Instantaneous deflection of light-weight concrete slabs

Abstract

Construction loading before the age of 28 d can have the most significant effects on the slabs, especially for multi-story structures. The changing properties of the young concrete complicate the prediction of serviceability design requirements also. An experimental investigation is performed on four simply supported Light-Weight Concrete (LWC) one-way slabs subjected to immediate loading at 14 d. Effects of aggregate type, loading levels and cracking moment together with the influences of ultimate moment capacity and service moment on the instantaneous deflection of slabs are studied. Comparison of the obtained results with predictions of existing models in the literature shows considerable differences between the recorded and estimated instantaneous deflection of LWC slabs. Based on sensitivity analysis of the effective parameters, a new equation is proposed and verified to predict the instantaneous deflection of LWC slabs subjected to loading at the age of 14 d.

Keywords

instantaneous deflection / light-weight concrete / expanded polystyrene / effective moment of inertia / cracking moment / moment capacity / service moment

Introduction

Instantaneous deflection (D_ins) is the fundamental parameter of the time-dependent and total deflection calculation of the Reinforced Concrete (RC) structures. It is the basis of time-dependent and total deflection calculation of RC structures. The ratio of long-term to the instantaneous deflection is a crucial parameter in serviceability considerations of the reinforced concrete structures. The existing models predict D_ins for Conventional Concrete (CC) beams considering the concrete properties at the age of 28 d [1] and do not cover other types of concrete and loading ages that are inevitable in most construction projects.

Although it is not verified by all experimental investigations, the existing design codes predict the long-term deflection of flexural elements as a multiple of the initial or instantaneous deflection. It might generally lead to an underestimated time-dependent and consequently, total deflection of RC member. Therefore, it may reduce the safety margin in the design process dramatically.

In recent decades, utilizing the chemical and mineral admixtures in concrete technology has introduced Light-Weight Concrete (LWC) as a reliable construction material. Using LWC may result in lighter elements and consequently smaller dimension that both decrease the total weight of the structure and the lateral load effects [2,3]. Therefore, the construction cost can be protected when applied to structures such as long span bridge and high-rise buildings [4,5]. Also, the better steel-concrete bond, better thermal insulation, durability performance, tensile strain capacity, and fatigue resistance make the LWC superior to CC [3,4].

Generally, two types of natural (pumice, diatomite, volcanic cinders, etc.) and artificial (perlite, clay, sintered fly ash, expanded shale, etc.) lightweight aggregates are used to supply LWC [6]. Access to lightweight aggregates such as expanded clay, shale, and slate, is limited in some locations and Expanded polystyrene (EPS) is commercially available worldwide [7]. EPS is a type of artificial lightweight aggregate with a very low density. It is a thermoplastic foam involving a gas phase in a polymer matrix and has high compressibility; therefore, it may provide little restraint to volume changes of the cement paste due to the applied load as well as the changes in the moisture content [8].

Deflection of flexural elements with low reinforcement ratio is highly sensitive to the age of concrete at loading and the shrinkage-restraint stress [9,10]. Modulus of Elasticity (MoE), loading value and load distribution and support condition are the principle factors in D_ins of one-way slabs. Also, continuity of sections along the span, i.e., variable cross-sections and hence, variable moment of inertia depending on the degree of cracking, strongly influence D_ins.

Considering the effect of mechanical properties on deflection behavior of RC structure, application of the existing models, for LWC should be handled with care. Partial replacement of normal-weight fine or coarse aggregate with EPS aggregate decreases the Compressive Strength (CS), MoE and density of LWC depending on the level of replacement [8]. However, the CS in this type of concrete is more sensitive to density changes than the variation of MoE [6–8]. The lower MoE in LWC results in a higher levels of instantaneous deflection, wider cracking and consequently, higher time-dependent and total deflection that cannot be predicted precisely by the existing models of the conventional concrete elements.

Mechanical properties of concrete are changing continuously from approximately liquid state to a visco-plastic material within a few hours, which is followed by further development into a hardened material with almost elastic properties. The cracking moment that has a considerable effect on time-dependent deflection can be reduced by shrinkage caused during the curing period before 28 d. The loading of young concrete structure requires using the distinct properties at the age of loading; in other words, either the model or the early age properties should be modified to get an accurate prediction. In LWC containing EPS aggregate, the CS and MoE at seven d is about 75% to 83% and 85% of those for 28 d respectively [6–8,11].

Flexural stiffness is changing with the cracking due to creep and shrinkage strains of concrete, and the loading age changes the predictable creep and shrinkage behavior. The higher the EPS aggregate content in the mix, the greater the drying shrinkage and creep development [8,11].

The main objectives of this study are:

(a) comparing the experimental values of instantaneous deflection of LWC slabs subjected to loading at the age of 14 d (D_ins−14). The LWC contains EPS beads as light-weight aggregate in the mix;

(b) comparing the development of MoE and CS with age in LWC mix containing EPS aggregate;

(c) comparing the recorded deflections with the predictions of existing models of effective moment of inertia (I_e) in codes of practice and empirical equations;

(d) investigating the effect of loading level, ratio of applied service moment (M_a) to cracking moment (M_cr) and ultimate bending capacity of section (M_u) on D_ins−14;

(e) modifying the best matching models of I_e to propose a new model for LWC slabs.

Effective moment of inertia

When an RC element cracks, its stiffness does not suddenly change to that of a section where the tension-concrete can be entirely disregarded [12]. In fact, the sections where the cracks are localized are separated by regions where the concrete in tension is uncracked with an increase of the stiffness. Behavior at the cracked section used to compute the cracked transformed moment of inertia (I_cr) is assumed to be linear elastic, and nonlinearity of the member stiffness is taken into account with I_e that models the transition from a gross (uncracked) moment of inertia (I_g) to I_cr. Figure 1 explains the deformation response of a simply supported slab in three stages of the uncracked, cracked and fully cracked section under sustained service loading.

A well-established method to predict the D_ins considering the loading of LWC elements before 28 d age still lacks in the literature. Different empirical models have been proposed to estimate I_e in the literature. The models proposed by Faza and Gangarao [13]; Yost et al. [14]; Rafi and Nadjai [15]; Alsayed et al. [16]; for fiber reinforced CC, Hall and Ghali [17] for CC and Fikry and Thomas [18] for both CC and fiber reinforced CC and Benmokrane et al. [19] for CC beam wrapped by fiber strips are examples of the existing models. However, more experimental studied are required to validate the proposed formulations and develop new models for LWC members.

Branson [20] introduced and developed the concept of using an effective moment of inertia to predict the curvature and deflections directly from elasticity theory. The concept then was verified by different researchers and organization mainly by implementing the idea of tension stiffening to apply for various types of structures, loading rates, section reinforced with fiber and FRP sheets and FRP bars. Table 1 shows the Eqs. (1‒18) of some verified models of I_eto enhance the prediction of Δ_ins in concrete members.

Research significance

In the absence of a reliable reference, the unknown effects of the loading before the full strength development of concrete, makes the deflection estimation of LWC slabs more complicated. Determination and limitation of time-dependent deflection by D_ins play a critical role in RC slab design. Hence accurate estimate of D_ins actively enriches the time-dependent deflection and consequently makes an improved structural and performance-based design.

The validity of the constitutive laws for the instantaneous and long-term behavior of LWC members should be verified, and the effect of EPS beads in the mix must be carefully checked. The results of this investigation provide an indicator to confirm the relevant studies. Furthermore, verification of the existing empirical equations for D_ins of LWC slabs will be assessed.

Experimental investigation

Test program

The experimental program consists of 4 LWC slabs to record the deflection and test specimens to measure the mechanical properties. The slabs were simply supported with identical span length and section dimensions (3.5 m length ´ 0.4 m width ´ 0.161 m height). Each slab specimen was moist cured for 14 d and then subjected to loading. Six 150 mm ´ 300 mm cylindrical specimens to measure CS and three 150 mm ´ 300 mm cylindrical specimens were used to measure MoE per each age. The specimens for CS and MoE tests were kept covered in a controlled chamber at (20±2) °C for 24 h until demoulding. The relative humidity of the environment was maintained at 50% during the tests. Then, the specimens were placed in water pre-saturated with lime at 20 °C. The compressive strength test was performed according to AS 1012.14 [21] and ASTM C39 [22] instructions, and the loading rate of the machine on the cylinders was 0.3 MPa/s until failure. MoE tests confirmed the instructions in AS 1012.17 [23] and ASTM C469 [24]. Figure 2 shows EPS beads in fresh and hardened LWC.

Materials

Natural aggregates

Two types of coarse and fine natural aggregates were used in this study. The crushed Dunmore latite with the maximum size of 10 mm was used as the coarse aggregate. In this type of LWC, a blend of 50% coarse and 50% fine sand provided the optimum mix, so equal weight proportion of Pepper-three P coarse sand with a maximum size of 5 mm and washed Kurnell Natural River sand fine aggregates were used in the mixture. Australian standard, AS-1141 [25] along with the Road and Maritime Services (RMS) regulations [26,27], were applied to sampling and testing of the aggregates. The properties of Pepper-three P sand; washed Kurnell natural river sand and crushed Dunmore latite are presented in Tables 2 and 3, respectively.

Artificial light aggregate

The low-density EPS beads were used as light-weight aggregate to reduce the density of concrete. BST aggregate is a commercial name for the used spherical-shaped polystyrene beads with hydrophilic type chemical coating. The percentage passing of the BST beads on 4.75 mm and 2.36 mm sieves were 100% and 90% respectively. Bulk and particle density of the beads were 35 and 67 kg/m³ respectively.

Admixtures

Lignosulfonates-based Water-Reducing Admixtures (WRA), confirming AS-1478.1 [38] was used in EPS-LWC to achieve the required workability.

Cement

EPS-LWC mixture contains the Shrinkage Limited (SL) cement confirming AS-3972 (2010) [39]. The chemical, physical and mechanical properties of the used SL type cement are presented in Table 4. These properties are confirming the specifications defined in AS-2350 (2006) also [40].

Mix proportions

To supply a structural concrete with low density in the range of 2000 kg/m³, the consistent materials used and the corresponding mixture proportions are presented in Table 5.

Load configuration and arrangement

Totally four LWC slabs with idem mix design in the concrete laboratory of University of Technology Sydney (UTS) were investigated. All slabs internally reinforced with 4 N12 deformed steel bars. As shown in Fig. 3, the slabs are similarly simply supported having length 3500 mm, depth 161 mm and width 400 mm. The side and bottom cover of concrete from the bar centroid are 40 and 25 mm respectively.

All slab specimens were subjected to different gravity loads, consisting of self-weight plus superimposed sustained loads via carefully constructed and arranged concrete blocks supported off the top (of the specimens). To provide the required loading, rectangular concrete blocks of predetermined size and weights were cast and weighed before the commencement of the test. Slab specimens were uniformly loaded by the concrete blocks using wooden timbers as loading pads. The wooden pads, having length 450 mm, depth 30 mm and width 100 mm were placed in equal intervals to provide uniformly distributed loading on the slabs. The deflection at mid-span of the slab specimen was recorded by Linear Variable Differential Transformer (LVDT). Figure 4 shows the arrangement of concrete blocks on the top surface of each slab to achieve the desired sustained load level and the position of LVDTs under the slabs.

Analysis of the experimental results

Development of mechanical properties with age

The existing models give under or overestimated values of the mechanical properties of concrete. Besides, the models to predict the properties of LWC in the ages other than 28 d are very rare in the literature. Therefore exact values of CS and MoE of the mix have been measured by test specimens in laboratory conditions at two ages of 14 and 28 d. Figure 5 compares the measured values and development of CS and MoE with age. There are about 11% and 9% increment of CS and MoE of LWC with age, respectively.

Difference between elastic and instantaneous deflection

The instantaneous deflection was recorded by LVDT installed at midspan, immediately after loading of the slabs at the age of 14 d. Table 6 shows the calculated elastic deflection (D_e) of slabs by 14 and 28 d MoE under different values of uniformly distributed load (W_a). The ultimate bending capacity of section (M_u), service moment due to gravity and applied loading (M_a) and the ratio of M_a/M_u are also given in Table 6.

Equations 19 and 20 are utilized to determine the elastic deflection of slabs under two levels of loading values and M_cr, respectively.

(20){{\displaystyle \Delta }}_e={\displaystyle \frac{\mathrm{5}}{\mathrm{384}}}\times {\displaystyle \frac{{{\displaystyle W}}_a\cdot l^{\mathrm{4}}}{{{\displaystyle E}}_c\cdot {{\displaystyle I}}_g}}

(21){{\displaystyle M}}_{cr}={{\displaystyle I}}_{cr}\times {{\displaystyle f}}_r

where; W_a, l, E_c, I_g, I_cr and f_r are uniformly distributed load (KN/m), span length (m), MoE of concrete (KN/m²), gross moment of inertia (m⁴), cracking moment of inertia (m⁴) and modulus of rupture (KN/m²), respectively.

Considering the instantaneous deflection as the elastic deflection is a source of errors in the design of concrete structures. Figure 6 shows the ratio of recorded instantaneous deflection to the elastic deflection at the ages of 14 d (D_ins/D_e) ₁₄ and the ratio of recorded instantaneous to the elastic deflection at the age of 28 d (D_ins−14/D_e−28) in the LWC slabs. The ratio of (D_ins/D_e) ₁₄ in the slabs varies between 1.58 and 2.39. Comparing the recorded deflection at 14 d (D_ins−14) with the estimated elastic deflection at 28 d (D_e−28), the ratio of D_ins−14/D_e−28 shows a discrepancy between 1.73 and 2.62. In fact, the increased flexural stiffness at the age of 28 d decreases the elastic deflection and consequently enhances the ratio of D_ins−14/D_e−28. The average ratio of

(D_ins/D_e)₁₄ and D_ins−14/D_e−28 are 1.98 and 2.17 respectively.

Loading effect

Since the slabs are identical in terms of the dimensions, reinforcement and support condition, the relative ratio of applied load to the flexural capacity of section (M_a/M_u) is a core factor in D_ins of lightly-reinforced concrete structures at the critical section. This ratio becomes more important for the loading of the slabs before 28 d age. Figure 7 compares the effect of M_a/M_u on the ratio of (D_ins/D_e) ₁₄ in slabs. The applied service load is about 30 and 40 percent of the ultimate section capacity in each pair of slabs with the same mix design.

The increasing rate of D_ins−14 under two levels of M_a/M_u ratio is different from the variation of (D_ins/D_e) ₁₄ ratio. Unexpectedly the effect of amplified M_a/M_u ratio by 10 % on D_ins−14 is considerably different in these identical slabs. Increasing M_a/M_u ratio by 10% has resulted about 80 to 100% higher D_ins−14 in slabs. In other words, the deflection of young (before the age of 28 d) LWC slabs is significantly sensitive to the loading levels and variation. Figure 8 compares the D_ins−14 of slabs under two levels of M_a/M_u ratio.

Together with the M_a and M_u effects, cracking moment (M_cr) is another critical parameter in deflection calculations of slabs. Notwithstanding the effects of creep and shrinkage, M_cr causes the first cracking in the tensile zone and changes the section behavior to non-elastic. Since M_cr and M_u are dependent characteristics of reinforced concrete section, they may have similar influences on D_ins/D_e ratio. Besides, the ratio of M_a/M_u is limited to 30 to 40% in each pair of slabs in this study. In fact, it looks like the application of predefined relationship between M_a and M_u in this study. According to Fig. 9 the (D_ins/D_e) ₁₄ ratio is changing by variation of M_cr/M_u and M_a/M_u ratios; however, it is noticeably more sensitive to the ratio of M_cr/M_a compared to M_a/M_u ratio. It is evident from Fig. 9 that the M_a/M_u and M_cr/M_a ratios have the increasing and decreasing effects on (D_ins/D_e) ₁₄, respectively.

The value of the applied M_a is very close to M_cr in LWC−3 and LWC−4, in which, the elastic behavior of the section is expected. However due to creep and shrinkage effects and initiation of cracking in tension zone; D_ins is about 60% higher than the estimated elastic deflection.

Proposed analytical model

Existing models of I_e have been utilized to predict the D_ins−14 of LWC slabs. Among the existing models, some equations are developed for particular conditions like point loading and the concrete members with FRP bars and near surface mounted FRP strips; however, their predictions have been compared with D_ins−14 values in this study to observe any compatibility in between. Estimated values of I_e from Eqs. (1–18) have been utilized to predict the D_ins−14 of slabs. The recorded D_ins−14 in slabs have been compared and evaluated with these predicted deflections in Fig. 10. Surprisingly there are considerable differences between the predicted and experimental data in two levels of the applied M_a/M_u ratio. Existing models give a better prediction at the higher loading level (M_a/M_u=0.4), and overestimate the D_ins−14 by about 100% in the lower loading level (M_a/M_u=0.3).

Equation (7) gives the most uncertain prediction in both loading levels, while, the estimated D_ins−14 by Eqs. (1), (16) and (17) are out of the range of the actual deflections; and there is no distinct well-matching relationship with the recorded data in those equations.

In this study, the presented equations well cover the individual intrinsic or extrinsic variables such as the properties of concrete and the support conditions in the I_e prediction. Besides, the Eq. (8), as the basic model of the relevant studies and Eq. (3), give the most accurate prediction to the recorded experimental data. Therefore, by modifying the models in Eqs. (3) and (8) as the basis of the new model, it is tried to include the effect of the time-dependent elasticity, M_u and M_cr in accompany with M_a into the proposed model, to obtain a better estimation of D_ins in LWC slabs.

Including the impact of the time-dependent elasticity ratio of the concrete and M_cr/M_u ratio are the principal modifications to adjust the basic model with the recorded data of LWC slabs deflection.

To include the effect of the loading before 28 d, the ratio of MoE at the loading age to MoE at the age of 28 d is implemented in the proposed model. Equations (3) and (8) are originally developed for beams. Equation (3) predicts the I_e in two distinct ranges of 1≤M_cr/M_a≤3 and M_cr/M_a>3. However, there is no limit for M_cr/M_a in Eq. (8). The ratio of M_cr/M_a varies between 0.76 and 1.01 while M_cr/M_u ratio varies in the range of 0.27 to 0.32 in this study. Consequently, the proposed equation is recommended to apply to M_cr/M_a≤3.

Equation (22) is proposed to predict I_e in LWC slabs. The required coefficients and limitations in the application of the model are illustrated in Table 7. This type of LWC is widely used in Australia; hence, the maximum I_e is recommended to agree with the limitations of AS-3600-09 [32].

(22){{\displaystyle I}}_e={{\displaystyle I}}_{cr}\times {{\displaystyle \left({\displaystyle \frac{{{\displaystyle M}}_{cr}}{{{\displaystyle M}}_a}}\right)}}^{\mathrm{2}}+\left[(\mathrm{1}-\beta )+{\displaystyle \frac{{({{\displaystyle M}}_{cr})}^{\mathrm{1.7}}}{({{\displaystyle M}}_a\times {{\displaystyle M}}_u)}}\right]\times {{\displaystyle I}}_g

Excluding the area of steel bars from the section, underestimates the I_g by 17%. Therefore, the transformed section method is utilized to minimize the error in the calculation of I_g of the uncracked section in the slabs.

Figure 11 compares the recorded D_ins−14 with the prediction of the proposed model in Equation (22). The proposed model well predicts the I_e values; consequently, the predicted D_ins−14 of LWC slabs is in agreement with the experimental data.

The developed models in the literature, including the models in Eqs. (1–18), are applied to the limited conditions of the material properties or structural behavior. Numerical investigations and the relevant supporting experiments can enhance the efficiency of the developed models and extend their range of application. It is worth to mentioning that the model in this study is developed from an experimental data of four LWC slabs and verified by predictions of 18 models of conventional concrete members. There is no similar study of LWC slabs in the literature to strengthen the basis of the proposed model; and more experimental investigations are required to evaluate its efficiency for different types of light-weight aggregate. A parallel numerical study by the authors investigates the instantaneous and time-dependent deflection and cracking of the LWC slabs. The results of the numerical analysis to verify the experimental results will be published in the future.

Concluding remarks

The following conclusions can be drawn from the current study:

(1) Instantaneous deflection of LWC at the age of 14 d can be predicted by the proposed equation in this study with high accuracy. Expanded Polystyrene bead (commercially known as BST) is used as light-weight aggregate in the LWC mix.

(2) The ratio of M_a/M_u strongly affects the instantaneous deflection of the LWC slabs. Increasing the ratio of M_a/M_u up to 10%, caused about 80% to 100% higher D_ins−14 in the slabs. This load increment resulted about 140% growing of the ratio of (D_ins/D_e) ₁₄ in the same section of LWC slabs.

(3) Taking the elastic deflection equal to the instantaneous deflection causes significant errors in deflection calculations and the design of concrete structures. This conclusion is valid even for M_cr≥M_a.

(4) The estimated elastic deflection of slabs at 14 d is about half the instantaneous deflection.

(5) M_cr is a crucial parameter in the instantaneous deflection calculation of the slabs. However, its interaction with M_u and M_a in the section, affects the deflection calculations also.

(6) The ratio of M_cr/M_u is implemented to propose a model to predict I_e in LWC slab subjected to uniformly distributed loading. The model showed good agreement with the experimental instantaneous deflection data.

References

[1] Mazzotti C, Savoia M. Long-term deflection of reinforced self-consolidating concrete beams. ACI Structural Journal, 2009, 106(06): 772–781

[2] Yoğurtcu E, Ramyar K. Self-compacting lightweight aggregate concrete: Design and experimental study. Magazine of Concrete Research, 2009, 61(7): 519–527

[3] Güneyisi E, Gesoğlu M, Booya E. Fresh properties of self-compacting cold bonded fly ash lightweight aggregate concrete with different mineral admixtures. Materials and Structures, 2012, 45(12): 1849–1859

[4] Mazaheripour H, Ghanbarpour S, Mirmoradi S H, Hosseinpour I. The effect of polypropylene fibers on the properties of fresh and hardened lightweight self-compacting concrete. Construction & Building Materials, 2011, 25(1): 351–358

[5] Vakhshouri B, Nejadi S. Mix design of light-weight self-compacting concrete. Case Studies in Construction Materials, 2016, 4: 1–14

[6] Xu Y, Jiang L, Xu J, Li Y. Mechanical properties of expanded polystyrene lightweight aggregate concrete and brick. Construction & Building Materials, 2012, 27(1): 32–38

[7] Trussoni M, Hays C D, Zollo R F. Fracture properties of concrete containing expanded polystyrene aggregate replacement. ACI Materials Journal, 2013, 110(5): 549–557

[8] Sabaa B, Ravindrarajah R S. Engineering properties of lightweight concrete containing crushed expanded polystyrene waste. In: Materials Research Society, 1997, Fall Meeting, Symposium MM, Advances in Materials for Cementitious Composites. 1997

[9] Scanlon A, Bischoff P H. Shrinkage restraint and loading history effects on deflections of flexural members. ACI Structural Journal, 2008, 105(4): 498

[10] Vakhshouri B. Time-dependent bond transfer length under pure tension in one way slabs. Structural Engineering and Mechanics, 2016, 60(2): 301–312

[11] Li Y, Liu N, Chen B. Properties of lightweight concrete composed of magnesia phosphate cement and expanded polystyrene aggregates. Materials and Structures, 2015, 48(1-2): 269–276

[12] Achintha P M, Burgoyne C J. Moment-curvature and strain energy of beams with external fiber-reinforced polymer reinforcement. ACI Structural Journal, 2009, 106(1): 20–29

[13] FazaS, Gangarao  H. Pre-and post-cracking deflection behaviour of concrete beams reinforced with fibre-reinforced plastic rebars. In: Proceedings of the First International Conference on Advance Composite Materials in Bridges and Structures (ACMBS-I). Sherbrooke: Canadian Society of Civil Engineers, 1992

[14] Yost J R, Gross S P, Dinehart D W. Effective moment of inertia for glass fiber-reinforced polymer-reinforced concrete beams. ACI Structural Journal, 2003, 100(6): 732–739

[15] Rafi M M, Nadjai A, Ali F, Talamona D. Aspects of behaviour of CFRP reinforced concrete beams in bending. Construction & Building Materials, 2008, 22(3): 277–285

[16] Alsayed S, Al-Salloum Y, Almusallam T. Performance of glass fiber reinforced plastic bars as a reinforcing material for concrete structures. Composites. Part B, Engineering, 2000, 31(6): 555–567

[17] Hall T, Ghali A. Long-term deflection prediction of concrete members reinforced with glass fibre reinforced polymer bars. Canadian Journal of Civil Engineering, 2000, 27(5): 890–898

[18] Fikry A M, Thomas C. Development of a model for the effective moment of inertia of one-way reinforced concrete elements. ACI Structural Journal, 1998, 95(4)

[19] Benmokrane B, Chaallal O, Masmoudi R. Flexural response of concrete beams reinforced with FRP reinforcing bars. ACI Structural Journal, 1996, 93(1): 46–55

[20] Branson D E. Instantaneous and Time-Dependent Deflections of Simple and Continues Reinforced Concrete Beams. HPR Report, No.7 Part 1. Alabama Highway Department/U.S. Bureau of Public Roads, 1965

[21] AS-1012.14, Methods of Testing Concrete—Method for Securing and Testing Cores from Hardened Concrete for Compressive Strength. Standards Australia, 1991

[22] ASTM-C39, Standard Test Method for Compressive Strength of Cylindrical Concrete Specimens. American Society of Testing and Materials, 2000

[23] AS-1012.17-97, Methods of Testing Concrete—Determination of the Static Chord Modulus of Elasticity and Poisson’s Ratio of Concrete Specimens. Standards Australia, 2014

[24] ASTM-C469/C469M-14, Standard Test Method for Static Modulus of Elasticity and Poisson’s Ratio of Concrete in Compression. American Society of Testing and Materials, 2000

[25] AS-1141, Methods for Sampling and Testing Aggregates. Standards Australia, 2011

[26] T235. R., Aggregate Least Dimension. Roads AND MARITIME SERVICES (RMS) NSW, Australia, 2006

[27] T239. R., Aggregate fractured faces. Roads and Maritime Services (RMS) NSW, Australia, 2006

[28] Bischoff P H. Deflection calculation of FRP reinforced concrete beams based on modifications to the existing Branson equation. Journal of Composites for Construction, 2007, 11(1): 4–14

[29] Alshaikh A H, Al-Zaid R. Effect of reinforcement ratio on the effective moment of inertia of reinforced concrete beams. ACI Structural Journal, 1993, 90(2): 144–149

[30] ACI-318-08. Building Code Requirements for Structural Concrete and Commentary. ACI Committee, American Concrete Institute, International Organization for Standardization, 2008

[31] ACI-435, Control of Deflection in Concrete Structures, ACI 435R-95. American Concrete Institute, 2000

[32] AS-3600-09, Concrete Structures. Standards Australia, 2009

[33] Bischoff P H, Gross S P. Equivalent moment of inertia based on integration of curvature. Journal of Composites for Construction, 2010, 15(3): 263–273

[34] Bischoff P H, Paixao R. Tension stiffening and cracking of concrete reinforced with glass fiber reinforced polymer (GFRP) bars. Canadian Journal of Civil Engineering, 2004, 31(4): 579–588

[35] CEBFIP-MC90. CEB-FIP Model Code (MC-90).London: Thomas Telford Ltd., 1993

[36] Newhook J. Reinforcing Concrete Structures with Fibre Reinforced Polymers. ISIS Canada: Design Manual No.3, The Canadian Network of Centres of excellence on intelligent sensing for innovative structures, 2001

[37] ACI-224. Cracking of Concrete Members in Direct Tension. In: ACI Journal Proceedings. Farming Hills: American Concrete Institute, 1986

[38] AS-1478.1, Methods for Sampling and Testing Aggregates, Particle Size Distribution-Sieving Method. Standards Australia, 2000

[39] AS-3972, General Purpose and Blended ceMents. Standards Australia, 2010

[40] AS-2350, Methods of testing Portland and Blended Cements. Standards Australia, 2006

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