6

solesudaneseUrban and Civil

Nov 25, 2013 (3 years and 10 months ago)

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Steel Design

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5.Beams
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1






*

Beams




AISC Definition of Beam and Plate Girder

1.
Beam


y
w
F
E
70
.
5
t
h


-

all rolled shapes


Some built
-
up

sections


2. Plate Girder


y
w
F
E
70
.
5
t
h



-

some built
-
up sections









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Bending of symmetrical Sections

y
yy
x
xx
S
M
S
M
f



xx
M
; Bending Moment about x
-
axis

yy
M
; Bending Moment about y
-
axis

y
x
S
S
,
; Section Modulus

x
y
y
y
x
x
C
I
S
C
I
S


,










Behavior of Laterally Stable Beams

Laterally stable beam
s

= Laterally supported beams

= beams which are stable against Lateral
Buckling.


x

y

C
x

C
y


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From the stage (b)

y
x
y
n
F
S
M
M



From the stage (d)

y
p
n
ZF
M
M






Modulus
plastic
ydA
Z




Shape factor

S
Z



= for normal I
-
shape

beams

18
.
1
09
.
1






䝥湥牡汬礠獡祩湧Ⱐ
p
M

楳‱i


杲敡瑥爠
瑨慮
y
M

景爠f
-
獨慰攠扥慭s


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plastic hinge

-

A po
int in a beam where
M

reaches
p
M

-

behaves as a hing
e

-

May lead to collapse of the beam





Failure mechanism



For statically determinate beams



景牭慴楯渠潦⁡⁰污獴楣⁨楮来敡湳f
晡楬畲f





For statically indeterminate beams



牥摩獴物扵瑩潮映浯浥湴猠慦瑥爠瑨攠
景牭慴楯渠潦⁰污獴楣⁨楮来



瑨牥攠桩湧
敳
牥慬映灬慳瑩挩⁩渠愠獰慮e
汥慤⁴漠晡楬畲l




M
P


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W
T

= W
0

+ W
1


Distributed Loads

applied before the

first plastic hinge

forms












Example






6
2
12
2
3
bd
d
bd
C
I
S
y
x
x








4
2
)
4
(
)
2
(
2
bd
d
b
d
Z



5
.
1
4
6
6
4
2
2



bd
bd



+


=

W
0

W
1

x

d

b


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Things to be considered in designing
beams


1. Yielding Strength or Plastic Strength

2. Local buckling

(FLB, WLB)

3. Lateral
-
torsional buckling

(LTB)

4. Failure(collapse) mechanism




Design of beams

1. Compact Section

2. Non
-
compact Section



-

Partially compact section

3. Slender Element Section




Compact Section


y
y
b
p
b
n
b
M
5
.
1
ZF
M
M







-

No local buckling before the section
reaches to the plastic strength

-

p






limiting value for compact section



shoul
d be applied to both the flange

and web





take

worst one as the section



classification


to prevent

working
-
load
deformation


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Non
-
Compact Section



坨敮⁦污湧W⁩猠湯湣潭灡捴

p
p
r
p
r
p
p
n
M
M
M
M
M


















)
(


r

㬠䱩浩瑩湧⁢⽴⁦潲⁴桥;
湯湣潭灡捴n
獥捴楯渠

r
y
F
F
E
83
.
0


r
M
㬠牥獩摵慬潭敮


X
r
y
S
)
F
F
(





r
F
;

牥獩摵慬⁳瑲敳r




㄰歳椠景爠桯t
-
牯汬敤




ㄶ⸵歳椠景爠睥汤敤

y
p
F
E
38
.
0






坨敮⁷敢⁩W潮捯浰慣

p
p
r
p
r
p
p
n
M
M
M
M
M


















)
(


where

y
r
F
E
70
.
5



y
p
F
E
76
.
3



X
y
r
S
F
M




摩晦敲敮琠d牯洠瑨攠捡獥映晬慮来r
湯湣潭灡捴


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Slender Section

r




p
cr
cr
n
M
M
SF
M


















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*

Lateral Buckling




Elastic Lateral Buckling


w
y
2
b
y
b
cr
C
I
L
E
GJ
EI
L
M















where


b
L

= unbraced length (in.)


G

= shear modulus = 11,200ksi


J

= torsional constant (in.
4
)

w
C

= warping constant (in.
6
)





The

moment corresponding to first yield





X
L
r
S
F
M
smaller


X
yw
r
yf
S
F
,
F
F





X
r
y
S
F
F



for nonhybrid cases.




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r
L
; unbraced length at
r
cr
M
M


p
L
;
unbraced length at which no LTB occurs

until pla
stic moment =
y
y
F
/
E
r
76
.
1





from
r
cr
M
M

, we obtain





2
r
y
2
r
y
1
y
r
F
F
X
1
1
F
F
X
r
L







2
EGJA
S
X
X
1




2
X
y
w
2
GJ
S
I
C
4
X










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LTB str
e
ngth

1)

When
p
b
L
L




p
n
M
M


------
-------------------



2)

When
r
b
p
L
L
L

















p
r
p
b
r
p
p
n
L
L
L
L
)
M
M
(
M
M
--------



3) When
r
b
L
L



cr
n
M
M


------------------------







Moment Gradient effect


b
C
-
factor


to be multiplied to Eq


o爠

,


C
B
A
max
max
b
M
3
M
4
M
3
M
5
.
2
M
5
.
12
C







max
M

=

absolute value of the maximum


moment within the unbraced length

(including the end points)



A
M

= absolute value of the moment at the


quarter point of the
unbraced length

B
M

= absolute value of the moment at the


midpoint of the unbraced length


C
M

= absolute value of the moment at the


three
-
quarter point of the unbraced

length


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5.Beams
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Steel Design

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19












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Steel Design

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*

Plastic Analysis



In most cases,
u
M

is obtained from elastic
analysis
.


But if the shape is compact



楦i
y
y
2
1
pd
b
r
F
E
M
M
076
.
0
12
.
0
L
L





























1
M

㴠獭慬汥爠潦⁴桥⁴睯⁥湤潭敮瑳

景爠瑨攠畮扲慣敤⁳敧浥湴


2
M

㴠污牧敲映瑨攠瑷漠敮搠浯浥湴=

景爠瑨攠畮扲慣敤⁳敧浥湴





2
1
M
/
M

㴠灯獩瑩癥⁷桥渠牥癥牳攠捵牶慴畲e

















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*
Summary



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* ASD

S
S
F
M
F
S
M
f
n
b
b
)
.
(




F.S = 1.67




Compact Section



S
F
F
S
S
F
SF
F
y
y
b
.
.





1)For I
-
shaped section bending about x
-
x
axis

y
y
b
F
F
F
66
.
0
67
.
1
1
.
1




2)F
or I
-
shaped section bending about y
-
y
axis

y
b
F
F
75
.
0





67
.
1
5
.
1
.
y
y
F
S
F
F



y
F
9
.
0

y
y
b
y
F
F
F
F
66
.
0
75
.
0
9
.
0










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Noncompact Section





y
y
y
y
b
F
F
S
S
F
SF
S
S
F
M
F
6
.
0
67
.
1
.
.









偡牴楡汬礠䍯浰慣琠卥捴楯m

䱩湥慲⁩湴敲灯污瑩潮⁢整睥敮⁃潭灡捴…L
乯湣潭灡捴⁳散瑩潮N


ㄩ⁆潲⁉
-
獨慰敤⁢敮摩湧⁡扯畴⁸
-
砠慸楳










b
F

y
F
66
.
0



y
F
6
.
0

p


r


y
p
F
65



c
yf
r
k
F
/
95




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2)For I
-
shaped bending about y
-
y axis





卬敮摥爠䕬敭湴⁓散瑩潮

楮⁡捣潲摡湣攠睩瑨⁁卄⁁灰敮摩砠䈮i







b
F

y
F
75
.
0

y
F
6
.
0

p


r





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Shear in Rolled Beams


It
VQ
v


Q
: 1st moment of area of the shaded area
about the N.A.



ave
rage shear stress
w
w
v
dt
V
A
V
f




Ex)
v

of W24

㤴Ⱐ
kips
200

V



N.A

9.06
5

0.51
5

0.875

24.31

I=2700 in
4


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1) stress at a junction of flange and web





3
in
9
.
92
2
875
.
0
31
.
24
875
.
0
065
.
9









Q


a) at flange









ksi
76
.
0
065
.
9
2700
9
.
92
200


v

b) at web









ksi
4
.
13
515
.
0
2700
9
.
92
200


v


2) at
N.A.



3
2
in
7
.
125
8
.
32
9
.
92
2
875
.
0
2
31
.
24
515
.
0
9
.
92












Q











ksi
1
.
18
515
.
0
2700
7
.
125
200


v




18.
1

0.7
6

13.
4


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average shear





ksi
16
515
.
0
31
.
24
200


v
f




Distribution of
V

to flange and web
assuming the linear distribution of
v

on
flange









kips
6
2
875
.
0
065
.
9
76
.
0
2
1


flange
V

kips
194
6
200



web
V

control for shear resistance



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Nominal shear strength
n
V

in rolled beams



in LRFD, for
y
w
F
E
45
.
2
t
/
h



w
yw
V
n
A
F
V
6
.
0






乯⁷敢⁢畣歬楮朠潣捵牳



奩敬摩湧映敮瑩牥⁷敢⁡牥愠捯湴牯汳



䙲潭⁥F
敲杹 摩獴潲瑩潮⁴桥潲y

y
y
y
y
F
F
F
6
.
0
58
.
0
3









in ASD,


y
V
w
v
F
F
A
V
f
4
.
0












y
y
y
V
F
F
F
35
.
0
67
.
1
58
.
0
67
.
1







y
F
4
.
0


for
yw
w
F
t
h
380
/







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Web shear strength considering web shear
buckling

















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*

Block shear




a. When
nv
u
nt
u
A
F
6
.
0
A
F






nt
u
nv
u
nt
u
gv
y
n
A
F
A
F
6
.
0
A
F
A
F
6
.
0
R








(AISC Equation J4
-
3a)

b. When
nv
u
nt
u
A
F
6
.
0
A
F






nt
u
nv
u
gt
y
nv
u
n
A
F
A
F
6
.
0
A
F
A
F
6
.
0
R








(AISC Equation J4
-
3b)







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*

Deflection


-

Serviceability Requirement






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*

Design of Beam


1.

Compute
u
M


2.

Select a shape

a.

Assume Section
-

compute strength


and Revise as necessary

b.

Use design chart

3.

Check Shear strength

4.

Check Deflection















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* Concentrated

Loads Applied


to Rolled Beams



Three categories to be considered

1) Local web yielding

(Determine
N
)

2) Web inelastic buckling (Crippling)

(Determine
N
)

3) Sidesway Web buckling


1. Local web yielding

Cri
tical Section : toe of the fillet


a distance k from the
top face of the top
flange


k

Critical Section


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A) LRFD (
0
.
1


)

1)

for interior loads ;


w
yw
n
t
F
k
N
R
5



2) for end reaction ;


w
yw
n
t
F
k
N
R
5
.
2







B) ASD (F.
S.=1.5)


1)

for interior loads















.
.
66
.
0
5
S
F
F
F
k
N
t
R
f
y
y
w
c


2) for end reaction



y
w
c
F
k
N
t
R
f
66
.
0
5
.
2





R

R

k

N

k

2.5k

N

2.5
k

2.5k


Steel Design

I

Tel.(02) 3290
-
3317

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-
5166

Prof. Kang,YoungJong

5.Beams
-
47






2. Web Crippling


A) LRFD (
75
.
0


)


1) For interior loads (Concentrated Load
acts a d/2 or more from member end)

w
f
yw
5
.
1
f
w
2
w
n
t
t
EF
t
t
d
N
3
1
t
80
.
0
R


























2) For
end reaction (Concentrated Loads
acts less than d/2 from member end)


2
-
a) for
2
.
0

d
N

w
f
yw
5
.
1
f
w
2
w
n
t
t
EF
t
t
d
N
3
1
t
4
.
0
R

























2
-
b) for
2
.
0

d
N

(long bearing)

w
f
yw
5
.
1
f
w
2
w
n
t
t
EF
t
t
2
.
0
d
N
4
3
1
t
4
.
0
R





























B) ASD


Allowable Service concentrated load =
n
R
5
.
0



Steel Design

I

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-
5166

Prof. Kang,YoungJong

5.Beams
-
48






3. Sidesway Web buckling

-

rarely occurs in rolled sections

-

can

influence the design of plate girder


A) LRFD (
85
.
0


)


1) When Comp. flange is restrained against
rotation

a) for




3
.
2
/
/

f
b
w
b
L
t
h



















3
2
3
/
/
4
.
0
1
f
b
w
f
w
r
n
b
L
t
h
h
t
t
C
R

b) for




3
.
2
/
/

f
b
w
b
L
t
h


n
R
no limit


2) When Comp. flange is not restrained
against rotation

a) for




7
.
1
/
/

f
b
w
b
L
t
h


















3
2
3
/
/
4
.
0
f
b
w
f
w
r
n
b
L
t
h
h
t
t
C
R

b) for




7
.
1
/
/

f
b
w
b
L
t
h


n
R
no limit


Steel Design

I

Tel.(02) 3290
-
3317

Fax. (02) 921
-
5166

Prof. Kang,YoungJong

5.Beams
-
49






Where


b
L

largest laterally unbraced length along
either flange at the point of load


f
b

flange width


h

clear distance between flanges less than
fillet or corner radius for rolled
shape


r
C
960,00
ksi

when
y
u
M
M



480,000
ksi

when
y
u
M
M


at the location of the force, ksi





Special Note for design of bearing

Nominal Strength of Concrete



6
.
0
85
.
0
1
'





A
f
P
c
p

1
A
= area of steel concentrically bearing
in a concrete support