M assachusetts In stitute of Technology
1. I n tro d u c tio n . L e t us c o n sid e r a th in -w a lle d box b eam of w eb h e ig h t 2h a n d c o v e r s h e e t w id th 2w w h ich is b e n t in su c h a w a y t h a t on e of th e c o v e r sh e e ts is in te n sio n w h ile th e o p p o site c o v e r s h e e t is in c o m p re ssio n (F ig. 1). In e le m e n ta ry b e a m th e o ry th e a s s u m p tio n is m a d e t h a t th e n o rm a l s tre s s in th e c o v e r sh e e ts d o cs n o t
II
crv1
\- h —h - h - ]
M + i M . d x _ J X V + 4 ^ d x
d x
Fig. 1. Sketch of spanwisc element of box beam with doubly symmetric cross section.
v a ry in th e d ire c tio n acro ss th e sh e e t. B ecau se of th e s h e a r d e fo rm a b ility of th e c o v e r sh e e ts th is a s s u m p tio n of e le m e n ta ry b e a m th e o ry is o fte n serio u sly in e rro r fo r w id e b ea m s. I n a e ro n a u tic a l e n g in e e rin g th is effect is k n o w n u n d e r th e n a m e of s h e a r lag.
In re c e n t p a p e r s,12 s h e a r lag in box b e a m s h a s been a n a ly z e d b y a n a p p lic a tio n
* Received Feb. 22, 1946.
1 E. Reissner, L east work solutions of shear lag problems. Journal of the Aeronautical Sciences, 8, 284- 291 (1941).
of th e th e o re m of le a s t w o rk w hich is th e b asic m in im u m p rin cip le fo r th e stresses.
T h e p re s e n t p a p e r c o n ta in s an a p p lic a tio n to th e p ro b lem of s h e a r lag of th e th e o rem of m in im u m p o te n tia l e n erg y , w hich is th e b asic m in im u m p rin cip le for th e strains.3 I t is sh o w n t h a t a p p lic a tio n of th e th e o re m of m in im u m p o te n tia l e n e rg y to th e p re s
e n t p ro b le m le ad s to sim p le r a n d m o re gen eral re su lts th a n th e a p p lic a tio n of th e th e o re m of le a s t w o rk . W h ile th e le a st-w o rk m e th o d fu rn ish e s th e stre sse s in box b e a m s w ith no c u t-o u ts , a p p lic a tio n of th e m in im u m -p o te n tia l-e n c rg y m e th o d f u r nishes, in a sim p le r m a n n e r, th e stresses in b e a m s w ith o u t o r w ith c u t-o u ts . I t also fu rn ish e s b e a m d eflections, a n d is e q u a lly c o n v e n ie n t fo r b e a m s s u p p o rte d in s t a t i ca lly d e te r m in a te o r in s ta tic a lly in d e te rm in a te m a n n er.
A p p lic a tio n , in th e m a n n e r d escrib ed below , of th e m in im u m -p o tc n tia l-e n e rg y p rin c ip le to th e p ro b le m of b e n d in g of th in -w alled box b e a m s le ad s to a d iffe re n tia l e q u a tio n fo r th e b eam d eflectio n w hich is a g en e ra liz atio n of th e re la tio n 2" = — M / E I ; th is d iffe re n tia l e q u a tio n c o n ta in s an a d d itio n a l te rm p ro p o rtio n a l to th e fo u rth d e r iv a tiv e of 2 w h ich ta k e s in to a c c o u n t th e s h e a r d e fo rm a b ility of th e c o v e r sh eets.
As th e o rd e r of ¿h e d iffe re n tia l e q u a tio n in th is th e o r y is h ig h e r th a n th e o rd e r of th e d iffe re n tia l e q u a tio n of e le m e n ta ry b e am th e o ry , b o u n d a ry c o n d itio n s a p p e a r in a d d i
tio n to th o se of e le m e n ta r y b eam th e o ry . T h e se a d d itio n a l b o u n d a ry c o n d itio n s arc d iffe re n t fo r b e a m s w ith c u t o u ts a n d fo r b e a m s w ith o u t c u t o u ts.
T h e m a n n e r of a p p lic a tio n of th e re su lts o b ta in e d in th e p re s e n t p a p e r is show n b y so lv in g e x p lic itly th e follow ing fo u r ex am p les.
1. S im p ly s u p p o rte d b eam . L o ad d is tr ib u te d ac c o rd in g to a cosine law .
2. C a n tile v e r b e a m w ith u n ifo rm load d is trib u tio n . C o v e r sh e e ts fixed a t th e s u p p o rt.
3. C a n tile v e r b e a m w ith u n ifo rm load d is trib u tio n . C o v e r sh e e ts n o t fixed a t th e s u p p o rt.
4. B eam w ith b o th e n d s b u ilt in. U n ifo rm load d is trib u tio n .
F o r th e sa k e of sim p lic ity , it is a ssu m e d in w h a t follow s t h a t th e cross sectio n s of th e b e a m s a rc re c ta n g u la r a n d d o u b ly sy m m e tric a l. I t also is a ssu m e d t h a t th e re is no c o n tin u o u s v a ria tio n of cro ss-se c tio n a l p ro p e rtie s.
T h e a u th o r b eliev es t h a t th e w a y in w hich th e p rin c ip le of m in im u m p o te n tia l e n e rg y is h e re a p p lie d to th e p ro b le m of s h e a r lag will p ro v e useful in o th e r p ro b lem s of s tr u c tu r a l m ech a n ics. A s a n e x am p le of such fu tu re a p p lic a tio n , th e th e o ry fo r co m b in e d to rsio n an d b e n d in g of b e a m s w ith o pen o r closed cross se c tio n s is m e n tio n e d .
2. F o rm u la tio n a n d s o lu tio n of p ro b le m . In th e follow ing, w e a n a ly z e a box b eam of d o u b ly sy m m e tric a l r e c ta n g u la r cro ss se ctio n , co m p o sed of c o v e r sh e e ts, sidew ebs a n d flanges. A given d is trib u tio n of lo ad s is a p p lie d to th e sidew ebs, a c tin g no rm al to th e p la n e of th e c o v e r sh e e ts (F ig. 1). T o th is load d is trib u tio n th e re c o rre sp o n d s a d is trib u tio n of b e n d in g m o m e n ts M ( x) . T h e sp an w ise c o o rd in a te b e in g x, let y be th e c o o rd in a te in th e p la n e of th e c o v e r sh e e ts p e rp e n d ic u la r to th e x d ire c tio n , a n d z(x) th e d eflectio n of th e n e u tra l a x is of th e b eam .
5 F. B. Hildebrand and E. Reissner, Least work an a lysis of the problem of shear lag in box beams, N.A.C.A. Technical N ote No. 893 (1943).
3 For a formulation of these theorems see for instance 1. S. Sokolnikoff and R. I). Specht, M athe
m atical theory of elasticity, McGraw-Hill Book Co., Inc., New York, 1946, pp. 275-287.
270 ERIC R EISSN E R [Vol. IV, No. 3
T h e p o te n tia l e n e rg y of th e b e n t b e a m m a y b e c o n sid e re d a s c o m p o sed of th re e p a rts . T h e first p a r t is th e p o te n tia l e n e rg y of th e load sy ste m . T h is m a y be w r itte n in th e form
n - •• • d2z
/
M ( x ) — dx,id~z (1)
dx-th e in te g ra l b e in g e x te n d e d o v e r dx-th e e n tire le n g dx-th of dx-th e b e a m.4 T h e seco n d p a r t is th e s tr a in e n e rg y of sid ew eb s a n d flanges! T h is m a y be w r itte n in th e form
1 r / d " z Y
n» =
7 J E I " ( - ¡ ¿ ) dx’(2)
th e q u a n tity d e n o tin g th e p rin c ip a l m o m e n t of in e rtia of th e tw o sid ew eb s a n d flanges.
T h e th ird p a r t is th e s tra in e n e rg y of th e tw o c o v e r sh eets. If it is a ssu m e d t h a t th e n o rm a l s tr a in s in th e ch o rd w ise d ire c tio n in th e sh e e ts m a y b e n e g le c te d , a s d is cussed in th e referen ce given in F o o tn o te 1, th e n th e s tra in e n e rg y of th e tw o sh e e ts is given b y th e in te g ra l
H. = y
f J
2t[Eex + Gy'Jdxdy, (3)w h ere th e q u a n tity t d e n o te s th e c o v e r s h e e t th ic k n e ss, a n d w h ere E a n d G a re th e e ffectiv e m o d u li of e la s tic ity a n d rig id ity . S p a n w ise n o rm a l s tr a in ex a n d s h e a r s tr a in 7 a re th e n ex p ressed in te r m s o f th e sp a n w ise sh e e t d is p la c e m e n t u a s follow s
d u du
«* = -— , 7 = — ■ (4)
dx d y
T h e th e o re m of m in im u m p o te n tia l e n e rg y s ta te s t h a t th e to ta l p o te n tia l e n e rg y
i r = n , + n w + n , (5)
b eco m es a m in im u m fo r th e c o rre c t d is p la c e m e n t fu n c tio n s u a n d z, if o n ly su c h d is
p la c e m e n t fu n c tio n s a re c o m p a re d w h ich s a tis fy all c o n d itio n s of s u p p o r t a n d c o n tin u ity im posed on th e d isp la c e m e n ts.
D ire c t a p p lic a tio n of th is c o n d itio n b y m e a n s of th e c a lc u lu s o f v a ria tio n s le ad s to a p a rtia l d iffe re n tia l e q u a tio n fo r u a n d to a c o m p le te sy ste m of b o u n d a ry c o n d i
tio n s. I n w h a t follow s, a n o rd in a ry d iffe re n tia l e q u a tio n fo r th e b e a m d e fle c tio n z a n d b o u n d a r y c o n d itio n s fo r i t a re o b ta in e d in s te a d . T h is is d o n e b y m a k in g a s u ita b le a p p ro x im a tio n fo r th e s h e e t d is p la c e m e n ts u a n d b y a p p ly in g th e ru les of th e c a lcu la- of v a ria tio n s to th e r e s u lta n t a p p ro x im a te ex p ressio n fo r th e p o te n tia l e n e rg y fu n c tio n .
A re a so n a b le a s s u m p tio n fo r th e sp an w ise s h e e t d is p la c e m e n ts is
- ±4[£+(1-5)t,(#
«(*,
y)= ±
h— + 1 - ) t/(r) . (6)
* Eq. (1) implies that the beam is supported in such a manner that the end forces and moments can do no work. This restriction shortens the developments slightly.
T h e seco n d te rm on th e rig h t of E q . (6) re p re se n ts th e c o rre c tio n d u e to s h e a r lag.
In s te a d of th e v a n ish in g ch o rd w ise v a ria tio n of th e s h e e t d is p la c e m e n ts of e le m e n ta ry b e a m th e o ry , we now a ssu m e a p a ra b o lic v a ria tio n . T h e re la tiv e m a g n itu d e of th e fu n c tio n U is a m e a su re for th e m a g n itu d e of th e s h e a r lag effect. T h e form of th e c o rre c tio n is su ch t h a t c o n tin u ity of th e d isp la c e m e n ts alo n g th e flanges, t h a t is along y = ±-w, is p reserv e d .
D e n o tin g d if fe re n tia tio n w ith re s p e c t to £ b y p rim e s, w e o b ta in th e follow ing e x p ressio n s fo r th e s tr a in s in th e sh e e ts from E q s. (6) a n d (4 ):
= + li 2h y 7 = + — — U.
w w
+1
-
#M
_ * \ w2/ (7)
( 8 )
O n th e b asis of E qs. (7) a n d (8) th e follow ing exp ressio n for th e s tra in e n e rg y of th e sh e e ts is o b ta in e d :
" - / / " • ' { * [ * " + ( ' “ M + c [ ~ 7 In E q . (9) th e in te g ra tio n w ith re sp e c t to y is c a rrie d o u t. S e ttin g
I , = 4 w t l l 2, I = I s + I n ; we h a v e
I f f 8 4 G 4 )
l l s = - E l A ( z ' y - + r _ ( U r + - z " U ’ + T — 1 / 4
2 J \ l a 3 jh 3w- )
d x .
(9)
(10)
(11)
S u b s titu tin g E q s. (11), ( |2 ) a n d (1) in to E q . (5), w e o b ta in th e follow ing expression fo r th e p o te n tia l e n e rg y of th e sy ste m
II = J jy £/(s")2 + A/V'jrf#
f 1 (8 4 G 4 )
+ j - E i ' ^ {Uy + - z'’ u ' + - ~ u j d x. (1 2)
D iffe re n tia l e q u a tio n s a n d b o u n d a ry c o n d itio n s for s a n d U are o b ta in e d b y m a k in g
511 = 0. (13)
T h u s , w ith Xi a n d x« d e n o tin g th e e n d s of th e in te rv a l of in te g ra tio n ,
272 ERIC R EISSNER [Vol. IV, No. 3
A s 5s" a n d 5 U are a r b itr a r y in th e in te rio r of th e in te rv a l (xi, xi) th e te rm s m u ltip ly in g th e m m u s t v a n ish . T h is gives th e follow ing tw o d iffe re n tia l e q u a tio n s
2 I , M
z" + U' + ---= 0, (15)
E I S
5 I E l
5 G U 5
2 E w i + 4
= 0. (16)
T h e in te g ra te d p o rtio n of E q . (14) d efines th e b o u n d a ry a n d tra n s itio n c o n d itio n s for th e fu n c tio n U. A t a sectio n w h ere th e s h e e t is fixed, bU —0 a n d
U = 0. (17)
A t a sectio n w h ere th e s h e e t is n o t fixed a n d c o n s e q u e n tly bU is a r b itr a r y ,
E I , [ U ' + | s " ] = 0. (18)
T ra n s itio n s c o n d itio n s fo r a d ja c e n t b a y s w ith d iffe re n t stiffn ess a re :
V and E I , \ U ' + f s " ] co n tin u o u s. (19) T h e a b o v e b o u n d a ry a n d tr a n s itio n c o n d itio n s are in a d d itio n to th o se im p o sed on s a n d M in e le m e n ta ry b ea m th e o ry , as m a y be verified b y re p e a te d in te g r a tio n b y p a r ts of th e te rm c o n ta in in g 5s" in th e in te g ra l of E q . (14).
3. T h e m o d ified b e a m e q u a tio n a n d its b o u n d a ry c o n d itio n s. B y e lim in a tin g th e q u a n tity U from E q s. (15) to (19), we o b ta in a sy ste m of re la tio n s c o n ta in in g th e b eam d e flectio n z only.
T h e d iffe re n tia l e q u a tio n fo r s is d e riv e d b y d iffe re n tia tin g E q . (16) a n d s u b s t i t u t ing U' from E q. (15). T h e re follow s
M E T2 / ’ M \ " I . ; z" -\---ÎC2 — — ( s H--- ) — z
E l G L 5 \ E l ) 37
= 0 . ( 20)
W h e n th e s h e a r d e fo rm a b ility of th e sh e e ts is n e g le c te d , t h a t is w h e n it is assu m ed t h a t G = « , E q . (20) re d u c e s to th e w ell k n o w n re s u lt of e le m e n ta ry b eam th e o ry .
E q u a tio n (20) m a y b e w r itte n in th e a lte r n a te form
2 E / 5 I , \ s IV M 2 E M "
z "---( 1 --- ) --- = ---+ --- (21
5 C, \ 6 7 / w l E l 5 G w*EI
W ith th e h elp of E q s. (15) a n d (16), th e b o u n d a ry c o n d itio n (17), w h ich h olds w hen th e sh e e t is a tta c h e d to th e s u p p o rt, is tra n s fo rm e d in to
/ 5 7 ,\ M '
( 1---) z ' ,, + = 0. (22)
V
6 7 / E lS im ila rly , th e b o u n d a r y c o n d itio n (18), w h ic h h olds w hen th e s h e e t is n o t a tta c h e d to th e s u p p o rt, b ecom es
T h e c o n tin u ity c o n d itio n s (19) m a y be tra n sfo rm e d in an a n alo g o u s m a n n e r.