## 414 Plastic Design

Plastic analysis and design is permitted only for steels with yield stress not exceeding 65 ksi. The reason for this is that steels with high yield stress lack the ductility required for inelastic deformation at hinge locations. Without adequate inelastic deformation, moment redistribution, which is an important characteristic for PD, cannot take place.

In PD, the predominant limit state is the formation of plastic hinges. Failure occurs when sufficient plastic hinges have formed for a collapse mechanism to develop. To ensure that plastic hinges can form and can undergo large inelastic rotation, the following conditions must be satisfied:

• Sections must be compact. That is, the width-thickness ratios of flanges in compression and webs must not exceed 1p in Table 4.8.

• For columns, the slenderness parameter 1c (see Section 4.4) shall not exceed 1.5K where K is the effective length factor and Pu from gravity and horizontal loads shall not exceed 0.75Ag—y.

• For beams, the lateral unbraced length Lb shall not exceed Lpd where

—or doubly and singly symmetric I-shaped members loaded in the plane of the web

—y and for solid rectangular bars and symmetric box beams

5000 + 3000(Ml/Mp) 3000 ry Lpd = F ry — F (4 :i47)

Fy Fy

In the above equations, Ml is the smaller end moment within the unbraced length of the beam, Mp is the plastic moment (= ZxFy) of the cross-section, ry is the radius of gyration about the minor axis, in inches, and Fy is the specified minimum yield stress, in ksi.

Lpd is not defined for beams bent about their minor axes, nor for beams with circular and square cross-sections because these beams do not experience lateral torsional bucking when loaded.

### 4.14.1 Plastic Design of Columns and Beams

Provided that the above limitations are satisfied, the design of columns shall meet the condition 1.7FaA > Pu where Fa is the allowable compressive stress given in Equation 4.16, A is the gross cross-sectional area and Pu is the factored axial load.

The design of beams shall satisfy the conditions Mp > Mu, and 0.55Fytwd > Vu where Mu and Vu are the factored moment and shear, respectively. Mp is the plastic moment capacity, Fy is the minimum specified yield stress, tw is the beam web thickness, and d is the beam depth. For beams subjected to concentrated loads, all failure modes associated with concentrated loads (see Sections 4.5.1.1.3 and 4.5.1.2.3) should also be prevented.

Except at the location where the last hinge forms, a beam bending about its major axis must be braced to resist lateral and torsional displacements at plastic hinge locations. The distance between adjacent braced points should not exceed lcr given by

where ry is the radius of gyration about the weak axis, M is the smaller of the two end moments of the unbraced segment, and Mp is the plastic moment capacity.

M/Mp is taken as positive if the unbraced segment bends in reverse curvature and is taken as negative if the unbraced segment bends in single curvature.

### 4.14.2 Plastic Design of Beam-Columns

Beam-columns designed on the basis of plastic analysis shall satisfy the following interaction equations for stability (Equation 4.149) and for strength (Equation 4.150):

Pu Mu

where

Pu = factored axial load

Pcr = 1.7FaA, Fa is defined in Equation 4.16 and A is the cross-sectional area

Pe = Euler buckling load = n2EI/(KT)2

Cm = coefficient defined in Section 4.4

Mu = factored moment

Mp = plastic moment = ZFy

Mm = maximum moment that can be resisted by the member in the absence of axial load = Mpx if the member is braced in the weak direction

= {1.07-[((/ry)y'/Fy]/3160}Mpx< Mpx if the member is unbraced in the weak direction I = unbraced length of the member ry = radius of gyration about the minor axis Mpx = plastic moment about the major axis = ZxFy Fy = minimum specified yield stress, ksi

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