STRESSES IN PRESTRESSED CYLINDER
T- TUhT 67 - 2
1. Find the effective axial load for the beam-column
This procedure considers singly and doubly symmetric beam-columns: members subject- ed combined axial compression and bending about one or both principal axes. The combi- nation of compression with flexure may result from (either)
(a) A compressive force that is eccentric with respect to the centroidal axis of the column, as in Fig. 42a
(b) A column subjected to lateral force or moment, as in Fig. 42b (c) A beam transmitting wind or other axial forces, as in Fig. 42c Interaction Formulas:
The cross sections of beam-columns must comply with formula (Hl-Id) or (Hl-Ib)9
whichever is applicable.
For (P11AkP11) ^ 0.2
Pu » / M/x Muv \
—u— + £. / ^L- + u^- \ < i o (Hl-Ia)
^0Pn 9 \<M4c ^AfJ ^ }
FOT (P^0P11) < 0.2
PU ( M11x Muv \
^r
2(J)0Pn+
\ QbMnhr^r +
x QbMny /-^r) - L0 (H1 ' lb)
For beam-columns:
Mcc> M^ = required flexural strengths (based on the factored loads) including second-order effects, kip-in or kip-ft
FIGURE 42. Combined compression and flexure.
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, B1 and B29 and added to approximate the actual second-order factored mo- ment M11.
M14 = B1Mn^B2M1, (Hl-2) As shown in Fig. 44, B1 accounts for the secondary P - 8 effect in all frames (includ- ing sway-inhibited), and B2 covers the P - A effect in unbraced frames. The analytical ex- pressions for B1 and B2 follow.
''-(dfe)*
1-
0 (H1'
3)where P11 is the factored axial compressive force in the member, kips P = ^EI
' (*02 [8.i}
Original frame = nonsway frame + sway frame
for Mn, fa M"
FIGURE 43. Frame models for Mnt and M7,.
where K= 1.0, / is the moment of inertia (in4) (cm4) and / is the unbraced length (in) (cm) (Both 7 and / are taken in the plane of bending only.)
The coefficient Cm is determined as follows.
(1) For restrained beam-columns not subjected to transverse loads between their sup- ports in the plane of bending
M1
C111 = 0.6-0.4-j- (Hl-4)
M2
where M1IM2 is the ratio of the smaller to larger moment at the ends of the portion of the member unbraced in the plane of bending under consideration. If the rota-
FIGURE 44. Illustrations of secondary effects, (a) Column in braced frame; (b) Column in unbraced frame.
tions due to end moments M1 and M2 are in opposite directions, then M1IM2 is negative; otherwise M1IM2 is positive.
(2) For beam-columns subjected to transverse loads between supports, if the ends are restrained against rotation, Cm = 0.85; if the ends are unrestrained against rotation, Cw=1.0.
Two equations are given for B2 in the AISC LRFD Specification:
B1= l- JT (Hl-5)
! v p MfM 1
*
P\S,HL)
or
«i-—b: <"">
' - X T T
where ^P14 = required axial strength of all columns in a story (i.e., the total factored gravity load above that level), kips
Xc/fc = translational deflection of the story under consideration, in 2/JT= sum of all horizontal forces producing A0,,, kips
L = story height, in
SP]e = summation of Pe for all columns in a story.
Values of Pe are obtained from Eq. [8.1], considering the actual K and / of each column in its plane of bending. Equation (H 1-5) is generally the more convenient of the two formu- las for evaluating B2. The quantity A0,//, is the story drift index. Often, especially for tall buildings, the maximum drift index is a design criterion. Using it in Eq. (H1-5) facilitates the determination of B2.
For columns with biaxial bending in frames unbraced in both directions, two values of BI (Bix and Bly) arc needed for each column and two values of B2 for each story, one for each major direction. Once the appropriate B1 and B2 have been evaluated, Eq. (Hl-2) can be used to determine M11x and Muy for the applicable interaction formula.
Preliminary Design
The selection of a trial W shape for beam-column design can be facilitated by means of an approximate interaction equation given in the AISC LRFD Manual Bending moments are convened to equivalent axial loads as follows.
PUJB = Pu + M^m + MuymU [8.2]
where />„ eff is the effective axial load to be checked against the Column Load Table in Part 2 of the AISC LRFD Manual, Pw M11x, and Muy are as defined in interaction formulas (Hl-Id) and (Hl-Ib) (PU9 kips; M11x, Muy, kip-ft); and m and U are factors adapted from the AISC LRFD Manual
Once a satisfactory trial section has been selected (i.e., P11^ the tabulated (^0Pn), it should be verified with formula (Hl-Id) or (Hl-Ib).
For a braced frame, K = 1.0 for design; KjLx = KyLy = 1.0 x 15 ft. Select a trial W14 shape using Eq. [8.2].
Pu^Pu + M^m+MuymU
For a W14 with KL = 15 ft m = 1.0 and U= 1.5. Substituting in Eq. [8.2], we obtain PUJB = 8°0 + 200 x 2.0 + O = 1200 kips (5338 kN)
In the AISC Column Load Tables (p. 2-19 of the LRFD Manual) if Fy = 36 ksi (248 mPa) and KL = 15 ft (4.57 m), Qfn = 1280 kips (>Pu>eff = 1200 kips) for a W14 x 159.