The purpose of this study was to examine several existing time dependent moduli of elasticity (MOE) of concrete and evaluate their effect on various prestress losses in typical bridge girders. Towards this end, a method proposed by Dischinger (1939a, b, c) for a variable MOE was considered herein, along with a couple of other methods from the literature. The MOE from Dischinger’s method varies with the concrete creep function. The current American Association of State Highways and Transportation Officials (AASHTO) specifications consider the concrete MOE to remain constant through the life of a structure (AASHTO 2010). AASHTO’s calculation of the concrete MOE is based on the unit weight and the 28 day compressive strength of concrete. The two additional methods considered herein were the American Concrete Institute (ACI) 209 Model Code (1992) and CEB-FIP 1990 Model Code (2010). A realistic concrete MOE that varies with concrete age is likely to result in more precise estimation of various prestress losses in concrete girders, resulting in more realistic girder design, prestress estimation, camber calculations and bridge deck design/construction.
Equations for variable MOE of concrete considering types of mineral admixtures and coarse aggregates were developed previously (Nemati 2006). Valuable data and general trends in concrete strengths, creep coefficient and MOE for typical Florida concrete were generated through another study (Tia et al. 2005). Yazdani et al. (2005) developed concrete MOE models based on aggregate classes in Florida and a variable concrete strength. There have been no other studies in the past in which the effects of a variable and time dependent concrete MOE on prestress losses were investigated.
The Model Code 2010 1,2 constitutes a significant step forward with respect to basing design on more physical and more comprehensive models. With regards to the punching shear provisions, an in-depth review of the previous versions of th e code (Model Codes 78 3 and 90 4) was performed. The b Model Code for Concrete Structures 2010 A Critique othe Comments on Finite Element Analysi s Introduction The following extracts from the fib website explain the history of the organisation and the rational of their 2010 publication: In section 7.11.2.2 of this document a description of the Finite Element Method is provided.
1.1 AASHTO Approach
The AASHTO regulates highway bridge design in the United States. Currently, all bridge design in the state of Texas is performed in accordance with the AASHTO LRFD (Load and Resistance Factor Design) 2007 specifications (TxDOT 2012). The constant MOE of concrete as specified by AASHTO Equation 5.4.2.4-1 (in U.S. units), reproduced in its SI form in Eq. (1).
$$ E_{c} = 0.043 times w_{c}^{1.5} times sqrt {f_{c}^{'} } $$
where Ec is the concrete MOE (MPa), wc is the unit weight of concrete (kg/m3), and f′c is the 28 day compressive strength of concrete (MPa).Equation (1) is valid for concrete with unit weights in the range of 1,442 kg/m3 (0.90 lbs/ft3) and 2,483 kg/m3 (155 lbs/ft3). The Texas Department of Transportation (TxDOT) considers this equation valid for 28 day compressive strengths up to 58.6 MPa (8.5 ksi) (TxDOT 2005). For the purposes of this work, the unit weight of concrete was taken as 2,403 kg/m3 (150 lbs/ft3). The MOE for prestressing strands was taken as 196.5 GPa (28,500 ksi) per AASHTO LRFD Section 5.4.4.2. AASHTO LRFD Equation 5.9.5.1-1 expresses the prestress loss in girders, as follows:
where ( Updelta f_{pT} ) is the total loss, ( Updelta f_{pES} ) is the loss due to elastic shortening, and ( Updelta f_{pLT} )is the losses due to long-term shrinkage and creep of concrete, and steel relaxationThe loss due to elastic shortening of concrete is given by AASHTO Equation 5.9.5.2.3a-1, as follows:
$$ Updelta f_{pES} = frac{{E_{p} }}{{E_{ci} }} times f_{cgp} $$
where Ep is the MOE of prestressing steel, Eci is the MOE of concrete at transfer, and fcgp is the sum of concrete stresses at the center of gravity of prestressing tendons due to the prestressing force at transfer and the self-weight of the member at the sections of maximum moment.The long-term loss is given by AASHTO Equation 5.9.5.3-1, and is reproduced in Eq. (4). In this equation, the first term corresponds to creep loss, the second term to shrinkage loss and the third to relaxation losses.
fpi is the prestressing steel stress prior to transfer, f′ci is the specified concrete compressive strength at time of prestressing, Aps is the area of prestressing steel, Ag is the gross cross sectional area of girder, γh is the correction factor for relative humidity, γst is the correction factor for specified concrete strength at transfer, and Δfpr
is the estimation of relaxation loss, taken as 16.6 MPa (2.4 ksi) for low relaxation strands.
1.2 Dischinger Method
Franz Dischinger (1887–1953) was a well-known German civil and structural engineer who was responsible for the development of the modern cable-stayed bridge. He is known for his work in prestressed concrete and, in 1939, published a theory called “Elastic and Plastic Distortions of Reinforced Concrete Structural Members and in Particular of Arched Bridges” (1939a, b, c). Dischinger showed that the MOE is a function of time since the creep of concrete is also a function of time. Dischinger’s evaluation of the change in concrete MOE with time was based on a creep coefficient determined from laboratory tests. He proposed the following equation for concrete MOE:
where Eot is the modified MOE at time t, Eo is the initial MOE, and ( psi_{t} ) is the creep coefficientAASHTO specifies a concrete creep coefficient (AASHTO Equation 5.4.2.3.2-1), as follows:
ks is the factor for effect of volume to surface ratio, kf is the factor for the effect of concrete strength, khc is the humidity factor for creep, ktd is the time dependent factor, V/S is the volume to surface ratio, t is the time (days), and ti is the age of concrete at time of load application (days).
In this study, the creep coefficient from Eq. (8) was calculated and used as the basis for the Dischinger Model (Eq. (7)).
1.3 ACI 209 Method
The ACI 209 Model Code (1992) specifies a time dependent concrete MOE based on a time dependent 28 day compressive strength. The time variable compressive strength (ACI Eq. 2.1) is as follows:
where f′c(t) is the compressive strength at time t; t is the time in days; (f′c
)28 is the 28 day compressive strength; and a and β are constants depending on curing and cement type, respectively.The values for a and β are reproduced in Table 1. For this study, Type I cement and moist cured concrete were assumed. When girders are manufactured, the concrete is normally steam cured to allow for quick turnaround. Both steam and moist curing were checked herein to note the difference in the values for the MOE. The difference in MOE was not significant for either curing type. ACI Equation 20.25 for the variable MOE is given in Eq. (14).
where Ec(t) is the MOE of concrete at age t days (MPa), w is the unit weight of concrete (kg/m3), and f′c(t) is the compressive strength at time t in days (MPa, Eq. (13)).
Cement type
Curing
Duration
I
III
ts (days)
Strength
a
4.0
2.3
Moist
1.0
0.7
Steam
β
0.85
0.92
Moist
0.95
0.98
Steam
1.4 The CEB-FIP Method
The CEB-FIP Model Code (2010) was initially published in 1978 and since then has impacted national codes in many countries. ACI and other well-known codes have referenced the CEB-FIP Code in their publications. The CEB-FIP gives time dependent concrete MOE is given in CEB-FIP Eq. 2.1-57, and presented in the following:
$$ E_{ci} left( t right) = beta_{E} (t)E_{ci} $$
where Eci(t) is the concrete MOE at an age of t days, Eci is the concrete MOE at an age of 28 days, and βE(t) is a coefficient depending on the age of concrete (t days); it is given by CEB-FIP Eq. 2.1-58 and is as follows:
where t is the age of concrete (days), t1 is 1 day, s is a coefficient which depends on the type of cement; s = 0.20 for rapid hardening high strength cements, 0.25 for normal and rapid hardening cements and 0.38 for slowly hardening cements. In this study, normal hardening cement was assumed. The CEB-FIP Code accounts for maturity of the concrete by allowing the time in days to be adjusted for temperature. In this study, this temperature effect on concrete maturity was not considered. The example bridge location for this study (as described later) was assumed to be in a stable environment with the temperature range per season to remain fairly constant.