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Prismatic folded plate analysis using finite strip-elements
COMPUTER METHODS IN APPLIED MECHANICS AND ENGINEERING 76 (1989) 101-118 NORTH-HOLLAND
PRISMATIC FOLDED PLATE ANALYSIS USING FINITE STRIP-ELEMENTS
Bruce W. GOLLEY Department of Civil Engineering, University College, University of New South Wales, Campbell, ACT 2600, Australia
William A. GRICE Royal Australian Engineers, Command and Staff College, Fort Queenscliff, Queenscliff, Victoria, 3225, Australia Received 22 January 1988 Revised manuscript received 3 January 1989 A procedure combining some advantages of the finite element method and finite strip method is presented for the analysis of prismatic folded plate structures and box girders. Displacement functions within strip-elements are combinations of polynomials used with conventional finite elements plus displacement functions used with simply supported finite strips, which are products of truncated trigonometric series and polynomials. At strip-element ends, kinematic boundary conditions are exactly satisfied, while force boundary conditions are approximately satisfied as natural boundary conditions. The equations partially uncouple for all combinations of boundary conditions enabling any number of terms in trigonometric series to be taken without increasing computer core requirements, essentially preserving a major advantage of the finite strip method used with simply supported structures. Unlike the finite strip method, when free edges occur at strip ends the proposed method converges to exact values with an increasing number of strip-elements and trigonometric terms. Two examples are considered to demonstrate the effectiveness of the technique.
~. Introduction
Prismatic folded plate structures may be analysed using a variety of techniques. When component plates are thin, it is assumed that in-plane deformation is governed by plane stress theory, while out-of-plane deformation is governed by thin plate theory. Goldberg and Leve [1] obtained analytical solutions to the governing equations for single span prismatic folded plates when two opposite edges were simply supported. The finite strip method [2] has been used to analyse folded plates and is particularly suited to single span structures when two opposite edges are simple supports, as the stiffness equations uncouple into smaller sets of equations, which is computationally efficient. With other boundary conditions, this uncoupling does not occur, and a major advantage of the method is lost. In addition, free edge boundary conditions are not satisfied using the finite strip method, and convergence to wrong values occurs, although the errors may be small. Rectangular flat shell finite elements have been used to analyse a variety of structures [3] but involve large systems of equations. 0045-7825/89/$3.50 © 1989, Elsevier Science Publishers B.V. (North-Holland)
102
B.W. Golley, W.A. Grice, Prismatic folded plate analysis using finite strip-elements
In this paper, a method is proposed for the analysis of single-span prismatic folded plates which combines some advantages of finite strips and finite elements. Polynomial displacement functions used with conventional finite elements are combined with those used with simply supported finite strips. The method is similar to a technique proposed by Wang and Zhang [4] which combines polynomial and trigonometric functions to approximate displacements. They called the technique the finite strip-element method, and the term is used in this paper. Unlike the functions used by Wang and Zhang, however, the trigonometric functions considered in this paper are such that partial uncoupling of stiffness equations occurs. The proposed technique has a number of advantages when compared with other methods for analysing this class of problem, including the following: (1) The stiffness equations partially uncouple, enabling any number of trigonometric coefficients used in displacement functions to be treated without increasing computer core requirements. (2) Only trigonometric and polynomial functions are used in displacement functions. All integrations are performed analytically, and computational problems arising with hyperbolic functions with large arguments which occur in the finite strip method are avoided. (3) Fcee edge boundary conditions are approximately satisfied in the normal manner of the displacement method. Convergence to exact values with increasing numbers of trigonometric terms and finite strip-elements occurs.
2. Assumptions and notation
In-plane deformation and out-of-plane deformation of each component plate in the folded plate structure are assumed to be governed by plane stress theory and thin plate bending theory respectively. Strain energy due to these combined effects uncouples in fiat plates, and hence a plane stress finite strip-element is first presented, followed by a plate bending finite strip-element. Displacement functions in these cases are such that they may be combined to form a compatible fiat shell finite strip-element suitable for folded plate analysis. Throughout this paper, Timoshenko's notation for stresses and stress resultants [5, 6] is adopted.
3. Middle surface displacement functions
A typical finite strip-element is shown in Fig. 1. The three displacement components, and rotation about the X-axis of the ith nodal line are approximated by M
4
Umf,.(~) , m=l
m=l
M
4
v'(x) = ~ Vimsin a,,,~ + ~ V-'m L ( t : ) , m=l
m=l
(1)
(2)
B.W. Golley, W.A. Grice, Prismatic foldedplate analysis using finite strip-elements
i
103
b i
IZ,w Ng. 1. Typical strip-element.
M
w'(x) = E Winsin Ofm ~
4
4" ~
m=l m
O'(x) =
~ O~mSinam~ + m=l
W- i~ j m• ( ~ ) ,
(3)
-' Omfm(~ ),
(4)
m=l 4
~ m=l
where u, v and w are displacements in the X, Y and Z directions respectively, a m = m~r, = x/a, a is the strip-element span, 0 = c3w/Oy and fm(~:) (m = 1, 2, 3, 4) are the polynomials f l ( ~ ) - 1 - ~, f 2 ( ~ ) - ~, f3(~:)- ~(1 - ~:) and f4(~)= ~:(1 + ~:)(1 - 2~). The coefficients U~, 0 ~ , . . . , O~ are in general to be determined. Within the strip-element, the displacement of the middle surface is approximated by
u(x, y) = u'(~)(1 - ~.) + uJ(~)n,
(5)
v(x, y) - v'( ~ )(1 - ,7) + v~( ~ ),l ,
(6)
w(x, y) = w~( {~)h~(~) + bO~(~:)h2(v/) +
w/( ~:)h3(v/)+ bOJ( ~:)h4(v/),
(7)
where ~,= y/b, b is the strip.element width, hl(~/)= 1 - 3 7 / 2 + 2~/3, h2(~/)= ~"/ - 272 +~/3, h3(~/) -- 37/2 - 27/3 and h4(~/) = -7/2 + 7/3. For bending moments at strip-element ends to converge to exact values, it is necessary to approximate nodal line displacements w and 0 with polynomials of at least cubic order, accounting for the order of the polynomials in the expressions for wi(x) and Oi(x). For d!splacement compatibility in the folded plate case, the same polynomial order is required for v'(x) and w'(x), and hence v~(x) contains a cubic polynomial. Although u~(x) only requires a quadratic polynomial for convergence to exact results, the cubic polynomial is taken for convenience. In this paper, the in-plane displacements u(x, y) and v(x, y) are taken as linear functions in y for brevity, although quadratic and cubic functions have been considered by Grice [7] by adding one and two internal nodal lines respectively.
104
B.W. Golley, W.A. Grice, Prismatic folded plate analysis using finite strip-elements
4. P l a n e stress s t r i p . e l e m e n t
4.1. Sn'ffness matrix and load vector development
The in-plane displacements u(x, y) and v(x, y) ((5) and (6)) may be written in matrix form as
(8) where u = { u(x, y) v(x, y)}t U
:
{U
t
t ..
1 U2
t
"Um
.
U m = {UimVim Ujm V mj}t
(9) t
t
(10)
.. UM) ,
(11)
,
(12) (13)
N - [ N ~ N ~ . . . N ~ . . . NM], Nm=
( 1 - */) cos a m 0
0
*/cos am~:
(1 -,/)sin a,,, ~:
0
0 ] 71sin am s~ /
(14)
and
"(1 (1 (1 (1
S
£
__.
-
*/)ft (~:) '7)/2(~) r)A(~) r)A(~) 0 0 0 0
,f,(/~)
0 0 0 0
(I -n)f,(~) (I - */)f2(~) (I - ,)f3(~e) (1 - ~)f4(~:)
*/f2(~) */f3(se) */f,(se)
0 0 0 0
0 0 0 0
Ttf,(~) *//2(~:) */f3(~:) */f4(~:)
(15)
The superscript 'e' refers to in-plane behaviour. Strains
E=
ey
Yxy
= d~u ,
(16)
B.W. Golley, W.A. Grice, Prismatic folded plate analysis using finite strip-elements
where
105
#
~x o d
a
o
e
(17)
o___o #y ax are obtained as
e = [B" ~ ' ]
0
(18)
'
where (19)
B ~ = [B 1 B 2 - - . B ~ . . . B ~ ] , £
£
B~m=dNm ,
(20)
~ =d~ e "
(21)
The total potential energy of the strip-element,//e, may be written in the form:
(22) where
¢~fo s"Br dx d y ,
(23)
k e = t j0
"--'tf~fo,,"E"'dxdy, g'-tf~fo'"E"'dxdy, E= E~,* :]
[
E* Exy Ex*y
(24) (25)
,
(26)
p~ = fbo fo N~'q dx dy ,
(27)
~ = f~ f~ ~l~'qdx dy ,
(28)
q = { qx(x, Y) qy(x, y)},
(29)
0
0
G~,
$
t is the strip-element thickness, E*, E y, E*y and Gxy are orthotropic properties relating
106
B.W. GoUey, W.A. Grice, Prismatic folded plate analysis using finite strip-elements
stresses and strains under plane stress conditions, and q,,(x, y) and qr(x, y) are applied forces per unit area acting in the X and Y directions, respectively. The matrix k ~ may be written in the partitioned form: m
kll
0
0
k
""
0
..
0
k~2 " ' "
0
"'"
0
k;mm
...
0
£
0
0
...
o
o
. . . .6 . .
(30)
; kMM
The matrix k" is the stiffness matrix of a simply supported finite strip given by Cheung [8]. Submatrices k ~ are null matrices for m # n. The matrix k ~ may be written in the partitioned form:
/?=
-
(31)
k" ~m
The vector p~ may be written in the partitioned form: t
e
q
Pt i ,
Pm
(32)
D i
,PM, Elements of p~, and/~ may be determined for any loading by simple integration. 4.2• Application of plane stress strip-element
Although of limited use on its own, the application of the plane stress strip-element under a variety of boundary conditions is discussed in this section. Under plane stress conditions, the following common boundary conditions may occur on edges parallel to the Y-axis: Simple support (S): o"x = v = O, Clamped support (C): u = v = O, Free edge (F): o"x = "% = O. Thus six combinations of boundary conditions need consideration, namely SS, SF, SC, FF, FC and CC.
B.W. Golley, W.A. Grice, Prismatic folded plate analysis using finite strip-elements
107
4.3. SS case
The finite strip procedure is the most computationally efficient in this case. Thus if all polynomial coefficients are set to zero, the displacement functions automatically satisfy the S boundary conditions on x = 0 and a. The global stiffness equations obtained from the strip-element total potential energy are of the form: =
e
Kll 0•
0 K22 .
6
6
•
.
o
b
"'" ". ' "
0 0.
"'" "•" .
'
0 !.
Ic
"'"
~
...
'pe
,,, e2. e
Um
(33)
Pm
••
6
~
~ "~'=' "~
mm "'"
..•
g~
U1 U. g2
•
...
K;~,
.b~,
e~,
where U~ contains the trigonometric coefficients for the ruth harmonic for all nodal lines• Known values of the trigonometric coefficients are incorporated by modifying (33). These modified equations may be solved as the uncoupled sets of equations g
e
K,.mUm = Pm ,
m ----"1, 2 , . . • ,
(34)
M .
Alternatively, the polynomial coefficients l?il and I72 ( i - 1, 2 , . . . , I, where I is the total number of nodal lines) may be set to zero, satisfying the kinematic boundary condition v = 0 on x = 0 and x = a. The global equations obtained from the strip-element total potential energy are then of the form: •,,
8
Kll
0
."
0
K"22
''"
•
"
0
...
0
"'"
•
i
o
l
l
'V~
'°~,1
0
K2
v~
P;
•
.
... .
• .
.
.
.
o l
•
•
•
. •
...
.
K 1
• .
6
- - £ "
0
,,
(35)
l
G
.
.
K
M
where Ug is a vector of polynomial coefficients for all nodal lines• Solution of these equations leads to the force boundary condition ~rx = 0 on x = 0 and x = a being approximately satisfied as a natural boundary condition• Although the equations do not uncouple, partial uncoupling occurs• From the ruth block of equations, namely K~mV~ + ~ ( ] 8 = p~ , US,,, may be eliminated• Repeating this procedure for m = 1, 2 , . . . , equations
~,'~,=~",
(36) M leads to the reduced
07)
which may be solved for the unknown polynomial coefficients. Backsubstitution in (36)
B.W. Golley, W.A. Grice, Prismatic folded plate analysis using finite strip-elements
108
enables the unknown trigonometric coefficients to be determined. Any number of trigonometric coefficients M may be considered without increasing computer core requirements. While inefficient for solving the SS case, the form of equations occurs with the SF and FF cases, and hence is presented in full in this section. 4.4. FF case
No polynomial coefficients are set to zero, other than those required to satisfy kinematic boundary conditions on nodal lines. The global equations are of the same form as (35) and may be solved using the same procedure. In this case, the force boundary conditions crx = %y = 0 on x = 0 and x = a are approximately satisfied as natural boundary conditions. 4.5. FC case
For the case where the boundary x = 0 is clamped, the displacement v is set to zero on x = 0 by setting the polynomial coefficients l?i~ = 0 for i = 1, 2 , . . . , I. To set u to zero on x = 0 it is necessary to set u ' ( 0 ) = 0 for i = 1 , 2 , . . . , I. This requires incorporating into the global stiffness equations the constraint equations M si's
i
- i
2., U r n + U 1 = 0 ,
i=1,2,...,I.
m=|
(38)
These constraint equations may be incorporated using Lagrange multipliers, leading to tht~ global equations Kt,
0
'"
0
'"
0
0
K~ ::
"'
0
...
0
.
i
6
6
•
,
,
|
...
,
ic m m
b
...
" •
•
i
6
G2.
P:_
b ,
*
(39)
J
... .
:2
m
•
6
,
6 .
Gf
.
.
... .
.
K M
i 1
C,"*
0
O:
where A is a vector of Lagrange multipliers and G~, and G~ are constraint matrices. Following the partitioning shown, the same solution procedure as described for (35) may be followed. 4.6. SF, SC and CC cases
Equations similar to (35) are obtained for the SF case, while for the SC al~d CC cases, the global equations are of the same form as (39). In all cases, the equations partially uncouple, and any number of trigonometric coefficients may be considered without increasing computer core requirements. It should be noted that all boundary conditions may be treated using a single computer program by setting up equations in the form of (39), with constraint equations including the
B.W. Golley, W.A. Grice, Prismatic folded plate analysis using finite strip-elements
109
clamped conditions on both x = 0 and x = a. Depending on the actual boundary conditions, trigonometric coefficients, polynomial coefficients or Lagrange multipliers are then set to zero prior to solving the equations. Mixed boundary conditions may also be treated [7].
$. Plate bending finite strip.elements
5.1. Stiffness matrix and load vector development A plate bending finite strip-element which combined the displacement functions of the BFS plate bending finite element [9] and Cheung's simply supported plate bending finite strip [10] was described by Golley et al. [11]. However the presence of the twist term in the BFS element is not convenient for folded plate analysis, and hence an alternative form is presented here. The transverse displacement w(x, y) from (7) may be written in matrix form
w(x, y)=[N~ N~]{W} ,
(40)
W~-" {W I W2"'" Wtm "'° WM}t ,
(41)
Wm ~- {Wim o i m W/m OJm}t
(42)
where
,
__. {W1-i W2-iW3-iW4-i(~l-i 02-i 1~3-i(~4-iwl-j W2-Jw3-JW4-JOI-! (~2 -j (~3 -j 04} -j t , N" = [N; N ; . . . N~ =
sin am~:[h,(r/) bh2(rl) h3(vl) bh4(vl)]
N" =
[A(¢)h1(vl)A(~:)hl(vl) f3(/~)h,(vl)
(43) (44)
(45)
and f4(¢)h,(~)
bf;(¢)h2(vl)bf2(¢)h2(n)
bf3(~)h2(vl) bf4(~)h2(n) f,(~)h3(v/) f2(~:)h3(v/) f3(g)h3(v/) f4(~:)h3(n) bfl( ~ )h40?) bf2( ~ )h4(n ) bf3( g )h4(n) bf4( ~ )h4(n) ] .
(46)
The superscript ' r ' refers to out-of-plane behavlour. Oeneralised strains
¢32W Ox2 K=
032w #y2 =d~w, 2 02w/
Ox OyJ
(47)
B. W. GoUey, W. A. Grice, Prismatic folded plate analysis using finite strip-elements
110
where t~2
-
Ox 2
02 d~ =
(48)
Oy 2
02
are obtained as
(49) where I¢
B e -. [B 1 B2""
(50)
B~n • • • B M ] ,
B~, = d~Nm,
(51) (52)
The total potential energy of the strip-element, H e, may be written in the form
1I ~ = ~ [w' #']
- [w*
(53)
where
k" ffifo f; ."DS" dxdy ,
(54)
x* - f fo .* '." * dxdy,
(55)
K'---fffo,*'D,"dxdy,
(s6)
D=
(57)
[o,o 0] DI Dy 0
o"
ffi =
0
f;fo*' fofo**
Dxy
N qz(x, y) dx dy
(ss)
N qz(x, y) dx dy,
(59)
Dx, Dy, D~ and Dxy are orthotropic plate constants and qz(x, y) is the force per unit area in the Z direction. The matrices k" and/~" are of the same form as the partitioned forms given for k ~ and/~ in (30) and (31).
B.W. Golley, W.A. Grice, Prismatic folded plate analysis using finite strip-elements
111
5.2. Application of bending finite strip-element Three boundary conditions may occur on edges parallel to the Y-axis, namely Simple support (S): M x = w = O, Clamped support (C): w = w,x = O, Free edge (F): M x - Vx = O . The treatment of these boundary conditions parallels the plane stress cases treated earlier. By setting all polynomial coefficients to zero, the SS boundary conditions are exactly satisfied and uncoupled equations of the form of (33) occur. In other cases, kinematic boundary conditions may be exactly satisfied and force boundary conditions approximately_ satis~ed, as natural boundary conditions. Thus to satisfy the condition w ffi 0 on say x ffi 0, W'~ and O'~ are set to zero for i = 1, 2 . . . 1. To satisfy the condition Ow/Ox = 0 on say x - 0 requires the incorporation of the two constraint equations M
~, amW~m- I~il + I ~ + I ~ + W~ = 0
(60)
mffil
for i - 1, 2 , . . . , I .
M mffil
Global equations of the form of (35) arise if there are no clamped edges, and of the form of (39) if one or more edges are clamped.
6. Folded plate application A typical folded plate structure is shown in Fig. 2. The displacement and rotation about the X-axis of nodal line i are approximated by M
u"(x) = ~
4
U~" cos am~ + ~
mffil M
Vmfm(~),-"
4
v'i(x) - ~ V~" sin Otmf + ~'. V,,fm'~(~),m--1
m~-I
M
4
-- E w2 m ffi l
M
+ E Wmfm( -" ~) ,
(63) (64)
m ffi l
4
O"(x) = ~'~ O~" sin am~ + ~'~ Omfm(~)'-" m=l
(62)
mffil
(65)
m----1
The stiffnes:~; matrix and load vector for a typical strip-element shown shaded in Fig. 2 may be obtained in terms of the global parameters U~, 0 ~ . ' - O , ~ using standard transformations,
112
B.W. GoUey, W.A. Grice, Prismatic folded plate analysis using .finite strip.elements
Fig. 2. Typical folded plate structure.
noting that m
mm
U,,,
1
0
0
0
Us
v"
0
cos/3
sin/3
0
v,i i --m
0
-sin/3
cos/3
0
,
0
0
0
1
I t;)q
ti
,,
(66)
IW m
where /3 is the angle between the strip-element an the Y'-axis. The same transformation matrix relates the local and global polynomial coefficients. To satisfy the condition u - 0 on x - 0 , the constraint equations M
2 u:~ + m'l
0;' =0
(67)
must be satisfied for i = 1, 2 , . . . , I. To satisfy the condition u = w = 0 on x = 0, as occurs with both S and C boundaries, V-,i #~i and O~i are set to zero for i = 1 2, " ' ' , I. The rotation boundary condition Owl Ox = 0 on x - 0 requires the constraint equations I'
'
M m=l
to be satisfied for i = 1 , 2 , . . . , 1. In addition, for all nodal lines i, where coplanar stripelements meet, the single constraint equation ~ . v ~ - P ", + V'2'+ . . . . V;'+ I/'4` sin/3~ -1
-
~w21
~ , + w;'+ O;' + g,
cos g = 0,
(69)
B.W. Golley, W.A. Grice, Prismatic folded plate analysis using finite strip-elements
113
where/3 i is the angle between the coplanar strip-elements meeting at nodal line i and the Y'-axis, must be satisfied. For all nodal lines i where adjoining strip-elements are noncoplanar, two constraint equations must be satisfied, namely M
X "mV"m-- ~'1' +
~;' + #;' + # 7 = 0
(7O)
m=l
and M
E amW~- W'/+ W~ + W'3'+I,T/~:
0.
(71)
m=l
The constraint equations are included in the formulation using Lagrange multipliers, as with the plane stress and plate bending cases.
7. Examples The folded plate shown in Fig. 3 was analysed by dividing a half of the structure into six (I = 7) and twelve (1 = 13) strip-elements as shown, and taking the number of trigonometric coefficients M = 5 and 10. Young's modulus was taken as 200000 MPa and Poisson's ratio was 0.3. A quarter of the structure was also analysed using a 10 by 15 mesh of square semiloof elements [12]. The vertical displacement of the centre of the structure wc and at a free edge on the centreline w E are summarized in Table 1, while moments per unit length Mx and My and normal forces per unit length Nx and Ny along the centreline are shown in Figs. 4-8 for various values of I and M.
|
~4oo
|
|
;!_ 500
l
-,-
-t-,
12
-,
Xy Fig. 3. Folded plate with FF boundaries. Dimensions in mm.
B.W. GoUey, W.A, Grice, Prismatic folded plate analysis using finite strip-elements
114
Table 1 Example 1 - Displacements w E and w c (mm). Multiplier = 103. Finite element solution w E - 0 . 6 7 7 7 and w c = 4.545 Total number of nodal lines, I M (1)
w E, I = 7 (2)
wE, I = 13 (3)
wc,l = 7 (4)
w c, I = 13 (5)
5 10 15 20
0.6754 0.6787 0.6786 0.6787
0.6748 0.6783 0.6782 0.6782
4.524 4.543 4.541 4.541
4.527 4.545 4.543 4.543
,.
,
i.
i
.
1=7
_
~M-5
f
M-IO
+
FEM
B Z
X
=E
q~
q'l
I
/
....
/
!';./ ,j
j.. F 0
......
t25
250 x
,
375
500
(mm)
Fig. 4. Centreline moment M~ in example 1, 1 = 7.
1"13
I
M-5
~
..... M.IO + FEM
&
7 0
/
!
I
I
i25
250
375
x (mm) Fig. 5. Centreline moment Mx in example 1, 1 = 13.
I
500
B.W. Golley, W.A. Grice, Prismatic folded plate analysis using finite strip-elements
115
m
....
(D f
+
E •Z. . , .
,//FEM
J'~"
qq,
~r
0
I
I
I
t25
250
375
x
500
(mm)
Fig. 6. Centreline moment My in example 1.
In the second example, the cantilever folded plate shown in Fig. 9 was analysed. In this case, Young's modulus was taken as 25000 MPa and Poisson's ratio was 0.15. The structure was subjected to dead load only, with a gravitational force of 24 kN/m 3. A half of the plate was divided into 3, 6 and 9 (I = 4 , 7 and 10) strip-elements of equal width. A half of the structure was also analysed by using a 10 by 12 mesh of semiloof finite elements. The vertical displacement at D is summarised for various values of I and M in Table 2. The normal force per unit length, Nx, at the clamped support is shown in Fig. 10 for M - 5 for different values of I. Tables 1 and 2 indicate that deflections determined using the proposed method agree well with other numerically determined values. In the first example where bending effects
0
ql'
M-5
Ill2
¢'U
....
,= Z
Z
Z- 13
///
0 X I
0 ¢'U I
I
0
I
I
I
t25
250
375
x
(mm)
Fig. 7. Centreline normal force Nx in example 1.
500
116
B.W. GoUey, W.A. Grice, Prismatic folded plate analysis using finite strip-elements
M=5
E
+ FEM
Z
Z
0 O N I
o
° ?
4
--
I
0
I
t25
250 x
I
375
(mm)
Fig. 8. Centreline normal force Ny in example 1.
t=l
AJ J J J / J B C 'JJJ~JJJJJ~JJJJJJjJ~jjJJJ~
11 D Fig. 9. Cantilever folded plate. Dimensions in mm.
Table 2 Example 2 - Deflection, w (mm) at location D. Finite element solution = 0.2666 Total number of nodal lines, 1 M (1)
4 (2)
5 10 15 20
~.2581 0.2585 0.2586 0.2586
,
7 (3)
10 (4)
0.2633 0.2637 0.2637 0.2638
0.2647 0.2651 0.2652 0.2652
.,,
500
B. W. GoUey , W. A. Grice, Prismatic folded plate analysis using finite strip-elements i,
Od
117
•,
M-5
o
x
=
o o
,w,I
\ _
J/
~
~"
I 0 N Od I
V "
- f
....
I'4
~
I
- ....
I=lO
+
-"r
FEM
I
0
1
750
I
t500
y'
2250
.3000
(am)
Fig. 10. Normal force N~ at clamped support in example 2.
predominate, convergence with increasing M is faster than in the second example where in-plane effects are dominant. Moments and normal forces are accurately predicted using the proposed method. In a number of cases, curves with M - 5 are indistinguishable from those with M = 10, and hence only the curves for M = 5 are shown. In all cases, results had essentially converged taking M = 10. Unlike the finite strip method, the normal moment and normal force at the free edge approach zero with increasing numbers of strip-elements and trigonometric coefficients.
8. Conclusion
Prismatic folded plate structures may be accurately analysed using the proposed finite strip-element procedure. Stiffness equations partially uncouple with all boundary conditions enabling any number of trigonometric terms to be considered, without increasing computer core requirements. As with the displacement finite element method, force boundary conditions are approximately satisfied as natural boundary conditions.
References [1] J.E. Goldberg and H.L. Leve, Theory of prismatic folded plate structures, IABSE Publications 17 (1957) 59-86. [2] Y.K. Cheung, Folded plate structures by finite strip method, ASCE J. Struct. Div. 95 (12) (1969) 2963-2979. [3] O.C. Zienkiewicz, The Finite Element Method (McGraw-Hill, London, 1977). [4] H. Wang and J. Zhang, The finite strip-element method, in: G. Yagawa and S.N. Atluri, eds., Computational Mechanics '86 (Springer, Tokyo, 1986) 1-151-I-156. [5] S.P. Timoshenko and S. Woinowsky-Krieger, Theory of Plates and Shells (McGraw-Hill, Neg York, 1959). [6] S.E Timoshenko and J.N. Goodier, Theory of Elasticity (McGraw-Hill, New York, 1970).
118
B.W. Golley, W.A. Grice, Prismatic folded plate analysis using finite strip-elements
[71 W.A. Grice, Static analysis of thin plates using the finite strip-element method, M.E. Thesis, University College. University of New South Wales, 1987
[81 Y.K. Cheung, Finite Strip Method in Structural Analysis (Pergamon, Oxford, 1976). [91 F.K. Bogner, R.L. Fox and L.A. Schmit, The generation of interelement compatible stiffness and mass
matrices by the use of interpolation formulae, in: Proc. Conf. on Matrix Methods (Wright-Patterson Air Force Base, Dayton, OH, 1965) 397-443. [10] Y.K. Cheung, Finite strip method in the analysis of elastic plates with two opposite simply supported ends, Proc. Inst. of Cir. Engrs. 40 (5) (1968) 1-7. [11] B.W. Golley, W.A. Grice and J. Petrolito, Plate bending analysis using finite strip-elements, ASCE J. Struct. Engrg. 113 (1987) 1282-1296. [12] B.M. Irons, The semiloof shell element, in: D.G. Ashell and R.H. Gallagher, eds., Finite Elements for Thin Shells and Curved Members (Wiley, New York, 1976) 197-222.
PRISMATIC FOLDED PLATE ANALYSIS USING FINITE STRIP-ELEMENTS
Bruce W. GOLLEY Department of Civil Engineering, University College, University of New South Wales, Campbell, ACT 2600, Australia
William A. GRICE Royal Australian Engineers, Command and Staff College, Fort Queenscliff, Queenscliff, Victoria, 3225, Australia Received 22 January 1988 Revised manuscript received 3 January 1989 A procedure combining some advantages of the finite element method and finite strip method is presented for the analysis of prismatic folded plate structures and box girders. Displacement functions within strip-elements are combinations of polynomials used with conventional finite elements plus displacement functions used with simply supported finite strips, which are products of truncated trigonometric series and polynomials. At strip-element ends, kinematic boundary conditions are exactly satisfied, while force boundary conditions are approximately satisfied as natural boundary conditions. The equations partially uncouple for all combinations of boundary conditions enabling any number of terms in trigonometric series to be taken without increasing computer core requirements, essentially preserving a major advantage of the finite strip method used with simply supported structures. Unlike the finite strip method, when free edges occur at strip ends the proposed method converges to exact values with an increasing number of strip-elements and trigonometric terms. Two examples are considered to demonstrate the effectiveness of the technique.
~. Introduction
Prismatic folded plate structures may be analysed using a variety of techniques. When component plates are thin, it is assumed that in-plane deformation is governed by plane stress theory, while out-of-plane deformation is governed by thin plate theory. Goldberg and Leve [1] obtained analytical solutions to the governing equations for single span prismatic folded plates when two opposite edges were simply supported. The finite strip method [2] has been used to analyse folded plates and is particularly suited to single span structures when two opposite edges are simple supports, as the stiffness equations uncouple into smaller sets of equations, which is computationally efficient. With other boundary conditions, this uncoupling does not occur, and a major advantage of the method is lost. In addition, free edge boundary conditions are not satisfied using the finite strip method, and convergence to wrong values occurs, although the errors may be small. Rectangular flat shell finite elements have been used to analyse a variety of structures [3] but involve large systems of equations. 0045-7825/89/$3.50 © 1989, Elsevier Science Publishers B.V. (North-Holland)
102
B.W. Golley, W.A. Grice, Prismatic folded plate analysis using finite strip-elements
In this paper, a method is proposed for the analysis of single-span prismatic folded plates which combines some advantages of finite strips and finite elements. Polynomial displacement functions used with conventional finite elements are combined with those used with simply supported finite strips. The method is similar to a technique proposed by Wang and Zhang [4] which combines polynomial and trigonometric functions to approximate displacements. They called the technique the finite strip-element method, and the term is used in this paper. Unlike the functions used by Wang and Zhang, however, the trigonometric functions considered in this paper are such that partial uncoupling of stiffness equations occurs. The proposed technique has a number of advantages when compared with other methods for analysing this class of problem, including the following: (1) The stiffness equations partially uncouple, enabling any number of trigonometric coefficients used in displacement functions to be treated without increasing computer core requirements. (2) Only trigonometric and polynomial functions are used in displacement functions. All integrations are performed analytically, and computational problems arising with hyperbolic functions with large arguments which occur in the finite strip method are avoided. (3) Fcee edge boundary conditions are approximately satisfied in the normal manner of the displacement method. Convergence to exact values with increasing numbers of trigonometric terms and finite strip-elements occurs.
2. Assumptions and notation
In-plane deformation and out-of-plane deformation of each component plate in the folded plate structure are assumed to be governed by plane stress theory and thin plate bending theory respectively. Strain energy due to these combined effects uncouples in fiat plates, and hence a plane stress finite strip-element is first presented, followed by a plate bending finite strip-element. Displacement functions in these cases are such that they may be combined to form a compatible fiat shell finite strip-element suitable for folded plate analysis. Throughout this paper, Timoshenko's notation for stresses and stress resultants [5, 6] is adopted.
3. Middle surface displacement functions
A typical finite strip-element is shown in Fig. 1. The three displacement components, and rotation about the X-axis of the ith nodal line are approximated by M
4
Umf,.(~) , m=l
m=l
M
4
v'(x) = ~ Vimsin a,,,~ + ~ V-'m L ( t : ) , m=l
m=l
(1)
(2)
B.W. Golley, W.A. Grice, Prismatic foldedplate analysis using finite strip-elements
i
103
b i
IZ,w Ng. 1. Typical strip-element.
M
w'(x) = E Winsin Ofm ~
4
4" ~
m=l m
O'(x) =
~ O~mSinam~ + m=l
W- i~ j m• ( ~ ) ,
(3)
-' Omfm(~ ),
(4)
m=l 4
~ m=l
where u, v and w are displacements in the X, Y and Z directions respectively, a m = m~r, = x/a, a is the strip-element span, 0 = c3w/Oy and fm(~:) (m = 1, 2, 3, 4) are the polynomials f l ( ~ ) - 1 - ~, f 2 ( ~ ) - ~, f3(~:)- ~(1 - ~:) and f4(~)= ~:(1 + ~:)(1 - 2~). The coefficients U~, 0 ~ , . . . , O~ are in general to be determined. Within the strip-element, the displacement of the middle surface is approximated by
u(x, y) = u'(~)(1 - ~.) + uJ(~)n,
(5)
v(x, y) - v'( ~ )(1 - ,7) + v~( ~ ),l ,
(6)
w(x, y) = w~( {~)h~(~) + bO~(~:)h2(v/) +
w/( ~:)h3(v/)+ bOJ( ~:)h4(v/),
(7)
where ~,= y/b, b is the strip.element width, hl(~/)= 1 - 3 7 / 2 + 2~/3, h2(~/)= ~"/ - 272 +~/3, h3(~/) -- 37/2 - 27/3 and h4(~/) = -7/2 + 7/3. For bending moments at strip-element ends to converge to exact values, it is necessary to approximate nodal line displacements w and 0 with polynomials of at least cubic order, accounting for the order of the polynomials in the expressions for wi(x) and Oi(x). For d!splacement compatibility in the folded plate case, the same polynomial order is required for v'(x) and w'(x), and hence v~(x) contains a cubic polynomial. Although u~(x) only requires a quadratic polynomial for convergence to exact results, the cubic polynomial is taken for convenience. In this paper, the in-plane displacements u(x, y) and v(x, y) are taken as linear functions in y for brevity, although quadratic and cubic functions have been considered by Grice [7] by adding one and two internal nodal lines respectively.
104
B.W. Golley, W.A. Grice, Prismatic folded plate analysis using finite strip-elements
4. P l a n e stress s t r i p . e l e m e n t
4.1. Sn'ffness matrix and load vector development
The in-plane displacements u(x, y) and v(x, y) ((5) and (6)) may be written in matrix form as
(8) where u = { u(x, y) v(x, y)}t U
:
{U
t
t ..
1 U2
t
"Um
.
U m = {UimVim Ujm V mj}t
(9) t
t
(10)
.. UM) ,
(11)
,
(12) (13)
N - [ N ~ N ~ . . . N ~ . . . NM], Nm=
( 1 - */) cos a m 0
0
*/cos am~:
(1 -,/)sin a,,, ~:
0
0 ] 71sin am s~ /
(14)
and
"(1 (1 (1 (1
S
£
__.
-
*/)ft (~:) '7)/2(~) r)A(~) r)A(~) 0 0 0 0
,f,(/~)
0 0 0 0
(I -n)f,(~) (I - */)f2(~) (I - ,)f3(~e) (1 - ~)f4(~:)
*/f2(~) */f3(se) */f,(se)
0 0 0 0
0 0 0 0
Ttf,(~) *//2(~:) */f3(~:) */f4(~:)
(15)
The superscript 'e' refers to in-plane behaviour. Strains
E=
ey
Yxy
= d~u ,
(16)
B.W. Golley, W.A. Grice, Prismatic folded plate analysis using finite strip-elements
where
105
#
~x o d
a
o
e
(17)
o___o #y ax are obtained as
e = [B" ~ ' ]
0
(18)
'
where (19)
B ~ = [B 1 B 2 - - . B ~ . . . B ~ ] , £
£
B~m=dNm ,
(20)
~ =d~ e "
(21)
The total potential energy of the strip-element,//e, may be written in the form:
(22) where
¢~fo s"Br dx d y ,
(23)
k e = t j0
"--'tf~fo,,"E"'dxdy, g'-tf~fo'"E"'dxdy, E= E~,* :]
[
E* Exy Ex*y
(24) (25)
,
(26)
p~ = fbo fo N~'q dx dy ,
(27)
~ = f~ f~ ~l~'qdx dy ,
(28)
q = { qx(x, Y) qy(x, y)},
(29)
0
0
G~,
$
t is the strip-element thickness, E*, E y, E*y and Gxy are orthotropic properties relating
106
B.W. GoUey, W.A. Grice, Prismatic folded plate analysis using finite strip-elements
stresses and strains under plane stress conditions, and q,,(x, y) and qr(x, y) are applied forces per unit area acting in the X and Y directions, respectively. The matrix k ~ may be written in the partitioned form: m
kll
0
0
k
""
0
..
0
k~2 " ' "
0
"'"
0
k;mm
...
0
£
0
0
...
o
o
. . . .6 . .
(30)
; kMM
The matrix k" is the stiffness matrix of a simply supported finite strip given by Cheung [8]. Submatrices k ~ are null matrices for m # n. The matrix k ~ may be written in the partitioned form:
/?=
-
(31)
k" ~m
The vector p~ may be written in the partitioned form: t
e
q
Pt i ,
Pm
(32)
D i
,PM, Elements of p~, and/~ may be determined for any loading by simple integration. 4.2• Application of plane stress strip-element
Although of limited use on its own, the application of the plane stress strip-element under a variety of boundary conditions is discussed in this section. Under plane stress conditions, the following common boundary conditions may occur on edges parallel to the Y-axis: Simple support (S): o"x = v = O, Clamped support (C): u = v = O, Free edge (F): o"x = "% = O. Thus six combinations of boundary conditions need consideration, namely SS, SF, SC, FF, FC and CC.
B.W. Golley, W.A. Grice, Prismatic folded plate analysis using finite strip-elements
107
4.3. SS case
The finite strip procedure is the most computationally efficient in this case. Thus if all polynomial coefficients are set to zero, the displacement functions automatically satisfy the S boundary conditions on x = 0 and a. The global stiffness equations obtained from the strip-element total potential energy are of the form: =
e
Kll 0•
0 K22 .
6
6
•
.
o
b
"'" ". ' "
0 0.
"'" "•" .
'
0 !.
Ic
"'"
~
...
'pe
,,, e2. e
Um
(33)
Pm
••
6
~
~ "~'=' "~
mm "'"
..•
g~
U1 U. g2
•
...
K;~,
.b~,
e~,
where U~ contains the trigonometric coefficients for the ruth harmonic for all nodal lines• Known values of the trigonometric coefficients are incorporated by modifying (33). These modified equations may be solved as the uncoupled sets of equations g
e
K,.mUm = Pm ,
m ----"1, 2 , . . • ,
(34)
M .
Alternatively, the polynomial coefficients l?il and I72 ( i - 1, 2 , . . . , I, where I is the total number of nodal lines) may be set to zero, satisfying the kinematic boundary condition v = 0 on x = 0 and x = a. The global equations obtained from the strip-element total potential energy are then of the form: •,,
8
Kll
0
."
0
K"22
''"
•
"
0
...
0
"'"
•
i
o
l
l
'V~
'°~,1
0
K2
v~
P;
•
.
... .
• .
.
.
.
o l
•
•
•
. •
...
.
K 1
• .
6
- - £ "
0
,,
(35)
l
G
.
.
K
M
where Ug is a vector of polynomial coefficients for all nodal lines• Solution of these equations leads to the force boundary condition ~rx = 0 on x = 0 and x = a being approximately satisfied as a natural boundary condition• Although the equations do not uncouple, partial uncoupling occurs• From the ruth block of equations, namely K~mV~ + ~ ( ] 8 = p~ , US,,, may be eliminated• Repeating this procedure for m = 1, 2 , . . . , equations
~,'~,=~",
(36) M leads to the reduced
07)
which may be solved for the unknown polynomial coefficients. Backsubstitution in (36)
B.W. Golley, W.A. Grice, Prismatic folded plate analysis using finite strip-elements
108
enables the unknown trigonometric coefficients to be determined. Any number of trigonometric coefficients M may be considered without increasing computer core requirements. While inefficient for solving the SS case, the form of equations occurs with the SF and FF cases, and hence is presented in full in this section. 4.4. FF case
No polynomial coefficients are set to zero, other than those required to satisfy kinematic boundary conditions on nodal lines. The global equations are of the same form as (35) and may be solved using the same procedure. In this case, the force boundary conditions crx = %y = 0 on x = 0 and x = a are approximately satisfied as natural boundary conditions. 4.5. FC case
For the case where the boundary x = 0 is clamped, the displacement v is set to zero on x = 0 by setting the polynomial coefficients l?i~ = 0 for i = 1, 2 , . . . , I. To set u to zero on x = 0 it is necessary to set u ' ( 0 ) = 0 for i = 1 , 2 , . . . , I. This requires incorporating into the global stiffness equations the constraint equations M si's
i
- i
2., U r n + U 1 = 0 ,
i=1,2,...,I.
m=|
(38)
These constraint equations may be incorporated using Lagrange multipliers, leading to tht~ global equations Kt,
0
'"
0
'"
0
0
K~ ::
"'
0
...
0
.
i
6
6
•
,
,
|
...
,
ic m m
b
...
" •
•
i
6
G2.
P:_
b ,
*
(39)
J
... .
:2
m
•
6
,
6 .
Gf
.
.
... .
.
K M
i 1
C,"*
0
O:
where A is a vector of Lagrange multipliers and G~, and G~ are constraint matrices. Following the partitioning shown, the same solution procedure as described for (35) may be followed. 4.6. SF, SC and CC cases
Equations similar to (35) are obtained for the SF case, while for the SC al~d CC cases, the global equations are of the same form as (39). In all cases, the equations partially uncouple, and any number of trigonometric coefficients may be considered without increasing computer core requirements. It should be noted that all boundary conditions may be treated using a single computer program by setting up equations in the form of (39), with constraint equations including the
B.W. Golley, W.A. Grice, Prismatic folded plate analysis using finite strip-elements
109
clamped conditions on both x = 0 and x = a. Depending on the actual boundary conditions, trigonometric coefficients, polynomial coefficients or Lagrange multipliers are then set to zero prior to solving the equations. Mixed boundary conditions may also be treated [7].
$. Plate bending finite strip.elements
5.1. Stiffness matrix and load vector development A plate bending finite strip-element which combined the displacement functions of the BFS plate bending finite element [9] and Cheung's simply supported plate bending finite strip [10] was described by Golley et al. [11]. However the presence of the twist term in the BFS element is not convenient for folded plate analysis, and hence an alternative form is presented here. The transverse displacement w(x, y) from (7) may be written in matrix form
w(x, y)=[N~ N~]{W} ,
(40)
W~-" {W I W2"'" Wtm "'° WM}t ,
(41)
Wm ~- {Wim o i m W/m OJm}t
(42)
where
,
__. {W1-i W2-iW3-iW4-i(~l-i 02-i 1~3-i(~4-iwl-j W2-Jw3-JW4-JOI-! (~2 -j (~3 -j 04} -j t , N" = [N; N ; . . . N~ =
sin am~:[h,(r/) bh2(rl) h3(vl) bh4(vl)]
N" =
[A(¢)h1(vl)A(~:)hl(vl) f3(/~)h,(vl)
(43) (44)
(45)
and f4(¢)h,(~)
bf;(¢)h2(vl)bf2(¢)h2(n)
bf3(~)h2(vl) bf4(~)h2(n) f,(~)h3(v/) f2(~:)h3(v/) f3(g)h3(v/) f4(~:)h3(n) bfl( ~ )h40?) bf2( ~ )h4(n ) bf3( g )h4(n) bf4( ~ )h4(n) ] .
(46)
The superscript ' r ' refers to out-of-plane behavlour. Oeneralised strains
¢32W Ox2 K=
032w #y2 =d~w, 2 02w/
Ox OyJ
(47)
B. W. GoUey, W. A. Grice, Prismatic folded plate analysis using finite strip-elements
110
where t~2
-
Ox 2
02 d~ =
(48)
Oy 2
02
are obtained as
(49) where I¢
B e -. [B 1 B2""
(50)
B~n • • • B M ] ,
B~, = d~Nm,
(51) (52)
The total potential energy of the strip-element, H e, may be written in the form
1I ~ = ~ [w' #']
- [w*
(53)
where
k" ffifo f; ."DS" dxdy ,
(54)
x* - f fo .* '." * dxdy,
(55)
K'---fffo,*'D,"dxdy,
(s6)
D=
(57)
[o,o 0] DI Dy 0
o"
ffi =
0
f;fo*' fofo**
Dxy
N qz(x, y) dx dy
(ss)
N qz(x, y) dx dy,
(59)
Dx, Dy, D~ and Dxy are orthotropic plate constants and qz(x, y) is the force per unit area in the Z direction. The matrices k" and/~" are of the same form as the partitioned forms given for k ~ and/~ in (30) and (31).
B.W. Golley, W.A. Grice, Prismatic folded plate analysis using finite strip-elements
111
5.2. Application of bending finite strip-element Three boundary conditions may occur on edges parallel to the Y-axis, namely Simple support (S): M x = w = O, Clamped support (C): w = w,x = O, Free edge (F): M x - Vx = O . The treatment of these boundary conditions parallels the plane stress cases treated earlier. By setting all polynomial coefficients to zero, the SS boundary conditions are exactly satisfied and uncoupled equations of the form of (33) occur. In other cases, kinematic boundary conditions may be exactly satisfied and force boundary conditions approximately_ satis~ed, as natural boundary conditions. Thus to satisfy the condition w ffi 0 on say x ffi 0, W'~ and O'~ are set to zero for i = 1, 2 . . . 1. To satisfy the condition Ow/Ox = 0 on say x - 0 requires the incorporation of the two constraint equations M
~, amW~m- I~il + I ~ + I ~ + W~ = 0
(60)
mffil
for i - 1, 2 , . . . , I .
M mffil
Global equations of the form of (35) arise if there are no clamped edges, and of the form of (39) if one or more edges are clamped.
6. Folded plate application A typical folded plate structure is shown in Fig. 2. The displacement and rotation about the X-axis of nodal line i are approximated by M
u"(x) = ~
4
U~" cos am~ + ~
mffil M
Vmfm(~),-"
4
v'i(x) - ~ V~" sin Otmf + ~'. V,,fm'~(~),m--1
m~-I
M
4
-- E w2 m ffi l
M
+ E Wmfm( -" ~) ,
(63) (64)
m ffi l
4
O"(x) = ~'~ O~" sin am~ + ~'~ Omfm(~)'-" m=l
(62)
mffil
(65)
m----1
The stiffnes:~; matrix and load vector for a typical strip-element shown shaded in Fig. 2 may be obtained in terms of the global parameters U~, 0 ~ . ' - O , ~ using standard transformations,
112
B.W. GoUey, W.A. Grice, Prismatic folded plate analysis using .finite strip.elements
Fig. 2. Typical folded plate structure.
noting that m
mm
U,,,
1
0
0
0
Us
v"
0
cos/3
sin/3
0
v,i i --m
0
-sin/3
cos/3
0
,
0
0
0
1
I t;)q
ti
,,
(66)
IW m
where /3 is the angle between the strip-element an the Y'-axis. The same transformation matrix relates the local and global polynomial coefficients. To satisfy the condition u - 0 on x - 0 , the constraint equations M
2 u:~ + m'l
0;' =0
(67)
must be satisfied for i = 1, 2 , . . . , I. To satisfy the condition u = w = 0 on x = 0, as occurs with both S and C boundaries, V-,i #~i and O~i are set to zero for i = 1 2, " ' ' , I. The rotation boundary condition Owl Ox = 0 on x - 0 requires the constraint equations I'
'
M m=l
to be satisfied for i = 1 , 2 , . . . , 1. In addition, for all nodal lines i, where coplanar stripelements meet, the single constraint equation ~ . v ~ - P ", + V'2'+ . . . . V;'+ I/'4` sin/3~ -1
-
~w21
~ , + w;'+ O;' + g,
cos g = 0,
(69)
B.W. Golley, W.A. Grice, Prismatic folded plate analysis using finite strip-elements
113
where/3 i is the angle between the coplanar strip-elements meeting at nodal line i and the Y'-axis, must be satisfied. For all nodal lines i where adjoining strip-elements are noncoplanar, two constraint equations must be satisfied, namely M
X "mV"m-- ~'1' +
~;' + #;' + # 7 = 0
(7O)
m=l
and M
E amW~- W'/+ W~ + W'3'+I,T/~:
0.
(71)
m=l
The constraint equations are included in the formulation using Lagrange multipliers, as with the plane stress and plate bending cases.
7. Examples The folded plate shown in Fig. 3 was analysed by dividing a half of the structure into six (I = 7) and twelve (1 = 13) strip-elements as shown, and taking the number of trigonometric coefficients M = 5 and 10. Young's modulus was taken as 200000 MPa and Poisson's ratio was 0.3. A quarter of the structure was also analysed using a 10 by 15 mesh of square semiloof elements [12]. The vertical displacement of the centre of the structure wc and at a free edge on the centreline w E are summarized in Table 1, while moments per unit length Mx and My and normal forces per unit length Nx and Ny along the centreline are shown in Figs. 4-8 for various values of I and M.
|
~4oo
|
|
;!_ 500
l
-,-
-t-,
12
-,
Xy Fig. 3. Folded plate with FF boundaries. Dimensions in mm.
B.W. GoUey, W.A, Grice, Prismatic folded plate analysis using finite strip-elements
114
Table 1 Example 1 - Displacements w E and w c (mm). Multiplier = 103. Finite element solution w E - 0 . 6 7 7 7 and w c = 4.545 Total number of nodal lines, I M (1)
w E, I = 7 (2)
wE, I = 13 (3)
wc,l = 7 (4)
w c, I = 13 (5)
5 10 15 20
0.6754 0.6787 0.6786 0.6787
0.6748 0.6783 0.6782 0.6782
4.524 4.543 4.541 4.541
4.527 4.545 4.543 4.543
,.
,
i.
i
.
1=7
_
~M-5
f
M-IO
+
FEM
B Z
X
=E
q~
q'l
I
/
....
/
!';./ ,j
j.. F 0
......
t25
250 x
,
375
500
(mm)
Fig. 4. Centreline moment M~ in example 1, 1 = 7.
1"13
I
M-5
~
..... M.IO + FEM
&
7 0
/
!
I
I
i25
250
375
x (mm) Fig. 5. Centreline moment Mx in example 1, 1 = 13.
I
500
B.W. Golley, W.A. Grice, Prismatic folded plate analysis using finite strip-elements
115
m
....
(D f
+
E •Z. . , .
,//FEM
J'~"
qq,
~r
0
I
I
I
t25
250
375
x
500
(mm)
Fig. 6. Centreline moment My in example 1.
In the second example, the cantilever folded plate shown in Fig. 9 was analysed. In this case, Young's modulus was taken as 25000 MPa and Poisson's ratio was 0.15. The structure was subjected to dead load only, with a gravitational force of 24 kN/m 3. A half of the plate was divided into 3, 6 and 9 (I = 4 , 7 and 10) strip-elements of equal width. A half of the structure was also analysed by using a 10 by 12 mesh of semiloof finite elements. The vertical displacement at D is summarised for various values of I and M in Table 2. The normal force per unit length, Nx, at the clamped support is shown in Fig. 10 for M - 5 for different values of I. Tables 1 and 2 indicate that deflections determined using the proposed method agree well with other numerically determined values. In the first example where bending effects
0
ql'
M-5
Ill2
¢'U
....
,= Z
Z
Z- 13
///
0 X I
0 ¢'U I
I
0
I
I
I
t25
250
375
x
(mm)
Fig. 7. Centreline normal force Nx in example 1.
500
116
B.W. GoUey, W.A. Grice, Prismatic folded plate analysis using finite strip-elements
M=5
E
+ FEM
Z
Z
0 O N I
o
° ?
4
--
I
0
I
t25
250 x
I
375
(mm)
Fig. 8. Centreline normal force Ny in example 1.
t=l
AJ J J J / J B C 'JJJ~JJJJJ~JJJJJJjJ~jjJJJ~
11 D Fig. 9. Cantilever folded plate. Dimensions in mm.
Table 2 Example 2 - Deflection, w (mm) at location D. Finite element solution = 0.2666 Total number of nodal lines, 1 M (1)
4 (2)
5 10 15 20
~.2581 0.2585 0.2586 0.2586
,
7 (3)
10 (4)
0.2633 0.2637 0.2637 0.2638
0.2647 0.2651 0.2652 0.2652
.,,
500
B. W. GoUey , W. A. Grice, Prismatic folded plate analysis using finite strip-elements i,
Od
117
•,
M-5
o
x
=
o o
,w,I
\ _
J/
~
~"
I 0 N Od I
V "
- f
....
I'4
~
I
- ....
I=lO
+
-"r
FEM
I
0
1
750
I
t500
y'
2250
.3000
(am)
Fig. 10. Normal force N~ at clamped support in example 2.
predominate, convergence with increasing M is faster than in the second example where in-plane effects are dominant. Moments and normal forces are accurately predicted using the proposed method. In a number of cases, curves with M - 5 are indistinguishable from those with M = 10, and hence only the curves for M = 5 are shown. In all cases, results had essentially converged taking M = 10. Unlike the finite strip method, the normal moment and normal force at the free edge approach zero with increasing numbers of strip-elements and trigonometric coefficients.
8. Conclusion
Prismatic folded plate structures may be accurately analysed using the proposed finite strip-element procedure. Stiffness equations partially uncouple with all boundary conditions enabling any number of trigonometric terms to be considered, without increasing computer core requirements. As with the displacement finite element method, force boundary conditions are approximately satisfied as natural boundary conditions.
References [1] J.E. Goldberg and H.L. Leve, Theory of prismatic folded plate structures, IABSE Publications 17 (1957) 59-86. [2] Y.K. Cheung, Folded plate structures by finite strip method, ASCE J. Struct. Div. 95 (12) (1969) 2963-2979. [3] O.C. Zienkiewicz, The Finite Element Method (McGraw-Hill, London, 1977). [4] H. Wang and J. Zhang, The finite strip-element method, in: G. Yagawa and S.N. Atluri, eds., Computational Mechanics '86 (Springer, Tokyo, 1986) 1-151-I-156. [5] S.P. Timoshenko and S. Woinowsky-Krieger, Theory of Plates and Shells (McGraw-Hill, Neg York, 1959). [6] S.E Timoshenko and J.N. Goodier, Theory of Elasticity (McGraw-Hill, New York, 1970).
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B.W. Golley, W.A. Grice, Prismatic folded plate analysis using finite strip-elements
[71 W.A. Grice, Static analysis of thin plates using the finite strip-element method, M.E. Thesis, University College. University of New South Wales, 1987
[81 Y.K. Cheung, Finite Strip Method in Structural Analysis (Pergamon, Oxford, 1976). [91 F.K. Bogner, R.L. Fox and L.A. Schmit, The generation of interelement compatible stiffness and mass
matrices by the use of interpolation formulae, in: Proc. Conf. on Matrix Methods (Wright-Patterson Air Force Base, Dayton, OH, 1965) 397-443. [10] Y.K. Cheung, Finite strip method in the analysis of elastic plates with two opposite simply supported ends, Proc. Inst. of Cir. Engrs. 40 (5) (1968) 1-7. [11] B.W. Golley, W.A. Grice and J. Petrolito, Plate bending analysis using finite strip-elements, ASCE J. Struct. Engrg. 113 (1987) 1282-1296. [12] B.M. Irons, The semiloof shell element, in: D.G. Ashell and R.H. Gallagher, eds., Finite Elements for Thin Shells and Curved Members (Wiley, New York, 1976) 197-222.