4.1 Strength evaluation for benchmark specimen J2

The strength of every member of the benchmark specimen, J2, is calculated and summarized in Table 4. In the calculations, however, the design values for the material mechanical properties were used: The compressive strengths of the concrete in the existing frame and wing walls were 21 and 24 N/mm², respectively, and the yield stresses of the reinforcement were 1.1 times the nominal values (295 × 1.1 = 325 N/mm² for D10 and 345 × 1.1 = 380 N/mm² for D16). For the determination of the ultimate strength and failure mode of the frame by Equation 3, conversions to an equivalent moment at the joint node when every member has reached its ultimate strength (values in Table 4) are shown in Table 5. The values in parentheses in Tables 4 and 5 are results based on the tested material properties shown in Tables 2 and 3, which are discussed later for comparison with the test results.

The beam and column ultimate flexural strengths of M_bu and M_cu were calculated according to the Japanese standards 19 using Equations 8 and (9), respectively. As shown in Figure 13, although no axial force was applied to the top of the upper-story column (left column in the figure), the lower-story column was subjected to a fluctuating axial force that resulted from the shear force of the beam. Consequently, in calculating the flexural strength of the lower-story column, the fluctuating axial force at the ultimate failure mechanism was considered; thus, Equations 9a and 9b were applied under compression and tension, respectively. The nodal flexural strength of the column was taken as the sum of the aforementioned strengths of the upper and lower columns (Equation 3), giving 173 and 191 kN·m in the positive and negative loading directions, respectively. Because the shear strengths of the beam and columns were greater than their flexural strengths, they are not discussed here.

(8)

(9a)

(9b)

Here, a_t is the gross area of the tensile longitudinal rebar; σ_y is the yield stress of the longitudinal reinforcement; d is the effective depth of the beam; N is the column axial force (positive for compression and negative for tension); b is the column width.

As previously stated, this study applied the Shiohara method to evaluate the joint strength. The moment capacity at joint hinging failure was calculated using Equations 1 and 2 17. Moreover, to compare with the ultimate shear strength from the Japanese design guidelines 1, the joint shear strength V_ju was also evaluated by Equation 10.

(10)

Here, κ is a joint shape factor (0.7 for exterior joints); ϕ is a factor dependent on the presence of orthogonal beams (0.85 for a joint without these beams); F_j is the nominal value for calculating the joint shear strength (F_j = 0.8F_c^0.7); and D_j is the effective joint depth, taken to be the horizontal embedding depth of the beam longitudinal bars.

The shear strength of the joint was converted to the joint moment _jM using Equation 11b, which was transformed from Equation 11a.

(11a)

(11b)

Here, T_b is the tensile force of the beam longitudinal rebar; V_c is the column shear force; L_b is the span length of the beam; j is the distance between the compressive/tensile force couple at the beam critical section (i.e., 7/8d); and L_c is the column height.

According to Table 5, it is estimated that the strength of the joint is the lowest for J2, indicating a joint failure mode; however, the shear strength (127 kN·m) is much higher than the moment capacity (+98, −101 kN·m).

4.2 Validation of the strengthening effectiveness by installing wing walls

In the following validation procedure, the calculations for the flexural strength of the column with a wing wall were omitted because the beam flexural strength was lower than the column flexural strength even for J2; thus, the column flexural strengths of J2 are shown also for the strengthened specimens in Table 5. The joint shear strengths of the strengthened specimens were evaluated using the effective depth of a column with a wing wall for D_j and the concrete compressive strength of the existing frame for F_c (F_j) in Equation 10.

The compressive and tensile forces at the boundaries between the beam and wing walls were calculated by common bending analyses for the critical sections of the upper and lower columns with wing walls based on Navier’s hypothesis. As an example, the results for J2-W2 under positive loading are shown in Figure 16, in which a neutral axis depth of x_n, curvature of φ, and compression on the concrete in the wing wall of C_c are illustrated. In the analysis, the concrete was regarded as elastic with a Young’s modulus of E_c calculated by Equation 14 1. The steel of the anchors was regarded as a perfect elasto-plastic material with a Young’s modulus of 2.0 × 10⁵ and a yield stress of 325 N/mm² (=1.1 × 295). The forces of the anchors, T_si and C_si, were obtained by their strains, which could be calculated by multiplying φ and x_n.

(14)

Here, γ is the bulk density of concrete (=24 kN/m³).

• Step 4
  Evaluating the ultimate moment capacity of the strengthened joint.

Figure 17(a) shows the actions of C_c, C_si, and T_si acting on the beam for the strengthened specimen with two walls, J2-W2. The ultimate moment capacity of the joint M_ju,r was calculated by Equation 15a, which was developed from Equation 5b. For J2-W1, as only one wing wall was installed on the lower column, the compressive and tensile actions did not work together, but rather, only one of them worked depending on the loading direction, as shown in Figure 17(b) and (c). Consequently, the joint’s strengths were calculated using Equations 15b and 15c for positive and negative loading, respectively,

(15a)

(15b)

(15c)

where C_si is the compression from the ith beam anchor; m is the total number of beam anchors under compression (the number of anchors in the compressive zone in the compressed wall); T_si is the tension from the ith beam anchor; n is the total number of beam anchors under tension (the number of anchors in the tensile zone in the pulled wall); and l_c/l_ci/l_ti is the distance to the joint node from C_c/C_si/T_si.

For J2-W2, because _jM_bu,r (=167 kN·m) is lower than M_ju,r (=201 and 202 kN·m under positive and negative loading, respectively) as well as _jM_cu (=173 and 191 kN·m under positive and negative loading, respectively), it was validated that the strengthening was sufficient to form an ideal beam yielding mechanism. In the case of J2-W1 under positive loading when the wing wall was pulled, as shown in Figure 17(b), the beam was still stronger than the joint (167 > 144 kN·m), so it was anticipated that the joint might fail prior to the connected beam and columns. Moreover, under negative loading when the wing wall was compressed, the strengths of the beam and joint were approximately equivalent, indicating the uncertainty of the failure mechanism.

In practical strengthening design, the aforementioned calculations for J2-W1 should be reconsidered because the joint failure was unlikely to be prevented. In this study, however, the design was completed for both strengthened specimens to verify the reliability of the earlier calculations.