The test results validated the feasibility and effectiveness of strengthening interior beam‒column joints via the wing wall installation method. In the design for practical engineering strengthening, it is essential to first assess whether the joints of the existing structure require strengthening. If strengthening is deemed necessary, the extent of reinforcement must be determined; specifically, the design parameters of the wing wall should be established. The strengthening design chart can be illustrated as Fig. 16.

Necessity determination for strengthening existing interior beam‒column joints

Structural details of the interior beam-column, such as dimensions, concrete strength, and reinforcement arrangement, must be thoroughly investigated and compared with the specifications of the relevant design codes^36,37,38,39. If the existing details do not comply with these specifications, for instance, if stirrups are not provided in the joint region, as observed in the target building of this study, a strength evaluation shall be conducted as follows.

Based on the calculation and comparison of the strengths of the beams, columns, and joints of the existing frame structure, a method for determining the necessity of strengthening the interior beam-column joints of the existing structure is proposed. The distributions of the flexural moment and shear force to an interior joint and its connected beams and columns under seismic action are shown in Fig. 17. By assuming that the beam length is l_b, the half-length of the beams, i.e., the distance from the centre of the joint to the inflection point of the beam, is l_b/2.The height of the columns is l_c, and the half-height of the columns is l_c/2.

Based on the parameters of the dimensions, reinforcement, and material strengths, the flexural strengths of the left beam M_{bu, l}, the right beam M_{bu, r}, the upper column M_{cu, t}, and the lower column M_{cu, d} can be calculated according to the formulas given in the design code for concrete structures⁴¹.The nodal moment at the flexural strength of the critical section of the left and right beams, _jM_{bu, l}and _jM_{bu, r},can be calculated by Eqs. (2) and (3), respectively. The nodal moment at the flexural strength of the critical section of the upper and lower columns,_jM_{cu, t} and _jM_{cu, d},can be calculated according to Eqs. (4) and (5), respectively.

Existing design codes^36,37 provide formulas for calculating the strengths of beam-column joints. However, these formulas are applicable only to structures that comply with code specifications, and are not suitable for non-conforming structures—such as the target building in this study, where no stirrups are placed in the joint region. Shiohara proposed an innovative flexural resistance model for beam-column joints⁴² and introduced a method for calculating the flexural strength M_ju⁴³. This method takes into account structural details such as the amount of stirrups in the joint, thereby enabling accurate strength evaluation even in cases where no stirrups are present.

According to the SHIOHARA model⁴³, the equations for calculating the strength reduction factor β_j, which is defined as the ratio of the flexural strength of the joint to the nodal moment at flexural hinging at the adjacent beam end, have been provided in the literature⁴³ for interior, exterior and corner joints. Because this study focuses on interior joints, the reduction factor for interior joints is given in Eq. (6). Consequently, the ultimate strength at joint hinging failure is given in Eq. (7).

Here, ξ_r is a reduction factor depending on ξ in Table 3; ξis the aspect ratio, i.e., D_b/D_c; D_b is the depth of the beam;D_c is the depth of the column; A_t is the sectional area of the effective tensile reinforcement in the beam cross-section;f_y is the yield stress of longitudinal reinforcement steel in the beam; b_j is the effective width of the joint, which is the width of the column or the beam when their widths are the same; F_c is the compressive strength of the concrete; A_j is the gross sectional area of the shear reinforcement in the joint crossing the vertical plane; and f_jy is the yield stress of the joint shear reinforcement steel.

The ultimate strength M_fu of the partial frame composed of the non-strengthened interior joint of the connected beams and columns is determined by the minimum values of _jM_{bu, l} + _jM_{bu, r}, _jM_{cu, t} + _jM_{cu, d} and M_ju, as illustrated in Eq. (8). By comparison, the failure mode of the frame can be determined:

(1) if the quantity _jM_{bu, l} + _jM_{bu, r} is minimized, the corresponding failure mode is expected to be beam hinging. This mechanism is considered ideal, as it aligns with the design objectives of codes such as ASCE³⁹, and therefore does not require strengthening.

(2) if _jM_{cu, t} + _jM_{cu, d} is the minimum, the failure mode should be column hinging, which needs to be strengthened; and (3) if M_ju is the minimum, the failure mode of the frame should be joint hinging, which does not satisfy the seismic design concept of”strong joints and weak members”, and it is necessary to strengthen the joints.

Calculation of the required strengthening amount

For structures that need strengthening for joints according to the judgement method presented in Chap. 4.1, the minimum strengthening demand M_need can be calculated according to Eq. (9) by taking the objective that the joint hinging strength M_ju is equal to that of beam hinging (_jM_{bu, l} + _jM_{bu, r}).

Determination of the wing wall parameters

Forces between beams and the wing walls

From the stresses of the beam anchors recorded during the test, as illustrated in Fig. 15, it can be found that the compressed wing wall exerts a compressive force C_w on the beam and that the tensile wing wall produces a pull force T_w on the beam through the anchors, as shown in Fig. 18.

Mechanical model for strengthening interior joints by wing walls

The compressive force C_w is mainly contributed by the concrete of the wing wall, and the pull force T_w is produced by the anchor bars connecting the wing wall and the beam. In the authors’ previous study³³, a strengthening model in which the accurate joint hinging strength can be calculated considering the contributions of both the compressive and tensile forces was developed. However, the calculations are too complex and need to be simplified for practical design. Compared with the compressive force C_w, which is mainly produced by the concrete of the wing wall, the tensile force T_w, which is produced by the anchor bars, is significantly less. If T_w is neglected, the hinging strength of the strengthened joint will be underestimated. However, a strengthening design will lead to improved safety. By neglecting T_w, the force distribution around the joint region is shown in Fig. 19.The strengthening demand M_need is provided by the moment produced by the force C_w to the joint centre of o.By assuming that the distance from the action point of the force to the joint centre is l_cw, the moment produced by C_w should be C_w·l_cw.This moment direction is the same as the joint hinging strength M_ju. Hence, the pull force C_w improves the joint strength by C_w·l_cw.If the improvement is greater than the strengthening demand M_need, i.e., Eq. (10) is satisfied, the strengthening objective can be successfully achieved.

Methods for determining the wing wall parameters

As shown in Fig. 20, the parameters of the wing wall to be determined for the strengthening design include the amount of column anchors, beam anchors, longitudinal reinforcement and transverse reinforcement; the concrete strength f_c; the wall thickness b_w; and the wall length l_w.The amount of column and beam anchors is determined in accordance with the detailing requirements for anchors in design codes, and the gross sectional area of the longitudinal and transverse reinforcements in the wing wall is greater than that of the beam and column anchor bars, respectively, to ensure that the post-installed wing wall has sufficient strength.The concrete strength F_c is set to be greater than that of the existing structure.The wing wall thickness b_w is taken to be an integer less than that of column b_c, as shown in Fig. 20. Thus, the wing wall length is the only parameter that needs to be determined.

The objective of strengthening is to ensure that the structure forms a beam-end hinging mechanism. By installing wing walls, the cross section of the column changes from square to T-shaped, as shown in Fig. 21. Under the flexural moment M_{c, bu}, at the critical section of the upper column, the longitudinal reinforcement in the column is tensile, and the concrete on the outer edge of the wing wall is under pressure. When the T-shaped section reaches the fully plastic state, the pull force C_w from the wing wall can be calculated according to the force equilibrium by Eqs. (11) and (12), i.e., the compressive force in the concrete is equal to the resultant force of the tensile stresses in the longitudinal reinforcement.

where T_y is the yield force of the longitudinal reinforcement in the column, N is the axial force to the column, f_y is the yield stress of the reinforcement, and A_s is the gross area of the longitudinal reinforcement in the column.

The relative depth of the concrete compression zone x can be obtained from Eq. (13).

where α is the reduction factor of the concrete strength at the equivalent rectangular stress block.

The distance l_cw from C_w to the joint centre can be calculated by Eq. (14).

By associating Eq. (14) with Eq. (9)~(12), Eq. (15) can be obtained. According to Eq. (15), the wing wall length l_w should satisfy Eq. (16).

In summary, all the parameters of the wing wall are determined: the number of anchors in the columns and beams, the amount of longitudinal and transverse reinforcement of the wing wall, the concrete strength, and the thickness and the length of the wing wall.The wing wall can be designed based on the proposed theoretical analyses and calculations.

Rationality confirmation of the wing wall design method

To verify the correctness of the proposed strengthening theory, the test results are evaluated based on the above design method. A comparison of the calculated and test results is shown in Table 3.

For the non-strengthened specimen Z1, the joint hinging strength M_ju is the smallest, and according to Eq. (8), the evaluated strength M_fu of the structure is 89.08kN.m. It is determined that the failure mode of the structure is joint hinging, and the structure needs to be strengthened.The maximum strength recorded in the test was 81.13 kN.m, and the failure mode was also joint hinging failure.Hence, the results from the theoretical evaluation are in agreement with the test results.

For the strengthened specimen Z1-W, the nodal moment is 214.13 kN.m at the flexural strength of the critical section of the T-shaped column after installing the wing wall. The nodal moment at the flexural strength of the critical section of the beam, i.e., when the beam is hinged at the wing wall face, is 122.41 kN.m. Based on the proposed strengthening mechanical model, the joint hinging strength M_{ju, r} after installing the wing walls can be calculated by Eq. (16).

By associating Eq. (17) with Equations (11) to (14), the strength of the strengthened joint is calculated to be 176.52 kN.m. The joint strength is improved by 87.44 kN·m due to the compressive force C_w, as expressed in Eq. (17). If the tensile force T_w from the beam anchors, according to the previous accurate method³³ and illustrated in Fig. 18, is also considered, the calculated joint strength would further increase by 24.22 kN·m, based on the yield stress of the anchors. However, this tensile contribution accounts for only 27.7% of the compressive contribution provided by C_w. Therefore, neglecting the tensile contribution of the wing wall anchors represents a conservative simplification that favors safer design, albeit with a minor trade-off in quantitative accuracy.

Comparing the calculated strengths of the beam, column and joint of the strengthened specimen, it can be seen that the strength of the beam is the lowest. Hence, the strength of the specimen, M_{fu, r}, is controlled by the beam (122.41 kN.m), and the failure mode of the structure is beam hinging, which approximately agrees with the test result of 127.17 kN.m.