## 8 Recent Advances in the Design of IPB Braced Frame
### Discussion of Modes of Buckling
Local-level uncertainties are (a) uncertainties involved in predicting rigidity of the end connection toward the bending in the presence of the different support conditions, (b) imperfection of the brace, gusset plate, and knife plate due to the manufacturing process, welding distortion, or mishandling during transportation, (c) residual stress of the gusset plate and knife plate (due to the welding and manufacturing process), (d) residual stress of the brace member, and (e) additional force on the gusset plate due to the frame action at the beam to column junction. Similarly, the global-level uncertainties are (a) different levels of imperfection in the brace member along the different planes, (b) the influence of the axial force on the bending capacity of the gusset plate and knife plate, (c) the influence of the load reversely on the buckling direction, (d) uncertainties involved in predicting buckling of closely spaced modes, and (e) the effect on the partial yielding of the gusset plate and knife plate on the stability of the brace member. The variability involved in the sensitivity of these parameters is also one of the major concerns for the stability assessment of the frame. Figure 6: Different clearance zones associated with the gusset plate. The first three modes of buckling of the IPB braced system obtained from the linear buckling analysis of the FEM model are shown in Figure 8. Some of the parameters destabilize the members and reduce their buckling loads, whereas some other parameters delay their buckling. The first two buckling modes for the IPB brace system are very closely spaced, which requires consideration of these uncertainties in evaluating the buckling mode such that the most destabilizing effect can be generated for the undesirable failure mode. It can be seen that the primary hinge, i.e., at the mid of the brace member, is common for both modes of buckling. As stated earlier, the bending capacity of the hinge plate is primarily responsible for the direction of buckling. The modes of buckling and the stability of the hinge plate are shown in Figure 9. Because both modes are in different planes, the  
Figure 8: Modes of buckling. Figure 7: List of uncertainties involved in predicting buckling directions. interaction of the hinges does not occur. The imperfection associated with the brace and the hinge plate introduces additional bending moments on the plate. It is very difficult to developed analytical formula to evaluate the plan of buckling due to the uncertainties involved in the IPB braced frame system. [PERSON] and [PERSON] [39; 40; 41] proposed a simplified analytical equation incorporating the uncertainties empirically using the multivariate regression analysis, as shown in Equation (1). \[\text{Mode of Buckling}=\frac{M_{c}^{k}\times\left[\left(\frac{L_{b}}{L_{k}}\right) ^{1.73}\times\left(\frac{L_{eff,k}}{l_{k}}\right)^{0.09}\times\left(\frac{L_{ eff,g}}{l_{g}}\right)^{0.55}\times\lambda^{-1.63}\right]}{M_{c}^{g}}\right\} \quad<1\;\rightarrow\;\text{IPB mode} \tag{1}\]  
where \(M_{c}^{k}\) and \(M_{c}^{g}\) are the reduced flexural capacities of knife plate and gusset plate, respectively, \((L_{b}/L_{k})\) is the ratio of the effective length of the plane of mode, \((L_{eff,k}/t_{k})\) is the knife plate effective slenderness ratio, \((L_{eff,g}/t_{g})\) is the gusset plate effective slenderness ratio, and \(\lambda\) is the normalized brace global slenderness ratio. Table 2 shows the comparison of the predicted and observed mode of buckling from past experimental studies. It is observed that the recently developed limit state criterion reliably evaluates the mode of buckling and prevents premature failure of the braced frame system.