parameters and models

1. Introduction The strength and durability of the concrete used in reinforced concrete structures have been increasingly related to technological developments. Today, the material known as high performance concrete has been continuing its development. Definition of the high performance concrete has been changing with time, geographical area and production technology. For instance, in the 1950s, concrete which had at least 35 MPa compressive strength was called high performance concrete. In the 1960s concrete which had a compressive strength between 41 and 52 MPa was produced commercially in the US. In the early 1970s concrete which had 62 MPa compressive strength was produced [1–4]. Recently, concrete which has 80–100 MPa compressive strength has been used in reinforced concrete and prestressed concrete structures. High performance concrete

∗

about 250 MPa compressive strength has been able to produced by using high strength aggregate. A lot of investigations are verified to define the stressstrain relationship and equivalent stress block parameters in the design of high performance concrete members. Some of these investigations are published by Ibrahim and MacGregor [5], Swartz et al. [6] Kaar et al. [7], Schade et al. [8], Attard et al. [9], Azizinamini et al. [10], and Wee et al. [11]. Although high performance concrete is commonly used, its properties are not known as much as the properties of ordinary concrete in the design of reinforced concrete sections. Therefore, in this study the parameters used to design reinforced concrete section in flexure, have been stated using experimental data. High performance concrete is produced using high strength aggregate and cement, water, mineral and chemical admixtures. Generally, superplasticizers or high-range water-reducer admixtures, etc., are used as chemical admixtures; silica fume, flay ash, ground slag, slag cement etc., are used as mineral admixtures.

Corresponding author. E-mail address: [email protected] (S. Pul).

0141-0296/03/$ - see front matter  2002 Elsevier Science Ltd. All rights reserved. doi:10.1016/S0141-0296(02)00172-4

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Nomenclature AS: b: c: d: C: Ea: Fs: fc: Fs: fyd: k: k 1, k 2, ⑀c : ⑀co: ⑀cu: ⑀s: ⑀su:

Area of steel in tension Breadth of section Neutral axis depth Effective depth of concrete section Compressive force Modulus of elasticity of concrete Stress-strain curve area Compressive strength of concrete Tension force in reinforcement Yield strength of steel Constant k3: Rectangular stress block parameters Strain of concrete Strain at the peak stress Ultimate strain of concrete Strain of steel Ultimate strain of steel

2. Determination of stress-strain model 2.1. Experimental study The model was obtained from the results of an experimental study, which are made in Structure and Materials Laboratory of Karadeniz Technical University, on eight series of high performance concretes [12]. Portland cement with a characteristic compressive strength of 42 MPa, superplasticizer called ASTM C 494 Type 12 and silica fume admixtures were used in the production of high performance concretes. The gradation of aggregate is given in Table 1. Experiments were verified on eight series standard cylinder (150 mm×300 mm) specimens produced using different water-cement ratios. Each series includes 12 specimens. The concrete mix proportions is given in Table 2. Uniaxial compressive tests on produced high performance concrete were performed invariable loading rate of 0.15 MPa/s. In these tests, two strain-gauges (Type TML-PL90) were located on the mutual faces of the each specimen in order to measure Table 1 Gradation of aggregate Size (mm)

Percent of total aggregate weight

0.00–0.25 0.25–0.50 0.50–1.00 1.00–2.00 2.00–4.00 4.00–8.00 8.00–16.00

8 7 7 10 15 23 30

vertical and horizontal strains under compression. Measurement lengths of these gauges were 90 mm. Strains under compression for each 0.566 MPa stress were recorded. These were used to draw σ–⑀ diagrams. Mechanical properties and mix proportioning of high performance concretes produced using granite aggregate are given in Table 2. As it is known stress-strain curve is affected by several variables and defining just one valid curve for each concrete is not possible. But, to explain concrete behaviour models which determine the stress-strain relation of the concrete is needed. Thus, a lot of concrete models are proposed by several researchers. Some of these are, Hognestad [13], Kent and Park [14], Sheik and Uzumeri [15], Roy and Sozen [16], Sargin [17], Saatcioglu and Razvi [18], Muguruma et al [19], Collins et al. [20], and Hsu and Hsu [21]. σ–⑀ model produced by Hognestad is one of the most commonly used for ordinary concrete (Fig. 1). It is assumed that the part of the σ–⑀ curve until the parabola peak is second order parabola and the falling branch is linear in this model. In this study, some regulations [22] were made on the Hognestad’s model for high performance concrete using σ–⑀ relation obtained from eight series high performance concrete specimens given in Table 2. In these regulations, the area of stress-strain curve, total compressive force and application point of the total compressive force are the same as the curves obtained by experimental studies. The stress-strain curve for ordinary concrete has been defined by Hognestad is given below; sc ⫽ fc

冋 冉 冊册 2ec ec ⫺ eco eco

2

(1)

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Table 2 Mechanical properties and mix proportioning of the concretes Mix no.

fc (MPa)

Ec (MPa)

103⑀cu (mm/mm)

Aggregate ( Water (kg/m3) dmax ⫽ 16mm) (kg/m3)

CementitousWater(kg/m3) cementitous Ratio

Silica fume in cementitous (kg/m3)

Superplasticizer (kg/m3)

H1 H2 H3 H4 H5 H6 H7 H8

60 66 70 75 81 84 89 94

33,500 35,200 36,000 37,500 38,000 39,400 40,700 43,000

2.40 2.45 2.55 2.60 2.80 2.55 2.70 2.75

1704 1716 1730 1769 1796 1809 1821 1835

550 550 550 550 550 550 550 550

50 50 50 50 50 50 50 50

11 11 11 16 16 22 22 22

193 187 182 165 154 149 143 138

0.35 0.34 0.33 0.30 0.28 0.27 0.26 0.25

to get more linear curve for high performance concrete. Hognestad’s, modified, and experimental stress-strain curves are shown in Fig. 2. 3. Determination of stress block parameters

Fig. 1. σ–⑀ curve for ordinary concrete proposed by Hognestad [13].

High performance concrete specimens are fractured suddenly and brittle when they reach ultimate stress under uniaxial compression. Thus, to define the falling branch of σ–⑀ curve is very difficult. In the regulations on the Hognestad Model, it is assumed that ultimate strain equals strain in the maximum stress (⑀co ⫽ ⑀cu) in this study. Thus, these equations given below are valid for ⑀cⱕ⑀cu. If k(⑀c / ⑀cu) is written instead of 2⑀c / ⑀cu and (k⫺ 1)(⑀c / ⑀cu)2 is written instead of (⑀c / ⑀cu)2 in Eq. (1), modified Eq. (7) is obtained for high performance concrete as

冉

冉 冊冊

ec ec sc ⫽ fc k ⫺(k⫺1) ecu ecu

2

(2)

Where; k ⫽ 2⫺[(fc⫺40) / 70] (60 MPaⱕfcⱕ94 MPa)

(3)

⑀cu, proposed in CEB[23], is modified according to experimental results as follow; ecu ⫽ [2.2 ⫹ 0.015(fc⫺40)]10⫺3(60MPaⱕfc

(4)

ⱕ94MPa) Stress-Strain curve of high performance concrete are more linear than ordinary concrete’s. So, Eq. (3) is used

It is known that real stress distribution in compressive area at a cross section is the same as stress-strain curve in uniaxial compression. But stress-strain curve is affected by a lot of variables. So, it is not possible to suggest a certain stress-strain curve for concrete. The area of stress-distribution and centre of gravity of this area is more important than the geometry of the stress distribution for equilibrium equation in reinforced concrete design. So rectangular stress block, which is suggested by ACI-318[24], is preferred for ease of calculation of area, centre of gravity and others (Fig. 3). The design method given for ordinary concrete by ACI-318 can be valid for high strength concrete. But k1, k2 and k3 rectangular stress block parameters should be obtained for high strength concrete again. So stress block parameters used in calculations have been developed using a modified Hognestad model for high performance concrete in this study. These parameters proposed by some codes and authors are given in Table 3. In this study, a new stress block model is constituted to obtain stress-strain parameters for calculation for high performance concrete. The area and centre of gravity of the curve obtained from Eq. (2) must be calculated to define stress distribution in a compressive area of a section of high performance concrete. Therefore, if Eq. (2) is integrated for the variable ⑀c, (in Eq. (5).) the area of stress-strain curve (Fa), is calculated by Eq. (6).

冕冋 冋

ecu

Fa ⫽

fc k

0

冉 冊册 册 2

ec ec ⫺(k⫺1) ecu ecu

e2c e3c Fa ⫽ fc k ⫺(k⫺1) 2 2ecu 3ecu

dec

ecu

,

0

(5)

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Fig. 2. Hognestad’s Modified Hognestad and experimental stress-strain curves.

Thus; 1 Fa ⫽ fcecu(k ⫹ 2). 6

(6)

Theoretical stress-strain curves developed in this study were divided into trapezoidal areas to determine of application point (the centre of gravity of σ–⑀ curve) of compressive force. Unit wideness of these areas are ⌬⑀c ⫽ 10⫺5 in axis ⑀ (Fig. 4). The block parameters, that describe rectangular stress distribution model this study, are calculated in equations below;

k2 ⫽ (ec⫺x) / ec

(7)

k1 ⫽ 2k2

(8)

k3 ⫽ Fa / fck1ec

(9)

Where x is strain in Fig. 3. It is supposed that unit strain distribution in a cross section in flexure is linear to calculate the neutral axis depth (Fig. 3). This assumption shows that the neutral axis depth, c, can be easily determined by Eq. (10). c ⫽ ecd / (ec ⫹ es)

(10)

In this study, the parameters were calculated in Eqs.

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Fig. 3.

375

Experimental and equivalent rectangular stress blocks and linear strain distribution [24,25].

Table 3 Rectangular stress block parameters Reference

k1( ⫽ 2k2)

k3

ACI-318 [24] Eurocode-2 [25] CSA-94 [26] NZS-95 [27] Attard and Steward [9] I˙brahim and MacGregor [5] Bing, et al. [28] Azizinamini, et al. [10]

1.09⫺0.008fc(0.85ⱖk1ⱖ0.65) 0.9⫺fc / 500 0.97⫺0.0025fcⱖ0.67 (ACI-318) 1.0948f⫺0.091 ⱖ0.67 c 0.95⫺fc /4000ⱖ0.70 (ACI-318) (ACI-318)

0.85 0.85 0.85⫺0.0015fcⱖ0.67 1.07⫺0.004fc(0.85ⱖk3ⱖ0.75) 0.6470f0.0324 ⱖ0.58 c 0.85⫺fc / 800ⱖ0.725 0.85⫺0.004(fc⫺55)ⱖ0.75 0.85⫺0.0073(fc⫺69)ⱖ0.6

k1 ⫽ ⫺0.0012fc ⫹ 0.805

(11)

k3 ⫽ ⫺0.002fc ⫹ 0.964

(12)

k1k3 ⫽ ⫺0.0024fc ⫹ 0.762

(13)

In these equations, for k1, k3, and k1k3 the correlation coefficients are 0.990, 0.985 and 0.999 respectively. Fig. 5 shows a comparison of the proposed and the other rectangular stress block parameters.

4. Conclusions From this study the following conclusions can be drawn;

Fig. 4.

σ–⑀ curve divided into trapezoidal areas.

(7)–(9) for the high performance concrete of which strengths are between 60 and 94 MPa. Linear regression analyses were made between these parameters and compressive strength. Equations obtained are given below.

앫 Rectangular stress block parameters used in ordinary concrete members cannot be used safely for high performance concrete members. New stress block parameters have been obtained from the experimental study. 앫 The model proposed by Hognestad for ordinary concrete is modified for high performance concrete. As a summary, using experimental stress-strain curves for high performance concrete, a new stress-strain model was defined. Then equivalent rectangular stress block parameters were defined using the modified stress-strain

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[7]

[8]

[9] [10]

[11]

[12]

[13]

[14] [15]

[16]

[17] [18] Fig. 5. Relationships between rectangular stress block parameters and concrete strength.

model for high performance concrete. These parameters are in a harmony with values obtained from experimental stress-strain curves. Therefore these parameters are able to use in design of the high performance concrete members.

[19]

[20]

[21]

[22]

References [23] [1] ACI Committee 363-R84, State-of-the art report on high strength concrete, ACI Journal July-August 1984, 364-410. [2] Nilson AH. Design Implications of Current Research on HighStrength Concrete, High Strength Concrete ACI SP-87, Detroit, MI, 1985, 85-118. [3] Swamy RN. High-strength concrete—material properties and structural behavior, High Strength Concrete ACI SP-87, Detroit, 1985, 119-46. [4] Larrard F, Mailer Y. High performance concrete, Second National Concrete Conference, Istanbul, Turkey, 1991,76-113. (in Turkish, translated from French). [5] Ibrahim HHH, MacGregor JG. Modification of ACI rectangular stress block for high-strength concrete. ACI Structural Journal 1977;94(1):40–8. [6] Swartz SE, Narayan Babu HD, Periyakarupan N, Refai TME.

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