Civil
Engineering
at JOHNS HOPKINS UNIVERSITY
Towards a Direct Strength Method for
Cold-Formed Steel Beam-Columns
Structural Stability Research Council
Orlando, Florida
May 2010
Y.Shifferaw1 , B.W.Schafer2
(1),(2) Department of Civil Engineering- Johns Hopkins University
Acknowledgments
• National Science Foundation
Overview
• Introduction
• Basis of DSM: yield and elastic critical
buckling
• Finite element collapse analysis in the P-M
space
• Direct Strength Method preliminaries for local
and distortional buckling in the P-M space
• Conclusion
• Future research
Introduction
• Current and postulated beam-column design
approaches
DSM anchor pts
DSM
Yieldanchor pts
Yield
Dis cr
Dis cr
Interaction
Interaction
70
70
P
P cr , d
60 cr , d
Py
60
50
50
P
Pn
n
40
40
Postulated  n curve
Postulated
 nM
curve
for all P and
ratios
for all P and M ratios

 cr , d
cr , d

y

 y
 n
30
30
20
20
Postulated  for a
Postulated
 for
given P and
Maratio
given P and M ratio
n
10
10
0
0 0
0
50
50
M
M n
n
100
100
My
150
150
M
M cr , d
cr , d
200
200
Sections considered
1.75 in
1 in.
250
1.625 in.
0.5 in.
3.625 in.
6 in.
0.5 in.
1 in.
2 in.
Eave Strut
1.625 in.
Channel
t=0.08in.
fy=55.9 ksi
DSM basis: major axis yielding and elastic
buckling
362Cmajor
2
First yield
Loc cr
1.5
Dist cr
1
P/P
y
0.5
0
-0.5
-1
-1.5
-2
-2
-1.5
-1
-0.5
0
Mx /Mx,y
0.5
1
1.5
2
DSM basis: minor axis yielding and elastic
buckling
362Cminor
2
First yield
Loc cr
1.5
Dist cr
1
P/P
y
0.5
0
-0.5
-1
-1.5
-2
-2
-1.5
-1
-0.5
0
Mz /Mz,y
0.5
1
1.5
2
DSM basis: biaxial yielding and elastic
buckling
362Cbiaxial
2
First yield
Loc cr
1.5
Dist cr
1
0
x
M /M
x,y
0.5
-0.5
-1
-1.5
-2
-2
-1.5
-1
-0.5
0
Mz /Mz,y
0.5
1
1.5
2
Finite element modeling
• Objective
To study combined P-M collapse loads in CFS
beam-columns for local and distortional limit
states.
• Method
– Material and geometric nonlinear analysis in
ABAQUS using S9R5 shell element models;
geometric local and distortional imperfections
considered
– Models generated from purpose-built Matlab code
Local FE
Major axis local for channel
1
FE Analysis
Ultimate Bounding Surface
0.9
0.8
0.7
P/P
y
0.6
0.5
0.4
0.3
0.2
0.1
0
-1.5
-1
-0.5
0
Mx /Mx,y
0.5
1
1.5
Major axis local for eave strut
1
FE Analysis
Ultimate Bounding Surface
0.9
0.8
0.7
P/P
y
0.6
0.5
0.4
0.3
0.2
0.1
0
-1.5
-1
-0.5
0
Mx /Mx,y
0.5
1
1.5
Distortional FE
Minor axis distortional for channel section
1
FE Analysis
Ultimate Bounding Surface
0.9
0.8
0.7
P/P
y
0.6
0.5
0.4
0.3
0.2
0.1
0
-1.5
-1
-0.5
0
Mz /Mz,y
0.5
1
1.5
Minor axis distortional for eave strut section
1
FE Analysis
Ultimate Bounding Surface
0.9
0.8
0.7
P/P
y
0.6
0.5
0.4
0.3
0.2
0.1
0
-1.5
-1
-0.5
0
Mz /Mz,y
0.5
1
1.5
Overview
• Introduction
• Basis of DSM: yield and elastic critical
buckling
• Finite element collapse analysis in the P-M
space
• Direct Strength Method preliminaries for local
and distortional buckling in the P-M space
• Conclusion
• Future research
 


M nd  M y  M p  M y 1   d


  dy







2




Preliminary DSM beam-column strength
prediction
LOCAL
if   0.776,
 n   ne
if   0.776,
0.4
0 .4


 cr    cr 

 
 y
 n  1  0.15



 y    y 



where
 y
  
  cr
 




0 .5
Local DSM vs major axis strength bounds for channel
DSM vs Strength Bounds-362Cloc,major
1
FE-Loc
DSM anchor pts
0.9
Y ield
Loc cr
0.8
DSM proposed
Interaction
0.7
P/P
y
0.6
0.5
0.4
0.3
0.2
0.1
0
-1.5
-1
-0.5
0
Mx /Mx,y
0.5
1
1.5
Local DSM vs minor axis strength bounds for channel
DSM vs Strength Bounds-362Cloc,minor
1
FE-Loc
DSM anchor pts
0.9
Y ield
Loc cr
0.8
DSM proposed
Interaction
0.7
P/P
y
0.6
0.5
0.4
0.3
0.2
0.1
0
-1.5
-1
-0.5
0
Mz /Mz,y
0.5
1
1.5
Local DSM vs major axis strength bounds for eave
DSM vs Strength Bounds-E loc,major
1
FE-Loc
DSM anchor pts
0.9
Y ield
Loc cr
0.8
DSM proposed
Interaction
0.7
P/P
y
0.6
0.5
0.4
0.3
0.2
0.1
0
-1.5
-1
-0.5
0
Mx /Mx,y
0.5
1
1.5
Local DSM vs minor axis strength bounds for eave
DSM vs Strength Bounds-E loc,minor
1
FE-Loc
DSM anchor pts
0.9
Y ield
Loc cr
0.8
DSM proposed
Interaction
0.7
P/P
y
0.6
0.5
0.4
0.3
0.2
0.1
0
-1.5
-1
-0.5
0
Mz /Mz,y
0.5
1
1.5
 


M nd  M y  M p  M y 1   d


  dy







2




Preliminary DSM beam-column strength
prediction
DISTORTIONAL
if d  0.673c,
 nd   y
if d  0.673c,

 cr
 nd  1  0.22a d
 y

where




0.5b
 cr
 d
 
 y




0.5b
y
0.5
 y 

d  
  cr 
 d
a  (1.136)^ (2  , b  (1.2)^ (2  , c  (0.834)^ (2  
  angular direction in radians in the P  M space
Distortional DSM vs major axis strength bounds
for channel
DSM vs Strength Bounds-362Cd,major
1
FE-Dist p
FE-Dist n
0.9
DSM anchor pts
Y ield
0.8
Dist cr
DSM proposed
0.7
Interaction
P/P
y
0.6
0.5
0.4
0.3
0.2
0.1
0
-1.5
-1
-0.5
0
Mx /Mx,y
0.5
1
1.5
Distortional DSM vs minor axis strength bounds
for channel
DSM vs Strength Bounds-362Cd,minor
1
FE-Dist p
FE-Dist n
0.9
DSM anchor pts
Y ield
0.8
Dist cr
DSM proposed
0.7
Interaction
P/P
y
0.6
0.5
0.4
0.3
0.2
0.1
0
-1.5
-1
-0.5
0
Mz /Mz,y
0.5
1
1.5
Distortional DSM vs major axis strength bounds
for eave
DSM vs Strength Bounds-E d,major
1
FE-Dist p
FE-Dist n
0.9
DSM anchor pts
Y ield
0.8
Dist cr
DSM proposed
0.7
Interaction
P/P
y
0.6
0.5
0.4
0.3
0.2
0.1
0
-1.5
-1
-0.5
0
Mx /Mx,y
0.5
1
1.5
Distortional DSM vs minor axis strength bounds
for eave
DSM vs Strength Bounds-E d,minor
1
FE-Dist p
FE-Dist n
0.9
DSM anchor pts
Y ield
0.8
Dist cr
DSM proposed
0.7
Interaction
P/P
y
0.6
0.5
0.4
0.3
0.2
0.1
0
-1.5
-1
-0.5
0
Mz /Mz,y
0.5
1
1.5
Conclusion
• Under combined loading the assumptions in linear
interaction equations are invalidated in CFS members
due to
– Un-symmetric shapes of common CFS sections
– Consideration of cross-section stability
• Finite element models for local and distortional
models are developed to examine load-bending
collapse envelopes.
• Preliminary Direct Strength Method design
expressions for beam-columns in local and
distortional buckling as a function of elastic section
slenderness are established and compared with the FE
models developed.
• Significant efficiency in the proposed DSM approach
in comparison with traditional design.
Future work
• Incorporation of recently proposed inelastic
bending provisions
• Further preliminary studies including global
buckling
• Beam-column tests
• Comprehensive FE parametric study
• Formal DSM proposals for beam-columns
?