U ti n51 gn ira eh S )g ni dn e B dn a( no isr oT dn a

(1)

U ti n 5 1 g

n ir a e h

S a n d T o r s i o n ( a n d B e n d i n g )

a

P lu A. Lagac © e 2001

(2)

T I

M - 1 26 0 .

s u h

T ,r f e a w h a v e c o n c e n rt a t e d n o e t h b e n d i n g f o s h e ll m

a e

b s. H o w e v e ,r i n t h e g e n e r a l c a s e a b e a m s i s u b j e c t e d : t o

• axial load, F

• bending moments, M

• shear forces, S

• torque t(orsional moments ,) T s

e l p m a x

E fo general aerospace she ll beam structures ll e

g n it c e n n o

c nodes

ti n

U 51 - 2

a F ll, 20 20

e r u g i

F 1 5.1

c ri A

a

(3)

e z il a e d

I et h cross-sec iton fo et h she ll beam oint ot w pa trs:

• P sa tr that ca rry extensional s rtess, σx x (and thus et h bending da n l

a i x

a loads)

• Pa trs that ca rry shear s rtess σx s (and thus et h shear loads and )

s e u q r o

t

o w

T examples a n… gai

• h igh aspect r oa it w hing w ti semi-monocoque cons rtuc iton e

t o

N s:

u q o c o n o

m e nco s rtuc iton

• – lla ni eo n piece w tihout internal rfaming

• – rfom French “coque” meaning “eggshell”

• – “mono” = eo n piece i

m e

s -monocoque

– s rtessed ns ki cons rtuc iton tiw h internal rfamework – s t lli have “eggshell” ot ca rry shear s rtesses, σx s

– internal rfamework ot ca rry axial s rtess, σx x

ti n

U 51 - 3

•

a

(4)

ll a F , 02 2 0

n o it a t n e s e r p e

R fo wing semi-monocoque construc iton b

ir b

e

w + s e g n a

lf = spar

b e w e

z il a e d

I st hi sec iton :a s n

o it a z il a e d

I fo wing semi-monocoque construc iton

ti n

U 51 - 4

T I

M - 1 26 0 .

e r u g i

F 1 5.2

s e g n a lf

skin

s r e n e ff it s

e r u g i

F 1 5.3

a

(5)

l a F l, 02 2 0

→ Skins da n webs ea r assumed ot ca rry lo y n shear s rtess σx s

→ Flanges da n st irngers ea r assumed ot ca rry lo y n axial s

s e rt

s σx x

e c a p

S hab tiat

e r u g i

F 15.4 Representa iton fo space hab tiat semi-monocoque n

o it c u r t s n o

c

s e g n a lf spar

ll a

w st fifeners e

z il a e d

I sa rf o wing:

a

P lu A. Lagac © e 2001 T

I

M - 1 26 0 .

•

ll a w

ti n

U 51 - 5

(6)

ll a F , 02 2 0

n o it a z il a e d

I fo space hab tiat semi-monocoque construc iton

r e t u

O ns dki a n wa lls ea r assumed ot ca rry o nly shear s rtess σx s

s e g n a l

F da n s it ffeners ea r assumed ot ca rry lo y an a l xi s rtess σx x

e z y l a n

A these cross-sec itons sa a beam under combined bending, shea ,r d

n

a torsion. Ut ilize tS . Venant assump itons:

1. There ea r enough closely spaced ir sgid ir ob t preserve et h shape f

o et h cross-sec iton r( o enough s it ffness ni et h internal bracing ot o

d such)

2. T he cross-sec itons ea er rf oe t warp to - fu o -plane tr

a t

S ot develop et h basic equa itons yb looking ta et h m ost basic case:

a

I

M - 1 26 0 .

e r u g i

F 1 5.5

→

ti n

U 51 - 6

(7)

e l g n i

S C e ll “ B o x B e a m ”

R fo geometry fo single c xe ll b eo b ma

s u l u d o

m -weighted centroid of e

g n a

lf da n s it ffener area used s

a o irgin n

w o d k a e r

B et h problem…

) a

( Axial Bending Stresses: Each lfange/s it ffener sh a some area d

e t a i c o s s

a hw ti dti a ti n car ires axial s rtess o nly (assume σxx si t

n a t s n o

c w tihin each lfange/s it ffener area)

a

P lu A. Lagac © e 2001 U 5n ti 1 - 7

e r u g i

F 1 5.6

(8)

T I

M - 1 26 0 .

e h

T axial s rtess si ed u o only t bending (and axial force fi that exists --

e v a e

l ta zero rf wo no ) da sn i therefore independent fo et h twis itng since et h g

n i

w si erf e ot warp (except near root -- tS . Venant assump itons)

* F , i n d M , S T m rf o s t a it c s t a y a n c r o s s - s e c it o n x f o e t h b e a m

r e d i s n o

C et h cross-sec iton:

e r u g i

F 15.7 Representa iton fo cross-sec iton fo xb o beam a e r

A associated hw ti r

e n e ff it s / e g n a

lf i = Ai

d n i

F et h modulus-weighted cen rtoid ( tN e: o lfange/s it ffeners m eay b e

d a

m mrfo di fferent mate irals)

a

a F ll, 20 20

ti n

U 51 - 8

(9)

e s o o h

C some sa xi system ,y z (convenience says eo n might e

s

u a “corne ”r fo et h beam) d

n i

F et h modulus-weighted nce rt oid loca iton: y ^* =

i

z ^* =

= msu over number fo lfanges/s it ffeners) r

e b m u

n = n

( tN e: fI o lfanges/s it ffeners ea r made fo et h e

m a

s mate ira ,l remove et h aste irsks) d

n i

F et h moments fo iner ita hw ti reference ot et h coordinate m

e t s y

s hw ti o irgin ta et h modulus-weighted cen rtoid

∑

^{A zi}

∑

^{A yi}

∑

^{A yi}^{* iz}

ti n

U 51 - 9

•

i *

∑

^A

∑

^{A y i}

*

∑

^{A iz}

∑

^Aⁱ

(

i *

*

∑

n i =1

* *² i

* *

i

Iy = I_z = Iyz =

a

*

(10)

ll a F , 02 2 0

• F eind t h s rtesses ni each lfange yb using et h equa iton previously e

p o l e v e

d d:

- E1 f2y - E1 f3z - E1 α ∆T �

� 0 rf oo n axial force

lli W

( od na example fo st nhi i rectia iton)

r a e h

S stresses: assume et h skins da n webs ea nr t hi such that et h r

a e h

s s rtess si constant through the ri thickness . e

s

U et h concept fo “shear olf w” previously developed: ]

h t g n e l/

e c r o F

[

s s e n k c i h t r a e h

s s rtess (ca lled st ehi t h shear n

a tl u s e

r t ni et h case fo torsion) k

o o

L ta et h example cross-sec iton da n label et h j“oints” da n “skins”

a

I

M - 1 26 0 .

) b (

F^T^O^T

� �

σ _x_x = E � �

E1 A ^*

s

q = σ t x

r a e h slf ow

ti n

U 51 - 01

(11)

R fo joints, skins, da n shear lfows ni s

s o r

c -sec iton fo xb o beam

k o o

L ta et h equ ilib irum fo nj t : oi n

o it a t n e s e r p e

R fo skins da n st irnger da n associated loads d

n

a shear lfows ta joint 1

n i k s r

e g n ir t s

n i k s

ti n

U 51 - 11

1

e r u g i

F 1 5.9

a

e r u g i

F 1 5.8

(12)

ll a F , 02 2 0

e t o

N s:

• T he st irnger o nly car ires axial load

• T nhe s ki car ires o nly shear wlf o

•� T he shear wlf to a et nh “ d” fe o et nh s ki (where ti si )”

t u c

“ m eust b et h same sa ta et h edge et( h s

s o r

c -sec iton c . u )t T shis i ed ou t equ ilib irum (σx y = σyx)

y l p p

A equ ilib irum:

⇒ - P +

q1 - q6 = -

e r o

M genera lly note tha :t

a

I

M - 1 26 0 .

• = 0

q1 d x - q6 d x = 0

ti n

U 51 - 21

∑

^Fx

x d + dP

P + dx

P d

x

⇒ d

(13)

A en gl that ns ki comes oint joint doesn t ’ ma tter since q along e

g d

e si always ni x-d riec iton n

R fo skins ta joint coming ni ta ya n angle

2. St irnger alone gives a ss ti cont irbu iton

e r u g i

F 15.11 Representa iton fo st irnger isolated ta oj int

a

. 1

e r u g i

F 15.10

P d dx

ti n

U 51 - 31

(14)

ll a F , 02 2 0

fI shear lfows “i o” nt join ,t sti cont irbu itons si ni ht e nega itve x-

; n o it c e ri

d fi shear lfows “o of” ut join ,t sti cont irbu iton si ni et h v

it i s o

p e x-d riec iton

e r u g i

F 15.12 Representa iton fo shear lfowing (lef )t oint da n ( irgh )t o fut o t

n i o

j

g n i d d

A lla st :hi u p - qin

q_o _tu -

e s

U st nhi i general

a

P d

x d

⇒

+ qo tu = 0 P d

x d

ti n

U 51 - 41

qin = -

T I

M - 1 26 0 .

. 3

(15)

ll a F , 02 2 0

r o

F a more comp ilcated join ,t eu s superposi iton

e r u g i

F 15.13 Representa iton fo joint w tih mul itple skins

--> Need na expression rf P -- o sta tr w tih:

= Aσ

: g n it a it n e r e ff i d

dσ xx

dx

e c n i

s ea r conside irng st irngers h

ti

w a un fiorm cross-sec iton

t n i o

J Equi ilb irum

ti n

U 51 - 51

I

M T - 16. 02

x

P x

⇒ = A dA

+ σ dx

x x

P d

dx

= 0 t

s o

M general sc e: a

q

out

- q

in

=

a

- A dσ dx

^x^x

(16)

T I

M - 1 26 0 .

r e d i s n o

C a simpler case:

• Iyz = 0

• Mz = 0 w

o n

k tha :t

x x

My z x

d ^� Iy �

z A

Iy

r a e h s

( resu tlan )t t

n e m o m

( fo area about )y o

S rf so t hi case, et h joint equ ilib irum equa iton becomes:

a

ll a F , 02 2 0

ti n

U 51 - 61

c ir t e m m y s

( sec iton)

My z - Iy

σ =

� - �

dMy

dx

� �

= A

= -

Sz

⇒ dP d

dx dP dx ll

a c e

R tha :t dMy

dx =

z

A = Q_y

(17)

• Symmet irc sec iton

• Mz = 0

w o

N have na equa iton rf eo t h equ ilib irum fo shear s rtesses ta et h joints. r

a e h

S s rtesses a irse ed ou t ot w reasons:

• Shear resu tlant

• Twis itng

n

I general have ta ya n cross-sec iton:

tI si convenient ot break pu et h problem oint ot w separate problems:

a

Q

y

S

z

q

out

- q

in

=

I

y

ti n

U 51 - 71

(18)

ll a F , 02 2 0

) 1

( “Pure Shear” ( “2) Pure Twist”

r a e h

s resu tlant acts t

a shear center os e

r e h

t si on twis itng

--> Solve each problem separately, then da ed t h resu tls (use )

n o it i s o p r e p u

s

o it i d n o

C n: T ohe t w force systems S( z, T a Snd z

’

^{, T}

’

⁾^{m e}^u^s^t ^b

t n e ll o p i u q

e

n o it a r t s n o m e

D fo equipo llence of force systems

ti n

U 51 - 81

T I

M - 1 26 0 .

◊

e r u g i

F 15.14

a

(19)

ll a F , 02 2 0

d = distance mrfo where shear resu tlant a octs t shear center S = S_z^′

T ′ - S_zd = T u

f e r a

c l: s ign could eb + ro - depending upon n

o it c e ri

d Sz si moved! n

R of posi itve da n nega itve cont irbu iton fo Sz

o

t torque

n i

H t: A pdd u torques about eil fn o Sz ac iton ni each case. y

e h

T m eust b et h same!

(⇒ d sh a magn tiude da n sign)

ti n

U 51 - 91

= same ⇒

= as m e ⇒

z z

e r u g i

F 15.15

a

I

M - 1 26 0 .

n if e

D e:

∑

^F

∑

^T

(20)

T I

M - 1 26 0 .

n o it u l o

S procedure :

n e v i

G • sec iton proper ites

• loading [T(x ,) Sz( x ]) d

n i

F : • shear s rtesses ( lfows) ( n joints)

• shear center

⇒ n( + )1 va irables r

e d i s n o

C “Pure Shear” case )

a Apply joint equ ilib irum equa iton ta each joint t

o

N e: n joints lliw yield n-1 independent n

o it a u q

e s. (one si dependent since t eh n

o it c e

s si closed) )

b U se Torque Boundary Condi iton

= T

ap lpeid

s i h

T si torque equivalence, n ot equ ilib irum o

D st hi about et eh il fn o ac iton fo Sz

ti n

U 51 - 02

a F ll, 20 20

. 1

a

l a n r e t n

∑ T

i

(21)

ll a F , 02 2 0

: n e h T

Tap lpeid = Sz d

qi (moment mar )i (s kinlength)i

d e if i c e p

S on stwi t (Pure Shear Case ,) os apply et h No t

s i w

T Condi iton ll

a c e

R mrfo Torsion Theory:

∫

^{τ ds}

: e r e H

: d n a

⇒

a

I

M - 1 26 0 .

c)

∑

l a n r e t n

∑

^Ti ⁼

dα dx

τ

= 2 A G

= 0 q

= t

ds = 0

ti n

U 51 - 12

∫ q t

(22)

ll a F , 02 2 0

n - 1 equa itons 1 equa iton 1 equa iton

n + 1 equa itons f n + 1 or va irables g

n i v l o

S these gives:

• q’s ed ou t “Pure Shear” case

• d

--> when complete, check :v ia

∑ Internal Shear loads = App iled Shear l

a t n o z ir o H

( & ver itca )l r

e d i s n o

C “Pure Torsion” case )

a Apply joint equ ilib irum equa iton ta each joint t

o

N e: again, n joints eg n-1 iv equa itons e

c n i

S on shea :r qo tu - qi n = 0

ti n

U 51 - 22

T I

M - 1 26 0 .

s i h

T gives: ) a ) b c)

. 2

a

(23)

ll a F , 02 2 0

e s

U Torque Boundary Cond tii on

= Tap lpeid

Tap lpeid = T + o r - S_zd

d n u o

f ni p 1 a tr

n - 1 equa itons 1 equa iton

• Need: n q s’ e

v l o

S these rf q so ’ ed ou t “Pure Torsion” case

m u

S resu tls rf “o Pure Shear” da n “Pure Torsion” cases d

l u o c

( eu qs is, qiT: qi ed ou t pure shear = qis

qi ed ou t pure torsion = qiT)

ti n

U 51 - 32

l a n r e t n

∑

^Ti

⇒

. 3

a

I

M - 1 26 0 .

) b

s i h

T gives: ) a ) b

(24)

ll a F , 02 2 0

q t lli

w

( eb impo trant ot determine de lfec iton)

lli w

( og over sample problem fo handout 6# ni rectia iton)

:r o

f Unsymmet irc She ll Beams

• Cannot make simpl fiying assump itons (use equa itons o

c up ilng bending ni y da )n z

• S ee handout B# 4

o t e u d ( s e s s e rt s e h t t a h t w o

N b e n d i n g , s h e a r a n d t o r s i o n ) e

r

a d e t e r m i n e d , p r o c e e d o t d if n t h e …

a

I

M - 1 26 0 .

w o

N have: shear lfows t

e g

( shear s rtesses :v ia ) o

l ca iton fo shear center

τxs =

ti n

U 51 - 72

(25)

N O I T C E

S PROPERTIES g

n i d n e

B S it ffness, IE

∑

^{A y}²

∑

^{A z}²

∑

^{A y z}

)

b Shear S it ffness, AG (have tn o done st hi before)

r e d i s n o

C et h de lfec itons ∆v da ∆w rn f a o segment ∆x hw lti o y n shear s

e c r o

f Sy da Sn z ac itng ta et h shear center

e r u g i

F 15.18 Representat ni fo o she ll beam segment w tih only shear s

e c r o

f ac itng ta et h shear center

a

)

a (as before)

Iz = Iy = I_yz =

*

ti n

U 51 - 82

(26)

ll a F , 02 2 0

n a

C express et h shear lfows ni each member sa cont irbu itons ed ou t Sy da Sn z :

q )( = qs y ( Ss) y + qz )( Ss z

: e r e h

w

= shear wlf eo d ou t Sy fo u n ti magn tiude

= shear wlf eo d ou t Sz fo u n ti magn tiude --> oT determine ∆v da ∆n w ti s, i b oest t eu ns a Energy Method tI nc ea b shown tha :t

A =

∫

A =

qy (s ) qz (s )

q_y

s t d

yy 1 2

( )

∫

qz 2

( )

1 _d_s

s d

ti n

U 51 - 92

z z

t qy q1 z

t

z y

MIT - 16. 02

(27)

ll a F , 02 2 0

l a n o i s r o

T S it ffness, JG y

l s u o i v e r

P ws a tha :t

r o

f( closed sec iton)

s a

( tj )us d id

= shear wlf eo d ou t T fo u n ti magn tiude ,

n e h

T using st nhi i et h above:

T q

G2 A t :

g n ill a c e

R

dα T

d x G J

⇒

∫

a

I

M - 61 . 02

) c

dα dx

:t e L

q t d s

∫

= 1 2 AG

q = q (s ) T : e r e h

w q (s)

=

∫

d s

dα dx

=

r o

f( closed sec iton)

ti n

U 51 - 03

2 A q

t d s

J =

(28)

ll a F , 02 2 0

, g n i z ir a m m u

S ot dif en t h de lfec itons: n

i a t b

O et h sec iton proper ites ,I( E G )A, G dJ a en t ah loc it fon o e

h

t shear center e

s o p m o c e

D load oint moments, shears ta shear cente ,r da n e

u q r o

t about shear center d

n i

F i(ndependen lty) bending, shea irng, da n twis itng de lfec itons t

u o b

a eil fn o shear centers (elas itc axis) m

u

S de lfec itons ot obtain total ed lfec iton .

1 . 2

. 3

. 4

e r u g i

F 15.19

a