2
Bending members (beam)Outlines
Introduction
Resistance of cross-section of beams
Lateral torsional buckling of beams
Local buckling of plate element in beams
Deflection of beams
3
Bending members (beam)introduction
state of loading- bending moment- shear force- axial force- combination of above
- bending about one axis- bending about both axes (bi-axial bending)
roof purlin crane beam
4
Bending members (beam)introduction
boundary conditions and spans
simply-support both-end fixed cantilever
multi-span continuous beam
structural system and load transfer
roof panel →secondary beams →main beams →columns → foundations
5
Bending members (beam)introductionsection types
solid-web section(normal / light gauged)
castellated beam
composite beam
6
Bending members (beam)introductionsection types
solid-web section
open-web section
non-uniform section(height, width, strength)
laced built-up section(truss)
7
Bending members (beam)introduction: failure modes
strength failure (cross-section resistance)yield, fracture, fatigue
Lateral-torsional-buckling (LTB) of beam
Local buckling of plates in beam
flexural-torsional buckling
flange: compressionweb: compression, bending and shear
excessive deflection
Less rigidity of bending
8
Cross-sectional resistance of beamsbending moment resistance (1)
assumption- perfect elasto-plastic model- cross section remains plane during bending
σ
ε
yf
yεdistribution of normal strain and stress
y1 f<σ y1 f=σ y1 f=σ
y2 f=σ
y1 εε <
y2 εε <
y1 εε = y1 εε >
y2 εε =
y1 εε >
y2 εε > ∞⇒2ε
∞⇒1ε
9
Cross-sectional resistance of beamsbending moment resistance (2)
Criteria 1: yielding at extreme fibre
exx MM ≤ dxn
x fWM
≤=σ
Criteria 2: yielding on full section
x
px
ex
pxpx W
WMM
==γ
pxx MM ≤d
pxn
x fWM
≤
Criteria 3: yielding on partial section
exxx γ≤ MM dxnx
x fW
M≤
γpxx1 γγ <<
elastic net (effective) section modulus
plastic net section modulus
elastic net section modulus
shape factor
plastic adaptation factor
10
Memory: tension members with bendingconcept of full plastic moment
Assumption of stress distribution- subjected to bending only, no tension- stress at each point reach yield point- yield point under tension and
compression is the same
yyypx )( fyAyAyfAyfAM −−++−−++ +=+=
0=N 0yy =− −+ fAfA −+ = AA
−−++ += yAyAWpx ypxpx fWM =
y1 f=σ
y2 f=σ
x
y+A
−A −y+y
Plastic neutral axis
The plastic neutral axis divides the cross section into two equal areas, and it may not coincide with the centroidal axis
(Full) plastic moment
plastic modulus of a section:
11
Review of section modulusshape factor & plastic adaptation factor of cross-section
shape factor
plastic adaptation factor
12
Cross-sectional resistance of beamsbending moment resistance (3)
Bi-axial bending:
Criteria 2: yielding on full section
Criteria 3: yielding on partial section
Criteria 1: yielding at extreme fibre
yx ,MM
dyn
y
xn
x fWM
WM
≤+=σ
dpyn
y
pxn
x fWM
WM
≤+
dyny
y
xnx
x fW
MW
M≤+
γγ
13
Cross-sectional resistance of beamsshear resistance
tISV
tISV
y
yx
x
xy +=τ
shear stress:
shear stress under bi-axial shear forces
shear resistance
mechanics of material:tI
SV
x
xy=τ
w
y
AV
=τapproximate value: (for I-shape or channel shape only)
vdf≤τ
algebraic or vector add?
Note: gross section or net section?
14
Cross-sectional resistance of beamstransverse force resistance
local stress due to the transverse force
dwz
ft
Fc ≤
⋅=σ
z
a
z y R5 {2 }a h h= + +
Fyh :distance from load surface to
the upper edge of the effective web width
— place, thickness and width of stiffeners— strength and stability of stiffeners and web around
w15tw15t
(height of rail)
Stress check
Design of stiffeners
15
Cross-sectional resistance of beamstransverse force resistance
local stress due to the transverse force
illustration of yh
16
Cross-sectional resistance of beamsstate of stresses and equivalent stress
state of combined stresses (SCS)
2c
2c
2zs 3τσσσσσ +⋅−+=
— stress state of existence of two or more stress components at same point / same load condition
Criteria of elasticityCriteria of partially plastic adaptation
— sign of stress component: tension positive and compression negative
d1zs f⋅≤ βσ 11 =β11 >β
maxσmaxτ
2max
2maxzs 3τσσ += ?
Which section has SCS?
Which point has SCS?
Equivalent stress of beam
practical design equation
17
Cross-sectional resistance of beamsdesign procedure of beam strength
Analytical model of the structure
Internal force diagram under different load casesLoading pattern and value, boundary conditions
Ascertain the computing point
Ascertain the section to be checked
Strength check
Bending moment and shear forces
Calculate the sectional properties
Calculate the nominal stress and equivalent stress
18
Lateral-torsional buckling of beamspreparation: shear centre
concept of shear centre
hbtIbhV
M z ××=22f
x
y
eVM z y= x
f22
4Ithbe =
20
Lateral-torsional buckling of beamspreparation: free torsion
free torsion1. shear stress on section due to free torsion2. uniform torsional angle along the member
'θtk GIM =
tItM z=τ tA
M z
02=τ
∑=
=n
iiitbI
1
3t 31
∫=
ts
AI d4 2
0t
r0
δ
Tr0
δ
T
open-section closed-section
21
Lateral-torsional buckling of beamspreparation: restrained torsion
''2yyω 5.0 θhEIhMB −==
restrained torsion and warping
θhu 5.0=''''
y 5.0 θφ hu ==''
y''
yy 5.0 θhEIuEIM −=−=
'''yy 5.0/ θhEIdzdMV −==
'''2yyω 5.0 θhEIhVM −==
24/5.0 2f
32yω htbhII ≈=
'''ωω θEIM −=
θ
ft
b
h
u
Tω MMM k =+
L
x
y'θtk GIM =
Example:cantilevered I-shape beam under end torsional moment
''ωω θEIB −=
22
Lateral-torsional buckling of beamsdifferential equations for elastic LTB of beam
equilibrium and deflection of LTB of beam
00''
x =−+ θNxNvvEI00
''y =−+ θNyNuuEI
0)( '20 =−+ θRNr
'0
'0
''''ω uNyvNxGIEI t +−− θθ
zy
xMxM
0x''
x =+ MvEI
0x''
y =+ θMuEI
0'x
''''ω =+− uMGIEI tθθ
zx
xM
xM
v
u'u
x
y
uv
θ
xM
xM
θxM
'xuM
Simply-supported beamEqual bending momentSecond-order, small deflection
differential equations: flexural-torsional buckling
23
Lateral-torsional buckling of beamssolution of LTB for simply-supported beam with uniform bending
0x''
y =+ θMuEI0'
x'
t'''
ω =+− uMGIEI θθ (6-46c)Substituting Equ.(-b) into Equ.(-c), we get differential equation about
0)/( y2''
tIV
ω =−− θθθ EIMGIEI x
)/sin( znC πθ =boundary conditions: 0''''
00 ==== ll θθθθ
(6-47)
(6-46b)
θ
(a)
Substituting Equ.(a) into Equ.(6-47), we get
0sin)(y
2x
2
22t
4
44ω =⋅⋅−+
znCEIMnGInEI πππ (b)
Then we have
)1(ω
2
2t
y
ω2y
2
crxx EIGI
IIEI
MMπ
π+=⇒ (6-48)
24
Lateral-torsional buckling of beamscritical bending moment
critical bending moment
)1(ω
2
2t
y
ω2y
2
crx EIGI
IIEI
Mπ
π+=
wf ,,, tthb
Assume web thickness is small but have enough stiffness to keep its shape, then we have the same eqution of critical bending moment
crxM
,25.024 y2
f23
2ω I
hthb
hI
=⋅
=
)1(ω
2
2t
2y
ω2y
2crx
1 EIGI
hIIEI
hMN
ππ
+=≈
Flange width, height, thickness of flange and web for I-shape
1N
,039.02 ≈E
Gπ 22
22f
f23
23f
ω
2t 1624
32
hbt
thbbt
II
=≈
2y
2
1 64.0EI
Nπ
≈if 122
22f ≈hb
t2y
2
5.0EIπ
>?
25
Lateral-torsional buckling of beamscritical bending moment with different boundary and loading
Effect of boundary conditions
])(
[)( ω
2
2yt
2ω
2y
y
ω2
y
y2
crx EIGI
IIEI
Mπ
μμμ
μπ
+=Table 6-2 in pp.163
ωy ,μμ
ocrx1crx MM ⋅= β
: critical bending moment for beam under uniform bending
ocrxM
0.11 =β 13.11 =β 35.11 =β 65.21 =β
Effect of loading pattern
26
Lateral-torsional buckling of beamscritical bending moment with effects of section type and loading
Effect of section types and loading point
a
])1()([ω
2
2t
y
ω2y32y322
y2
1crx EIGI
IIBaBa
EIM
πββββ
πβ +++++=
yB
2β
—distance from load point to shear centre
—parameter indicating asymmetric degree for section
∫ −+= 022
xy )(21 ydAyxyI
B
— bending:0; UDL:0.46; CL at mid-span 0.55
3β — bending:1; UDL:0.53; CL at mid-span 0.40
load
deflection
S
Same direction with deflection for load to S,a<0
loaddeflection
S
Reverse direction with deflection for load to S,a>0
27
Lateral-torsional buckling of beamsparameters affect the LTB of beams
Critical bending moment
])1()([ω
2
2t
y
ω2y32y322
y2
1crx EIGI
IIBaBa
EIM
πββββ
πβ +++++=
])(
[)( ω
2
2yt
2ω
2y
y
ω2
y
y2
crx EIGI
IIEI
Mπ
μμμ
μπ
+=
—— rigidity of section—— distance of lateral supports—— section types (width of compressive flange)—— loading pattern (bending moment)—— loading point—— boundary conditionsHow about the initial imperfection?
28
Lateral-torsional buckling of beamselasto-plastic lateral-torsional buckling of beams
)1(ω
2
2t
y
ω2y
2
crx EIGI
IIEI
Mπ
π+=
Elastic LTB
})(])[(1{
)()()(
ω2
2t
y
ω2y
2
crxt
t
t
tt
EIKGI
EIEIEI
Mπ
π ++=
Elasto-plastic LTB using tangent modulus theory
• short beam• large residual stress
29
Lateral-torsional buckling of beamsultimate capacity of beam with initial imperfections
xMxM
,u θ
xM
crxM
30
Lateral-torsional buckling of beamsdesign equations
dy
y
xb
x fWM
WM
≤+ϕ
Design equation for beams
yxbyxy
crxex
ex
crxcrxx fWfW
fM
MMMM ϕσ
===≤
dxb
x fW
M≤⇒
ϕ : stability coefficient for beamsbϕ
xW
←bi-axial bending
: gross section modulus
Conditions no need to check LTB of beams1. A rigid decking is securely connected to compressive flange of beams
How about the bottom flange under compression?2.The max ratio of unsupported length to flange width is less than values
listed in Table 6-3 in pp167
21
b 2x y
4320 23514.4
yb b
y
tAhW h f
λϕ β η
λ
⎡ ⎤⎛ ⎞⎢ ⎥= ⋅ ⋅ + + ⋅⎜ ⎟⎢ ⎥⎝ ⎠⎢ ⎥⎣ ⎦
31
Local buckling of plates in beamsintroduction
Local buckling of plates in beamflange: compressionweb: compression, bending and shear
22
2
tcr )()1(12 b
tEkυ
πχψσ−
=
Critical local buckling stress
32
1. ← flange having larger stiffness in beams
2. ← outstand flange of I-section, b is flange outstand
← flange supported along both edges of box-sectionb is the supported part in both edges
Local buckling of plates in beamscritical local buckling stress of flange
2
2
2
2
cr )1(12 btEk ⋅
−⋅=
μπχσ
1=χ
425.0=k
stress distribution of flange in beam
- shear stress is small- normal stress is nearly uniform in flange
critical local buckling stress
0.4=k
33
Local buckling of plates in beamscritical local buckling stress of web
local buckling of plate under uneven compressive stress
2
2
2
2
cr )1(12 btEk ⋅
−⋅=
μπχσ
χ,
)5.01/(4 α−=k
tb,
crσ : max. compressive stress( ) while plate buckles
: height and thickness ofplate (web)
maxσ
minσ
maxσ)474.01/(1.4 α−=k
26α=k
max
minmaxσ
σσα −=
3/20 ≤≤ α4.13/2 ≤< α
44.1 ≤< α
k : stability factor of plate 24,2 == kα =1.61ww , th
max crσ σ→
critical local buckling stress of plate simply supported at 4 edges
34
Local buckling of plates in beamscritical local buckling stress of web
local buckling of plate in shear
τ
why plate buckles in shear?
35
Local buckling of plates in beamscritical local buckling stress of web
critical local buckling stress of plate in shear
τ
2minmax )/(
434.5 +=k
χ =1.24
min
max
2
2
2
2
cr )1(12 btEk ⋅
−=
μπσ
2min
2
2
2
cr )1(12tEk ⋅
−=
μπτ
2
w
w2
2
cr )()1(12 h
tEkμ
πχτ−
⋅=
2w )/(434.5 ahk +=
4)/(34.5 2w += ahk
while 1/w ≤ah
while 1/w >ah
wh
a
critical local buckling stress of web of I-section in shear
36
Local buckling of plates in beamscritical local buckling stress of web
local buckling and critical local buckling stress of plate under local compressive stress
cσcrc,c σσ ⇒
2 2c
cr c,cr cr
( ) ( ) 1σ σ τσ σ τ
+ + ≤
2 2c
cr c,cr cr
( ) ( ) 1σ σ τσ σ τ
+ + ≤
Equ.(6-67)
2 2c
cr c,cr cr
( ) ( ) 1σ σ τσ σ τ
+ + ≤ Equ.(6-67a)
2c
cr c,cr cr
( ) 1σ σ τσ σ τ
+ + ≤ Equ.(6-68)
Equ.(6-68a)
21crc, )100(
htC=σ
critical local buckling stress of plate under combined stress
37
Local buckling of plates in beamsdesign criteria of preventing local buckling of plates
Criteria 1: critical local buckling stress is larger than yield point
ycr f>σ
ybcr f⋅> ϕσ
σσ >cr
Criteria 2: critical local buckling stress is larger than critical overall buckling stress of member
Criteria 3: critical local buckling stress is larger than actual stress in plate
Discussion: which criteria is the severest for local buckling prevention?
38
Local buckling of plates in beamsmethod to prevent plates from local buckling
how to promote the critical local buckling stress?
1. Modify boundary condition2. Modify width-to-thickness ratio
2
2
2
2
cr )1(12 btEk ⋅
−=
μπσ
- increase thickness- setup stiffeners- which is better?
transverse stiffener
longitudinal stiffener
short stiffener
decrease the actual stress in members (criteria 3)Increase the height of section, thus makes the decrease of actual stress is faster than that of critical local buckling stress
39
Local buckling of plates in beamsstiffeners
transverse stiffener
longitudinal stiffener
short stiffener
40
Local buckling of plates in beamsdesign against local buckling of I-section
outstand flange under compression
consider the residual stress and initial imperfection,
y
2
2
2
cr 95.0)1(12
fbtEk
≥⎟⎠⎞
⎜⎝⎛
−=
υπσ
yy2
52
y2
2 2358.1895.0)3.01(121006.2425.0
95.0)1(12 fffEk
tb
=×−
×××≤
×−≤
πυπ
yy2
52
y2
2 23513)3.01(12
1006.225.0425.0)1(12 fffEk
tb
=×−
××××≤
×−≤
πυηπ
yy2
52
y2
2 2354.15)3.01(12
1006.25.0425.0)1(12 fffEk
tb
=×−
××××≤
×−≤
πυηπ
consider the residual stress and initial imperfection, plus partial plasticity
41
Local buckling of plates in beamsdesign against local buckling of I-section
web under bending, shear, local compression, respectively
1. subjected to bending moment
y
2
2
52
cr )3.01(121006.22461.1 f
ht
w
w ≥⎟⎟⎠
⎞⎜⎜⎝
⎛−
××××=
πσy
235174ft
h
w
w ≤
Q/A: how about for the unsymmetric I-section with larger compressive flange?
2. subjected to pure shear
y
235104ft
h
w
w ≤yvy
2
2
52
cr 58.0)3.01(12
1006.2)134.5(24.1 ffht
w
w =≥⎟⎟⎠
⎞⎜⎜⎝
⎛−
×××+×=
πτ
3. subjected to local compression
y
2
crc, 100166 fht
w
w ≥⎟⎟⎠
⎞⎜⎜⎝
⎛××=σ
y
23584ft
hw
w ≤
1.61, 170, restrained
1.23, 150, unrestrained
a / h=2.0
a / h=2.0
42
Local buckling of plates in beamsdesign against local buckling of I-section
design procedure for real steelwork (no local buckling allowed)
1. For hot-rolled sections
2. For welded built-up sections
3. subjected to local compression
No need to check
y
23580ft
h
w
w ≤ O.K.yf
f 23515ft
b≤
yf
f 23515ft
b≥ modify flange section
y
23580ft
h
w
w ≥ modify web section, or setup stiffeners
yy
235)150(17023580ft
hf w
w ≤< setup transverse stiffeners
y
235)150(170ft
hw
w ≥ setup transverse, longitudinal stiffeners, plus short stiffeners if necessary
43
1)()( 2
crc,
c2
crcr≤++
σσ
ττ
σσ
Local buckling of plates in beamsdesign against local buckling of I-section
design procedure for real steelwork (no local buckling allowed)
4. ascertain the space of stiffenerstransverse stiffeners longitudinal stiffenersstability check for each grid
Method:Equ.(6-76)-(6.86)
ww 25.0 hah ≤≤
w1w 25.02.0 hhh ≤≤ a
1h
1)()(crc,
c2
cr
2
cr≤++
σσ
ττ
σσ
1b
1t
5. design of stiffenersdimension, requirement of strength, rigidity and stability
44
Local buckling of plates in beamsdesign of bending members using post-buckling strength
Mechanism of post-buckling strength in bending— subjected to bending moment
— subjected to shear force
45
Local buckling of plates in beamsdesign of bending members using post-buckling strength
design principles— elastic design allowing local buckling— subjected to static load
— subjected to bending momentconcept of effective sectionbending resistance on effective section, Equ.(6-87~90) pp177
design method
— subjected to shear forceconcept of tension field and truss-beamdiscount the shear strength, Equ.(6-91~94) pp178-179
— subjected to combined bending and shearEqu.(6-95~100) pp179-180
47
Local buckling of plates in beamsapplication of post-buckling strength
National Stadium bird nest
1000×1000×20×20
600×600×10×10
48
Local buckling of plates in beamslimit of width-to-thickness ratio and design method of beam
width-to-thickness height-to-thicknessof flange of web
plastic design 9 70
partially plastic design 13 (150)
elastic design 15 (170)
design using (20) 250~300post-buckling strength
49
deflection of beamscalculation of elastic deflection and rigidity
uniformly-distributed-load (UDL)
EIqL
3845 4
=δ
EIFL48
3=δ
EIML10
2≈δ
[ ], [ ] /1200 ~ /150L Lδ δ δ≤ =
q
F
nF
concentrated load at mid-span (CL)
multiple concentrated load (MCL)
requirement of rigidity
51
Design of bending memberssummary
Selection of sectionCalculation of section resistance (strength)
Calculation of width-to-thickness ratio, setup stiffeners, design of stiffenersCalculation of beams using post-buckling strength
Calculation of overall stability
Calculation of deflection
normal stress, shear stress, local compressive stress, combined stressUsing net section, except shear stress check
Calculation of local buckling of plates
Need to check the overall stability?Ascertain the critical LTB bending momentUsing gross section