The non-linear flexural-torsional behaviour of straight slender elastic beams with arbitrary cross sections

The non-linear flexural-torsional behaviour of straight slender

elastic beams with arbitrary cross sections

Citation for published version (APA):

Erp, van, G. M., Menken, C. M., & Veldpaus, F. E. (1988). The non-linear flexural-torsional behaviour of straight slender elastic beams with arbitrary cross sections. Thin-Walled Structures, 6(5), 385-404.

https://doi.org/10.1016/0263-8231(88)90019-5

DOI:

10.1016/0263-8231(88)90019-5

Document status and date: Published: 01/01/1988 Document Version:

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Thin-Walled Structures 6 (1988) 385--404

The Non-linear Flexural-Torsional Behaviour of

Straight Slender Elastic Beams with

Arbitrary Cross Sections

G. M. van E r p , C. M. M e n k e n & F. E. V e l d p a u s Department of Mechanical Engineering, Eindhoven University of Technology,

Eindhoven, The Netherlands (Received 25 July 1987; accepted 19 May 1988)

A BS TRA C T

A potential energy fimctional ]or the non-linear flexural-u~rsional behaviour o f straight slender eht~tic beams with arbitrary cross sections is derived. The result is generally applicable u~ situations where the strains are small and the Bernou-Ili hypotheses are valid. It is shown that special theories can be derived from the general expressions in a consi.~tent mariner, by neglecting some specific terms.

NOTATION ff

lal_

A A A c A . K A - B ~ , i , ~ eq Vector Length of

Inner product of two vectors Cross product of two vectors Dyadic product of two vectors U n d e f o r m e d cross sectional area Second-order tensor

Conjugate of A

Inner product of a tensor and a vector Inner product of two tensors

Unit vectors Unit base vector

Cartesian components of the Green-Lagrange strain tensor 385

Thin-Walled Structures 0263-8231188l$03.50 O 1988 Elsevier Science Publishers Ltd,

386 (;. ,U. van Erp, C M. M e n k e n . F. E, Vehtpatts F H I, 12. L L [)i, Pi qi s S lls. V,, 14.' s X l ' , g . F E Xi S u b s c r i p t s S 0

Deformation gradient tensor Higher-order torsion constant

Polar moment of inertia about the shear centre

Moments of inertia about the v and z axes respectively Length

Cartesian component of force vector

Cartesian component of distributed load vector

Radial distance of a point on the cross section to the shear centre Arc length

Surface

Displacement components of the shear centre in the directions e ~ , e 2, e3 respectively

Coordinate along beam axis

Coordinate along the principal centroidal axes

Coordinate along 9 and ~. axes through the shear centre Geometric constants Warping constant Strain Curvature components Curvature vector Gradient operator Shear centre Undeformed state 1 INTRODUCTION

During the past 15 years several articles ~-~3 on non-linear flexural-torsional behaviour of beams have been published. In the majority of these articles, a potential energy functional in terms of displacements and rotations is used, which is mostly derived in the following manner.

First, a displacement field containing the components of the rotation matrix is determined; then this field is used to calculate the non-linear strain components and, finally, these strains are used to determine the potential energy functional.

Deriving the potential energy functional in this manner, without intro- ducing various approximations, is almost impossible, because both the components of the rotation matrix and the strain expressions in terms of displacements are lengthy and complicated in their exact form. Therefore most of the articles on non-linear flexural-torsional behaviour of beams

Non-finear flexural-torsional behaviour of elastic beams 387

restrict themselves to a special class of deformation as a consequence of the approximations made.

In this paper, a coordinate-free dyadic notation is used to avoid approximations concerning the magnitude of the deflections and rotations. This enables the derivation of a potential energy functional and curvature expressions which are generally applicable to situations where the strains are small and the Bernoulli hypotheses are valid.

The special theories of the articles mentioned can be derived easily from these general expressions by neglecting specific terms.

Rotations in non-linear beam theory are mostly described in terms of Euler angles or modified Euler angles. This results in a 'geometric torsion' expression which is asymmetric in the transverse displacement components and therefore often leads to confusion. In this report, the rotations are described in a special way, resulting in a 'geometric torsion" expression which is skew-symmetric in these components, as would be expected.

2 T H E G E N E R A L POTENTIAL E N E R G Y F U N C T I O N A L Assuming the strain energy density to be a quadratic functional of the Lagrangian strain tensor components, e~/= (1/2)(u~./+ u/.,+ tth.iUh./) (i. j, h = 1, 2, 3), and taking into account the condition e22 = e33 = - / / e t l . the potential energy functional (see Ref. 14) may be written as

7rt[iT] = ~ [E~, +4G~, +4O~3]dV- poq, u, d V - P0,u, dS (1)

where i7 is the displacement field, Vo is the volume of the undeformed body, S ° is the part of the boundary where the loads are prescribed, p0 is the mass density, q, represents the body forces per unit mass, P~ are the prescribed external loads, E is Young's modulus and G is the shear modulus. In this equation the summation convention is applied, which means that repetition of an index in a term denotes a summation with respect to that index over its range.

3 K I N E M A T I C S OF S T R A I G H T S L E N D E R ELASTIC BEAMS A straight slender prismatic beam of length L is shown in Fig. l(a). The beam is made of a homogeneous isotropic linear elastic material. Each material point in this beam is described by a rectangular Cartesian system of coordinates (x, y, ~,). The coordinate x coincides with the elastic axis of the

388 G. M. van Erp. C. M. Menken. F. E. VeMpaus

i ~ , ~ _

Fig. 1. (a) Beam with coordinate system. (b) Cross section.

b e a m , defined as the line which connects the shear centres of the cross sections of the beam. If the shear centre is not a material point of the cross section, as is often the case for thin-walled open sections, it is still considered to follow all the deformations of the cross section, as if it was a real material point of that cross section. It is assumed that before deformation the elastic axis is a straight line. With x representing length along this axis, it can be r e p r e s e n t e d by

-r0 = x e l ; xE[0, L] (2)

In the u n d e f o r m e d state the cross section is oriented such that -~., and -~3 are parallel to the principal axes. T h e position of an arbitrary material point, b e f o r e d e f o r m a t i o n , is given by

~0(x, y, ~) = 7o(X) + ~0(Y, ~) (3)

w h e r e if0 = Y~z + ~'~3, while y and ~' d e n o t e length along the ~2 and -~3 axes. Besides the coordinate axes 9, ~, a second set of coordinate axes is defined in the cross section, parallel to y, ~ but with the origin located at the centroid (Fig. l ( b ) ) , y , ~, and y , z are related by

= (y - ys) and ~ = (z - zs) (4)

y~, Z~ are the y and z coordinates of the shear centre.

A f t e r d e f o r m a t i o n , the position vector of a material point is given by ~(x, y, ~,). In o r d e r to determine Y the following assumptions are made:

- - T h e total deformation of the beam can be considered as the result of two successive motions: first, a rigid translation and rotation of each cross section due to bending and warping free torsion; next, a warping displacement perpendicular to the displaced cross sections.

- - T h e cross section does not distort in its plane during deformation. - - S h e a r d e f o r m a t i o n due to transverse forces can be neglected.

Non-linear flexural-torsional behariour of elastic beams 389

I G(0)

Fig. 2. Beam after warping free motion.

T h e position vector 2 can now be expressed as ~(x, y, ~) = 7(x) + ~(x, y, ~) +f(x. y. ~.)7~ w h e r e

(5)

a ( x . y, z) =

+ 73(x);

7 = ~(x) r e p r e s e n t s t h e beam axis in the d e f o r m e d configuration; i 2, i 3 are unit vectors parallel to the principal axes of the cross section after the warping free m o t i o n ;

f(x, y~, ~.) i,

represents small normal warping displacements (Tt = i~ • i~).

If the d i s p l a c e m e n t of a point on the elastic axis is described by its c o m p o n e n t s u . v . wsin the directions e~. e_,. e3 respectively. ~ can be expressed as

7 = (x + u~)-~ + v~-~2 + w~-~3

(6)

T h e unit t a n g e n t vector to the d e f o r m e d elastic axis at a point Q can be o b t a i n e d f r o m

d7

(Q) = ~ ( Q ) (7)

w h e r e s is the arc length along the d e f o r m e d elastic axis. Differentiating 7 with respect to x instead o f s gives

d7 d7 ds

391) G. M. van Erp, C. M. ,Uenken. F, E. Vehtpaus

w h e r e the prime ( ' ) means differentiation with respect t o x . and

I d7 [ ,'-~ w~'2] ': l

Es = ~ - 1 = [ ( l + u ' ) : + t s + - (9)

If shear d e f o r m a t i o n due to transverse forces is neglected, the unit tangent vector ~'(Q) will be perpendicular to the cross section in every point

O.

l I ~ - 12 s l 3 (10)

T h e triad

el, e2,

e3 can be transformed into the triad i,. i._. i 3 by means of a rigid translation and rotation.

T h e rigid rotation can be described by an orthogonal tensor R:

R'-~k = t k ( k = 1 , 2 , 3 ) ( 1 1 )

R = R(x) ; RC.R = R-R c = I ; detR = 1

w h e r e I is the unit tensor and R c is the conjugate of R.

To analyse the beam deformation, that is to define the flexural and torsional curvatures of the beam axis. the derivatives of Tk with respect to s are studied:

d T k _{- -} d R _{o e k} _ ₌ d R .RC.T, ( 1 2 )

ds ds ds

T h e orthogonality p r o p e r t y o f R implies that (dR/ds) • R c is a skew tensor. a n d t h e r e f o r e eqn (12) m a y also be written as

aT,_ _ --aR.Rc-L = .Tk

(13)

ds ds

w h e r e ~ is the axial vector of (dR/ds) • R c.

A c c o r d i n g to the classical definition) 5 the torsional and flexural beam curvatures are defined as the c o m p o n e n t s of the vector ff with respect to the local basis i,, i2. i3:

-fi = x] i, + xzi2 + xai3 (14)

w h e r e X, is the torsional curvature and X-" and X3 are the flexural curvatures. DifferentiatingT, (k = 1, 2, 3) with respect to x instead o f s yields

JTk

as

dx dx ds - (1 + e~)~.-:" t , = × . t k - - ' ( 1 5 ) w h e r e ~ = (1 + e~)~.

Non-linear flexural-torsion,d behaviot,r o f elmtic beams 391

Combination of eqns (14) and (15) yields

= ( 1 + e~)[Xt it + X'- i: + X3 i3 ] ( 1 6 )

The deformation process can be described completely in terms of the deformation gradient tensor F, which is given by

V = (~,,-/)c (17)

The gradient operator with respect to the undeformed configuration can be written as

_ (3 il ?)

q,,

= e , ~ +3,

oy

+-~3~ (IS)

Substitution of eqns (4) and (5) into (17) yields F = [ E'7'+(~'~)+'~f71+f(~'7')0x ~'1

+ '~f 7,-& + '~f7,-~3 + R

ay Oz (19)

where R = (7~, +72~z +73~3).

The Green-Lagrange strain tensor is defined by

E = ~ ( F c . r - I) ( 2 0 )

The components of E which are relevant in beam theory are (see Section 2) ell = -elOEO-e, = ~ [ ( F - - ~ t ) - ( F - - ~ ) - 1]

e12 = 'el t E t e 2 = / [ ( F o e l )"

(Fo-~2)]

(-.2 1) e13 = -et " E ' - e 3 = ½[(F'-et)" (V*-e3)]

Substitution of eqn (19) into (21) yields the expressions for e , . el_- and et3. ~ Before proceeding, attention is focused on the warping function f. In the case of thin-walled open sections, the normalized warping displacements are mostly described in terms of the so-called sectorial area

co(n,p).

In this paper, however, preference is given to the equivalent, but more general, Saint Venant warping function ~b(y.z). If Xt is chosen as the warping amplitude, the functionfcan be .written as

f(x,y,

z) = ×l(x)th(y, z) (22)

(The consequences of choosing Xt as the warping amplitude are discussed in Section 10 where an alternative approach is proposed.)

~9_ G. ,U. van Erp, C. AI. Menken, F. E. Vehtpau.s

Besides the Bernoulli hypotheses, no assumptions have been introduced so far. In the case of small strains, however, strains may be neglected c o m p a r e d with unit,,'. Applying this type of approximation to eqn (21) and substituting eqn ('~'~).. .yields ~

1 /~Xi e~t = ~, + ( -5:X~ + ~,X2) + ~ ( f + ~.:)X~ "- - - ~ b (23) _ & t " e,, = _~ - ~x, + x, ;-GZy ]

(24)

i (

;,,/, )

<3 = ~ Yx, + x, ;-77:, (25)

Using relation (4), e~ may also be written as 1 , , ('~X, ell = g - vX~ + ZXe +-q r - x i + ,b (26) " " _ FIX w h e r e = E,+Y~X~-Z, X2 and , = (.~.2+ ~:) 4 S T R A I N E N E R G Y

In b e a m t h e o r y the strain energy is given by (see eqn (1))

U = -~[Ee-~ + 4Ge~2 + 4Gd3]dV (27)

Since v and = are coordinates along the principal axes, the following identities hold"

amb&4 = 0

(28) w h e r e A is the area of the cross section.

Substituting eqns (23)-(26) into eqn (27) yields

U = 7_ EA-42 + El,. x~- + Els X3 + G J x f + E F \ ax + E H x ;

Non-linear flexural-torsional behaviour of elastic beams 393 w h e r e h = z-dA ;

I =;A ?-'aA ;

H = ~ ~4dA = [3 = f a y2dA ; F = ~a ~ 2 dA = e~ + Ys ~ ( 3 - - Zs X2 • [32 =7-, ( Y 2 + Z - ) z d A - 2 z ~ = 1---[ z r - d A :

( y - + ~ ) y d A - 2 y , = ~'3 y~"dA ; [3, = -~- (y" z-)qJdA ; _es = _{[( 1 + u~)2 + vs" +} " W~2]1/2 _{- 1}

In Ref. 13 a general non-linear beam theory is also derived. The strain e n e r g y expression obtained is the same as in this paper, except for the last t e r m o f eqn (29) which is missing. This term however has to be taken into a c c o u n t in the case of asymmetric sections.

5 T H E P O T E N T I A L E N E R G Y F U N C T I O N A L

T h e total potential energy is the sum of the strain energy U and the potential of the loads f~

rr = U + F t (30)

It is a s s u m e d that the beam is subjected to dead weight surface tractions

p~-dl + Pz-e2 + p3-d3 at both ends (x = 0 and x = L) and q2~2,

q3e3

per unit

length.

Besides these 'external' tractions the beam is loaded by a force q-e3 acting at the centroid, representing the weight of the beam.

T h e potential of the loads m a y be written as

]

f0

~'~-~- E Pi uidA -- [q2u2+q3u3+'qu3(c)]dx (31)

i = l x=O:x=L

w h e r e u3(c) is the displacement c o m p o n e n t of the centroid in the ~3 direction.

394 G. M. van Erp, C. M. M e n k e n . F. E. Veldpaus

The displacement vector/7 of a material point B is given by

K(B) = Y ( B ) - Y o ( B )

(32)

The c o m p o n e n t s of ~(B) in the directions

e., e,_,

e3 can be expressed (see Ref. 14) as

Ut = ~ + Y R t 2 + 7 , R I 3 + x I ~ R ~ I

II 2 = I.' s q'- v ( R 2 2 - l ) + ~R_, 3 + X t ~ b R 2 t ( 3 3 )

it3 = "~'s + .9R32 + ~-(R33 -- 1) + X l l / / R 3 t

where ~ = us - y~ R~, - z~

RI3

and Rij are the components of R with respect to

el, 8"., 83 (Rq = -(i • ij). T h e terms X~tOR,.~ and XL~IIR3t in eqn (35) may be neglected according to the assumption of small strains. ~

Combination o f eqns (29) and (31) yields

7r = 5 EA~g + El, ~(~_ + El3 X5 + GJ?(~ + EF \ &r + EHx~

- [ P I ( U s - y ~ R I : - z~Rt3)+ P,_v~ + P3w~ + M,_RI3 +(-M3)RI,_

+ BXI Rll + ( - Mc)R.,3 +

Mt3R32 + (R,_,_ -

1) fA p2vdA (34) + (R33- 1)

;A p3~dA]~=o

L

-

[q2v~+q=.~(R::-

l) + ( -mt2)R_,3 +

q3w~+mt3R32

+ q3 ~. (R33 - 1) + ~(w~ - - y s R 3 2 - ( R 3 3 - 1 ) 2 s ) ] d x where P1 = y~. pldA

M2 = ;4 (ptz)dA

;

M,, = YA (-p2~.)dA ;

rot2 P-" = Ya P2 dA M3 = f a ( - p l y ) d A M~3 = f a (P3 Y) dA ( - q : ~. ) : rn,3 = q3 9

; P3= f A p 3 d A

• B = YA (p~O)dA

N o n - l i n e a r f l e x u r a l - t o r s i o n a l b e h a v i o u r o f elastic b e a m s 395

E q u a t i o n (34) includes the non-linear contributions resulting from the m o v e m e n t of the points of application of the loads. This contribution, which can have a significant influence on the behaviour of the beam. has to be t a k e n into account, it.,_,

In the case of thin-walled o p e n sections, the average displacement of the cross section is normally written as -''t~

"ff = ttD -- YD Rl2 -- ~'D RI3 (35)

w h e r e D indicates the sectorial origin. In this paper, however, the following expression is preferred

= tq - y ~ R , , . - zsRI3 (36)

It can readily be seen that both eqns (35) and (36) represent the d i s p l a c e m e n t of the centroid as no warping occurs. Since eqn (36) results in simpler expressions, preference is given to it.

Constitutive equations for the normal force N and the bending m o m e n t s M., a n d .'~3 a b o u t the centroidal axes in the d e f o r m e d state are obtained bv integration over the cross section of the normal stress and its m o m e n t s . T h e result is

1 / , , )

M , = E I 2 ( x 2 _{+ "-B-'X;)} ' " (37)

/~3 = E/3(X3 ' . -~fl3X~) :

T h e b i m o m e n t acting on the cross section in the d e f o r m e d state is

B = fA (~o-,)dA (38)

w h e r e o'~ is the normal stress in the d e f o r m e d state. Integration of this expression yields

B = E F x I , ~ + ~ E F ~ , x ~ (39)

w h e r e ,x m e a n s differentiation with respect to x.

C o n s i d e r i n g the integrand of the strain energy U (29) as a function of~,__X,, X2 a n d XLx, it is readily verified that the constitutive expressions of Nt, M,., M3 a n d B are the partial derivatives of the integrand with respect to ~, X2, ×3 a n d XLx respectively.

396 G. M. van Erp, C. M. Menken, F. E. Veldpaus

Let Mz~ be the partial derivative of the integrand of U with respect to X~. The constitutive equation for MI~ may then be written as

Mis = Mt~ - B. x (40)

where Mls is the torsional moment with respect to the shear centre. Calculating Mtx and B~ and substituting the result in eqn (40) yields

Mls = GJx~ - EFXt.xx + x,(EI(~ + EI,_fl,_X: - E/3/~3X3) + 2EHx~

(41)

An alternative expression for M~ which is often used 1"L''t6 is given in eqn (42) and is obtained by solving eqn (37) forT, X-" and X3 and substituting into eqn (41):

Mis = GJxl - EFxt.xx + Xt ( - ~ N + flz~lz-fl3~13 )

+~_Ex, 4 H - - I , B ~ - 1 3 ~ ~

(42)

Equation (42) is the same as the general non-linear differential equation for torsion obtained in Refs 11-13. The differential equations for bending which are used in Ref. 12 do hot contain the terms with/32 and/33 (eqn (37)), but these terms should be taken into account if the cross section is asymmetric.

6 T H E R O T A T I O N T E N S O R R

To express the strain energy and load potential in terms of displacements. the rotation tensor R is studied more closely.

The rigid rotation which transforms the triad et, e2, e3 at a material point Q of the undeformed axis into the triad il, i.,, i3 at Q of the deformed axis is represented by a rotation tensor R. In beam theory this rotation is often described in terms of so-called modified Euler angles. The rotation tensor is then broken down as

R = R ~ . R o o R , , (43)

where R,, Ra, Rv are respectively rotations of magnitude a,/3, 3' about the reference axes ~t, e2, e3. This method, however, has some disadvantages (to be discussed at the end of this section) and therefore an alternative method is used in this paper.

Non-linear flexural-torsional behaviour of elastic beams 397

A n axis with u n i t v e c t o r ~, perpendicular to 7t and 7t, is chosen as rotation axis. W h e n ~t and it are inclined at an angle ~b, ~ can be written as

-7.

= (sin~b)-~-~l • 1 l (44)

A rotation tensor Q. representing a rotation, of an arbitrary vector ~, over an angle y , about an axis with unit vector ~ can be expressed as ~4

Q . ~ = (cosy)~ + (1 - cos~/)-~" 7 + (sin,/)-~* 7~ (45) T h e rotation tensor Rt, which maps Ft on it. can thus be written as R t ' a = (cos$)~ + (1 - c o s ~ b ) ~ . ~ + (sin~b) ~ . (46) w h e r e cos ~b = ~, • it and (sin qb)~ = F t . tt.

W h e n ~ i s m a p p e d o n it by Rt, Rt.~2 and R t - ~ j will generally not coincide with i2 and i 3 respectively. Coincidence o f R t • e2 and Rt "e3 with 7_, a n d 73 can be achieved by an additional rotation a about 7t. It is. however. also possible to rotate-~2 and e3 over an angle 13 about et first such that, if R~ is applied to the rotated basis, R, °~_,* and R t °-e~' coincide with i, and i 3.

F r o m a kinematical point of view both approaches are equivalent but, since the second a p p r o a c h leads to simpler expressions, preference is given to it. T h e rotation about ~t, o v e r an angle/3, is represented by a rotation tensor R_,. T h e rotation tensor R = R~ ° R2 may be written (see Ref. 14) as

R . ~ = [(cos$)l --el it + (1 - cosqb)~]-[(cos/3)~ + (sinfl)-~,* ~] + it-e, "~ (47) W h e n strains are neglected c o m p a r e d with unity, the following c o m p o n e n t s Rij o f R with respect to et, e:, e3 are obtained:

1 + u" - v; cos/3 - w; sin/3 - w" cos/3 + v~ sin/3 ]

v~ (l+u~)cosO+b(w;2cos#-w'v~sin#) -(l+u~)sinO+b(-w~v'cosl~l-w'ZsinB)l w" (l+u~)sin~+b(-w~v~cosfl+v'Zsinfl) (l+u~)cosfl+b(v;2cos~+w'v~sinfl) .J (48) w h e r e (1 + u~) = cosqb = x / ( l - vg z - w "2) and b = , 2 l - c o s ~ I - X / ( 1 - v ~ -w~ 2) t-) t,) sin2~ v s- + W s"

398 G. AI. van Erp, C. M. Alenken. F. E. I,'ehlpaus 7 C U R V A T U R E S

T h e r o t a t i o n t e n s o r R consists o f two successive r o t a t i o n s (R = R, . R . ) . R c • ( d R / d . r ) • K m a y t h e r e f o r e also be written as

R c o d R o ~ = RCoRCo ( d R , °R .dR" ]

cLr _ ~---~--

:+R,-~r'/o~/

(49)

= ( R C . ~ + ~ ) . ~

w h e r e / 7 , a n d ~ are the axial vectors o f R c . (dR,/d,r) a n d R ~ . (dR_,/d.t) r e s p e c t i v e l y .

T h e axial v e c t o r k o f a skew tensor QC. ( d Q / d x ) , with Q given as

Q . ~ = (cosy)-~ + (1 - cos-y)-e~.~ + (siny)-~,~ c a n be w r i t t e n (see Ref. 14) as

d~ ,~ + s m y - - ~

~

= -~ + ( l - c o s y ) (501

W i t h e q n (50) the following c u r v a t u r e expressions (see Ref. 14) can be d e r i v e d

1 - cos 05 ( ,t'~ v~' - w~' v" ) ]

Xi = /3'-~ sin2b _]

t - cos,b

X2 = cos/3 -w~ sinZbcos~b ( I"s ~'s l*$'s + 14 s . . . 2w;, ) ]

1 - cos 6 ] + s i n/3 v~" + s i n'- ~b cos ~b (v~ 2 v~' + w" w" v~ ) (51 ) 1 - cos~ . . . ] X3 = cosB v~'+ s i n 2 6 c o s ~ (v'2v, +w~w, v') 1 - COS ~b + s i n B w~'+ sin'- -/ cos 6 "1 ,F , t t ¢ 12 pc | (I,s ~s Ws +yes Ws )

_]

w h e r e

coscb = (I - I'~: - w~:) ''2 and sin2~ = (v~ 2 + w "2)

Non-linear flexural--torsional behaviour o f elastic beams 399

terms of modified Euler angles. If these angles are expressed in terms of transverse displacements, )¢~ may be written (see Ref. 7) as

., ,, ,,, )(

,,, ., I's ~ s 14s !

X l ~-" tiff + I"s 14"s + - - - . - - . ~ 2 ,9 - -

W~2)1/'2_

1 - w ~ " 1 - v ~ "

(52)

where ~ is an Euler angle. Equation (52) is asymmetric in v~ and w~. That this may lead to confusion is shown in Ref. 11.

If the m e t h o d of this paper is used to describe the rotations, Xt is given by

1 - ( 1 - v ~ - - w " ) t/-

= ,, [ v " w" - w " v ' ] (53)

X~ ~ ' + vs- + w "2

This equation is skew-symmetric in v~ and w~ as would be expected.

In Ref. 13 a second-order beam theory is derived in a totally different m a n n e r from that in this paper. The expression derived for Xt is

= +~(v~ w s - w , v;) (54)

If in eqn (53) (1 - v "2 - w'2) ~2 is approximated by 1 - ½v '-~ - ½w'2. the same expression is obtained.

Substitution of the curvature expressions eqn (51) and rotation matrix c o m p o n e n t s eqn (48) into the potential energy functional eqn (34) renders a functional in terms of the displacements u , v , w~ and the rotation/3. The only approximation applied so far is the replacement of terms of order (1 + E) by unity, in accordance with the assumption of small strains. The expressions are therefore valid for beams exhibiting arbitrary large deflections and rotations.

8 S P E C I A L T H E O R I E S

In m a n y practical problems the magnitude of the deflections and rotations is limited and expressions as accurate as eqns (50) and (53) are not needed. T h e r e f o r e simplified expressions are derived in the following by neglecting higher o r d e r terms.

8.1 Beams exhibiting moderate deflections and large rotations

T o d e t e r m i n e the order of magnitude of the terms that should be retained in the case of m o d e r a t e deflections, a side-step is made at this point, and the classical case of a beam loaded by a bending m o m e n t in its plane is studied.

400 G. M. van Erp, C. M. Menken, F. E. Veldpaus

R =

I

V ( 1 - v '2) v' ~ / ( l - v - v ' '2) O ] 0

0 0 l

-i~ = (1 + u ' ) ~ + v'-~2

cos0 = -e1"71 = (1 - v'-')I/2---,sin0 = v'

dO (55) C O S 0 - - = V" ctr dO X = - - = v " ( l - v ' 2 ) -t/2 d x

Fig. 3. Beam loaded by a bending normal.

E q u a t i o n s (55), which can be found in many standard text books on applied m e c h a n i c s , represent the exact expressions w h e n strains are neglected c o m p a r e d with unity. In the linear theory, the term cos 0 is put equal to unity; this results in

R =

1 - v ' 0

v' 1 0

0 0 1

and X = v" (56)

T o distinguish the theory of m o d e r a t e deflections from the linear theory, at least o n e extra term should be taken into account in the approximation of cos 0. This results in

cos0 = 1-½v'" (57)

Substitution of this expression into (55) yields

R = I ,'l 1 --:~ IVt2

0 0

and X = v " + ~ v ' ' v ' 2 (58)

F r o m this it can be concluded that, in the case of m o d e r a t e deflections, t e r m s quadratic in the derivatives of the deflections should be retained c o m p a r e d with unity.

A p p l y i n g this type of approximation to the rotation matrix and curvature expressions derived in this paper yields

Non-linear flexural-torsional behaviour o f elastic beams 401 R = i 1 - .,. v,- - ~ w , " i i • r "P w; - v~ c o s / 3 . - w~ sinfl (1 - [ v;:)COS/3 - ~ w ; v', sin 8 (1 - ~ w~Z )sin fl - ½ w; v~ c o s fl - w~cosB + v~ sin/3 -(1 - ~v;")sinl3 - t.,.w~ v" cos[3 (1 - [w;")cosB + ~w" v; sinB rr ,tt ,t Xt = / 3 ' + ~ ( w ~ v , - ~ , ~ Ls) Xz = cosfl[ - w~' - "(w~ 2 w~' + ~'~ v~' w~ )] + sinfl[v;' + [(v~ 2 v" + w" w~' t'~ )] t t t I s , r r , r t ~ X3 = cosfl[v~' + ~(v; 2 v;' + ws w~ v ' ) ] + sinfl[w~' + ~(v~ ,~ ns + wV w;')] (59)

T h e strain e~ is now given by

t p2 ~ = u" + Iv': + ~w~

T h e rotation matrix and curvature expressions for a b e a m loaded by bending in its plane in the case of m o d e r a t e deflections eqn (58) can be obtained from eqns (59) and (60) by putting/3 and ws equal to zero.

T h e rotation matrix and curvatures for problems with m o d e r a t e deflec- tions and m o d e r a t e rotations can be obtained from eqns (59) and (60) when in these expressions cos/3 and sin/3 are replaced by

cosfl = l - ~ f l 2 and sinfl = / 3 - A f t 3 (61)

F o r p r o b l e m s with m o d e r a t e deflections and small rotations cos/3 and sin/3 m a y be replaced by

cosB = 1 and sin/3 = / 3 (62)

8 . 2 B e a m s e x h i b i t i n g s m a l l deflections a n d large r o t a t i o n s

F o r this specific class of problems only terms linear in the derivatives of v, and ws have to be retained in the rotation matrix and curvatures. This implies

R =

I

v~ 1 -v~cos/3 - w~ sin/3 cos/3 -w~cos/3 + v~sin/3 - s i n B

w~ sin/3 cos/3 (63) )~1 = /3t X: = - cosBw~' + sin/3v~' X3 = cos/3v~' + sin/3w;' ~s = ll; (64)

402_ G. M. van Erp, C. M. Menken, F. E. Veldpaus

Comparison of eqns (61), (62), (65), (66) with the results of Ref. 11 shows that, due to the neglect of the geometric torsion in that paper, the results are only valid for small deflections and large rotations, acc'ording to the terminology of this paper.

The rotation matrix and curvatures for problems with small deflections and moderate rotations can be obtained from eqns (63) and (64) when cos/3 and sin/3 are replaced by

COSfl = 1--½/~ 2 and sin/3 =/3-~,/33 (65)

If small deflections and small rotations are considered, only first-order terms in fl,/3' and the derivatives of vs and w, have to be taken into account. Equations (48) and (51) then reduce to the following well known forms:

R = 1 -v~ -w; v', 1 -/3 L w~ /3 1 and X-" = - w~' (66) X3 = v~' t Es = /~s 9 C O M P A R I S O N WITH O T H E R RESULTS

During the last 15 years several articles dealing with the non-linear flexural- torsional behaviour of beams have appeared in the literature. Almost all these articles are limited to a special class of deformations and/or beams. In Ref. 14 it is shown that these special theories can be derived from the general expressions of this paper in a consistent manner, by neglecting specific terms.

10 W A R P I N G

In the case of slender beams, the warping displacements are mostly written

a s

f ( x , y , z) = Xl(x)O(Y, z) (67)

However, by postulating that the amplitude of the warping displacement equals X~, it is also postulated that, at a section where warping is prevented, the strains e~2 and e~3 have to vanish too (eqns (24) and (25)). This is certainly not the case in reality and should in general not be assumed, since these

Non-linear flexural-torsional behaviour of elastic beams 403 strains are proportional to the stresses o-~, and o'~3. To overcome this p r o b l e m , an expression similar to the one proposed by Reissner m7 for the case of non-uniform linear torsion can be used:

f(x, y, z) = g(x)dj(y,z) (68)

where g(x) is a function yet to be determined.

Reissner ~7-t9 showed that, in the case of non-uniform linear torsion of thin-walled beams with open cross sections, the practical improvement gained by working with eqn (68) instead of eqn (67) will in general be negligible. For non-uniform linear torsion of thin-walled beams with closed or partly closed cross sections, however, the more accurate eqn (68) leads to results which are quite different from what would follow from a use of eqn (67). In the case of beams with an arbitrary cross section undergoing both bending and non-uniform torsion, eqn (68) must also be expected to lead to m o r e accurate results than eqn (67). The influence of this alternative warping approach is at present being investigated at the Eindhoven University of Technology.

11 C O N C L U S I O N S

A potential energy functional for non-linear flexural-torsional behaviour of straight elastic beams with arbitrary cross sections has been presented. The functional is generally applicable to situations where the strains are small and the Bernoulli hypotheses are valid. It has been shown that the potential energy expressions for special theories can be derived from this general expression in a consistent manner, by neglecting specific terms.

A C K N O W L E D G E M E N T This work was supported by research grants made Netherlands Technology Foundation (STW).

available by The

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