Finite Element Analysis of Fiber Reinforced Concrete Eccentric Anchorage Zone in Prestressed Girders

4.1. Geometric Modeling and Boundary Conditions

The anchorage plate, wedge plate and concrete block were modeled using three-dimensional solid elements. To represent the reinforcements, parts of deformable “wire” were employed, where a wire is depicted as a line in ABAQUS/CAE. The model’s bottom surface was fixed in the Y direction, while an axial load was applied through a specified vertical displacement on the anchorage heads (wedge plates), using an amplitude function (smooth step). The analysis was conducted in Step-1 (Dynamic, Explicit), following the initial step of simulation setup.

4.2. Material Modeling

Once the geometric input was completed, the next step was to input the material properties into ABAQUS. The forthcoming sub-sections present a comprehensive account of the procedures undertaken during this phase.

4.2.1. Fibrous Concrete

The model of Concrete Damage Plasticity (CDP), a feature in ABAQUS, was employed to characterize the response of concrete under various loading conditions. This model incorporates concrete degradation and considers both primary failure modes: crushing under uniaxial compression and cracking under uniaxial tension. To define the plasticity behavior of concrete, several parameters were necessary, including the concrete plastic potential eccentricity, the dilation angle (ψ), the ratio of compressive stress in the biaxial state to that in the uniaxial state (fbo / fco), the shape factor of the yielding surface (K), and the viscosity parameter. The required CDP parameters used in this study are provided in Table 2 [16-20].

Table 2.

CDP parameters	Value
Dilation angle (ψ)	31°
Eccentricity	0.1
biaxial/uniaxial ratio (fbo / fco)	1.16
K	0.667
Viscosity parameter	0

For the stress-strain relationship of the fibrous concrete in compression, a strain-hardening methodology was employed in this research. The material model initially responds linearly up to 30% of the compressive strength of fiber-reinforced concrete, transitioning to a parabolic stress-strain curve until the effective stress reaches the compressive strength. After this threshold, the material was assumed to behave with perfect plastic until crushing took place [20-22]. The relationship is defined by the following equations [21, 23, 24]:

(1)

(2)

(3)

where ε_pf is the strain at peak stress (f’_cf), f’_c is the cylinder compressive strength of plain concrete, ε_cuf is the maximum compressive strain, E_c is the elastic modulus of plain concrete, v_f is the fiber volume ratio, l_f is the fiber length and d_f is the fiber diameter.

The following equation was used for the tensile stress-strain relationship of fibrous concrete [21, 25-27]:

(4)

where ε is the tensile strain at stress f_t, ε_tf is the strain at peak stress, f’_tf is the tensile strength of fibrous concrete, and ß is a fiber parameter, which for hooked fibers can be calculated from the following equation [28]:

(5)

where R.I is a reinforcing index that is equal to w_f l_f /d_f;wf represents the ratio of the fibers' weight to the concrete weight. The following equations are used to calculate ε_tf and f’_tf [23]:

(6)

(7)

where f’_t and ε_t are the tensile strength and cracking strain of plain concrete, respectively, N_f is the number of fibers per unit area, which is equal to ƞ_o(4v_f/πd_f²), ƞo represents the orientation factor, which can be assumed to be 0.5 [23].

Concrete degradation in tension is determined by the damage parameter d_t, defined as the ratio of the tensile stress after cracking to the ultimate tensile stress [18]. Additionally, a compressive stiffness recovery factor of w_c=0.8 was applied, assuming that the compressive stiffness is largely restored when the crack closes as the load transitions from tension to compression.

4.3. Element Types and Mesh Generation

As illustrated in Fig. (4), all the solid members (anchorage plates, wedge plates and concrete) were discretized utilizing the first-order hexahedral elements. The C3D8R element was used for this purpose, which consists of eight nodes and incorporates reduced integration and hourglass control. These elements are known for providing accurate results while keeping computational costs at a minimum for three-dimensional analyses. The reinforcement was modeled using a two-node, three-dimensional truss element designated as T3D2. A general element size of 100 mm was employed in this analysis [19].

5.1. Effect of Fiber Volume Ratio

Table 3 presents the ultimate load for all specimens, organized to highlight the influence of fiber volume ratio on their performance. The specimens are grouped based on the bar size used for spiral/tie reinforcement. The last column of the table provides the percentage comparison of the ultimate load for fiber-reinforced specimens relative to those without fiber reinforcement within each group. Incorporating fibers had a pronounced impact, as the ultimate load of specimens increased by 30% to 50% with a fiber volume ratio of 2%, compared to the non-fiber-reinforced counterparts. The highest observed enhancement, as the effect of using 2% of fibers, was a 50% increase in ultimate load (from 9,987 kN for A-00-00-00 to 14,948 kN for A-20-00-00), and occurred in the group without any spiral or tie reinforcement. The bridging effect of steel fibers enhances the tensile strength of various sections in this group, which lack conventional reinforcement. The table also presents that most of the improvement in the ultimate load of the specimens is for a volume ratio of 1.5% of fibers, as the ultimate load increased between 26% to 47% for this volume ratio. On the other hand, using fibers with a volume ratio of 0.5% resulted in only a slight increase in the ultimate load, ranging from 1% to 8%. This indicates that the optimal fiber volume ratio for enhancing the ultimate load of the specimens examined in this study lies between 1% and 1.5%.

In an examination to test whether the fibers can completely replace reinforcement using the spiral/tie reinforcement, a comparison has been made between the fiber-reinforced specimens without spiral/tie reinforcement with the control specimen that has spiral/tie reinforcement but without fibers. The specimen A-00-20-10 has no fibers and is reinforced with spirals and ties with a bar size of 20 mm and 10 mm, respectively, and is taken as a control specimen for this comparison. The ultimate load for this specimen is 17,799 kN. The first comparison is made to check if the fibers can replace either the spiral or ties and the second comparison is to check if fibers can replace the entire reinforcement, which includes both spiral and ties. To test if the steel fibers can completely replace reinforcement using spiral reinforcement, specimen A-10-00-10 is used, which is fiber fiber-reinforced specimen with a volume ratio of 1%, with a tie bar size of 10 mm, and without spiral reinforcement. The value of ultimate load for this specimen is 17,311 kN which is close to the ultimate load of the control specimen. Meanwhile, specimen A15-00-10, which is similar to A-10-00-10, but with a fiber volume ratio of 1.5%, has an ultimate load of 19,009 kN, which is more than the ultimate load of the control specimen. These findings suggest that the optimal fiber volume ratio for maximizing the ultimate load of the tested specimens is between 1% and 1.5%.

Similarly, and to test if the steel fibers can completely replace reinforcement using tie reinforcement, A-10-20-00 is used, which is a fiber-reinforced specimen with a volume ratio of 1%, with a spiral bar size of 20 mm, but without tie reinforcement. The ultimate load for this specimen is 17,091 kN, which is close to the ultimate load of the control specimen. Specimen A15-20-00 is similar to A-10-20-00, but with fiber volume ratio of 1.5%, and has an ultimate load of 18,010 kN, which is more than the ultimate load of the control specimen. Hence, steel fibers with a volume ratio of 1%-1.5% can completely replace reinforcement using the tie reinforcement of the specimens included in the study.

To check if the fibers can completely replace reinforcement using the entire reinforcement (spiral and tie), specimen A-20-00-00 is compared with the control specimen. Specimen A-20-00-00 has no spiral or tie reinforcement but is fiber-reinforced with a volume ratio of 2%. The ultimate load of this specimen is 14,948 kN, representing 84% of the ultimate load of the control specimen. This means using up to 2% of steel fibers for the studied specimen cannot replace the entire reinforcement, but 84% of the ultimate load of a reinforced specimen can be reached.

Fig. (5) shows the load-vertical displacement curves for the specimens across different groups. During the initial linear phase, the specimens displayed similar responses, but then some different behavior after that range. Specimens reinforced with fibers exhibited flatter load-displacement curves, indicating a more ductile failure mode compared to those without fibers.

5.2. Effect of Spiral Reinforcement

Table 4 presents the influence of spiral reinforcement on the ultimate load of the specimens in different groups. All the specimens in this table have the same tie bar size of 10 mm. The final column of the table shows the percentage of the specimens' ultimate load relative to the ultimate load of the case without spiral reinforcement for each group. The ultimate load of the specimens tends to increase as the spiral bar size increases. With a 20 mm spiral bar size, the load increased by 21% to 31% compared to the specimens without this reinforcement. The lowest observed increase (21%) occurred in the group of specimens without fiber reinforcement. For the various specimen groups, increasing the spiral bar size from 16 mm to 20 mm resulted in an ultimate load increase ranging from 0% to 11%.

Fig. (6) shows the effect of spiral reinforcement on the load-vertical displacement curves of the specimens. During the initial linear phase, the specimens displayed similar responses and then a close behavior after that range. Regardless of the bar size of the spirals, the value of vertical displacement at ultimate load is more for the groups with fiber reinforcement.

5.3. Effect of Tie Reinforcement

Table 5 presents the influence of tie reinforcement on the ultimate load of the specimens in different groups. All the specimens in this table have the same spiral bar size of 20 mm. The final column of the table shows the percentage of the specimens' ultimate load relative to the ultimate load of the case without tie reinforcement for each group. The specimens’ ultimate load increases with an increase in tie bar size. Compared to the case of specimens without tie reinforcement, the ultimate load increased in the range between 24% (from 14,313 kN for A-00-20-00 to 17,799 kN for A-00-20-10) and 32% (from 17,091 kN for A-10-20-00 to 22,597 kN for A-10-20-10) by using tie bar size of 10 mm. This increase in the ultimate load is attributed to the tie's ability to effectively resist tensile stresses in the general zone. This reinforcement prevents the formation of brittle tensile cracks, thereby enhancing both the load-carrying capacity and the structural integrity of the specimens under loading. Again, the minimum increase of the ultimate load for this case (24%), as the result of using tie reinforcement, was for the group of specimens without fiber reinforcement.

Fig. (7) shows the effect of tie reinforcement on the load-vertical displacement curves of the specimens. During the initial linear phase, the specimens displayed similar responses and then some different behavior after that range. Specimens with tie reinforcement exhibit flatter load-displacement curves compared to those without ties. This behavior reflects enhanced energy dissipation and ductility attributed to the tie reinforcement's ability to restrain crack propagation effectively. Positioned in the general zone, tie reinforcement resists tensile stresses in this region, thereby mitigating the risk of brittle tensile failure.

5.4. Bursting Stresses

Fig. (8) shows the contours for bursting stresses in the elastic stage at the model surfaces in both the long and short directions for two specimens: A-00-16-10 (without fibers) and A-15-16-10 (with a volume ratio of 1.5% of fibers). In the longitudinal direction, bursting stresses spread across the area between the two anchorage devices. Furthermore, these stresses are primarily concentrated in the general zone, and are distributed in a depth approximately equal to the length of each respective direction. These stress results align with previous studies [1, 5, 19, 30], which reported similar distributions and locations. The bursting stress patterns were nearly identical for all specimens, suggesting that the variables included in the study had no influence on this stress type within the elastic range. Only results for these two specimens are shown in Fig. (8), and the other specimens demonstrated a similar distribution of these stresses.

5.5. Cracking Pattern

The ABAQUS program lacks a specific tool for displaying crack extent in the model; however, parameters such as logarithmic strain, plastic strain, or tensile damage can serve as indicators of crack formation. In the current study, the tensile damage parameter (DAMAGET) was employed to indicate the propagation of cracks, with results for a typical specimen shown in Fig. (9) from two perspectives. In this specimen, the red areas indicate that tensile damage is in the range of 58%–64% in certain elements. It is evident that cracks in both directions start from the anchorage devices and are primarily concentrated at the top of the specimens, within the local zone. Additionally, a horizontal crack is visible on the side away from loading, attributed to tension caused by load eccentricity. Notably, the crack patterns in this study align closely with those reported in previous studies [4, 19, 31].

5.6. Validation of the FEA Model

To verify the FEA model and assess its accuracy, the results for two specimens from the current study were compared with the ultimate load capacity prediction from Roberts' equation [29], which is based on experimental work. Roberts’ work was a part of the study carried out in the early 1990s at the University of Texas at Austin with backing from the National Cooperative Highway Research Program (NCHRP) [1, 29, 30], which is regarded as a foundational study in the field of anchorage zones. In applying this equation, the specimens from the current study were modeled with two anchorage zones, each with an effective supporting area of 400x400 mm. As shown in Table 6, the ultimate load results from the current study align closely with those predicted by Roberts' equation, with a ratio ranging from 1.16 to 1.17. The slightly higher ultimate load in the current study is likely due to the larger effective supporting area in the FEA model compared to that used in Roberts' equation calculations. Additionally, both studies indicate a minimal effect on the ultimate load by raising the size of the bar for the spiral reinforcement from 16 mm to 20 mm. This limited effect is due to the maximum lateral confining strength, capped at 8.3 MPa in Roberts' equation, beyond which increasing the spiral bar size has no impact. Similarly, the FEA results from the current study show only a slight increase in ultimate load between the two specimens in Table 6 when the spiral bar size is increased from 16 mm to 20 mm, which is consistent with Roberts' equation results and further validates the FEA model.