When casting prestressed structures, it is normal to use concrete with higher strength than what is used in regular reinforced structures. A reason for this is that the relatively large forces acting on the prestressed concrete section demands large dimensions to withstand the pressure. A high strength makes it possible to minimize the dimensions, and thereof minimize the self-weight.
The high strength also keeps the structure from having large deformations. Concrete classes B35 – B55 are mostly used, but in some cases higher strength is needed [11] [10].
To achieve a concrete which provides as little creep and losses as possible, special mix design and composition are chosen. By avoiding large creep and losses, it is possible to reduce the loss of effective prestressing.
By using a firm concrete with low water-to-cement-relation, corrosion in the prestressing tendons is avoided.
3.1.1 COMPRESSIVE STRENGTH
To describe the compressive strength of concrete, the characteristic cylinder strength fck at 28 days is used. EC2 states that this compressive strength is taken as the strength where there is a 95% or more of which test results do not fail. Corresponding mechanical properties are given in EC2 [13] [15].
If it is necessary, the compressive strength at a specific time t different from 28 can be found as following [13]:
𝑓𝑐𝑘(𝑡) = 𝑓𝑐𝑚(𝑡) − 8(𝑀𝑃𝑎) 𝑓𝑜𝑟 3 < 𝑡 < 28 𝑑𝑎𝑦𝑠 3.1 𝑓𝑐𝑘(𝑡) = 𝑓𝑐𝑘 𝑓𝑜𝑟 𝑡 ≥ 28 𝑑𝑎𝑦𝑠 3.2
According to EC2 one can determine the mean compressive strength of concrete fcm(t) at age t from the mean strength fcm at age 28 days as shown in equation 3.3. This value depends on type of cement, temperature and curing conditions [13].
𝑓𝑐𝑚(𝑡) = 𝛽𝑐𝑐(𝑡)𝑓𝑐𝑚 3.3
where
𝛽𝑐𝑐(𝑡) = 𝑒𝑥𝑝 {𝑠 [1 − (28 𝑡 )
0,5
]} 3.4
The factor s depends on the cement strength class, and t is given in days.
To determine the value for the design compressive strength fcd, the characteristic cylinder strength is divided with a safety factor, as following
𝑓𝑐𝑑= 𝛼𝑐𝑐𝑓𝑐𝑘 𝛾𝑐
3.5 where c is the partial safety factor for concrete set to 1,5, and cc is a coefficient which takes the long-term effects on the compressive strength into account. cc is normally set to 0,85 [13].
For concrete class B45 the design compressive strength is equal to 𝑓𝑐𝑑 =0,85∗45𝑀𝑃𝑎
1,5 =
25,5𝑀𝑃𝑎. From this, stress-strain relation can be used to design the cross-section.
𝜎𝑐 = 𝑓𝑐𝑑[1 − (1 − 𝜀𝑐 𝜀𝑐2)
𝑛
] 𝑓𝑜𝑟 0 ≤ 𝜀𝑐 ≤ 𝜀𝑐2 3.6 𝜎𝑐 = 𝑓𝑐𝑑 𝑓𝑜𝑟 𝜀𝑐2 ≤ 𝜀𝑐 ≤ 𝜀𝑐𝑢2 3.7 where
n is the exponent according to table 4.1 in EC2
c2 is the strain reaching the maximum strength, = 2.0*10-3 according to Figure 3.1
cu2 is the ultimate strain, = 3.5*10-3 according to Figure 3.1
3.1.2 TENSILE STRENGTH
EC2 states that the axial tensile strength fct is the highest stress the concrete can withstand when subjected to centric tensile loading [13]. As testing for uniaxial tensile strength are difficult to perform, a splitting tensile strength fct,sp is found by testing to determine an approximate value [11]. This approximate value is found by equation 3.8.
𝑓𝑐𝑡 = 0,9𝑓𝑐𝑡,𝑠𝑝 3.8
The tensile strength fctm(t) develops with time and depends on highly on the curing and drying conditions and the dimension of the structure, and is assumed equal to:
𝑓𝑐𝑡𝑚(𝑡) = (𝛽𝑐𝑐(𝑡))𝛼𝑓𝑐𝑡𝑚 3.9 where
fctm is the mean tensile strength of the concrete, found in Figure 3.1
cc(t) follows from equation 3.4
= 1 for t < 28
= 2/3 for t 28
Figure 3.1 Strength and deformation characteristics for concrete [11]
3.1.3 DEFORMATIONAL PROPERTIES
When looking at the deformational properties, the once of most interest are the elastic moduli, creep and shrinkage deformation.
3.1.3.1 Elastic moduli
The elastic modulus is a value which tells us something about the stiffness of the concrete. With higher elastic modulus, the stiffness increases. Deformations due to elasticity are highly dependent on the composition of the concrete, especially the aggregates.
The secant modulus, also called modulus of elasticity or Young´s modulus, are shown in Figure 3.2 as Ecm and is defined as the ratio between the applied stress and the corresponding strain which occurs within the elastic limit. Values for Ecm between c=0 and c=0,4fcm are given in Figure 3.1 for concrete with quartzite aggregates. For concrete with aggregates such as limestone and sandstone, the value of Ecm are to be reduced by 10% and 30% respectively. The value is to be reduced by 20% when using basalt aggregates [13] [16].
Figure 3.2 Idealised stress-strain relationship for concrete in uniaxial compression [11]
3.1.3.2 Creep coefficient
Deformation due to creep occurs after a load is applied to the structure. With time the deformation of the concrete gradually increases, an may reach a value as high as three to four times the immediate elastic deformation. P. Bhatt states that “creep is defined as the increase of strain with time when the stress is held constant” [15].
The total creep deformation of concrete cc(,t0) due to constant compressive stress c applied at time t0 is calculated as in equation 3.10 [13].
𝜀𝑐𝑐(∞, 𝑡0) = 𝜑(∞, 𝑡0)𝜎𝑐 𝐸𝑐
3.10 Where
(,t0) is the creep coefficient. This value is related to the tangent modulus Ec that can be taken as 1.05 Ecm. If the compressive stresses which are subjected to the concrete are less than 0,45fck(t0) at an age t0, the tangent modulus can be taken from Figure 3.2 [13].
3.1.3.3 Shrinkage
The shrinkage of concrete is affected by the same parameters as the creep coefficient, such as the ambient humidity, compressive strength, element dimensions and composition of concrete.
The total shrinkage strain cs is made up by two components, the plastic shrinkage strain and the drying shrinkage strain. The plastic shrinkage appears during the hardening of the concrete, whereas the drying shrinkage develops slowly due to loss of water in the concrete [13] [15].
The total shrinkage strain is found as following:
𝜀𝑐𝑠 = 𝜀𝑐𝑑+ 𝜀𝑐𝑎 3.11
where
c is the drying shrinkage strain
c is the autogenous shrinkage strain (plastic strain)
3.1.3.4 Thermal stress
Changes in temperature may have consequences for exposed structures, as large internal forces occur if the deformations are prevented. Heating or cooling of parts of a structure creates thermal gradients that induces stresses. For a structure of length L that rests on a frictionless surface, the raise in temperature T gives an increase in length of L=T*T*L. This gives the corresponding thermal strain T:
𝜀𝑇 = 𝛼𝑇∗ ∆𝑇 3.12
where T is the coefficient of thermal expansion which from EC2 is set to 10*10-6/oC. If the deformations are prevented, a compressional stress of 𝜎 = 𝐸 ∗ 𝜀𝑇 occurs [2][11] [13].
3.1.4 MATERIAL PARAMETERS
Material properties for Bagn bridge are given in Table 3.1.
Table 3.1 Material properties for concrete Concrete B45
Characteristic compressive strength fck 45 MPa
Mean axial tensile strength fctm 3.8 MPa
Mean elastic modulus Ecm 36 GPa
Characteristic cylinder strength after 28 days fcck 36 MPa
Coefficient cc 0.85
Partial factor of safety c 1.5
Weight of unreinforced concrete Wc 2400 kg/m3