.
13.122 Design Process
ksi :=
1. MATERIALS
1000⋅lbf
2
lton := 2240⋅lbf
in
For HTS Components:
σ Y := 47⋅ksi
σ max := 21.28⋅ksi
σ max = 9.5
lton
3
E := 29.6⋅10 ⋅ksi
2
ν := .30
in
G :=
E
2⋅ ( 1 + ν )
For HY-80 Components:
σ Y2 := 80⋅ksi
σ max := 23.52⋅ksi
σ max = 10.5
lton
3
E := 29.6⋅10 ⋅ksi
2
ν := .30
in
2. GEOMETRY
y :=
y IB + y OB
2
z :=
section spacing = frame spacing = a
zIB + zOB
B :=
2
longitudinal position from AP := x
( yIB − yOB) 2 + (zIB − zOB)2
Assume for 1st design iteration:
y NA :=
Select number of stiffeners,N so
B
b :=
⋅in
that 23in<b<28in:
N+ 1
yD
2
3. PRIMARY STRESS
B in this context is maximum
beam at the DWL
not plate breadth.
lton 
lton
2.5
2.5
M bH := − 0.000457⋅L ⋅B⋅
B
M bS := 0.000381⋅L ⋅B⋅
B

2.5
2.5
ft 
ft

σ DHM :=
σ DSM :=
−M bH⋅ ( y D − y NA)
σ KHM :=
Iyy
−M bS⋅ ( y D − y NA)
σ KSM :=
Iyy
−M bH⋅ ( y K − y NA)
Iyy
−M bS⋅ ( y K − y NA)
Iyy
Or for 1st iteration before having Iyy:
σ DHM := σ max
σ KHM := −σ max
At NA for external shell:
σ DSM := −σ max

σ KSM := σ max
τ := 0⋅psi
 σ DSM 
 σ DHM 
 σ TNA := .5⋅max
 σ KHM 
 σ KSM 
σ CNA := − .5⋅max

Below neutral axis,
external shell:
Above neutral axis,
external shell:
Above neutral axis,
internal decks:
)
(
σ T := σ TNA + σ KSM − σ TNA ⋅ 
 yNA 
) y
(
σ T := σ TNA + σ DHM − σ TNA ⋅ 
σ T := σ DHM⋅
y NA − y 

y − y NA
y − y NA

D − y NA 
σ C := σ DSM⋅
y D − y NA
(
)
(
) y
σ C := σ CNA + σ KHM − σ CNA ⋅ 
y NA − y 
 y NA 
σ C := σ CNA + σ DSM − σ CNA ⋅ 

y − y NA
D − y NA 
y − y NA
y D − y NA
 σ T 
σ MAX := max
Below neutral axis,
internal decks:
σ T := σ KSM⋅
y NA − y
σ C := σ KHM⋅
y NA
1
y NA − y

 −σσ C 
y NA
notes_design_process.mcd
4. LOCAL LOADS:
Sea and weather; choose largest
Passing waves:
HWV( y
) := y DWL + .55⋅ L⋅ ft − y
(4-1)
 π  − y ⋅cos π 

 6   6


Heel:
HH( y ) :=  y DWL + zz⋅tan
Green Seas:
y D + 4⋅ ft −


HGS(y) := max y D + 8⋅ ft
⋅  x −
2⋅

L


Waveslap:
HWS := 7.82⋅ft



L
− y 
2

y
(4-2)
(4-3)
(4-4)
Independent
Dead Load:
Live Load:
Damage:
HDL := 1.72⋅
ft
in
⋅tt
t = plate thickness
(4-5)
(4-6)
HLL := 2.37⋅ft
   y + z⋅tan
 π    π  
 D z  − y ⋅cos  


HDAM(y) := max 
 6   6


yD − y


2
(4-7)
notes_design_process.mcd
5. PLATE LIMIT STATES
- assume values for b,t; refine t as required so all gRL<1
σ ax := σ MAX
τ xy := 0
E⋅( t)
D :=
3
(
12⋅ 1 − ν
)
(5-1)
2
PSPBT -Panel Serviceability Plate Bending Transverse σ bx := 0⋅psi
σ x := σ ax + σ bx
.5⋅  σ x − σ y
(
σ VM :=
2

b 

 σ y := σ by
σ by := − .5⋅p⋅ 

t 
)2 + (σ y)2 + (σ x)2 + 3⋅ τ 2
γ S := 1.25
γRPSPBT := γ S⋅
σ VM
(5-2)
σY
PSPBL - Panel Serviceability Plate Bending Longitudinal σ bx := .34⋅p⋅ 
b
2
σ x := σ ax + σ bx
t
σ y := σ by
(5-3)
.5⋅  σ x − σ y
(
σ VM :=
σ by := 0⋅psi
)2 + (σ y)2 + (σ x)2 + 3⋅ τ 2
γRPSPBL := γ S⋅
σ VM
σY
- PCMY - Panel Collapse Membrane Yield (or PFMY)
σ x := σ ax
σ VM :=
σ y := 0⋅psi
.5⋅  σ x − σ y
(
γ C := 1.5
)2 + (σ y)2 + (σ x)2 + 3⋅ τ 2
- PFLB - Plate Failure Local Buckling
2
σ axcr := −4⋅
π ⋅D
2
b ⋅tt
(5-4)
b
ks := 5.35 + 4⋅  
 a
γRPCMY := γ C⋅
σ VM
σY
Use Equation 12.4.7. (or PCLB)
2
2
τ := 0⋅psi
τ cr := −ks⋅
π ⋅D
2
b ⋅tt
Rs :=
τ
τ cr
(5-5)
interaction
formula:
Rc := 1 − Rs
2
σ o := Rc⋅σ
σ axcr
RPFLB :=
3
σC
σo
γRPFLB := γ S⋅R
RPFLB
notes_design_process.mcd
6. STIFFENED PANEL LIMIT STATES
HSW := DEPTH − TSF
Aw := (DEPTH − TSF)⋅TSW
TSW
Af := BSF⋅TSF
TSF
As := Aw + Af
Combination of plate and stiffeners (from p287, equation 8.3.6 in text):
TSF
d := DEPTH −
2
+
A := As + Ap
 A Aw 
−
+ Af⋅A
Ap
4 
3
2
C1 :=
( A)
Aw


 Aw

+ Ap

+
A
 2
p
2
y p := d⋅  1 −
y f := − d⋅
A


A


cf := y f − .5⋅TSF
TSF
σ bx :=
−( M bcen⋅cf)
I
2
(y is to mid-point)
- PYTF - Panel Yield Tension Flange
σ ax := σ T
2
I := A⋅d ⋅C
C1

L

12 
M bend := − p⋅ b⋅
cp := .5⋅t + y p
(6-1)
L := a
Aw⋅ 
t
Ap := b⋅tt
M bcen := p⋅ b⋅
L
2
(6-2)
2
12
γ S := 1.25
σ x := σ ax + σ bx
RPYTF :=
σx
γRPYTF := γ S⋅R
RPYTF
σY
(6-3)
- PYCF - Panel Yield Compression Flange
σ ax := σ C
σ bx :=
−( M bend⋅cf)
σ x := σ ax + σ bx
I
- PYTP - Panel Yield Tension Plate
σ ax := σ T
σ ay := 0.0⋅psi
τ xy := 0.0⋅psi
σ VM :=
σ bx :=
−( M bend⋅cp)
(
σ ay := 0.0⋅psi
τ xy := 0.0⋅psi
σ VM :=
σ bx :=
)2 + σ y2 + σ x2 + 3⋅ τ xy2
−( M bcen⋅cp)
.5⋅  σ x − σ y
(
)
2
2
RPYTP :=
σ x := σ ax + σ bx
σ VM
σY
σ y := σ ay + σ by
(6-5)
γRPYTP := γ S⋅R
RPYTP
usually much smaller than PYCF, plate closer to NA
2
+ σ y + σ x  + 3⋅ τ xy
4
(6-4)
γRPYCF := γ S⋅R
RPYCF
assume
for simplicity: σ by := 0⋅psi σ x := σ ax + σ bx
I
2
σY
- usually much smaller than PYTF, plate closer to NA
- PYCP - Panel Yield Compression Plate σ ax := σ C
−σ
σx
assume
for simplicity: σ by := 0⋅psi
I
.5⋅  σ x − σ y
RPYCF :=
RPYCP :=
σ VM
σY
σ y := σ ay + σ by
(6-6)
γRPYCP := γ S⋅R
RPYCP
notes_design_process.mcd
- PCCB - Panel Collapse Combined Buckling.
 Ae Aw 
−
+ Af⋅be⋅tt
4 
 3
Aw⋅ 
let:
be := brat⋅b
b
Ae := As + be⋅tt
Ie
ρ e :=
γ x :=
Ae
(
)
2
γx
 B

 ⋅

Cπ := min  a 2⋅ ( 1 + 1 + γ x)  


1


 be⋅t + As 
(
12
1
3
Cπ ⋅L⋅t
2
π ⋅E
σ ecr :=
 Cπ⋅L 

 ρe 
)
3
⋅ HSW⋅TSW + TSF⋅BSF
BSF
i
be :=
2
⋅σ
σ ecr
3
(6-7)
(6-8)
(
ρ e⋅ 3⋅ 1 − ν
2
be
)
2
b
=
Check,
must
equal brat
(6-9)
be := b starting point
can also use:
be := root( be − br( be) , be)
γRPCCB := γ C⋅
 b⋅ t + As 
- PCSB - Panel Collapse Stiffener Buckling
Isz :=
2
Ie := Ae⋅d ⋅C
C1
2
Ae
12⋅ 1 − ν ⋅Ie
be⋅tt
σ axcr := −
C1 :=


Isp := d ⋅  Af +
σC
(6-10)
σ axcr
Aw 
3
(6-11)

1
3
J :=
3
BSF⋅TSF + (HSW)⋅TSW
TSW
Cr :=
3
3
 ⋅d
 TSW  b
1 + .4⋅ 
for m=1:
Isp +
(
σ at2 :=
 σ at1 
 σ at2 
)
4
2


D⋅ Cr⋅ E⋅ Isz⋅ d
4⋅ D⋅Cr⋅b

⋅ G⋅ J + 4⋅
+
3 
2
b
2⋅ Cr⋅b ⋅t 
π

π
(6-12)
(6-13)
(6-14)
1
Isp +
σ at := −min
4
2
2
2
2

π ⋅E⋅Isz⋅d
4⋅D
D⋅Cr⋅ a + b 

⋅ G⋅ J +
+

3 
2
2
a
2⋅ Cr⋅b ⋅t 
π ⋅b

π
for m=2:
t
1
σ at1 :=
 4⋅ D⋅Cr 
m := ⋅ 
2
π 
 E⋅ Isz⋅d ⋅bb 
a
1
4
recalling that; if 1< m < 2, value obtained
is conservative
5
γRPCSB := γ C⋅
σC
σ at
(6-15)
notes_design_process.mcd
- PCSF - Panel Collapse Stiffener Flexure
Use limiting mode (Mode I, II, or III). See text Sec 14.2.
Rule of thumb for eccentricity of welded panels:
a. Mode I Compression failure of flange
I
ρ I :=
A
ζI
R :=
2
a
λ I :=
( ζ I) 2
−
4
π ⋅ρ
ρI
σY
⋅
E
( λ I)
(δo + ∆ I)⋅yf
µ I :=
( ρ I) 2
limit state
1 − µI
−
η I :=
σ axu := −R
R⋅ σ Y
2
a
RPCSF1 :=
σ ax
σ axu
(6-16)
σ ax := σ C
750
M o := 0⋅lbf ⋅in
(Point E of fig. 14.2):
Beam column parameters:
∆ :=
δ o := 0⋅ in
M o⋅y f
(6-17)
∆ I := −∆
∆
ζ I := 1 − µ I +
II⋅σ Y
γRPCSF1 := γ C⋅R
RPCSF1
1 + ηI
( λ I) 2
(6-18)
b. Mode II Compression failure of plate:
For maximum moment and center deflection assume simply supported beam:
q := p⋅b
b
q⋅aa
M o :=
2
δ o :=
8
5⋅ q⋅a
4
(6-23)
384⋅E⋅II
determine failure stress using plate parameters
β :=
b
t
σY
⋅
E
btr := T⋅bb
σ F :=

2.75
ξ := 1 +
T := .25⋅  2 + ξ −
(β )2
(ξ)2 −

 τ 
⋅σ Y⋅ 1 − 3⋅ 
T
 σY 
T − .1
10.4 

(6-19)
(β )2 
2
(6-20)
For combination (from equation 8.3.6 in text) and transformed plate:
 Atr Aw 
−
+ Af⋅A
Aptr
4 
 3
Aw⋅ 
Aptr := btr⋅tt
Atr := As + Aptr
Aw
y ftr := −d⋅
2
C1tr :=
( Atr)
Aw


+ btr⋅t

2
y ptr := d⋅  1 −
Atr


+ btr⋅t
Atr
2
ρ tr :=
(6-21)
Itr := Atr⋅( d) ⋅C
C1tr
2
Itr
λ :=
Atr
a
π ⋅ ρ tr
⋅
σF
E
(6-22)
Correction for load eccentricity:
h := SCG +
∆ p := h⋅ As⋅ 
t
2
1
 Atr
−
1
A
η p := ∆ p⋅
y ptr
(ρ tr)
(6-24)
2
Beam column with σF and transformed geometry
η :=
(δo + ∆ )⋅yptr
( )
µ :=
M o⋅y ptr
2
ρ tr
limit state
σ axtru := −R
R⋅ σ F
σ axu :=
Atr
A
σF
Itr⋅σ
⋅σ
σ axtru
ζ :=
1− µ
1 + ηp
RPCSF2 :=
6
+
σ ax
σ axu
1 + ηp + η
( 1 + η p) ⋅ ( λ ) 2
R :=
ζ
2
−
γRPCSF2 := γ C⋅R
RPCSF2
(ζ)2
4
−
1− µ
(6-25)
(1 + η p)⋅ (λ )2
(6-26)
notes_design_process.mcd
c. Mode III tension failure in flange:
To determine intersection at G:
a
λ GH :=
µ GH :=
π ⋅ρ
ρ tr
σY
⋅
E
−M oG⋅y ftr
RGHneg :=
(ζGH)2
−
2
4
5⋅ M oG⋅a
1 + η pGH
+
2
η pGH := ∆ p⋅
48⋅E⋅II
1 − µ GH
ζ GH :=
Itr⋅σ Y
ζ GH
δ oG :=
Assume value for MoG and iterate
(MoG>Mo; fails in Mode I or II)
y ftr
(ρ tr)
η GH :=
2
(δoG + ∆ )⋅yftr
(ρ tr)
2
(6-27)
1 + η pGH + η GH
−
(1 + η pGH)⋅ (λ GH)2
1 − µ GH
RGHpos :=
(1 + η pGH)⋅ (λ GH)2
ζ GH
2
(ζGH)2
+
4
+
1 − µ GH
(1 + η pGH)⋅ (λ GH)2
For Mode II at point G:
η p := ∆ p⋅
y ptr
(ρ tr)
η :=
2
(δoG + ∆ )⋅yptr
(ρ tr)
µ :=
2
RII :=
Adjust MoG until these are equal:
ζ
2
M oG⋅y ptr
σF
Itr⋅σ
(ζ)2
−
ζ :=
4
−
1− µ
1 + ηp
+
1 + ηp + η
( 1 + η p) ⋅ ( λ )
2
1− µ
( 1 + η p) ⋅ ( λ )
σ axtruck := RGH⋅σ Y
(6-28)
(6-29)
2
σ axtru := −RII⋅σ
σF
(6-30)
For line GH:
From Section 16.1 plastic moment:
CP1 :=
Ap − Aw − Af
2⋅ Ap
g
Zw := Aw⋅(.5⋅HSW + g)
CP2 := ( CP1) − CP1 + .5
2
CP2
Zp := Ap⋅t⋅C
g := CP1⋅tt
ZP := Zf + Zw + Zp
Zf := Af⋅(HSW + g + .5⋅TSF)
M P := σ Y⋅Z
ZP
(6-31)
limit state
σ auG := σ Y⋅
Atr
A
⋅R
RGH
σ au :=
MP − Mo
M P − M oG
7
⋅σ
σ auG
γRPCSF3 := γ C⋅
σC
σ au
(6-32)
notes_design_process.mcd
7. GIRDER BENDING CALCULATIONS
BG :=
B1 + B2
tG :=
2
HGW := DEPTH − TGF
For plate:
Ap := BG⋅ttG
LG := 2⋅aa
q := p⋅B
BG
Aw := (DEPTH − TGF)⋅TGW
TGW
(7-3)
Af := BGF⋅TGF
TGF AG := Aw + Af
(7-4)
TGF
2
+
cf := y f − .5⋅TGF
TGF
 A Aw 
−
+ Af⋅A
Ap
4 
3
Aw⋅ 
tG
C1 :=
2
2
I := A⋅( d) ⋅C
C1
2
A
For combination of plate and girder
(from p287, equation 8.3.6 in text):
 Aw

 2 + Ap
y f := − d⋅
A


(7-5)
Aw


+ Ap

2
y p := d⋅  1 −
A


cp := .5⋅tG + y p
Considering shear lag (Fig 3.36):
BGe := .75⋅B
BG
(7-2)
A := AG + Ap
For Girder:
d := DEPTH −
(7-1)
bulkheads and stanchions)
B1 + B2
Loads: Calculate at girder location, then:
Geometry:
LG = unsupported length (between
B1⋅t1 + B2⋅tt2
Ape := BGe⋅ttG
 Ae Aw 
−
+ Af⋅A
Ape
4 
 3
Aw⋅ 
Ae := AG + Ape
C1e :=
2

LG 
M bend := − p⋅ BG⋅
12 

Bending moments:
2
(7-6)
Ie := Ae⋅( d) ⋅C
C1e
2
Ae
M bcen := p⋅ BG⋅
LG
2
(7-7)
12
- GYCF - Girder Yield Compression Flange
σ ax := σ C
σ bx :=
−( M bend⋅cf)
σ x := σ ax + σ bx
Ie
RGYCF :=
−σ
σx
γRGYCF := γ S⋅R
RGYCF
σY
(7-9)
- GYCP (GYBP) - Girder Yield Compression Plate
- usually much smaller than GYCF and state of stress in plate more complex; we will not use
(7-8)
- GYTF - Panel Yield Tension Flange
σ ax := σ T
σ bx :=
−( M bcen⋅cf)
σ x := σ ax + σ bx
Ie
- GYTP - Girder Yield Tension Plate σ ax := σ T
σ ay := 0.0⋅psi σ bx :=
τ xy := 0.0⋅psi σ VM :=
.5⋅  σ x − σ y
(
)
( )
+ σy
σY
(7-10)
γRGYTF := γ S⋅R
RGYTF
assume
for simplicity: σ by := 0⋅psi
Ie
2
σx
usually much smaller than GYTF, plate closer to NA
−( M bend⋅cp)
2
RGYTF :=
( )
+ σx
8
2
 + 3⋅ τ xy
2
RGYTP :=
σ x := σ ax + σ bx
σ VM
σY
σ y := σ ay + σ by
(7-11)
γRGYTP := γ S⋅R
RGYTP
notes_design_process.mcd
8. FRAME BENDING CALCULATIONS
BF := a
LF = unsupported length (between bulkheads, supported girders,
deck edge, turn of bilge, continuous decks)
tF := t
Loads: Calculate at frame panel location, then:
Geometry:
HFW := DEPTH − TFF
For plate:
Ap := BF⋅ttF
(8-2)
q := p⋅B
BF
Aw := (DEPTH − TFF)⋅TFW
TFW
Af := BFF⋅TFF
TFF
AF := Aw + Af
d := DEPTH −
TFF
2
+
 A Aw 
−
+ Af⋅A
Ap
4 
3
Aw⋅ 
tG
C1 :=
2
2
I := A⋅( d) ⋅C
C1
(8-5)
Aw


+ Ap

2
y p := d⋅  1 −
A


(8-6)
2
A
For combination of plate and stiffeners
(from p287, equation 8.3.6 in text):
cf := y f − .5⋅TFF
TFF

 Aw
 2 + Ap
y f := − d⋅
A


cp := .5⋅tF + y p
Considering shear lag (Fig 3.36):
 Ae Aw 
−
+ Af⋅A
Ape
4 
 3
Aw⋅ 
BFe := .75⋅B
BF
Ape := BFe⋅ttF
Ae := AF + Ape
2

LF 

M bend := − p⋅ BF⋅
12 

C1e :=
M bcen := p⋅ BF⋅
LF
2
Ae
σ by :=
−( M bcen⋅cf)
γ S := 1.25
RFYTF :=
σy
σ by :=
−( M bend⋅cf)
Ie
σ y := σ ay + σ by
- FYTP - Frame Yield Tension Plate
σ ax := σ C
σ ay := 0.0⋅psi σ by :=
τ xy := 0.0⋅psi
σ VM :=
−( M bend⋅cp)
(
RFYCF :=
σ ay := 0.0⋅psi σ by :=
τ xy := 0.0⋅psi
σ VM :=
−( M bcen⋅cp)
(
)
RFYTP :=
γRFYCF := γ S⋅R
RFYCF
(8-10)
σ y := σ ay + σ by
2
2
2
+ σ y + σ x  + 3⋅ τ xy
9
σ VM
σY
γRFYTP := γ S⋅R
RFYTP
usually much smaller than FYCF, plate closer to NA
assume
for simplicity: σ bx := 0⋅psi
Ie
.5⋅  σ x − σ y
σY
(8-11)
)2 + σ y2 + σ x2 + 3⋅ τ xy2
2
−σ
σy
usually much smaller than FYTF, plate closer to NA
- FYCP - Frame Yield Compression Plate
σ ax := σ T
(8-9)
- consider when say<0
assume
for simplicity: σ bx := 0⋅psi σ x := σ ax + σ bx
Ie
.5⋅  σ x − σ y
γRFYTF := γ S⋅R
RFYTF
σY
- FYCF - Frame Yield Compression Flange
σ ay := 0⋅psi
(8-7)
(8-8)
12
σ y := σ ay + σ by
Ie
2
Ie := Ae⋅( d) ⋅C
C1e
2
- FYTF - Frame Yield Tension Flange
σ ay := 0⋅psi
(8-3)
(8-4)
A := AF + Ap
For Frame:
(8-1)
RFYCP :=
σ x := σ ax + σ bx
σ VM
σY
σ y := σ ay + σ by
(8-12)
γRFYCP := γ S⋅R
RFYCP
notes_design_process.mcd
9. FRAME COLLAPSE PLASTIC HINGE (FCPH)
flange
From Section 16.1: (can also use Hughes Table 16.1
breadth of plate is "effective breadth
web
Aw := tw⋅h
hw
Ap := tp⋅bbpe
Af := tf⋅bbf
γ C := 1.5
AT := Ap + Aw + Af
plate
(9-1)
if Ap>Aw+Af; i.e. Ap>At/2 => g (distance from top
of plate to PNA) is in plate
Ap − Aw − Af
g :=
2⋅b
bpe
(9-2a)
centroid of upper half
centroid of lower half
 bpe⋅gg
tf 
 hw



+ Aw⋅ 
+ g  + Af⋅  g + hw +
2
 2
 2


y 1 :=
2
y 2 :=
tp − g
2
(9-3a)
AT
2
plastic section modulus, if Ap>At/2
ZP1 :=
AT
2
⋅ ( y 1 + y 2)
(9-4a)
______________________________________________________
if Ap<Aw+Af; i.e. Ap<At/2 => g is is web
g :=
Af + Aw − Ap
(9-2b)
2⋅ttw
centroid of upper half


Af⋅  hw − g +
y 1 :=
tf 
2
centroid of lower half
+
tw⋅ ( hw − g)
2


Ap⋅  g +
2
y 2 :=
Af + Aw − g⋅ tw
tp 
2
2
+
tw⋅g
2
(9-3b)
Ap + g⋅ tw
plastic section modulus, if Ap<At/2
ZP2 :=
AT
2
⋅ ( y 1 + y 2)
(9-4b)
Calculate FCPH ---------------------------(9-5)
AT


ZP := if  Ap >
, ZP1 , ZP2
2


ZP
M P := σ Y⋅Z
RFCPH :=
M bcen
Mp
RFCPH
γRFCPH := γ C⋅R
10
(9-6)
notes_design_process.mcd
10. COLUMN COLLAPSE BUCKLING (CCB):
π⋅  d − dI
4
Tube Geometry:
I :=
1+ η
λ
2
1+ η
.25⋅ 1 +

λ
2
2
−

1
λ
2
2

(10-1)
γ C := 1.5
4
I
ρ :=
Le
λ :=
A
π ⋅ρ
ρ
⋅
σY
(10-2)
E
 Lcol 
 ρ 
η := α⋅ 
α := .002



−
A :=
Le := .7⋅L
Lcol
Eccentricity Ratio:



π⋅  d − dI
2

64
for a pinned column:
R := .5⋅ 1 +
4
σ ult := R⋅σ
R Y
(10-3)
σ a :=
 σa 
P
(10-4)
γRCCB := γ C⋅ 
 σ ult 
A
11. GIRDER BUCKLING CALCULATIONS
- GCCP1, pinned/center:
Le := L
L := 2⋅aa
α := .0035



R := .5⋅ 1 − µ +
1+ η

(λ )2 
−
η := α⋅



.25⋅ 1 − µ +
I
rc :=
L
ρ
+
1+ η
λ
2
δ o :=
cp⋅A
A
q⋅ L ⋅cp
µ :=
rc

I
ρ :=
(11-1)
A
column slenderness
parameter:
8⋅ I⋅ σ Y
1− µ
σ a := σ C
2
λ
−
4
384⋅E⋅II
2
δo
2
5⋅ q⋅( L)
(
σ ult := − R⋅σ
R Y
)
λ :=
Le
π ⋅ρ
ρ
γRGCCP1 := γ C⋅
rc :=
I
.556⋅q⋅L
δ o :=
A
cp⋅A



R := .5⋅ 1 − µ +
1+ η
λ
2
−

q⋅ L ⋅cp



.25⋅ 1 − µ +
Le
λ :=
24⋅ I⋅
I σY
1+ η
λ
2
2
−
π ⋅ρ
ρ
1− µ

4
λ
2
I
ρ :=
384⋅E⋅II
2
µ :=
σY
E
σa
σ ult
(11-2)
- GCCP2, clamped/center:
Le := .577⋅L
L
⋅
η := α⋅
A
Le
ρ
+
δo
rc
σY
⋅
(11-3)
E
σ a := σ C
(
σ ult := − R⋅σ
R Y
)
γRGCCP2 := γ C⋅
σa
σ ult
- GCCF, clamped/end:
Le := .423⋅L
L
rc :=
−I
A
cf⋅A
δ o :=
.444⋅q⋅L
2
µ :=
−q⋅L ⋅cf
λ :=
12⋅ I⋅
I σY
2
4
ρ :=
384⋅E⋅II
Le
π ⋅ρ
ρ
⋅
η := α⋅
A
Le
ρ
+
δo
rc
(11-4)
σY
E
1+ η
1+ η
1− µ


R := .5⋅ 1 − µ +
− .25⋅ 1 − µ +
−
σ a := σ C


2
2
2
λ
λ
λ




11
I
(
σ ult := − R⋅σ
R Y
)
γRGCCF := γ C⋅
σa
σ ult
notes_design_process.mcd