Validated three-dimensional finite element modeling for static behavior of RC tapered columns

3.1 Compressive behavior

Different models have been proposed to describe the stress–strain relationship of concrete in compression. In 1971, Kent and Park modified the second degree Hognestadʼs model, and the ascending part of the modified model is given by Eq. (2), while the descending part is a straight line whose slope is governed by the concrete strength [16]:

where ε o is the strain at the peak point on the curve and f c ' is the compressive strength. This model was used to define the compressive behavior of concrete in this study with ε o taken as 0.002. The linear part was taken up to a stress of 0.45 f c ' . The modulus of elasticity of concrete (E _o) is taken equal to the secant modulus of elasticity; it is the slope of the straight line drawn from the origin to the point on the curve at which the stress is 0.45 f c ' . A concrete crushing strain of 0.0038 was adopted in agreement with the range in the ACI code commentary (0.003–0.004), at which the strength is developed for members of normal concrete [1]. The inelastic strain was calculated according to the following equation [12]:

3.2 Tensile behavior

The FE analysis results for RC concrete structures are highly sensitive to the user-defined parameters of tension stiffening. These parameters are the ultimate tensile stress (f _t) and ultimate cracking strain (ε _ucr). The post-failure strain softening of the tensile stress, demonstrated in Figure 1b, can be approximated to linearly varying from the maximum tensile stress (f _t) at failure strain (ε _cr) to zero at the ultimate total strain. Thus, the ultimate total strain is the total strain at which a complete loss of tensile strength is reached. According to Reddiar [16], this strain is one-tenth that corresponding in compression (ε _cu), and ε _cu is evaluated according to the following equation:

At any point, the inelastic cracking strain ε _incr is evaluated according to the following equation [12]:

The Abaqus analysis user’s manual [13] estimates a total cracking strain of about 10ε _cr. The ultimate cracking strain ε _ucr adopted in the current modeling is taken equal to 10ε _cr and the tensile strength was taken equal to the modulus of rupture 0.62 f c ' .

Alkloub et al. [6], Sinaei et al. [17], and Lee et al. [18] have used the same value of the ultimate cracking strain for the modeling of the tensile behavior of concrete in Abaqus. To define the tensile strength of concrete into Abaqus, Sinaei et al. [17] and Ahmed [19] have adopted 0.61 f c ' and 0.6 f c ' , respectively.

As mentioned in the Abaqus analysis user’s manual, increasing the values of the tension stiffening parameters enables to obtain FE solutions. In contrast, using too little values results in a local cracking failure which causes an overall unstable behavior. Therefore, to obtain ε _cr, f _t was divided by E _o or 4,700 f c ' whichever is smaller.

4.1 Model No. 1

This model is for the tapered long column labeled “MTUO-04” and tested by Brant [7]. The column was tested under a uniaxial eccentricity of 25 mm causing bending about the major axis. The geometry of the column, reinforcement, and material properties are as in Table 2. The eight-node brick element (C3D3R element) was used to model concrete, while the two-node linear 3D element (T3D2 element) was used for the steel reinforcement. The 3D solid elements of concrete, the reinforcement model, and the idealized column with the end conditions, are shown in Figure 2.

Figure 2

(a) Concrete elements, (b) reinforcement model, and (c) idealized column.

The eccentric load was applied at the stronger end while the column specimen was in a horizontal orientation, and a suitable supporting rig was used to balance the gravity load. Three steel meshes were used at each end to strengthen the regions of transferring the action and reaction forces. Such meshes were also involved in the FE model with a configuration similar to that used in the experiment, as shown in Figure 3. To avoid high distortions and stress concentration, two elastic solid plates were modeled at both ends. Their contact with the concrete was simulated with tie constraints.

Figure 3

(a) Mesh (part module), (b) mesh (assembly module), (c) whole base pinned, and (d) partially restrained base area.

The need for the bearing plate at the base of the column is that not the whole base area was restrained against motion. In case of 3D solid elements, even permitting the rotational degrees of freedom not clamped, restraining the translational degrees of freedom at all of the supported area results approximately in a fixed support (Figure 3c). As a result, the model will be notably stiffer than the actual structure. Therefore, the base area was partially hinged as shown in Figure 3d.

With multiplying by 0.8 modifier, the cube strength of concrete was converted to 33 MPa. Tension stiffening parameters are f _t = 0.62 33 = 3.56 MPa and ε _ucr = 1.2 × 10⁻³. The modulus of elasticity E _o and Poisson’s ratio are 28,710 MPa and 0.18, respectively. Throughout the study, steel reinforcement behavior was considered bilinear elastic-full plastic with E = 2 × 10⁵ MPa, Poisson’s ratio of 0.3, and inelastic strain of 0.004. The concrete plasticity parameters were used as those in Table 1. The meshing was performed choosing a size of 60 mm, and the nonlinear analysis was achieved. To verify the adequacy of the mesh, the mesh was refined with a size of 50 mm and the analysis was repeated. The load–deflection curve with the 50 mm size, as shown in Figure 5c, is closer to the experimental result with an imperceptible variation. Thus, this size was adopted.

With an activated geometric nonlinearity. The nonlinear Static Riks analysis was performed. The load–displacement relationship of the experimental work, computerized analytical solution by Brant, and the nonlinear FE solution from Abaqus are shown in Figure 4a.

Figure 4

(a) Load–mid-span deflection and (b) load–deflection for different end conditions.

The ultimate load obtained from the experimental work is 1,597 kN. The ultimate load in the FE analysis is 1,599 kN. The FE analysis well predicted the load–displacement relationship. The failure difference between the numerical and experimental displacements is due to that the dial gauges were removed when the applied load reached about 80–85% of the expected failure load. At this stage, a large deformation occurred at a constant level of load. Hence, the displacement increase in the numerical result is realistic, and it represents the unrecorded large displacement that occurred after the removal of the dial gauges.

No tensile cracks appeared before failure and the failure was brittle (compression failure) with tensile cracks at the opposite face of the beam initiated after crushing. This behavior can be noticed in the crushing pattern (DAMAGEC) and the cracking pattern (DAMAGET), shown in Figure 5a and b, respectively. Thus, the failure mode of the FE model agrees with the experimental work in which the failure was crushing of concrete at 0.76 m below the column mid-height.

Figure 5

(a) Compression damage, (b) tension damage, and (c) mesh verification.

The maximum experimental strain measured in column MTUO-04 is about 2.3 × 10⁻³, which corresponds a stress in steel reinforcement of 460 MPa. This value of stress in the steel bars is shown in Figure 6c. The maximum value of stress in the steel (460 MPa) reveals the accordance between the numerical and experimental results that no steel yielding occurred before the failure. The higher stresses in steel bars located at the concave side compared to those in the convex, are intuitionally expected and they reveal the effect of bending due to primary eccentricity as well as p-delta secondary moments. The strain values above 2.3 × 10⁻³ are not expressive because they occurred only in some elements at the bottom end due to the stress concentration.

Figure 6

(a) Deformed shape, (b) plastic strains, and (c) stress in steel reinforcement.

Figure 6a shows the deformed shape at the last load increment. The plastic strains in the circled region (Figure 6b), along with the crushing pattern, indicate the location where concrete crushed or is intending to crush. A path was drawn across the cross-section area of the upper end of the column starting from the convex side to the concave one, as shown in Figure 7a. The variation in the compressive stress in the y-direction (S ₂₂) along the drawn path is as shown in Figure 7b. Furthermore, the agreement with the analytical solution by Brant confirms the accuracy of the FE solutions. The region of high stress is the periphery of the point at which the load was applied.

Figure 7

(a) Path across the top end of the column and (b) S₂₂ variation along the path.

As shown in Figure 2c, the upper end of the column, at which the load was applied, was modeled as a hinged roller; u 1 = u 3 = 0 , and it was free to move in y-direction. The condition of hinged–hinged roller appears not practical and a margin of safety had been taken during the test to prevent the loss of the overall stability of the column being compressed under a high load. Thus, some rotation restriction existed during test at the stationary end (smaller end). It was noticed that restraining all the bottom cross-section area leads to a structural behavior notably stiffer than the actual behavior, as can be shown in Figure 4b. On the other side, modeling of the bottom end as a real hinge makes the structure significantly weaker than in fact, and increasing the tension stiffening or other parameters do not make the result exceeds the lower curve in Figure 4b (hinged base). Therefore, modeling of an elastic bearing plate at this end and pining it partially from about the centroid (Figure 3d) makes the end able to rotate but with a rotational stiffness. At the pinned nodes, the translational degrees of freedom were restrained ( u 1 = u 2 = u 3 = 0 ) . This condition for the bottom end lead to a realistic simulation for the actual support.

The tensile strength of concrete had not been introduced in the experimental study presented by Brant to be used in the numerical modeling. On the other hand, since both Model 1 and 2 were tested under axial loads with the existence of flexural stresses, it is reasonable to use the modulus of rupture of concrete as a tensile strength.

4.2 Model No. 2

This model is for the column labeled “MDUO-03,” which was also tested by Brant. It is tapered in two directions. Its larger end, where the eccentric load was applied, is 300 mm × 300 mm and the smaller end is 200 mm × 200 mm. Length and details of reinforcement are the same as in Model No. 1. The end conditions are the same as for column MTUO-04. The FE model is shown in Figure 8.

Figure 8

(a) Whole model, (b) reinforcement, (c) bearing plate and wire mesh, and (d) application of the eccentric load to the column.

The cube compressive strength is 46.4 MPa and yield stress of reinforcement is 475 MPa. The column was tested under an eccentricity of 60 mm. Strengthening with steel meshes of 8 mm diameter was done, as for column MTUO-04. These meshes were also inserted in the numerical model. Bearing plates with only elastic properties were modeled to transfer the forces at the two ends. The plasticity parameters for CDP are the same as in Table 1. The cube strength was converted to 37 MPa. The defined parameters for tension stiffening were f _t = 3.77 MPa and ε _ucr = 1.318 × 10⁻³. Types of FEs are the same as those for Model No.1. Taking the nonlinear geometry in account, the Static Riks solver was used to perform the nonlinear static analysis. The load–mid-span deflection and the deformed shape are shown in Figure 9a and b, respectively.

Figure 9

(a) Load–deflection relationship of Model No. 2, (b) deflected shape, (c) plastic strains in concrete, and (d) stress in steel reinforcement.

As mentioned by Brant, the ultimate load of column MDUO-03 was 1,565 kN. Hence, the failure load from the FE solution, which is 1,554 kN, indicates the acceptance of the FE solution. Figure 10 shows the patterns and locations of the concrete crushing (DAMAGEC) and tensile cracking (DAMAGET). The stress in steel reinforcement (Figure 9d) is below the yield stress, which is the case in the test. Also, the plastic strain in the red colored region in Figure 9c indicates the location of the crushing of concrete. Consequently, the FE model well predicted the failure mode.

Figure 10

Damages in Model No. 2: (a) compression damage and (b) tension damage.

4.3 Model No. 3

This model is for the column, labeled “C3,” tested by Maliki and Mahmood [8] under an axial concentric load. The geometry of the specimen, reinforcement, and material properties are given in Table 3.

In the FE modeling, the column was hinged at the bottom ( u 1 = u 2 = u 3 = 0 ) and free at the top. A uniform pressure of 30 MPa was applied. As for the two previous models, the modulus of rupture was used as the value of the tensile strength, thus f _t = 3.22 MPa. The ultimate cracking strain ε _ucr is equal to nine times this value divided by 23,500 (the secant modulus of elasticity according to the curve). Hence, it is 1.37 × 10⁻³. The plasticity parameters are the same as those in Table 1. Figure 11 shows the FE model of the column, the stress in the y-direction, and the stress in steel reinforcement. From Figure 11b and also from Figure 12b, the average developed stress in concrete is 23.23 MPa, which is 0.86 f c ' . It is in line with the load carrying capacity of concrete in compression members (0.85 f c ' A c , where A c is the concrete area).

Figure 11

(a) FE Model No. 3, (b) stress in the y-direction, and (c) stress in steel reinforcement.

Figure 12

(a) Load versus axial displacement and (b) input and output stress–strain diagrams.

The experimental ultimate load was 740 kN, while the FE analysis resulted in an ultimate load of 756 kN. Hence, the ultimate load capacity in the FE model is very close to the experimental result and, as a result, it is acceptable. The plastic strains, in Figure 10b, refer to the locations where crushing starts. It is reasonable and expected that the concrete crushing starts at the top end.

From the load–displacement curves shown in Figure 12a and also from the displacement in Figure 13a, the numerical model appears stiffer than the tested member. One of the possible interpretations of this deficiency is that, even the maximum value is the same, the actual stress–strain curve of the column concrete is flatter than that defined into the Abaqus. In other words, the modified Hognestad’s model, which was used in the modeling, does not efficiently describe the actual compressive behavior of the tested member.

Figure 13

(a) Axial deformation at the last load increment, and (b) equivalent plastic strain, and (c) load–displacement relationships.

The tensile strength of the concrete material had not been introduced within the material properties in the study presented by Al-Maliki and Mahmood [8]. Therefore, it can only be estimated. According to Mehta [2] and as well as the Abaqus analysis user’s manual [13], the tensile strength of concrete is 7–10% of its compressive strength. Using the upper limit of the range, the tensile strength of the concrete material of Column C3 is 2.7 MPa. This value was used in the numerical solution and the analysis was repeated. It can be shown, from Figure 13c, that the column becomes slightly stiffer and the difference between the numerical and experimental load–deflection relationships slightly increases. Thus, using a tensile value below 3.22 MPa is not beneficial for the analysis results.