Analytical Studies 1 Rigid-Plastic Analysis

Jensen and Braestrup11_{presented the first analytical study that attempted to provide a theoretical basis}

for the existence of a linear relationship between ultimate pullout load and compressive strength. They assumed that concrete obeys the modified Mohr<Coulomb failure theory (sliding or separation possible) and that the extracted cone has the shape of the idealized conic frustum. The analysis assumed “rigid- plastic” behavior and that the normal and shearing stresses were distributed uniformly on the failure surface. It was concluded that, if the friction angle of the concrete equals one-half of the apex angle and if the tensile strength is a constant fraction of the compressive strength, there is a proportional relationship between the ultimate pullout load and compressive strength. The analysis has been criticized as not providing a true behavioral prediction because the conclusions are a direct result of the underlying assumptions rather than from a rigorous assessment of the true behavior during the test.12–14

3.3.2.2 Nonlinear Finite Element Analysis

In 1981, Ottosen was the first to use the finite-element method to analyze the state of stress and to attempt to determine the failure mechanism of the pullout test.15_{He used nonlinear material models, a three-}

dimensional failure criterion, and a smeared cracking approach to follow the progression of failure with increasing pullout load. The pullout test geometry developed by Kierkegaard-Hansen was used. The analysis considered the concrete a homogenous material, i.e., the presence of individual coarse aggregate particles was not modeled.

A significant finding of Ottosen’s analysis was that, at about 65% of the ultimate load, a series of circumferential cracks had developed extending from the edge of the insert head to the bearing ring. Despite the circumferential cracks, additional load could be sustained by a highly stressed narrow band, or “strut,” extending from the insert head to the bearing ring. Ultimate failure was attributed to “crushing,” or compressive failure, of the concrete within this strut. Ottosen concluded that this was the reason for the good correlation between pullout strength and compressive strength.

Ottosen’s analysis demonstrated that the pullout test subjects the concrete to a highly nonuniform, triaxial state of stress. Within the compression strut, Ottosen found that the state of stress was predom- inantly biaxial-compression “occasionally superposed by small tensile stresses.”15_{Because of the tensile}

stresses, Ottosen concluded that the tensile strength of concrete had a secondary influence on the ultimate pullout load. He showed that, because the ratio of tensile strength to compressive strength decreases with increasing strength of concrete, the ratio of pullout strength to compressive strength would be expected to decrease for increasing concrete strength. This would explain the previous observations of Malhotra and Carette6_{and Richards.}7

Yener and Vajarasathira16 _{performed a plastic fracture analysis using the finite element method.}

Cracking was assumed to occur perpendicular to the direction of maximum tensile strain. “Crushing” failure was defined to occur if the maximum strain was compressive when an element cracked. It was noted that the high shearing stresses within the region between the insert head and bearing ring cause high tensile stresses, which result in circumferential cracking that defines the eventual failure surface. The analysis predicted that circumferential cracking began to form at the corner of the insert head at about 25% of the ultimate load. The circumferential crack propagated toward the bearing ring, but was arrested by high compressive stresses at about 50% of ultimate load. Another crack initiated at the corner of the insert head and propagated toward the bearing ring, so that at 70% of the ultimate load the trumpet-shaped frustum was completely formed. At this stage, the frustum was prevented from pulling out completely by frictional resistance due to high radial compressive stresses acting at the juncture of the frustum and the main body, just below the bearing ring perimeter. Additional load could be applied until crushing occurred around the perimeter of the frustum. Thus, while the crack patterns were similar to those in Ottosen’s analysis, a different ultimate load carrying mechanism was hypothesized.

3.3.2.3 Linear-Elastic Finite Element Analysis

Stone and Carino17_{also performed finite-element analyses for pullout tests with apex angles of 54$ and}

70$. Because their analyses were linear-elastic, the results are applicable only until the formation of cracks. In agreement with Ottosen, Stone and Carino found that the pullout test subjects the concrete to a complex three-dimensional state of stress. High compressive stresses exist within the “strut” region between the bearing ring and insert head. Figure 3.9 shows the principal stress trajectories for the two test configurations that were analyzed (because of symmetry only one-half of the specimens are shown). For the 70$ apex angle, there is close agreement between the compressive stress trajectories and the trumpet-shaped fracture surface observed in the companion test specimen.12 _{Note that the principal}

tensile stresses act perpendicular to the compressive stress trajectories. For the 54$ apex angle, the compressive stress trajectories and failure surface are less curved. The agreement between the directions of the stress trajectories and the shape of the failure surface led Stone and Carino to conclude that tensile strength is likely to play a greater role in the pullout test than was proposed by Ottosen.

The nonuniformity of the stresses along the idealized conic frustum is illustrated in Figure 3.10, which shows the variation of the principal stresses for the 70$ apex angle at about 20% of the ultimate load.17

There are very high tensile and compressive stresses at the edge of the insert head due to the stress concentration effect at the sharp corner. The tensile stresses decrease with distance from the insert head and become compressive near the bearing ring. The compressive stresses are high near the insert head and near the bearing ring, and they are nearly uniform in the middle region of the idealized frustum. Thus, in contrast to Ottosen’s conclusion, the frustum is subjected to significant tensile stresses. 3.3.2.4 Fracture Mechanics Analysis

Another finite-element analysis used the principles of nonlinear fracture mechanics to gain an under- standing of the failure mechanism.18_{This analysis used a discrete cracking model with special elements}

to represent the behavior of cracked concrete. The pullout test configuration had the following geometry:

D = 61 mm (2.4 in.); d = h = 25.4 mm (1 in.); and apex angle (2_) = 70$.

The nonlinear, fracture mechanics analysis revealed that two crack systems develop during the course of the pullout test. The first (primary) crack is a circumferential crack that initiates at the edge of the insert head and propagates into the concrete at angle of about 60$ with the load axis. The first crack begins at a low pullout load because of the large tensile stress concentration at the insert head (Figure 3.10), and it is arrested as it penetrates a region of low tensile stress below the bearing ring. Tensile and shearing stresses continue to be carried across the first crack because of the small crack-opening dis- placement. According to the analysis, the formation of the first crack results in high tensile stresses in the region between the insert head and the bearing ring. Thus, a second (or secondary) crack initiates at a point between the insert head and bearing ring, and it propagates in two directions toward the ring and the insert head.

FIGURE 3.9 Tension and compression stress trajectories before formation of cracks and the approximate shape of the conic frustums after ultimate load.17

Head

Reaction ring

A Apex angle = 70$

Failure surface Tensile stress trajectory Compressive stress trajectory

Head

Reaction ring

B Apex angle = 54$

Figure 3.11 shows the deformed shape (highly exaggerated) of the finite-element model after the second crack has propagated toward the ring and insert head. It is apparent that the second crack defines the shape of the conical fragment that is eventually extracted from the concrete. The ultimate load could not be determined in the analysis because the computer program would not permit the formation of a crack at the highly compressed node at the corner of the bearing ring. It was postulated that the failure surface would be formed completely by cracking of the final ligament between the insert head and the secondary crack tip. This final crack propagation would be primarily a shear failure along a surface inclined at a small angle to the load axis.

The results of the analysis were compared with experimental crack patterns observed in sectioned pullout test specimens that had been loaded to various fractions of ultimate strength.18_{For example,} Figure 3.12 shows the crack pattern in a specimen loaded to ultimate pullout load. The primary (first) and secondary (second) crack systems are clearly visible, and it is seen that the crack trajectory at the insert head is nearly parallel to the load axis. This figure also shows that the trumpet-shaped failure surface is completely formed; yet the conic frustum is not separated from the base concrete. The reason for this behavior is explained in the next section.

Ballarini et al.19 _{reported on the results of a linear-elastic fracture analysis of a two-dimensional (as}

opposed to axisymmetric) pullout test. A perfectly elastic, brittle material was assumed. Some of the conclusions of their analysis are as follows:

• Cracking begins at the edge of the insert as a tensile crack (as opposed to shearing).

• Initial cracking begins at an angle of about 75$ with respect to the load axis, and the initial cracking is stable.

With increasing load, the crack propagates toward the bearing ring at a decreasing angle with respect to the load axis.

Experiments using mortar specimens and different pullout configurations verified the analytical pre- dictions. It was found that, by expressing ultimate load in terms of fracture toughness, differences between the correlations for various test configurations were greatly reduced. Although not directly applicable to actual pullout tests, this study provided some insight into the crack propagation process.

FIGURE 3.10 Variation of principal stresses along a line extending from insert head to bearing ring (70$ apex angle).17 1.0 0.8 0.6 0.4 0.2 0.0 1

Relative Distance – Insert to Ring

Stress (MPa) P σc σt σc σt 0 -1 -2 -3 -4