Concrete Creep Strain Estimation Formula and Design Values of Concrete by Considering the Strength Development

Abstract

This paper presents an estimation of creep coefficients to investigate design values of creep coefficients for different types of cement. A formula for estimating compressive strength during loading, which is used to calculate the creep strain used in estimating the creep strain coefficient, was formulated based on the existing formula for estimating compressive strength. First, the design values of creep coefficients for effective prestress calculations are presented. The values were calculated assuming cement types and standard conditions for prestressed concrete (PC) bridges. Then the authors confirm the effect of the creep coefficients on the verification of simple PC girder.

1. Introduction

In recent years, there has been an increase in the use of concrete admixtures to address on the one hand concerns about the difficulty of obtaining high quality aggregate, and on the other to ensure the durability of structures. In particular, concrete with low water-cement ratio has been used in PC structures to ensure relatively high design compressive strength. As a result, the total alkali content of concrete increases. Consequently, there is growing interest in materials which can be used in response to this, such as the use of concrete mixed with fly ash or blast-furnace slag powder.

The 2004 edition of the Design Standard and Commentary for Railway Structures (Concrete Structures) (hereafter “the 2004 Railway Standard”) [1] and the 2017 edition of the JSCE Standard Specifications for Concrete (Design Edition) (hereafter “the 2017 JSCE Standard”) [2] provide formulas for calculating creep strain per unit stress. These formulas were formulated mainly for ordinary Portland cement, and their applicability to mixed cements has not been clarified.

This paper reports on the formulas and design values of creep coefficients taking account into cement types. In estimating the creep coefficient, a compressive strength estimation equation for loading was formulated based on the existing compressive strength estimation equation to be used in calculating the creep strain. Then, design values of creep coefficients for effective prestress calculations are presented. These were calculated assuming the type of cement and standard conditions of PC bridges. Finally, the influence of the increase or decrease of the design values on the verification of simple PC girders is confirmed.

2. Application of creep strain formula to mixed cement concrete

2.1 Application of the creep strain calculation equation to mixed cement concrete

Equation (1) shows the creep strain calculation equation in the 2004 Railway Standard [1] and the 2017 JSCE Standard [2].

  

$$\epsilon {\text{'}}_{\mathit{cc}}(t,t\text{'})/\sigma {\text{'}}_{\mathit{cp}}=\frac{4W(1-\mathit{RH}/100)+350}{12+f{\text{'}}_c(t\text{'})}\cdot {\log }_e(t-t\text{'}+1)$$(1)

where, ε'_cc(t,t')/σ'_cp: creep strain per unit stress (×10⁻⁶/(N/mm²)) at age t (days) for concrete first loaded at age t' (days), W: unit water content of concrete (kg/m³), RH: relative humidity (%), t' and t: effective ages of concrete at and during loading (days), f '_c(t'): effective age of concrete at loading t' (days).

Equation (1) is mainly applicable to ordinary Portland cement (hereinafter “N”) and has been applied mutatis mutandis to early strength Portland cement (hereinafter “H”). Therefore, based on previous reports [3][4], the differences in creep properties of different cement types, such as blast-furnace cement (Class B) (hereinafter “BB”) and fly ash cement (Class B) (hereinafter “FB”), were investigated. Figure 1 shows the experimental values of unit creep strain [5]. The values in Fig. 1 are normalized by a function of f '_c(t') in order to take into account the difference in the compressive strength f '_c(t') and Young's modulus at the time of loading among the specimens. The unit water volume W is 175 kg/m³ and the relative humidity RH is 60%. Although variations were observed between test cases, no clear trend was observed by admixture replacement ratio. Therefore, it is considered that (1) can be applied to mixed cements by appropriately considering f '_c(t').

Fig. 1　Effect of admixture mixing ratio on unit creep strain [5]

2.2 Compressive strength estimation formula for loading

Equation (2) expresses the compressive strength of concrete using N, H, BB and FB at the age of t days. The equation refers to the compressive strength estimation equation used for verification against temperature cracking which appears in the 2007 JSCE Standard Specification for Concrete [Design Edition] (hereinafter referred to as the 2007 JSCE Standard [6]). Equation (2) expresses the compressive strength estimation equation for the verification of temperature cracking. It is noted that the equation refers to the compressive strength estimation equation for the verification of temperature cracking in the Standard Specifications for Concrete [Design Chapter] of the Japan Society of Civil Engineers (JSCE) enacted in 2007 (hereinafter the “2007 JSCE Specifications”) [6].

  

$$f{\text{'}}_c(t)=\frac{t}{a+bt}d(28)f{\text{'}}_c(28)$$(2)

where a, b: constants, d(28): rate of increase of compressive strength at 91 days of age relative to 28 days of age, f '_c(28): compressive strength of concrete at 28 days of age (N/mm²).

The values of the constants a, b, and d(28) in (2) were determined so that they can be expressed with the same degree of accuracy as the compressive strength estimation equations [2][7] in the 2017 JSCE Standard, Part 6: Verification against Temperature Cracking. Table 1 shows the values of the constants a, b, and d(28) determined with reference to the results of the least-squares method and the estimating equations in the 2007 JSCE Specifications [6]. Figure 2 shows the calculated values by (2). From Fig. 2, the differences in strength development for each cement type can be confirmed. Figure 3 shows a comparison of the calculated values with the previously published experimental data [7] to [11]. Figure 3 shows experimental data of compressive strength measured at several ages between 1 and 91 days of age. As shown in Fig. 3, although FB is replaced by N with fly ash by 15% to 22% [5] the calculated values generally evaluate the actual strength. The calculated values exceeded the actual strength in some experimental data with W/B=40%-50% for FB and W/B=30%-40%-50% for BB, and the difference was slightly larger in some data. However, the variation in the prediction accuracy of the previous compressive strength estimation equation for N shown in Fig. 3(a) was comparable to the variation in the prediction accuracy of the previous equation.

Table 1　Coefficients used in equation (2) [5]

Type of cement	a	b	d(28)	f 'c(28)
Ordinary Portland Cement	4.5	0.95	1.11	−20+30(C/W)
Early strength Portland cement	1.7	0.98	1.04	−15+30(C/W)
Blast-furnace cement Class B	6.2	0.93	1.15	−10+25(C/W)
Fly ash cement Class B	6.2	0.93	1.15	−25+30(C/W)

Fig. 2　Strength development of concrete calculated by Equation (2)

Fig. 3　Comparison of compressive strength between previously published experimental [5] data and calculation data

2.3 Effects of kind of cement on creep strain and creep coefficient

Figure 4 shows the creep strain per unit stress for the same W/B and age at start of loading. The trend in the increase in unit creep strain is dependent on cement type. Figure 5 shows the creep coefficients calculated from (1) and (2). Here, the creep coefficient was calculated from the compressive strength at loading, based on the relationship between compressive strength and Young's modulus, to obtain Young's modulus at loading, E_ct, and then calculated by (3).

  

$$\phi (t,t\text{'})=({\epsilon \text{'}}_{\mathit{cc}}(t,t\text{'})/{\sigma \text{'}}_{\mathit{cp}})・E_{\mathit{ct}}$$(3)

where φ(t,t'): creep coefficient at age t (days) of concrete first loaded at age t' (days), ε'_cc(t,t')/σ'_cp: creep strain per unit stress at age t (days) of concrete first loaded at age t E_ct: Young's modulus of concrete (N/mm²) at the effective age of t' (days) at the time of loading.

Fig. 4　Unit creep strain and cement type

Fig. 5　Creep factor and cement type

The difference in creep coefficients at the same age of the cement type is due to the difference in strength development shown in Fig. 2. In the construction of PC structures, the compressive strength at the introduction of prestress is generally controlled and the creep modulus is calculated by (3). If the compressive strength at the time of prestressing is the same, the creep coefficient is expected to be about the same regardless of the type of cement.

3. Review of design values for calculating creep coefficients

3.1 Review of design values for prestress force calculation

In conjunction with the review of (2), the design values of the creep coefficient were also reviewed. The design values were determined assuming a prestressed concrete structure under standard conditions. The relative humidity of the top and bottom surfaces of the girder was calculated as the top (dry repeatedly) and bottom (always dry) surfaces of the member since the calculation of long-term deflection of a PC girder requires appropriate consideration of the relative humidity of the top and bottom surfaces of the girder. However, when the effects of creep are modeled in terms of forces, such as the variation of the prestressing force of a girder, the effects due to the difference in creep coefficients between the top and bottom surfaces are small. Therefore, the creep coefficient was calculated as the average value of the cross section as the design value of the creep coefficient used in the prestressing force calculation.

The design values of the creep coefficient were considered as follows. Equation (1) is constructed based on the creep test results of concrete specimens. In the construction of (1), the creep coefficient can be calculated by arbitrarily setting the age of the material at which loading begins, t'. However, creep tests generally do not set the initial loading age to 28 days or 3 months, and there is room for further study on the accuracy of the prediction equation when the loading age increases. In (1) and (2), for example, φ = 2.7 when loading age t' = 28 days. As shown in Table 2, the design value for 28 days of material age in the 2004 Railway Standard is φ = 1.5, which is about 1.8 times larger than the value in the 2004 Railway Standard.

Table 2　Design values of creep coefficient of concrete for effective prestressing calculation

Source/method of study	Type of cement	Age of concrete at introduction of prestress, age of concrete at loading (day)
		4	7	4-7	14	28	90	365
The 2004 Railway Standard	-	-	-	2.7	1.7	1.5	1.3	1.1
Equations (1) to (4)*	N	-	3.1	-	2.5	2.2	1.8	1.4
	H	2.9	2.5	-	2.3	2.0	1.7	1.3

*N: Ordinary Portland cement (blast-furnace cement Class B and fly ash cement Class B are also applicable), H: High strength Portland cement

The design values of the 2004 Railway Standard have been used in the construction of PC and PRC girders, and no problems have been reported, at least in terms of the calculation of the amount of prestress reduction. Therefore, it was decided to use the design values, rather than the prediction formulas, to be consistent with the 2004 Railway Standard. In other words, it is necessary to examine the accuracy of the design values together with the equation for calculating not only the creep coefficient but also the amount of reduction in prestress.

Therefore, the design values of creep coefficients were calculated by (4) based on (1) to (3) and the commonly used Whitney rule, referring to the study [12] in the JSCE Standards enacted in 2017. This is calculated as the residual creep from the time of each loading to the design service life in relation to the creep at the time of prestressing as the design creep value for the age of the material at the time of each loading.

  

$$\phi (t,{t\text{'}}_i)=\phi (t,{t\text{'}}_1)-\phi ({t\text{'}}_i,{t\text{'}}_1)$$(4)

where, φ(t,t'_i): creep coefficient at age t (days) of concrete first loaded at age t'_i (day), where t'₁ is 7 days (N, BB, FB) or 4 days (H). The value of t was set to t = 38,000 (days) to ensure a service life of 100 years, which is a rather large value considering the time until the start of in-service.

As shown in Table 2, the creep coefficient of early-strength cement is slightly smaller than that of ordinary cement when loaded at relatively early ages (Fig. 4). On the other hand, the creep coefficients of blast-furnace cement Class B and fly ash cement Class B were almost the same as those of ordinary cement (Fig. 4), suggesting that the values of ordinary cement should be applied as design values. As a result, the creep coefficient increases in the range of 0.3 to 0.8, but the creep curve does not differ significantly from the 2004 Railway Standard.

3.2 Influence of design values on verification of simple PC and PRC girders

The effects of shrinkage strain ε'_cs and creep factor φ on the verification results of each performance item for PC and PRC girders were investigated. Table 3 shows the main test design conditions of the girders for which trial designs were conducted. One example, each of PC-T girders, PRC-T girders, PC-box girders, PRC-box girders, and PC-through girders were selected. ε'_cs = 200 × 10^-6 and φ = 2.7 [1] were used as the base values. The verification results were checked when the values of ε'_cs and φ were increased or decreased by a factor of 0.5 and 2.0, respectively. The design standard strength of the concrete is 40 N/mm² (34 N/mm² when introduced) with a maximum water cement ratio of 50%.

Table 3 Trial design conditions

Design conditions	PC			PRC
	T-shaped girder	Box girder	Through girder	T-shaped girder	Box girder
Number of lines	1	2	1	2	2
Bridge length	24 m	31.3 m	31 m	38 m	34 m
Type	Shinkansen	Conventional line	Conventional line	Conventional line	Shinkansen
Design speed	260 km/h	90 km/h	110 km/h	160 km/h	260 km/h

Figure 6 shows the calculation results. Here, safety (failure), recoverability (damage), and “Assumptions for verification,” which are the determining factors in the design, are presented. In “Assumptions for verification,” the stress levels of concrete and steel were examined, and it was confirmed that the compressive stress σ'_c at the edge of concrete due to permanent action is less than 40% of the design compressive strength f '_cd of concrete, and the tensile stress σ_p of PC steel due to variable action is less than 70% of the design tensile strength f_pud. The tensile stress σ_p of the PC steel due to the variation is less than 70% of the design tensile strength f_pud.

Fig. 6　Results of trial design of various PC and PRC girders (Upper: with different amount of shrinkage strain, lower: with different creep coefficient)

Regardless of the girder type, the shrinkage strain ε'_cs and creep factor φ have a small influence on the verification results. In particular, for girders, the sensitivity of ε'_cs and φ is not found in safety (failure), which is the determining factor in design. In addition, in the case of the “verification assumption,” which is verified using the steel stresses, the verification results varied slightly, but there was no increase in the verification values to the extent that it became a determining factor. Thus, in the design of simple PC girders, the increase or decrease of shrinkage strain ε'_cs and creep factor φ has a small effect on the verification results. In other words, as shown in Table 2, although the design values of the creep coefficients increased from the 2004 Railway Standard, the effect on the verification values for the different girders is considered to be small.

4. Conclusions

(1) Based on the previous experimental results showing that the effect of cement type on unit creep strain was relatively small, it is considered that creep of mixed cement concrete can be generally expressed by considering the difference in compressive strength development of concrete in the creep strain calculation equation of the 2004 Railway Standard.

(2) Based on the existing compressive strength estimation equation, the compressive strength estimation equation under loading was formulated for the calculation of creep strain. Creep coefficients for effective prestress calculations considering cement types are presented.

(3) In the trial design of simple PC girders and PRC girders, the effects of the shrinkage strain ε'_cs and the creep coefficient φ on the verification results were confirmed. Although the design values of creep coefficients were increased in this report, the effects on the verification results of these girders were confirmed to be small.

References

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• [11]  Fukagawa, M., Nakamura, N., Masuda, Y. and Abe, M., “Mechanical properties and durability of high-strength concrete using classified fly ash,” Proc. of Japan Concrete Institute, Vol. 19, No. 1, pp. 205-210, 1997 (in Japanese).
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Authors

	Ken WATANABE, Ph.D. Senior Chief Researcher, Head of Concrete Structures Laboratory, Structures Technology Division Research Areas: Design and Maintenance of Concrete Structures
	Mami NAKAMURA, M.E. Assistant Senior Researcher, Concrete Structures Laboratory, Structures Technology Division Research Areas: Design and Maintenance of Concrete Structures