Finite element analysis of two-dimensional shear flexible frame structures: Nonlinear analysis

COMPUTER METHODS IN APPLIED MECHANICS AND ENGINEERING 67 (1988) 55-68 NORTH-HOLLAND

FINITE ELEMENT ANALYSIS OF TWO-DIMENSIONAL SHEAR FLEXIBLE FRAME STRUCTURES: NONLINEAR ANALYSIS

Dimitrios KARAMANLIDIS University of Rhode Island, Department of Civil Engineering, Kingston, R1 02881, U.S.A. Received 13 January 1987

Two curved beam/column elements are developed for the nonlinear pre- and postbuckling analysis of free-form frames and arches undergoing arbitrarily large deflections (displacements and rotations). For both elements, the formulation of the pertinent matrices is pursued on the basis of modified versions of the variational theorem due to Hellinger and Reissner with the effects of transverse shear deformation and bending-stretching coupling included. Numerical results are presented for a number of well-selected test problems which demonstrate that both elements represent reliable and highly accurate tools for the numerical solution of geometrically nonlinear structural mechanics problems.

Nomendature A

C D exx E E I K0 K~ Kt r u

cross-sectional area of the curved beam, configuration, inverse constitutive matrix of the curved beam, linear part of axial strain, Young's modulus of the material of the curved beam, constitutive matrix of the curved beam, moment of inertia of curved beam, linear part of tangent stiffness matrix, geometric stiffness matrix, tangent stiffness matrix, residual force vector, displacement vector at a generic point of the curved beam,

WEF

z(x)

element nodal displacement vector, work by external forces, function describing the curved centroidal line of the beam.

axx axial strain at a generic beam point, axz transverse shear strain at a generic beam point, /3 vector of generalized stress coefficients, e strain vector at a generic point of the curved beam, •/~,, nonlinear part of axial strain, K bending strain. A(.) symbol for incremental quantities, (").x derivative with respect to x.

~. Introduction In recent years, the development of beam/column elements that can be used to numerically solve nonlinear problems involving large displacements, but small strains, has become the 0045-7825/88/$3.50 ~) 1988, Elsevier Science Publishers B.V. (North-ltoUand)

56

D. Karamanlidis, Shear flexible frame structure: Nonlinear analysis

subject of increasing interest. Based on a consistent virtual work approach, Argyris et al. [1] proposed a method (that has meanwhile been adopted by others) of developing the geometric stiffness matrix of a straight beam in space having a total of twelve kinematic degrees of freedom. With the same matter deals also a recent paper by Stein [11] where, however, unlike in [1] emphasis was placed not on the development of a particular element but rather on the mathematical foundations of finite elasticity beam problems. The development of consistent finite deflection so-called Timoshenko beam theories has been pursued in the recent papers by Hasegawa et al. [5], Sheinman [8], and the present author [7]. References [1, 11] are representative for the vast majority of currently existing geometrically nonlinear beam models for they both consider the standard assumed displacement finite element variant only. On the other hand, recent work by Noor and coworkers [2-4] and by this author [12-14] has clearly demonstrated that when applied to geometrically nonlinear problems, so-called mixed finite element models have to offer certain advantages from both a theoretical and a computational standpoint. For example, within the framework of the mixed formulation it is a relatively easy matter to develop simple and accurate curved elements capable of exactly representing the rigid-body and constant strain modes (patch test). Achieving the same goal in the case of assumed displacement olements is almost impossible. The purpose of this paper is to further explore the potential of properly formulated mixed element models for theanalysis of geometrically nonlinear flee-form arches and flames. More specifically, on the. basis of previous work by this author [12-14] two curved beam/column elements have been developed using modified versions of the variational theorem due to Hellinger and Reissner [15] as discretization bases. In the case of the first element, the displacement vector Au = [Au, Aw, A0J t and the stress resultant vector Ao-= [aN, AQ, AM] t are treated as independent variables whereas in the case of the second element the displacements Au, Aw, and A0 along with the shear strain increment AeQ have been approximated independently from each other. For both elements, a stiffness matrix is obtained, therefore, their implementation into the library of any existing general-purpose finite element code does not pose any difficulties whatsoever.

2. Mathematical formulation

2.1. Kinematics Let a curved plane beam be considered which undergoes large displacements (Fig. 1). In describing its nonlinear behavior the following assumptions are made: (i) The current configuration C tN) is used as reference configuration ('updated Lagrangian' description). (ii) The beam is shallow (cf. [15]) before and after deformation. (iii) Plane beam sections before deformation remain plane after deformation with shear deformation effects being inc:, :ded ('Timoshenko beam' theory). ,,. (iv) Deformations in thickness direction are negligible. (v) Strains are small. Then, the geometric strain displacement relations of the beam can be formulated as follows (cf. [7]) for more details):

D. Karamanlidis, Shear flexible frame structure: Nonlinear analysis

.

57

z{

Fig. 1. Shallow beam element undergoing large deformations.

A a ~ = Ae~. + Anx~ + ~'- A~,

(la)

Aa~z = A0 + A w , . ,

(lb)

Aexx= Au,x + z,~ Aw~,

(2a)

A~xx --"

(2b)

where ,

{

"(AW,x)2,

In the foregoing, Aa.. and Aa.z are incremental Green strains [15]; ~ denotes the distance P o e (Fig. 2); z --- z ( x ) is a function describing the curved beam geometry; and, ('),x indicates differentiation with respect to the local coordinate x.

C

~o~

Fig. 2. Kinematic variables.

i"

58

D. Karamanlidis, Shearflexible frame structure: Nonlinear analysis

2.2. Variational bases

Let now a free-form geometrically nonlinear two-dimensional frame be considered (Fig. 3) that has been represented as an assemblage of (n) shallow curved elements and denote by: 1 the distance between the two nodes of a single element; Au = [Au, Aw, A0J t and A [ r = [AN, AQ, AMJ' the incremental displacement and stress vectors at a generic element point, respectively; and E A , El, G A s the longitudinal, flexural, and shear stiffnesses, respectively. In this case, the variational theorem due to Hellinger and Reissner [15] reads as follows:

q'rR(A~,Aor)~---~n{--f/ r(AN)2 L

(AQ)2

+ 26A

+

(AM)2] dx4-f/N-~(~Wx) 2dx 2El

+ £ [AN" (5u x + z x" Aw,x) + AQ" (AO + Aw,x) + AM. AO,xI dx} - (WEF) + (correction terms) = stationary,

(3)

where (WEF) stands for the work by externally applied distributed and/or concentrated forces while the correction terms account for inherent linearization errors (e.g., 'equilibrium imbalance' or 'compatibility mismatch', cf. [16]). The stationarity of the functional in (3) is contingent to the a priori satisfaction of the geometric boundary and interelement continuity conditions ('essential' constraints). In accord with the fundamental work by Prager [17] and Buffer [18] no essential constraints whatsoever need be fulfilled by the stress resultants. It is worth noticing that by taking the variation of the functional in (3) with respect to the stress vector A¢, the following Euler-Lagrange equations ('natural' constraints) are obtained:

\

!

°S"

"...\ ! /i

\ Fig. 3. Shallowbeam element: nodal degrees of freedom and forces.

D. Karamanlidis, Shear flexible frame structure: Nonlinear analysis

8AN:

AN - - ~ - + Au,x + z~ Aw,~ = 0,

59

(4a)

~Q

GA---:~+ 40 + Aw~ = 0 ,

8AQ: S~M:

AM

--~

(4b)

+ A0.~--0.

Oc)

These equations indicate that in the case of a finite element model developed on the basis of (3) the compatibility between strains calculated from the stress resultants, i.e., AeQ_] =

r__I./EA j

o

L-

: o lraN-I

-q I' -Q-i .... .- TgrJL-s-

(5a)

Ae = D Ao',

(Sb)

and those calculated from displacement field, i.e., Ae~x = Au,x + z,~. Aw~,

(6a)

Aaxz = A0 + A w ~ ,

(6b)

A,, = Ao,,

(6c)

is fulfilled a posteriori only. Based on previous experiences [12-14] and for reasons to become apparent in the sequel, in this paper not (3) itself but rather two modified versions thereof have been utilized towards the development of geometrically nonlinea~ shear flexible beam elements. The first of these two modified variational theorems is obtained from (3) by constraining the stress resultants so as to a priori fulfill the conditions listed below: AN~ • 0 ,

(7a)

AM x - AQ = 0 ,

(7b)

AQ,x + (AN-z.x). x = 0.

(7c)

The pe~inent variational theorem reads then as follows:

[ (AN)2

(AQ)2

(AM)2 ]

+ [AN-Aa + (AQ + AN. z , ) " A - + AM" A0JS} - (WEF) + (correction terms) = stationary.

(8)

60

D. Karamanlidis, Shear flexible frame structure: Nonlinear analysis

Introducing in (8) appropriate trial functions for the independent variables Au and Ao-, i.e., Au = N A a ,

Ao" = P/I,

(9a,b)

where N is the matrix of displacement shape functions, P is the matrix of stress trial functions, A~ the element nodal displacement vector, and/3 the vector of generalized stress coefficients, yields ¢rmm(Aa,/3)---~

(_ ½#t/.//l

+ ½ a~tK~ Aa + / l t G a ~ )

n

- (WEF) + (correction terms) = stationary.

(10)

In the foregoing,

½~tH/l = fl [ (AN)2 + (AQ) 2 2GA--'-'-~ +

2EA

(AM) 2 ] 2~

dr,

(11a)

u a ~ a ~ = fb N. ½(Aw~)2 , dx ,

1 A~tirT

(ilb)

#'G Aa = [aN. Au + (AQ + AN. Z D" aw + AM. A0]'o.

(11c)

Owing to the fact that for two separate elements t h e / l ' s are independent from each other, they can be eliminated elememwise from (10) leading to a displacement-like formulation, i.e., *rmR1(A~) -- ~ ½ ASt(Ko + g,,) At~ - (WEF) + (correction terms) = stationary, n

(12)

where

Ko=GtH-1G.

(13)

A second modified version of the variational theorem due to Hellinger and Reissner, (3), can be obtained as follows. First, the stress resultant vector, Ao', is eliminated from (3) by virtue of (5) leading to the formulation given below:

~(Aw.,,) d x -

+

G A s • A e o • (A0

[EA'Ae

+ Aw,x) +

N'(Au,,,

+ z,,.Aw.,,)

E I . A e M • A0,,]

- (WEF) + (corection terms) = stationary.

dx} (14)

D. Karamanlidis, Shear flexible frame structure: Nonlinear analysb

61

Preceding equation is then further constrained by imposing the subsidiary conditions

Ae N = Ae~x ,

A e M ffi AK

(15)

upon the independent variables Au and Ae. The pertinent variational equation reads as follows:

qrmR2(Aa,A£O)----~ { -- ft½GA~'(AeQ)2dx + ~ N" ½(Aw,x)2dx +

ft½EA" (Au,,~ + z,x" Aw.x)2 dx

+ fl GAs Aeo" (A0 + Aw,x) dx + ft ½El(AO,,~)2dx} (16)

- (WEF) + (correction terms) = stationary.

Introduction in (16) of pertinent trial functions for the independent variables Au and AeQ, i.e., Au

--

N

AeQ = p~,

A~,

(17)

where/3 ! is constant, and subsequent element-by-element elimination of the/3~'s yields

( °:

)

~'mR2(A~) ~-~"~ ~ Aat /~'0"~"m s ggt +/~o A~ A

-

(WEF) + (correction terms) = stationary,

(18)

where

½A~tl~oA~ --~t½EA'(Au,: + z,x"AW,,,)2dx+ £ ½EI(AO:)2dx

(19a)

.j,tA~ --f,(A0 + Aw..) dx,

(19b)

(19c) It is noted for the record that in a previous paper [21] tihe author showed that (16) can be viewed as the variational basis of a well-known heuristic method of developing nonlocking shear flexible element,; for bending analysis along the lines of the standard assumed displacement finite element method. This method is known in the pertinent literature as the 'substitute shape function approach" (cf. [22]). More specifically, the element stiffness matrix obtained from (19a), (19b) can be seen (cf. [6]) to be identical to the one derived in a rather complicated way in a recent paper by Whitcomb [23].

D. Karamanlidis, Shear flexible frame structure: Nonlinear analysis

62

3. Step-by-step solution precedure For both finite element models presented in the previous section, the final matrix equation that is obtairled after the individual elements have been put together to form the structure at hand has exactly the same form as the equation one would have obtained in the case of the standard displacement formulation, viz. Kt A~ -- AAPref + r ,

(2o)

where K, is the tangent stiffness matrix, A represents a dimensionless proportionality factor, Pref is a suitably chosen reference load vector, and r denotes the pseudo-force or residual force vector. The standard procedure of solving (20) consists in imposing in each solution step incremental load changes (represented by AA) and solving for A~, provided that Kt and r have been calculated on the basis of the data from previous steps. It is a well-known fact that the procedure just described fails to give reasonable results as soon as the vicinity of a critical point in the load-deflection diagram is approached. Recently, an increased interest has been focused on the development of so-called continuation algorithms having the capability of handling the problem just stated (cf. [19] for an excellent review of the current state-of-theart). In this paper, based on previous experiences [9, 14, 20] the so-called energy incrementation scheme was chosen as continuation algorithm. A detailed description of the method has been presented previously [9], therefore, only the major points should be summarized here. For a general nonlinear problem represented by (20) the incremental energy by the externally applied forces can be written down in the form AS - ' ( A P +

0 t Aa.

(21)

Assuming proportional load changes, the vectors AP and Aa can be scaled as follows:

Ae---- A e o,,

(22a)

A~ = AAx + y .

(22b)

Introduction of (22) into (21) yields a quadratic equation in terms of AA, viz. a . ( A ~ ) 2 -.I- b . ( L ~ ) + C --- 0 ,

where

(23)

a --- ptre f x ,

(24a)

b "- rtx + Ptre f y ,

(24b)

c = rty - AS.

(24c)

It is seen from preceding equations that by imposing in each solution step incremental energy changes AS, the pertinent load changes AA can be determined from (23). Regarding the

D. Karamanlidis, Shear flexible frame structure: Nonlinear analysis

63

question of how to choose the 'right' solution when solving the quadratic (23), the same remedies are applicable that have been suggested in conjunction with so-called 'arc-length' incrementation algorithms (cf. [9, 19]).

4. Applieations The two elements proposed herein were implemented in the MELIlqAgeneral-purpose finite element code [10] and in order to demonstrate their usefulness extensive test studies were carried out. Results are presented here for the following cases: (i) a fully fixed beam under a concentrated load at nu'dspan, (ii) a shallow circular arch subject to uniform pressure load, and (iii) a deep circular arch under a concentrated force at apex.

4.1. Fully fixed beam First, the geometrically nonlinear behaviour of a fully fixed beam subject to a concentrated force at midspan was studied. The descriptive data for the problem are given in Fig. 4. The problem was solved by means of both elements presented herein. Figure 4 shows in form of load-deflection diagrams the numerical results obtained for different shear modulus values and for different element meshes. In all cases, the loading was applied in 100 self-correcting incremental steps without iterations. On the basis of these results, the hybrid stress element presented herein appears to be quite efficient: the solution obtained by using just two elements for half the beam agrees very well with the converged (exact) solution. Using a mesh of four elements for half the beam, the hybrid stress element solution coincides with the converged one whereas the solution obtained by the hybrid strain element is slightly in disagreement but still more than acceptable. It is worth mentioning that the same example was solved in a recent paper by Noor and Peters [4] using various two- and three-noded mixed and reduced/selective integration displacement models. A direct comparison of the performances of the elements proposed herein and those developed in [4] although desirable does not seem feasible due to the lack of pertinent data in [4] concerning the number of solution increments, iteration steps, etc. In [4] the best results were obtained by the two-noded MD2 element and by the three-noded MD3 element. Four and two elements were necessary, respectively, in o~der to obtain the converged (exact) solution for the midspan deflection w. Based on this, the conclusion seems in order that the two-noded hybrid stress element proposed herein is at least as efficient as the MD2 and MD3 elements developed in [4].

4.2. Shallow circular arch The large-deflection behaviour of a fully fixed shallow circular arch subject to a uniformly distributed normal pressure load (cf. Fig. 5) was studied next. The problem is very similar to the shallow arch considered in [4] except for the loading which consisted of a concentrated force at apex. Owing to the symmetry of the problem, one half of the structure was divided into five elements. The problem was solved by using both the hybrid stress and hybrid strain elements

64

D. Karamanfidis, Shear flexible frame structure: Nonlinear analysis

8d

L

{: I

: 0508

-

,..

I

h :3'17Sx10"3

'b=l~

--

'

"

• =.,m.

i~

o. O.IO

,.

ii'

~ : I' "~

I..7

O, IO

"

,,=/"

/ ~O.'GO 0.~1

i

i

'

1,20

I.~0

I.lO

G ===' J 2.10

2.40

2.70

/

~ O,~m 0 . 1

3.00

G=10342*10'°J-

/O.IO

O,lO

1.20

I.~

I, w~

2. I0

2.40

2.70

3.00

deflection parameter 6 (:IO0*w/L)

deflection p a r a m e t e r ~ (=100*w/L) 8 i

' " x : C '

R

~x

¢~

~) 4 X

8

i) E 'L Z t~ APX Z

,;

I

]fl

I. . . .

o

'

::

I

I

I

/

i,

] S

][

* t ]

,<

i

i

I

-

~ mx

o

,~z

~,

,

,,,w-

.

~ L~,~ _ ,.

f~ "

•

;~'~

G -

8.~-'-:-,10v

• G - e.m84*lO'

.j~-

X G - 1.791S*10v

-

~.

~ --

x G ~~s~*~0'

~

~

*

[ *--

1--

G n 0 . ~ 7 i ; i 0v

o 0 00

0.30

0.00

0. IO

I.I

1.90

I.O0

~.10

2.40

l . PO

deflection p a r a m e t e r ~ (=lO0*w/L)

3.00

0.00

I, I0

4,10

7,|

I,|

II,|

ll, g

ll. lO

ILl

21,i0

24,00

deflection p a r a m e t e r ~ (=lO0*w/L)

Fig. 4. Fully fixed beam subjected to a concentrated force at midspan: comparison of numerical results.

proposed herein. In addition, the relationship between shear deformation effects and nonlinear structural response was also studied. Figure 5(a) shows in form of load-deflection diagrams for the case G = oo the results obtained by the hybrid stress and hybrid strain elements in comparison with each other and with the converged (exact) solution. For both elements, the accuracy of the obtained numerical solutions appears to be excellent. The same is true for the case where shear deformation effects were included in the analysis (cf. Fig. 5(b)).

D. Karamanlidis, Shear flexible flame structure: Nonlinear analysis

65

R •

h

=

b= H=

22 1 3 3

,.~ii

o ;I _f I~

! I

0.oo o.:so OAIO o.lo

I t.2o

X i..m

x']fla't4 aat.rldal,'~ i.m

2.lo

2.qo

~.?o

3.00

deflection parameter 6 (=w/H)

it •

e

}_,."

/

2

J

.9.o 8

J

iO~O.3O

o

O.OO O.m

I.~0

I.SJO _B.IIO :P.IO ~'.40 ~t.70 3.00

deflection parameter 6 (=w/H)

deflection parameter 6 (=w/H)

Fig. 5. Fully fixed shallow arch under normal pressure: load-deflection diagrams.

In contrast to this, all Of the two-noded elements developed in [4] reportedly gave results which were in considerable disagreement with the converged solution when a mesh of four elements for half the arch was used. The converged solution was obtained in [4] by using two MD3 three-noded elements. Finally, the purpose of Fig. 5(c) is to illustrate how the nonlinear load-deflection characteristic of the structure under consideration changes depending on the transverse shear modulus G.

66

D. Karamanlidis, Shearflexible frame structure: Nonlinear analysis

4.3. Symmetric buckling of a deep circular arch

The final problem considered herein in order to assess the performance of the two element models proposed by the author is that of the symmetrical buckling and postbuckling of a hinged-hinged deep circular arch under a concentrated force at apex. Because of the symmetry of the problem only one half of the structure was modeled (cf. Fig. 6(a)). Figure 6(a) shows in form of load-deflection diagrams the results obtained by using both elements proposed herein in comparison with each other as well as with the converged (exact) 2P~

S ~

s

l

I Pree =

2"56x105 1

.~i-.-q" - - - - l i

"

KdBI/ I llllllllln

i~

l

l

l

~'"*,%,,~1 G=025x10~-

? al, o

&

%

Lqn

\

,<

o

L.

[

_.-

"

ill

•

~

I l t r l a D i.10elementl

~

imrmm

8

,i.' 0,~10

|

IUrldn}" 5 elements --

I

I

0,40 • O. IO

deflection

i no

I.~0

parameter

I. ~HOI I . l O

I . I10

'(

|,00

7,~D

iO.lO

14.40

l il.O0

hybltd 8t,nltn J I I 10.80 ~IS.~O ~ . 1 0

parameter

I .=n,.. Ji~.40 N . O 0

w

8

I II

.

d

I

I

xX

I

//" -.~'---,,-~ - '-'~ II O=O.125xlO' I

\

\\

9 b

•

1, - - h ~ m l d ~ I

l

]at

A

?

I"4

K

d

II

--"

3.GO

deflection

,&

8

O0

~ (=w/H)

S dl

8. °

¢ o ~ u r i t d moluUon I

I I I

O. i n

o

\ /,/

i/ .

--G .....

.

. x

- 0,60"10v G - 0.8*10 v w/o sheer elfech,

"~ .,,,

LI (D ,,0a (P

,I

i oe~

" " - ~ - ~-'/ I

"i oo ~.7o ~.4o ,'i.m 0~,.eo le.m n . m zs.m n.eo 33.~o p.~" deflection parameter w

"i

O0

3.~ tl

7,qO

deflection

II.I0

14,00

IO.S

parameter

5"J.20 a . l O

21t.JO I ] , X

3T.O0

w

Fig. 6. Comparisonof numerical results for simply supported deep circular arch under a concentrated force at apex.

D. Karamanlidis, Shear flexible frame structure: Nonlinear analysis

67

solution. In accordance with the results obtained for the previous examples, it is seen that the solution obtained by means of the hybrid stress element is in excellent agreement with the converged one. Even in the case where a relatively coarse mesh consisting of five elements per half the arch w-as chosen, the accuracy is more than acceptable for all practical purposes. On the other hand, the hybrid strain element appears to be not as accurate as the hybrid stress element. Finally, the results plotted in Figs. 6(b), 6(c), 6(d) aim at showing how the nonlinear response of the deep circular arch depends on the actual value for the transverse shear modulus G.

5. Conclusions Two alternative finite element models were developed for the analysis of free-form beams and arches undergoing arbitrarily large elastic deformations (displacements and rotations). Both elements are based on modified versions of the variational theorem due to Hellinger and Reissner. Also, in both cases transverse shear effects have been included in the formulation. Based on extensive numerical experiments, the first element (called hybrid stress element) was found to be reliable, computationally efficient, and very accurate. The performance of the second element (called hybrid strain element) was also found to be acceptable although not as good as in the case of the hybrid stress element. Finally, it is worth mentioning that the hybrid stress element was found to compare most favorably with mixed elements proposed in the recent paper by Noor and Peters [4].

Appendix A. Trial functions For the development of the two shear-flexible beam elements presented herein the following trial functions were chosen:

Hybrid stress element: Aw, AN, AQ, AM,

linear, constant, quadratic, cubic.

No approximations need be made regarding Au and A0 in the interior of an element.

Hybrid strain element: Au, Aw, A0, AeQ,

linear, linear, linear, constant.

¢

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D. Karamanlidis, Shear flexible frame structure: Nonlinear analysis

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