Beam Bending Formula

7/29/2019 Beam Bending Formula

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Explanatory o t e s

Ge n e r a l

Al l w e i g h t s and measures shown o n i n v o i c e i l l b e g o v e r n e d b y s t a n d a r d s o f t h e

r e s p e c t i v e s p e c i f i c a t i o n s s o o f f . e r e d

Carehas been t a k e n t o e n s u r e t h a t a l l d a t a and i n f o r m a t i o n h e r e i n i s f a c t u a l andt h a t n u m e r i c a l v a l u e s a r e a c c u r a t e . T o t h e b e s t o f our k n o w l e d g e , a l l i n f o r m a t i o n

c o n t a i n e d I n t h i s handbook I s a c c u r a t e a t t h e t i m e o f p u b l i c a t i o n . C o n t i n e n t a l

Hardware P t e L t d assume no r e s p o n s i b i l i t y f o r e r r o r s i n o r m i s i n t e r p r e t a t i o n ft h e I n f o r m a t i o n c o n t a i n e d i n t h i s handbook o r i n I t s u s e .

Ma t e r i a l

S e c t i o n s

The s t r u c t u r a l components r e f e r r e d t o i n t h i s handbook a r e o f s t e e l t oB S 4 3 6 0 , ' W e l d a b l e s t r u c t u r a l s t e e l s ' a n d / o r i t s r e l a t e d e q u i v a l e n t s .

The u n i v e r s - 4 1 bOMikrid. columns and t e e s c u t t h e r e f r o m , t h e j o i s t s ,

. X

c h a n n e l s , d g i v e a r i n g p i l e s and r o l l e d t e e s a r e g e n e r a l l y a s l i s t e d i nU

B S 4 : P a r t V u n i v e r s a l b e a r i n g p i l e s and r o l l e d t e e sP ! 9 8 0 t s , Ui n r e g u l a r d u B c a r e a d .

NA . i f i e u n i v e

. 5 2 ' t a p e - ed f l "

C o l u m , q c o r e - is e r i a l s i z e 3 $ 6 m - . * .

J I -f,

i s s e c t i o n , i s t e d s e p a r a t e l y , i s r o l l e d i n t h e0 -M a p p r o p r i a t e o n e t o be used a s6m ' . I t i.

F t s.An h i o n r s e d i o n t h a n t h o s e r o l l e d i s

Dimensional n i t s

The d i m e n s i o a - f i f U t - t i o n i ' . . ' a r e g i

c a l c u l a t e d p r o p e r t i e s ( c e n t . 6 i d a l

L i p p , moments o f i n e r t i a , eW,

c e n f i n i d t t L ý ( q c b i ) . - ' U i i t w , A S d r f a i e e r e

ra n c e s on d i m e n s i o n s a n d

Other u n i t s

r e s (mm) a n d t h e

e c t i o n a l a r e a s , r a d i i o f

i d p l a s t i c m o d u l i ) a r e g i v e n i n

i n square metres M2) . F o r

ference should be made o

T h e - u n i t s o f forcemass and a c c e l e r a t i o n a r e t h o s e o f t h e S y s t e m e

I n t e r n a t i o n a l ( S I ) . They a r e t h e Newton ( N ) , t h e kilogramme k g ) a n d t h e

m e t r e p e r s e c o n d p e r second m / s 2 ) s o t h a t 1 N - 1kgx I m A 2 . Thea c c e l e r a t i o n due t o g r a v i t y v a r i e s s l i g h t l y f r o m p l a c e t o p l a c e and f o r

c o n v e n i e n c e a ' s t a n d a r e v a l u e o f 9 . 8 0 6 6 5 m/s2has b e c o m e e n e r a l l y

a c c e p t e d i n s t r u c t u r a l e n g i n e e r i n g . With t h i s c o n v e n t i o n , t h e f o r c ee x e r t e d by a m a s s o f l k g u n d e r t h e a c t i o n o f g r a v i t y i s t h e ' t e c h n i c a l u n i t '

o f 9 . 8 0 6 6 5 N . I n t h e same way 9 . 8 0 6 6 5 k i l o n e w t o n s ( k N ) i s t h e f o r c e

e x e r t e d by m a s s o f 1 t o n n e ( 1 0 0 0 k g ) under r a v i t y , and 1 k N t h e f o r c e

f r o m a mass o f 0 . 1 0 2 t o n n e .



EXPLANATORY NarES

Comparable s p e c i f i c a t i o n s and c h e m i c a l a n a l y s e s

S p e c i f i c a t i o n s

S t m c t u r a l s e c t i o n s

Qu a l i t y G r a d e

B S4360 43A 62300184000

SO A 72500193000

T h e f o l l o w i n g s p e c i f i c a t i o n s a r e n o r m a l l y r e a d d y - i v a

'i l'a b l e , b u t a r ) y o f f

w i l l depend upon a c c e p t a n c e o f f u l l s p e c i f i c i t i b " i " - , ; d !.

f l s a n d / o r a n y_ e t ao t h e r s p e c i f i c a t i o n s n o t l i s t e d b e l o w .

Te n s i l e s t r e n g t h

l b r i n 2 N 1 m m 2

Di n 1 7 100 R S t . 3 7 . 2 5

KWMM2

M i n Y i e l d s t r e s s

I b l i n z N / m M 2

430/580 44/59 38 500 265 27

490/640 5 0 1 6 5 50000 345 3 6

- , . . 4 0 0 / 5 5 0 4 1 1 5 6

450 m i n 46 min

36000 250 2 5

50000 345 36

.WMM2.

2 327

36

3601510 3 6 1 5 2 32 500 225430/580 44159 38 500 265t 4 4 . 2 6 2

S t 5 2 . 3 7

j K 0 , 2 1 1 0 1 1 S S 4 1

S S 50

J I S G 3106 SM 1

SM 0

50000 345

4021510 ' S

490/608

U . - - . 4 0 2F , /510

No t e s : S e c t i o n s v e r % n c h (16mm) u p 6 ' , a n d

A l l f i g u r e s a r e a p p r o x i m a t e and

O t h e r s p e c i f i c a t i o n s a r e o b t a i

'Ve g . UN I 7070-72" 1 '

F A

4 9 0 7 6 o 8

34000 235 24

40000 275 28

34000 235 24

45000 31 4 3 2

0 0 3 / 0 . 1 0 0 . 0 0 3 / 0 . 1 0 - -

) I S 3101 ( 1 9 7 6 ) ' S S 4 i

S S so

) I S 3106 ( 1 9 7 6 ) SM 41 A 0 . 2 5

SM 41 B 0 . 2 2

SM 41 C 0 . 1 8

- 2.5xC in

035 0.60-1.20

0.35 1 . 4 0

0 . 0 4 0 0 . 0 4 0

0. 0 4 0 0.040

0.040 0.040

- C u 0 . 2 0 ( m

- Cu 0 . 2 0 (

SM 0A 0 . 2 2 0.55 1 . 5 0 0.040 0.040 -

SM 08 0 . 2 0 0.55 1 . 5 0 0.040 0.040 -

SM 5 0C 0 . 1 8 0.55 1 . 5 0 0.040 0.040 -



EXPLANATORY NOTFS

N"

l

+ V ý ' , I ' ý ! ' '

De s i g n o f b e a m s

De s i g n f o r m u l a e f o r beams

0

i ý. ,

N o t a t i o n ý ' : , , ý ,

L , ý . . ý ' ' " l e n g t h of s p a n ýi n m i l l i m e t r e s . : ý h : ý ' ý : < ý ; ý , ý , . . . , . -

" W ý ý ý = t o t a l d i s t r i b u t e d o r p o i n t l o a d i n Ne w t ¢ h s : a t ý ' : ý ý .- ýW a 2 - p o i n t l o a d i n Newtons. ý ý ý ; '

y .

, . :

; , , ý ; ý . ý. , . . . ý:.,.ý'ý`s hýýaYim__um ending m o m e n t n Newton m i l l i m e t r e s .

E - r e s u l t a n t o f p o i n t l o a d s R Newtons. ý ý , '

ýs . R ý , e t c . - r e a c t i o n a t A , B , o r C t c , i n Newtons. ý ,F - shearing o r c e i n Newtons. .

m - a p p l i e d m o m e n t n Newton m i l l i m e t r e s .

Mx ý ý ý ý - bending moment n N e w t o n m i l l i m e t r e s

a t d i s t a n c e X f r o m t h e l e f t hand s u p p o r t A . ý '

a x _ - d e f l e c t i o n i n m i l l i m e t r e s .

._ . , ý x f'"ýPe m aaýans.

d e r t h e l o a d i n Newton m i l l i m e t r e s .ý . . . ^m i l l i m e t r e s .

i o n i n m i l l i m e t r e s : :

e f l e c t i o n i n m i l l i m e t r e s .

a n s :. 1 x 0 s N l m m z. . .

., ý ý

e r t i a ' o f uniform s e c t i o n b e a m i n m m ý. .

.

r d , . t e f t hand su p p a ti ' i cu i i r i of x t e r n a l l o a d s ý ý

. ,

r d .

d i a g r a m s ) when c a u s i n g

tans are i v e n , but the s i g n s d e p e n d

wh i c h s e c t i o n i s b e i n g c o n s i d e r e d ,

R a i n e d , b Y in s p e c t i o n .

. ,

. W h e r e space p e r m i t s , g e n e r a l equations o r Mx n d x .

. ' ý a t a ny o i n t o f t h e b e a m , , and a l s o t h e e q u a t i o n t o t h e

e l a s t i c l i n e (ý, hav e been i n c l u d e d . . ý .

. , ý ý ý Va l u a o r . S l o p e . T h e s e m a y be u s e d I n e v a l u a t i n g h e

. ý . , . . . . ý . . angle o f ý r o t a t i o n f o r , r ubber i e a r l n g s a n d ý s i m i t a r ý . ,

' ' ý " c o n s t r u c t i o n a l e l e m e n t s . '

'S

.

ý - - ..' u or,.... ný ..." . ý " . ý .



F ý(PLANAT O R Y NOTES

ý Mmr

_ _ c

a t X rom A 2X3L+ L 3 )

r

- W(4X3-6X2L+L3)

L

j ý N ( I - r)

Span - L

T o t a l uniform l o a d - W . .

- Wý- R B2

Mma x -8L

a t m i d - s p a n- 5 WL3

A m a x38 4 EI

i ý - i s - WLý2 4 E 1

{ S

' r :

Mx -2L

(L-X)

ý ( -2ý(

X3-

2 4 E I Lx

. w : '

' ý Y . " :

r -------ý----?--1

Simply supported b e a m ' U n i f o r m l o a d o n f u l l span

R ^L.

ý L i

r --------------ý

Aii i i i i a i i i a i a i i i i a i a i i i i i i a i i i i i

ý ---- i' x

u . iý, ý ý ý ý ý ý f ý ý ý ý' ý il l l

r ( a + 0 . 5 r b )

. W 6 E I ý(Lx-az-Lb(I-r)l

ý ,

!i i i I l l u u

.f

ý ,ý ý

q ý ct _

L_i

' i

_ iR e

_ ý_

ý ý I I I I Il l ý i ý MýMx

, - ý(3L3-4Xz)

' ý ý a t X rom A '. , between A 8 x -ý Z(16X4-40X3Lz+25Lý)

a nd c e n t r e ,

i x

ý ý

i

e

_ipe

---- ;-;-

Span ' ; - L

To t a l l o a d ý - W

ý-ý ý 3W

2.

M m a x -6L

a t m i d - s p a ný

a -m a x

W

ýA a

-5 L 2i ý - i9ýý

- 9 -ýZ ( 1 6 X ý-24XzL 2 + 5 L 4)



Simply supported beam P o i n t l o a d a t mid-span

L___w LSpan = L

T Po i n t l o a d = W

RA M.

4R ,

at mid-span

6maxI WLI

1 48 El

-------- RA= R B 2 2 -1 2

'A= 8

M,

a t X fr o m

A

b e t w e e n A 6 X

a n d c e n t r e i

ix

Simply supported beam P o i n t l o a d a t any p o s i t i o n

RA CL _ _

A

a W bSpan -

Poi n t oad - W

RA-Wb Ra - Wa

L L

. 1

-IRS- J

MmaxWab

a t CL

u n d e r l o a d1 6 C

Wa2b2

WL2

1 6 E I

WX2

WX 3 1 _ 2 - 4X2)4 8 E I

W L2 - 4 X ' )1 6 E I

3 E I L

W b ( L + b ) ; 1 . - W8b(L+a)6 E I L 6 E I L . ,

Wh e n a > b W a b ( L + b ) j i ( - - L - + b )6 m a x 2 7 E I Li s a t

X from AI

X3

Simply supported b e a m ' T"equal symmetrical p o i n t l o a d s

r "

a W b W

A

M AL

R ,

I - - fI MM&

_ .A

Span - L

T w o p o i n t l o a d s , each - W

R A- R B = W

M, , , , , o v e r l e n g t h b = wa

b m a x a t mid-span . Wa 3 1 - 1 - 4 a l )2 4 E I

B under e i t h e r l o a d = Wa(3L-4a)6 E I

iA-iB - W a ( L - a )2 E I

I f a-b- , 6 - 2 3 WLI3 m a x 648 E l



EXPLANATORY NOTES

A '

Beam i x e d a t both ends Un i f o r m l o a d on f u l l span

IB

N

S pan = L

T o t a l uniform l o a d = W

Wi 1 . 1 1 2

B e a m f i x e d a t both ends P o i n t l o a d a t mid-span

M , W L I - 6LX , 6 V)1 2 L

a t XWX2(L-W

f r o m A 2 4 E I L

WX V-3LX+2V)1 2 E I L

a t 0 . 2 1 1 L from e i t h e r endM -

ME=0

2 _ Span L

\ 1I-

cR .-R W

A BI I P o i n t l o a d

.- W

RAf

X_

MA

11

iR,

r

MA=MI_WL

1 2

i m - - W L

a t mid-span ý '2 4

MA=MI3

3

6maxVVL

3 8 4 E I

2

WL

8

Mc W

a t . m i d - s p a n8

WL 3

1 9 2 E I

M X W 4 X - Q.a t X from A

between 6 x WX(3L-4X)

A and C4 8 E I

l i x ".WX L - 2 X )8 E 1

a t 0.25L from e i t h e r end MD=ME-0

B e a m f i x e d a t b o t h ends P o i n t l o a d a t a n y p o s i t i o n

a W b C p a n

A .98 P o i n t l o a d WC P

W b 2 ( L + 2 a ) =IW2 ( L + 2 b )

-1 RA-

L 3R B

U

I - _ _ _ _ ___ - - - - -

RAf

: x; R ,

MA. _Wab2 M.- - Wa b

L2 L2

a t C , u n d e r l o a d , M c =

a t X from A

between

A and C

2Wa2b2

L3

Wab2+Wb2(L+ 2 a ) X

L2

L3

Wb 2 X 2 f 3 L a - ( L + 2 a ) X ]6 E I L I

Wb2X[2La- (L+ a ) X ]

2 E I L I

2Wa3b2

3 E I ( L + 2a)2

when a>bihe maximum e f l e c t i o n i s a t X = 2LaL + 2 a



NOTES

C a n t i l e v e r Uniform l o a d S p e c i a l c a s e : Uniform

on p a r t o f span l o a d on f u l l span

w

A PM= D

' I a 8 ý b i CL

S pan = L Span L b

a C 0

Uniform l o a d - W Uniform l o a d W

RA = w R Aw

MA = -W ( a +j2 MAB A L L

2 ) 2

i C = D -- ! A - L ( 3 a ' + 3ab + b 2 ) i , W L I6 E I 6 E I

6 DWL3

8 E I

W a l + 1 8 a 2 b + 1 2 a b 2 + b - I + 4 c ( 3 a 2 + 3ab + b2) )

2 4 E I

C a n t i l e v e r T r i a n g u l a r l o a d S p e c i a l c a s e : T r i a n g u l a r

on p a r t o f span l o a d on f u l l span

A

RA

r. - .

MA i

i - . - .

fl m

pl p ' ý "

S pan L Span L a

b 0

T r i a n g u l a r l o a d -W T r i a n g u l a r l o a d W

RAI

= w RA w

MA- wa

M _WL

3 3

6 CWe L +!2 6c

WL3

1 5 E I ( 4 ) 1 5 0

is - i c - waz i cWL 2

: 1 2 E I . 1 2 E I

C a n t i l e v e r P o i n t l o a d - - . . , S p e c i a l c a s e - P o i n t l o a d

a t any p o s t on a t f r e e e n d

w Span = L Span L

A Cb

a b I i Po i n t l o a d - w P o i n t l o a d

L . . . . . . . . . .L i RA - w R A

M, - ýWa M

a

0

ww

-WL

b c . W z L+ ý s c - W L I3 E I ( 2 ) 3 E I

i s - i c -W a 2

i c _WL2

2 E I 2 E I



EXPLANATORY N C 7 T E S

r-R A i

L_

z . _

Propped c a n t i l e v e r U n i t o r m lo a d o n f u l l span

r -------------7ý x

D

5 L

M .

- --+ _ _ý. ý . , ý ý ý . ý ý Mo

e

1Re

i ý ý ý

ý ý <_ ý

ý _ _ . e_._ _ . _ . y

S pan = L

T o t a l uniform l o a d = W

R , , = 81N

R a= 8W

a t A Mmax = -WL8

a t 8L f r o m A Mo =ý28WL

a t 0 . 5 7 8 5 L f r o m A S ma x =WL;

a t B

a t x 2 X z )

trnm A

Span a L

P o i n t l o a d = W

at4L f r o m A , Mc = 0

Pr o p p e d a n t i l e v e r P o i n t t o a d a t mid-span

w? - - ý - ý - H - - - ý - - _ ý

Propped c a n t i l e v e r : . - - ý . = P o i n t l o a d a t a n y p o s i t i o n

, ý ý

R ý® ý6W

i 8

1 8 5 E I

= W L 24 8 E I

Mx =SL ( L 2- 5 L X +4 X 2 )

S X8 E IL (3L 2-5 L X +

4 8 E I L

at A Mmax = - ' '-W

a t m i d - s a n ' c 'p ' ) ý ý . . : ý 3 2

under l o a d

a t 0.5528L from A , ' & ý

i x = WX (6L2-15LX+8X2)

1 6

5WL

WL31 0 7 E 1

a t B i B . , = WLz3 2 E I

a t ýt

L from A , M , - . m 0

r ' - ýR^ - . _ ,

M ý t . _ . .

W b

AL C

?B

ý ! - - ý - - ý - - - ý - - - - - - - ý - . a

ý . _ .SR e

i i

S pan - L


R , b ( 3L 2- b 2 ) R e Wa2(2L+b)

2L3e

2 L ;

Mn_ _ Wab(L+ 1 M W a 2 b ( 2 L + )2L2 c i 2 L 3

i 8 -Wa264 E I L

Absolute max e f l e c t i o n

i s under the l o a d

w h e n a -b ,(2-0.5858L

When a > ý 2

ma x d e f l e s i i o n i s between

A and C

When a < b ý 2m a x d e f l e c t i o n i s between

C a n d B

W L 3S m a x m a x

1 0 2 E 1

S max-Wa3b,( L + b ) 3

3 E 1 ( 3 L z - b 2 ) z

S m a x =Wa 2 b ý b

6 E I 'VZ L + b

a t n = a L 3 L z b 2 from A , M =0



P r o p p e d cant

a

ý a ýý ý ý ý

^ ý C f f I F i r m r m y -

t 7aGnýn

_ c

Propped c antilever

>Wfoment a p p C i e d a t a n y

S p a n = L A p p l i e d r r i o m c n t =

_ ý_Ma L

2 L Z h` miý = r n a ( 2b- a )

R,,=- C Z a

-* , c Uniform l o a dl B ý a

- -----ý-- ---ýRA=-3ýa

M = " " `a

- 3(Lz-f x ) , -n=_m+MA

-4.4 2 2 6 5 m

0. 5 7 73 5 m

3mab

L '

L

r r ý

2 L'ý L

a s f o r

Uniform C o a d on l e n g t h b e y o n d prop

f l e c t i o r r a t C

g a t i v e d e f i

max

_ - - Wa2

Wa x yB E I

-ý . ý a n e g = -

Wx a

S4Ei

S l o p e a t C

a t X=

P r o p p e d eantilever

L ý ® a ý- ------

-- - --rS '

u n i i i u i i i i n u i n u i i m u r r

Po i n t t o a d a t f r e e ° e

Span = L F u l l l e n g t h = 5


Rn =-32R ý

= ýL ý5 + 2ý

MA =ýa

M a = - Wa

D e f l e c t i o n a t C

M a x . negative deflection

a t X=ýL

S l o p e a t

= i F u l l l e n g t h = S

b n e h

W z

4E 1

WLza

27E1

= 4E 1 (S + a )

a t X=3

from A , M =



EXPLANATORY N C ) T E S

S imply supported b e a m S i n g l e concentrated m o v i n g l o a d

x

22mmmm.,

M aximurn P o s i t i v e S h e a r a t any s e c t i o n occurs w h e n t h e

l o a d i s immediately o t h e r i g h t o f t h e s e c t i o n . S i m i l a r l y ,

Maximum N e g a t i v e Shear occurs w h e n t h e l o a d i s t o t h e

l e f t . F o r a s e c t i o n d i s t a n c e X from A :

Maximum B e n d i n g Moment t a n y s e c t i o n occurs when he

load i s over the s e c t i o n . For a section distance X 6m :

I L _ _ _ X ýL

T h e A b s o l u t e Maximum B ending

Moment nd D e f l e c t i o n occur

under t h e l o a d a t m i d - s p a n ,

Maximum end s l o p e a t A ccurs

w i t h t h e l o a d a t X = 0A2 2 6 5 L

f r o m A .

( L - b i ) 2

' ý7

S imply supported b e a m T w o concentrated m o v i n g l o a d s

L 4 _ L ; ýD

L

L -b 2 2

4 L

- t

A s s u m e : W, >

0,0 64 1 5 N - L 'E l

W2= n r ,

F i x e d D i s t a n c e b=mL

b , = __ýý_2b n m b , = r n - n m ) L

Maximum R e a c t i o n a t A and Absolute Maximum P o s i t i v e

Shear occur w h e n W , i s immediately o t h e r i g h t o f A :

P . - D-,;f;,,- f 7 -%A/ , %A / L-Uý -dx __ - -- - - . - I

F o r a s e c t i o n d i s t a n c e X from A :

X< - b

Po s

L

N o t e : 1 . Fo r R B m , , , interchange values o f W, a n d W2 i n th e

f o r m u l a f o r R A m , ,

2 . F o r Negative h e a r , i n t e r c h a n g e W , and W i n

formulae f o r P o s i t i v e S h e a r , measuring X f r o m

towards A .

3 . i f m>-E- a l c u l a t e R 8 m , x and N e g a t i v eI - n

Shear v a l u e s f o r W , o n l y a s s i n g l e l o a d .

Absolute Maximum Bending Moment ccurs under

w h e n t h a t l o a d and t h e r e s u l t a n t f b o t h l o a d s a r e

e q u i d i s t a n t from mid-span ( s e e l o a d i n g d i a g r a m ) :

M m a x m a x ý ( L - b r ,

4 L

B

I f m , th e Maximum B e n d i n g Moment t a n y

section occurs u n d e r one o f th e l o a d s . Fo r a s e c t i o n

distance X from A :

X < ( I - ) W n ) <

a x u n d e r W , = ( L - b , - X ) X u nder W, X±2) L - X )

L



I 1 ,,

1 2

A

-i-i- I iI

I L-bo? , -2- 7 , M 1 1 i O * i .

Simply supported beam T w o concentrated m o v i n g l o a d s ( c o n t i n u e d )

L

f -- _ L _ - - - -

w' 2

A

IL i i -mi

W ,

-

- - I -

M, , I

n

I f m > , t h e Maximum Bending Moment t any s e c t i o n

always occurs under W , ( t h e h e a v i e r l o a d ) , whether W i s

o n o r o f f the s p a n .

F o r a s e c t i o n d i s t a n c e X from A :

X < ( I - m )

1 ý 4m a x

- (L- , - X)X

L (I- m ) < , X

Mrr.x ( L- X ) XW,

I f n <m < t h e Absolute Maximum Bending

n

Moment ccurs under W , wi t h W o n t h e s p a n .

I f n <m > n t h e Absolute Maximum Bending

n

Moment ccurs under W , a t mid-span w i t h W o f f t h e s p a n .

N o t e : When the two l o a d s a r e e q u a l (W, = W2 n d n = z )

t h e c r i t i c a l v a l u e o f Fn . 0 . 5 8 5 8 .

n

Simply supported b e a m s c a r r y i n g s e v e r a l m o v i n g concentrated l o a d s

T h e Maximum R e a c t i o n a n d t h e Maximum T h e Maximum Bending Moment d u e oShear d u e t o s e v e r a l m o v i n g concentrated s e v e r a l m o v i n g concentrated l o a d s occursl o a d s occur t o n e support w i t h o n e o f the under o n e o f t h e l o a d s when h a t l o a d andl o a d s a t t h a t s u p p o r t . T h e l o c a t i o n producing t h e g r a v i t y centre f a l l l o a d s are e q u i d i s t a n tthe Absolute Maximum must be f o u n d b y from mid-span. T h e Absolute Maximum mustt r i a l . be determined b y t r i a l .



EXPLANATORY NOTES

St r u c t u r e p r o p e r t i e s

F ormulas f o r g e o m e t r i c a l p r o p e r t i e s o f s e c t i o n s

Cr o s s - s e c t i o n a r e a . 4 = f A dA

F i r s t moment Sý=fAydA, S s,=fAxd.4

i = ýv - S .P o s i t i o n o f c e n t r o i d Y=-

A A

ý i b _ ý

XMo m e n t o f i n e r t i a

Product o f i n e r t i a

J,=fAy'dA, J, = f Ax'd .4

J . , , = A x y dA

Po l a r moment f i n e r t i a J p = f A r dA= , + I ,

Radius o f g y r a t i o n

P o l a r radius f g y r a t i o n i p=rip

AT

y

y4 4

S e c t i o n modulus Z. 1 Z2

. y l

Tr a n s i t i o n o f a x i s&=S.-Ad, S,=Aý

J. =J. +Aj, 2j.,,=jXy+jýý

Sx ' ýSzcosa-Sysina

R o t a t i o n o f a x i s J.'= . s in2a+ JVCOSI a+ J . y s i n

J . , + J" = . + , = p

Y

Y 2

Centroid Axis

y

e : ý i v X



o m e t r i c a l p r o p e r t i e s o f s e c t i o n s

Section Centroid a x i s

from e d g e : y

cm

r"-b-f. .

r t e - .

'ý

" S t .

/I i ,/ / i

. ý . ¢ ý . , ; . . ý ; ,.

Y =

' L ý nlw-?

ý'-n-ý"ý ý4 , .

ý - - - 6

: : ,

: ý : : ,-

r e0,924R.'

r ,

_

2

h

c m '

_h '

12

_ h '1 2

ý ' ' H. : ý ýi 12(M-h')

z . : ý ' s a

Mo m e n t o f i n e r t i a : ) S e c t i o n m o d u l u s : Z

r ' . : ý ; .

- . "ý 0.82 R

. . . y; =2 h ' ý ý. 3; .

Yý3

Y -2 a+b ( . ý ý " ý ý ý . : , a = + 4ab+6= ý

3 + b ' ' . h ý ý . :'-ý":ý 36 a+b)h

. . , . . .

T

R =2 4 R'- r ' )

R ` - ý

2 (Rý-ra) sing 8( B + s i n B ý c o a B ) .

3 ( Rý- r = ) ý B_2(R'-ý)= s i n = B .

9 ( R = - r s I ý B

0 . 9 9 03 l

' ' 12

. ' ý ý ' .

' ý " 0.6381 R ', ý ' ý + i ? :

. , . .

. ' S ý

c m;

ý

6

JZha

12

6 F!ý-hý)

aDa

3 2

0 . 6 ' 90 6 R °

Z ý=

T

Z 12

__a=+4ab+b= _Z i

12 2 a + b )h

Bf(s - h a

BN

A Rý-ý

4 R

Around Upper S e c t i o n

Around_ _

J

ýM ý o a

ZR-Y

Aound _ J

ýýZ

Y-r'cosBS e c t i o n

Aroundý

.0.6364 t " Upper 0.8303 t ý

S e c t i o n



EXPLANATORY NOTES

Di s t r i b u t i o n d i a g r a m o f s h e a r s t r e s s i n beam s e c t i o n s

S e c t i o n Shear s t r e s s diagram

I

r . .

Z . . ,

TIMIX =

Tm a x

3 Q - 3 Q

2 b h 2 A

r M l I x = . -q = Q

4 b h 8 A

2 r ' - J

roar 4 Q 4 Q. I-- - .

-=T A ; r r '

3 (BH - b h 2 ) (BH-bh) Qr n l a x = T

2(BH'-bh) (B-6) A



and d e f l e c t i o n f beams

0

1

No . Load c o n d i t i o n Bending m o m e n t R e a c t i o n f o r c e D e f l e c t i o n

and s h e a r s t r e s s

2M

I - ý -

f-

L . . I

Mý M. =M.

R A =

RA= f a _ b

R A=Pbll Ra=Pallf r o m h t so n C W W W .

d c a m . - A 1 1 1 1 - 4 0 1 1

d c MýýF,I "

d... -M , , ( I - f - b )

d ,..- ' V X(g@3+J$S2&+aim

1 2 . 6 1 + 3 6 3 + 1 2 . 4 + 1 2 . 6 . + 4 6 2 , ý

9 PJU De f l e c t i o n o f span

c i 7. c e n t e r i s c a l c u l a t e d

JWCMP&(h-1-2C) R.-R"J i m b y a p p l y i n g f o r m u l a

i Li g i v e n i n N o . 8 t oR A

a >CR .

MD-L - ( A - 1 - 2a -P(

b+2.)

II e ach o a d .

8 - P I

. 4hP. P

.

1 ,

RAF-1

2 3P13- RA=RB=P d D3 " - 6 4 8 E I

I n c a s e o f beam s u b j e c t t o d i s t r i b u t e d l o a d s , l o a d s y m b o l , W, i n d i c a t e s t h e t o t a l l o a d o f t h e beam.' F o r e x a m p l e , I n c a s e o f a beam s u b j e c t t o u n i f o r m l y i s t r i b u t e d l o a d , w , W q u a l s wt where i ss p a n l e n g t h .

i JU d-

A , .-

d-WM bm nk--A d o num f U"41

r n m r i k i c k r r h u -- --



EXPLANATORY NOTES

No. Load condition Bending moment R e a c t i o n f o r c e D e f l e c t i o n

a n d shear s t r e s s

1 1

1 2

P P P

MC=M8=D 16=5piAA AN 4 12

RAP

An

R A =RB2

d-53PI"

1296EI

1 9

20

W W fT -Lz; -> 1

RA RM

N "--JP- T t

R o - R e - I V 1 1 1 ' 7 . 1 1 - 4 0 1 - 4 0 12 WEi

m- \W

RA = R i r-- i

M A R A

-MA=-MB=MC=Lt MA=Jb=f8 2

I f i

d-

I d-

dmm-P1 3

1 92EI

I n c a s e - o f beam s u b j e c t t o d i s t r i b u t e d l o a d s , l o a d s y m b o l , I N , I n d i c a t e s t h e t o t a l l o a d o f t h e b e a m .

f b r e x a m p l e , i n c a s e o f a beam s u b j e c t o u n i k i r m l y i s t r i b u t e d l o a d , w , W q u a l s v 4 where i s

s p a n l e n g t h . & I



N o . Load c o n d i t i o n Bending m o m e n t . . . R e a c t i o n f o r c e D e f l e c t i o n

and s h e a r s t r e s s ' -

I n c a s e o f beam s u b j e c t t o d i s t r i b u t e d l o a d s , l o a d s y m b o l , W, i n d i c a t e s t h e i t o t a l l o a d o f t h e beam.F o r e x a m p l e , i n c a s e o f a beam s u b j e c t t o u n i f o r m l y d i s t r i b u t e d l o a d , w , W q u a l s w i , where i ss p a n . l e n g t h .



EXPLANATORY NOTES

Wo r k i n g s t r e s s e s

I

C ombined t r e s s e s

D e f l e c t i o n c o e f f i c i e n t s

T h e v a l u e s g i v e n i n t h e s a f e l o a d t a b l e s a r e b a s e d on t h e a l l o w a b l e

s t r e s s e s i n BS 4 9 ' a s f o l l o w s :

T h i c k n e s s f m a t e r i a l Grade A l l o w a b l e s t r e s s e s NImM2)o f s t e e l

(mm) B e n d i n g S h e a r ( l ) B e a r i n g

Up t o a n d i n c l u d i n g 40 43 165 100 190

O v e r 40 150 90

Up o a n d i n c l u d i n g 65 so 2 30 140 260

O v e r 65 Y , / 1 . 5 2 ( 2 )

Up o a n d i n c l u d i n g 40 5 5 2 8 0 170 320

O v e r 40 2 6 0 160

N o t e s : 1 ) On n s t i f f e n e d w e b . ( 2 ) Ys- y i e l d ' s t r e s s , t o b e a g r e e d w i t h t h e

m a n u f a c t u r e r and n o t t o exceed 350NIm M 2 T h i s a p p l i e s o n l y t o u n i v e r s a l column

s e c t i o n s and p pound s e c t i o n s n d , i n t h i s handbook a f f e c t s c e r t a i n In i v e r s a l coluvw"Muspitly f o r which a v a l u e o f Ys- 325N/mM 2 h a s been

a s s u m e d .

t

I

we b s h a v e b e e n c a l c u l a t e d i n2 8 a ( i ) o f B S 4 4 9 , n a m e l y W-pctB

t r a t e d l o a d i n N , b u t t a b u l a t e d i n k N .

u t s a s g i v e n i n BS 4 4 9 , Clause 3 0 a .

3 ) / t , i n which

r o o t f i l l e t s

i n g s , i n w h i c h

o f s i m p l y s u p p o r t e d beams

la t eh s h a l l n o t be t a k e n a si m p l y s u p p o r t e d beams

f o r a n i n t e r m e d i a t e b e a r i n g o v e r

u s , u n l e s s t h e web i s s t i f f e n e d .

0 IMMMOMM he d i r e c t i o n b e a r i n g s t r e s s a t t h er o o t o f t h e w e b s l i m i t e d t o t h e V a l u e s g i v e n i n B S 4 4 9 , T a b l e 9 and

i n c l u d e d i n t h e I

The l e n g t h o f w e b r e s i s t i n g b r u s h i n g i s ' d e t e r m i n e d o n h e ,a s s u m p t i o n t h a t t h e l o a d i s d i s p e r s e d t h r o u g h t h e f l a n g e and t h e b e a r i n g

a n d / o r f l a n g e p l a t e a t an a n g l e - o f 3 0 0 ( B S 4 4 9 , C l a u s e 2 7 e )

C l a u s e s 1 4

W e r e b e n d i n g and s h e a r s t r e s s e s , o r b e a r i n g , b e n d i n g and s h e a r s t r e s s e s ,a r e c o - e x i s t e n t , t h e b e a m s h o u l d b e ' C h e c k e d i n accordance i t h B S 4 4 9 ,

C l a u s e 1 5 t o B S 449 i n c l u d e s t h e r e q u i r e m e n i t h a t t h e m a x i m u m d e f l e c t i o n



d u e o l o a d s o t h e r t h a n t h e w e i g h t o f t h e s t r u c t u r a l f l o o r s o r r o o f , s t e e l w o r k

and c a s i n g , i f a n y , s h a l l n o t exceed 1 / 3 6 0 t h o f t h e s p a n . F o r a s i m p l y

supported beam, t h e u n i f o r m l y d i s t r i b u t e d l o a d W i n kN t o produce t h i s

d e f l e c t i o n , i s :

W

384EI _ _ CI

°_ _

5 x 60Lz 1000

w h e r e C = d e f l e c t i o n c o e f f i c i e n t t a b u l a t e d f o r span L i n m

E = m o d u l u s of e l a s t i c i t y = 2 1 0 0 0 k N / c m z (= 210000N/mmz)

I = moment f i n e r t i a o f beam, i n c m '

T h e l o a d W w i l l be l e s s t h a n t h e t a b u l a r l o a d i f t h e span exceeds

1 2 . 1 7 , 16.97 or 10.00 times t h e beam d e p t h f o r grades 5 0 , 43 or 55 s t e e l

r e s p e c t i v e l y . F o r such c a s e s , i t m a y be n e c e s s a r y t o c o n f i r m t h a t t h e t o t a l

l o a d i s w i t h i n t h e c a p a c i t y o f t h e b e a m and t h a t t h e l o a d s t o be

considered o r d e f l e c t i o n purposes do n o t exceed W °.

T h e t a b l e b e . l o w g i v e s l i m i t i n g v a l u e s o f t h e span/depth a t i o f o r u n i f o r m l y

loaded simply supported b e a m s f o r d i f f e r e n t g r a d e s o f s t e e l . f o r v a r i o u s

r a t i o s o f W d / W t where Wd = l o a d considered o r d e f l e c t i o n purposes and

W t = t o t a l l o a d on beam. I f t h e a p p r o p r i a t e span/depth a t i o i s exceeded,

then t h e r e l e v a n t d e f l e c t i o n w i l l exceed 1 / 3 6 0 t h o f t h e span u n l e s s t h e

bending s t r e s s is r e d u c e d .

Ma t e r i a l L i m i t i n g v a l u e s o f span t o depth r a t i o f o r W d / W ,

1 . 0 0 . 9 0. 8 0 . 7 0 . 6 0 . 5

G r a d e 50 1 2 . 1 7 1 3 . 5 3 1 5 . 2 2 1 7 . 3 9 20.29 2 4 . 3 5

G r a d e 4 3 1 6 . 9 7 1 8 . 8 6 2 1 . 2 1 24.24 2 8.28 33.94G r a d e 5 5 1 0 . 0 0 1 1 . 1 1 1 2 . 5 0 1 4 . 2 9 1 6 . 6 7 20.00

T hese a r e t h e s a f e u n i f o r m l y d i s t r i b u t e d l o a d s ( i n c l u d i n g s e l f w e i g h t )

which can.be c a r r i e d by s i m p l y . supported b e a m s w i t h adequate l a t e r a l

r e s t r a i n t t o t h e compression f l a n g e . They have t h e r e f o r e g e n e r a l l y been

c a l c u l a t e d i n accordance w i t h t h e a l l o w a b l e bending s t r e s s e s g i v e n i n

BS 4 4 9 , T a b l e 2 and i n c l u d e d i n t h e t a b l e on page 2 2 .

T h e l o a d s a r e p r i n t e d i n t h r e e d i f f e r e n t t y p e f a c e s t o draw a t t e n t i o n

t o p a r t i c u l a r r i t e r i a which m a y a f f e c t t h e l o a d - c a r r y i n g c a p a c i t y o f a

beam. B o l d , i t a l i c and o r d i n a r y t y p e f a c e s have been u s e d i n t h a t

sequence f o r maximum l a r i t y and t h e s i g n i f i c a n c e o f each i s e x p l a i n e d

b e l o w .

Bold f a c e . Loads p r i n t e d i n t h i s t y p e a r e g r e a t e r t h a n t h e w e b

b u c k l i n g c a p a c i t y o f t h e b e a m ( U B , j o i s t o r channel) a l o n e . I f s u f f i c i e n t

a d d i t i o n a l c a p a c i t y i s n o t p r o v i d e d by t h e b e a r i n g ( s e e page 2 2 ) , w e b

s t i f f e n e r s w i l l be n e c e s s a r y t o r e a l i s e t h e f u l l web a p a c i t y .

i t a l i c f a c e . Loads p r i n t e d i n t h i s t y p e a r e w i t h i n t h e b u c k l i n g

c a p a c i t y o f t h e u n s t i f f e n e d w e b and produce a maximum e f l e c t i o n o f

l e s s t h a n 1 / 3 6 0 t h o f t h e s p a n .

Ordinary f a c e . Loads p r i n t e d i n t h i s t y p e . produce a maximum

' AIADC DT[ t7r%



EXPLANATORY NOTES

B e a m s without l a t e r a l support

n . Such c a s e s s h o u l d t h e r e t o

r ' D e f l e c t i o n c o e f f i c i e n t s .

I n a d d i t i o n t o t h e c r i t e r i a covered b y t h e u s e o f d i f f e r e n t t y p e f a c e s ,two o t h e r m a t t e r s r e q u i r e c o m m e n t ,

examined a s i n d i c a t e d above L i l l

d e f l e c t i o n e x c e e d i n g 1 / 3 6 0 t h

( a ) S h e a r c a p a c i t y . W h e r e t h e s h e a r c a p a c i t y o f t h e U n s t i f f e n e d w e b i s

l e s s t h a n t h e bending c a p a c i t y o f a b e a r n , t h e s a f e l o a d i s c a l c u l a t e d

on t h e a l l o w a b l e a v e r a g e s h e a r s t r e s s g i v e n i n BS 4 4 9 , T a b l e 1 1 , and

i n c l u d e d i n t h e t a b l e on page 2 2 . Such c a s e s a r e marked t i n t h e t a b l e s ,

t h a t t h e l e n g t h o f s t i f f b e a r i n g and t h i c k n e s s o f f l a n g e p l a t e o r p a c k i n g

( i f a n y ) p r o v i d e a s u f f i c i e n t a d d i t i o n a l l e n g t h o f w e b i n b e a r i n g ( s e epage 2 2 ) ,

exceed t h e b e a r n c o m p o n e n t o f t h e d i r e c t b e a r i n g c a p a c i t y o f t h e w e

a t i t s j u n c t i o n w i t h t h e f l a n g e . The d e s i g n e r s h o u l d t h e r e f o r e e n s u r eI

( b ) Web r u s h i n g . Many f t h e l o a d s t a b u l a t e d , i n t h e v a r i o u s t y p e

The a l l o w a b l e c o m p r I e s s i v e s t r e s s d u e t o bending about t h e x - x a x i s f o ru n i v e r s a l beams a n d columns, j o i s t s and channels s t h e l e s s e r o f t h e

v a l u e s O f Pb , g i v e n i n T a b l e s 2 and

'

3 a , b o r c (depending on t h e g r a d e o fs t e e l ) i n B S 449. The a l u e s i n ' F a b l e 2 a p p l y when the c o m p r e s s i o n flange

i s SO S u p p o r t e d that l a t e r a l i n s t a b i l i t y i s obviated. When th e c o m p r e s s i o n

O u t l a t e r a l r e s t r a i n t and having h e ends o f t h e c o m p r e s s i o n . f l a n g e

T a b l e 2 i s g i v e n a s L , w i t h t h e s a f e l o a d t a b l e s .

which m a y be u

T a b l e 3 a p p l i e s . Th

f l a n g e i s unsupported o r

E x a m p l e : F i n d the a l l o w a b l e s t r e s s ,Pb , ,

a nd t h es a f e

u n i f o r m l y

i s t r i b u t e d l o a d , W, f o r a 533 = 210 UB 82 i n g r a d e 4 3 s t e e l spanning 7 m

E f f e c t i v e l e n g t h = 0.85 x 7 00 = 595cm ( s e e BS 449, Cla use 2 6 )

p a r t i a l l y r e s t r

Li f f i c i e n t l y S u p p o r t e d l a t e r a l l y , t h e appropriate

D / T = 4 0 .

u m e f f e c t i v e l e n g t h o f compression f l a n g e

hout r e d u c t i o n o f t h e a l l o w a b l e bending s t r e s s e s i n

The s a f e u n i f o r m

From BS

d i s t r i b u t e d l o a d , W= f Z / L

= (8x1 19X 7

= 245kN

103/7000)N

L a t e r a l i n s t a b i l i t y i s n o t a c r i t e r i o n f o r t h e s e s e c t i o n s w h e n b e n t

about t h e y - y a x i s a n d t h e a l l o w a b l e s t r e s s e s i n T a b l e 2 a p p l y ,

N o t e ý T h e allowable s t r e s s e s i n B S 449, Table 2 are given o r ) p a g e 22 a n d Tables

3 a , b a n d c o f t h e standard are r e p r o d u c e d o r ) the following p a g e s ,



NOTES

and s a f e l o a d t a b l e s

3a o f BS 4 4 9 , allowable t r e s s Pb c i n bending ( N / m m 2 ) f o r b e a m s o f G r a d e 43 s t e e l

1r, D / T

10 15 2 0 25 30 35 40 . 50

90 165 165 165 165 165 165 1 6 5 1 6 5

9 5 16 5 1 6 5 1 6 5 16 3 1 6 3 163 163 163

1 00 16 5 1 6 5 1 6 5 1 5 7 1 5 7 1 5 7 1 5 7 1 5 7

1 05 16 5 1 6 5 1 6 0 15 2 1 5 2 152 1 5 2 15 2

110 165 1 6 5 1 5 6 14 7 14 7 14 7 14 7 1 47

115 1 6 5 1 6 5 1 5 2 1 4 1 14 1 1 4 1 1 4 1 1 4 1

1 2 0 16 5 1 6 2 1 4 8 13 6 13 6 13 6 13 6 13 6

1 3 0 165 155 139 126 126 126 126 126

140 165 149 130 1 1 5 1 1 5 1 1 5 1 1 5 1 1 5

150 165 143 122 104 104 104 104 104

160 163 136 1 1 3 95. 94 9 4 94 9 4

170 159 130 104 9 1 85 8 2 8 2 8 2

180 1 55 1 24 96 87 80 7 6 7 2 7 1

190 1 5 1 118 93 8 3 7 7 7 2 68 6 2

200 147 1 1 l 89 80 7 3 6 8 64 5 9

210 1 4 3 , 105 87 7 7 7 0 6 5 6 1 5 5220 1 3 9 , 99 84 7 4 67 6 2 5 8 5 2

230 134 9 5 8 1 7 1 64 5 9 5 5 4 9

240 130 9 2 78 69 6 1 5 6 5 2 47

250 126 90 7 6 66 5 9 5 4 5 0 4 4

260 122 88 74 64 57 5 2 4 8 4 2

270 1 1 8 86 72 6 2 5 5 5 0 4 6 40

280 1 1 4 84 70 6 0 5 3 4 8 44 3 9

290 1 1 0 8 2 68 5 8 5 1 4 6 4 2 3 7

300 106 8 0 66 5 6 49 4 4 4 1 3 6

I n t e r m e d i a t e v a l u e s m a y b e obtained b y l i n e a r i n t e r p o l a t i o n .

t e : F o r m a t e r i a l s o v e r 40 mm t h i c k t h e s t r e s s s h a l l n o t exceed 1 5 0 N / m mz .



EXPLANATORY NOTES

o f stanchions a n d s t r u t s

T able"I 7a o f BS 449. A l l o w a b l e s t r e s s p , o n gross section for c o m p r e s s i o n fo r G r a d e 4 3 s t e e l

P , (N/mM2) o r G r a d e 4 3 s t e e l

0

0 1 5 5 155 154

1 0 1 5 1 1 5 1 15 0

20 147 146 146

30 143 1 42 14 2

40 139

50 133 133 132

6 0 12 6 1 2 5 1 2 4

70 115 114 113

8 0 10 4 1 02 1 0 1

90 9 1 9 0 89

1 00 79 78 77

11 0 6 9 6 8 6 7

120 6 0 5 9 5 8

130 5 2 5 1 5 1

140 4 6 4 5 4 5

150 4 0 40 3 9

1 60 36 35 3 5

1 7 0 3 2 32 3 1

1 8 0 29 28 28

1 9 0 26 26 25

2 00 24 23 23

2 1 0 2 1 2 1 2 1

220 2 0 1 9 1 9

230 1 8 1 8 1 8

240 1 7 1 6 1 6

250 1 5

300 1 1

350 8

AXIAL ST R ESSES

Th e s a f e loads on s t a n c h i o n s a n d

s t r u t s h a v e b e e n calculated i n

a c c o r d a n c e w i t h the allowable

a x i a l s t r e s s e s specified i n BS449 ,

C l a u s e 3 0 , a n d shown i n the tables

t h a t follow:

3 4 5 6 7

1 5 4 153 153 153 152 152 1 5 1

150 149 149 148 148 148 , 14 7

1 46 145 145 144 144 144 143

14 2 1 4 1 1 4 1 1 4 1 140 140 139

137 137 136 136 1 36 135 134

1 3 1 13 0 1 30 129 128 127 126

1 2 3 1 2 2

1 1 2 1 1 1

100 9 9

8 7

7 6 7 5

1 2 1 120 119

110 108 1 07

97 96 95

85 84 83

74 7 3 72

118 117

106 105

94 92

8 1 80

7 1 70

6 6 6 5 6 4 6 3 6 2 6 1 6 1

5 7 5 6 5 6 5 5 5 4 5 3 5 3

5 0 4 9 4 9 4 8 48 4 7 4 6

44 4 3 4 3 4 2 4 2 4 1 4 1

3 9 3 8 3 8 3 8 3 7 3 7 3 6

3 5 3 4 34 3 3 3 3 3 3 3 2

3 1 3 1 30 3 0 30 2 9 2 9

2 8 2 8 2 7 2 7 2 7 2 6 2 6

2 5 2 5 2 5 2 4 2 4 2 4 2 4

2 3 2 3 2 2 2 2 2 2 2 2 2 2

2 1 2 1 20 20 20 20 20

1 9 1 9 1 9 1 9 1 8 1 8 1 8

1 8 1 7 1 7 1 7 1 7 1 7 1 7

1 6 1 6 1 6 1 6 1 6 1 6 1 5

I ntermediate a l u e s m a y b e obtained b y l i n e a r i n t e r p o l a t i o n .

Documents

Beam Bending Formula