Effects of shrinkage on the behavior of structural elements

grouped into two main groups: the first is related to the environmental conditions, such as relative humidity, ambient temperature and wind velocity, whereas the latter is related to the intrinsic characteristic of the concrete material, such as the content and type of aggregates, the water and cement content as well as the presence of additives.

As far as environmental conditions are concerned, the major role is played by the relative humidity; a low ambient humidity produces larger gradients near the drying surfaces, thus increasing the drying rate.

As regards the intrinsic characteristic of the concrete mix, aggregates influence a lot drying shrinkage. They are inert, due to their low permeability, so they restrict the overall deformations of the material, providing restraining actions to the cement paste that undergoes drying shrinkage. In particular higher aggregate concentration and higher aggregate stiffness result in lower shrinkage strains. Greater water and cement concentrations produce instead higher shrinkage deformations. In case of high water content drying shrinkage increases due to larger amount of evaporable water; whereas higher cement content results in a lager fraction of cement paste in concrete which represents the shrinking part of the material. Considering the water to cement ratio, reducing this value produces a reduced porosity of the cement paste, leading in turn to lower shrinkage strains. Moreover, also the inclusion of additives in the concrete admixture influences the drying shrinkage potential, since they modify the microstructure of the cement paste, as well as the pore structure.

3.2.2 Effects of shrinkage on the behavior of structural elements

indeed related to the moisture gradient that is not uniform within the member.

The drying front migrates indeed gradually from the exposed surface to the interior of concrete elements. It follows that the tendency to shrink is largest on the external surfaces due to rapid moisture loss and lower in the inner parts of the element, furthest from the drying surface. Since the higher shrinkage strains on the external surfaces are restrained by the lower ones of the inner points, a non uniform strain field arises within the member, termed as differential shrinkage. This phenomenon gives rise to the development of an internal state of stress to restore strain compatibility, so as to ensure that sections remain plane.

The induced eigen-stresses consist of tensile stresses near the external surfaces, while the inner core is compressed, as can be seen in Figure 3.1.

Figure 3.1 Differential drying shrinkage and induced stresses within a cross-section [124]

It is worth noting that also creep is involved in the process; since the internal stresses induced by shrinkage develop gradually with time, they are relieved by creep. Because of the high magnitude of external shrinkage strains the relief caused by creep in not often enough to prevent cracking of concrete. Tensile stresses near the drying surface may indeed exceed concrete tensile strength, leading to surface cracking before the application of any load and giving rise to possible aesthetic, durability and serviceability problems. In particular, the degree of restraint to shrinkage, the extensibility and strength of concrete in tension, as well as creep effects play a major role.

It must be underlined that differential shrinkage due to unsymmetrical drying may even cause warping in the concrete member. Moreover, in case of significant external restraints provided to shrinkage deformation, high tensile stresses develop and wide cracks can be often observed.

3.2.2.2 Reinforced elements

Structural concrete elements are usually reinforced with steel bars. They provide restraint to concrete shrinkage due to the bond action.

Considering a member reinforced with a single centered bar (see Figure 3.2), it can be observed that as the concrete shrinks, the reinforcement becomes compressed and imposes an equal and opposite tensile force to concrete. Due to this constraint to concrete shrinkage, the average shortening of the member is reduced, assuming the compatibilized value ε^mi. By following the formulation proposed in [119] and with reference to an uncracked tension member supposed characterized by a linear elastic behavior and subjected to a uniform shrinkage strain ε^sh, the average member strain ε^mi due to shrinkage can be evaluated as:

) 1 (

1 ε ρ

ε^mi= ^sh +n , (3.1)

being n the modular ratio (Es/Ec) and ρ the reinforcing steel ratio (As/Ac). It is worth noting that shrinkage strain ε^sh is assumed as negative.

Equation (3.1) is calculated for the condition of zero axial load (i.e. N=0) by by substituting the compatibility equations:

sh c

mi ε ε

ε = +

s

mi ε

ε =

(3.2)

into the equilibrium equation:

s s s c c c s

c N A E A E

N

N= + =

ε

ε

being ε^{c and} ε^s respectively the concrete and the steel strains caused by stresses.

Moreover, the appearance of tensile stress in concrete equal to σ^{c = -(Ec} ε^{sh n} ρ)/(1+nρ) leads to a reduction in the loadrequired to crack the

member. It results:

( )









 + +

= ρ

ρ ε

n n f

N E

N ^sh

ct c sh cr

,

cr 1 1 (3.4)

being Ncr,sh and Ncr the cracking load respectively for a shrunk and a non-shrunk member.

Section Elevation Strain Stress

Figure 3.2 Strains and stresses on an uncracked section of an uniaxial tension member, due to an hyopotized uniform shrinkake strain ε^{sh [132]}

In case of high shrinkage strain the member may crack prior to loading. If the element cracks the initial shortening caused by shrinkage, will be reduced compared the value derived from Equation (3.1) and more refined approaches should be applied.

It is worth noting that the reported equations represent only a simplified formulation, reported to better highlight the phenomenon. However, this approach is not exhaustive. The concrete is indeed characterized by a nonlinear behavior and by differential shrinkage, due to the uneven tendency to shrink of points at different distances from the drying surface, as explained in §3.2.2.1.

Moreover, since shrinkage induced stresses developed gradually with time, also creep effects influence the development of the stress fields. It follows that creep cannot be neglected in calculation.

Passing now to the description of the behavior of RC elements subjected to shrinkage with reinforcement not symmetrically placed in the depth of the section, it can be stated that a shrinkage-induced curvature takes place (see Figure 3.3).

Shrinkage, in an unsymmetrically reinforced concrete member, like a RC beam, produces indeed deflections, also of significant magnitude, before the application of any external load. While concrete shrinks, it compresses the bonded rebars that impose equal and opposite tensile forces on concrete at the level of the reinforcements. Differently from an element symmetrically reinforced, in this case the resulting tensile force acts at some eccentricity to the centroid of the concrete cross-section, resulting in a gradual warping of the member.

The shrinkage induce curvature (in sign and magnitude) depends on the amount and position of the reinforcements. Considering for example a RC beam if the amount of bottom (tensile) reinforcement is greater than the top (compressive) one a positive deflection takes place, on the contrary for heavy top reinforcements a negative (hence, upward) deflection appears.

Figure 3.3 Strains and stresses in a singly reinforced beam, due to an hypnotized uniform shrinkage strain ε^{sh [117]}

As already observed for symmetrically reinforced concrete elements, cracking may occur prior to loading if the induced tensile stresses overcome the tensile strength of concrete. It is worth noting that shrinkage warping may significantly increase in case of cracked elements. Moreover, as in case of symmetrically reinforced members, creep effects and differential shrinkage should be always considered in simulations.

The importance of considering shrinkage becomes even more pronounced in case of externally restrained RC members. External restraint to shrinkage is very common in RC structure, due to the presence of supports of the structural elements or to the connection of the elements to other part of the structure.

These restraints prevent the element to shorten or warp, causing internal stresses and deformations that, in a statically indeterminate structure, result in additional internal actions. As a matter of fact, the reactions of a restrained indeterminate member change because of shrinkage, leading to a possible significant redistribution of moment and shear, as well as to the appearance of tensile stresses that may cause premature cracking.