chapter 2 bending shearing tension.pdf

7/29/2019 Chapter 2 Bending shearing tension.pdf

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SUEJECT (Dale IPage I

BENDING, SHEARING AND TENSION

A beam which is subject to a combination of bending moment,shearing and tension force, wi l l carry out certain deformations ,deforma tion which can be divided in thre e components .

I

/ A BENDING- -------I

0 SHEARING- --------



C

______-__LONGATIONJCONTRACTION___________

SUBJECT

BENDING, SHEARING A ND TENSION

3 = p o \ ~ h o N S ~ * O= 0,3 F O R 5%

B e i n g a b l e t o d e t e r m i n e t h e f o r c e s a c t i n g o n a l oa de d be a m is jt h e r e f o r e a ba s ic know l e dge , no t onl y be c a us e a h a t c h c o v e r c a n b e !looked upon as a b e a m , a l s o d u e t o t h e fact t h a t m o s t of t h e d e t a il s jW e de s i gn c a n be analysed by us ing the beam the or^ ,

Date

S ~ g n

HES/ID

1 1t !I II i4 -I i --N.A

1I Ik(l+Eu

Page

2.2No

u-1 : -

E a c h of t h i s t y p e of de f o r m a t i on , will c r e a t e s t r e s s e s . S t r e s s e swhich individually o r t o g e t h e r will d e t e m i n e t h e requi red sec t ion . ?/-

C---. --GI!S t r e s s e s a r e a lways propor t iona l a ga i n s t t h e d e f o r m a t i o n . -

E = YOUMGS H O b u C

C = fhFFtA hoa"L

2 iIIIIj

E I

I I

- I I- 2(1+ J ) i I



( BASIC BEAM THEORY----------------I

SUBJECT

BENDING, SHEAR ING AND TENSION

a

S p l i t t h e S e a m at d i s t a n c e XoI

'

For equi l ib r ium, s o m e f o r c e s a n d m o m e n t s m u s t e x i s t o n t h e c u ts u r f a c e .

f!

Dale

S~g n

HES/ID

If we t h e n r e rn a v e a s m a l l cube of the beam , a? t h e d i s t a n c e Xo, t h ef o l l o w i n g s t r e s s e s w il l o c c u r .

Fage

2.3

No



SUBJECT 1Date 1 Page

Th e fo l lowing re la t ionships ex i s t :

BENDING, SHEAR ING AND TENSION

= T"a"

F o r i n s t a n c e , t h a t m e a n s if w e r e p l a c e t h e c u b e b y a p a r t of a

beam's webb, i t is e a si ly s e e n t h a t ;

I 2.4 1

IIn order to so lve a b e a m p r ob le m t h e ap p l ie d f o r c e m u s t b e js e p a r a t e d i n a c c o r da nc e w i th t h e f ol low ing :

S~gn .

HES/ID

TENSION =

BENDING =

No

S H E A R =

TORSION =

A c t i ng t h r ough C.O.G.

Along (pr inc ipa l ) ax es round n eut ra l a x i s (N.A.)

T h ro u gh s h e a r c e n t r e (S.C.)

A r ound t o rs i on c e n t r e



If the bendingaround C.O.G.,expressed by:

SUBJECT


BENDING AN D TENSION-----------------takes place around principal axes and the tensionresult ing stresses , at point 1, in X-direction ca n be

I expressed by:i

Date

S~gn

HES/ID

A*.erne f ibre of the sect ion can therefore be

Page

2.5

No

BENDING A N D TENSION------- ----------

C = P Sin OL

Q = P Cos oiQ i

=Q Sin 6Q 2 = Q Cos p



.

SUBJECT


T22 -&+ y /

t Y +'1

I

.eI T

J:II 'b el I i

Date

S ~ g n

HES/ID

II

I jTs I

Ii

0 I 1II

I; My caused by Q2 s + I

jI Mz caused by 0,s +It

I Tz caused by Q 2

!2 hT F o l ~ '

Ty ailsed by Ql a Xo '6rzf f

Page

2.6

No



ASSY MMETRICAL BENDING.....................SUBJECT

BENDING, SHEARING A N D TENSION%

Assymetrical bending is when the bending moment is act ing aroundoth er ax es than principal.

I Exam ple, showing principal ax es for differ ent sectio ns.

Date

S~gn

HES/ID

IC a s e (D) s of special interest as a bending mo ment M y will c r e a t e is t res s es and deformation as s e e n below.

Page

2.7

No

The s ect ion is supposed to move free in y-direction.



I SHEARING--------

,-

I If a s h e a r f o r c e s h ou ld c r e a t e s h e a r s t r e s s e s o nl y, i t m u s t act

t h r ou g h t h e s h e a r c e n t r e (S.C.).---------

SUBJECT


If i s i s a c t i o n s o m e w h e r e e l s e i t will c a u s e a co m b i n a t i o n o f s h ea r i n gan d to r s ion .

S.C. f o r d i f f e r e n t sec t ions .

Dale

Sign

HES/ID

IB u t e v e n if s h e a r f o r c e i s a c t i n g t h r o u g h t h e S.C., s h e a r s t r e s s e s a r en o t u n i fo r m a lo n g t h e s e c t i o n ( a s s e e n i n T a b l e I) .

Page

2.8

NO

C o n n e c t e d t o t h i s i s o f t e n m e n t i o n e d " s he a r a r e a " m a t h e m a t i c a l l y'------- ,

i t i s d e f i n e d as: ,

o r in words:

Iy x b--------S~

m o m e n t of i n e r t i a

wid th of s e c t i o n a t C.O.G. I

;

S t a t i c m o m e n t of t h e p a r t a b o v e o r be low C.O.G. !(wi th r e s p e c t t o C.O.G.)

A n a r ea , i f d iv id ed by t h e s h ea r f o r ce , g i v e s a s h ea r s t r e s s of t h es a m e m a g n i tu d e a s a c t u a l l y o c c u rs .



I EFFECTIVE STRESS--------------

SUBJECT


I n t he no r m a l de s i gn it is a l w a y s w i s e t o i n c l u d e in t h e d es ig n s a f e t ya g a i n s t p l a s ti c d e f o r m a t i o n , t h e r e f o r e , t h e q u e st io n of t h e t o t a ls t r e s s s i t ua t i on is a l w a ys i m por t a n t .

As long as t h e s t r es s e s a r e a c t i n g i n o n ly o n e d i re c ti o n i t is e a sy t oc a l c u la t e t h e t o t a l s t r e s s e s (o r e f f e c t i v e s tr es se s) .------

Date

S ~ g n

HESfID

B ut as soon as t h e s t r e s s e s o c c u r i n m a n y d i r e c t io n s it is m o r ed i f f ic u l t . T he m os t c om m onl y u se d t he o r y f o r t h i s s i t ua t i o n is

c a l l e d VON MISES.--------

Page

2.9

NO

I F o r a s it ua ti on w ith t w o d i r ec ti on a l s t r e ss e s ( r a n d y ) it reads :



T h e a t t ach ed p ag es a r e v e r y u s ef ul f o r d e t e rm in in g f o r ces , m o m en t s

and def lec t ions fo r d i f f e ren t ty pes of beams .

PageSUBJECT

FORMULAS FOR BEAMS

In o r d e r t o t ak e f u ll ad v an t ag e of t h e i r p o t en t i a l i t i s im p o r t an t t or e m e m b e r :

Date

I a. Loads can a lway s b e ad d ed t o each other .

~ ~ g n

HES/ID

b. The ro ta t ion is 0 at a c lam ped suppor t.

NO

I c.For a beam which exte nd s beyond a s imply suppor ted point , the

I r o t a t io n i s eq u a l on b o th s i d es of t h e s up p or t.I

/c. For small angles: t a n d = o(

Units used in th e tabIe:

PI.. n = fo rce

= force(1engt h

' Ini = radiansi



2.5!~?~y !j=:OPTED FI T BOTH ENDSC

-



a s *

5 LO



SUBJECT 1Date 1PageTABLE /

DISTRIBUTION OF SHEAR STRESSES S ~ g n I NO 1

T = Shear Force

1

e = Sc :51 (2 )

f - br, + hr,) h/ (at, + 2/11,+b f l )

T* = b(h - f ) 1,!2I, rh

7 ,- br6+ th),/24, I,,

lZ

J+7

]E-,c;,;

I- -- - -

I t ! s )' = r



I U D L M = constant x q x ~2 Point Load M = constant x P x L

LPPNo.

SUBJECT rBENDING MOMENT A N D REACTION FORCES FOR ACONTINUOUS BEAM U'ITH EQUAL LENGTH AND 1

SECTION MODULES I N A L L SPAN I

I R = constant x q x L R = constapxx P

73m-w~ Co*-uG?& I R E R L ~ I O N FORLEIHDt7-8 \ he-k

I I M a Mac M a MC MD \ A RC RD RE

1 2 5 P , 4 ~ I

Date

Sign.

bEc




A hatchcover beam looks l ike the sketch, exposed to a UDL OF 4,O

TIM*, L = l o r n

,

What s i z e o f weld i s required b e t w e e n topplate and web, allowableshear stress is 50 ~ / r n r n ~

SUBJECT

goo. tob1

! SOLUTIONj --------

Date

S~gn

HESIID

Page

2.10

NO.



I EXAMPLE 2---------

SUBJECT


I Find maximum bending mo men t and shear forc e for th i s beam.

/SOLUTION

~ R ~ L F--------i Table for continuous beam, 2 span

Date

S ~ g n

HES/ID

Page

2.1 1No



EXAMPLE 3------SUBJECT

BENDING, SHEARING A N D TENSION

What is the minimum ?.+ we can allow on this ramp cover, i f wewould like to keep the def l ec t io n t o m a x im um 100 mm duringhoisting.

1S ' g n ~ ~ ~ / ~ ~

Date Page

2.12 iNO




SUBJECT

I EXAMPLE 4---------

Date

I I

What wil l ma xi m um re be at 14 span, on the hatchcover beam inEx. 1.

SOLUTION--------

Page

2.13 iS~gn

HES/IDNO. 1I



I EXAMPLE 5---------SUBJECT

BENDING, SHEARING A N D TENSION

C u t o u t is n e e d e d f o r a l in k m e ch an i s m i n t h e h a t ch co v e r b ea m inEx. 1. C al cu l a t e r eq u i red web th i ck n es s . Th e c u t o u t is 250 mrn

high and 300 mm long.

NOTE-hat a b o u t s t r e s s c o n c e n t r a t i o n a n d s eco n d a ry m o m en t .

Date

S~gn ES/ID

S OL UT IO N: ( T h e c u t o u t i s a s su m e d t o b e c l o s e t o t h e s u p po r t)--------

Page2.14

NO

l

a. Due t o d ec rea s ed a r e a

I

b. S t r e s s c o n c e n t r a t i o n f a c t o r :

Ex t r ap o l a t i o n f r o m a t t a c h e d g r a p h , g i ve s:

c. Secondary m o m e n t

,F T I

. - f



EXAMPLE 5 (continued)---------- .& = g ~ 3p .

r

- 6 3 6 E 6 =f f@ r f E $+ 2 1 , 2 3 F 6

/ f' f

1 5 2 6 6 cmL = 2 7 nm

I

SUBJECT


Date

S~gn

HES/ID

page

2.15 1NO 1

chapter 2 bending shearing tension.pdf

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