flexural edited.ppt

7/23/2019 flexural edited.ppt

1/49

Team TeachingStructural Design

Civil Engineering Department2010


2/49

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


3/49

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


4/49

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 "


5/49

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 "=


6/49

(

)

htw (x x

RA

A B

cy

*a+

*b+


7/49

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 "=


8/49

A B

A B

A B

A B

A B

f,-y

f%-y

f%-y

f%-y*a+

*b+

*c+

*d+

Bending (oment


9/49

A B

A B

(p

(oment


10/49

h

tw

-y

-y

%Ac.-y

T%At.-y

!lastic neutral axis

a


11/49

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+


12/49

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 =


13/49

A B

Bending (oment

A B

*a+

*b+


14/49

/'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"


15/49

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
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16/49

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


17/49

A B

Bending (oment

A B

*a+

*b+

/eflection

0oad

irst

yield1

2

3

4

5

0

0


18/49

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


19/49

Table 6.1 Width-Thickness Parameters(*)

:lement p r

Flange

3eb

;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;


20/49

p

rp

r

7ompact shape

>oncompact shape

&lender shape


21/49

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"


22/49

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


23/49

$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


24/49

6o

'nstability

'nelastic

0TB

7lastic

0TB

0b

(n

(p

(r

ompact

shapes


25/49

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 )( =


26/49

$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


27/49

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


28/49

, 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"


29/49

$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


30/49

$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%)


31/49

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


32/49

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


33/49

)n

9.;9y.Aw 9.;9y.Aw-----------

h8tw

y

*h8tw+2

y

yy 2;9yy h8tw

1199

1199 13:9

=94999Aw


34/49

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


35/49

&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=


36/49

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"


37/49

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


38/49

+00

L

290

L

8A0L


39/49

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"


40/49

Fig%re 1"+2 $ype 8 Beam Bearing Plates

htwx x

RA

A B

*a+

*b+

B 6t

*c+


41/49

Fig%re 1"++ $ype 2 bearing plates

tw

RA

A B

66

?

?

d

R


42/49

(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"


43/49

>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=


44/49

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


45/49

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=


46/49

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


47/49

$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


48/49

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


49/49

flexural edited.ppt

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