Handbook of Civil Engineering Calculations

Pu = required compressive strength (based on the factored loads), kips

<$><Pn = design compressive strength, kips (kN)

(pbMnx, <j>bMny = design flexural strengths, kip-ft (kNm)

<pc = resistance factor for compression = 0.85

(pb = resistance factor for flexure = 0.90

The subscript jc refers to bending about the major principal centroidal (or jc) axis; y refers
to the minor principal centroidal (or j/) axis.

Simplified Second-Order Analysis
Second-order moments in beam-columns are the additional moments caused by the axial
compressive forces acting on a displaced structure. Normally, structural analysis is first-
order; that is, the everyday methods used in practice (whether done manually or by one of
the popular computer programs) assume the forces as acting on the original undeflected
structure. Second-order effects are neglected. To satisfy the AISC LRFD Specification,
second-order moments in beam-columns must be considered in their design.
Instead of rigorous second-order analysis, the AISC LRFD Specification presents a
simplified alternative method. The components of the total factored moment determined
from a first-order elastic analysis (neglecting secondary effects) are divided into two
groups, Mnt and Mlt.

1. Mnt—the required flexural strength in a member assuming there is no lateral transla-
   tion of the structure. It includes the first-order moments resulting from the gravity
   loads (i.e., dead and live loads), calculated manually or by computer.


2. Nit—the required flexural strength in a member due to lateral frame translation. In a
   braced frame, Mlt = O. In an unbraced frame, Af/rincludes the moments from the lateral
   loads. If both the frame and its vertical loads are symmetric, Mlt from the vertical loads
   is zero. However, if either the vertical loads (i.e., dead and live loads) or the frame
   geometry is asymmetric and the frame is not braced, lateral translation occurs and Af/,
   =£ O. To determine Mn (a) apply fictitious horizontal reactions at each floor level to
   prevent lateral translation and (b) use the reverse of these reactions as "sway forces" to
   obtain Aflt. This procedure is illustrated in Fig. 43. As is indicated there, Mlt for an un-
   braced frame is the sum of the moments due to the lateral loads and the "sway forces."

Once Mnt and Af/, have been obtained, they are multiplied by their respective magnifi-

cation factors, B 1 and B 29 and added to approximate the actual second-order factored mo-

ment M 11.

M 14 = B 1 Mn^B 2 M 1 , (Hl-2)

As shown in Fig. 44, B 1 accounts for the secondary P - 8 effect in all frames (includ-

ing sway-inhibited), and B 2 covers the P - A effect in unbraced frames. The analytical ex-

pressions for B 1 and B 2 follow.

''-(dfe)*

1

0 (H1

'

3)

where P 11 is the factored axial compressive force in the member, kips

P = ^

EI

' (*0^2 [8.i}