Transcript 2006 ASCE/SEI Structures Congress

Manual and Inelastic-Analysis Based
Design of Partially-Restrained Frames
Using the 2005 AISC Specifications
By
Christopher M. Foley, PhD, PE
Marquette University
Milwaukee, WI
John Schaad, MS, EIT
Jezerinac, Geers, & Associates
Dublin, OH
2006 ASCE-SEI Structures Congress
St. Louis, MO
May 18-20
2006 ASCE/SEI Structures Congress
St. Louis, MO
May 18-20
1
MOTIVATION FOR PRESENTATION
AISC Specifications are becoming more liberalized in the designer's favor
and are beginning to allow software capabilities to be exploited.
There are new demands on the structural engineer to understand phenomena that
software is now able to consider.
Focus can now be on SYSTEM BEHAVIOR rather than members or
components and designing for target behavior is possible.
How do we teach these concepts and specification developments to students?
It would be very beneficial to have a manual methodology to get starting sizes
for inelastic analysis-based design.
Address Wooten's Third Law of Steel Column Design - Corollary Number 2:
"The computer renders obsolete the necessity of rationalizing
and simplifying problems - or even of understanding them" (Wooten 1971)
2006 ASCE/SEI Structures Congress
St. Louis, MO
May 18-20
2
TYPICAL FRAMING PLAN
Roof Level:
B
A
3@10 ft.
D
C
30 ft.
30 ft.
Superimposed DL: 63 psf
• Comp. Slab:
46 psf
• Ceiling:
2 psf
• Fireproofing:
3 psf
• MEP systems:
12 psf
Superimposed LL: 30 psf
E
30 ft.
30 ft.
1
Floor Level:
Superimposed DL: 83 psf
• Comp. Slab:
46 psf
• Ceiling:
2 psf
• Fireproofing:
3 psf
• MEP systems:
12 psf
• Partitions:
20 psf
Superimposed LL: 50 psf
30 ft.
2
3
30 ft.
Steel Framing: 5 psf
Cladding: 25 psf (wall area)
Wind, WL: 20 psf
4
Steel Material: A992
N
PR Connections
Flexible (pin) Connections
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May 18-20
X-Braced Bay
3
BASE ANALYTICAL MODEL
B
A
D
C
E
WR
15
PIR
PER
PIF
F
RkR PE
WF
PLR
PLF
RkF
15
Rkbp
30
30
30
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May 18-20
30
4
BILINEAR CONNECTION MODELS
M
M
F
0.50  M pb
R
0.50  M pb
10
EI z
L
F
0.50  M pb
FLOOR BEAM
10 

EI z
L

R
0.50  M pb
M
0.75  M pc
ROOF BEAM
10 
EI z
L

0.75  M pc
2006 ASCE/SEI Structures Congress
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May 18-20
BASE PLATE
5
AISC APPENDIX 7 - Preliminary Design
Member design can be greatly streamlined if the following constraints on
member selection are included.
• Choose a target interstory drift to meet target 2nd order sway amplification:
  HL
1
B2 
 1.15   H  0.153  
 P
P



u , nt
H
u , nt

1
 0.85  HL



Implies that behavior is "nearly" linear up to first hinge formation.
• Choose member sizes to avoid stiffness reduction;
Pu
 0.50   b  1.00  EI *  EI
Py
• Choose member sizes to avoid P   effect;
Pu
 0.15  B1  1.00
PeL
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May 18-20
6
AISC APPENDIX 1 - Preliminary Design
Local Buckling:
• Flanges;
• Webs in Combined flexure and axial compression;

Pu
Pu 
h
 0.113
 276.02  0.328  
Py
tw
Py 

Pu
 0.113
Py
bf
2t f
 9.15

Pu 
h
 29.94   2.10    35.88
tw
Py 

Stability and Nonlinear Geometric Effects
Pu
 0.675
Py
Lateral-Torsional Buckling
(column members)
(beam members)
r 
 ry 
y min
Lb

113.43
min

Lb

 M1  
0.12

0.076


    580 
 M 2 

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May 18-20
7
AISC APPENDIX 1 - Preliminary Design
Moment capacity in presence of axial loading
Pu
0.9 Pny
Assume column bent in reverse curvature and the
inflection point is at 2/3 column height:
1512 
 ry min  0.12  0.076  0.5   580  1.96


Therefore, if ry   ry min then;
P
9
M cap  1  u
8  0.9 Pny
Pu
0.9 Pny
M cap
0.20
M cap


P


M


1  u
pc

 0.9 Pny
 1 Pu
 1 
 2  Pny


  M pc



P
   M pc   0.90  0.45 u
 Pny



  M pc

Mu
0.9M n
0.9 M n
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May 18-20
8
LOADING COMBINATIONS
ASCE 7 - 02 Strength Limit State (with corrections):
1.4  DL
0.2% notional loads
1.2  DL  1.6  LL  0.5  LLr
0.2% notional loads
1.2  DL  0.5  LL  1.6  LLr
0.2% notional loads
1.2  DL  0.5  LL  0.5  LLr  1.6  WL
ASCE 7 - 02 Serviceability Limit State
DL  LL
DL  0.5L  0.7W
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9
DESIGN ASSUMPTIONS
The following assumptions are made in the design:
• Unbraced lengths for columns were taken as the story height.
• The compression flange for beams in positive flexure is fully braced.
• The compression flange for girders subjected to negative flexure is
braced at column lines and at beam lines.
• Compression forces in beams is negligible.
• Columns are pin-pin for minor axis bending.
• Beams are non-composite with floor system.
2006 ASCE/SEI Structures Congress
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May 18-20
10
MECHANISM 1 - BEAM STRENGTH
R
uI
P
R
uI
P
Plastic Hinges:
F
uI
P
- beam hinge
F
uI
P
- connection hinge
Plastic hinges form in beams
indicating SCWB behavior.
Pu
Loading Combinations:
Pu
1.4  DL
1.2  DL  1.6  LL
Simplified Gravity
Load Analysis:
M pb
M uss  b M n
M uss
M
M M pb
2006 ASCE/SEI Structures Congress
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 M pb  M M pb
M uss
b M n 
1  M 
11
BEAM SERVICEABILITY
Moment Diagram - Kotylar (1996)
P
P
k mb
k mb

L
a
a
M
1
2 EI
1
kmb L
 8a 2  
M m  Pa 1  2 
L 

8 Pa 3 
Me 
L2
Moment-Area Principle Yields
Pa
CL 
 3L2  4a 2  24a 2  
24EI
2006 ASCE/SEI Structures Congress
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May 18-20
12
BEAM DESIGN
Strength:
Assume connection strength at 50% of the plastic moment capacity:  M  0.50
Assume that the beams are bent in reverse curvature: M1 M 2  0.50
Compute the ultimate simply-supported beam moment.
Compute the required strength of the PR beam.
Using the unbraced length  Lb  10 establish:  ry min
Select a beam for strength considerations.
Serviceability:
Assume that beam connection stiffness results in PR behavior:   10
Check total, DL and LL deflections at mid-span.
Check that connections do not exceed yield moment at service level loads.
Adjust beam size as required.
Beams Selected:
W16x40 (roof)
W21x55 (floor)
2006 ASCE/SEI Structures Congress
St. Louis, MO
May 18-20
13
MECHANISM 2 - Gravity and Notional Loading
Gravity and notional loading are assumed to be applied in a proportional
manner - therefore, a combined mechanism is targeted.
10
  30
N
R
u
15  15
15
20
R
PuIR PuI
F
PuIF PuI
PuER
PuEF
PuLR
PuLF
Plastic Hinges:
- beam hinge
- connection hinge
N uF

Plastic hinges form
in beams indicating
SCWB behavior.
Loading combinations applied in a non-proportional manner (e.g. gravity load first
and then lateral load to failure) will likely result in a sway mechanism forming.
2006 ASCE/SEI Structures Congress
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May 18-20
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MECHANISM 3 - Gravity and Lateral Loading
Lateral and gravity loading combinations are applied in a proportional manner therefore, a combined mechanism is targeted.
10
  30
H
R
u
15  15
15
20
R
PuIR PuI
F
PuIF PuI
H uF
PuER
PuEF
PuLR
PuLF
Plastic Hinges:

- beam hinge
- connection hinge
Plastic hinges form
in beams indicating
SCWB behavior.
Previous beam design indicates that plastic hinges will NOT form when gravity
loading at the following magnitude is applied:
1.2  DL  0.5   LL  LLr 
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May 18-20
15
SIZING FOR TARGET MECHANISMS
SCWB criteria can be used to help ensure assumed targeted mechanisms form.
M 
cap col ,2
M 
 2M  M p 
cap col ,2
bm
W16  40
W16  40
W16  40
W16  40
W 21 55
W 21 55
W 21 55
W 21 55
bm
M 
M 
cap col ,1
  M cap 
col ,2
 2M  M p 
cap col ,1
2006 ASCE/SEI Structures Congress
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May 18-20
  M cap 
col ,2
 M  M p 
bm
 M  M p 
bm
16
FRAME FOR FURTHER EVALUATION
The framework shown below is the system that will be used for displacement
evaluation.
W16  40
W16  40
W16  40
W16  40
W12  58
W12  58
W 21 55
W12  58
W 21 55
W12  58
W 21 55
W12  58
W 21 55
W12  58
W12  58
W12  58
W12  58
W12  58
We will now check the columns and/or beams to ensure second-order effects
are "small" and the frame is serviceable with respect to interstory drift.
2006 ASCE/SEI Structures Congress
St. Louis, MO
May 18-20
17
SUB-ASSEMBLAGE DISPLACEMENTS
Using modifications to the work of Englekirk (1994), displacement expressions
for interior, exterior and column base segments can be generated.
Interior Sub-Assembly
i 
Vi  h1  h2 
2
12 E
  L   h1  h2  6 E 




Ic
Rk 
 Ib
h1
R
Ic
Ib
e 
6E
2
h h 
R  Vi   1 2 
 L 
Ve
Ib
Partially Restrained Column Bases
Vh13
 cb 
3EI c
 3EI c 
 1 

R
h
kb
1


h h 
R  Vi   1 2 
 L 
L 2
L 2
  L   h1  h2  6 E 




I
2
I
R
c
k 
 b
Ib
Ic
Vi
Ve  h1  h2 
Rk
Rk
h2
Exterior Sub-Assembly
Vi
Rk
Ic
h1
Ic
h2
Ve
L 2
For simplicity, we will assume inflection points at 1/2 first story height and
mid-height of the second story columns.
2006 ASCE/SEI Structures Congress
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May 18-20
18
"SMALL" 2nd ORDER EFFECTS AND SERVICEABILITY
Exterior and Interior Columns: W12  58
2nd Floor Beam: W 21 55
Assume inflection points at mid-heights: h1  h2  7.5  90
Connection and base plate stiffness:
Rk ,beam  10
EI
 918,333 k  in rad
L
RkI,bp  10
EI
 765,278 k  in rad
L
2006 ASCE/SEI Structures Congress
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May 18-20
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"SMALL" 2nd ORDER EFFECTS - STRENGTH L.S.
1.2  DL  1.6  LL  0.5  LLr with notional loading
 H  4.47 k
 P  1,666 k
u , nt
 H max  0.074 at 1st story
L  180
i 
 icb
e 
1.118 180 
  360  180  6(29, 000)  1



   0.046
12(29, 000)  (1,140) (475) (918,333)  2
2
(1.118)(90)3  3(29, 000)(475) 

 1 
 0.02

3(29000)(475) 
(765, 278)(90) 
(0.559) 180 
 ecb
6(29, 000)
4.47 k
 0.559 k
8
Vi  1.118 k
Ve 
2
  360 
180 6(29, 000)  1



   0.04
 (1,140) 2(475) (918,333)  2
I
tot
 0.07
OK
E
tot
 0.06
OK
(0.559)(90)3  3(29, 000)(475) 

 1 
 0.02

3(29000)(475) 
(765, 278)(90) 
2006 ASCE/SEI Structures Congress
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May 18-20
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"SMALL" 2nd ORDER EFFECTS - STRENGTH L.S.
1.2  DL  0.5   LL  LLr   1.6  W without notional loading
 H  32.4 k
 P  1,368 k
u , nt
 H max  0.65
at 1st story
L  180
i 
 icb 
e 
 ecb
8.10 180 
  360  180  6(29, 000)  1



   0.33
12(29, 000)  (1,140) (475) (918,333)  2
32.4 k
 4.05 k
8
Vi  8.10 k
Ve 
2
 3(29, 000)(475) 
(8.10)(90)
 1 
 0.23
3(29000)(475) 
(765, 278)(90) 
I
tot
 0.56
OK
E
tot
 0.37
OK
3
(4.05) 180 
  360 
180  6(29, 000)  1



   0.26
6(29, 000)  (1,140) 2(475) (918,333)  2
2
(4.05)(90)3  3(29, 000)(475) 

 1
 0.11
3(29000)(475) 
(765, 278)(90) 
2006 ASCE/SEI Structures Congress
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May 18-20
21
SERVICEABILITY LIMIT STATE
DL  0.5   LL  LLr   0.7  W without notional loading
 H  14.18 k
 P  1,176 k
u , nt
 H max  0.33
at 1st story
L  180
i 
 icb
3.55 180 
  360  180  6(29, 000)  1



   0.14
12(29, 000)  (1,140) (475) (918,333)  2
2
(3.55)(90)3  3(29, 000)(475) 

 1 
 0.10

3(29000)(475) 
(765, 278)(90) 
e 
 ecb
14.2 k
 1.78 k
8
Vi  3.55 k
Ve 
(1.78) 180 
  360 
180  6(29, 000)  1



   0.11
6(29, 000)  (1,140) 2(475) (918,333)  2
I
tot
 0.28
OK
L 400  0.45
2
E
tot
 0.16
OK
(1.78)(90)3  3(29, 000)(475) 

 1 
 0.05

3(29000)(475) 
(765, 278)(90) 
2006 ASCE/SEI Structures Congress
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22
FRAME FOR MECHANISM ANALYSIS
The framework shown below is the system that resulted from the serviceability and
2nd order effects evaluation.
W16  40
W16  40
W16  40
W16  40
W12  58
W12  58
W 21 55
W12  58
W 21 55
W12  58
W 21 55
W12  58
W 21 55
W12  58
W12  58
W12  58
W12  58
W12  58
We will now simply ensure that the member sizes chosen will result in the
targeted mechanisms forming at levels higher than the strength limit state
combinations.
2006 ASCE/SEI Structures Congress
St. Louis, MO
May 18-20
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MECHANISM 2 - Combined Mechanism
1.2  DL  1.6  LL  0.5  LLr
PR  NuR
PF  NuF
1.2  DL  0.5  LL  1.6  LLr
PR  NuR
PF  NuF
1.2  DL  0.5   LL  LLr   1.6  WL
PR  H uR
PF  HuF
10
  30
PR
15  15
PF
15
Py  Ag Fy
20
M p   M pb
Py  Ag Fy
R
PuIR PuI
M p,con  0.5   M pb 
F
PuIF PuI

Py   Pny
M p,col  min 0.75   M p,col , M cap,col 
2006 ASCE/SEI Structures Congress
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EVALUATION OF DESIGN USING INELASTIC ANALYSIS
The frame designed preliminarily using the preceding procedure was checked
using MASTAN2 (Ziemian and McGuire).
The following loading combinations were evaluated:
1.2  DL  1.6  LL  0.5  LLr
DL  LL
1.2  DL  0.5  LL  1.6  LLr
DL  0.5   LL  LLr   0.7  W
1.2  DL  0.5   LL  LLr   1.6  W
The yield surface of MASTAN2 was manipulated as follows:
P
 Pny
1.0
Default MASTAN2 yield surface
connects end points.
 Pny  Fy ,col  Ag
Z x  Fy ,col  min  M p ,base ,  M p ,col 
1.0
M
M
or
 M p,base  M p,col
2006 ASCE/SEI Structures Congress
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May 18-20
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EVALUATION OF DESIGN USING INELASTIC ANALYSIS
1.2  DL  1.6  LL  0.5  LLr   
  ult  2.11
Node 15
Elem. 35
0.100
2.5
2.0
0.075
Element 35 - Right
P/Py

1.5
0.050
1.0
Node 15 - Horiztonal
0.025
0.5
0.0
0.0
0.5
1.0
1.5
2.0
Displacement (in.)
2.5
3.0
0.000
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1
M/Mp
2006 ASCE/SEI Structures Congress
St. Louis, MO
May 18-20
26
EVALUATION OF DESIGN USING INELASTIC ANALYSIS
1.2  DL  0.5  LL  1.6  LLr 
  ult  2.08
Elem. 39
Node 15
Elem. 35
0.05
2.5
0.04
2.0
Element 35 - Right
Element 39 - Right

P/Py
1.5
0.03
0.02
1.0
Node 15 - Horiztonal
0.5
0.0
0.00
0.01
0.25
0.50
Displacement (in.)
0.75
1.00
0.00
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1
2006 ASCE/SEI Structures Congress
St. Louis, MO
May 18-20
M/Mp
27
EVALUATION OF DESIGN USING INELASTIC ANALYSIS
1.2  DL  0.5   LL  LLr   1.6 W   
Elem. 39
  ult  2.04
Node 15
Elem. 35
Elem. 100
0.5
2.5
0.4
2.0

P/Py
1.5
1.0
0.3
ALR = 1.0
0.2
Element 35 - Right
Element 39 - Right
Element 100 - Bottom
Node 15 - Horiztonal
0.5
0.1
0.0
0
2
4
6
Displacement (in.)
8
10
0.0
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1
M/Mp
2006 ASCE/SEI Structures Congress
St. Louis, MO
May 18-20
28
SERVICEABILITY EVALUATION OF DESIGN
DL  LL
No connections hit the yield moment
Vertical beam deflections were well below acceptable thresholds
DL  0.5   LL  LLr   0.7  W
1.2
1is
0.28
No connections hit yield
moment at service levels.
Applied Load Ratio
1.0
0.8
Typical limit on interstory drift is:
180
is 
 0.45
400
0.6
0.4
1st InterStory Drift
2nd InterStory Drift
Roof Level Drift
0.2
0.0
0.0
0.1
0.2
0.3
0.4
Displacement (in.)
2006 ASCE/SEI Structures Congress
St. Louis, MO
May 18-20
29
CONCLUDING REMARKS
An approximate methodology for sizing members within the context of AISC
Appendices 1 and 7 has been outlined.
The method has been shown to be relatively accurate given its overall simplicity.
The advantage of the approach is that it focuses on "system" behavior while
maintaining flexibility to consider beam-columns, beams, and connections.
The formulas for displacement have been show to be accurate for preliminary
design purposes and they provide the engineer with significant problem feel.
Small multiple-story multiple-bay frames can be sized using the procedure to
control second-order effects and the resulting designs have significant reserve
strength.
The best use of the methodology would be to demonstrate the important
provisions in the new AISC (2005) specifications in a simplified manner so that
algorithms for computer implementation can be developed.
2006 ASCE/SEI Structures Congress
St. Louis, MO
May 18-20
30
REFERENCES
Englekirk, R. (1994) Steel Structures - Controlling Behavior Through
Design, John Wiley & Sons, Inc., New York, NY.
Foley, C.M. and Schinler, D. (2003) "Automated Design of Steel Frames Using
Advanced Analysis and Object-Oriented Evolutionary Computation",
Journal of Structural Engineering, 129 (5), pp. 648-660.
Kotylar, N. (1996) "Formulas for Beams with Semi-Rigid Connections"
Engineering Journal, AISC, Fourth Quarter, pp. 142-146.
Wooten, J. (1971) "Wooten's Third Law and Steel Column Design",
Engineering Journal, 2nd Quarter, pp. 1-3.
2006 ASCE/SEI Structures Congress
St. Louis, MO
May 18-20
31
Extra slides showing detailed computations
to follow this slide.
2006 ASCE/SEI Structures Congress
St. Louis, MO
May 18-20
32
SERVICE LOADING
Gravity Loading
Roof
R
E , DL
P
Floor

 15    30 

  68 psf  5    25 psf     

 2   2 

 15 
  25 psf     5   8,850 lbs
 2 
 30 
PEF, DL   88 psf  5    25 psf 15   

 2 
  25 psf 15  5   14,100 lbs
 30 
PER, LL   30 psf  5  
  2, 250 lbs
 2 
 30 
PEF, LL   50 psf  5  
  3,750 lbs
 2 
 30 
PIR, DL   68 psf 10  

 2 
 15 
  25 psf    10   12,075 lbs
 2 
 30 
PIF, DL   88 psf 10  

 2 
  25 psf 15 10   16,950 lbs
 30 
  30 psf 10  
  4,500 lbs
 2 
 30 
PIF, LL   50 psf 10  
  7,500 lbs
 2 
R
I , LL
P
2006 ASCE/SEI Structures Congress
St. Louis, MO
May 18-20
33
SERVICE LOADING (continued)
Wind Loading
Floor
Roof
 90 
WF   20 psf 15  

 2 
 13,500 lbs
 15  90 
WR   20 psf   

 2  2 
 6,750 lbs
Leaning Columns
Floor
Roof
PLR, DL   68 psf  60 120 
 15    1 
  25 psf  60     2    
 2   2 
 256,050 lbs
1
PLR, LL   30 psf  60 120    
2
 108,000 lbs
PLF, DL   88 psf  60 120 
1
  25 psf  60 15  2    
2
 339,300 lbs
1
PLF, LL   50 psf  60 120    
2
 180,000 lbs
2006 ASCE/SEI Structures Congress
St. Louis, MO
May 18-20
34
AISC APPENDIX 7 - DIRECT ANALYSIS
Column Nominal Strength
Pn, in plane computed using Kin plane  1.0
Out of plane strength defined in usual manner.
Design Analysis Requirements
Analysis must incorporate geometric nonlinearity: P   and P  
and construction/erection imperfections.
• AISC amplification factors allowed if reduced stiffness is used;
B2 
1
1
 Pnt   H
 0.85  HL
B1 
Cm
P
1 r
Pe1
Interstory drift,  H , and axial force demands computed using;
EI   0.8  b  EI
*
P  P
 b  4  r 1  r

 Py  Py
2006 ASCE/SEI Structures Congress
St. Louis, MO
May 18-20

 
 
Pr
 0.50
Py
35
AISC APPENDIX 7 (continued)
Design Analysis Requirements (continued)
• P   effects can be omitted
 B1  1.00 
when;
Pr  0.15  PeL
• Computer software capable of conducting geometrically nonlinear
analysis is allowed.
• Story out-of-plumb imperfections must be included through notional loading;
Ni  0.002  Yi
Out-of-plumbness can be directly inserted into the analytical model (1/500).
• If second-order amplification is less than 50% notional loads need only
need only be applied with gravity load combinations.
2006 ASCE/SEI Structures Congress
St. Louis, MO
May 18-20
36
AISC APPENDIX 1 - INELASTIC ANALYSIS AND DESIGN
Material Limitations:
The yield strength of members shall not exceed 65 ksi.
Local Buckling:
• Webs in Combined flexure and axial compression;
Pr
 0.125
0.9  Py
h
E
 3.76 
tw
Fy
 2.75  Pr 
 1 

 0.90  Py 
Pr
 0.125
0.9  Py
h
E
 1.12 
tw
Fy

Pr 
E
  2.33 

1.49


0.90

P
Fy

y 

• Flanges;
bf
2t f
 0.38 
E
Fy
2006 ASCE/SEI Structures Congress
St. Louis, MO
May 18-20
37
AISC APPENDIX 1 (continued)
Stability and Nonlinear Geometric Effects
• First order (mechanism) analysis can be used provided second-order
effects are considered. Second order inelastic analysis is permitted.
• Sufficient rotational ductility in columns preserved through limiting axial
load levels;
Pr  0.90   0.75  Fy Ag   0.675  Py
Lateral-Torsional Buckling
Lb
E
 4.71
r
Fy
(targeted for column members)
 M1   E
Lb 
  0.12  0.076 
   
ry 
M
 2    Fy



(targeted for beam and beamcolumn members)
2006 ASCE/SEI Structures Congress
St. Louis, MO
May 18-20
38
AISC APPENDIX 1 (continued)
Axial Capacity, Moment Capacity and Combined Forces
Pr  Pc  0.90  Pn
Pr
0.9 Pn
M r  M c  0.90  M p  0.9  1.6  M y 
Pr
8  Mr
 
0.9 Pn 9  0.9 M p

  1

 Mr
Pr

2  0.9 Pn   0.9 M p

  1

Pr
 0.20
0.9 Pn
Mr
 0.90
0.9
M
p
2006 ASCE/SEI Structures Congress
St. Louis, MO
May 18-20
Mr
0.9 M p
39
BEAM DESIGN - STRENGTH
Assume connection strength at 50% of the plastic moment capacity:  M  0.50
At the strength limit state, beams are bent in reverse curvature and ratio of
end moments is 0.50.
Floor Beams
Roof Beams
Pu1  1.4 12.08   16.9 k
Pu1  1.4 16.95   23.7 k
Pu 2  1.2 12.08   1.6  4.5   21.7 k 
Pu 2  1.2 16.95   1.6  7.5   32.3 k 
M uss   21.7 k 10  217 k  ft
b M n req
M uss   32.3 k 10  323 k  ft
M uss
323

M


 b n req
1  M  1  0.5
M uss
217


1  M  1  0.5
 215.3 k  ft
 144.7 k  ft
Lb  10
(negative bending)
10 12 

 ry min  0.12  0.076  0.5  580   1.31


Try W14x30 b M n  M p  177 k  ft
r 
y min
 1.49
Try W16x36 b M n  M p  240 k  ft
2006 ASCE/SEI Structures Congress
St. Louis, MO
May 18-20
r 
y min
 1.52
40
BEAM DESIGN - SERVICEABILITY (continued)
Assume that beam connection stiffness results in PR behavior:   10
Check total, DL and LL deflections at mid-span.
Roof Beams
Floor Beams
W14x30
P  16.95  7.5  24.4 k
P  12.08  4.5  16.6 k
kmb
 29,000 ksi   291 in4 
EI z

 10 
L
360
 234,417 k  in rad

1
 0.91
2 EI
1
kmb L
W16x36
a  120
 CL tot  0.16
kmb

 29,000 ksi   448 in4 
EI z

 10 
L
360
 360,889 k  in rad
1
 0.83
2 EI
1
kmb L
a  120
 CL tot  0.42
 7.5 


 CL LL 
  0.42   0.13
 24.4 
 CL LL
 4.5 

  0.16   0.04
 16.6 
 CL DL
 16.95 
 12.08 


 CL DL 

  0.42   0.29
  0.16   0.12
 24.4 
 16.6 
2006 ASCE/SEI Structures Congress
St. Louis, MO
May 18-20
41
BEAM DESIGN - SERVICEABILITY (continued)
Ensure end moments are less than connection yield moment at service loads.
Roof Beams
Floor Beams
W14x30
P  16.95  7.5  24.4 k
P  12.08  4.5  16.6 k
1

 0.91
2 EI
1
kmb L
W16x36
a  120

1
 0.83
2 EI
1
kmb L
a  120
8(16.6)(120)3 (0.91)
Me 
 134.3 k  ft
2
(360) (12)
8(24.4)(120)3 (0.83)
Me 
 180.0 k  ft
2
(360) (12)
M M p  0.5177   88  134 NG
M M p  0.5 240  120  180 NG
connection yield moments may be exceeded at service
Revise to W16x40
Revise to W21x55
ry  1.57  1.31
ry  1.35  1.31
b M n  M p  274 k  ft
b M n  M p  473 k  ft
2006 ASCE/SEI Structures Congress
St. Louis, MO
May 18-20
42
SIZING FOR TARGET MECHANISMS (continued)
Gravity loading combinations will be used (axial load in columns greatest).
r 
K y L  15
y min
 1.96
Interior 2nd Story Column
Exterior 2nd Story Column
 Pu 2 max  3 21.7  65.1 k
 Pu 2 max  14.2  21.7  35.9 k
M 
M 
cap col ,2
 274 k  ft
cap col ,2
W12x58
W8x40
Exterior 1st Story Column
Interior 1st Story Column
 Pu1 max  22.9  32.3  55.2 k
 Pu 2 max  11.8  16.8  28.6 k
 Pu1 max  332.3  96.9 k
 Pu 2 max  316.8  50.4 k
M 
cap col ,1
  M cap 
W12x58
 137 k  ft
col ,2
 473 k  ft
M 
cap col ,1
  M cap 
col ,2
 236 k  ft
W8x40
2006 ASCE/SEI Structures Congress
St. Louis, MO
May 18-20
43
COLUMN DESIGN - Out-of-Plane Buckling
Only first-story columns and gravity load combinations will be checked at this point.
K minor  1.0
Lu  15 ft
Exterior Column: W8x40
Interior Column: W12x58
1.4  DL
1.4  DL
Pu  3  16.9 
Pu  12.4  16.9 
 3   23.7   121.8 k
 19.7  23.7   72.7 k
1.2  DL  1.6  LL  0.5  LLr
Pu  3  16.8 
 3   32.3  147.3 k
1.2  DL  0.5  LL  1.6  LLr
Pu  3   21.7 
 3   24.0   137.1 k
c Pny  527 k  147 k OK
1.2  DL  1.6  LL  0.5  LLr
Pu  11.8  16.8 
  22.9  32.3  83.8 k


1.2  DL  0.5  LL  1.6  LLr
Pu  14.2  21.7 
 18.8  24.0   78.7 k
c Pn  299 k  84 k OK
W12x58
W8x40
2006 ASCE/SEI Structures Congress
St. Louis, MO
May 18-20
44
COLUMN DESIGN - Cross-Section Stability Checks
Exterior Column: W8x40
Interior Column: W12x58
1.2  DL  1.6  LL  0.5  LLr
Pu  3  16.8  3   32.3  147.3 k
1.2  DL  1.6  LL  0.5  LLr
Pu  11.8  16.8   22.9  32.3  83.8 k
Py  Ag Fy  850 k
Py  Ag Fy  585 k
bf
bf
2t f
 7.82  9.15
2t f
Pu 147

 0.17  0.675
Py 850
Pu
84

 0.14  0.675
Py 585

P
h
 27.0  29.94  2.10  u   57.8
tw
Py 

W12x58
 7.21  9.15

Pu 
h
 17.6  29.94  2.10    58.7
tw
Py 

W8x40
2006 ASCE/SEI Structures Congress
St. Louis, MO
May 18-20
45
FLOOR-LEVEL SUBASSEMBLAGES
R
h1
Ic
Ib
Ib
L 2
L 2
Ve
Ic
Rk
Ib
h h 
R  Vi   1 2 
 L 
Ic
Vi
h2
Rk
Rk
h2
h1
Vi
h h 
R  Vi   1 2 
 L 
Ve
Ib
R
Ic
Rk
Ve
Ic
h1
Ic
h2
Ve
L 2
L 2
2006 ASCE/SEI Structures Congress
St. Louis, MO
May 18-20
46
COLUMN DESIGN - Out-of-Plane Buckling
Only first-story columns and gravity load combinations will be checked at this point.
K minor  1.0
Lu  15 ft
Exterior Column: W8x35
Interior Column: W10x45
1.4  DL
1.4  DL
Pu  3  16.9 
Pu  12.4  16.9 
 3   23.7   121.8 k
 19.7  23.7   72.7 k
1.2  DL  1.6  LL  0.5  LLr
Pu  3  16.8 
 3   32.3  147.3 k
1.2  DL  0.5  LL  1.6  LLr
Pu  3   21.7 
 3   24.0   137.1 k
c Pn  332 k  147 k OK
1.2  DL  1.6  LL  0.5  LLr
Pu  11.8  16.8 
  22.9  32.3  83.8 k


1.2  DL  0.5  LL  1.6  LLr
Pu  14.2  21.7 
 18.8  24.0   78.7 k
c Pn  260 k  84 k OK
W10x45
W8x35
2006 ASCE/SEI Structures Congress
St. Louis, MO
May 18-20
47
COLUMN DESIGN - Cross-Section Stability and LTB Checks
Exterior Column: W8x35
Interior Column: W10x45
1.2  DL  1.6  LL  0.5  LLr
Pu  3  16.8  3   32.3  147.3 k
1.2  DL  1.6  LL  0.5  LLr
Pu  11.8  16.8   22.9  32.3  83.8 k
Py  Ag Fy  665 k
Py  Ag Fy  515 k
bf
bf
2t f
 6.47  9.15
2t f
 8.1  9.15
Pr 147

 0.22  0.675
Py 665
Pr
84

 0.16  0.675
Py 515

h
P
 22.5  29.94  2.10  r   56.29
tw
Py 


h
Pr 
 20.5  29.94  2.10    57.99
tw
Py 

r 
y min

180
 1.59  2.01
113.43
r 
y min

180
 1.59  2.03
113.43
W10x45
W8x35
2006 ASCE/SEI Structures Congress
St. Louis, MO
May 18-20
48
LOADING COMBINATIONS
1.4  DL
Roof Beam Loads
Floor Beam Loads
PuER  1.4 8.85  12.4 k
PuEF  1.4 14.1  19.7 k
PuIR  1.4 12.1  16.9 k
PuIF  1.4 16.9  23.7 k
Leaning Columns
PuLR  1.4  256.1  358.5 k
PuLF  1.4 339.3  475.0 k
Notional Loading (applied laterally)
NuR  0.002   2 12.4  1116.9  358.5  1.14 k
NuF  0.002   2  12.4  11  16.9   358.5
 0.002   2  19.7  11  23.7   475.0  2.69 k
2006 ASCE/SEI Structures Congress
St. Louis, MO
May 18-20
49
LOADING COMBINATIONS (continued)
1.2  DL  1.6  LL  0.5  LLr
Roof Beam Loads
Floor Beam Loads
PuER  1.2 8.85  0.5  2.25  11.8 k
PuEF  1.2 14.1  1.6 3.75  22.9 k
PuIR  1.2 12.1  0.5 4.5  16.8 k
PuIF  1.2 16.9  1.6  7.5  32.3 k
Leaning Columns
PuLR  1.2  256.1  0.5 108.0  361.3 k
PuLF  1.2 339.3  1.6 180.0  695.2 k
Notional Loading (applied laterally)
NuR  0.002   2 11.8  1116.8  361.3  1.14 k
NuF  0.002   2  11.8  11  16.8  361.3
 0.002   2  22.9  11  32.3  695.2  3.33 k
2006 ASCE/SEI Structures Congress
St. Louis, MO
May 18-20
50
LOADING COMBINATIONS (continued)
1.2  DL  0.5  LL  1.6  LLr
Roof Beam Loads
Floor Beam Loads
PuER  1.2 8.85  1.6  2.25  14.2 k
PuEF  1.2 14.1  0.53.75  18.8 k
PuIR  1.2 12.1  1.6  4.5  21.7 k
PuIF  1.2 16.9  0.5 7.5  24.0 k
Leaning Columns
PuLR  1.2  256.1  1.6 108.0  480.1 k
PuLF  1.2 339.3  0.5180.0  497.2 k
Notional Loading (applied laterally)
NuR  0.002   2 14.2  11 21.7   480.1  1.49 k
NuF  0.002   2  11.8  11  16.8  361.3
 0.002   2  18.8  11  24.0   497.2  3.09 k
2006 ASCE/SEI Structures Congress
St. Louis, MO
May 18-20
51
LOADING COMBINATIONS (continued)
1.2  DL  0.5   LL  LLr   1.6  WL
Roof Beam Loads
Floor Beam Loads
PuER  1.2 8.85  0.5  2.25  11.8 k
PuEF  1.2 14.1  0.53.75  18.8 k
PuIR  1.2 12.1  0.5 4.5  16.8 k
PuIF  1.2 16.9  0.5 7.5  24.0 k
Leaning Columns
PuLR  1.2  256.1  0.5 108.0  361.3 k
PuLF  1.2 339.3  0.5180.0  497.2 k
Notional Loading (applied laterally)
Second-order effects will be "small" and thus, no notional loading.
Factored Wind Loading
HuR  1.6  6.75  10.8 k
HuF  1.6 13.5  21.6 k
2006 ASCE/SEI Structures Congress
St. Louis, MO
May 18-20
52
FLOOR-LEVEL - EXTERIOR
b
Ve
b 
Ic  
h1
Ve  L  h1  h2 
2
6 EI b
Ib
Rk  
h2
R
Ve
c
 con
Ve
Ve
Ic  
Ib  
Ib  
Rk  
Ve
Rk
R
Ic
Ve
Ve  h1  h2 
3
c 
R
12 EI

2006 ASCE/SEI Structures Congress con
St. Louis, MO
c
May 18-20

Ve  h1  h2 
2
Rk
53
FLOOR-LEVEL - INTERIOR
b
b 
Vi
R
Vi  L  h1  h2 
Ic  
h1
2
12 EI b
Ib
Rk  
h2
R
Vi
 con
c
Ib  
Rk  
c 
Vi  h1  h2 
3
Ic
Ic  
Ib  
Rk
con 
12 EI c
2006 ASCE/SEI Structures Congress
St. Louis, MO
May 18-20
Vi  h1  h2 
2
2 Rk
54
STORY STIFFNESS
Interior Sub-Assembly
Ki 
Exterior Sub-Assembly
Vi
12 E

i   L   h1  h2  6 E 
2



  h1  h2 
Ic
Rk 
 Ib
Ke 
Ve
6E

 e   L   h1  h2  6 E 
2



   h1  h2 
2Ic
Rk 
 Ib
Concrete floor diaphragm provides displacement compatibility. This
leads to relationship between interior and exterior shear;
i   e
Vi
2
  L   h1  h2  6 E 
  L   h1  h2  6 E 




V





e 
Ic
Rk 
2I c
Rk 
 Ib
 Ib
Portal frame assumptions regarding shear distribution met when;
 L h1  h2   L h1  h2 
Vi
 Ve   
 

2
I
I
2Ic 
c
 b
  Ib
2006 ASCE/SEI Structures Congress
St. Louis, MO
May 18-20
55