buckling multiplier (stress, load, or moment). distortional. later-torsional. local. length of a half sine wave. Finite Strip Analysis and the Beginnings of the Direct Strength Method. Toronto, July 2000 AISI Committee on Specifications. Overview. Introduction Background
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buckling multiplier (stress, load, or moment)
distortional
later-torsional
local
length of a half sine wave
Finite Strip Analysisand the Beginnings of the Direct Strength Method
Toronto, July 2000
AISI Committee on Specifications
property
give E, G, and v
nodes
give node number
give coordinates
indicate if any additional support exist along the longitudinal edge
give applied stress on node
elements
give element number
give nodes that form the strip
give thickness of the strip
lengths
give all the lengths that elastic buckling should be examined at
these shape functions are
also known as [N]
Plane stress Kuv comesfrom these strain-displacement relations
Bending Kwq comesfrom these strain-displacement relations
where {d} is the nodal displacementsthe same shape functions as before are used,therefore [G] is determined through
partial differentiation of [N].
SS
SS
SS
SS
width = 6 in. (152 mm)
thickness = 0.06 in. (1.52 mm)
E = 29500 ksi (203000 MPa)
v = 0.3
Hand Solution:
= 10.665 ksi
2.44
0.84
8.44
0.059
comp. lip
comp. flange
web
2.44
0.84
8.44
0.059
lip
flange
web
* this k value would be fine-tuned by AISI B4.2
local buckling
distortional
local
half-wavelength
buckling multiplier
distortional
2.44
0.84
8.44
0.059
lip
flange
web
* this k value would be fine-tuned by AISI B4.2
Hand Analysis
Compression
Lip = 56.6 ksi
Flange = 62.4
Web = 5.2
Bending
Lip = 56.6
Flange = 62.4
Web = 31.3
Finite strip analysis
Compression
Local = 7.5 ksi
Distortional ~ 20
Bending
Local = 40
Distortional = 52
*these calculations include long column interaction, to ignore this interaction replace Pne with Py
*these calculations include long column interaction, to ignore this interaction replace Pne with Py
*these calculations include long column interaction, to ignore this interaction replace Pne with Py
*these calculations include long column interaction, to ignore this interaction replace Pne with Py
Mne = My since “braced” = 113 in-kips
Mnl = 89 in-kips
Mnd = 86 in-kips
Mn = 86 in-kips, distortional controls even though elastic critical is 30% higher
*these calculations include long column interaction, to ignore this interaction replace Pne with Py
1.6
1.4
1.2
local buckling controlled
distortional buckling controlled
1
n
/F
u
0.8
strength F
0.6
local strength curve
0.4
0.2
distortional strength curve
0
0
1
2
3
4
5
6
7
8
.5
slenderness of controlling mode (F
/F
)
n
c
r
Example strength & deflection calc. completed at fy
Example strength & deflection calc. completed at fa
It is anticipated that degradation of gross properties (i.e, Ag Ig) due to local/distortional/overall buckling may be approximated in the same manner as the degradation in the strength, e.g.,
Old
New
(0.50,0.00)
(2.44,0.0)
(2.44,0.84)
(0.50,1.50)
(0.00,2.00)
(2.44,7.60)
(0.00,8.44)
(2.44,8.44)
*these calculations include long column interaction, to ignore this interaction replace Pne with Py
“typical” C
Pcrl = 6.6 kips
Pcrd = 17.7 kips
Pcre = 25.7 kips
Py = 44.2 kips
Pne = 21.6 kips
Pnl = 12.2 kips
Pnd = 14.9 kips
Pn = 12.2 kips
“nestable” C
Pcrl = 11.4 kips
Pcrd = 24.3 kips
Pcre = 22.6 kips
Py = 43.4 kips
Pne = 19.4 kips
Pnl = 13.7 kips
Pnd = 15.8 kips
Pn = 13.7 kips
0
0
0
0
1
2
defining a single unloaded strip fixed at the 1 edge
and attached at the 2 edge will add the above elastic
stiffness to the solution wherever the 2 edge is attached
to the member. Remember, this stiffness is along the
length of the member (the length of the strip)
in order to make an unloaded strip you will have to create
a very short dummy element near the 2 edge because loadingis defined at the nodes not the elements
3.43
2.44
2.44
3.0
0.84
0.97
0.81
3.95
8.44
8.44
0.083
4
0.059
0.059
C_rack
0.060
15.93
C_nolip
C
4
L
4
0.068
5.0
1.0
0.060
C_deep
4
2.06
2.44
L_lip
0.25
10.0
8.44
0.075
0.059
11.93
6
H
Z
Z_deep
plate
0.070
drawings not to scale, all dim. in inches
consider solution for a simply supported plate with a stress gradient
displaced shape
internal energy
external work
total potential energy
variation and solution