Post-yield behavior of a beam with partial end fixity

Int. J . mech, ,.qel. Pergamon Press. 1974. Vol. 16, pp. 385-388. Printed in Great Britain

POST-YIELD BEHAVIOR OF A BEAM WITH PARTIAL END F I X I T Y * PHILIP G. H O D G E , JR. Department of Aerospace Engineering and Mechanics, University of Minnesota, Minneapolis, Minnesota, U.S.A.

(Received 30 July 1973, and in revisedform 14 November 1973) S u m m a r y - - A simply supported beam can undergo a relative axial displacement proportional to the induced axial force. Complete formulas and curves are obtained for transverse displacements of the order of the beam thickness. 1. I N T R O D U C T I O N

CONVENTIONAL t r e a t m e n t of b e a m problems, either elastic or plastic, assumes t h a t the ends of the b e a m are free to m o v e laterally so t h a t there are no induced axial stresses. H a y t h o r n t h w a i t e 1 showed t h a t a b e a m whose ends were fully fixed could s u p p o r t a load m a n y times the conventional yield-point load. J o n e s ~ has p r e s e n t e d an analysis of a b e a m which was subject to a prescribed axial displacement at the end. Obviously the conventional case a n d H a y t h o r n t h w a i t e ' s results can be regained as special cases of J o n e s ' development. I n his p r e s e n t a t i o n of specific results, J o n e s assumed t h a t the axial end displacement was proportional to the square of the m a x i m u m transverse displacement. H o w e v e r , it is not evident to w h a t real physical situation this assumption corresponds. The purpose of the present n o t e is to re-examine J o n e s ' work with a more physical basis. To this end we consider a rectangular beam which is simply s u p p o r t e d in a prescribed rig (Fig. l ( a ) ) . I t is assumed t h a t the rig is p u r e l y elastic so t h a t (Fig. l(b) ) F = KU, (1) where K can be d e t e r m i n e d either b y an analysis of the rig or b y simple direct experiment. The b e a m material is rigid-perfectly plastic with yield stress %. The yield condition for positive m o m e n t is 3 re+n2= 1, (2) where m = M / M o = M / 2 B a o g ~, n = N / N o = N / 4 B a o g . (3) 2. A N A L Y S I S The beam will remain motionless until the load reaches the yield-point value of 2Mo/L. At this point a yield hinge will form in the center with the rest of the beam remaining rigid. Since along a straight section the axial force is constant and the moment * This investigation was sponsored by the Office of Naval Research. 28 385

386

PHILIP G, HEDGE, JR.

r 2B

/////////////// . P/Z

(o)

L

L-U

~}'~ u o

~,,,z

(b)

FxG. ]. Beam with partial end fixity. (a) Beam m rig; (b) static and kinematic relations. varies linearly, the only yield hinge will remain at the center and the half-beam will remain straight. As shown in Fig. l(b), the extension A a n d the rotation 0 of the yield hinge are defined by W 2 + ( L - U ) 2 = ( L + A ) 2, tan 0

W/(L-

(4)

U). t

The analysis is best presented in terms of the small parameter (5)

h = H/L.

We consider vertical displacements up to the order of the thickness so that W / L is of order h and U / L is of order h 2 or less. Therefore, we define dimensionless kinematic quantities by w = W / H , u = U L / H 2, ~ = A N o / M o (6) and write (4) in the form ;~ = h[(w ~ -

2u) + h = w = ( u - w2/4) + O(M)],

0 = hw[1 + h ~ ( u - w ~ / 3 ) +O(M)].

t

(7)

The statics of the beam are also illustrated in Fig. l(b) and lead to the equilibrium equations m = p-2w.f-h~pu,

n = cos 0 ( f + t h p t a n 0), /

(8)

where f = F/No,

p = PL/2Mo.

(9)

I t follows from (1) that u = .f/k,

(to)

where we have defined the dimensionless rig constant k =- h K / 4 B a o.

(]l)

Post-yield b e h a v i o r of a b e a m w i t h partial end fixity

387

Therefore, we m a y write (8) in t h e f o r m m = p - 2fw-h2p.f/k,

1

.f+ ½h2 w ( p - $ w ) + O(h'),

n

(12)

J

w i t h o u t regard for t h e order of m a g n i t u d e of k. I f n < 1, t h e " s t r a i n - r a t e v e c t o r " w i t h components (~, ~) m u s t be n o r m a l to the p a r a b o l a (2) in stress space, a hence = 2n& (13) F o r n = 1, t h e stress p o i n t is at a corner of the t o t a l yield curve a n d (13) is n o t required. 5.4

I

[

;

i

;

I

t

J //I

3.2 3.0 2.8 2.6

P

2.4 2.2

2.0 1.8 1.6

1.4

1,2

1.0 0

I 0.2

0.00

[

I

I

f

I

I

0.4

0.6

0.8

1.0

1.2

1.4

I

1.6

1.8

w

FIG. 2. L o a d - d e f l e x i o n curves. 3. S O L U T I O N F o r simplicity of exposition we shall keep only leading t e r m s in t h e r e m a i n d e r of t h e d e v e l o p m e n t , t h u s allowing errors of order h ~. A n obvious p e r t u r b a t i o n scheme m a y be used if greater accuracy is desired. W i t h this simplification, we replace (12) b y m=p-2wf,

n=

fi

(14a, b)

F o r n < 1, s u b s t i t u t i o n of (7), (10) and (14b) in (13) leads to k w - d n / d w = k'n.

(15)

Once (15) is solved, (14a) a n d (2) can be c o m b i n e d to g i v e p as a function o f w . F o r n = 1, it follows f r o m (2) t h a t m --- 0 whence p is o b t a i n e d directly from (14a). Therefore, t h e complete solution is g i v e n b y if

n < 1,

(16a)

388

PHILIP G. HODGE, JR.

then

n = (e -kw + k w - 1)/k,

(16b)

p = l + 2 w n - - n ~,

(16c)

p ~ 2w.

(16d)

else

F o r small values of k, equations (16) can be e x p a n d e d in a series in kw which leads to p = l+kwa+0(k2).

(17)

E v i d e n t l y bhe limiting value is the yield point load p = 1 corresponding to no axial restraint. F o r large k, the first t e r m in (16b) becomes negligible and we are led to p = l+w~+O(1/k~).

(is)

The limiting value here corresponds to H a y t h o r n t h w a i t e ' s solution for full end fixity. REFERt~NCES 1. R . M. HAYTHORNTHW~TTE, Engineering 183, 110 (1957). 2. N. JONES, Int. J. mech. Sei. 15, 547 (1973). 3. P. G. HODGE, JR., Plastic Analysis of Structures, c h a p t e r 7, McGraw-Hill, N e w Y o r k (1959).