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

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  • 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


    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.


    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 sup- ports 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 con- tinuous 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 dis- tributed 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 determi- nation 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.


    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 character- ized 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



    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 displace- ments. 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 mem- bers, 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 inter- mediate 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 con- tinuous 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. wop; 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

    VS = I& + VJ 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*psu (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 hori- zontal 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


    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 inter- mediate 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.


    !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:~ 12 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:;]



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

    #;A; = W;Hz + (Z;/2E) + K;

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


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

    AC = (n x n)-matrix

    A: = (n x n)-matrix

  • Analysis of 3D rigid-jointed rectangular plexus frame 853

    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 sub- matrices 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)


    - - &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.


    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...


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