Civil
Engineering
at JOHNS HOPKINS UNIVERSITY
Notes on
AISI Design Methods for Sheathing Braced Design
of Wall Studs in Compression
B.W. Schafer
report to AISI-COFS Design Methods Committee
April 2008
Overview
• Part of the AISI-COFS funded project on the design
of sheathed walls with dis-similar sheathing.
• Presentation developed from a report of the same
name
• Project updates, including full report available at
www.ce.jhu.edu/bschafer/sheathedwalls
• Report covers
– Design methods via 1962, 1980-2004, 2007 AISI
– Examination of sheathing stiffness ‘k’
– Initial summary of known demands and limit states on a
sheathed wall stud in compression
AISI Design Methods
1962 AISI
Design
Manual
1980 to
2004 AISI
Spec.
2007
AISI-COFS
Wall Stud
Standard
(S211)
Winter’s method
discrete spring model
Simaan and Peköz
shear diaphragm model
“simplified”
discrete spring model
Basic notation
I1, r1, Pcr1: strong-axis buckling perpendicular to the plane of the wall
I2, r2, Pcr2: weak-axis buckling parallel to the plane of the wall
e: initial imperfection
1962 AISI
Design
Manual
1962 AISI Specification
• “The safe load-carrying capacity of a stud may be computed on
the basis that the wall material or sheathing (attached to the stud)
furnishes adequate lateral support to the stud in the plane of the
wall, provided the wall material and its attachments to the stud
comply with the following requirements:”
1962 AISI Specification
1962 AISI
Design
Manual
• “The safe load-carrying capacity of a stud may be computed on
the basis that the wall material or sheathing (attached to the stud)
furnishes adequate lateral support to the stud in the plane of the
wall, provided the wall material and its attachments to the stud
comply with the following requirements:”
fastener spacing, a,
must be less than
8EI 2 k
amax1  2 2
A fy
a max 2
Lr2

2r1
1962 AISI Specification
1962 AISI
Design
Manual
• “The safe load-carrying capacity of a stud may be computed on
the basis that the wall material or sheathing (attached to the stud)
furnishes adequate lateral support to the stud in the plane of the
wall, provided the wall material and its attachments to the stud
comply with the following requirements:”
fastener spacing, a,
must be less than
8EI 2 k
amax1  2 2
A fy
a max 2
Lr2

2r1
1962 AISI Specification
1962 AISI
Design
Manual
• “The safe load-carrying capacity of a stud may be computed on
the basis that the wall material or sheathing (attached to the stud)
furnishes adequate lateral support to the stud in the plane of the
wall, provided the wall material and its attachments to the stud
comply with the following requirements:”
fastener spacing, a,
must be less than
8EI 2 k
amax1  2 2
A fy
a max 2
Lr2

2r1
fastener-sheathing stiffness, k, must be at least:
k min 
f y2 aA2
240 ,000 ,000 I 2
1962 AISI Specification
1962 AISI
Design
Manual
• “The safe load-carrying capacity of a stud may be computed on
the basis that the wall material or sheathing (attached to the stud)
furnishes adequate lateral support to the stud in the plane of the
wall, provided the wall material and its attachments to the stud
comply with the following requirements:”
fastener spacing, a,
must be less than
8EI 2 k
amax1  2 2
A fy
a max 2
Lr2

2r1
fastener-sheathing stiffness, k, must be at least:
k min 
f y2 aA2
240 ,000 ,000 I 2
fasteners must have at
least this much strength:
Fmin 
keP
2 EI 2 k / a  P
Fastener spacing limit #1
1962 AISI
Design
Manual
• Design basis: amax1 requires that weak-axis buckling of the stud,
including contributions from the wall stiffness k, that is
developed from fasteners at spacing a, is greater than or equal to
the squash load of the column. Pcr 2 k @ a ,KL 2  L   Af y
fastener spacing, a,
must be less than
8EI 2 k
amax1  2 2
A fy
a max 2
Lr2

2r1
Fastener spacing limit #1
1962 AISI
Design
Manual
• Design basis: amax1 requires that weak-axis buckling of the stud,
including contributions from the wall stiffness k, that is
developed from fasteners at spacing a, is greater than or equal to
the squash load of the column. Pcr 2 k @ a ,KL 2  L   Af y
fastener spacing, a,
must be less than
8EI 2 k
amax1  2 2
A fy
a max 2
Lr2

2r1
 id L2
Pcr 2
2

PE

Winter (1960) solution for
column on a stiff, continuous,
foundation
PE
convert continuous
foundation stiffness, ,
to discrete springs, k,
and set Pcr2=Afy
solve for
a, you get
Af y
 EI 2 L
2
2

2

2k a L2
 2 EI 2 L2
Fastener spacing limit #2
1962 AISI
Design
Manual
• Design basis: amax2 requires that weak-axis buckling of the stud
over a length of 2a (twice the fastener spacing) must be greater
than the strong-axis buckling over the entire length.
Pcr 2 k  0,KL2  2a   Pcr1 KL1  L 
fastener spacing, a,
must be less than
8EI 2 k
amax1  2 2
A fy
a max 2
Lr2

2r1
Fastener spacing limit #2
1962 AISI
Design
Manual
• Design basis: amax2 requires that weak-axis buckling of the stud
over a length of 2a (twice the fastener spacing) must be greater
than the strong-axis buckling over the entire length.
Pcr 2 k  0,KL2  2a   Pcr1 KL1  L 
fastener spacing, a,
must be less than
8EI 2 k
amax1  2 2
A fy
a max 2
Lr2

2r1
Equating Pcr1 and Pcr2 implies
KL 1  KL 2
r1
r2
Insure weak-axis buckling over 2a does not control
solve for
a, you get
L 2a

r1 r2
1962 AISI
Design
Manual
minimum stiffness k
• Design basis: kmin is the same as the amax1 design check.
Pcr 2
2

PE

 id L2
Af y
PE
 EI 2 L
2
2

2

2k a L2
 2 EI 2 L2
(from Winter 1960)
fastener-sheathing stiffness, k, must be at least:
sub in for E and
, solve for k
k min 
f y2 aA2
240 ,000 ,000 I 2
Pcr 2 k @ a ,KL 2  L   Af y
fastener strength
1962 AISI
Design
Manual
• Design basis: forces developed in an imperfect, but stiff
continuous foundation under the design load P, should be
carried by the fasteners, plus Winter adds some interesting
empirical corrections.
Winter (1960) defines forces developed in a stiff continuous foundation supporting an imperfect column
sreq  d o
 id
1   id  act
Fmin  sreq a 2
  2k a
Fmin  e
kid
1  kid k act
fasteners must have at
least this much strength:
Fmin 
keP
2 EI 2 k / a  P
fastener strength (continued)
1962 AISI
Design
Manual
Fmin
kid
e
1  kid k act
again, Winter uses the solution for buckling of a column on a stiff continuous foundation,
this time to find the ideal stiffness id (kid):
Pcr 2 2

PE 
 id L
 id  2kid a
2
PE
which results in
and
PE   2 EI 2 / L2
Pcr 2  P

kid  P 2 2EI 2 a

2
after substitution and rearranging, we find
Fmin 
ek act P
2 2 EI 2 a k act
fasteners must have at
least this much strength:
k id  P
this ends, rational derivation and Winter introduces empiricism
Fmin 
keP
2 EI 2 k / a  P
fastener strength (continued)
Fmin 
ek act P
2 2 EI 2 a k act
k id  P
1. assume kact=kid only under first denominator term
2. remove 2 under the radical in first term as well
3. remaining kact = k
results in
fasteners must have at
least this much strength:
Brace forces resulting from
Example 11 of 1962 design manual
note kact=80kid
Fmin 
keP
2 EI 2 k / a  P
fastener strength (continued)
1962 AISI
Design
Manual
Fmin 
ek act P
2 2 EI 2 a k act
k id  P
1. assume kact=kid only under first denominator term
2. remove 2 under the radical in first term as well
3. remaining kact = k
2% “rule”
0.02
Fmin( P)
fasteners must have at
least this much strength:
0.015
P
Fmin2( P)
results in
Brace forces resulting from
Example 11 of 1962 design manual
note kact=80kid
0.01
P
Fmin 
0.005
0
0
2000
4000
6000
P
8000
1 10
4
keP
2 EI 2 k / a  P
1962 AISI
Design
Manual
Summary of 1962 Specification
• The fastener-sheathing stiffness must insure the
following condition is met
Pcr 2 k @ a ,KL 2  L   Af y
• The fastener-sheathing strength must insure the
following condition is met
Fmin  e
k
k 2 k id  1
where
kid  P 2 a 8EI 2
• In addition to insure adequate performance in the face
of potential defects
Pcr 2 k  0,KL2  2a   Pcr1 KL1  L 
• If the above conditions are met Pcr=Pcr1 (strong-axis).
1962 AISI
Design
Manual
Critique of 1962 Specification
• The fastener-sheathing stiffness must insure the
following condition is met
Pcr 2 k @ a ,KL 2  L   Af y
useful, but arbitrary, does not even
insure that strong axis controls
• The fastener-sheathing strength must insure the
following condition is met
Fmin
k
e
k 2 k id  1
couched in something theoretical, but in
the end empirical, not wholly consistent
with current approaches
• In addition to insure adequate performance in the face
of potential defects
Pcr 2 k  0,KL2  2a   Pcr1 KL1  L 
arbitrary, realistic?
• If the above conditions are met Pcr=Pcr1 (strong-axis).
Underlying theory not directly applicable, no torsional-flexural buckling
check, to my knowledge none of us have ever even done a k test!?
1980 to
2004
AISI
Spec.
1980 to 2004 AISI Specification
• Also known as the Simaan and Peköz method,
or the shear diaphragm model, or the “D4
method”. Existed from 1980 to 2004 in Spec.
Section D4. Abandoned in favor of a return to
Winter’s method, more or less.
• Why was the method
abandoned?
• What can we learn
from the “mistakes”?
1980 to
2004
AISI
Spec.
Practical limitations of D4 method
• “The design expressions are complex. The design expressions do
not give credit to the presence of supplementary steel bridging
which is typically installed in order to align members and to
provide necessary structural integrity during erection and in the
completed structure. Provided there is adequate steel bridging,
the imperfect sheathing approach in Section D4 (a) can produce
a lower capacity than an all steel approach. The most popular
sheathing, gypsum wallboard, is seen by some as too moisture
and load cycle sensitive to act as a reliable structural brace for the
service life of a structure. Other restrictions in Section D4 (a) are
for the most part impractical for typical use.” (Trestain 2002).
1980 to
2004
AISI
Spec.
Theoretical limitation of the D4 Method
• For the most part Winter’s method may
conceptually be understood as a column
supported by discrete springs:
• The D4 method has never been presented with
an equivalent mechanical model, instead it is
always described as a summation of energies
(column bending + shear distortion of
diaphragm), what is underneath the hood?
1980 to
2004
AISI
Spec.
Consider the shear energy term
L
1
2 0

Q z  dz
shear energy
Ds 
“shear” angle
du  z 
 z  
  y z 
dz
2
1980 to
2004
AISI
Spec.
Consider the shear energy term
L
1
2 0

Q z  dz
shear energy
Ds 
“shear” angle
du  z 
 z  
  y z 
dz
2
equivalence with rotational spring foundation:
Ds 
L
1
2 0

Q y dz  D 
2
L
1
2 0

 y 2 dz
1980 to
2004
AISI
Spec.
Consider the shear energy term
mechanically
equivalent
model to shear
diaphragm
L
1
2 0

Q z  dz
shear energy
Ds 
“shear” angle
du  z 
 z  
  y z 
dz
2
equivalence with rotational spring foundation:
Ds 
L
1
2 0

Q y dz  D 
2
L
1
2 0

 y 2 dz
considering
discrete
fasteners
1980 to
2004
AISI
Spec.
D4 method summary
• The method was abandoned for numerous
practical reasons, and now can be seen to have a
serious theoretical limitation
• However, lots of great and complicated
mechanics in the D4 method. Torsional-flexural
buckling is treated thoroughly (even for dissimilar and one-sided sheathing).
• The role of shear in deforming the sheathing is
real, but have to be careful with how that
actually braces the stud
2007
AISI-COFS
Wall Stud
Standard
(S211)
2007 AISI-S211 Wall Stud Standard
• “Wall stud assemblies using a sheathing braced design shall be
designed assuming that identical sheathing is attached to both
sides of the wall stud and connected to the bottom and top
horizontal members of the wall to provide lateral and torsional
support to the wall stud in the plane of the wall.”
• “Both ends of the stud shall be connected to restrain rotation
about the longitudinal stud axis and horizontal displacement
perpendicular to the stud axis.” Further, in B1.2(b) it is
prescribed that the global buckling load of a stud, with fasteners
spaced distance “a” apart shall be determined ignoring any
sheathing contribution (i.e. k = 0) over a distance of 2a, i.e.:
Pcr k  0,KL x  L ,KL y  2a ,KL t  2a 
2007
AISI-COFS
Wall Stud
Standard
(S211)
Pcr by AISI-S211-07
• It is assumed that the sheathing provides enough stiffness that Pcr
ignoring the sheathing over a length equal to twice the fastener
spacing, a, is always less than Pcr considering the sheathing:
 0 ,KL y buckling
 2a  across a
defective fastener
 0,KLx  L ,KLt  2a 
assumed

minbuckling
Pcryofkthe@stud
a ,KL y  L 
engaging all fasteners
PcrTF k  0,KLx  L; k
Pcr by AISI-S211-07
2007
AISI-COFS
Wall Stud
Standard
(S211)
• It is assumed that the sheathing provides enough stiffness that Pcr
ignoring the sheathing over a length equal to twice the fastener
spacing, a, is always less than Pcr considering the sheathing:
 0 ,KL y buckling
 2a  across a
defective fastener
 0,KLx  L ,KLt  2a 
min P k  0 ,KL   2a 
cry
y
PcrTF k  0,KLx  L ,KLt  2a 
assumed

assumed

minbuckling
Pcryofkthe@stud
a ,KL y  L 
engaging all fasteners
min
Pcry
PcrTF k  0,KLx  L; k
k @ a ,KL  L 
y
PcrTF k  0,KLx  L; k @ a ,KLt  L 
Pcr by AISI-S211-07
2007
AISI-COFS
Wall Stud
Standard
(S211)
• It is assumed that the sheathing provides enough stiffness that Pcr
ignoring the sheathing over a length equal to twice the fastener
spacing, a, is always less than Pcr considering the sheathing:
 0 ,KL y buckling
 2a  across a
assumed
defective fastener
 0,KLx  L ,KLt  2a 
min P k  0 ,KL   2a 
cry
y
PcrTF k  0,KLx  L ,KLt  2a 

assumed

minbuckling
Pcryofkthe@stud
a ,KL y  L 
engaging all fasteners
min
Pcry
PcrTF k  0,KLx  L; k
k @ a ,KL  L 
y
PcrTF k  0,KLx  L; k @ a ,KLt  L 
validity of assumption
depends on the stiffness
k, as k0 definitely not
a valid assumption
2007
AISI-COFS
Wall Stud
Standard
(S211)
AISI-S211-07 Commentary
• Wall Stud Standard provides only minor
guidance, primarily wall stud design is left to
rational analysis.
• However, the commentary provides one such
rational analysis method, relying primarily on the
2% rule for fastener demands.
• Let us revisit the classic derivation to better
understand the implications of the 2% rule.
2007
AISI-COFS
Wall Stud
Standard
(S211)
2% rule in AISI-S211-07 Commentary
• Consider the basic derivation (supplementing the commentary)
2007
AISI-COFS
Wall Stud
Standard
(S211)
2% rule in AISI-S211-07 Commentary
• Consider the basic derivation (supplementing the commentary)
Finally solving for the bracing force:
note, 1%P, not 2%P, but brace force is
a function of k and do can be higher or lower
2007
AISI-COFS
Wall Stud
Standard
(S211)
Comparison of 1962 and 2007 methods
1962 AISI
Design
Manual
Analyze any design
AISI 2007 (S211-07)
Pcr  minPcry , PcrTF 
where
Pcry k  0 ,KL y  2a 
PcrTF k  0,KLx  L ,KLt  2a 
and
2%P for fasteners
Prescribed failure mode
AISI 1962
Pcr  Pcrx KLx  L  *
subject to
Pcry k  0,KL y  2a   Pcrx KL x  L 
Pcry k @ a ,KL y  L   Af y
and
~2%P for fasteners
* It is important to note that AISI 1962 did not
include torsional-flexural buckling and PcrTF<Pcrx
though as torsional resistance is increased PcrTF
will asymptote to Pcrx.
Looking towards new methods..
Limit states (abridged)
• Stud
– Global buckling (F, T, FT – including sheathing)
• should the stud be checked assuming defective fastener?
– Local buckling (probably ignore sheathing)
– Distortional buckling (probably including sheathing)
• should the stud be checked assuming a defective fastener?
• Connections
– Stud-fastener-sheathing connection
– Track-fastener-sheathing connection
– Stud-to-track connection
• Sheathing
• Construction loads (requires all-steel check)
Limit states (abridged)
state of development...
• Stud
– Global buckling (F, T, FT – including sheathing)
• should the stud be checked assuming defective fastener?
– Local buckling (probably ignore sheathing)
– Distortional buckling (probably including sheathing)
• should the stud be checked assuming a defective fastener?
• Connections
– Stud-fastener-sheathing connection
– Track-fastener-sheathing connection
– Stud-to-track connection
• Sheathing
• Construction loads (requires all-steel check)
Up to date work on bracing a stud
Sheathing k
y
 ( y )  G ( y )
x( y )  sin(
y
L
)
d x( y ) 
y
 ( y) 
  cos(
)
dy
L
L
G 
y
 ( y) 
 cos(
)
L
L
w
x
Sheathing k
y
F( y )  (  2 1 )  wt / n
 ( y )  G ( y )
F ( y)
k
x( y )
x
w
k  2π
Gwt  a 
sin  
Ln
 2L 
When sheathing k matters?
• If ksheathing<10kfastener... probably matters
ktotal    
1
1
kfastener

1
ksheathing  
kfastener
0.8
ktotal (  )
fastener
0.6
sheathing
kfastener
0.4
0.2
0
0.1
1
10
ksheathing (  )
kfastener
100
Getting past flexure...
• Support of the stud exists for other DOF as well, and in some
cases preliminary characterization has been done
Overview
• Part of the AISI-COFS funded project on the design
of sheathed walls with dis-similar sheathing.
• Presentation developed from a report of the same
name
• Project updates, including full report available at
www.ce.jhu.edu/bschafer/sheathedwalls
• Report covers
– Design methods via 1962, 1980-2004, 2007 AISI
– Examination of sheathing stiffness ‘k’
– Initial summary of known demands and limit states on a
sheathed wall stud in compression
All for now, Thank you...
Current Modeling
Stress distribution at Peak Load (Gray color represents yielding - connections)
Lab – General View 1
Lab – CFS specimens for scale