The analysis of a three-dimensional rigid-jointed rectangular plexus frame

Pergamon

Compurers & Srrwrures Vol. 49. No. 5. pp. 84-66, 1993 0 1994 Elsetier Science Ltd

Printed in&eat Britain. 0045-7949/93 $6.00 +O.OO

THE ANALYSIS OF A THREE-DIMENSIONAL RIGID-JOINTED RECTANGULAR PLEXUS FRAME

P. D. ANDR~OTAKI-PANAYOTOLJNAKOU

Department of Architecture, Section of Applied Mechanics, National Technical University of Athens, 5 Heroes of Polytechnion Avenue, Zographou GR-157 73, Athens, Greece

(Received 2 July 1992)

Abstract-The investigation of a three-dimensional rigid-jointed rectangular plexus frame under a general static loading is presented. Since the matrices involved in the exact elasticity solution are of maximum dimensions equal to the square of the number of the nodes of the frame, the proposed methodology is more convenient in comparison to existing methods because it requires less memory space and computer coding.

1. INTRODUCTION

Until recently, the analysis of a spatial or planar multi-storey and multi-column rectangular plexus frame has attracted the interest of many investigators. Lustgarden [I], Clough et al. [2], Weaver and Nelson [3], Cervenka and Gerstle [4], Gluck [S] and many others, based on the flexibility or stiffness approach and using several digital computer methods and iteration techniques, have analysed such structures, mainly considering the simply supported straight member that connects two consecutive nodes of the frame. This has meant that the resulting matrices were of very large dimensions. Other researchers [6-g] used the continuous straight beam on elastic supports in order to analyse planar rigid-jointed plexus frames under a general co-planar or out-of-plane loading.

In this paper we extend the idea of using a continuous straight beam on elastic supports as a basic structure for the static analysis of a three-dimensional (3D) rigid-jointed frame in the case: (a) of a general geometric construction; (b) of a general external static loading, consisting of concentrated and distributed forces and couples; (c) when the vertical displacements of the columns are either known or elastic, and (d) when the effects of internal bending and torsional moments are taken into consideration in the analysis.

The resulting system is uncoupled and solved, furnishing closed-form formulae for the determination of all the redundants of the structure. The maximum dimensions of the matrices involved are (mm x mm), where m is the number of storeys, n is the number of columns in the direction of the width and 7 the number of columns in the direction of the length of the structure. Finally, an application of the developed solution is presented, demonstrat- ing the correctness and the potentialities of the proposed method.

2. PRELIMINARIES

Consider Fig. 1 the 3D rectangular rigid-jointed plexus frame, consisting of three kinds of rectangular plane frames, namely:

(a) verticalframesp(p=1,2 ,..., s ,..., z),inthe direction of the width of the whole structure, parallel to (Ox, Oz)-plane;

(b) vertical frames q (q = 1,2, . . . , p, . . . , n), in the direction of the length of the 3D frame, parallel to (Oy, Oz)-plane, and

(c) horizontal frames r (r = 1,2,. . . , u, . . , m), formulating the storeys of the structure, parallel to the (Ox, Oy)-plane.

The intersection of two different plane frames creates a centroidal axis of their common beam and the intersection of three plane frames of different orientation is one rigid joint of the 3D-rectangular plexus frame.

Based on this geometric construction the spatial frame may also be considered to consist of three different types of beams i, k, j parallel to the Ox, Oy and Oz axes, respectively. The horizontal i-beam, as a member of the vertical p-frame, can be specified by the integer number i = 1,2, . . . , u, . . . , m and the p = s symbol for the frame. The horizontal k-beam, as a member of the vertical q-frame, can be characterized by the number k=1,2 ,..., e ,..., m and the q = p symbol showing the frame where it belongs. Finally, the vertical j-beam, as a member of the vertical p-frame, is determined by the axial number j = 1,2, . . . ) p, . . . , n and the symbol p = s of the plane frame of the structure. Another way of number- ing the i-, k- andj-beams is to use i or k orj, followed by an ordered pair of subscripts i,, kap, jsP, showing the intersection of the plane frames r = o with p = s,

r = u with q = p and p = s with q = p, that creates the i-, k- or j-beam, respectively. The numbers for beams and frames increase according to the positive direction of the Ox, Oy, Oz axes (Fig. 1). Each node

849

850 P. D. ANDRIOTAKI-PANAYOTOLJNAKOU

Fig. 1. Typical building frame.

of the spatial frame is characterized by an ordered triad of integers, aps, indicating the common point of intersection of the three different kinds of plane frames which have already been mentioned: r = Q, q = p and p = s, respectively.

Summing up, the 3D plexus frame consists of a number of -mr incompressible horizontal i-beams in the direction of the Ox-axis, -mn incompressible horizontal k-beams in the direction of the Oy-axis, and nt vertical incompressible j-beams fixed on the ground in the direction of the Oz-axis, so that a number of mm rigid joints can be formulated.

The soil may undergo vertical elastic displacements. The structure is subjected to a generic 3D loading consisting of distributed and concentrated intermediate or nodal forces and couples. All these quantities are analysed to 3D coordinate systems parallel to the Oxyz axis, which is related to the undeformed structure (Fig. 1).

azis, respectively, where cr, p and s are integer numbers while i, i and k remain constant symbols.

The flexural rigidities of the horizontal and vertical members of the structure may vary from one span to another, but they remain constant within each span. We also consider in every cross-section of the members, a 3D centroidal coordinate system (123) parallel to the general Oxyz one that coincides with the principal axes of inertia of the cross-sections.

The superscript T denotes the transpose of a matrix or vector, E is the modulus of elasticity and G symbolizes the shear modulus of elasticity of the medium.

2.1. Notations for the i = cr- or j = p-beam of the p = s-plane frame

I& and I:: are moments of inertia of the members [ups, cr(p + l)s] and [aps, (a + l)ps] about the t-axis (t = 1,2,3), respectively

I:; = I$ /a&, IS: = Ib”, /(a&s)2, IT = I&/(a&S)3, (t = 1,2, 3)

I*‘” = 1$:/a: $, fv Z$i” = Z,‘d’/(a,*S)2, Io*dll” = I,‘d”/(ab*S)3, (t = 1,2, 3)

a$ = l/a&,, a;;)2 = l/aziS.

The span-length between two consecutive PzP is the external force applied on the ups-node in nodes [aps - a(~ + I)$], [aps - ap(s + l)] and the direction of the t-axis (t = 1,2,3), m& is the [ups -(a + l)ps] is denoted by a&, a&, and external couple applied on the ups-node about the

Analysis of 3D rigid-jointed rectangular plexus frame 851

t-axis (t = 1,2,3), H& is the external intermediate axial force of the a&-span.

is the axial external resultant force of the i = a-beam, Hz: is the external intermediate axial force of the a $-span.

is the external resultant force of the j = p-beam.

Mfb”, M$ (t =2, 3), M$‘s, M:tpS”’ (t=l,2)

are the internal bending moments, referring to the left, right, upper, and sub-cross-sections of the node ups about the t-axis, caused by the intermediate external loading acting on both fixed end members [aps, a(p + l)s], or [aps, (a + l)ps], respectively.

direction of the t-axis, respectively. Nf:, N$, N*“*3s, N$3S are the internal axial forces of the left!, right, upper, or sub-cross-sections of the node ups, respectively. Z$, Z*‘” (t=1,2,3) are the external moments about th:t-axis at the aps-node, previously internal between the i,,, kaP and jSP continuous basic structures of the 3D frame. X&, X*” (t = 1,2,3) are the reactions in the direction of tee t-axis, of the i,, or jlP continuous beams of the p =s-frame. $&, $zF (t = 1,2,3) are the slopes of the cross-section aps about the t-axis. wo”p; w *IS (t = 1,2,3) are the displacements of the cross- seztion ups in the direction of t-axis. r#~t” is the soil reaction in the direction of t-axis at the j = p-column of the frame p = s. ups is the translational spring constant of the (m + l)ps-support of the j = p-beam of p = s-frame.

2.2. Notations for the k = o-beam of the q = p-plane frame

According to the definitions already determined in Sec. 2.1, we introduce the corresponding symbols for the k = CJ continuous on r-elastic supports beam of the q = p plane frame

are the actions in the direction of the t-axis referring to the left, right, upper, and sub-cross-sections of the node ups, caused by the intermediate members [ups, a@ + l)s] or [ups, (a + l)ps], respectively

V’S = I& + V’J” “P UP #p (r=l,3)

v*” = v;y + V$” “P

(r = l,2).

M$, M$ (t = 2,3), M,*pU.“, M$+

are the internal bending moments about referring to the left, right, upper, or sections of the node ups, respectively.

(t = 1,2)

the t-axis, sub-cross-

Q f;;, Q$’ (t = 2,3), Q;y, Qo*ps’u (t = 1, 2)

are the internal shear forces of the left right, upper, or sub-cross-sections of the ups-node in the

2.3. Notation for the 30 plexus frame

For a generic vector, referring to all the nodes of the frame, we simplify the notation as follows:

Moreover, the use of an overbar with vectors or matrices shows that the corresponding quantity refers to all the i-beams (of the p-frames) of the structure. The use of an asterisk and an overbar with vectors or matrices denotes that this quantity corresponds to all the j-beams of the 3D frame, while a tilde shows that the vectors or matrices include all the k-beams of the structure.

As mentioned in the Introduction, the 3D plexus frame is analysed to an equivalent set of: (i) mr horizontal continuous beams i, parallel to the Ox-axis lying on n elastic supports; (ii) mn horizontal continuous beams k, in the direction of the Oy-axis resting on ? elastic supports, and (iii) PIT vertical continuous beams j, parallel to the Oz-axis, on m + 1 elastic supports, with the (m + 1)~s

852 P. D. ANDRIOTAKI-PANAYOKWNAKOU

support being a fully fixed end (or a fixed end with is the tridiagonal symmetric (n x n) matrix, similar elastic vertical displacement). to A:.

All i-beams are subjected to the given intermediate external loading on the (0 1, 03)- and (0 1, 02)- plane, as well as to the nodal moments mo”, + Z& and

Rf = [Mi;; - M$]

the nodal external forces P& (t = 1,2,3). The j-beams are subjected to the given inter- =[_M?:” M:f_M;F . Mgy

mediate general loading on the (0 1, 03)- and the (02, 03)-plane, as well as to the nodal moments is the vector of the external intermediate loading of Z:; (t = 1,2,3). dimensions (1 x n).

The k-beams are subjected to the given intermediate external loading being analysed on the (02, 03)- and (01,02)-plane and the .?!i”, nodal Ef=[P$+ V&+LY&,(M$_,-Mf;;)

moments (t = 1,2,3). + c(;;(M;f;+, - M$)]

3. ANALYSIS is the vector of dimensions (I x n), similar to R,$.

3.1. i-Beams loaded on the plane (01,03) (Fig. 2~)

If we consider the i,, straight beams, continuous on i=[l 1 ..’ I]

n elastic supports and loaded by forces and couples on (0 I,0 3)-plane, according to Figs 2(a, b), we can is the unit (1 x n)-vector. ~2 is the resultant axial write the following matrix expressions force of the i = u-beam.

where

!C=rlc,%?l $2 .‘. $ $1 = (1 x n) vector

*F, Wz, Zf, Z$, X:, X2, m:, rn$ = (1 x n) vectors being similar

r 21:~ 1’2” 0 . 01

to I++$ formulated

0 0

0 0

which is the tridiagonal symmetric (n x n)-matrix.

Bc = [Ib’.‘,” (Z$_ , - I$) -z$y

is the tridiagonal (n x n)-matrix, being formulated similarly to A$.

l-p = (B$)‘, X;JT=~;

is the tridiagonal symmetric (n x n) matrix, similar to A?.

Af;' = [-I:!," _ , (1;;; ~ 1 + 1;;) -I:;]

(1)

(2)

Forallthei=a-beams(a=1,2,...,m)ofthe generic p = s-plane frame the following matrix

equations result

#;A; = W;H”z + (Z;/2E) + K;

X-j = $;C; + W;D; + E; (3)

$;A;=(Z;/G)+K:,

where based on eqns (l), (2) and on Sec. 2

AC = (n x n)-matrix

A: = (n x n)-matrix


Hf = 3BF = (n x n)-matrix Wr, &,X3,X,;, $,, Z,, SE;, K,,, I$ are (1 x mn~)-vec- tom being formulated similarly to &.

Cc = 6ET$ = (n x n)-matrix

D: = 12EAf = (n x n)-matrix

Kf=(m$+Rf)/2E= (1 x n)-vector

KF=mF/G = (1 x n)-vector

A, =

0 0 ... A;

is a diagonal (T x r)-matrix with elements the #/;=[ni” ny ... nkq = (1 x m)-vector (mn x mn) submatrices A; (s = 1,2,. . . ,7) and the

mn x mn)-nul submatrices 0. The final dimensions of

!&=r+: JI?2 ... *?I *g ti% ... 4% A, are (mnr x mn?).

.. $:I tK% ... Cl (4) RZ, C2, D,, A,

is a (1 x mn) vector of the slopes about the 2-axis of all the nodes of the p = s-frame, W;, P2, Ki, X;, E;, Xi, $j , 2; , K; are the (1 x mn) vectors being formulated similarly to I&.

A; = . . . . . . . . . . . . . . . . . . . . . . . . . . . .

= (mm x mm)-matrices similar to A,, (6)

J 0 ... 0

0 J ..’ 0 J= . . . . . . . . . . . . . . . . . . . . . . .

0 0 ... J

is a diagonal (T x T)-matrix, with elements the (m x mn) submatrices J and the (m x mn) nul submatrices 0. The final dimensions of J are (mr x mm).

is the diagonal (m x m)-matrix with elements the (n x n)-submatrices A: and the (n x n)-nul subma- rt*=Ml=b7l’ .‘. v!P! vi’ trices 0. The final dimensions of Ai are (mn x mn), Hi, C;, D2, A; are (mn x mn) matrices similar to Ai. . . 12 . . . 0-m tl;’ . . rl;] i 0 ... 0

0 i ... 0 J = I 1 . . , . . . . . . . . . . . . . . . . . . .

0 0 ..’ i

is the diagonal (m x m)-matrix with elements the unit (1 x n)-vectors i and the nul (1 x n) vectors 0. The final dimensions of J are (m x mn).

For all the i beams of the p = 1,2, . . . , s, . . . ,7 plane frames, one may write, according to eqns (3) and (4) and the notation introduced in Sec. 2, the following elasticity equations

is a (1 x mr)-vector for the resultant external axial forces of all the i-beams of the 3D frame.

3.2. i-Beams loaded on the plane (01,02) (Fig. 26)

For all the i-beams (i = 1,2, . . . , o, . , m) of the generic p = s plane frame, loaded horizontally, one can write the following matrix equations, analogous to the first two equations of (3)

$; A; = W”,H; + (Z;/2E) + p3

(7) X;=$;C;+B’“,D;+E;,

- - &A, = IV, H2 + (Z2 /2E) + x2

- - x,=$2C2+B’,D2+j?2

X,JT= tjl

&A, = (Z,/G) +%,

where the vectors and matrices in eqns (7) are of the same dimensions and of the same formulation as the corresponding eqns (3) and (4) except the subscripts

(5) of the elements: (i) of the vectors $, Z, E, K, m, R and of the matrices A, H, C, D are ‘3s’ (instead of ‘2s’), (ii) of the vectors W, X are ‘2s’ (instead of ‘3s’) while (iii) the elements P and V of the vectors E: are ‘2s’ (instead of ‘3s’).

in which

$L, . . , r/G,,, kL, . . . , r&L,] = (1 x mnz)-vector.

854 P. D. ANDRIOTAKI-PANAY~IWNAKOU

For s=1,2,... , t, eqns (7) result in the follow- For all the vertical columns of the frame ing expressions analogous to the first two equations (p=l,2,... , n) the following matrix equations are of (5) formulated

~Js = Ct;R + &/2E) f J& (8)

$2+sA:s = W:sH,*” + (Z:“/2E) + K:”

J&=&C,+ p$,+E,, X:’ = +:“C;’ + WFD;’ + E;

where the matrices and vectors are of the same (11)

dimensions and are formulated similarly to the Xrs J*T = q:” + #”

corresponding eqns (5) and (6). $$“A:‘= Z:‘/G,

3.3. j-Beams loaded on the (01,03)-plane (Fig. 4a, 6)

For a generic j-beam of the p = s frame the where according to the equation previously devel-

following equations, corresponding to (l), are valid, Oped for the i = a-beam considering the beam continuous on a number of (m + 1) elastic supports, while the (m + 1, p)s soil A:" = (m x m)-matrix

support is a fully fixed end H+& = 3B*” = (m x m)-matrix P P

+,*LA;Z” = 3 Wp+‘sBp*2s + (ZtL’/2E) + (R,*2”/2E) C;” = 6ET:& = (m x m)-matrix

Xp*” = ,E,;,I-;” + 12EW:‘“A;2.’ + Ep*&

where

(9) D:” = 12EA,*& = (m x m)-matrix

A*3s = (m x m)-matrix P

Kp*& = (R,*“/2E) = (1 x m)-vector

W*lS z*zs X*lr x*31 **3.c z*3s _ P’ P’ P’ P’ P’ P -

vectors of dimensions (1 x m), of an analogous to $,*” formulation

Ap*> = [I,*$ .2(I,zlT,p -I- ILT*42S) IZi*d”]

:

21*‘” . . . IP

ZW25 IP

0 0 . . . 0

I*!25 IF 2(r:dh + If;“) [*‘2” . . . 0 . . .

2P 0 =

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

0 0 0 * . . l:yp 2(z:?,,p + rg")

= tridiagonal symmetric matrix of dimensions (m x m)

Bp*& = [1,*“2;,0 (Z,*!?.p - 1::“) - 1,*6’7 = tridiagonal (m x m)-matrix, being formulated similarly to ArL’

rFzr = (B,*&)T

A;*= = [I;:;$ - (Z,*T!‘yp + Z$“LF) Izrh] = tridiagonal symmetric matrix of dimensions (m x m), analogous to Ap+Z’

A*3S _ _ Zf3S P -[ (I_ I-p (Zi3: ,,p + Z$) -Z$j = tridiagonal symmetric matrix of dimensions (m x m) (10)

R*” = [MO*pY’b - M:;&] = [ _ ,$di (M;;32s - &f$*S) . . . (Mz’$& - M,$ZP)] = (1 x t,q)-vector of the P

external intermediate loading

Ep+& = [ V$” + a,*?, ,(M~*;,, - MT;&) + LZ$‘~~(MI,‘:~~ - M;;“)] = (1 x m)-vector of the external intermediate loading of the j = p-beam

i* = [l 1 .. I]= (1 x m)-unit vector

,;‘s = resultant axial force of the i = p-beam of the frame

$I;,$ = vertical reaction of the fully fixed end of the j = p-column.


= (1 x mn)-vector of the slopes of the deflection of the nodes ups about 2-axis

= (1 x mn)-vectors being formulated in the same way as vector $:” (12)

r A;2 0 . . . 01 0

A:“= A;” . . . 0

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

0 0 ’ ” A;" J is the diagonal (n x n)-matrix, with elements the (m x m) submatrices A,*” and the (m x m)-nul submatrices 0. The final dimensions of A:” are (mn x mn), A?, Hr, Cy, D:“, are (mn x mn)- matrices, similar to A!“,

i* O* . . . O*

. . . . . . . . . . . . . . . . . . . . . . . . . . . . O* O* . . i*

is the diagonal (n x n)-matrix with elements the unit (1 x m) vectors i* and the nul (1 x m) vectors O*. The final dimensions of J* are (n x mn).

For all the j-beams of all the plane frames p=1,2 ,..., s )...) 7 one can write the following matrix elasticity equations

where

is the (1 x mnr)-vector of the slopes of the deflection of the nodes aps about the 2-axis, @, ii:, 8, @,

p, X:, R, E$ are (1 x mnr)-vectors with the same as vector 9: formulation,

A$’ 0 ... 0

@’ = [Af2] =

I 1 0 A;’ ... 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

0 0 ..’ Aj’

is the diagonal (7 x r)-matrix with elements in the main diagonal the (mn x mn) submatrices A?, and the (mn x mn) nul submatrices 0.

The final dimensions of @ are

(mn7 x mn7) (14)

R:, c:, w, A:

= (mn7 x mnr)-matrices similar to AZ

I:+$:=[rl:“+9:“1

= [tl :31 + 4 :31 . . . nn*3I + bn*31,

n:32 + 4~32. . . l;32 + 4~32, . . .

= (1 x nr )-vector

J* 0 ... 0

J* = 0 J* ... 0

[ 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

0 0 ... J*

is the diagonal (7 x 7)-matrix with elements in the main diagonal the (n x mn) submatrices J* and the rest (n x mn)-nul submatrices 0. The final dimensions of J* are (n7 x mnt).

3.4. j-Beams loaded on the (02,03)-plane (Fig. 4~)

For all the j-beams of a generic frame p = s loaded on the (02,03)-plane, according to Fig. 4(c), one can write the following equations, which are similar to the first two equations of (11)

$:“A:“= W:“H:” + (Zf’“/2E) + K:

(1% Xp=$:“C:‘+ WrD:‘+ ET”.

All the matrices and vectors in eqns (15) can be formulated similarly to the corresponding matrices and vectors of eqns (11) and (12), except the superscripts of the elements: (i) of the vectors $ *, Z*, R +, m*, E* and the matrices A*, H+, C+, D* are ‘1s’ (instead of %‘), (ii) of the vectors W*, X* are ‘2.7’ (instead of ‘ls’), and (iii) the element V of the vector E:‘, is ‘2s’ (instead of ‘1s’).

856 P. D. ANDRIOTAKI-PANAYOKIUNAKOU

If we consider all the plane frames p = 1,2 ,...) s ,... , T we can write two more matrix equations analogous to the first two equations of (13)

$:A:= @R:+(Z:/2E)+R:

(16) X:=$:e:+ firs:+E;c.

The dimensions and the formulation of the matrices and vectors in eqns (16) are the same to the corresponding of eqns (13) and (14), if we also take into account the previous notation about the change of superscripts.

3.5. k-Beams loaded on the (02,03)-plane (Fig. 3a)

We consider the generic k-beam, of the plane frame q = r as continuous and lying on 7 elastic supports. The loading is supposed to be co-planar with the frame. Therefore, according to Figs 3(a, b) and following the same procedure as Sec. 3.1, similar to eqns (l)-(4) the equations are only valid if only we take the following into account.

1. In equations analogous to eqns (1) and (2), all the superscripts s, 2, 1 are substituted by p, 1, 2, and the subscripts p - 1, p by s - 1, s.

2. In equations analogous to eqns (3) and (4) the superscript s and the subscripts 2, 1 are submitted by p and 1, 2 respectively.

3. Vectors of dimensions (1 x n), (1 x mn) and matrices of dimensions (n x n), (m x mn), (mn x mn) of eqns (l)-(4) must have, in the case of k-beams, the dimensions (1 x T), (1 x m7) and (5 x T), (m7 x mz), respectively.

Therefore, for all the k = 1,2, . , Q, . . , m horizontal beams of all the q-plane frames (q = 1,2 ,..., p ,..., n), loaded on the (02,03) plane, the following final system of elasticity equations result, written in a matrix form analogous to (5) and following the notation of Sec. 2

(17)

where &=[Ij/tll ‘. . ti1h ... $I,, ‘.. $!d,ti12, . . . *12r *!I221 . *t** . . . *t,, . . J/t,, . . ’ *’ $!,,,,I is a (1 x mnz)-vector of the slope of th?deflection about the l-axis, pJ, .&, g2, R2, .?,, R, (8,) R2 are (I x mnz)-vectors, being formulated as a $,-vector and h, , R, , cl, b, , A, are diagonal (n x n)-matrices with elements in the main diagonal

the (mr x mr) submatrices Al;, Hf , Cp, D$‘, A$ and the rest (mr x m7)-nul submatrices 0.

The final dimensions of the matrices are

(mn7 x mnr) (18)

3 0 ..’ 0

3= 0 3 “. 0

: 1 . . . . . . . . . . . . . . . . . . . . . . . . 0 0 .‘. 3

is a diagonal (n x n)-matrix with elements in the main diagonal the (m x m7) submatrices 3 and the nul (m x m7) submatrices. The final dimensions of 3 are (mn x mnz). i 0 ‘.. 0 3= 0 i “’ 0

I 1 . . . . . . . . . . . . . . . . . . . . . . . . . 0 0 “’ i is the diagonal (m x m)-matrix with elements in the main diagonal the unit (1 x 7)-vectors I’and the rest of the elements the nul (1 x r)-vectors 0. The final dimensions of 3 are (m x m7).

&=[q$]=[fl;l . s; . . Ir;’ . . .

is a (1 x mn)-vector of the resultant intermediate external axial forces in the direction of 2-axis for the 3D frame.

3.6. k-Beams loaded on the (01,02)-plane (Fig. 36)

In a similar manner to that used previously, one can write for all the k-beams of all the q=l,2 )...) p ,...) n frames, loaded on the horizontal plane, two more equations in matrix form similar to eqns (8) for the horizontally loaded i-beams

$,A, = win, + (.&/2E) + K (19)

where

h=[$:$:...W

= w:,, . . . e,, . I .4%, ” $I,,?

*:x . . *:2,. *iI,, *12r . .

*:“I . *. d4.r~ . .4G”, . ” 4Ll (20)


is a (1 x mm)-vector of the slopes of the deflection of the nodes about the 3-axis, w,, z,, x,, &, E, are (1 x mn~)-vectors being similarly to $r formulated, &, fi,, c,, b, are diagonal (n x n) matrices with elements in the main diagonal the (m7 x mr) submatrices A$, HP, C$, D$ and the rest (mr x mz) nul submatrices 0. The final dimensions of A,, fi,, c, and f), are (mm x mnr).

4. DERIVATION AND SOLUTION OF THE ELASTICITY MATRIX SYSTEM

In order to find the compatibility conditions and the equilibrium equations valid for all rigid joints of the 3D frame we have to compare them to the vectors for the generalized forces and displacements of the supports of the basic structures already introduced in Sets 3.3-3.6. But, since the order of the elements of these vectors is different for i-, i- and k-beams, we must adapt the order of the elements of i- and k-beams to that of i-beams and vice versa.

For this purpose, if m is the number of storeys, n the number of columns in the direction of Ox, and t the number of columns in the direction of Oy -axis, we formulate the submatrix E*’ (0 = 1,2,. . . , n) such that

~*‘=[t$~] (i=l,2 ,..., mn), (i=l,2 ,..., m)

&:=I fori=(j-l)n+8 (21)

.5$=0 fori#(i-l)n+O.

Now, let E* be the following (mn x mn)-matrix

e* = [c*1 e*z . . . E*n].

Finally we define

(22)

c* 0 ... 0

e* = 0 E* ... 0 (23) . . . . . . . . . . . . . . . . . . . . . . . . . . . .

0 0 ... f*

which is a diagonal (r x r) matrix with elements in the main diagonal the submatrices E*, while the rest of the elements are the (mn x mn) nul submatrices 0. The final dimensions of e* are (mm x mm).

Now, we denote by

<2’ $22 . . i2”

8s (24) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1 5’1 4’2 . . . 5”

a (r x n) matrix with elements in the (mn x mr) submatrices @‘, where the upper right indices k and I represent the serial number for the line and the

column in g, respectively. The final dimensions of g are (rmn x nmr).

The submatrices 2” of the previous matrix are formulated according to the following law

P=[E$ (i=l,2 ,..., mn),(j=l,2 ,..., m7),

(k = 1,2, . . . , T), (I = 1,2, . . . , n)

$=l fori=(m-l)n+Iandi=(m-I)r+k

Ct=O fori#(m-l)n+Zandi#(m-l)r+k,

m=l,2 ,..., 112. (25)

After the formulation of e*- and &matrices, it is obvious that for a generic vector of j- and k-beams the following relations hold true

$* = +e*, 4 = $g. (26)

Also, we derive that

J; = $*e*‘, $ = $S’,

while from (26) and (27) we deduce

(27)

$* = pCgTe* or J = $*e*%. (28)

So, using eqns (26)-(28) one may change the order of any vector of i-, i- or k-beams to the desired one.

As was emphasized in Sec. 2, the i-, j- and k-beams were considered incompressible. This assumption must be used in order to diminish the dimensions of the displacement vectors of the rigid joints of the frame.

It is obvious that

Is _ is WV - W.JI 9 $$s = &s

“P IP ’ $f = $2P al 1

$3P _ -3p #S - WISP op

w*ls = W*ls sl 9

w*3s = w*3s UP Ip . (29

Accordingto(29)fors=l,2 ,..., m,p=l,2 ,..., n, and s=l,2,... , T, the following vectors can be formulated

@; = [W;; W;; . . . ~$1 = (1 x m)-vector

5; = [w;; w;; . . . tvi:] = (1 x n)-vector

ri$ = [G;e $4 . . . IV,,] = (1 x m)-vector ‘ZP

(30)

@=[#p c;p . . . r@] = (1 x r)-vector

- ** - WI - [r@S W&u . . . w$] = (1 x m)-vector

- *s _ w3 -[wfY wp . . . w 7,‘” = (1 x n)-vector

858 P. D. ANDRIOTAKI-PANAYOT~LJNAKOU

The remaining vectors PSI Wp and vy were already determined in Sets. 3.2, 3.4 and 3.6, and we can only introduce the vectors

G,i = WS, = (1 x mn)-vector;

Gf = WY = (1 x mr)-vector;

r?y = W:’ = (1 x mn)-vector. (31)

Equations (30) and (31) are valid for a generic p = s and q = p plane frame. By now, for all the plane frames of the structure, namely for s = 1,2, . . . , t andp=l,2,..., n, these relations give

I& = [$ 6,: . . . S]] = (1 x m7)-vector

G2 = ri$ = WS, = (1 x mn)-vector

t& = [G; 0: . . G;] = (1 x nr)-vector

3, = Ctf = WY = (1 x mr)-vector

C2 = [s; *; . . . a:] = (1 x mn)-vector (32)

Et, = [Sit: 6; . . Gt;] = (1 x nr)-vector

tist:=[@Tt:’ WI -*I . . . d:‘] = (1 x m7)-vector

*F s a? = Wp = (1 x mn)-vector

GT=[Gt:’ Wj -*I . Glj:‘] = (1 x nr)-vector.

The new vectors introduced in eqns (32) for the displacements of the nodes of the whole structure are of smaller dimensions in comparison to the previously existing vectors. So, we must relate these two categories of displacement vectors. For this purpose, we formulate convenient matrices as follows:

E= [E c ... c]

is a (1 x m)-matrix with elements of the (n x n) unit submatrices E. The final dimensions of E are (n x mn).

8 = [tl 8 ‘. e]

is a (1 x m)-matrix with elements of the (7 x 7) unit submatrices 8. The final dimensions of 8 are (7 x mr).

E* =[E* E* . . E*]

is a (1 x n)-matrix with elements of the (m x m) unit submatrices E*. The final dimensions of E* are (m x mn).

e*=p* t-j+ . . . e*]

is a (1 x 7)-matrix with elements of the unit (mn x mn) submatrices 8*. The final dimensions of 8* are (mn x mnr).

E 0 ... 0

EC OE...O

[ 1 (33) . . . . . ..f.................

0 0 ... E

is a (7 x 7) diagonal matrix with elements of the (n x mn) submatrices E, and the (n x mn) nul submatrices 0. The final dimensions of E are (nt x mn7).

8=[, e .'. e]

is a (1 x n)-matrix with elements the unit (m7 x mr)

submatrices 8. The final dimensions of 8 are (mr x mn7). 8 0 ..’ 0 8= 0 8 ‘.. 0

I 1 . . . . . . . . . . . . . . . . . . . . . . . 0 0 ‘.. 8 is a (n x n) diagonal matrix with elements the (7 x m7) submatrices 8 and the (7 x mr) nul submatrices 0. The final dimensions of 8 are (n7 x mnr).

E* 0 ... 0

. . .

E* =

I 1 ..” . . ...” . . . . . . . . . . . . .9..

0 0 ... E*

is a (7 x 7) diagonal matrix with elements the (m x mn) submatrices E* and the (m x mn) nul submatrices 0. The final dimensions of E* are

(mt x mnr).

Now, using the matrices mentioned previously we can write the following matrix relations connecting the displacement vectors of different dimensions for the p = s and the q = p generic plane frame

W”=$GJ W;=GS I I 9 2, w;=Gt;E,

wp = wf, wp = w$3, wp = w$@ (34)

WY = $TSE* , Wp = @y, WFs = EfsJ*.

Also we may write for the 3D structure that

W,=s,J, W2=@7@*, w,=&E,

Fv, = @,dp, Iv2 = c23, w3 = &B, (35)

fl=+fE*, m=~+;e*, @=@:J*.


0

P 2

Fig. 2. (a) Geometry and sign convention of a horizontal basic structure, loaded on the (0 1,03)-plane. (b) Geometry and sign convention of a horizontal basic structure, loaded on the (0 1,02)-plane.

Fig. 3. (a) Geometry and sign convention of a basic structure, loaded on the (02,03)-plane. (b) Geometry and sign convention of a horizontal basic structure, loaded on the (0 1,02)-plane.

860 P. D. ANDRIOTAKI-PANAYOTOUNAKOU

According to the already defined positive direc- q2 = -,Jrgr= $;e*r

tions of the nodal vectors for generalized forces and displacements of the considered continuous basic Jj = &gr = @e*‘.

structures (Figs 14), and through eqns (27) to (28), one can write the following relations, express- where 0 represents the nul (1 x mn~)-vector.

ing the equilibrium equations and compatibility Now, based on eqns (5) (8) (13) (16) (17) and

conditions (20), and using also (26) and (37) we can derive the following 18-matrix equations, which together with

X,+XIer+qe*r=O the six equilibrium matrix relations (36) consist of the elasticity system for the 3D frame being com-

X,C-R,+X:e*%=O posed of 24 matrix equations with unknowns $,, x,,p,&,Zi, 2:. pi((i=1,2.3)and the w,, @?

&e* + _&Ce* - X: = 0 and $: vectors

2,+&CT--_e*r=O

-22+.RrgT-~e*T= 0

Zj+&P+2:e*r=0

(36)

and

W, = -p,gT= -we*’

W, = W,iF= @e*’

+, =$,$r= -$Te*’ (37)

Fig. (b)

D ‘““\ 1

ICI UP s * 3s i 3s JI UP UP 5;;

a+l, ps

m+l,ps Z+VGZG?m

13

@a) (4b) (k)

4. (a) Geometry and sign convention of a vertical basic structure loaded on the (0 1,03)-plane. Geometry and sign convention of a vertical basic structure, loaded on the (02, O!)-plane. (c) Geometry and sign convention of a vertical basic structure subjected to torsional loadmg.


pJ*F=q:+$:

(G/2E)$,e*A: = (2/2E) (38)

(-G/2E)$,i& = (5/2E) + (G/2E)if,

ij,ed, = - W, &I, + (2J2E) + R,

x, = $,eC, - Iv, cl), + &.

In order to decouple the previous elasticity system we follow the following procedure.

1. We multiply some of eqns (38) on the right-hand side by convenient matrices, namely: the 1 lth by (-e*r), the 13th and 17th by (CT), the 7th and 10th by (e*‘), the 16th by (-eF), the 8th by (e*‘J’), the 18th by (SFJF), the 12th by (e*FSJF), the 6th by (a’), the 2nd by (e*J*‘), the 15th and 9th by (- l), the 14th by (gFe*J*F), and

2. We add by parts two of the resulting equations with one of the remaining eqns of (38), or three of the resulting equations. More specifically

in which

a, = (G/ZE)& + e*@e*‘+ gAA,gF

= (mnr x mnr)-matrix

a2 = A, + e*@e*‘+ (G/2E)i%,CF

= (mnr x mnr)-matrix

a, = A, + (G/2E)e*@e*‘+ 8&OF

= (mn7 x mn7)-matrix

g, = e*Cfe*F Jr,

g, = gC,e’ Jr = (mn7 x mt)-matrices

g, = -e*CFe*rg3T,

g, = C,dF = (mnr x mn)-matrices

g, = EC, iFe*J*‘,

a=c,e*J*T= (mn7 x n7)-matrices

b, = - 3SFe*Rf e*’ = (mn x mnz)-matrix,

c, = l%R,e = (n7 x mnr)-matrix

dl = - Je*R:e*‘= (mr x mnt)-matrix,

c7 = ER, = (nr x mnz)-matrix

d2 = - JiS,F = (mt x mnr)-matrix,

b, = Ji?f& = (mn x mnr)-matrix

the 4th together with the 11th and the 13th the 1st together with the 7th and the 16th the 5th together with the 10th and the 17th the 3rd together with the 8th and the 18th the 6th together with the 12th and the 15th the 2nd together with the 9th and the 14th.

(40)

d, = J(e*i5fe*7 + 6B,Z)JF = (mr x mT)-matrix

After some algebra and through relations (35) we b3 = -.%‘(D, + e*D:e*l)tG3’= (mn x mn)-matrix

derive the following, which are equivalent to the elasticity system (38) cg = -E(D, + gb,P)e*J*F= (n7 x nr)-matrix

k, = (G/2E)R, - @‘e*T+ R,c=,

k, = R, + @e*‘- (G/2E)Rzi+‘= (1 x mnr)-vectors

k, = K3 + R,gF = (1 x mn7)-vector,

(39) k - -& -E~e*FJF+E,~FJT=(l x mr)-vector 4-

k,=&-(E,+E~e*‘)~JF=(l xmn)-vector

k, = tj: + @ - (E, + E, gr)e*J*F = (1 x nr)-vector.

862 P. D. ANDRIOTAKI-PANAYOKWNAKOU

In order to solve the system (39) we consider two cases.

Case (a). W3 is a known or nul vector. From the 4th and 5th of (39) we can derive

(41) c2 = (iha + i&4)& -WC.

Substitution of (41) into the first three equations in (39) gives

hf,+$,f,=f,

Jizhl+$3h,=f,

3A + $2j2+ ihi3 = f3,

where

f, =a, -g3b;‘b,,

f2 = -&b;‘b’ = (mnz x mnt)-matrices

h’=a2-g,d;‘d,,

h2 = -g2d;‘dl = (mnz x mn7)-matrices

j’= -g3b;‘hr, i2 = -g’d;‘d2,

j, = a3 - g2d;‘d2 - &b;‘b,

= (~77 x mnr)-matrices;

(42)

d, = tC3c1 -t k, - k,b;‘b, = (1 x mnt)-vector

e2 = d,c, + k2 - k,d;‘d, = (1 x mnr)-vector

e3=k,-(kqd;‘dz+kSb;‘b2)

= (1 x mnr )-vector. (43)

Solving (42), we determine the vectors $, , q2, i&3 through the formulae

q3= AB-I, 3, = L,f;’ - AB-‘f,f;‘,

q2 = lzh;’ - AB-‘h2h;‘, (49

in which

A =t,-d,f;‘j, - d2h;‘j, = (1 x mn7)-vector

B = j, - f,f;‘j, - hzh;‘j, = (mnz x mnt)-matrix.

(45)

Also, the sixth equation of (39) through the last equation of (40) gives the vector #, expressing the soil vertical reactions, as follows:

4: = $‘g, + q296 + (E, + E,C3e*J*T - I,C, - tj:.

(46)

Case (b). c3 is linearly depending on the soil vertical reaction # .

In this case, one further matrix relation is valid

where

+: = -t?,v, (47)

v = [$tl+ “PI

is a diagonal (nt x nr) matrix with elements the

%I+ ‘)p translational spring constants of the (m + 1)~s supports. From eqns (46) and (47) one may easily derive that

~3=$‘m’+J2m2+~,,

in which

ml = g5(e3 - W’,

(48)

m, = g,(c, - V)-’ = (mnr x mnz)-matrices

e4 = {(kT2 + E,gT)e*J*‘- #:} (49)

(c, -V)-’ = (1 x nr)-vector.

Now, from the two first equations of (39), through the new relation (48) we deduce that

$,a: +$,a: = P,b, +k:,

(50) $,a:+Ji2a4*=~,dl+k~,

where

ay=a’-m,c,, a: = -m).c,, a: = -m,c,,

a4 *=a*--m2c2 (51)

= (mnt x mm)-matrices

k:=k,+d,c,, k; = k, + f4c2

= (1 x mnt)-vectors.

Equations (SO) together with the third, fourth and fifth equations of (39), after some algebra, lead to

Slf:+ihf:+$3rf=e

$‘h: +&h: + $,h: = t’; (52)

S’j’ + $7i7 + q3i3 = d3,


in which 5. NUMERICAL APPLICATION

ff=a:--g,b;‘b,, c=a:,

q = -&b;‘b, = (mm x mm)-matrices

hf=a:, hf =a: -g,d;‘d,,

(53) h: = -g,d;‘d, = (mnt x mnr)-matrices

8: = k: - k,d;‘d, = (1 x mnr)-vectors.

From the decoupling of eqns (52) we can derive closed-form formulae for the determination of $r-, $r-, $,-vectors. Also, eqn (48) gives vector I,, while relation (47) leads to the determination of the vertical soil reaction @ for the case of elastic foundation.

In order to illustrate the correctness and the potentialities of the developed methodology we consider a symmetric, fixed in the ground, rectangular 3D plexus frame (Fig. 5). We apply two different types of loading; one is vertical symmetric, consisting of a constant continuously distributed loading q along all the horizontal beams of the structure. The other is considered antisymmetric, consisting of horizontal concentrated forces P in the direction of the @-axis, applied on the nodes, and of horizontal continuously distributed loadings q’ and 2q’, applied along the horizontal beams i of the first (p = 1) frame of the structure, in the direction of the @-axis. The effects of such types of loading on the deformation of a symmetric structure are predictable.

The data of the problem under consideration are

Now, for both cases, since $, , &, $i3 and G, are known, one may easily derive G,, wirz vectors from eqns (41). Later, through relations (35) and making use of the second, sixth, eighth, 12th, 14th and 18th of the elasticity system (38) the unknown vectors X,, x2, 8:, K:, xz, and 8, can be immediately determined. Moreover, the first three equilibrium equations (36) gives the X,, 8, and Xf vectors, respectively. Finally, it is obvious that through the first, fourth, fifth, seventh, tenth, 1 lth, 13th, 16th and 17th equations of (38) we derive the zi, Z,+, and zi (i = 1,2,3) vectors. So, all the structure redundants are calculated and the static analysis of the structure can be completed by finding the internal generalized forces at the right, left, upper or sub-cross-section of the rigid joints, using the well-known formulae.

I*” = ps = 1 a0 UP ’ P3’= 1 a0 . 81

(a = 1,2), hs = L&3) (54)

1*2s = I*‘” = 21 3P 3P ’ I*” = 3.61 3P (P,S = 1,2,3)

G = Q.4E, p = G/2E, w:=o; P = 0.4q’a.

Based on these data, eqns (2) and (lo), and on the corresponding eqns of Sets 3.2, 3.4-3.6, we formulate the matrices related to the i-, j- and k-beams of dimensions (3 x 3).

Fig. 5. A symmetric 3D plexus frame under symmetric and antisymmetric loading.

864 P. D. ANDRIOTAKI-PANAYOTOUNAKOU

Because of the symmetry of the structure the following relations hold true

Since the numerical formulation of the above matrices is immediate, we only give here the construction of two of them, namely

-1 -1 0 B2” = B3” = fib = fib = _

0 c 0 (1

-1 1 0

Since all the frames have the same geometrical characteristics we can easily formulate the diagonal matrices of dimensions (9 x 9), which correspond to the previous submatrices (55), according to relations (4) and (12) and analogous equations of Sets 3.2, 3.4 and 3.6. For example

In the same manner we construct the diagonal (27 x 27) matrices for all of the 3D frame through relations (6) (14) and the analogous equations of Sets 3.2, 3.4 and 3.6, i.e.

H; 0

R,=A,=f-I,=H,= [ 0 H;

0 0

0

0

Hi I Also, matrices J, J and J* are found from which 3, J and J* can be constructed, i.e.

We must now formulate

eqns (21)-(25) e* and e, according to

0 0

e* = 0 e* 0

i 1 0 0 c* c* = [E*’ E*2 ,*‘I c,, e1: Cl? e = I?’ 2: c22 [ 1 531 ~32 ,33

Submatrices L *” (0 = 1,2, 3) are of dimensions (9 x 3) and the non-nul elements of these matrices

are

*1 _ *I - *I _ *2 - *2_ *2 _ *3 tll -642 -t73 -t21 -ES2 -EM -t31

= *3 - *3 - 662 - 693 - 1.

Submatrices iZk’ are of dimensions (9 x 9) and the non-nul elements of these matrices are

Also, we need the formulation of the (9 x 27)-matrix E, according to relations (33) namely

E 0 0

E= 0 E 0

i 1 0 0 E

100100100

I 0 0 1 0 0 1 0.

001001001 1 Through relations (40), (43) and (45), and using

an appropriate computer program, we can calcu- late all the ai, b,, c,, d, (i = 1, 2, 3) g, (i = 1, 2, 3,4) and fi, h, (i = 1, 2), j,(i = 1, 2, 3) and B matrices, related to the geometrical characteristics of the structure.

Again using eqns (2) (4) (6), (lo), (12). (14) and the corresponding relations of Sets 3.2, 3.4 and 3.6, we can formulate the vectors expressing the external loading. So, we write the following.

865

(a) Symmetric loading.

Analysis of 3D rigid-jointed rectangular plexus frame

The non-nul vectors are

Rz = a, = (qa2/6E) [ 1 0 -1 1 0 -1 1 0 -1 1 0

~&G,=qcc[l 2 1

The calculations show

1 -1 1 0 -1 1 0 -1 1 0 -1 1 0 -11

1211211211211

2 1 1 2 1 1 2 1 1 2 I].

that vector A is equal to zero. Consequently, $3 = 0 and therefore $, = t,f;' , - -

$, = /?h;' . The numerical results for the slopes $, and & give

$1 = b&l> +: = [a $; = -t,b; = (qa3/6EZ)[a a a b b b c c c], $2 = Ml?

$:=$i=$i=(qa3/6EZ)[a 0 -a b 0 -b c 0 -cl,

where

a = 0.2947368, b = 0.1157895, c = 0.1263158.

Equations (41) show, through the program calculations and together with the previous results, that the displacement vectors iir, , d, are equal to zero. All these results are in agreement with those expected for this symmetric loading.

(b) Antisymmetric loading. Now, the non-nul vectors are

& = (q’a2/6E)[l 0 -1 2 0 -2 2 0 -2 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 O]

E3 =q’a = [1.4 2.4 1.4 2.4 4.4 2.4 2.4 4.4 2.4 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 01.

The calculations of the program firstly provide vector $,

$j =(q’a3/EZ)[u, 0 -u, a, 0 -a2 u3 0 -a, b, 0 -b, 6, 0 -b,

b, 0 -b3 c, 0 -c, c, 0 -c* cj 0 -c,]

where

a, = 0.152080, aI = 0.205692,

b, = 0.003964, b, = -0.004588,

c, = 0.058329, c2 = 0.054065,

u3 = 0.151967

b, = - 0.008371

cj = 0.026019.

Then vector 7, is found to be equal to

$,=[$;I, @=1,2,3)

tif = tif =Wa31W~, a2 ai h b2 b, cl c2 ql,

where

a, =0.1292654, u,=0.1275013, 6, =0.274726,

6, = 0.303995, c, = 0.340158, c2 = 0.422452

and

$:= (q’a/Wa, a2 aI b, b2

where

a, = 0.090235, a, = 0.090401,

6, = 0.241898, c, = 0.279366,

b, CI ~2 c,l,

b, = 0.217782

c2 = 0.346948.

Vectors & and I, are equal to zero. Displacement vector I& of dimensions (1 x 9) is estimated as follows:

G2 = [@$I (p = 1,2,3),

#=@3 2

= (q’a4/EZ)[0.9058995 0.6699041 0.25699781

a: = (q’a4/EZ)[1.066123 0.8224351 0.33231861.

866 P. D. ANDRIOTAKI-PANAYOTOLJNAKOU

All these results are in agreement with those (c) The possibility of taking into account an elastic expected according to the deformation of the sym- foundation. metric structure subjected to this antisymmetric (d) The possibility of finding easily the effect of any loading. loading on any point of the structure or the soil

reactions. 6. CONCLUSIONS

In this investigation the static analysis of a 3D rectangular rigid-jointed plexus frame is achieved under a general spatial external loading. The frame, consisting of m storeys, n columns in the direction of the x and r columns in the direction of the y-axis, is analysed to a matrix system of 24 coupled equations with equal number of unknown vectors.

The system is decoupled and solved through closed-form formulae, considering the vector of the 4, vertical displacements of the nodes as known or elastic (elastic foundation). In the analysis the incom- pressibility of the beams and columns of the frame is 5. taken into account and the effect of shear and axial 6, internal forces was assumed negligible in comparison to that of bending and torsional moments.

Among the most important results of this research 7. one may list the following.

(a) By means of the matrix algebra, the brief formulation and closed-form solution of the problem under consideration.

(b) The reduction of the dimensions of the matrices involved to (mnr x mm) requiring less memory space and computer coding, in comparison to other existing methods.

8.

9,

REFERENCES

P. Lustgarden, Iterative method in frame analysis. J. Strut; Div., ASCE 89, 75-94 (1963). R. W. Clouah. I. P. Kina and E. L. Wilson. Structural analysis of muhistorey biildings. J. Struct. biv., ASCE 90, 19-34 (1964). W. J. Weaver and M. F. Nelson, Three-dimensional analysis of tier buildings. J. Strucf. Div., ASCE 92, 385404 (1966). V. Cervenka and K. H. Gerstle, Approximate lateral analysis of building frames. Report to AC1 Committee 422 (1969). J. Gluck, Lateral-load analysis of asymmetric multi- story structures. Proc. ASCE ST2 (1970). E. D. Panayotounakos, Static and dynamic analysis of plexus frames based on the matrix algebra. Technica Chronica l-2 (1966). D. E. Panayotounakos, C. P. Spyropoulos and J. N. Prassianakis, Interaction of multistorey and multicolumn rigid-jointed frames supported on an elastic foundation under static loading. Comput. Sfrucf. 26, 8.5-869 (1987). P. D. Andriotaki-Panayotounakou, Structural analysis of a one storey multicolumn cylindrical frame. Compuf. Swuct. 27, 379-391 (1987). P. D. Andriotaki-Panayotounakou, Three dimensional static analysis of a plane rigid-jointed plexus frame spatially loaded. Compuf. Strucf., to be published.

Documents

The analysis of a three-dimensional rigid-jointed rectangular plexus frame