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The following is concerned with a consistent one-dimensional treatment of theclass of beam problems dealing with the plane deformation of originally plane beams.Our principal result is a system of non-linear strain displacement relations which isconsistent with exact one-dimensional equilibrium equations for forces and momentsvia what is considered to be an appropriate version of the principle of virtual work

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Page 1: On One-dimensioON ONE-DIMENSIONAL LARGE-DISPLACEMENT FINITE-STRAIN BEAM THEORY nal Large-displacement Finite-strain Beam Theory Reissner

Journal of Applied Mathematics and Physics (ZAMP) Vol. 23, 1972 Birkh~iuser Verlag Basel

On One-Dimensional Finite-Strain Beam Theory" the Plane Problem 1) By Eric Reissner, Dept. of Applied Mechanics and Engineering Sciences, University of California, San Diego, La Jolla, California, USA

Introduction

The following is concerned with a consistent one-dimensional treatment of the class of beam problems dealing with the plane deformation of originally plane beams. Our principal result is a system of non-linear strain displacement relations which is consistent with exact one-dimensional equilibrium equations for forces and moments via what is considered to be an appropriate version of the principle of virtual work.

Having a consistent system of equilibrium and strain displacement equations it is further necessary to stipulate, or rather to establish by means of an appropriate set of physical experiments, an associated system of constitutive equations. We discuss the nature of this aspect of the problem, including a solution of its linearized version, but without arriving at the solution of the general problem.

The principal novelty of the present results is thought to be a rational incorpo- ration of transverse shear deformation into one-dimensional finite-strain beam theory. A case may he made that the theory, with this effect incorporated, is of a more harmonious form than the corresponding classical theory, where account is taken of finite bending and stretching, while at the same time it is postulated- following Euler and Bernoulli-that the transverse shearing strain is absent, with the corresponding force being a reactive force.

As an application of the general work a solution is given of the problem of circular ring buckling, including consideration of the effects of axial normal strain and of transverse shearing strain on the value of the classical Bresse-Maurice L6vy buckling load.

Kinematics of Beam Element

We consider an element ds of a one-dimensional beam with equations x = x (s) and y=y(s) before deformation. We designate the tangent angle to the beam curve by ~P0 and write cos ~o o = x' (s) and sin Cpo = y' (s), where primes indicate differentiation

1) A report supported by the Office of Naval Research and the National Science Foundation, Washing- ton, D. C.

Page 2: On One-dimensioON ONE-DIMENSIONAL LARGE-DISPLACEMENT FINITE-STRAIN BEAM THEORY nal Large-displacement Finite-strain Beam Theory Reissner

796 Eric Reissner Z A M P

with respect to s. We note that Cpo is also the angle between the normal to the beam

curve and the y-axis. Due to deformation the points x=x(s) and y=y(s ) of the undeformed beam

curve are changed to x(s)+u(s) and y(s)+v(s). We now assume that transverse elements which were originally normal to the beam curve do not necessarily remain so but end up enclosing an angle �89 ~ - Z with this curve. At the same time we designate

the angle enclosed by such an element and the y-axis by q~. We then have a geo- metrical situation as shown in Figure 1. We note in particular, in addition to the

. , , - - - x + u + ( x + u ) ' d s

-.- x~l+u(s)

y(s)+v(s) x(s) - - %

h "~-% y(s)

I f Figure 1 x

angle X, the relative change of length e of the beam curve element ds, and the change o f the angle ~00 into an angle q0, and we read from the deformed beam element, as

relations between Z, cO, e, u and v,

_ _ y'+v' (la, b) x'+u' = cos(~0 + Z), ~ = sin(q~ + Z). l + e l + e

Dynamics of Beam E|ement

We now consider the deformed beam element, with normal and shear forces N and Q and with a bending moment M, in accordance with Figure 2. Together with

this we assume force load intensities Px and py and a moment load intensity m, per unit of undeformed beam curve length, also in accordance with Figure 2.

We then read from Figure 2 as component equations of force equilibrium in the

directions of x and y,

(N cos q~-Q sin q0)' + p~= 0, (2a)

(N sin ~o+Q cos ~0)' + p y = 0. (2b)

Page 3: On One-dimensioON ONE-DIMENSIONAL LARGE-DISPLACEMENT FINITE-STRAIN BEAM THEORY nal Large-displacement Finite-strain Beam Theory Reissner

Vol. 23, 1972 On One-Dimensional Finite-Strain Beam Theory: the Plane Problem 797

O'~Q'dst~k M+M'ds

, ~ 2 _ _ * % ~ _ _

O mdsl /'% \

Figure 2

At the same time we obtain as equation of moment equilibrium

M ' + ( 1 +e)(Q cos z - N sin x ) + m = 0 . (3)

We note, for future use, the possibility of deducing from (2a, b) the relations

N'-qCQ+n=O, Q'+q/N+q=O, (2*a, b)

where n = Px cos cp + py sin (p and q = py cos (p - Px sin cp are components of load intensity in the directions of N and Q, respectively.

Constitutive Equations

We postulate that the material of the beam is elastic and that we have the existence of axial and transverse force strains s and 7 and of a bending strain ~, in

such a way that constitutive equations for beam elements may be written in the form

N=f~(~, 7, K), Q= fQ(~, ~, ~c), M = f~(~, ~,, ~). (4)

We are ignorant, at this point, not only in regard to the form of the functions f in (4), but also in regard to definitions for the components of strain e, 7 and tc which enter into the constitutive equations (4)2).

2) However, we expect that e ~ e, 7 ~ Z and ~c ~ q/-~o~, for sufficiently small strain.

Page 4: On One-dimensioON ONE-DIMENSIONAL LARGE-DISPLACEMENT FINITE-STRAIN BEAM THEORY nal Large-displacement Finite-strain Beam Theory Reissner

798 Eric Reissner ZAMP

Defining Equations for Strain

In order to obtain equations for strain we consider a virtual work equation of the form

S2

(N 6e+Q 67+M 6~c)ds Xl

82

= J (Px 6u+p, 6v+m&o) ds (5) Sl

+ [(N cos (p - Q sin ~o) 6u + (N sin p + Q cos ~o) 3v + M 6q~]~,

and we stipulate, as Principle of Virtual Work, that equation (5) be equivalent to the dynamic equations (2) and (3) in the interior of the interval (s 1, s2), given that be, 37 and 6K are appropriate expressions for virtual strains.

Since we know the form of the dynamic equations but do not at this point know expressions for virtual strains we use equation (5), in conjunction with (2) and (3), to deduce expressions for virtual strains.

Introduction of (2) and (3) into equation (5) gives a relation of the form

$2

(N 6~+O 67+M 6~c) ds Sl

82

= - j {(N cos (p -Q sin go)' 6u+(N sin rp+Q cos rp)' gv ~, (6)

+ [ M ' + (1 + e)(Q cos x - N sin )~)] 6q~} ds

+ [(N cos q0-Q sin q0) 6 u + . . - + M bq)]~,

and in this we may now consider N, Q and M as arbitrary differentiable functions ofs. In order to utilize (6) we integrate by parts, thereby eliminating all derivatives

of N, (2 and M as well as the boundary terms on the right. In this way we obtain

S2

j (N 6~+Q bT+M 6~c) ds SI

$2

= ~ [(N cos q0 - Q sin ~0) bu'+ (N sin ~0 + Q cos q~) 6v' (7) S1

+ M 6q0' - (1 + e) (Q cos z - N sin )~) 6qo-1 ds.

The arbitrariness of N, Q and M means that (7) implies the virtual strain dis- placement relations

6~ = (6u') cos q~ + (6v') sin qo + (~5 qo) (1 + e) sin X,

@ = (6v') cos qo - (6u') sin ~o - (6qo) (1 +e) cos X,

(8a)

(8b)

(9)

It remains to take the step from virtual strain displacement relations to actual

strain disNacement relations.

Page 5: On One-dimensioON ONE-DIMENSIONAL LARGE-DISPLACEMENT FINITE-STRAIN BEAM THEORY nal Large-displacement Finite-strain Beam Theory Reissner

Vol. 23, 1972 On One-Dimensional Finite-Strain Beam Theory: the Plane Problem 799

One of these actual strain displacement relations follows directly from equa- tion (9) in the form

~=~0'-~0o. (10)

A correspondingly simple derivation of expressions for e and y is clearly not possible through direct use of (8 a, b). Remarkably, we may obtain e and y by using (8 a, b) in conjunction with the geometrical relations (1). To do this we observe that equations (1) imply the following relations between virtual quantities

6 u' = (6 e) cos (q) + Z) - 6 (q9 + Z) (1 + e) sin (q~ + Z), (11 a)

6v' =(6e) sin (q~ +Z)+6(qo +Z) (1 +e) cos (q0 +Z). ( l lb)

We now use (11a, b) in order to eliminate 6u' and 6v' in (8a, b). In this way we obtain

6 e = (6 e) cos Z - (6)0(1 + e) sin Z, (12 a)

67 =(6e ) sin X + (6)0 (1 +e) cos Z- (12b)

The form of (12a, b) is such that we can now go from virtual strains to actual strains. The results are

e=(1 +e)cos Z - 1, 7=(1 +e) sin Z. (13a, b)

Having (13a, b) we can further express e and y in terms of u, v and cp. Intro- duction of (13 a, b) into (1 a, b) gives first

x' +u ' =(1 +a) cos (P-7 sin q),

y ' + v ' = ( 1 + ~) sin (p+7 cos ~o,

and then, by inversion

= (x' + u') cos q) + (y' + v') sin ~o - 1,

7 = (Y' + v') cos q) - (x' + u') sin ~0.

(14a)

(14b)

(15a)

(lgb)

We finally note the possibility of rewriting the moment equilibrium equation (3) somewhat more simply with the help of the strain components e and y as in (13), in the form

M ' + ( I + e ) Q - v N + m = 0 . (3*)

Observations on the Problem of Experimentally Derived Constitutive Equations

In order to see the nature of the problem of experimentally establishing the nature of the functions f in equations (4) we consider the problem of an originally straight beam, with x = s , y = 0 and ~00=0 , fixed at the end x = 0 and subject to given displacements u(a)= v a, v(a)= va and cp (a)= ~0 a at the other end. We assume

Page 6: On One-dimensioON ONE-DIMENSIONAL LARGE-DISPLACEMENT FINITE-STRAIN BEAM THEORY nal Large-displacement Finite-strain Beam Theory Reissner

800 Eric Reissner ZAM P

absent distributed loads and have then from equations (2a, b)

N c o s ~ o - Q s i n q ~ = X ~ , N s i n ~ o + Q c o s q ) = Y a , (16a, b)

where X. and Y. are two constants of integration the mechanical significance of which is evident.

To proceed further we consider the moment equation (3*) as a differential equation for (p, by writing

Q = Y a c o s q o - X ~ s i n ( o , N = Yasinqo+XaCOS(O, (17a, b)

and by considering the constitutive equations involving N and (2 partially inverted in the form

e = f~(N, Q, ~o'), 7 = f~(N, Q, (p') (18a, b)

so that M = f M ( e , 7, to)= f~(n, Q, q/)= g(~o, qo'). The resultant second-order equation for q~ must be solved subject to the boundary

conditions (p (0) = 0 and cp (a) = qo,, with which (p = qo (x; X~, Ya, q0o). Having (p we find u and v from (14a, b). The boundary conditions for u and v

are satisfied upon setting a

[(1 + e) cos q~ - ? sin ~o] dx = u a Aft a , (19 a) 0

a

[(1 + e) sin (0 + 7 cos qo3 dx = v a . (19 b) 0

We now measure Xo, Y, and M~ as functions of uo, v a, % , and of a, giving a set of three relations X ~ = f x o ( u a, v,, (pa; a), etc. The remaining task then is to deduce from the form of these three experimentally determined functions fxo, fYo and f u~

the form of the desired three functions fN, fe , fM in equations (4). The linear case. We consider a range of stresses and strains within which

{N Q M} = [C] {E 7 ~c} (20)

with a view towards determining the elements CN~,..., CM~ of the three by three

matrix [C]. From equations (17) follow the linearized relations

Q=Q_.,,=Y~, N = N . = X . , (21a, b)

and the moment equation (3*), again with boundary conditions q)(0)=0 and qo (a)= q)a, is reduced to

M' + Qa - ~ " 0. (22)

Equations (19) for the translational edge displacements become a a

j' e dx = u,, ~ (q) + 7) dx = va. (23 a, b) 0 0

Page 7: On One-dimensioON ONE-DIMENSIONAL LARGE-DISPLACEMENT FINITE-STRAIN BEAM THEORY nal Large-displacement Finite-strain Beam Theory Reissner

Vol. 23, 1972 On One-Dimensional Finite-Strain Beam Theory: the Plane Problem 801

In order to solve the problem as stated in (20) to (23) we partially invert (20) in the form

{g 7 M} = I-C*] {N a Qa (P'}, (24)

and write (22) in the form C}~ q/ '+ Qa=0, with solution

X O,X--X 2

~P= Pa ~ -+Qs 2C~t~ (25)

We then have further, from (24),

Ms = , , 1 (26) Gnu Ns+(CuQ-ga) Qs+ C*~ a -1 (Pa

and, upon making use of (23 a, b),

_ �9 Ns+ * Qa+ * G - C~N a C~q a C ~ qG, (27 a)

a3 * a2 (27b)

We now stipulate knowledge of a matrix [B], as a result of experiment, such that

{Ua ?)a (tOa} :" [B] { N s Qa Ms}. (28)

Having (26) to (28) we may then successively determine the elements of the matrix [C*] in terms of the elements of [B]. To see this we write

a C~vIN a-- 2 C*O_ a ~Ps - C* K Ns q 2 C~K Qa + ~ Ms, (26')

and have then from the relation ~0 a = B~u N s + B ~ Qs + B~M Ms that

a - B ~ , a - C * e a C* u _ B~M, (29) C ~ 2 C*~ -BYQ' C*

, from which CM~, C~Q and C}N follow in succession in terms of elements of [B]. We next introduce (26') into (27a, b) and compare the resultant relations with

corresponding relations in (28). In this way we obtain the remaining six elements C~N, etc. of the matrix [C*] in terms of the elements of [B].

Finally, having [C*] we find the elements of [C] by returning from (24) to (20).

Buckling of Circular Rings

As an application of the foregoing we consider the classical problem of in-plane buckling of a circular ring of radius R, subject to a uniform normal pressure p. We wish to obtain a buckling-load formula which incorporates the effects of (1) the symmetrical deformation of the ring prior to the onset of buckling, (2) axial strain associated with the buckling mode, (3) transverse shearing strain associated

Page 8: On One-dimensioON ONE-DIMENSIONAL LARGE-DISPLACEMENT FINITE-STRAIN BEAM THEORY nal Large-displacement Finite-strain Beam Theory Reissner

802 Eric Reissner ZAMP

with the buckling mode. We will be concerned, in particular, with the question of appropriate constitutive equations.

Inspection of Figure 2 indicates that for uniform normal pressure p, per unit of deformed beam curve, we have as expressions for the load intensity components q and n in the force equilibrium equations (2* a, b)

q = p ( l + e ) c o s z = p ( l + e ) , n = - p ( l + e ) s i n z = - p ? , (30a, b)

together with an absent moment load intensity m in equation (3*). We further have, with K as in equation (10) and with R d~0 o = ds, that q0' = R- 1 + ~c.

Therewith the equilibrium equations (2*a, b) and (3*) may be written in the form

N ' - ( R -1 +~c) Q - 7 p = 0 , (31 a)

Q'+(R -1 +to)N+(1 + ~) p---0, (31b)

M ' + ( I + ~ ) Q - y N = 0 . (31c)

In complementing (31) by constitutive equations we have no difficulty in deciding that suitable relations involving tc and e are of the form

M = D t c , e = C N . (32a, b)

In stipulating a relation involving ~ we find it necessary to concern ourselves with the question whether ? would be determined by the force Q tangential to the deformed cross section or by a force Q, normal to the deformed centerline. Evidently, we have Q, given in terms of Q and N by the relation Q, = Q cos z - N sin ;( or, approximately, by Q, = Q - N ?. If we stipulate that y = B Q , we arrive at a relation for , / in terms of Q and N, of the form ?=BQ/(1 +BN)3). If we use Q instead of Q, at the outset we have instead that 7 = BQ. We may subsume both relations to one of the form

Be (32c) Y= 1 + 2 B N '

and consider in the end the two limiting cases 2 =0 and 2 = 1. Having equations (31) and (32) we now consider the stability of the state

N = - P , ~ = - ~ p , Q = M = ? = ~ c = 0 (33)

for which, evidently, in view of (31 b) and (32b)

P = (1 - ~e) R p, ep = CP. (34)

We now write

N = - P + N 1, e = - ~p + q , (35)

and linearize (31) and (32) in terms of Q, M, 7, ~:, N1 and el.

3) This, t~gether with (3 2 b ), is e~ective~y equiva~ent t~ c~nstitutive equati~ns ~f the f~rm Q=(~'/B) + (~ ?~/ C) and N=(8/C)+(y2/2 C)-

Page 9: On One-dimensioON ONE-DIMENSIONAL LARGE-DISPLACEMENT FINITE-STRAIN BEAM THEORY nal Large-displacement Finite-strain Beam Theory Reissner

Vol. 23, 1972 On One-Dimensional Finite-Strain Beam Theory: the Plane Problem 803

Equations (31) become

N ; - R - 1 Q - p ? = O ,

Q' + R -1N 1 - P tc + p ~1 = 0 ,

M ' + ( 1 - e e ) Q + P ~ = 0 .

(36a)

(36b)

(36c)

Equation (32 a) remains as is and equations (32 b) and (32 c) become 4)

51 = CN 1, 7 = ( 1 - 2 BP) -1 BQ. (37a, b)

We now use (32a), (34) and (37a, b) to write (36a, b, c) as a system of equations for N1, Q and ~c, as follows

P B NI '- (1-~ 1 - CP 1--2BP) Q~-=O' (38a)

CP Q ' + ( l q 1 - c P ) N1 ~ - - P f c = 0 , (38b)

( BP ) Q = 0 . (38c) Dtr 1-CP-~ 1 - 2 B P

It is evident that (38 b), differentiated once, may be written with the help of (38 a) and (38c) as one second-order differential equation for Q.

Appropriate solutions, for a complete ring, will be of the form Q = cos n s/R where n =2, 3 . . . . . From this follows as the equation for possible values of P,

( )( 1 . . ) , rt 2 C P 1 -+ R2 ~ 14 1 - C P 1 - C P 1-2BP R 2

,( . . ) + ~ I-CP-~ l _ 2 B p =0.

(39)

Equation (39) may be written as a cubic equation for pR2/D, involving axial-strain and transverse shear-strain parameters k~= CD/R 2 and k~,=BD/R 2. We will here limit ourselves to a discussion of the case k~=0, with ky=-k, for which the cubic equation reduces to a quadratic of the form

k(pR2/D) PR2 (14 l_2k(pR2/D) )=n2-1 . (40) 1--2 k(PR2/D) ~ -D- - k(pR2/D)

The smallest positive value of P follows from this for n = 2. We consider in particular the cases 2 = 1 and 2 = 0.

4) We note the possibility that C and B, as well as D in equation (32a), may be considered to depend on ee.

Page 10: On One-dimensioON ONE-DIMENSIONAL LARGE-DISPLACEMENT FINITE-STRAIN BEAM THEORY nal Large-displacement Finite-strain Beam Theory Reissner

804 Eric Reissner ZAMP

When 2= 1 we have from (40), in agreement with a recent result by Smith and Simetses s).

PR 2 1 3D l + 4 k -1-4k+O(k2)" (40a)

When 2 = 0 the solution is

3D - 6k 14 ( k + l ) z 1 = ~ + 1 1 ( k + l ) 2 ~-..-

= 1 - 4 k + O(k2). (40 b)

For small k, say for k up to 2~, the values of PR2/3D given by the two different formulas differ by very little. For larger k equation (40a) is the conservative one, giving a larger shear correction effect than equation (40b). To illustrate, when k = �88 then (40 a) predicts that transverse deformability reduces the classical buckling load by 50 %, while (40b) predicts a reduction by 42 ~.

s) j. Eng. Mech. Div., ASCE 95, EM3, 559-569 (1969).

Summary The paper formulates a one-dimensional large-strain beam theory for plane deformations of plane

beams, with rigorous consistency of dynamics and kinematics via application of the principle of virtuai work. This formulation is complemented by considerations on how to obtain constitutive equations, and applied to the problem of buckling of circular rings, including the effects of axial normal strain and trans- verse shearing strain.

Zusammenfassung Das Ziel dieser Arbeit ist eine eindimensionale Theorie mit endlichen Dehnungen und Schub-

form~inderungen, fiir ebene Verformungen yon urspriinglich ebenen Balken. Das wesentliche der Theorie ist die genaue Vertr~iglichkeit der dynamischen und kinematischen Gleichungen, insoweit das Prinzip der virtuellen Arbeiten in Frage kommt. Die vorstehenden Entwicklungen sind vervollstiindigt durch Be- trachtungen iiber das Problem der Aufstellung yon Spannungs-Form~inderungsbeziehungen und durch eine Anwendung auf das Knickproblem des Kreisringes einschliesslich des Einflusses yon Axialdehnung und Schubverformung,

(Received: May 5, 1972)