# Consideration of second-order effects on plastic design of steel moment resisting frames

## Abstract

This work mainly aims to propose a new design procedure combining the benefits of the Performance-Based Plastic Design approach (PBPD) with a rigorous accounting of second-order effects. In fact, by exploiting the kinematic theorem of plastic collapse, second-order effects can be accounted for employing the concept of collapse mechanism equilibrium curve. The same tool constitutes the base of the Theory of Plastic Mechanism Control (TPMC) design approach. Besides, the paper reports a critical comparison between TPMC and PBPD, both having the scope to design structures exhibiting a collapse mechanism of global type. These two approaches are also compared with the refined PBPD where second-order effects are accounted for by the kinematic approach. Many steel moment resisting frames are designed according to PBPD, TPMC and refined PBPD and their performances have been compared on the bases of push-over analyses.

## Keywords

TPMC PBPD Global mechanism MRFs Plastic design## List of symbols

- \(dv_{k}\)
Vertical virtual displacement occurring at \(k\)th storey

- \(h_{{n_{s} }}\)
Top storey height

- \(h_{1}\)
First storey height

- \(h_{i}\)
Storey height

- \(h_{k}\)
\(k\)th storey height with respect to the foundation level

- \(B_{2}\)
Parameter to account for second-order effects (Eq. 14)

- \(C_{e}\)
Spectral acceleration in the energy balance formulation

- \(d\theta\)
Virtual rotation of column bases

- \(E\)
Energy occurring in an elastic system

- \(E_{e}\)
Earthquake input energy

- \(E_{p}\)
Energy to be dissipated by hysteresis

- \(F_{i}\)
Force acting at the ith storey

- \(F_{k}\)
Seismic force al \(k\)th

- \(g\)
Acceleration of gravity

- \(l_{i}\)
Bay span

- \(M\)
Building mass

- \(M_{b,i,k}\)
Plastic moment in the beam ends when plastic hinge has been developed

- \(M_{{b.jn_{s} }}\)
Plastic moment of \(j\)th beam at the top storey

- \(M_{b.jk}\)
Plastic moment of \(j\)th beam at \(k\)th storey

- \(M_{c} \left( h \right)\)
Moment in the column at the height \(h\) above the ground

- \(M_{c.i1}\)
Plastic moment of \(i\)th column of the first storey reduced due to the simultaneous action of the axial force

- \(M_{pbi}\)
Plastic moment of beams at the ith storey

- \(M_{pbr}\)
Required plastic moment of top storey beams

- \(M_{pc}\)
Required plastic moment of columns at the first storey

- \(M_{ux} \left( h \right)\)
Final bending moment

- \(n_{b}\)
Number of beams

- \(n_{c}\)
Number of columns

- \(n_{s}\)
Number of storeys

- \(P_{cg,i}\)
Gravity load on the column at \(i\)th floor level calculated using the seismic load combination

- \(q_{i,k}\)
Vertical load acting in the seismic load combination

- \(R_{\mu }\)
Reduction factor

- \(S_{v}\)
Spectral velocity of the expected seismic event

- \(S_{v.y}\)
Spectral velocity value corresponding to yielding

- \(T\)
Period of vibration

- \(V\)
Base shear

- \(V_{{n_{s} }}\)
Seismic design shear at the top storey

- \(V_{i}\)
Storey shear at the ith storey

- \(V_{k}\)
Seismic design shear at \(k\)th storey

- \(V_{lk}\)
Concentrated forces acting on columns at the kth storey coming from the inner bays of the column

- \(V_{n}\)
Storey shear at the top storey

- \(W\)
Seismic weight

- \(W_{e}\)
Virtual external work

- \(W_{i}\)
Virtual internal work

- \(W_{k}\)
Total vertical load acting at \(k\)th storey

- \(\Delta /L\)
Maximum target drift

- \(\alpha\)
Rigid plastic analysis collapse mechanism multiplier

- \(\alpha_{{i_{m} }}^{\left( t \right)}\)
First order collapse multiplier of undesired mechanisms

- \(\alpha_{0}^{\left( g \right)}\)
First order collapse multiplier of global mechanism

- \(\alpha_{0}\)
First order collapse multiplier of horizontal seismic forces

- \(\alpha_{coef}\)
Coefficient provided by Eq. (7)

- \(\beta_{i}\)
Shear proportioning factor

- \(\gamma\)
Slope of the mechanism curve accounting for second-order effects

- \(\gamma^{\left( g \right)}\)
Slope of the collapse mechanism equilibrium curve of global mechanism

- \(\gamma_{{i_{m} }}^{\left( t \right)}\)
Slope of the collapse mechanism equilibrium curve of undesired mechanisms

- \(\gamma_{m}\)
Modification factor based on Newmark and Hall studies (Newmark and Hall 1982)

- \(\delta_{i}\)
Step function

- \(\delta_{k}\)
Horizontal displacement occurring at the

*k*th storey- \(\delta_{u}\)
Ultimate design displacement

- \(\theta_{p}\)
Plastic rotation of dissipative members

- \(\lambda\)
Distribution coefficient lower than 1

- \(\lambda^{*}\)
Updating of the distribution coefficient \(\lambda\)

- \(\lambda_{i}\)
Distribution coefficient

- \(\mu_{s}\)
Structural ductility factor which is equal to target drift divided by yield drift

- MRFs
Moment resisting frames

- PBPD
Performance based plastic design

- TPMC
Theory of Plastic Mechanism Control

- \(\sum\nolimits_{k = 1}^{{n_{s} }} {M_{b,i,k} }\)
Sum of beams plastic moments belonging to the left bay

- \(\sum\nolimits_{k = 1}^{{n_{s} }} {M_{b,i - 1,k} }\)
Sum of beams plastic moments belonging to the right bay

- \(\sum\nolimits_{k = 1}^{{n_{s} }} {S_{i,k} } \delta_{k}\)
Axial forces at the \(k\)th storey in the \(i\)th column at the collapse state from the right bay

- \(\sum\nolimits_{k = 1}^{{n_{s} }} {S_{i - 1,k} } \delta_{k}\)
Axial forces at the \(k\)th storey in the \(i\)th column at the collapse state from the left bay

## Notes

## References

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