Steel Ch4 - Beams Movies

8/17/2019 Steel Ch4 - Beams Movies

1/80

68402 Slide # 1

esign of Beams for Flexure

Monther waikat

Assistant Professor

epartment of Building Engineering

An-Najah National Uniersit!

"#$%&' (tru)tural Design of Buildings **

"+$&%' Design of (teel (tru)tures"&,&,' Ar)hite)tural (tru)tures **


2/80

68402 Slide # 2

Introduction

Moment Curvature Response

Sectional Properties

Serviceability Requirements !e"lections

Compact$ %on&compact and Slender Sections

'ateral (orsional )uc*lin+

!esi+n o" )eams

Design of Beams for Flexure


3/80

68402 Slide # ,

Beams under Flexure

Members sub-ected principally

to transverse +ravity loadin+

. /irders important "loor beams$ide spacin+

. Joists less important beams$closely spaced

. Purlins roo" beams$ spannin+beteen trusses

. Stringers lon+itudinal brid+ebeams

. Lintels s1ort beams aboveindodoor openin+s


4/80

68402 Slide # 4

Design for Flexure

'imit states considered

. 3ieldin+

. 'ateral&(orsional )uc*lin+

. 'ocal )uc*lin+. Compact

. %on&compact

. Slender


5/80

68402 Slide #

Design for Flexure ./FD (pe)0

Commonly 5sed Sections

. I 7 s1aped members sin+ly& and doubly&symmetric

. Square and Rectan+ular or round SS

. (ees and !ouble 9n+les

. Rounds and Rectan+ular )ars

. Sin+le 9n+les

. 5nsymmetrical S1apes

:ill not be coveredin t1is course


6/80

68402 Slide # 6

(e)tion For)e- eformation

/esponse 1 Plasti) Moment

2M

P

3 A beam is astructural member

that is subjected

primarily to

transverse loads andnegligible axial

loads.

The transverse loads

cause internal SFand BM in the beams

as shown in Fig. 1

w P

V(x)

M(x)

x

w P

V(x)

M(x)

x

Fig. 1 SF ! BM in a SS Beam


7/80 68402 Slide # ;


/esponse 1 Plasti) Moment 2M

P

3

(1ese internal S< = )M cause lon+itudinal a>ial stresses

and s1ear stresses in t1e cross&section as s1on in t1e


8/80 68402 Slide # 8



P

3

Steel material "ollos a typical stress&strain be1avior as

s1on in


9/80 68402 Slide # B



P

3

I" t1e steel stress&strain curve is appro>imated as a bilinear

elasto&plastic curve it1 yield stress equal to σy$ t1en t1esection Moment & Curvature M&φ response "ormonotonically increasin+ moment is +iven by


10/80

68402 Slide # 0

Moment-4urature 2NE53

. )eam curvature φ is related to its strain and t1us to t1eapplied moment

EI

M

y

== ε

φ

φ

y y

(1)(1) (2)(2) (3)(3) (4)(4)


11/80

68402 Slide #

Moment-4urature 2NE53

x x F S M =

. :1en t1e section is it1in elastic ran+e

y x y F S M =

y x p F Z M = a A Z x )2(=

x x

xS

M

I

y M F ==

Where S is the elastic section modulus

. :1en t1e moment e>ceeds t1e yield moment My

. (1en

Where Z is the plastic section modulus 1.1 S≈

a


12/80

68402 Slide # 2


/esponse 1 Plasti) Moment 2

M

P

3

My

M p

A: Extreme fiber reaches εy B: Extreme fiber reaches 2εy C: Extreme fiber reaches εy!: Extreme fiber reaches "#εy E: Extreme fiber reaches i$fi$ite strai$

A

B C E!

Cur%ature& φ

' e c t i ( $ M ( m e

$ t &

M

σy

σy

σy

σy

εy

εy

σy

σy

σy

σy

σy

σy

2εy

2εy

5εy

5εy

10εy

10εy

A B C ! E

My

M p

A: Extreme fiber reaches εy B: Extreme fiber reaches 2εy C: Extreme fiber reaches εy!: Extreme fiber reaches "#εy E: Extreme fiber reaches i$fi$ite strai$

A

B C E!

Cur%ature& φ

' e c t i ( $ M ( m e

$ t &

M

σy

σy

σy

σy

εy

εy

σy

σy

σy

σy

σy

σy

2εy

2εy

5εy

5εy

10εy

10εy

A B C ! E

σy

σy

σy

σy

σy

σy

σy

σy

εy

εy

εy

εy

σy

σy

σy

σy

σy

σy

σy

σy

σy

σy

σy

σy

2εy

2εy

2εy

2εy

5εy

5εy

5εy

5εy

10εy

10εy

10εy

10εy

A B C ! E

Fig. % Mφ

response o& a beam section


13/80

68402 Slide # ,



P

3

Calculation o" MP Cross&section sub-ected to eit1er Eσy or & σy att1e plastic limit? See


14/80

68402 Slide # 4

Moment-4urature

. :1en t1e 1ole section is yieldin+ a plastic 1in+eill be "ormed

plastic hinge

. Structural analysis by assumin+ collapsin+ mec1anisms o"a structure is *non as FPlastic analysisG

. (1e plastic moment Mp is t1ere"ore t1e moment needed att1e section to "orm a plastic 1in+e


15/80

68402 Slide #



P

3

(1e plastic centroid "or a +eneral cross&section

corresponds to t1e a>is about 1ic1 t1e total area is

equally divided$ i?e?$ 9C @ 92 @ 92

. (1e plastic centroid is not t1e same as t1e elastic centroid orcenter o" +ravity c?+? o" t1e cross&section?

. 9s s1on belo$ t1e c?+? is de"ined as t1e a>is about 1ic1 9y @ 92y2?

c*,* = eastic -*A*

A"& y"

A2& y2Abut the c*,* A"y" = A2y2

y"

y2c*,* = eastic -*A*

A"& y"

A2& y2Abut the c*,* A"y" = A2y2

y"

y2


16/80

68402 Slide # 6



P

3

. is o" symmetry$ t1eneutral a>is corresponds to t1e centroidal a>is in t1e elastic

ran+e? oever$ at Mp$ t1e neutral a>is ill correspond to t1e

plastic centroidal a>is?

92 > yCEy2

9s s1on in


17/80

68402 Slide # ;



P

3

92 > yEy2 is called () t1e plastic section moduluso" t1e cross&section? Halues "or are tabulated "orvarious cross&sections in t1e properties section o"t1e 'R

∀φbMp @ 0?B0


18/80

68402 Slide # 8

Ex0 $0+ (e)tional Properties

!etermine t1e elastic section modulus$ S$ plastic sectionmodulus$ $ yield moment$ My$ and t1e plastic moment

MP$ o" t1e cross&section s1on belo? :1at is t1e desi+n

moment "or t1e beam cross&section? 9ssume 9BB2 steel?

"2 i$*

". i$*

" i$*

#* i$*

"*# i$*

F"

+

F2

t0 = #* i$*

"2 i$*

". i$*

" i$*

#* i$*

"*# i$*

F"

+

F2

"2 i$*

". i$*

" i$*

#* i$*

"*# i$*

F"

+

F2

t0 = #* i$*

,00 mm

mm

0 mm

2 mm

400 mm

400 mm


19/80

68402 Slide # B

9+ @ ,00 > E 400 & & 2 > 0 E 400 > 2 @ 800 mm2 9" @ ,00 > @ 400 mm2

9"2 @ 400 > 2 @ 0000 mm2

9 @ 0 > 400 & & 2 @ ,600 mm2

distance o" elastic centroid "rom bottom @

I> @ 400>2,2 E00002?&4?,2 E 0>,60,2 E,60020&4?,2 E ,00>,2 E400,B2?&4?,2 @ 0,?;>06 mm4

S> @ 0,?;>06 400&4?, @ B;;?>0, mm,

My&> @ @ 680?2 *%&m?

S x - elastic section modulus


y

1##(1## " / 2) .## 2# "#### "2*"1*

"3"## y mm

− + × + ×= =


20/80

68402 Slide # 20


distance o" plastic centroid "rom bottom @

y @ centroid o" top 1al"&area about plastic centroid

@ mm

y2 @ centroid o" bottom 1al"&area about plas? cent?

@ mm

> @ 92 > y E y2 @ B00 > 26?; E ?, @ 242400 mm,

> & plastic section modulus

"3"##1## 4##

2

22*.

p

p

y

y mm

∴ × = =

∴ =

py py py

py

"3"## "1* 4## 22*. / 222*. 2.*

4##

× − × − =

22*. / 2 ""*=


21/80

68402 Slide # 2


Mp&> @ > ,4406 @ 8,4?, *%?m

!esi+n stren+t1 accordin+ to 9ISC Spec? 8,4?,@ ;0?B *%?m

C1ec* @ Mp ≤ ? My

(1ere"ore$ 8,4?, *%?m J ? > 680?2 @ 020?, *%?m

& KL


22/80

68402 Slide # 22

Flexural efle)tion of Beams -

(eri)ea6ilit!

Steel beams are desi+ned "or t1e "actored desi+n loads?(1e moment capacity$ i?e?$ t1e "actored moment stren+t1

φbMn s1ould be +reater t1an t1e moment Mu causedby t1e "actored loads?

9 serviceable structure is one t1at per"orms

satis"actorily$ not causin+ discom"ort or perceptions o"

unsa"ety "or t1e occupants or users o" t1e structure?

. imum de"lection o" t1e desi+ned beam is c1ec*ed at t1eservice&level loads? (1e de"lection due to service&level loads

must be less t1an t1e speci"ied values?


23/80

68402 Slide # 2,

(eri)ea6ilit! /e7uirements

. Steel beams need to satis"y S'S in addition to 5'S

. Serviceability limit states are usually c1ec*ed usin+ non&"actoredloads?

. !e"lection under live loads s1all be limited to ',60

. !ead load de"lections can be compensated by camberin+ beams?

. S'S mi+1t also include limitin+ stresses in bottom or top "lan+es i""ati+ue is a concern in desi+n :ill be "urt1er discussed it1 plate

+irders?

. Standard equations to calculate de"lection "or di""erent load cases


24/80

68402 Slide # 24


(eri)ea6ilit!

(1e 9ISC Speci"ication +ives little +uidance ot1er t1an astatement$ FServiceability Design Considerations$G t1at

de"lections s1ould be c1ec*ed? 9ppropriate limits "or

de"lection can be "ound "rom t1e +overnin+ buildin+ code

"or t1e re+ion? (1e "olloin+ values o" de"lection are typical ma>?

alloable de"lections?

'' !'E''

. Plastered "loor construction ',60 '240

. 5nplastered "loor construction '240 '80

. 5nplastered roo" construction '80 '20

. !' de"lection 7 normally not considered "or steel beams


25/80

68402 Slide # 2


(eri)ea6ilit!

In t1e "olloin+ e>amples$ e ill assume t1at

local buc*lin+ and lateral&torsional buc*lin+ are

not controllin+ limit states$ i?e$ t1e beam section is

compact and laterally supported alon+ t1e len+t1?


26/80

68402 Slide # 26

Ex0 $0& - Defle)tions

!esi+n a B m lon+ simply supported beam sub-ected to5!' o" 6 *%m dead load and a 5!' o" 8 *%m live load?

(1e dead load does not include t1e sel"&ei+1t o" t1e

beam?

. Step *. Calculate t1e "actored desi+n loads it1out sel"&ei+1t?u @ ?2 ! E ?6 ' @ 20 *%m

Mu @ u '2 8 @ 20 > B2 8 @ 202? *%?m SS beam

. Step **. Select t1e li+1test section "rom t1e 9ISC Manual desi+ntables?

> @ Muφb060?B>,44 @ 64>0, select +1, x ", made "rom 9BB2 steel it1 φbMp @ 224 *%?m


27/80

68402 Slide # 2;


. Step ***. 9dd sel"&ei+1t o" desi+ned section and c1ec* desi+ns @ 0?,8 *%m

(1ere"ore$ ! @ 6?,8 *%m

u @ ?2 > 6?,8 E ?6 > 6 @ 20?46 *%m

(1ere"ore$ Mu @ 20?46 > B2 8 @ 20;?2 *%?m J φbMp o"

:6 > 26?-/

. Step *0. C1ec* de"lection at service loads? @ 8 *%m! @ '4 ,84 A I

> @ > 8 >0, > B4 ,84 > 200>2

! @ 2;?, mm N ',60 & "or plastered "loor construction

. Step 0. Redesi+n it1 service&load de"lection as desi+n criteria' ,60 @ 2 mm N '4,84 A I>

(1ere"ore$ I> N ,6?;>06 mm4


28/80

68402 Slide # 28


Select t1e section "rom t1e moment of inertia selection "romSection Property (ables 7 select +1, x $1 it1 *x @ 6>06 mm4

%ote that the ser&iceability design criteria controlled the design

and the section


29/80

68402 Slide # 2B

Ex0 $0, Beam Design

!esi+n t1e beam s1on belo? (1e un"actored dead and live loadsare s1on in 0 E ?6 > 40 @ 64 *%

Mu @ 5 '2 8 E P5 ' 4 @ 2BB?; E 44 @ 44,?; *%?m

#*. 5/ft* (dead load)"# 5ips (live load)

# ft*

" ft*

#* 5/ft* (live load)

40 *%0 *%m

*%m

4? m

B m


30/80

68402 Slide # ,0

Ex0 $0, Beam Design

Step **. Select +"1 x %% > @ 6,>0, mm,φbMp @ 0?B>6,>0,>,44000000 @ 48,?B *%?mSel"&ei+1t @ s @ 0?64 *%m?

Step ***. 9dd sel"&ei+1t o" desi+ned section and c1ec* desi+n

! @ 0 E 0?64 @ 0?64 *%m

u @ ?2 > 0?64 E ?6 > @ ,0?4 *%m

(1ere"ore$ Mu @ ,0?4 > B28 E 44 @ 4?8 J φbMp o":2 > 44?

-/ Step *0. C1ec* de"lection at service live loads?

Service loads

. !istributed load @ @ *%m

. Concentrated load @ P @ ' @ 40 *%


31/80

68402 Slide # ,

Ex0 $0, Beam Design

!e"lection due to uni"orm distributed load @ ∆d @ '4 ,84 AI!e"lection due to concentrated load @ ∆c @ P ', 48 AI

There&ore) serviceload de&lection d 2 c

∆ @ >>B4

>0B

,84>,>06

>200 E40>B,

>0B

48>,>06

>200∆ @ ,?4 E 8?; @ 22? mm' @ B m?

9ssumin+ unplastered "loor construction$ ∆ma> @ '240 @ B000240 @,;? mm

(1ere"ore$ ∆ J ∆ma> -/


32/80

68402 Slide # ,2

.o)al Bu)kling of Beam (e)tion

4ompa)t and Non-)ompa)t

Mp$ t1e plastic moment capacity "or t1e steel s1ape$ iscalculated by assumin+ a plastic stress distribution E or & σyover t1e cross&section?

(1e development o" a plastic stress distribution over t1e cross&

section can be 1indered by to di""erent len+t1 e""ects. Local buckling o" t1e individual plates "lan+es and ebs o" t1e

cross&section be"ore t1ey develop t1e compressive yield stress σy?

. Lateral-torsional buckling o" t1e unsupported len+t1 o" t1e beam member be"ore t1e cross&section develops t1e plastic moment Mp?

(1e analytical equations "or local buc*lin+ o" steel plates it1various ed+e conditions and t1e results "rom e>perimentalinvesti+ations 1ave been used to develop limitin+ slendernessratios "or t1e individual plate elements o" t1e cross&sections?


33/80

68402 Slide # ,,

M

M

M

M

Figure 3. 'ocal buc*lin+ o" "lan+e due to compressive stress s




34/80

68402 Slide # ,4

Steel sections are classi"ied as compact$ non&compact$ orslender dependin+ upon t1e slenderness l ratio o" t1e

individual plates o" t1e cross&section?

. Compact section i" all elements o" cross&section 1ave λ ≤ λp

. Non-compact sections i" any one element o" t1e cross&section 1as λp ≤ λ ≤ λr

. Slender section i" any element o" t1e cross&section 1as λr ≤ λ

It is important to note t1at

. I" λ ≤ λp$ t1en t1e individual plate element can develop and sustain σy "orlar+e values o" e be"ore local buc*lin+ occurs?

. I" λp ≤ λ ≤ λr $ t1en t1e individual plate element can develop σy at somelocations but not in t1e entire cross section be"ore local buc*lin+ occurs?

. I" λr ≤ λ$ t1en elastic local buc*lin+ o" t1e individual plate element occurs?




35/80

68402 Slide # ,

4lassifi)ation of (e)tions

. Classi"ications o" bendin+ elements are based on limits o" local buc*lin+

. (1e dimensional ratio l represents

. (o limits e>ist λp and λr

λp represents t1e upper limit "or compact sections

λr represents t1e upper limit "or non&compact sections

f

f

t

b

2=λ

wt

h=λ

f b

hwt

f t

compact P

λ λ ≤compact nonr P −≤≤ λ λ λ

slender r λ λ ≥


36/80

68402 Slide # ,6

ompact

!on"ompact

Slender

σ

y

C ( m p r e s s i % e

a x i a ) s t r e s s & σ

Effecti%e axia strai$& ε

ompact

!on"ompact

Slender

σ

y

C ( m p r e s s i % e

a x i a ) s t r e s s & σ

Effecti%e axia strai$& ε

Figure 4. Stress&strain response o" plates

sub-ected to a>ial compression and local

buc*lin+?

(1us$ slendersections cannot

develop Mp due to

elastic local

buc*lin+? %on&compact sections

can develop My but

not Mp be"ore local

buc*lin+ occurs?

Knly compact

sections can

develop t1e plastic

moment Mp?




37/80

68402 Slide # ,;

9ll rolled ide&"lan+e s1apes are compact it1 t1e"olloin+ e>ceptions$ 1ic1 are non&compact?

. :2>48$ :40>;4$ :4>BB$ :4>B0$ :2>6$ :0>2$ :8>0$:6> made "rom 9BB2




38/80

68402 Slide # ,8

4lassifi)ation of (e)tions

. (1e limits are f b

hwt

f t

f

f

t

b

2=λ

wt

h=λ

y

p F

E .*=λ

y

p F

E 3*#=λ

4*.33*#

−=

y

r F

E λ y

r FE#*=λ

WebFlange


39/80

68402 Slide # ,B

.ateral-8orsional Bu)kling

2.8B3

M

M

M

M

(a)

(b)

M

M

M

M

M

M

M

M

(a)

(b)

• (1e laterally unsupportedlen+t1 o" a beam&membercan under+o '() due tot1e applied "le>uralloadin+ )M?

Figure 5. 'ateral&torsionalbuc*lin+ o" a ide&"lan+e beamsub-ected to constant moment?


40/80

68402 Slide # 40


2.8B3

'() is "undamentally similar to t1e "le>ural buc*lin+ or"le>ural&torsional buc*lin+ o" a column sub-ected to a>ial

loadin+?

. (1e similarity is t1at it is also a bi"urcation&buc*lin+ typep1enomenon?

. (1e di""erences are t1at lateral&torsional buc*lin+ is caused by"le>ural loadin+ M$ and t1e buc*lin+ de"ormations are coupled int1e lateral and torsional directions?

(1ere is one very important di""erence? ial load causin+ buc*lin+ remains constant alon+ t1elen+t1? )ut$ "or a beam$ usually t1e '() causin+ bendin+moment M> varies alon+ t1e unbraced len+t1?

. (1e orst situation is "or beams sub-ected to uni"orm )M alon+ t1eunbraced len+t1? :1yO


41/80

68402 Slide # 4


2.8B3

Uniform BM

Consider a beam t1at is simply&supported at t1e ends andsub-ected to "our&point loadin+ as s1on belo? (1e beamcenter&span is sub-ected to uni"orm )M M? 9ssume t1atlateral supports are provided at t1e load points?

'aterally unsupported len+t1 @ 'b?

I" t1e laterally unbraced len+t1 'b is less t1an or equal to a

plastic len+t1 'P t1en lateral torsional buc*lin+ is not a

problem and t1e beam ill develop its plastic stren+t1 MP?

6 b

PP


42/80

68402 Slide # 42


2.8B3

Uniform BM

'p @ ?;6 r y > & "or I members = c1annels I" 'b is +reater t1an 'P t1en lateral torsional buc*lin+ ill

occur and t1e moment capacity o" t1e beam ill be

reduced belo t1e plastic stren+t1 MP as s1on in


43/80

68402 Slide # 4,

.ateral-8orsional Bu)kling 2.8B3

Uniform BM

Mn @ Mcr @

M n - moment capacity

'b - laterally unsupported length.

M cr - critical lateral-torsional buc(ling moment.) * + !a

* // !a

# y - moment of inertia about minor or y-axis 0mm1 2

3 - torsional constant 0mm1 2 from the Section !roperty 4ables.

$ w - warping constant 0mm5 2 from the Section !roperty 4ables.

(1is Aq? is valid "or A'9S(IC '() only li*e t1e Auler equation? (1is

means it ill or* only as lon+ as t1e cross&section is elastic and no

portion o" t1e cross&section 1as yielded?

2 2

2 2

y w

b b

EI E #$ % %

π π + ÷


44/80

68402 Slide # 44


Uniform BM

Fig. 16 #ateral Torsional Buc7ling 89ni&orm Bending:

Plastic

%o

Instability

Inelastic

'() Alastic

'()

6r

M ( m e $ t C a p a c i t y & M

$

7$braced e$,th& 6 b

M$ =

+2

2

2

2

b

w

b

y

%

E #$

%

EI π π

−

−−−=

pr

pb

r p pn % %

% % M M M M )(

M$ = M p

8x Fy= M p

'x (Fy 9 "#) = Mr

6 p 6r


$


$

M

$

7$braced e$,th& 6 b7$braced e$,th& 6 b

M$ =

+2

2

2

2

b

w

b

y

%

E #$

%

EI π π M$ =

+2

2

2

2

b

w

b

y

%

E #$

%

EI π π

+2

2

2

2

b

w

b

y

%

E #$

%

EI π π

−

−−−=

pr

pb

r p pn % %

% % M M M M )(

M$ = M p

8x Fy= M p

'x (Fy 9 "#) = Mr

6 p

(#*Fy)


45/80

68402 Slide # 4


Uniform BM

9s soon as any portion o" t1e cross&section reac1es t1eyield stress


46/80

68402 Slide # 46


Uniform BM

2

#

#

*#.*.""

*#4*"

++=

$c

hS

E

F

hS

$c

F

E r % x

y

x y

tsr

−−−−= pr

pbr p pn

% % % % M M M M )(. I" 'p ≤ 'b ≤ 'r $ t1en

. (1is is linear interpolation beteen 'p$ Mp and 'r $ Mr

. See


47/80

68402 Slide # 4;

Moment 4apa)it! of Beams

(u6je)ted to Non-uniform BM

9s mentioned previously$ t1e case it1 uni"orm bendin+moment is orst "or lateral torsional buc*lin+?


48/80

68402 Slide # 48



;b is alays +reater t1an ?0 "or non&uni"orm bendin+moment?

. ;b is equal to ?0 "or uni"orm bendin+ moment?

. Sometimes$ i" you cannot calculate or "i+ure out ;b$ t1en it can be

conservatively assumed as ?0? "or doubly and sin+ly symmetricsections

ma! - magnitude of ma!imum bending moment in Lb

" - magnitude of bending moment at #uarter point of Lb

$ - magnitude of bending moment at half point of Lb

C - magnitude of bending moment at three-#uarter point of Lb

. 5se absolute values o" M

#*1*2

*"2

max

max <+++

=c & A

b M M M M

M

Fl l (t th f 4 t


49/80

68402 Slide # 4B

Flexural (trength of 4ompa)t

(e)tions

#*1*2

*"2

max

max <+++

= & A

b M M M M

M

Mmax

MA

MB

MC' ()arter

' mid

' three"()arter

Moments determined between bracing

points


50/80

68402 Slide # 0

9alues of 4 6

$1


51/80

68402 Slide #


(u6je)ted to Non-uniform Bending

Moments

(1e moment capacity Mn "or t1e case o" non&uni"ormbendin+ moment

. Mn @ ;b > Mn "or t1e case o" uni"orm bendin+ momentQ ≤ M p

. Important to note t1at t1e increased moment capacity "or t1e non&uni"orm moment case cannot possibly be more t1an M p.

. (1ere"ore$ i" t1e calculated values is +reater t1an M p$ t1en you1ave to reduce it to M p


52/80

68402 Slide # 2



6r 6 p

Mr

M p

C b = "*#C b = "*2

C b = "*

M ( m e $ t C a p a c i t y &

M $

7$braced e$,th& 6 b

Figure 11. Moment capacity versus 'b "or non&uni"orm moment case

5ni"orm )M5ni"orm )M

%on&uni"orm )M%on&uni"orm )M

Cb

@ ?0 means uni"orm )M


53/80

68402 Slide # ,

(tru)tural Design of Beams

. Steps "or adequate desi+n o" beams Compute t1e "actored loads$ "actored moment and s1ear

2 !etermine unsupported len+t1 'b and Cb

, Select a :< s1ape and c1oose > assumin+ it is a compact section

it1 "ull lateral support

4 C1ec* t1e section dimension "or compactness and determine φbMn

5se service loads to c1ec* de"lection requirements

yb

) x

F

M Z

φ =

nb) M M φ ≤

ynb)

y pn

ZF M M

ZF M M

4*#=≤

==

φ


54/80

68402 Slide # 4

Ex0 $0$ Beam esign

5se /rade 0 steel to desi+n t1e beam s1on belo? (1eun"actored uni"ormly distributed live load is equal to 40

*%m? (1ere is no dead load? 'ateral support is provided at

t1e end reactions? Select :6 section?

21 ft*

!

" # $ips%&t.

6atera supprt / braci$,

'' @ 40 *%m@ 40 *%m

;? m;? m


55/80

68402 Slide #

Ex0 $0$ Beam esign

. Step *. Calculate t1e "actored loads assumin+ a reasonable sel"&ei+1t? 9ssume sel"&ei+1t @ s @ ?46 *%m?

!ead load @ ! @ 0 E ?46 @ ?46 *%m?

'ive load @ ' @ 40 *%m?

5ltimate load @ u @ ?2 ! E ?6 ' @ 6?8 *%m?


56/80

68402 Slide # 6

Ex0 $0$ Beam esign

m F E r % y y p "*""###

112#####1*1#.*"/.*" =

×==

mmS

I r

x

w y

ts "*13"#"2

"#."#"#*"

4.

=×

×××==

. Step ***. Select a ide&"lan+e s1ape. Compute > @ 462?,060?B,44 @ 4B,>06 mm,?. Select :6 > 0 steel section

. > @ 08>0, mm, S> @ ,2;>0, mm, r y @ 40?4 mm

. C @ 60>0B

mm6

Iy @ ?>06

mm4

@ 0?6,>06

mm4

.

.

.

2

#

#

*#.*.""

*#4*"

++=

$c

hS

E

F

hS

$c

F

E r % x

y

x y

tsr


57/80

68402 Slide # ;

Ex0 $0$ Beam esign

m %r

2.*3"*2"43"#"2

""#.*#

11*#

2#####

"###

"*134*"

.

=+××

××

×××=

3"*2""#.*#

43"#"2

2#####

11*#.*."

*#.*."

2

.

2

# =

××

×××+=

+

$c

hS

E

F x y

.10 @ ! & (


58/80

68402 Slide # 8

. Step *0. C1ec* i" section is adequate. Mu N φMn %ot KL

. Step 0. (ry a lar+er section?. 9"ter "e trials select :6 > 6; φMn @ 4B;?; N Mu -

. Step 0*. C1ec* "or local buc*lin+?λ @ )" 2(" @ ;?;T Correspondin+ λp @ 0?,8 A


59/80

68402 Slide # B

Ex0 $0: Beam esign

!esi+n t1e beam s1on belo? (1e concentrated liveloads actin+ on t1e beam are s1on in t1e


60/80

68402 Slide # 60

Ex0 $0: Beam esign

. Step *. 9ssume a sel"&ei+1t and determine t1e "actored desi+nloads'et$ s @ ? *%m

P' @ , *%

Pu @ ?6 P' @ 26 *%

u @ ?2 > s @ ?8 *%m

(1e reactions and bendin+ moment dia+ram "or t1e beam are

s1on belo?


61/80

68402 Slide # 6

Ex0 $0: Beam esign

s " '.1( $ips%&t.

"2 ft* 3 ft* "# ft*

)* $ips )* $ips

)+.+ $ips ,# $ips

,,'.+ $ip-&t. ,() $ip-&t.

A

B C

A B C


2". -2". - 2". -2". -"*3 -/m"*3 -/m

*# m*# m*. m*. m 2*1 m2*1 m2#4* 5-2#4* 5- 23* 5-23* 5-

1*4 5-*m1*4 5-*m "* 5-*m"* 5-*m


62/80

68402 Slide # 62

Ex0 $0: Beam esign

. Step **. !etermine 'b$ Cb$ Mu$ and MuCb "or all spans?

Span !b

/m0

Cb

Mu

/$-m0

Mu%C

b

/$-m0

AB *. "*. 1*4 12*3BC 2*1 "*#

(assume)1*4 1*4

C! *# "*. "* 124*2

It is important to note t1at it is possible to 1ave di""erent 'b andC

b values "or di""erent laterally unsupported spans o" t1e same

beam?

Cb 7 (able ,&


63/80

68402 Slide # 6,

Ex0 $0: Beam esign

.Step ***. !esi+n t1e beam and c1ec* all laterally unsupported spans

9ssume t1at span B; is t1e controllin+ span because it 1as t1e

lar+est Mu @ ;4?B060?B,44 @ 24,8>0, mm,

A&ter &ew trials select +"1 x ,4 &rom section property Table.

'p @ ?B4 m 'r @ ?;, m


64/80

68402 Slide # 64

Ex0 $0: Beam esign

(1us$ "or span 9)$ φbMn @ 8?8 *%?m N Mu & KLample demonstrates t1e met1od "or desi+nin+ beams it1several laterally unsupported spans it1 di""erent 'b and Cb values?


65/80

68402 Slide # 6

Ex0 $0" Beam esign

!esi+n t1e simply&supported beam s1on belo? (1euni"ormly distributed dead load is equal to *%m andt1e uni"ormly distributed live load is equal to ,0 *%m? 9concentrated live load equal to 40 *% acts at t1e mid&span?'ateral supports are provided at t1e end reactions and at

t1e mid&span? 5se /rade 0 steel? " 1.' $ips%&t.

"2 ft* "2 ft*

1' $ips

A

B

C

! " (.' $ips%&t.


1# 5-" 5-/m# 5-/m

*. m *. m


66/80

68402 Slide # 66

Ex0 $0" Beam esign

. Step *. 9ssume t1e sel"&ei+1t and calculate t1e "actored desi+n loads?'et$ s @ ? *%m

! @ E ? @ 6? *%m

' @ ,0 *%m

u @ ?2 ! E ?6 ' @ 6;?8 *%m

Pu @ ?6 > 40 @ 64 *%(1e reactions and t1e bendin+ moment dia+ram "or t1e "actored loadsare s1on belo?

+(.() $ips +(.() $ips

u " ).,( $ips%&t.

"2 ft* "2 ft*

1+ $ips

B

x M/x0 " +(.() x - ).,( x(%(

.1 5-

.*3 5-/m

*. m *. m2.*" 5- 2.*" 5-

M(x) = 2.*"(x) ; .*3(x)2/2


67/80

68402 Slide # 6;

. Step **. Calculate 'b and Cb "or t1e laterally unsupported spans?Since t1is is a symmetric problem$ need to consider only span 9)

'b @,?6 m$ M> @ 2;6? > 7 6;?8 >22

(1ere"ore$

M 9 @ M> @ 0?B m @ 22 *%?m & quarter&point alon+ 'b @ ,?6 m

M) @ M> @ ?8 m @ ,8; *%?m & 1al"&point alon+ 'b @ ,?6 m

MC @ M> @ 2?; m @ 4B8 *%?m & t1ree&quarter point alon+ 'b@ ,?6 m

Mma> @ M> @ ,?6 m @ 4?6 *%?m & ma>imum moment alon+ 'b @,?6 m(1ere"ore$ Cb @ ?,6

Ex0 $0" Beam esign


68/80

68402 Slide # 68

Ex0 $0" Beam esign

. Step ***. !esi+n t1e beam sectionMu @ Mma> @ 4?6 *%?m

'b @ ,?6 m$ Cb @ ?,6

Required > @ 4?6060?B,44 @ ;B>0, mm,

9"ter "e trials$ select :2 > ; steel section

'p @ ?46 m 'r @ 4?,; m

'p J 'b J 'r

φbMn @ 6BB *%?m N φbMp @ 6,B?, *%?mφbMn@ 6,B?, NMu KL


69/80

68402 Slide # 6B

Ex0 $0" Beam esign

. Step 0. C1ec* "or local buc*lin+?)" 2(" @ ;?8;T Correspondin+ λp @ 0?,8 A


70/80

68402 Slide # ;0

(hear 4apa)it!

3he shear capacity o& the beam is

nv) + + φ ≤ vw yn A F + .*#=

For 4-shaped sections5 the &actors ;2 and φ2 are &unctions o& the shear

buc$ling o& the eb and thus the ration h%t

yw

F

E t hif 21*2/ ≤

#*"=v

representing the case o& no eb instability.

9 @ dt

4*#=vφ

#*"=vφ


71/80

68402 Slide # ;

(hear 4apa)it!

4*#=vφ

y

vw

y

v

F

E * t h

F

E * *"/"#*" ≤<

( ) yw

vv

F t h

E*

2/

"*"=

3he second case represents inelastic eb buc$ling

y

vw

F

E * t h *"/ ≥

y

vw

F

E * t h "#*"/ ≤ #*"=v

3he last case represents elastic eb buc$ling

For all other doubly and singly sym. sections and channels except

round 6SS

4*#=vφ w

yvv

t h

F E *

"#*"=

3he &irst case represents the case o& no eb instability.

4*#=vφ

4*#=vφ


72/80

68402 Slide # ;2

(hear 4apa)it!

. 9 @ dt

. (1e eb plate buc*lin+ coe""icient$ *v$ is +iven. cept "or t1e stem o" tee s1apes$

*v @?2

. >

+

=2

2

2.##*&

w

v

t hhaor ha

ha*


73/80

68402 Slide # ;,

Beam Bearing Plates

esign o& a beam bearing plate ould re7uire chec$ing8

t

& !

1- Web 9ielding and Web crippling to determine

(- Bearing capacity to determine B

#- Plate moment capacity to determine t

AISC Specifications: Chapter K


74/80

68402 Slide # ;4

Beam Bearing Plates

When a bearing plate is used at beam end5 to limit states shall beconsidered

1. Web Yielin!

3his represents yield o& the eb at the 2icinity o& the &lange due

to excessi2e loading

* ! ≥w yn t F ! * , )*2( +=

CAS" 1: At S#pport

,

,

w yn t F ! * , )( +=

CAS" 2: Interior $oa

n , , φ ≤ #*"=φ


75/80

68402 Slide # ;

Beam Bearing Plates

Web crippling represent the possible buc$ling o& the eb

2. Web Cripplin!

n , , φ ≤ *#=φ

w

f y

f

w

wn t

t F E

t

t

d

!

t ,

+=

*"

2"1*#

CAS" 1: At S#pport

CAS" 2: Interior $oa

2*#≤d !

w

f y

f

wwn

t

t F E

t

t

d

! t ,

−+=

*"

22*#

1"1*# 2*#≥

d

!

w

f y

f

wwn

t

t F E

t

t

d

! t ,

+=

*"

2"3*#


76/80

68402 Slide # ;6

Ex0 $0; Beam esign

C1ec* t1e beam s1on in t1e "i+ure belo "or. S1ear capacity?. :eb yieldin+?. :eb cripplin+?. 9ssume t1e idt1 o" t1e bearin+ plate is 00 mm? 5se /rade 0

steel?

" 1.' $ips%&t.

"2 ft* "2 ft*

1' $ips

A

B

C

! " (.' $ips%&t.

1# 5-"# 5-/m2 5-/m

2 m 2 m

:6>26


77/80

68402 Slide # ;;

Ex0 $0; Beam esign

.Step *. (1e section used "rom A>ample 4?6 is :2>;?

(1e sel"&ei+1t s @ 0?8, *%m

! @ 0 E 0?,8 @ 0?,8 *%m

' @ 2 *%m

u @ ?2 ! E ?6 ' @ 2? *%m

Pu @ ?6 > 40 @ 64 *%(1e reactions and t1e bendin+ moment dia+ram "or t1e "actored loadsare s1on belo?

+(.() $ips +(.() $ips

u " ).,( $ips%&t.

"2 ft* "2 ft*

1+ $ips

B

x M/x0 " +(.() x - ).,( x(%(

.1 5-

2* 5-/m

2 m 2 m" 5- " 5-

Hu @ ,; *%


78/80

68402 Slide # ;8

Ex0 $0; Beam esign

. Step **. 1t @ 6?8

9ssume unsti""ened eb$ 1t J260$ *v @

9ssume unsti""ened eb$ 1t J260$ *v @

φHn @ 0?B0?6,44,BB6?4>0&, @ 4;4?4 *%N Hu

3*.111

2#####21*221*2 =


79/80

68402 Slide # ;B

Ex0 $0; Beam esign

. Step ***. :eb yieldin+ critical is support* @ B mm

φR @ 2?* E %2?>B E 00>,44>6?4000 @ ,24?; *%

φR N reaction @ ,; *% KL. Step *0. :eb cripplin+ critical is support

d @ ,BB mm t @ 6?4 mm t" @ 8?8 mm

%d @ 00,BB @ 0?2 N 0?2

w

f y

f

wwn

t

t F E

t

t

d

! t ,

−+=

*"

2 2*#1

"1*#


80/80

Ex0 $0; Beam esign

φR @ ;B *% N reaction @ ,; *% KL

*! ,n *23"#1*.

3*3112#####

3*3

1*.2*#

44

"##1"1*.1*#

*"2 =×××

−×+×= −

Steel Ch4 - Beams Movies

Documents

Transcript of Steel Ch4 - Beams Movies