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Team TeachingStructural Design
Civil Engineering Department2010
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A beam is a structural member that is subjected primarily to
transverse loads and negligible axial loads.
The transverse loads cause internal shear forces and bendingmoments in the beams
w P
V(x)
M(x)
x
w P
V(x)
M(x)
x
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Rolled shape and built-up cross-sections
h htw tw
Fyt
h
w
2550
Fyth
w
2550
Beams
Plate girder
where Fy is yield stress, MPa
Beam !late "irder
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For beams, the basic relationship between load eects and strength can be
written as
where
Mu! controlling combination o actored load moments
b! resistance actor or beams !0"#0
Mn! nominal moment strength
$he design strength b" Mnis sometimes called the design moment"
nbu MM "
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The bending stress at a given point can be found fromthe #exure formula$
where M% bending moment at the cross section y % perpendicular distance from the neutral
plane to the point of interest&'x % the moment of inertia of the area of the
cross section with respect to the neutral axis.
x
bI
yMf "=
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(
)
htw (x x
RA
A B
cy
*a+
*b+
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For maxim%m stress,
where c is the perpendic%lar distance rom thr ne%tral axis to the extreme
iber,
&x is the elastic section mod%l%s o the cross section"
$he two abo'e e%ation are 'alid as long as loads are small eno%gh so that
the material remains within its linear elastic range"
For str%ct%ral steel i the maxim%m stress, this means that max m%st not
exceed Fy, and the bending moment m%st not exceed
xxx S
M
cI
M
I
cMf ===
"max
xyy
SFM "=
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A B
A B
A B
A B
A B
f,-y
f%-y
f%-y
f%-y*a+
*b+
*c+
*d+
Bending (oment
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A B
A B
(p
(oment
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h
tw
-y
-y
%Ac.-y
T%At.-y
!lastic neutral axis
a
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From e%ilibri%m o orces,
tc
ytc
AA
FAFyA
TC
=
=
=
""
$he plastic moment Mp is the resisting co%ple ormed by the twoe%al and opposite orces
whereA! total cross*sectional area, mm2
a! distance between the centroids o the hal*areas, mm
Z=(A/2)a! plastic section mod%l%s, mm+
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a beam can be co%nted on to remain stable %p to the %lly plastic
conditions, the nominal moment strength can be ta-en as the plastic moment
capacity. that is,
/therwise,Mnwill be less thanMp"
s with a compression member, instability can be o'erall sense or it can be
local"
pn MM =
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A B
Bending (oment
A B
*a+
*b+
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/'erall b%c-ling is ill%strated in Fig%re 1"a" 3hen a beam bends, thecompression region (abo'e the ne%tral axis) is analogo%s to a col%mn and,
in a manner similar to a col%mn, will be b%c-le i the beam is slender
eno%gh" $he b%c-ling o the compression portion o the cross section is
restrained by tension portion, and the o%tward delection (lex%ral
b%c-ling) is accompanied by twisting (torsion)" $his orm o instability is
called lateral*torsional b%c-ling (4$B)"
4ateral*torsional b%c-ling can be pre'ented by lateral bracing o
compression one, preerable the compression lange, at s%iciently close
inter'als as shown in Fig%re 1"b" s we shall see, the moment strength
depends in part on the %nbraced length, which is distance between pointo lateral s%pport"
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M
M
M
M
(a)
(b)
M
M
M
M
M
M
M
M
(a)
(b)
""6hasil download p%
rd%e %ni'6beam b%c-l
ing"mpg
http://../hasil%20download%20purdue%20univ/beam%20buckling.mpghttp://../hasil%20download%20purdue%20univ/beam%20buckling.mpghttp://../hasil%20download%20purdue%20univ/beam%20buckling.mpghttp://../hasil%20download%20purdue%20univ/beam%20buckling.mpghttp://../hasil%20download%20purdue%20univ/beam%20buckling.mpghttp://../hasil%20download%20purdue%20univ/beam%20buckling.mpg -
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3hether the beam can s%stain a moment large eno%gh to bring it
to the %lly plastic condition also depends on whether the cross*sectional integrity is maintained " $his integrity will lost i one o
the compression elements o the cross section b%c-les" $his can be
either b%c-ling o compression lange, called lange local b%c-ling
(F4B), or b%c-ling o compression part o the web, called web
local b%c-ling (34B)" $his b%c-ling strength will depend on the
width*thic-ness ratio o the compression elements o the cross
section"
""6hasil download p%rd
%e %ni'6local b%c-ling"mpg
http://../hasil%20download%20purdue%20univ/local%20buckling.mpghttp://../hasil%20download%20purdue%20univ/local%20buckling.mpghttp://../hasil%20download%20purdue%20univ/local%20buckling.mpghttp://../hasil%20download%20purdue%20univ/local%20buckling.mpghttp://../hasil%20download%20purdue%20univ/local%20buckling.mpghttp://../hasil%20download%20purdue%20univ/local%20buckling.mpg -
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A B
Bending (oment
A B
*a+
*b+
/eflection
0oad
irst
yield1
2
3
4
5
0
0
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7%r'e 8 Beam %nstable beore irst yield.
7%r'e 2 Beam can be loaded past irst yield b%t not ar
eno%gh or ormation a plastic hinge and the res%lting plasticcollapse.
7%r'e + Beam can be loaded past irst yield b%t not ar
eno%gh or ormation a plastic hinge and the res%lting plastic
collapse.7%r'e 9 Beam can be loaded reached plastic collapse, %niorm
moment o'er %ll length o beam.
7%r'e 5 Beam can be loaded reached plastic collapse, 'ariable
bending moment (gradient moment) o'er %ll length o beam
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Table 6.1 Width-Thickness Parameters(*)
:lement p r
Flange
3eb
;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
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p
rp
r
7ompact shape
>oncompact shape
&lender shape
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beam an ail by reaching Mp and becoming %lly plastic, or it can be ail
by b%c-ling in one o the ollowing ways
8" 4ateral torsional b%c-ling (4$B), either elastically or inelastically.
2" Flange local b%c-ling (F4B), elastically or inelastically.+" 3eb local b%c-ling (34B) elastically or inelastically"
the maxim%m bending stress is less than the proportional limit when
b%c-ling occ%rs, the ail%re said to be elastic" /therwise, it is inelastic"
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For compact beams, laterally s%pported, &7 F8"8 gi'es the nominal
strength as
(&7 :%ation F8*8)
where
$he limit o 8"5My or Mp is to pre'ent excessi'e load deormationsand is satisied when
or
pn MM =
yyp MZFM 5"8" =
SFZF yy "5"8" 5"8S
Z
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$he moment strength o compact shape is a %nction o the %nbraced length
4b, deined as distance between points o lateral s%pport, or bracing", as
shown in the Fig%re 1"80"
AB
B
0b
0b
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6o
'nstability
'nelastic
0TB
7lastic
0TB
0b
(n
(p
(r
ompact
shapes
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pb LL pn MM =
rbp LLL )?)((@pr
pb
rppbnLL
LLMMMCM
=
rb LL wyb
y
b
bn CIL
EJGIE
LCM "")
"("""
2 +=
>o instability
nelastic 4$B
:lastic 4$B
where
Lb! %nbraced length (mm)
G! shear mod%l%s ! A0,000 MPa or str%ct%ral steel
J! torsional constant (mm9)
Cw! warping constant (mm1)"
Lateral -Torsional Buckling
SxFrFyMr )( =
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$he bo%ndary between elastic and inelastic b%c-ling
2
28 )(88)(
"ry
ry
r FFFF
ryL ++
=
2"""
8AJGE
S
x
=
2
2 )"
(9
JG
S
I
C x
y
w=
$he bo%ndary inelastic stability
y
y
pF
rL
A=
C!A
bMMMM
MC +9+5"2
5"82
max
max
+++=
M"ax! absol%te 'al%e o the maxim%m moment within the %nbraced length (incl%ding the end
point points), >*mm
MA! absol%te 'al%e o the moment at the %arter point o the %nbraced length, > *mm
M!! absol%te 'al%e o the moment at the midpoint o the %nbraced length, >*mm
MC! absol%te 'al%e o the moment at the three*%arter point o the %nbraced length, >*mm
Bending coeicient 7b
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B
B
0b%0
b%1.14
0b%082
b%1.39
082
0b%0b%1.32
0b%0
b%2.2:
(1 (2%(1
0b%082b%1.;:
a a
ABand /$ b%1.;:Bc$ b%1.99
*a+ *b+
*c+ *d+
*e+
*f+0ateral restraint
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, or the lange is non compact, b%c-ling will be inelastic,
andrp
))((pr
p
rppn MMMM
=
f
f
t
b
2=
y
pF
80=
ry
rFF =
+0
xryr SFFM )( =
0== #tr$##r$#%&ua'Fr MPa or rolled shapes"
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$he shear strength o a beam m%st be s%icient to satisy the relationship
where V% ! maxim%m shear based on the controlling combination o actored
loads, >
! resistance actor or shear ! 0"#0
n
! >ominal shear strength, >
n(u )) "
x
0
)
(
)
*a+
*b+
*c+
*d+fv
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$he shearing stress is gi'en by
where f! 'ertical and horiontal shearing stress at the point o
interest
! Vertical shear orce at the section %nder consideration * ! irst moment, abo%t ne%tral axis
I ! moment o inertia abo%t ne%tral axis
t! width o the cross section at the point o interest"
tI*)f(""=
twh
fv%)
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nd the nominal strength corresponding to this limit state is
w y nA F )" 10 " 0 =
$his will be the nominal shear strength pro'ided there is no shear b%c-ling
o the web, or
yw Fth 8800B
3here
Aw! area o the web ! &"tw, mm2
&! o'erall depth o the beam, mm
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For , there is no web instability, and
yw Fth 8800B
wyn AF) "10"0=
For , inelastic web b%c-ling can occ%r, and
ywy FthF 8+08800
w
y
wyn th
FAF)
8800"10,0=
For , the limit state is elastic web b%c-ling
2108+0 wy thF
2)(
000,#09
w
wn
th
A) =
where
Aw! area o the web !&+tw, mm2
&! o'erall depth o the beam, mm
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)n
9.;9y.Aw 9.;9y.Aw-----------
h8tw
y
*h8tw+2
y
yy 2;9yy h8tw
1199
1199 13:9
=94999Aw
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a coped beam is connected with bolts as in Fig%re 1"8A, there will be a
tendency or segment B7 to tear o%t" $he applied load in this case will be the
'ertical beam reactions, so shear will occ%r along line B and there will be
tension along B7"
h
tw
d
A
B
Fig%re 1"8A
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&7 C9"+, DBloc- &hear E%pt%re &trength, gi'es two e%ations or
the bloc- shear design strength
(&7 :%ation C9*+a)
(&7 :%ation C9*+b)
where
A,! gross area in shear (in Fig%re 1"8A, length B times
the web thic-ness), mm2
An! net area in shear, mm2
A,t!gross area in tension (in Fig%re 1"8A, length B7 times
the web thic-ness), mm
Ant! net area in tension, mm2
?""10"0@ ntu,(yn AFAF- +=
?""10"0@ ,tyn(un AFAF- +=
5"0=
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n addition to being sae, a str%ct%re m%st be ser'iceable"
ser'iceable str%ct%re is one that perorms in a satisactory manner,
not ca%sing any discomort or perceptions o %nsaety or the
occ%pants or %sers o the str%ct%re" For a beam, this %s%ally means
that the deormations, primarily the 'ertical sag, or delection, m%st
be limited" :xcessi'e delection is %s%ally an indication o a 'ery
lexible beam, and this can lead to problem with 'ibrations" $hedelection itsel can ca%se problems i elements attached to the
beam can damaged by small distortions" n addition, %sers o
str%ct%re may 'iew large delections negati'ely and wrongly ass%me
that the str%ct%re is %nsae"
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For the common case o simply s%pported, %niormly loaded beam s%ch
as that in Fig%re 1"20, the maxim%m 'ertical delection is gi'en by
Fig%re 1"20 Gelection simply s%pported beam
&ince delection is a ser'iceability limit state, not one o strength,
delection sho%ld always be comp%ted with ser'ice loads"
EI
Lw 9"
+A9
5=
A B
0
4
3>4%
5 w0
7'
w
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+00
L
290
L
8A0L
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Beam design entails the selection o cross*sectional shape that will ha'e eno%gh strength and
will meet the ser'iceability re%irements" $he design proced%re can be o%tlined as ollows
8" Model the str%ct%re. deine s%perimposed dead load and li'e load"
2" 7omp%te the actored load moment M%" $he weight o the beam is part o the dead load
b%t is %n-nown at this point" 'al%e may be ass%med, or the weight may be ignored
initially and chec-ed ater a shape has been selected"
+" &elect a shape that satisies this strength re%irement" $his can be done in one o two ways
a" ss%me a shape, comp%te the design strength, and compare it with the actored load
moment" Ee'ise i necessary" $he trial shape can be easily selected in only a limited
n%mber o sit%ations"
b" Hse the beam design charts in Part + o the man%al"
9" 7hec- the shear strength"
5" 7hec- the delection"
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Fig%re 1"+2 $ype 8 Beam Bearing Plates
htwx x
RA
A B
*a+
*b+
B 6t
*c+
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Fig%re 1"++ $ype 2 bearing plates
tw
RA
A B
66
?
?
d
R
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(8) Getermine dimension > so that web yielding and
web
crippling are pre'ented"
(2) Getermine dimension B so that the area B x > iss%icient to pre'ent the s%pporting material
rom being cr%shed in bearing"
(+) Getermine the thic-ness t so that the plate has
s%icient bending strength"
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>ominal strength or web yielding at the s%pport
t the interior load, the nominal strength
$he design strength is , where
wyn tF./- ")5"2( +=
wyn tF./- ")5( +=
n- "0"8=
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Geinition $he web crippling is b%c-ling o the web ca%sed by the
compressi'e orce deli'ered thro%gh the lange"
For an interior load, the nominal strength or web crippling is
w
fy
f
w
t
tF
t
t
&
.
t-n w
"
+8+5A
5"8
2
+=
For a load at or near the s%pport (no greater than hal the beam depth rom
the end), the nominal strength is
w
fy
f
wwn
t
tF
t
t
&
.t-
"+88#
5"8
2
+= 2"0
&
.or
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w
fy
f
ww
ttF
tt
&.t-n "2"0988#
5"8
2
+=
2"0>&.or
or
$he resistance actor or this limit state is
5"0=
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the plate co'ers the %ll area o the s%pport, the nominal strength
is
the plate does not co'er the %ll area o the s%pport,
where
fc0! 2A*day compressi'e strength o concrete, MPa
A1! bearing area, mm2
A2
! %ll area o s%pport, mm2
8I"A5"0 Af2 cp=
828I"A5"0 AAAf2 cp=
282 AA
area 2 is not concentric with 8, the 2 sho%ld be ta-en largest
concentric area that is geometrically to 8, as ill%strated in Fig%re 5"+9"
$he design bearing strength is , wherepc2 10"0=c
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$he a'erage bearing press%re is treated as a %niorm load on the bottom
o the plate, is is ass%med to be s%pported at the top o'er central width o2and length.as in Fig%re 1"+5" $he plate is considered to bend abo%t an
axis parallel to the beam span" $h%s, the plate is treated as a cantile'er o
span length n=(!32)/2and a width o."
RAA B
6d
R
tw
B
t
n n??
?
1@
Fig%re 1"+5
Bearing Plate
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From Fig%re 1"+1, the maxim%m bending moment in the plate per 8 width is
.!
n-nn
.!
-M uuu
"2
"
2"
2
==
$he nominal moment strength Mn is
9228
2tF
ttFM yyp =
=
-y
-y
!lastic neutral axisa
1@
t
.%-y*1xt82+
T%-y*1xt82+
Fig%re 1"+1 Plastic Moment capacity
o a rectang%lar cross section
upb MM &ince
uy Mt
F 9
"#0"02
y
u
F.!
n-t
""
"222"2 2
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