This means that if the temperature in a concrete beam varies with the depth y of the cross section, Young’s modulus also varies with the depth of the beam, 𝐸(𝑇) = 𝐸(𝑇(𝑦)) = 𝐸(𝑦), since it is dependent of the temperature. In turn, this makes the strain due to mechanical loading, equation (3.1), dependent of the temperature and the depth, given as;
𝜀𝛽 = 𝜎
𝐸(𝑇(𝑦)) (3.6)
3.3 Strain sensors
A sensor, transducer, converts a physical phenomenon into an electrical signal. There are different types of sensors for measuring strains. Here strain gauges and optical fibre sensors will be presented.
3.3.1 Strain gauge
A strain gauge usually consists of a wire or a metal foil between a flexible backing material such as thin paper or an epoxy-type plastic film [17]. The metal foil is placed in a pattern as shown in figure 3.1. The strain gauge is bonded on to a structural element. When the
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structure is exposed to forces, moments, pressure etc., the element will either elongate or contract. This causes the gauge length to elongate or contract as well. This change in the gauge length can either be used to calculate the strain in the structure indirectly by using equation (3.3) where the change in gauge length is divided by the initial gauge length.
More commonly today is to use an electrical resistance strain gauge, where the strain gauge is connected to an electrical circuit that can measure the change in resistance in the strain gauge. When the gauge length changes, the electrical resistance in the strain gauge also changes and it is then possible to find the strain with the expression for the gauge factor GF;
𝐺𝐹 =∆𝑅 𝑅⁄
𝜀 (3.7)
where R is the resistance of the strain gauge and ΔR is the change in resistance. The gauge factor and resistance of a strain gauge is given by the manufacturer and varies with the material used for the metal foil.
The number of desirable elements in a strain gauge, as shown in figure 3.1, is dependent on the stress-condition of interest. If there are stresses in only one direction, or only one direction is of interest, a single element foil gauge may be used. A two-element and three-element rosettes are used when there are stresses in two directions, for either known or unknown directions respectively.
Figure 3.1: Metal foil strain gauges. (a) single element, (b) two-element rosette, (c) three-element rosette.
The performance of a strain gauge is depending on the foil material. For good performance, the material should have a high gauge factor and high resistivity. If the number of loops in a strain gauge is increased while the gauge length is held constant, the resistance of the gauge is increased [18]. The material should preferably also have low temperature sensitivity. This is important because changes in the temperature cause changes in the resistivity of the material. The structural expansion or contraction due to temperature also causes strains that the strain gauge cannot separate from the mechanical loading.
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3.3.2 Optical fibre sensors
Optical fibres are very thin, often made from fused silica when used for structural monitoring and can transmit light over large distances. The two main parts of optical fibres are the core, which will transmit the light along the optical fibre, and a cladding which surrounds the core. The purpose of the cladding is to keep the light waves in the core and not let them disappear out. To obtain this function the cladding has a slightly lower refraction index than the core [19].
Compared to electrical resistance strain gauges which need to be connected to an electrical circuit, an optical fibre sensor uses light which hinders electromagnetic interference in influencing the measurements. Similar to the strain gauge, strain measurements by optical fibre sensors are also affected by changes in the temperature, both because of thermal expansion and because temperature influences the refraction index of the optical fibre.
3.3.2.1 Fibre Bragg gratings
Fibre Bragg gratings are a variety of optical fibre sensors where the surface of the optical fibre has been exposed to UV light [20]. This cause changes in the refractive index, and by sending pulses of UV light through the core of the optical fibre, the refractive index becomes periodic and enables the fibre Bragg grating to reflect one specific wavelength of light, called the Bragg wavelength λB;
𝜆𝐵 = 2𝑛𝑒𝑓𝑓Λ (3.8)
where neff is the effective refraction index and Λ is the spacing of the grating. Fibre Bragg gratings have the ability to measure both strain and temperature and it is possible to calculate the change in temperature ΔT and the strain ε in the optical fibre if the change in the Bragg wavelength is known as in [20];
∆𝜆𝐵= 𝜆𝐵[(𝛼𝑜𝑝𝑡+ 𝜉)∆𝑇 + (1 − 𝑃𝑒)∆𝜀] = 𝐾𝑇∆𝑇 + 𝐾𝜀𝜀 (3.9) where αopt is the thermal expansion coefficient, ξ is the thermo-optic coefficient and Pe is the photo-elastic constant for the optical fibre. KT and Kε are linear coefficients for temperature and strain respectively.
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4 Creep and shrinkage
Over time concrete are exposed to the long-term effects creep and shrinkage. When concrete is stressed for a long period of time the concrete will continue to be compressed beyond the instantaneous deformation at the moment of loading without there being added any new load. This additional deformation is called creep. When the concrete is drying out, it shrinks. This effect is called shrinkage and is not affected by the load applied to the concrete. Both creep and shrinkage are dependent of the concrete strength, the dimensions of the concrete element exposed to drying and the relative humidity [21].