Optimum Beam-to-Column Stiffness Ratio of Portal Frames under Lateral Loads By Pedro Silva, Ph.D., P.E. and Sameh S. Badie, Ph.D., P.E. An article in the March 2008 issue o STRUCTURE magazine discussed design optimization o portal rames, such as those shown in Figure 1, under gravity loads. This ollow-up article discusses similar design optimization under the inuence o lateral loads. It presents, both analytically and graphically, the procedure that is used to establish the design optimization. Since the drit ratio, defned as the ratio o the rame lateral deection to column height, δ, is one o the controlling actors in the design o portal rames subjected to lateral loads, the design optimization is perormed as a unction thereo. Design optimization is achieved by minimizing the sum o normalized column and beam moments o inertia necessary to satisy a prescribed drit ratio limit. The authors perormed their analyses using fnite element subroutines implemented in Matlab. Following the results, the authors present simple graphical procedures that can be used to illustrate the optimum beam-to-column stiness (I b/I c c ) ratios in the design o portal rames under lateral loads.
Effect on the Frame Behavior
n g i S e
Under lateral loads, rame design is highly dependent on the beam-to-column stiness ratio, beam-span-to-column-height ratio, and the column end supports.
s r e e n i
I axial deormations are ignored, using slope-deection methods it can be demonstrated that or a at rame with fxed supports, as shown in Figure 1(a), the lateral deection, ∆ , is:
D l a r u t c u r t
g n e l a r u t c u r t s r o f s e u s s i
n g i s e d n o s n o i s s u c s i d
S
Fixed Support Conditions
∆ f =
6A + 4K 6A + K
PL3c 24 EI c
C P
C A b,I b
B
P
D
A b,I b
θ
B
D
Lc Ac,Ic
Ac,Ic
Ac,Ic
A
A
E
(a)
Lc
Ac,Ic E
(b)
L b
L b
Figure 1: Portal rames under lateral loads (a) Flat rame, (b) Pitched rame.
For α=0 (corresponding to infnitely sti columns), the rame deection, ∆ , is our times higher than the deection or α= (corresponding to infnitely sti beams). This is the case because or α=0 the rame response reverts to that o two cantilever columns, and or α= it reverts to that o two propped cantilever columns. The results or α=0.74 are also shown in Table 1. In order to meet the drit ratio, δ, the column stiness is obtained by solving or ∆ δLc in Equation (1). Normalizing the required column stiness in terms o the variable EI c c /PL 2 c gives the ollowing result: L
c
=
6A + 4K 6A + K
brevity and illustrative purposes, only the results or κ =1.5 =1.5 are presented in Figure 2(a). In this fgure is shown the ratio λ c as a unction o α or dierent roo pitches. It is clear that as a unction o α, λ c is insensitive to variations in the roo pitch. This fgure also shows that as α increases, λ c reduces, which matches the results discussed in Table 1. The design optimization rule can be obtained in terms o summing the normalized ratios α+λ c. Once again, or κ =1.5, =1.5, the sum λ c+α as a unction o α is shown in Figure 2(b). Curves in this fgure clearly indicate that a minimum point exists; this point corresponds to the optimum beam-to-column stiness ratio, α. Curves in this fgure also show that the minimum point is nearly insensitive to the roo pitch. Consequently, the optimum beam-tocolumn stiness ratio, α, can be expressed in terms o the ollowing minimization rule:
1
: Fi Fixe xed d Su Supp ppor ortt 24D Equation (2)
In Table 1, the normalized column stiness is given in terms o the variable λ , showing once again that the required λ c or α=0 is 4 times higher than or α=. The authors developed expressions similar to those presented in Table 1 or other values o roo pitch, beamto-column I’s, α, and κ . However, or
*
A
3 K 1 d Fixed ed Sup Suppor portt = [L + A ]= 0 ⇒ A * = − K : Fix d A 4D 6 c
Equation (3)
Table 1: Deected shape or varying values o α = I b /I /I c c or κ = 1.00: Fixed rame. ≤ D Lc :
(a) α = I b /I c = zero
Fixed Fix ed Sup Suppor portt
Equation (1)
In Equation (1), α is the beam-tocolumn moment o inertia ratio, given by α=I b/I c c, and δ is the drit ratio designated as the ratio o lateral deection to co lumn height. This equation also includes an arbitrary beam-span-to-column-height ratio, designated herein as κ =Lb/Lc . The authors used this equation to develop Table 1, depicting the rame deection, ∆ , under three beam-tocolumn ratios, α=I b/I c c, and or a beamspan-to-column-height ratio o κ =1. =1.
∆ f
P
CL
Deflected shape
STRUCTURE magazine
4 PL3c
+
24 EI c +
34
4 24D
August August 2008 2008
(c) α = I b /I c = infin infinity ity (d) ∆ f
P
B
CL
A
A
L c
∆ f
P
B
$ f
(b) α = I b /I c = 0.74
B
CL
A
3
+
1.5515 × PLc 24 EI c
+
1.5515 24D
+
PL3c 24 EI c
+
1 24D
c
8.00 , s s e n f f 6.00 i t S n m u 4.00 l o C d e z i l 2.00 a m r o N0.00
8.00
Pitch = 0
λ
Pitch = 10 Pitch = 20 Pitch = 30
L A V O R P P A R U O Y R O F
6.00 α + c
λ
f 4.00 o m u S 2.00
C
B
0
0.4
0.8
1.2
1.6
D
0.00
2
A
Beam to Column I Ratio; α I b /I c
0
E
=
0.4
0.8
1.2
1.6
2
Beam to Column I Ratio; α I b /I c =
(a) K (Lb /Lc) =1.5, Fixed support
(b) K (Lb /Lc) =1.5, Fixed support
Figure 2: Results or fxed supports.
?
c16.00 λ , s s e n f 12.00 f i t S n m u 8.00 l o C d e z i l 4.00 a m r o N0.00
16.00
Pitch=0 Pitch=10 Pitch=20 12.00 Pitch=30 α
!
+ c
λ
C B
f 8.00 o m u S
4.00
D
0.00 0
1
2
3
4
5
E
A
Stiffness Ratio (I b /I c )
0
1
2
3
4
5
Beam to Column I Ratio; α I b /I c =
(a) K (Lb /Lc ) =1.5, Pinned support
(b) K (Lb /Lc) =1.5, Pinned support
Figure 3: Results or pinned supports.
Table 2: Deected shape or varying values o α=I b /I c or κ =1.00: Pinned rame.
(b) I b /I c = 1.8257
(c) I b /I c = infinity
∆ f
P
∆ f
P
B C L
B C L
Deflected shape A
A
5.0955 × PLc
4 PL3c
24 EI c
24 EI c
3
$ f
L c
+
! E T I S B E ? S W T I ? F S R E U O N R ? ? I O P D Y L T I T I E E F L D S S F A A I V A A U E R E T D R S Q O R R G C E N L N I L I D O A N O M C O T U P O A D O . L T E . E E . N G E V I G
+
5.0955
+
24D
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9 0 4 4 1 2 0 - 2 9 9 8 8 5 5 - 2 2 0 0 5 5 : : E X N A M F O O H C . P H C E T G N 1 E L 2 A 4 B E 2 T I 0 O L U 2 S 0 G 4 . , T W E Y E W K R T C W S U Y T T R N E E K B , I L E T L L S I E V S I W 4 U 0 O 3 L
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A D V E R T I S E M E N T - F o r A d v e r t i s e r I n f o r m a t i o n , v i s i t w w w . S T R U C T U R E m a g . o r g
Pinned support conditions
Following the same methodology, the authors developed expressions or rames with pinned supports. As beore, i axial deormations are ignored, or a at rame with pinned supports, the lateral deection, ∆ , is given by: 4
f
2K A
PL3c
D Lc :
24 EI c
Pinned Support
Equation (4)
The authors used Equation (4) to develop Table 2 , depicting the rame deection, ∆ , under α=1.8257 and infnity, and or an arbitrary κ . Clearly, in the condition o α=0 the rame is unstable; consequently, this case is not shown. In order to meet the required drit ratio, δ, the required column stiness is obtained by solving or ∆ δL, as depicted in Equation (4). Normalizing the required column stiness in terms o EI c /PL 2 c results in Equation (5). As beore, the authors developed expressions similar to those presented in Table 2 or other conditions and or brevity and illustrative purposes, only the results or κ =1.5 are presented in Figure 3(a). L
c
= 4 +
2K A
δ
24D
:
c
λ
, I n m u l o C d e z i l a m r o N c
0.00
Fixed w/δ =0.025
8.00
Fixed w/δ =0.050 Pinned w/δ =0.025
6.00
Pinned w/δ =0.050
4.00
2.00
0.00 0
1
2
3
0
Beam Span-to-Column Height Ratio;κ = Lb /Lc
(a) Normalized (A *
1
2
3
Beam Span-to-Column Height Ratio, κ
* / 6 ) D and A D (b) Normalized column stiffness, λ c
+ K
Figure 4: Optimum ratio charts or lateral loads.
Table 3: Design example results or a roo pitch o 20 .
Supports (1) Fixed
α
Using Figure 4(a) (2)
(A * + K / 6 ) *
A a)
Pinned Support
10.00
Fixed-Fixed Pinned-Pinned
t n r κ p o ( ; O m u p p t r 0.30 d l o u e C S o p z i p l o d u a t e S 0.20 n m m n d r a i o e P e x N B . i 1 F 0.10 . 2
Pinned
1
0.60
√ × δ ) o √ ∗ i 0.50 × α m t a ∗ u r + I α m ; 6 / 0.40 i t
D =0.177
D = 0.354
Using Figure 4 (b)
*
λ c
a)
I c
=
L c
PL2c
I b
=
4
*
A I c 4
b)
$ f
(3)
(4)
E s cm (in ) (5)
mm (in ) (6)
mm (in.) (7)
0.868
2.785
1.66 (398)
1.44 (346)
2.236
8.903
5.30 (1,273)
11.85 (2,846)
92.02 (3.62)
b)
4
4
Using Equations (1) and (4)
Equation (5)
For a rame with pinned supports, Figure 3(b) shows once again that there is an optimum value or design, corresponding to the minimum point on these curves. The optimization rule is mathematically expressed by Equation (6). *
A
d = [L + A ]= 0 ⇒ d A c
*
A
=
1
3K
6
D
:
Pinned Support
Equation (6)
Design Optimization Graphical approach
For rames with fxed and pinned supports, the results o normalizing Equations (3) and (6) in terms o (α*+κ /6) D and α* D , respectively, are presented in Figure 4(a). The results o substituting Equation (3) and Equation (6) into Equations (2) and (5) are presented in Figure 4(b). Figure 4(a) and the work developed in this article show that: 1) As κ =Lb/Lc increases, so does the optimum stiness ratio, α*. 2) In rames with pinned supports, the optimum stiness ratio is signifcantly higher than or a rame with fxed supports. 3) As the drit ratio increases, so does the required stiness ratio.
Design example using optimization rules
The authors used the charts in Figure 4(b) signifcantly to the rame response. In uto develop the design example presented in ture investigations, design optimizations this section. Design optimization ollows the will consider the ra me response dominated design calculations presented in Table 3. In by both gravity and lateral loads. this section, the rame has a roo pitch o 20 and consists o Grade 50 steel with E s =200 GPa (29,000ksi), an applied lateral load o Sameh S. Badie, Ph.D, P.E. is an Associate 889.6 kN (200 kips), and beam and column Proessor at the Civil & Environmental lengths o Lb=5.49 meters (18 eet) and Lc = Engineering Department o George 3.66 meters (12 eet), providing an aspect Washington University, Washington DC. ratio κ =1.50. Using a design drit ratio His research interests include analysis and δ=0.025 ( 1/40), the limiting lateral deection design o reinorced/prestressed concrete o the rame was ∆ δLc = 91.44 millimeters structures. Any comments or insights relating (3.6 inches). The computed values, shown to this article are encouraged and can be in Table 3 column (7), only slightly exceed sent to
[email protected] . this value, indicating that the design charts presented in Figure 4 can be used reliably to Pedro Silva, Ph.D,, P.E. is an Associate compute the optimum moments o inertia Proessor at the Civil & Environmental I c and I b. Engineering Department o George Washington University, Washington DC. His research interests include analysis and Future investigations design o structures subject to seismic and The design charts developed herein conblast loading. Any comments or insights sidered only rames under lateral loads. relating to this article are encouraged and The design optimizations based only on can be sent to
[email protected] . drit ratios may not necessarily apply to cases where gravity loads also contribute
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