3.2. Change in Stiffness of Columns with PBO–FRCM Reinforcement
As the columns were being tested, the longitudinal and transverse strains were measured by strain gauges arranged along the circumference of the columns at half of their height. Depending on the column type, different arrangements of strain gauges were adopted. In the reference columns C_C_0, C_C_16 and C_C_32, two vertical strain gauges, V0 and V2, and two horizontal strain gauges, H1 and H3, were used. Strain gauges V0 and H1 were located on the side where the force acted at the eccentricity (the more compressed side) (
Figure 5a). Six strain gauges were used in the case of columns C_1H, C_2H, C_1V1H and C_1V2H. Two vertical strain gauges, V0 and V5, were located in the plane of compression. The next four strain gauges, H2, H4, H7 and H9, measured circumferential strains (
Figure 5b).
The horizontal columns’ displacements (deflections) were measured by Linear Variable Differential Transformers (LVDTs, HBM Masstechnik, Darmstadt, Germany). The measurement span of the transducers was ±10 mm. The LVDTs were mounted on a separate steel frame while the measurement took place at half the height of the columns (
Figure 6) [
19]. The columns were tested until failure under monotonically increasing displacement. The load, strains and horizontal displacements were acquired with an automatic data acquisition system.
The curvatures of the specimens were calculated from Equation (7) on the basis of the measured maximum longitudinal strains
εv2,lim and
εv1,lim on the more and less compressed (tensioned) sides of the cross-section, respectively.
While analyzing the value of longitudinal strains
εv, in the confined columns with the PBO mesh only, with horizontal layout fibers over the main direction (C_1H and C_2H), failure was observed at a comparable level of strain (
Table 4). For the columns in the group C_1H, the limit compression strains amount to 2.736‰–2.962‰; the values in group C_2H amount to 2.827‰–3.200‰. In both columns groups that were loaded at the core limit, C_1H_32 and C_2H_32, at the failure stage, there occurred tension on the side opposite to the action of load. The presence of the longitudinal strengthening reduces the limit strains
εv2 of axially compressed columns at which point the destruction of the section occurs, which is fairly unfavorable. For instance, in the element C_1H_0, the strain
εv2,max = 2.736‰, and the additional longitudinal strengthening in the element C_1V1H_0, resulted in a decrease in these strains to
εv2,max = 2.392‰. In contrast, the strains for the elements C_2H_0 and C_1V2H_0 were recorded:
εv2,max = 3.200‰ and
εv2,max = 1.734‰, respectively. The impact of the longitudinal PBO mesh on the limit values of compression strains is evident in the element groups C_1V1H and C_1V2H. It is evident in both groups C_1V1H and C_1V2H that eccentrically compressed elements are capable of transferring considerably higher compression strains on the side of the action of force than axially compressed elements. In addition, the value of these strains rises jointly with the rise in eccentricity.
The bending moments at the instant of failure (
Mmax) were calculated from (8) on the basis of the maximum deflections
wmax (
Figure 7).
The bending stiffness of columns can be numerically analyzed with the use of Bernoulli’s hypothesis with or without eccentric load and additional reinforcements such as fiber materials. The additional longitudinal composite reinforcements contribute to the increasing bending stiffness directly and transverse composite reinforcements give confinement effect to increase stiffness. The axial stiffness should not be evaluated under the combination of axial force and bending moment.
Assuming that Bernoulli’s hypothesis is applicable in this case (plane section remains plane) and starting with the general dependence between the curvature of the specimen’s deformed axis (1/
r), bending moment
Mmax and bending stiffness
EI (9), the change in stiffness was analyzed depending on the type of strengthening of the column and the stress intensity in the latter.
The next three diagrams (
Figure 8,
Figure 9 and
Figure 10) show the change (decrease) in the stiffness of the analyzed columns depending on their stress intensity. The horizontal axis represents the ratio of column stiffness at failure
EI to initial column stiffness (
EI)
P=0 for the load eccentricity of, respectively, 0, 16 and 32 mm. The vertical axis represents the ratio of the ultimate force to the load capacity of the axially compressed column in a given group for the load eccentricity of 0, 16 and 32 mm. The broken line marks the trend in stiffness change.
In the case of column C_C_0 (most stressed), the stiffness at the point of failure amounts to 35% of the initial value (
Figure 8). For the unstrengthened columns loaded at the initial eccentricity of 16 mm and 32 mm (C_C_16 and C_C_32), which were put under less stress, the stiffness at the point of failure amounts to, respectively, 45% and 71% of the initial stiffness value. The elasticity modulus value of the “plain concrete” decreases until about 0.2
Ec,max before failure. The smaller decrease in stiffness of the reinforced concrete columns, than that resulting from the change in the elasticity modulus of the “plain concrete” itself, is evident due to the presence of the longitudinal reinforcement and the shape of the cross-section of the columns.
A similar trend in the change of stiffness is observed in the columns with a single layer (1H) of transverse composite reinforcement (
Figure 9). The addition of another layer (2H) of transverse composite reinforcement results in greater stiffness of the composite jacket, and so of the whole cross-section (
Figure 10). This is illustrated by the slope of the trend line in the two diagrams.
The stiffness of the composite jacket in these investigations is defined with the equivalent modulus of elasticity of the PBO–FRCM strengthening according to the following formula:
where
Ef is given in
Table 2 and
R is the radius of a circle with the circumference equals the circumference of a considering cross-section. For the considered columns with the square cross-section with the side length
a:
One should note here that in comparison to the reference columns, the decreases in load capacity were observed for columns C_1V2H_0 and C_1V2H_16 (
Table 5) [
18]. This is not surprising as it was caused by the increase in the stiffness of the columns due to the little-deformable composite jacket. The longitudinal composite reinforcement reduces the longitudinal deformability of the columns, which is rather disadvantageous. Stiffer transverse composite reinforcement reduces the ability of the columns to deform (deflect) in the bending plane. This observation applies particularly to axially compressed columns at a slight eccentricity. An analysis of the diagrams shows that the stiffness of the columns strengthened with PBO mesh on mineral mortar depends on the intensity of stress in the concrete confined by the composite jacket. The stress intensity depends on the eccentricity, which equals the sum of the initial eccentricity and the deflection of the column.
The next two diagrams (
Figure 11 and
Figure 12) and
Table 5 show the change in the stiffness of the columns as a function of the maximum (ultimate) force registered in the course of the tests. In
Figure 11, which illustrates the behavior of the columns strengthened only transversely, one can see that the introduction of one (1H) or two (2H) layers of transverse composite reinforcement results in an increase in the stiffness of the composite jacket. The stiffness of the columns in the state of the ultimate bearing capacity depends on the intensity of stress in the cross-section at the instance of failure. The stress intensity does not increase geometrically with the number of strengthening layers. The columns with longitudinal composite reinforcement behave completely differently (
Figure 12). In this case, the columns’ stiffness is determined by the presence of the longitudinal composite reinforcement. The lower stress intensity, in comparison with the specimens of type C_1H and C_2H, is accompanied by a reduction in the flexural rigidity of the columns. The application of composite reinforcement along the axis of the columns resulted in an increase in their longitudinal stiffness. Both types of columns: C_1V1H and C_1V2H show considerably greater ductility than the corresponding columns without longitudinal composite reinforcement C_1H and C_2H. This is reflected in the lower value of stiffness at failure at lower stress intensities, in comparison with the columns of type C_1H and C_2H.
The ductility of the columns in these investigations is defined as the ability to horizontally displace the columns, which is induced with the bending moments (eccentric load) what is presented in
Figure 13.
Mmax is the first-order moment. The slenderness ratio of the RC columns
λ <
λlim according to [
29].
The effect of the composite jacket in PBO–FRCM columns is closely connected with the variation in the elasticity modulus of the concrete (
Ec) due to the stress destruction of the concrete core. Microcracks develop in the concrete beyond the level of stress in the column at which Poisson’s ratio
ν is no longer a liner [
35]. As a result of the damage, the load-carrying surface area is reduced and consequently the stiffness of the member decreases. The next graphs (
Figure 14,
Figure 15,
Figure 16 and
Figure 17) show Poisson’s ratio versus eccentricity for the analyzed columns. One can see that beyond a certain stress value, Poisson’s ratio
ν quickly increases, which is due to the extensive microcracking of the concrete core. This stress level corresponds to 60–70% of the maximum (ultimate) force
Pmax observed during the tests.
As the stress further increases, the rate of volumetric changes begins to fall. The concrete is no longer a continuous body, undergoes disintegration and is held only by the external composite jacket. This situation lasts until the reinforcement at the end of the overlap of the PBO mesh starts to delaminate. In the columns strengthened only transversely, i.e., C_1H and C_2H, the Poisson ratio exceeds 0.5, and the volumetric strain assumes negative values. In the case of columns C_1V1H and C_1V2H, the effect of the longitudinal PBO mesh (reducing the compressive stress increment) is clearly visible and ratio ν < 0.5.
With regard to the variability of PBO–FRCM column stiffness, the variation in the ratio of transverse strain to longitudinal strain (Poisson’s ration ν), due to the destruction of the concrete inside the composite jacket should be taken into account in the standards.