Formula Sheets: Plastic Analysis | Design of Steel Structures - Civil Engineering (CE) PDF Download

 Page 1


GA TE CE 2026 F orm ula Sheet: Steel Structures - Connections,
Plate Girders, T russes, and Plastic Analysis
1. Connections - Simple and Eccen tric
• Bolt shear strength (LSD):
V
dsb
=
f
ub
·A
sb
v
3·?
mb
where:
– V
dsb
: Design shear strength of b olt (kN)
– f
ub
: Ultimate tensile strength of b olt (N/mm²)
– A
sb
: Nominal shank area of b olt (mm²)
– ?
mb
: P artial safet y factor for b olts (1.25)
• Bolt b earing strength:
V
dpb
= 2.5·k
b
·d·t·
f
u
?
mb
where:
– V
dpb
: Design b earing strength (kN)
– k
b
: Co e?icien t (dep ends on edge distance, pitc h, etc.)
– d : Bolt diameter (mm)
– t : Thic kness of connected plate (mm)
– f
u
: Ultimate strength of plate (N/mm²)
• W eld strength (LSD):
T
dw
=
0.7·t
w
·L
w
·f
u
v
3·?
mw
where:
– T
dw
: Design strength of w eld (kN)
– t
w
: Throat thic kness of w eld (mm)
– L
w
: Effectiv e length of w eld (mm)
– ?
mw
: P artial safet y factor for w elds (1.25 for shop w elds, 1.5 for site w elds)
• Eccen tric connection - Resultan t force:
R =
v
V
2
+(P ·e)
2
where:
– R : Resultan t force on b olt/w eld (kN)
– V : Shear force (kN)
– P : Axial force (kN)
– e : Eccen tricit y (mm)
2. B eam-Column Connections
• Shear capacit y of connection:
V
d
= min(V
dsb
,V
dpb
,V
dw
)
where:
– V
d
: Design shear capacit y (kN)
– V
dsb
, V
dpb
: Bolt shear and b earing strengths (kN)
1
Page 2


GA TE CE 2026 F orm ula Sheet: Steel Structures - Connections,
Plate Girders, T russes, and Plastic Analysis
1. Connections - Simple and Eccen tric
• Bolt shear strength (LSD):
V
dsb
=
f
ub
·A
sb
v
3·?
mb
where:
– V
dsb
: Design shear strength of b olt (kN)
– f
ub
: Ultimate tensile strength of b olt (N/mm²)
– A
sb
: Nominal shank area of b olt (mm²)
– ?
mb
: P artial safet y factor for b olts (1.25)
• Bolt b earing strength:
V
dpb
= 2.5·k
b
·d·t·
f
u
?
mb
where:
– V
dpb
: Design b earing strength (kN)
– k
b
: Co e?icien t (dep ends on edge distance, pitc h, etc.)
– d : Bolt diameter (mm)
– t : Thic kness of connected plate (mm)
– f
u
: Ultimate strength of plate (N/mm²)
• W eld strength (LSD):
T
dw
=
0.7·t
w
·L
w
·f
u
v
3·?
mw
where:
– T
dw
: Design strength of w eld (kN)
– t
w
: Throat thic kness of w eld (mm)
– L
w
: Effectiv e length of w eld (mm)
– ?
mw
: P artial safet y factor for w elds (1.25 for shop w elds, 1.5 for site w elds)
• Eccen tric connection - Resultan t force:
R =
v
V
2
+(P ·e)
2
where:
– R : Resultan t force on b olt/w eld (kN)
– V : Shear force (kN)
– P : Axial force (kN)
– e : Eccen tricit y (mm)
2. B eam-Column Connections
• Shear capacit y of connection:
V
d
= min(V
dsb
,V
dpb
,V
dw
)
where:
– V
d
: Design shear capacit y (kN)
– V
dsb
, V
dpb
: Bolt shear and b earing strengths (kN)
1
– V
dw
: W eld shear strength (kN)
• Momen t capacit y of connection:
M
d
=
f
y
·Z
p
?
m0
where:
– M
d
: Design momen t capacit y (kNm)
– Z
p
: Plastic section mo dulus of connection elemen ts (mm³)
– ?
m0
: P artial safet y factor (1.1)
• In teraction form ula for com bined shear and momen t:
(
V
u
V
d
)
2
+
(
M
u
M
d
)
2
= 1.0
where:
– V
u
, M
u
: F actored shear and momen t (kN, kNm)
3. Plate Girders
• Shear capacit y of w eb:
V
d
=
f
y
·d
w
·t
w
v
3·?
m0
where:
– V
d
: Design shear capacit y (kN)
– d
w
: Depth of w eb (mm)
– t
w
: Thic kness of w eb (mm)
• Momen t capacit y of plate girder:
M
d
=
f
y
·Z
p
?
m0
where:
– Z
p
: Plastic section mo dulus of girder (mm³)
• W eb buc kling c hec k:
d
w
t
w
= 67·?, ? =
v
250
f
y
where:
– ? : Slenderness parameter
• Stiffener spacing for shear buc kling:
a= 1.5·d
w
where:
– a : Spacing of transv erse stiffeners (mm)
2
Page 3


GA TE CE 2026 F orm ula Sheet: Steel Structures - Connections,
Plate Girders, T russes, and Plastic Analysis
1. Connections - Simple and Eccen tric
• Bolt shear strength (LSD):
V
dsb
=
f
ub
·A
sb
v
3·?
mb
where:
– V
dsb
: Design shear strength of b olt (kN)
– f
ub
: Ultimate tensile strength of b olt (N/mm²)
– A
sb
: Nominal shank area of b olt (mm²)
– ?
mb
: P artial safet y factor for b olts (1.25)
• Bolt b earing strength:
V
dpb
= 2.5·k
b
·d·t·
f
u
?
mb
where:
– V
dpb
: Design b earing strength (kN)
– k
b
: Co e?icien t (dep ends on edge distance, pitc h, etc.)
– d : Bolt diameter (mm)
– t : Thic kness of connected plate (mm)
– f
u
: Ultimate strength of plate (N/mm²)
• W eld strength (LSD):
T
dw
=
0.7·t
w
·L
w
·f
u
v
3·?
mw
where:
– T
dw
: Design strength of w eld (kN)
– t
w
: Throat thic kness of w eld (mm)
– L
w
: Effectiv e length of w eld (mm)
– ?
mw
: P artial safet y factor for w elds (1.25 for shop w elds, 1.5 for site w elds)
• Eccen tric connection - Resultan t force:
R =
v
V
2
+(P ·e)
2
where:
– R : Resultan t force on b olt/w eld (kN)
– V : Shear force (kN)
– P : Axial force (kN)
– e : Eccen tricit y (mm)
2. B eam-Column Connections
• Shear capacit y of connection:
V
d
= min(V
dsb
,V
dpb
,V
dw
)
where:
– V
d
: Design shear capacit y (kN)
– V
dsb
, V
dpb
: Bolt shear and b earing strengths (kN)
1
– V
dw
: W eld shear strength (kN)
• Momen t capacit y of connection:
M
d
=
f
y
·Z
p
?
m0
where:
– M
d
: Design momen t capacit y (kNm)
– Z
p
: Plastic section mo dulus of connection elemen ts (mm³)
– ?
m0
: P artial safet y factor (1.1)
• In teraction form ula for com bined shear and momen t:
(
V
u
V
d
)
2
+
(
M
u
M
d
)
2
= 1.0
where:
– V
u
, M
u
: F actored shear and momen t (kN, kNm)
3. Plate Girders
• Shear capacit y of w eb:
V
d
=
f
y
·d
w
·t
w
v
3·?
m0
where:
– V
d
: Design shear capacit y (kN)
– d
w
: Depth of w eb (mm)
– t
w
: Thic kness of w eb (mm)
• Momen t capacit y of plate girder:
M
d
=
f
y
·Z
p
?
m0
where:
– Z
p
: Plastic section mo dulus of girder (mm³)
• W eb buc kling c hec k:
d
w
t
w
= 67·?, ? =
v
250
f
y
where:
– ? : Slenderness parameter
• Stiffener spacing for shear buc kling:
a= 1.5·d
w
where:
– a : Spacing of transv erse stiffeners (mm)
2
4. T russes
• Axial force in truss mem b er (metho d of join ts):
F
i
=
?
F
x
·cos?+
?
F
y
·sin?
where:
– F
i
: Axial force in mem b er i (kN)
– F
x
, F
y
: F orces in x and y directions (kN)
– ? : Angle of mem b er with resp ect to horizon tal
• Design strength of truss mem b er (tension):
T
d
= min(T
dg
,T
dn
,T
db
)
where:
– T
dg
, T
dn
, T
db
: Design strengths for yielding, rupture, and blo c k shear (kN)
• Design strength of truss mem b er (compression):
P
d
=
?·f
y
·A
?
m0
where:
– ? : Strength re duction factor (based on slenderness, p er IS 800:2007)
5. Plastic Analysis - Beams and F rames
• Plastic momen t capacit y:
M
p
=
f
y
·Z
p
?
m0
where:
– M
p
: Plastic momen t capacit y (kNm)
– Z
p
: Plastic section mo dulus (mm³)
• Shap e factor:
S
f
=
Z
p
Z
e
where:
– S
f
: Shap e factor (dimensionless)
– Z
e
: Elastic section mo dulus (mm³)
• Collapse load for a simply supp orted b eam (UDL):
w
u
=
8·M
p
L
2
where:
– w
u
: Ultimate uniformly distributed load (kN/m)
– L : Span length (m)
• Collapse load for a fixed b eam (p oin t load at midspan):
P
u
=
4·M
p
L
where:
3
Page 4


GA TE CE 2026 F orm ula Sheet: Steel Structures - Connections,
Plate Girders, T russes, and Plastic Analysis
1. Connections - Simple and Eccen tric
• Bolt shear strength (LSD):
V
dsb
=
f
ub
·A
sb
v
3·?
mb
where:
– V
dsb
: Design shear strength of b olt (kN)
– f
ub
: Ultimate tensile strength of b olt (N/mm²)
– A
sb
: Nominal shank area of b olt (mm²)
– ?
mb
: P artial safet y factor for b olts (1.25)
• Bolt b earing strength:
V
dpb
= 2.5·k
b
·d·t·
f
u
?
mb
where:
– V
dpb
: Design b earing strength (kN)
– k
b
: Co e?icien t (dep ends on edge distance, pitc h, etc.)
– d : Bolt diameter (mm)
– t : Thic kness of connected plate (mm)
– f
u
: Ultimate strength of plate (N/mm²)
• W eld strength (LSD):
T
dw
=
0.7·t
w
·L
w
·f
u
v
3·?
mw
where:
– T
dw
: Design strength of w eld (kN)
– t
w
: Throat thic kness of w eld (mm)
– L
w
: Effectiv e length of w eld (mm)
– ?
mw
: P artial safet y factor for w elds (1.25 for shop w elds, 1.5 for site w elds)
• Eccen tric connection - Resultan t force:
R =
v
V
2
+(P ·e)
2
where:
– R : Resultan t force on b olt/w eld (kN)
– V : Shear force (kN)
– P : Axial force (kN)
– e : Eccen tricit y (mm)
2. B eam-Column Connections
• Shear capacit y of connection:
V
d
= min(V
dsb
,V
dpb
,V
dw
)
where:
– V
d
: Design shear capacit y (kN)
– V
dsb
, V
dpb
: Bolt shear and b earing strengths (kN)
1
– V
dw
: W eld shear strength (kN)
• Momen t capacit y of connection:
M
d
=
f
y
·Z
p
?
m0
where:
– M
d
: Design momen t capacit y (kNm)
– Z
p
: Plastic section mo dulus of connection elemen ts (mm³)
– ?
m0
: P artial safet y factor (1.1)
• In teraction form ula for com bined shear and momen t:
(
V
u
V
d
)
2
+
(
M
u
M
d
)
2
= 1.0
where:
– V
u
, M
u
: F actored shear and momen t (kN, kNm)
3. Plate Girders
• Shear capacit y of w eb:
V
d
=
f
y
·d
w
·t
w
v
3·?
m0
where:
– V
d
: Design shear capacit y (kN)
– d
w
: Depth of w eb (mm)
– t
w
: Thic kness of w eb (mm)
• Momen t capacit y of plate girder:
M
d
=
f
y
·Z
p
?
m0
where:
– Z
p
: Plastic section mo dulus of girder (mm³)
• W eb buc kling c hec k:
d
w
t
w
= 67·?, ? =
v
250
f
y
where:
– ? : Slenderness parameter
• Stiffener spacing for shear buc kling:
a= 1.5·d
w
where:
– a : Spacing of transv erse stiffeners (mm)
2
4. T russes
• Axial force in truss mem b er (metho d of join ts):
F
i
=
?
F
x
·cos?+
?
F
y
·sin?
where:
– F
i
: Axial force in mem b er i (kN)
– F
x
, F
y
: F orces in x and y directions (kN)
– ? : Angle of mem b er with resp ect to horizon tal
• Design strength of truss mem b er (tension):
T
d
= min(T
dg
,T
dn
,T
db
)
where:
– T
dg
, T
dn
, T
db
: Design strengths for yielding, rupture, and blo c k shear (kN)
• Design strength of truss mem b er (compression):
P
d
=
?·f
y
·A
?
m0
where:
– ? : Strength re duction factor (based on slenderness, p er IS 800:2007)
5. Plastic Analysis - Beams and F rames
• Plastic momen t capacit y:
M
p
=
f
y
·Z
p
?
m0
where:
– M
p
: Plastic momen t capacit y (kNm)
– Z
p
: Plastic section mo dulus (mm³)
• Shap e factor:
S
f
=
Z
p
Z
e
where:
– S
f
: Shap e factor (dimensionless)
– Z
e
: Elastic section mo dulus (mm³)
• Collapse load for a simply supp orted b eam (UDL):
w
u
=
8·M
p
L
2
where:
– w
u
: Ultimate uniformly distributed load (kN/m)
– L : Span length (m)
• Collapse load for a fixed b eam (p oin t load at midspan):
P
u
=
4·M
p
L
where:
3
– P
u
: Ultimate p oin t load (kN)
• Num b er of plastic hinges for frame collapse:
N
h
=R-1
where:
– N
h
: Num b er of plastic hinges required
– R : Degree of static i ndeterminacy
4