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Interaction of multi-storey and multi-column rigid-jointed frames supported on an elastic foundation under static loading
INTEFbWTION OF MULTI-STOREY AND MULTI-COLUMN RIGID-JOINTED FRAMES SUPPORTED ON AN ELASTIC FOUNDATION UNDER STATIC LOADING D. E. PANAYOT~UNAKOS,C. P. S~~~opot~u)s and J. N. PRASIANAKIS Department of Theoretical and Applied Mechanics, National Tcchni~ University of Athens, 184 Pat&ion Street, GR 112 57 Athens,Gnome
Abetract-In this investigation an exact matrix solution for the static analysis of a multi-stony and multi-column rectangular plexus frame on an elastic foundation is presented in the most general case of msponse and loading. Furthcrmorc, based on the formulation and proof of a convenientcorollary and
proposition, associated with the existing closed-formsolution of the matrix equation of the type AX+ XB= C, the problemof interactionbetween an elastic soil and the frame is completely solved. The entire procedure is illustrated by a comprchcnsive example concerning a four-rtorey and four4umn plexus frame supported on an elastic foundation. Several numcrkal results are also given. As the obtained closed-fotm fon@ac, giving the structure radtmdants. include matrices of dimensions equal to the number (m x n) of the nodesof the frame, the proposedsolutionmetl~oclology is more convenientin comparison
with those in the existingliterature,becauseit rcquins less memoryspace and coding.
1. ~ODU~ON
the Gayiey-Ha~lton theorem of the matrix algebra, the slopes, deflections and internal forces and couples The structural (static and dynamic) analysis of multi- of the nodes of the structure were calculated by storey and multi-column rectangular building frames, closed-form formulae. through digital computer methods, has attracted the In this investigation, as an extension of the interest of many researchers the last thirty years. methodology described in [8], the exact matrix solKani[l], Rubinstein [2], Soehrcns [3], Lustgarden [4], ution for the static analysis of a multi-storcy and Tcxcan [5l and others investigated such types of struc- multi-column plexus frame is presented in the most tures by using several iteration and relaxation tech- general case of response and loading (including tcmniques based on the flexibility, the stiffness or the pcraturc differences and different constant comgross joint and floor moments approaches, A more pressive forces acting at the free ends of each storey thorough ~v~ti~tion on the previous ~velopmcn~ and column, respectively). Factor, based on the was presented by Clough et al. [6,4, who analysed fo~ulation and proof of a ~nve~ent corollary and Iargc building frames through two different solution proposition associated with the exact solution of the techniques. In particular, in[6] two methods were previous matrix quation [S,91,the problem of interdescribed and compared; the fkst used a special form action between an elastic soil and the frame is of Gauss-Scidel iteration of the quilibrium quacompletely solved. The continuous horizontal and tions, while the second arranged the stiffness matrix vertical beams on elastic supports are selected as in the form of a tridiagonal system of submatrices, basic structures, while di!Tercnt rotational and transand then obtained the solution by means of a recur- lational springs are considered at the supports of each sion equation. column with the soil. The flcxural rigidities may vary Although all previous works arc suitable for corn- from one span to the next, but they are assumed to puter use, the consideration of a simply supported be constant within each span. Also, without loss of straight member, between two consecutive nodes of generality, the ratio of the flexuraI rigidity for each the frame, as the basic structure rcquircs matrices of in~~~atc span of a basic structure i = Q(orj s: p) dimensions much larger than the number (m x n) of over the same quantity for the corresponding span of the nodes of the structure. This disadvantage was the basic strncturc i=l (orj=l) is considered as confrontcd in[8], in which the author selected as constant. Furthcrmorc, in the analysis, the effcots of the basic structure the continuous straight beam on shearing and axial forces arc omitted. FinalIy; the elastic supports. The developed solution method- entire pmcedurc is illustrated by a comprehensive ology led to a matrix equation of the form example concerning a four-storcy and four-column AX+XB-C, in which the (mxn)-matrix X in- plexus frame and scvcral numerical results arc also cludes the structure rcdundants (slopes of the m x II- giVCll. . nodes), while the mmaining known matrices arc of Although the problem being examined is rigorous suitable dimensions. Through an exact solution of the and complicated, the Ilnal closed-form formulae givprevious equation, which was given in [S,9) based on ing the structure rcdundants can be easily and directly 855
856
D. E. PANAYOTOUNAK~
applied to the static and dynamic interacting analysis of an arbitrary plexus frame supported on an elastic foundation. Also, as the inserted matrices are of dimension qua1 to the number (m x n) of the nodes of the structure, the proposed solution methodology is more convenient in comparison with previous formulations, because it requires less memory and computer coding.
ef &.
horizontal i = u-beam lying on n - 2 intermediate simple elastic supports with translational spring constants Vy,,. , . ) u&,_ if in the y-direction; also, the terminal supports have translational spring constants u:,; u&l;u:,; I& in the y- and x-directions, as well as rotational spring constants e,,; e,, respectively (Fig. 2a). For the N: total compressive force acting at the free end of the beam, we have:
2. PRRLIMINARIES-NOTATIONS
Consider a rectangular multi-storey and multicolumn planar rigid-jointed plexus frame, being composedofi(i=l,2 ,..., e ,..., m)horixontaland j(j=1,2 ,..., p, . . . , n) vertical incompressible continuous straight beams, supported on an elastic foundation with translational and rotational spring constants different for each column. The structure is subjected to a generic co-planar loading consisting of dis~but~ and concentrated forces and couples, as well as to constant temperature differences between the limits of the cross-sections of each member. We suppose also tJrat different constant compressive forces P?: and I’$ act at the free ends of the storeys and the columns, respectively. A sketch of such a type of structure is shown in Fig. la. The flexural rigidities of the horizontal and vertical members may vary from one span to the next, but within each span these quantities remain constant. The frame can now be analysed to a set of i-horizontal and j-vertical continuous beams on elastic supports. We symbolize with (ffp) the span of the i = a-beam between the up and the a(~ + I)-nodes, and with (a~*) the cormsponding span of the j = u-beam between the up and (a + l)p-nodes (asterisks denote quantities corresponding to the vertical members). Fu~he~ore, we select as structure redundants the 2; and ?!; bending moments of the ij-nodes of the frame. Figure l(b and c) shows these redundants acting on the corresponding nodes of the basic structures. We note here that, in the following, quantities referring to the right, left, upper and sub-cross-section of the node will carry the upper right indices t, f, u and s respectively. Finally, without loss of generality, we suppose that the ratio of the flexural rigidity for an arbitrary span of the basic structure i = u (or j = p) over the same quantity for the corresponding span of the basic structure i = 1 (or j = 1) remains constant, i.e.:
in which n&, denotes the distributed axial force acting at the (ap)-span. Figure 3a shows the quilib&m of an intermediate node up under the action of the external couple m&, the external load Pip, the internal couples M$, and h4&‘,the shear forces Q&’ and Q’.;p’and the total vertical reaction X:,. In this figure w& denotes the displacement of the n-node in the y-direction. In Fig. 3@ and c) the quilibrium of the Q1 and on-nodes are presented, in which +:, and $:, are the slopes, and w:, and w& the displacements in the x-direction. Based on the previous figures and taking into account the in~rnp~~~~ty assumption of the beam, the following sets of equilibrium equations for the ap -, u 1- and (In-nodes result:
(3)
M:, = hf:[ -m:, = -e,,JI:r; XY #, = Q;,‘= I&W:,; XX 81--N$j = u:,w;,t
(4)
node an: ML= M&‘fm&==e,JrL; Yl m)rr- QA XY XL
~5
-Ngb’=
=
vY,wY,,
*
u&w&,,
(9
and wzr = (is=1 ,...,
m; j-l,...,n).
3. MATRIX ANALYSIS OF A STRAIGH’F’ CONTINUOUS REAM ON RLASTIC SUPPORTS
As the basic structure of the frame is the straight continuous beam, in our further discussion, we shall examine in principle the matrix analysis of such a
Also,
based
Wk.
(6)
on qn (2), one may have (7)
Introducing the last of qns (4) and (5) into qn (7), and taking into account relation (7) together with (6).
Static analysis of frames on elastic foundation
i=a
i=o
(mAI
(mtl)Z
(m+Wn
ImMiP
a
Fig. 1. (a) Typical building frame. (b) Geometry and sign convention of a horizontal basic structure. (c) Geometry and sign convention of a vertical basic structure.
we obtain:
we obtain n-linear quations which can be written in a matrix form as follows:
(u:, + u&)w:, = N:;
w;, = w”,
=
CA,
N:l(u:, + G,);
xx81 = JGA41 + G,>; Xx, = u”,N~l(u~,+ I.&).
I =
3WyA,2 + m:/2E + R.IZE + 1T,/2,
(9)
in which
(8)
Applying the principle of virtual work for any two consecutive spans of the beam and combining the results with the first of qns (3~(9, for all the nodes,
A 8.I
=
P;o(PI)), Vk-
1))-
J;opJr J;.,,l,
= t&diagonal symmetric matrix of dimensions
(n x n) with first element Zlb,, + (e,, /2E) and last element 2J;,,_ ,))+ (eJ2E);
D.E.
PANAYOTQLJNAKOS
et (11.
a Y
Fig. 2. (a) and (b) Geometry and sign convention of straight continuous beams on elastic supports.
Static analysisof frames on elastic foundation
859
C
b
Fig. 3. (a)-(c) Geometry and sign convention of an intermediate and two end-nodes of the frame.
A ..2'[-J~p-I))rJ~p-I))-J;bp),J;sp)lt 5 tri-diagonal matrix of dimensions (n x n)
with first element -J$,, and the last element J&n- 1,);
r: =ok,
** *, $k) = (1 x n)-vector of the slopes of the uj-nodes; (9a)
w’,) i= (1 x n)-vector of the vertical displacements of the nodes; m:=@tr:,,..., mk) = (1 x n)-vector of the external couples acting at the crj-nodes;
W{==(W$,,...,
R:P (-M:i’,
foregoing methodology and combining the wellknown expressions Y.1QV
T, - (&Art,,) t -J~,l,Ar(,,,+~.~~Ar.,,, . . = (1 x @-vector of temperature differences. We note here that, in the previous analysis, without loss of generality, the efTectsof shear and axial forces are neglected. On the other hand, based on the
a{H,_,jj(M$-
M%;_,,)
Q&’= I’&,’+ a~&.f$,,+ ,) - M&‘) with the second parts of eqns (g)-(5), we construct a set of 2n additional equations concerning the reactions of the elastic supports, which can be written in the following vectorial form:
h42; - Mii, . . . , Mg)
- (1 x n)-vector of terminal couples due to the external intermediate loading of the fullyfkedendspansofthebeam;
- V$+
Xi= 6E?:B,,r + 12EW&3 + r,;
(10)
x:=
(11)
WiYJ *
where Xi = (1 x n)-vector of the reactions;
= tridiagonal matrix of dimensions (n x n);
*,, , _ [ _ Jm (*I)*- Gt,-
1))+ J&Jl J&l,1
- tridiagonal symmetric matrix of dimensions
D. E. PANAYOTQUNAK~S el aI.
860
(n x n) with first element -J;:,, element -J& _ ,,,;
To = [P’,, + V:, + a;oI,(M2
-
and the last
- M;;), . . .
+ C, + a;,(, - ,,dW:, - 1l- Mg)]
= (1 x n)-vector;
&, rk,-
and
(17)
I)#?)9
L=r-j;;4,P,~
matrix.
In the previous relations (9a) and (12), 1 is the common coefficient of the thermal expansion; 4,) = ‘(‘.p,- G,, denotes the constant temperature difference between the limits of the cross-section of the (up)-span; E represents the modulus of elasticity, and a{,,,, is a coefficient equal to l/a,.,,,, where q,@, denotes the length of the (up)-span. Also, V$, v;; are the reactions of the simply supported member (UP) CC, = V$ + V&,‘),while J&,, .&, and J&,, are coefficients given by the relations
J&P)= Jtsph,,j ; J;b,, = J&a &,; Jr;,, = J~.dah;
(13) hp, = J~op~140p~ 9
where J(,+,,denotes the moment of inertia and &,, the height of the cross-section of the (up)-span. Finally, the superscript T represents the transpose of a matrix or vector. By now, solving eqn (9) for Pfvector and introducing the resulting new expression together with eqn (11) into relation (lo), we find the following equation for the II’:-vector:
(14)
The remaining vectors XY,, Y: can be evaluated through eqns (11) and (9) respectively. In the case when the supports of the beam are rigid, not on the same level, the FQvector is known, and, con~equently, the F;- and X$vecton result immediately through eqns (9) and (10). We consider now the second basic structure of the frame, namely the vertical continuous beam j = p, the (m + I)-support of which has translational and rotational spring constants Jib, ,,,,, &,+ ,I,, and re.spectively. The other supports of the beam a A\zple elastic with spring constants 8Y,(Fig. 2b). According to the previously developed methodology, the corresponding equations of eqns (9)-(11) for this case arc expressed as:
4,#-
lb)
+
k4Lb
{k-l,,-kP,L
k,,l
= tri-diagonal symmetric matrix of dimensipns (m + 1) x (m + l)+with first element -J;‘,,, and last element J;b,,; L
= r-k&-
I))9r;;;.-,,,, - k&L
$&)I
= tri-diagonal symmetric matrix of dimensizns (m + 1) x (m + 1) v$th first element and last element -J&,,; 4.2 = Q,2 = u-i-diagonal matrix (m+ 1) x (m + 1);
-
J$
of
dimensions
+;=(J&,...,J&, J&+,,P)T=(m+l)x of the slopes of the Q-nodes;
l-vector
IPi=(Z{,,...,$Y,,, ~~~+,,p)T=(m+l)xl-vector of the horizontal displacements of the ip -nodes; &,= (d;,,...,J&, &fm+,,JT=(m+l)x l-vector of the external couples acting at the ip -nodes; R, = (4&b’,
IV: = [3(m: + R, + IET,,)A; t B,, 2+ r,l
x [y’, - 12EA,,3- A$B,,,]-‘.
(16)
= u-i-diagonal symmetric matrix of dimensigns (m + 1) x (m + 1) pith first element U;,,, and last element W;,,, + [zCm+,,,/2E];
(12)
YY,= [u:] = (n x n)-diagonal
fi; + f,;
in which I =
...K
+ 12E&,
i;=+;rt;,
M:,‘),
Pi2 + V:, + a’;c,,(M:/ - M:i) + a&,(M$
?; = 6E4,,+;
$2; - n;riJ, . . . ( i+;nY+,J
= (m + 1) x l-vector of the terminal couples due to the external intermediate loading of the fully fixed-end spans of the continuous beam;
= (m + 1) x l-vector of temperature differences;
t,,+;=3X,,$;+&2E + 4/2E
+ &/2;
(15)
= (m + 1) x (m + l)-diagonal
matrix.
(18)
861
Static analysis of frames on elastic foundation
+
t
The
coefficients a’& J;,,,; Jh,,; JbpJ; *“’ ftapj and I,,, are determined as previously, namely as in the case of the horizontal continuous beam. We notice here that, for the (m + l)p-support of the foregoing basic structure, the following relation is valid:
in which * l CB(m+I)P+1); %l+I)p= *(m+I)P ? (m+ 1)~ * tt(m
=~(m+I)p(8(mp)Y+(m+r)p+2);
+ 1)~ =G(m+Il~
1; (mp) 8 (mtlb
As the previous plexus frame is constructed by j = 1, . . . , m-horizontal and j = 1,. . . , n-vertical continuous beams, we shall try to reduce the dimensions of qns (1 5)-( 17) in order to formulate matrices of dimensions qua1 to the number of the horizontal basic structures. For this p rpose, through the (m + l)p-component of the P ,-vector, as well as through qn (17), the (m + 1)ptomponent of the matrix equation (16) takes the form:
On the other hand, the combination of the last relation with the (m + l)p-component of eqn (15), after some algebra, leads to the following relation:
(224 Consequently, we succeed in giving the $i’,+ ,,,,displacement as a linear combination of the Ja-slope and the r?&displacement corresponding to the node mp of the frame. Introducing qn (22) together with eqn (21) into the (m + l&component of the matrix eqn (IS), as well as into the mp- and (m + l)pcomponents of the vectorial qn (17) one may obtain:
8 (m-h
kt~p=~~Ik~-l)p) + r&,-
I)P)- J&) - kp)B(m+
+&ip&r+l)P
4&I)p)
13;~m+l~p~~~+I*-~~m+l~pl/
ye@, l)p =
l IlP~(m+
I)P
&mtl?9= W%y,,(l
1;
+ 1Wkp,(l-
[3&l+ i)p(2c&&+ 1)p-
+ f&p)&ll+
111/
g(m+l)p 1;
L1 -3d@+l)p
l)p
I~~,}+I~E{~;;:~-,u~:.-,~P
+ kp,+
--&kp,i,-,,lkp~
in which
11- 3&l+
,,/6~&,)1.
(19)
~~m+,*=:~~+l)p~$+l,p.
B(m+l)p=
+&m+
kp,L3+d2
(24)
+L
+ Borit IJp)- 2.k&,E,+
,,lklg
&m+ ~jp) Ir/aQlp
+ Arap;
(25)
in which
d (m + Ilp = 8 (m+I)p[l(m+I)pjl(n+I*/2E +B
(aI+
I*l/kp,[l
c
-
3~@l+l,pGll+UPI;
e(,+ I)p= &“)/&r,p)
+ @(m+ ,)p/Wl;
?(m+I)p= kp)/Pkp)
+
la' &+ I)&7 = tcm+I)P mh
o%+,,/w1; 4(m+l)p
=
&4&,+
Throughout the previous calculations, the matrix system (15)-(17) takes the following reduced form:
up;
+ s ~1.+,)P=~~m+,,l~E+~,,+,,p12E+~~~~+~~12. (214
Furthermore, qn (20), based on relations (21) and (2la), is transformed to the equivalent ae+I*=t(P+,*f~+e(m+I*~Ylp+~(l.+IIP(
(22)
where
x = t&diagonal matrix of dimensions (m x m) Pv 1
resulting from A,, after the deduction of the last line and the substitution of the two* last eleqents of the q-line Ji@l- klp];2Vkn- I)$)-t Jimp,I+ J;mpIi by
x p,l
B(m+l)p- $JOn+ffp’ =
=specQw?
tri-diagmiI m trix of dimensions (m x m) resulting from f ,, 2afterthe deductionof the last line and the substitution of the tw Iast eleplents of* the *m-line by - J ;m- I)p)i J&u- I)p)- J&p)- J@&n + I)pI 3+ ) ;&I,t (Ill+I)&?;
BpS2= t&diagonal
matrix of dimensions (m x m) resulting from 8, 2after the deduction of the last line and the substitution *of the tpo last el$ments of the -tine by Jim _ I)pb;
Jh - UPI - J~w,B,sn+VP+ 9 &,&a. tG
L
acting at the nodes of the basic structure of the frame are (Fig. lb and c): (i) the bending moments m& + Z;,, and &,, which are applied at the q-nodes of the continuous beam i = u and j = p respectively; (ii) the loads X&; P$; X&, which are applied on the q-nodes of the continuous beam i = d (Fig, lb); from these forces P&,denotes the external load, while X&; X& are internal forces due to the cane&m of the i=a and j=p beams; and (iii) the loads .&; b&; .& acting at t#e up-nodes of the continuous beamsj = p (Fig. 1~);P& denotes the external load, while &,, g;,, are internal forces due to the connection of the i = u and j = p beams. We must underline here that, based on relations (I), the tri-diagonal matrices 4, ,, &,,, &,l, S,., of eqns (9) and (lo), as wellXas the corresponding matrices 1 ?,I? q I 2r &+,, B,* of eqns (27)-(29) become:
= t&diagonal matrix of dimensions (m x m) resulting from A,,1 after the deduction of the last line and the substitution of
= (m x. I)-vectors being constructed as the corresponding vectors of eqns (18) after the deduction of the last elements respectively;
Based on the previousiy developed equations, as well as on Fig. l(a-c), the matrix eqns (9), (10) and (27), (28) are transformed as follows. Continuous beam i = a:
+ R,f2E i- AT,/2
Bt=(m x I)-vector
resulting from R, after the deduction of the last element and thg subs$tution of *the m-element by MS MZ -t-2Et-J&4J,,, I)@ + 3kJ@n+ I)$
(32)
f,=(m x I)-vector
resulting from f# after the deduction of the last element;
rc, = (m x I)-vector resulting from Ifp after the deduction of the last element and the substitution of the m-element by iy, + k@ + d;,,, I]#&&
i)p- k$)
+ ~~~~(~~_ ,)p- tg,
Furthermore, the equilibrium equations of the beams i = CTand j = p give (Fig. lb and c):
-t6Ef-~~~b~+1~+~~~ljlrn+1fpl; and b = (m x m)-diagonal matrix resulting from $# after the deduction of the last line. (30) 4. MATRIX ANALYSIS OF THE PLEXUS FRAME
The plexus frame is composed by a set of i = m and j = n-continuous beams on elastic supports, So, the generalized quilibrium matrix equations of the structure can be obtained through a convenient coupling of eqns (9)~(11) and (27~(29)~ contenting the previous beams, by using indiipensable equilibrium _ _ _ and _ compatibility conditions. By now, the point loads
i
a-l in which
~~p-Ij;+kt,+,]p=o,
(37)
863
Static analysis of frames on elastic foundation
Also, because of the incompressibility assumption, one may obtain:
E=(l,..., 1) = (1 x m)-unit vector; JJx=(F&f, ,.., $;) = (1 x n)-vector of the common vertical displacements;
w:, = w: (for the i = o-beam), l. w,, = )$; (for the j = p-beam).
(41)
m eqns (32), (33) and (36) constiForu=1,2,..., tute m-matrix equations, concerning the equilibrium ofthei=1,2,..., m-beams, which can be written in a new general matrix form as follows:
E = [3:] = (n x n)-diagonal matrix of the vertical spring-constants;
= (m x m)-matrices; f = (?,J R = [&J; * = [?J = (m x n)-matrices. (49)
CY’A, = 3CWyAz + Z’/ZE
+ (m’ + R)/2E + AT/2
(42)
Xy= 6EC’PBr + 12ECWYA,+ I-
(43)
XXI = N’,
(44)
All these matrices are constructed based on the
corresponding vectors and matrices given by the relations (30). By now, the equilibrium equations for the g-nodes of the plexus frame are expressed by the relations:
in which X’=
_2Y;
xv
= &;
Z’=
_&,
(50)
C = [c,] = (m x m)-diagonal matrix; VZ= [I;];
WY= [Iv;,];
xy = [Xi,];
X” = [x:,1
while the corresponding compatibility conditions by the equations
Z’ = [iq];
= matrices of dimensions (m x n);
N’=(Nf,...,
WkSk-?,
N:)r = (m x I)-vector,
(51)
+y = W”JT-r
1
(52)
J = (1,. . . , l)T = (n x I)-unit vector, (a = 1,...,
m;j=l,...,
Y’ = +,
n);
A,; AZ;A,; B1
(53)
in which:
= matrices of dimensions (n x n),
m’ = PCJ R = [R,]; T = [T,,,]; r = [I-~,] = matrices of dimensions (m x n).
(45)
All previous matrices are constructed based on the corresponding vectors and matrices given by the relations (9a) and (12). On the other hand, for p = 1,. . . , n eqns (34), (35) together with (37) and (40) constitute n-matrix equations, concerning the equilibrium of the n-beams, which can be written in an analj-l,..., ogous general matrix form as follows:
w
= (m x n)-matrix, ky = 6E&#+
+ lZEX,+~
+F
(47)
w==(w;,...,
Wk)’ = (m x I>vector,
+=(lfi;,...,&i)=(I
WI
in which E I &] - (n x @-diagonal matrix; c = [&J; &y = [&$I; b=
[&I; $7 - [&I;
ix- [I;] = matrices of dimensions (m x n); fix =(N,,..., 1.
k:) = (1 x @-vector;
xn)-vector.
The elements of the matrices r and ? represent the uniform temperature differences of the corresponding spans (up) and (up)’ of the structure. The combination of the ftrst of relations (50) with eqns (44). (47) and (52), after some algebra, leads to the following equation:
Br+J+X,w4,,
(55)
which entail that only the matrices A, e, f, and A2 depend on the foregoing spring constants.
in which x, = 2$x, = (m x m)-matrix, k, = -(N’+
?J)/6E
+ 21, r EJ = (m x I)-vector,
!=trace(JTtJ)=E,+***+?“.
(55a)
In a similar manner, the combination of the second of relations (50) with eqns (43), (48) and (51) leads to: k’P’B,
+ &A, = K,,
(56)
in which Al = 2cA, - %/6E = (n x n)-matrix, K,= - (8’ + )Tf)/SE + 2&dA, c = trace(jTCb
= (1 x n)-vector,
= c, + - . - + c, .
(564
Finally, the third of eqns (50) together with eqns (42), (46) and (53) give: A’#” + Y’B = P,
(57)
5. FXACTSOLUTION OF THE MATRIX SYSTEM (55x Wh (sr) An exact matrix-solution of eqn (57), without increase of dimensions of the corresponding matrices, has been given in [8,9]. The advantage of this method is that the inverted matrices are of dimensions equal to (m x m) or (n x n), i.e. equal to the number of nodes of the structure, a fact that requires less memory space and computer coding. In the following, we shall briefly discuss the previous solution and, in the sequel, we shall formulate and prove a convenient corollary and proposition (associated with this solution) through which the problem of interaction between an elastic soil and the plexus frame is solved in a closed form. In the beginning, multiplying both members of eqn (57) on the left with A # 0 and on the right with B # 0 and subtracting the new resulting equations, we obtain: A’Y - YzB2 = P1;
where
symmetric matrix,
B = A,& = (n x n)-u-i-diagonal P = a + f&b
symmetric matrix,
A3Yz + Y’B’ = P,;
+ (m’ + R + &!-‘/2E
(60)
Following the same procedure as previously we lead to the reductional formula: + IC-‘(‘I + %$-r/2 AmY*+(-l)m+‘YzBm=P,,
= (m x n)-matrix,
e-3C-IX
P, = A2P - APB + PB’.
+ e WxJT = (m x n)-matrix,
a = - 3(TA,&’ + C-iXrr )
b = 3A&r
(59)
In the sequel, multiplying both members of eqn (59) on the left by A2 and on the right by Bz, adding the resulting equations, and, also using eqn (57), one may have:
A=C-Ii, = (m x m)-u-i-diagonal
P2 = AP - PB.
= (n x n)-tri-diagonal 2 = (m x m)-tri-diagonal
P ,=Am-1P-AP,_2B+(-l)m+rPBm-‘,
matrix, matrix.
(57a)
Equations (55), (56) and (57) constitute the final matrix system describing the equilibrium of the multistorey and muIticolumn rectangular plexus frame in the most general case of response and loading. One may observe that the elements of the matrices A, a, e, I,, A1 and k, depend on the elastic spring constants I (m+l)p,j~+,)rand~+,,,andconsequently on the elastic behavior of the soil. Furthermore, we must underline that, in civil engineering practice, the transverse loading as well as the temperature difference between the limits of the cross-sections and the uniform temperature differences of the spans of the structure are omitted. In this case, one may have the following simplifications:
(61)
where m is an integer number 2,3, . . . . We must underline here that in the foregoing set of eqns (59) to (61) one may also consider the matrix equation AT-VB”=Po,
(62)
in which A’; B” are the (m x m) and (n x n) unit matrices respectively, while PO is the (m x n)-null matrix. Consider now the characteristic polynomial of the quadrangular symmetric matrix A, i.e.: A(y) = p,y’
(i = sum. index 0, 1, . . . , m),
(63)
in which p, are constant coeikients. Multiplying both members of eqns (62), (57), (59), (60), . . . , (61) with adding by parts PO, PI, P29 P39 * * *. pm, respectively, the resulting Cm+ lkauations and usinn the wtll-
865
Static analysis of frames on elastic foundation
Gayley-Hamilton theorem of matrix algebra, we obtain the following matrix equation: known
Based on the previous corollary, as well as on relations (57a), the solution (67) for ‘Y’can be written as:
V[( - l)‘+ ‘ptBC] = p/P,
(64)
a direct (m x m)-inversion of the last we calculate the V-matrix by the closed
So, the matrix-equation (55), based on the last relation and after some algebra, leads to: W” = S-‘M,,
(65)
‘y’= (P,P,)@,
S-i = (lq8, Lj,e + X,)-l = (m x m)-matrix, M, = [k - k #(a)LiI - #,
in which
d~xWJ
= (m x I)-vector,
8 = [(- l)c+‘p,B1]-’ = quadrangular (m x n).
(69)
where
symmetric matrix of dimensions L, = &J
Corollary 5.1.
= (m x n)(n x n)(n x 1) = (n x I)-vector,
1P= coefficient = ITBq-‘L I>
The expansion
q=sum.index j = sum. index 1,2,. . . ,m
h(P) = (p,P,), can be transformed
‘,
(66)
-
(69a)
~T~A~~~f~~)ul~JJ,)= MI,
(70)
in which
q cm;
i g = e-S-‘B2 = (m x m)-matrix,
q = sum. index 1,2;. . . , m; j=sum.indexq,q+l,...,
m.
l!‘“{(lqbBq-i8B2 + A2)
where U,=(-l)p+‘pjAI-Q;
1,2 ,...,
By now, introducing the expressions for the Y-matrix and W”-vector given by eqns (68) and (69) into eqn (56), we tind:
in the form
b(P) = U,PP-
m.
(67)
y = L,IT = (n x n)-matrix, M2 = K2 - JTC[ &eS- ‘A, JT) + &r)]SB,
Proof From the second of the reductional we obtain the relation
formulae (61)
based on the last equation,
one may
,+(P)=p,P,=(-l)‘+‘p,A’-SPBq-’ 1,2 ,...,m;
= (m x I)-vector,
1, = coefficient = ECU,l
(j = 1,2,. . . , m; q = sum. index 1,2,. . . , j). Consequently, have:
= (1 x n)-vector, A, = k, - &a)L,
+(_])‘+‘A’-‘PI-1
(j=
+ fi(eW’J’?. (68)
j = sum. index 1,2,. . . , m. Through equation formula
#C(B)E f(a) + &#“b)
Y = ,@)a;
t = sum. index 0, 1,2,. . . , m
4 -sum.
q=sum.
index 1,2 ,...,
m.
It is necessary now to introduce proposition.
(70a) the following
Proposition 5.1.
index 1,2 ,...,
j).
On the other hand, for a different running of the indices q and j, the last expression for b(P) remains unchanged, but the q, j-indices take the values
The expansion of the expression
has the form
q = sum. index 1,2,...,m; j = sum. indexq,q+l,..., Consequently, obvious.
m.
the validity of eqns (66) and (67) is
where & are constant matrices of dimensions
coetkients (n x n).
and r/_i.,_i
are
D. E. PANAYOTQIJNAKCS et al.
866 Proof
Putting Q = JbX b = (m x n)-matrix
(71)
and using corollary 5.1, we obtain:
= UJ(U,QB’+
U,QB’ +. . .
. ..+U.QBm-‘hBO+... e. . +
+ U,QB’ +. . .
U,fi(U,QB”
. ..+U.QB”-‘)yBm-‘.
(72)
Through the substitutions U,/KJ, = A, = (m x m)-matrices,
foundation, according to the previously developed theory one may list the following steps in order to facilitate the user of digital computers: Srep 1. By the relations (70a), (45), (49), (55a), (56a), (69) and the corollary 5.1 we calculate the (1 x n)-vector M2, while through the same procedure, as well as proposition 5.1 a d formula (76), we calculate the (m x m)-matrix d . Furthermore, using eqn (75) we evaluate the (1 x n)-vector &Y Srep 2. Through Step 1 and the use of the second of eqns (69) we determine the (m x m)-matrix S-‘, while, at the same time, based on relations (69a) and the third equation of (69), we also obtain the (m x I)-vector M,. In the sequel, applying the first of eqns (69) we calculate the (m x I)-vector WX of the horizontal displacements. Step 3. From the results of Steps 1 and 2, also using the tinal formula (77), we determine the (m x n)-matrix ‘Y of the rotations of the nodes of the frame.
(i; i = 1,2, . . . , m), 6. NUMERICAL RESULTS AND DJSCUSSION
BkyB’ = E, = (m x m)-matrices, (k; I=O,l,...,
m-l),
the expansion (72) can be written as tc[Btc(Q)vI = A,/Qa,- I.,-, (i; j = sum. indices 1,2, . . . , m).
(73)
Multiplying the last equation on the left with the (1 x m)-vector J TC and taking into account the validity of relation jTCAoJ = ECU,&
= ~~ = consts,
one may have:
As an application of the proposed methodology in this paper we shall try to determine the nodal slopes and displacements of the four-storey and fourcolumn plexus frame shown in Fig. 4. The structure is subjected to uniformly distributed loadings, which vary from one span to the next, with the dimensionless ratio q2/q, = 2. Four concentrated unqual horizontal forces are also acting in the direction of the four respective horizontal beams of the frame. The flexural rigidities EJ vary from one span to the next, but they are assumed constant within each span. Also, the dimensionless ratio h/a is considered to be equal to 1. Based on relations (l), (9a), (12), (30) and (45) we formulate the following matrices of the geometrical characteristics of the frame:
Through the last formula, the vectorial eqn (70) takes the following simple form: &x= h4&,
(75)
-2
where AZ=6-i = [(l,bBq-’ - /+,bB’-‘gB’-‘)0B2 q=sum.index i,i=sum.indices
1,2 ,...,
+ AZ]-’
m
l,Z,...,m.
(76)
Finally, based on eqns (75) and (76) the exact matrix solution of eqn (68) becomes: ‘I’= #(a + &&b
+ eS-‘A4,~)6X
(77)
In conclusion, for the evaluation of the structure redundants of a plexus frame supported on an elastic
J a*
-2 2
0 l-l
0 0
Static analysis
j=l
j=2
ta
_t___
of frames on elastic foundation
867
j=4
j=3
i-lt h
i=2+ h
i-3+ h
i= 4+ h
20
+a
1
Fig. 4. Numerical example.
C = E = (4
x 4)-unit matrices.
(78)
As a first application we consider that the supports of the frame are rigid on the same level. Consequently, the &-vector becomes equal to zero, while the fundamental matrix equations (55) and (57) take, respectively, the following forms:
2/l
Ocl=Ix
2 1 0 0 1410 0141’ 0 0 1 2
1I r -1
1
r -1 where
7
0
07
-7
-11
In the beginning, according to the previously developed theory, we formulated the characteristic polynomial of the matrix El with the following five roots: po =
1329; p1= - 1248; p2 = 330; P3 =
-32;
p,= 1.
D. E. PANAYOT~UNAK~S er al.
868
In the sequel, based on eqn (65), we evaluated the matrix 8, i.e. - 18,009 -5016 8=
-30,912
-448 20,432
I
-5016 63,889
jy,=_
-5016
-448
-63,889
-20,432
--30,912 20,432
- -5016 18,0091 (82)
- 20,432 -
eqn (52) we calculate the (4 x 4) matrix % Yas follows:
“‘91 24EJ
’ X
i 0.332199 0
0.216128 0
0.147386 0
0.156280 0
Also, from relations (69) we find L, =(-0.000063,
(87)
O.OOOOO6, O.OOOOO6, -0.000063)T;
1., = -0.00012;
1, = -0.00034;
1, = -0.003264,
(83)
1, = -0.02624.
Through eqns (66), (67) and the second part of (69) one may obtain: i
q-1
’
As a second application we consider that the four-storey and four-column plexus frame shown in Fig. 4 is supported on four elastic supports with equal translational spring constants S. The dimensionless ratio a’J”/6EJ is taken equal to 0.75. In this case, the fundamental matrix equations (55), (56) and (57) are reduced to the following form:
m
~2YzJ +
2,W" =k,,
(88)
+
ih2 =K2,
(8%
JT'4'~,
--!
8,Y’ + ‘Pa, = a + a2W”JT + hba,,
PO)
-0.593066 -0.396498 -0.008605 J -0.005736 -0.036729 O.OOQ499-0.086673 0.288282 -0.067339 -0.036729 0.142863 -0.130010 -0.131741 -0.396498 o.fmO749 -0.101009
in whicithe
(4 x 4)-matrices 8,, f,, a,, a, a2 and the vector , are the same as in eqns (81), while the remaining matrices and vectors are given by:
3920 -432 3212 ’ -3:;; 32,052 3920 - -32,052 57.260 3096 3096 57,260 20,180 9232 I -9232 -20,180
r-2
2
0
07
(84) On the other hand, based on eqns (69) and selecting the dimensionless ratio aq,/P qua1 to 4, we have:
M , = &;
a i a,=4
(1 .oOOOOo,2.000000,
40
-5 1-5
1
04 i
0
0
2
2
- 3.OoOOOO, 4.OooOOO); II’“= -&(-0.332199, -0.147386,
-0.216128,
-0. 156280)T.
(85)
Finally, based on the formula (77). as well as on relations (85), the second part of qns (84) and on eqn (82), we evaluate the matrix V’ of the slopes of the nodes, i.e. p*
x
-- a’qt 8U
0.026378 -0.012976 0.620499 -0.614819 0.434861 -0:320695 0.351313 -0.508372 1.176598 - 1.532732 1.606571 -1.524232 1.912498 -2.641767 2.783818 -2.471136
’
(86)
while,
based
on
the second
of qns
(85) and
Now, from the 6fth of qns (70a) we evaluate the coefficients 1, (q = 1,2,3,4). while through the combination of the fourth and the third of qns (70a) with relations (88), (81), and the second of (84) and the corollary 5.1, we calculate the M,-vector. On the
869
Static analysis of frames on elastic foundation
other hand, from the first of eqns (70a) one may obtain the (4 x 4)-matrices /I and y, while, according to the proposition 5.1, one may also have the (4 x 4>matrices A,, Q,. So, the coefficients pY, included in the previous proposition, can be easily determined. By now, based on the above, as well as on eqn (76), we iind the matrix g-l, while, through the third of eqns (69) the vector M,. Furthermore, using eqn (75) and the first two parts of relations (69), we evaluate the #‘X and W-vectors,
i.e.
Comparing the solution results of relations (86) and (87), corresponding to a rigid soil, with the results of (93) derived from the case of an elastic soil, one may conclude that the effect of an elastic foundation on the upper structure is appreciable.
REFERENCES 1. G. Kani, Analysb of Mulriclorey Frames. Fred. Uog. Publ. Co., Ne\; Yo;k, N.Y. (19j6). 2. M. F. Rubinsteio. Multistorv. frame analvsis . bv dkital computers, Presented at the 2nd Conf. on Electronic Computation, AXE, Pittsburgh, PA (1960). 3. J. E. Soehrens, The electronic computer as a tool oo electronic computation. ASCE, Pittsburgh, PA (1960). 4. P. Lustgardeo~ Iterative method in f&e a&y&, Joumal of the Structural Division. ASCE. Vol. 89. ’ No !3T2,April, 1963, Z-94. . 5. S. S. Texcan, Discussion of “Simplified Formulation of Stiffness Matrices” by M. Wright, Journal ofthe SWUCrural Division, ASCE, Vol. 89, No ST6, December, 1963, pp. 445-450. 6. R. W. Clough, E. L. Wilson and I. P. King, Large capacity multistory frame analysis programs, Jou&l of the Struclural Division, ASCE, Vol. 89, No ST4, August, 1963, pp. 179-204. 7. R. W. Clough, I. P. King and E. L. Wilson, Structural analysis of multistory buildings, Journal of the Structural Division, ASCE, Vol. 90, No ST3, June, 1964, pp. 1e34. 8. E. D. Paoayotounakos, Static and dynamic analysis of plexus-frames based on the matrix-algebra, Technica Chronica, Vols l-2, Athens (I 966). 9. E. D. Paoayotounakos and N. Galidakis, Trfgerroste, Techica Chronica, Vols 425-426, Athens (1959). 10. R. W. Clough. Structural analysis by means of a matrix algebra program. ASCE Con/. on Electronic Computation, Kansas City, p. 109 (November 1958). 11. G. L. Rogers, Dynamics of Framed Structures. John Wiley and-Sons, inc., New-York, N.Y. (1959). 12. J. H. Argyris. Enerav Theoremsand StructuralAnalvsis. Butte&&s Publ&tioos, London, England (l&j.- ’ 13. C. H. Samson and H. W. Bergmann, Analysis of low-aspect-ratio aircraft structures, Journal of AeroSpace Sciences, Vol. 27, No 9, September, 1960, pp. 619-693. 14. S. S. Texcan, Moment equations for computer analysis of frames, Journal of the Structural Division, ASCE, Vol. 90, No ST3, June, 1964, pp. 35-53. 15. J. S. Przcmieoiecki, Theory of Matrix Structural Analyst&McGraw-Hill, N.Y. (1968). 16. Jennings, Matrix Computationfor Engineers and Scienfisrs, John Wiley and Sons, N.Y. (1978). II
8x = -&;
( - 24.294810, - 23.692039,
-23.786440, W”-- -$&
- 24.224697)
(-0.366388,
-0.192900,
-0.269922,
-0.173642).
(92)
Finally, using formulae (77), (51) and (52) we obtain the following solutions for the W, 4and W-matrices: y_
-0QI 24EJ
-0.308968 0.853872 -0.855832 0.272967 0.461652 -0.339069 1.740275 1.230110 - 1.927782 3.001748 1.945198 -3.382358
x
-0.227488 -0.246797 -0.307535 -0.453721
&&!L
24EJ
x
0.173642 0 i 0.366388 0 0.269922 0 0.192900 0
WY,0441 24E/
x
0 0 24.294810 0 0 0 0 23.692039 0 ’ 0 0 23.786440 0 24.224697 0 0 (93)
Abetract-In this investigation an exact matrix solution for the static analysis of a multi-stony and multi-column rectangular plexus frame on an elastic foundation is presented in the most general case of msponse and loading. Furthcrmorc, based on the formulation and proof of a convenientcorollary and
proposition, associated with the existing closed-formsolution of the matrix equation of the type AX+ XB= C, the problemof interactionbetween an elastic soil and the frame is completely solved. The entire procedure is illustrated by a comprchcnsive example concerning a four-rtorey and four4umn plexus frame supported on an elastic foundation. Several numcrkal results are also given. As the obtained closed-fotm fon@ac, giving the structure radtmdants. include matrices of dimensions equal to the number (m x n) of the nodesof the frame, the proposedsolutionmetl~oclology is more convenientin comparison
with those in the existingliterature,becauseit rcquins less memoryspace and coding.
1. ~ODU~ON
the Gayiey-Ha~lton theorem of the matrix algebra, the slopes, deflections and internal forces and couples The structural (static and dynamic) analysis of multi- of the nodes of the structure were calculated by storey and multi-column rectangular building frames, closed-form formulae. through digital computer methods, has attracted the In this investigation, as an extension of the interest of many researchers the last thirty years. methodology described in [8], the exact matrix solKani[l], Rubinstein [2], Soehrcns [3], Lustgarden [4], ution for the static analysis of a multi-storcy and Tcxcan [5l and others investigated such types of struc- multi-column plexus frame is presented in the most tures by using several iteration and relaxation tech- general case of response and loading (including tcmniques based on the flexibility, the stiffness or the pcraturc differences and different constant comgross joint and floor moments approaches, A more pressive forces acting at the free ends of each storey thorough ~v~ti~tion on the previous ~velopmcn~ and column, respectively). Factor, based on the was presented by Clough et al. [6,4, who analysed fo~ulation and proof of a ~nve~ent corollary and Iargc building frames through two different solution proposition associated with the exact solution of the techniques. In particular, in[6] two methods were previous matrix quation [S,91,the problem of interdescribed and compared; the fkst used a special form action between an elastic soil and the frame is of Gauss-Scidel iteration of the quilibrium quacompletely solved. The continuous horizontal and tions, while the second arranged the stiffness matrix vertical beams on elastic supports are selected as in the form of a tridiagonal system of submatrices, basic structures, while di!Tercnt rotational and transand then obtained the solution by means of a recur- lational springs are considered at the supports of each sion equation. column with the soil. The flcxural rigidities may vary Although all previous works arc suitable for corn- from one span to the next, but they are assumed to puter use, the consideration of a simply supported be constant within each span. Also, without loss of straight member, between two consecutive nodes of generality, the ratio of the flexuraI rigidity for each the frame, as the basic structure rcquircs matrices of in~~~atc span of a basic structure i = Q(orj s: p) dimensions much larger than the number (m x n) of over the same quantity for the corresponding span of the nodes of the structure. This disadvantage was the basic strncturc i=l (orj=l) is considered as confrontcd in[8], in which the author selected as constant. Furthcrmorc, in the analysis, the effcots of the basic structure the continuous straight beam on shearing and axial forces arc omitted. FinalIy; the elastic supports. The developed solution method- entire pmcedurc is illustrated by a comprehensive ology led to a matrix equation of the form example concerning a four-storcy and four-column AX+XB-C, in which the (mxn)-matrix X in- plexus frame and scvcral numerical results arc also cludes the structure rcdundants (slopes of the m x II- giVCll. . nodes), while the mmaining known matrices arc of Although the problem being examined is rigorous suitable dimensions. Through an exact solution of the and complicated, the Ilnal closed-form formulae givprevious equation, which was given in [S,9) based on ing the structure rcdundants can be easily and directly 855
856
D. E. PANAYOTOUNAK~
applied to the static and dynamic interacting analysis of an arbitrary plexus frame supported on an elastic foundation. Also, as the inserted matrices are of dimension qua1 to the number (m x n) of the nodes of the structure, the proposed solution methodology is more convenient in comparison with previous formulations, because it requires less memory and computer coding.
ef &.
horizontal i = u-beam lying on n - 2 intermediate simple elastic supports with translational spring constants Vy,,. , . ) u&,_ if in the y-direction; also, the terminal supports have translational spring constants u:,; u&l;u:,; I& in the y- and x-directions, as well as rotational spring constants e,,; e,, respectively (Fig. 2a). For the N: total compressive force acting at the free end of the beam, we have:
2. PRRLIMINARIES-NOTATIONS
Consider a rectangular multi-storey and multicolumn planar rigid-jointed plexus frame, being composedofi(i=l,2 ,..., e ,..., m)horixontaland j(j=1,2 ,..., p, . . . , n) vertical incompressible continuous straight beams, supported on an elastic foundation with translational and rotational spring constants different for each column. The structure is subjected to a generic co-planar loading consisting of dis~but~ and concentrated forces and couples, as well as to constant temperature differences between the limits of the cross-sections of each member. We suppose also tJrat different constant compressive forces P?: and I’$ act at the free ends of the storeys and the columns, respectively. A sketch of such a type of structure is shown in Fig. la. The flexural rigidities of the horizontal and vertical members may vary from one span to the next, but within each span these quantities remain constant. The frame can now be analysed to a set of i-horizontal and j-vertical continuous beams on elastic supports. We symbolize with (ffp) the span of the i = a-beam between the up and the a(~ + I)-nodes, and with (a~*) the cormsponding span of the j = u-beam between the up and (a + l)p-nodes (asterisks denote quantities corresponding to the vertical members). Fu~he~ore, we select as structure redundants the 2; and ?!; bending moments of the ij-nodes of the frame. Figure l(b and c) shows these redundants acting on the corresponding nodes of the basic structures. We note here that, in the following, quantities referring to the right, left, upper and sub-cross-section of the node will carry the upper right indices t, f, u and s respectively. Finally, without loss of generality, we suppose that the ratio of the flexural rigidity for an arbitrary span of the basic structure i = u (or j = p) over the same quantity for the corresponding span of the basic structure i = 1 (or j = 1) remains constant, i.e.:
in which n&, denotes the distributed axial force acting at the (ap)-span. Figure 3a shows the quilib&m of an intermediate node up under the action of the external couple m&, the external load Pip, the internal couples M$, and h4&‘,the shear forces Q&’ and Q’.;p’and the total vertical reaction X:,. In this figure w& denotes the displacement of the n-node in the y-direction. In Fig. 3@ and c) the quilibrium of the Q1 and on-nodes are presented, in which +:, and $:, are the slopes, and w:, and w& the displacements in the x-direction. Based on the previous figures and taking into account the in~rnp~~~~ty assumption of the beam, the following sets of equilibrium equations for the ap -, u 1- and (In-nodes result:
(3)
M:, = hf:[ -m:, = -e,,JI:r; XY #, = Q;,‘= I&W:,; XX 81--N$j = u:,w;,t
(4)
node an: ML= M&‘fm&==e,JrL; Yl m)rr- QA XY XL
~5
-Ngb’=
=
vY,wY,,
*
u&w&,,
(9
and wzr = (is=1 ,...,
m; j-l,...,n).
3. MATRIX ANALYSIS OF A STRAIGH’F’ CONTINUOUS REAM ON RLASTIC SUPPORTS
As the basic structure of the frame is the straight continuous beam, in our further discussion, we shall examine in principle the matrix analysis of such a
Also,
based
Wk.
(6)
on qn (2), one may have (7)
Introducing the last of qns (4) and (5) into qn (7), and taking into account relation (7) together with (6).
Static analysis of frames on elastic foundation
i=a
i=o
(mAI
(mtl)Z
(m+Wn
ImMiP
a
Fig. 1. (a) Typical building frame. (b) Geometry and sign convention of a horizontal basic structure. (c) Geometry and sign convention of a vertical basic structure.
we obtain:
we obtain n-linear quations which can be written in a matrix form as follows:
(u:, + u&)w:, = N:;
w;, = w”,
=
CA,
N:l(u:, + G,);
xx81 = JGA41 + G,>; Xx, = u”,N~l(u~,+ I.&).
I =
3WyA,2 + m:/2E + R.IZE + 1T,/2,
(9)
in which
(8)
Applying the principle of virtual work for any two consecutive spans of the beam and combining the results with the first of qns (3~(9, for all the nodes,
A 8.I
=
P;o(PI)), Vk-
1))-
J;opJr J;.,,l,
= t&diagonal symmetric matrix of dimensions
(n x n) with first element Zlb,, + (e,, /2E) and last element 2J;,,_ ,))+ (eJ2E);
D.E.
PANAYOTQLJNAKOS
et (11.
a Y
Fig. 2. (a) and (b) Geometry and sign convention of straight continuous beams on elastic supports.
Static analysisof frames on elastic foundation
859
C
b
Fig. 3. (a)-(c) Geometry and sign convention of an intermediate and two end-nodes of the frame.
A ..2'[-J~p-I))rJ~p-I))-J;bp),J;sp)lt 5 tri-diagonal matrix of dimensions (n x n)
with first element -J$,, and the last element J&n- 1,);
r: =ok,
** *, $k) = (1 x n)-vector of the slopes of the uj-nodes; (9a)
w’,) i= (1 x n)-vector of the vertical displacements of the nodes; m:=@tr:,,..., mk) = (1 x n)-vector of the external couples acting at the crj-nodes;
W{==(W$,,...,
R:P (-M:i’,
foregoing methodology and combining the wellknown expressions Y.1QV
T, - (&Art,,) t -J~,l,Ar(,,,+~.~~Ar.,,, . . = (1 x @-vector of temperature differences. We note here that, in the previous analysis, without loss of generality, the efTectsof shear and axial forces are neglected. On the other hand, based on the
a{H,_,jj(M$-
M%;_,,)
Q&’= I’&,’+ a~&.f$,,+ ,) - M&‘) with the second parts of eqns (g)-(5), we construct a set of 2n additional equations concerning the reactions of the elastic supports, which can be written in the following vectorial form:
h42; - Mii, . . . , Mg)
- (1 x n)-vector of terminal couples due to the external intermediate loading of the fullyfkedendspansofthebeam;
- V$+
Xi= 6E?:B,,r + 12EW&3 + r,;
(10)
x:=
(11)
WiYJ *
where Xi = (1 x n)-vector of the reactions;
= tridiagonal matrix of dimensions (n x n);
*,, , _ [ _ Jm (*I)*- Gt,-
1))+ J&Jl J&l,1
- tridiagonal symmetric matrix of dimensions
D. E. PANAYOTQUNAK~S el aI.
860
(n x n) with first element -J;:,, element -J& _ ,,,;
To = [P’,, + V:, + a;oI,(M2
-
and the last
- M;;), . . .
+ C, + a;,(, - ,,dW:, - 1l- Mg)]
= (1 x n)-vector;
&, rk,-
and
(17)
I)#?)9
L=r-j;;4,P,~
matrix.
In the previous relations (9a) and (12), 1 is the common coefficient of the thermal expansion; 4,) = ‘(‘.p,- G,, denotes the constant temperature difference between the limits of the cross-section of the (up)-span; E represents the modulus of elasticity, and a{,,,, is a coefficient equal to l/a,.,,,, where q,@, denotes the length of the (up)-span. Also, V$, v;; are the reactions of the simply supported member (UP) CC, = V$ + V&,‘),while J&,, .&, and J&,, are coefficients given by the relations
J&P)= Jtsph,,j ; J;b,, = J&a &,; Jr;,, = J~.dah;
(13) hp, = J~op~140p~ 9
where J(,+,,denotes the moment of inertia and &,, the height of the cross-section of the (up)-span. Finally, the superscript T represents the transpose of a matrix or vector. By now, solving eqn (9) for Pfvector and introducing the resulting new expression together with eqn (11) into relation (lo), we find the following equation for the II’:-vector:
(14)
The remaining vectors XY,, Y: can be evaluated through eqns (11) and (9) respectively. In the case when the supports of the beam are rigid, not on the same level, the FQvector is known, and, con~equently, the F;- and X$vecton result immediately through eqns (9) and (10). We consider now the second basic structure of the frame, namely the vertical continuous beam j = p, the (m + I)-support of which has translational and rotational spring constants Jib, ,,,,, &,+ ,I,, and re.spectively. The other supports of the beam a A\zple elastic with spring constants 8Y,(Fig. 2b). According to the previously developed methodology, the corresponding equations of eqns (9)-(11) for this case arc expressed as:
4,#-
lb)
+
k4Lb
{k-l,,-kP,L
k,,l
= tri-diagonal symmetric matrix of dimensipns (m + 1) x (m + l)+with first element -J;‘,,, and last element J;b,,; L
= r-k&-
I))9r;;;.-,,,, - k&L
$&)I
= tri-diagonal symmetric matrix of dimensizns (m + 1) x (m + 1) v$th first element and last element -J&,,; 4.2 = Q,2 = u-i-diagonal matrix (m+ 1) x (m + 1);
-
J$
of
dimensions
+;=(J&,...,J&, J&+,,P)T=(m+l)x of the slopes of the Q-nodes;
l-vector
IPi=(Z{,,...,$Y,,, ~~~+,,p)T=(m+l)xl-vector of the horizontal displacements of the ip -nodes; &,= (d;,,...,J&, &fm+,,JT=(m+l)x l-vector of the external couples acting at the ip -nodes; R, = (4&b’,
IV: = [3(m: + R, + IET,,)A; t B,, 2+ r,l
x [y’, - 12EA,,3- A$B,,,]-‘.
(16)
= u-i-diagonal symmetric matrix of dimensigns (m + 1) x (m + 1) pith first element U;,,, and last element W;,,, + [zCm+,,,/2E];
(12)
YY,= [u:] = (n x n)-diagonal
fi; + f,;
in which I =
...K
+ 12E&,
i;=+;rt;,
M:,‘),
Pi2 + V:, + a’;c,,(M:/ - M:i) + a&,(M$
?; = 6E4,,+;
$2; - n;riJ, . . . ( i+;nY+,J
= (m + 1) x l-vector of the terminal couples due to the external intermediate loading of the fully fixed-end spans of the continuous beam;
= (m + 1) x l-vector of temperature differences;
t,,+;=3X,,$;+&2E + 4/2E
+ &/2;
(15)
= (m + 1) x (m + l)-diagonal
matrix.
(18)
861
Static analysis of frames on elastic foundation
+
t
The
coefficients a’& J;,,,; Jh,,; JbpJ; *“’ ftapj and I,,, are determined as previously, namely as in the case of the horizontal continuous beam. We notice here that, for the (m + l)p-support of the foregoing basic structure, the following relation is valid:
in which * l CB(m+I)P+1); %l+I)p= *(m+I)P ? (m+ 1)~ * tt(m
=~(m+I)p(8(mp)Y+(m+r)p+2);
+ 1)~ =G(m+Il~
1; (mp) 8 (mtlb
As the previous plexus frame is constructed by j = 1, . . . , m-horizontal and j = 1,. . . , n-vertical continuous beams, we shall try to reduce the dimensions of qns (1 5)-( 17) in order to formulate matrices of dimensions qua1 to the number of the horizontal basic structures. For this p rpose, through the (m + l)p-component of the P ,-vector, as well as through qn (17), the (m + 1)ptomponent of the matrix equation (16) takes the form:
On the other hand, the combination of the last relation with the (m + l)p-component of eqn (15), after some algebra, leads to the following relation:
(224 Consequently, we succeed in giving the $i’,+ ,,,,displacement as a linear combination of the Ja-slope and the r?&displacement corresponding to the node mp of the frame. Introducing qn (22) together with eqn (21) into the (m + l&component of the matrix eqn (IS), as well as into the mp- and (m + l)pcomponents of the vectorial qn (17) one may obtain:
8 (m-h
kt~p=~~Ik~-l)p) + r&,-
I)P)- J&) - kp)B(m+
+&ip&r+l)P
4&I)p)
13;~m+l~p~~~+I*-~~m+l~pl/
ye@, l)p =
l IlP~(m+
I)P
&mtl?9= W%y,,(l
1;
+ 1Wkp,(l-
[3&l+ i)p(2c&&+ 1)p-
+ f&p)&ll+
111/
g(m+l)p 1;
L1 -3d@+l)p
l)p
I~~,}+I~E{~;;:~-,u~:.-,~P
+ kp,+
--&kp,i,-,,lkp~
in which
11- 3&l+
,,/6~&,)1.
(19)
~~m+,*=:~~+l)p~$+l,p.
B(m+l)p=
+&m+
kp,L3+d2
(24)
+L
+ Borit IJp)- 2.k&,E,+
,,lklg
&m+ ~jp) Ir/aQlp
+ Arap;
(25)
in which
d (m + Ilp = 8 (m+I)p[l(m+I)pjl(n+I*/2E +B
(aI+
I*l/kp,[l
c
-
3~@l+l,pGll+UPI;
e(,+ I)p= &“)/&r,p)
+ @(m+ ,)p/Wl;
?(m+I)p= kp)/Pkp)
+
la' &+ I)&7 = tcm+I)P mh
o%+,,/w1; 4(m+l)p
=
&4&,+
Throughout the previous calculations, the matrix system (15)-(17) takes the following reduced form:
up;
+ s ~1.+,)P=~~m+,,l~E+~,,+,,p12E+~~~~+~~12. (214
Furthermore, qn (20), based on relations (21) and (2la), is transformed to the equivalent ae+I*=t(P+,*f~+e(m+I*~Ylp+~(l.+IIP(
(22)
where
x = t&diagonal matrix of dimensions (m x m) Pv 1
resulting from A,, after the deduction of the last line and the substitution of the two* last eleqents of the q-line Ji@l- klp];2Vkn- I)$)-t Jimp,I+ J;mpIi by
x p,l
B(m+l)p- $JOn+ffp’ =
=specQw?
tri-diagmiI m trix of dimensions (m x m) resulting from f ,, 2afterthe deductionof the last line and the substitution of the tw Iast eleplents of* the *m-line by - J ;m- I)p)i J&u- I)p)- J&p)- J@&n + I)pI 3+ ) ;&I,t (Ill+I)&?;
BpS2= t&diagonal
matrix of dimensions (m x m) resulting from 8, 2after the deduction of the last line and the substitution *of the tpo last el$ments of the -tine by Jim _ I)pb;
Jh - UPI - J~w,B,sn+VP+ 9 &,&a. tG
L
acting at the nodes of the basic structure of the frame are (Fig. lb and c): (i) the bending moments m& + Z;,, and &,, which are applied at the q-nodes of the continuous beam i = u and j = p respectively; (ii) the loads X&; P$; X&, which are applied on the q-nodes of the continuous beam i = d (Fig, lb); from these forces P&,denotes the external load, while X&; X& are internal forces due to the cane&m of the i=a and j=p beams; and (iii) the loads .&; b&; .& acting at t#e up-nodes of the continuous beamsj = p (Fig. 1~);P& denotes the external load, while &,, g;,, are internal forces due to the connection of the i = u and j = p beams. We must underline here that, based on relations (I), the tri-diagonal matrices 4, ,, &,,, &,l, S,., of eqns (9) and (lo), as wellXas the corresponding matrices 1 ?,I? q I 2r &+,, B,* of eqns (27)-(29) become:
= t&diagonal matrix of dimensions (m x m) resulting from A,,1 after the deduction of the last line and the substitution of
= (m x. I)-vectors being constructed as the corresponding vectors of eqns (18) after the deduction of the last elements respectively;
Based on the previousiy developed equations, as well as on Fig. l(a-c), the matrix eqns (9), (10) and (27), (28) are transformed as follows. Continuous beam i = a:
+ R,f2E i- AT,/2
Bt=(m x I)-vector
resulting from R, after the deduction of the last element and thg subs$tution of *the m-element by MS MZ -t-2Et-J&4J,,, I)@ + 3kJ@n+ I)$
(32)
f,=(m x I)-vector
resulting from f# after the deduction of the last element;
rc, = (m x I)-vector resulting from Ifp after the deduction of the last element and the substitution of the m-element by iy, + k@ + d;,,, I]#&&
i)p- k$)
+ ~~~~(~~_ ,)p- tg,
Furthermore, the equilibrium equations of the beams i = CTand j = p give (Fig. lb and c):
-t6Ef-~~~b~+1~+~~~ljlrn+1fpl; and b = (m x m)-diagonal matrix resulting from $# after the deduction of the last line. (30) 4. MATRIX ANALYSIS OF THE PLEXUS FRAME
The plexus frame is composed by a set of i = m and j = n-continuous beams on elastic supports, So, the generalized quilibrium matrix equations of the structure can be obtained through a convenient coupling of eqns (9)~(11) and (27~(29)~ contenting the previous beams, by using indiipensable equilibrium _ _ _ and _ compatibility conditions. By now, the point loads
i
a-l in which
~~p-Ij;+kt,+,]p=o,
(37)
863
Static analysis of frames on elastic foundation
Also, because of the incompressibility assumption, one may obtain:
E=(l,..., 1) = (1 x m)-unit vector; JJx=(F&f, ,.., $;) = (1 x n)-vector of the common vertical displacements;
w:, = w: (for the i = o-beam), l. w,, = )$; (for the j = p-beam).
(41)
m eqns (32), (33) and (36) constiForu=1,2,..., tute m-matrix equations, concerning the equilibrium ofthei=1,2,..., m-beams, which can be written in a new general matrix form as follows:
E = [3:] = (n x n)-diagonal matrix of the vertical spring-constants;
= (m x m)-matrices; f = (?,J R = [&J; * = [?J = (m x n)-matrices. (49)
CY’A, = 3CWyAz + Z’/ZE
+ (m’ + R)/2E + AT/2
(42)
Xy= 6EC’PBr + 12ECWYA,+ I-
(43)
XXI = N’,
(44)
All these matrices are constructed based on the
corresponding vectors and matrices given by the relations (30). By now, the equilibrium equations for the g-nodes of the plexus frame are expressed by the relations:
in which X’=
_2Y;
xv
= &;
Z’=
_&,
(50)
C = [c,] = (m x m)-diagonal matrix; VZ= [I;];
WY= [Iv;,];
xy = [Xi,];
X” = [x:,1
while the corresponding compatibility conditions by the equations
Z’ = [iq];
= matrices of dimensions (m x n);
N’=(Nf,...,
WkSk-?,
N:)r = (m x I)-vector,
(51)
+y = W”JT-r
1
(52)
J = (1,. . . , l)T = (n x I)-unit vector, (a = 1,...,
m;j=l,...,
Y’ = +,
n);
A,; AZ;A,; B1
(53)
in which:
= matrices of dimensions (n x n),
m’ = PCJ R = [R,]; T = [T,,,]; r = [I-~,] = matrices of dimensions (m x n).
(45)
All previous matrices are constructed based on the corresponding vectors and matrices given by the relations (9a) and (12). On the other hand, for p = 1,. . . , n eqns (34), (35) together with (37) and (40) constitute n-matrix equations, concerning the equilibrium of the n-beams, which can be written in an analj-l,..., ogous general matrix form as follows:
w
= (m x n)-matrix, ky = 6E&#+
+ lZEX,+~
+F
(47)
w==(w;,...,
Wk)’ = (m x I>vector,
+=(lfi;,...,&i)=(I
WI
in which E I &] - (n x @-diagonal matrix; c = [&J; &y = [&$I; b=
[&I; $7 - [&I;
ix- [I;] = matrices of dimensions (m x n); fix =(N,,..., 1.
k:) = (1 x @-vector;
xn)-vector.
The elements of the matrices r and ? represent the uniform temperature differences of the corresponding spans (up) and (up)’ of the structure. The combination of the ftrst of relations (50) with eqns (44). (47) and (52), after some algebra, leads to the following equation:
Br+J+X,w4,,
(55)
which entail that only the matrices A, e, f, and A2 depend on the foregoing spring constants.
in which x, = 2$x, = (m x m)-matrix, k, = -(N’+
?J)/6E
+ 21, r EJ = (m x I)-vector,
!=trace(JTtJ)=E,+***+?“.
(55a)
In a similar manner, the combination of the second of relations (50) with eqns (43), (48) and (51) leads to: k’P’B,
+ &A, = K,,
(56)
in which Al = 2cA, - %/6E = (n x n)-matrix, K,= - (8’ + )Tf)/SE + 2&dA, c = trace(jTCb
= (1 x n)-vector,
= c, + - . - + c, .
(564
Finally, the third of eqns (50) together with eqns (42), (46) and (53) give: A’#” + Y’B = P,
(57)
5. FXACTSOLUTION OF THE MATRIX SYSTEM (55x Wh (sr) An exact matrix-solution of eqn (57), without increase of dimensions of the corresponding matrices, has been given in [8,9]. The advantage of this method is that the inverted matrices are of dimensions equal to (m x m) or (n x n), i.e. equal to the number of nodes of the structure, a fact that requires less memory space and computer coding. In the following, we shall briefly discuss the previous solution and, in the sequel, we shall formulate and prove a convenient corollary and proposition (associated with this solution) through which the problem of interaction between an elastic soil and the plexus frame is solved in a closed form. In the beginning, multiplying both members of eqn (57) on the left with A # 0 and on the right with B # 0 and subtracting the new resulting equations, we obtain: A’Y - YzB2 = P1;
where
symmetric matrix,
B = A,& = (n x n)-u-i-diagonal P = a + f&b
symmetric matrix,
A3Yz + Y’B’ = P,;
+ (m’ + R + &!-‘/2E
(60)
Following the same procedure as previously we lead to the reductional formula: + IC-‘(‘I + %$-r/2 AmY*+(-l)m+‘YzBm=P,,
= (m x n)-matrix,
e-3C-IX
P, = A2P - APB + PB’.
+ e WxJT = (m x n)-matrix,
a = - 3(TA,&’ + C-iXrr )
b = 3A&r
(59)
In the sequel, multiplying both members of eqn (59) on the left by A2 and on the right by Bz, adding the resulting equations, and, also using eqn (57), one may have:
A=C-Ii, = (m x m)-u-i-diagonal
P2 = AP - PB.
= (n x n)-tri-diagonal 2 = (m x m)-tri-diagonal
P ,=Am-1P-AP,_2B+(-l)m+rPBm-‘,
matrix, matrix.
(57a)
Equations (55), (56) and (57) constitute the final matrix system describing the equilibrium of the multistorey and muIticolumn rectangular plexus frame in the most general case of response and loading. One may observe that the elements of the matrices A, a, e, I,, A1 and k, depend on the elastic spring constants I (m+l)p,j~+,)rand~+,,,andconsequently on the elastic behavior of the soil. Furthermore, we must underline that, in civil engineering practice, the transverse loading as well as the temperature difference between the limits of the cross-sections and the uniform temperature differences of the spans of the structure are omitted. In this case, one may have the following simplifications:
(61)
where m is an integer number 2,3, . . . . We must underline here that in the foregoing set of eqns (59) to (61) one may also consider the matrix equation AT-VB”=Po,
(62)
in which A’; B” are the (m x m) and (n x n) unit matrices respectively, while PO is the (m x n)-null matrix. Consider now the characteristic polynomial of the quadrangular symmetric matrix A, i.e.: A(y) = p,y’
(i = sum. index 0, 1, . . . , m),
(63)
in which p, are constant coeikients. Multiplying both members of eqns (62), (57), (59), (60), . . . , (61) with adding by parts PO, PI, P29 P39 * * *. pm, respectively, the resulting Cm+ lkauations and usinn the wtll-
865
Static analysis of frames on elastic foundation
Gayley-Hamilton theorem of matrix algebra, we obtain the following matrix equation: known
Based on the previous corollary, as well as on relations (57a), the solution (67) for ‘Y’can be written as:
V[( - l)‘+ ‘ptBC] = p/P,
(64)
a direct (m x m)-inversion of the last we calculate the V-matrix by the closed
So, the matrix-equation (55), based on the last relation and after some algebra, leads to: W” = S-‘M,,
(65)
‘y’= (P,P,)@,
S-i = (lq8, Lj,e + X,)-l = (m x m)-matrix, M, = [k - k #(a)LiI - #,
in which
d~xWJ
= (m x I)-vector,
8 = [(- l)c+‘p,B1]-’ = quadrangular (m x n).
(69)
where
symmetric matrix of dimensions L, = &J
Corollary 5.1.
= (m x n)(n x n)(n x 1) = (n x I)-vector,
1P= coefficient = ITBq-‘L I>
The expansion
q=sum.index j = sum. index 1,2,. . . ,m
h(P) = (p,P,), can be transformed
‘,
(66)
-
(69a)
~T~A~~~f~~)ul~JJ,)= MI,
(70)
in which
q cm;
i g = e-S-‘B2 = (m x m)-matrix,
q = sum. index 1,2;. . . , m; j=sum.indexq,q+l,...,
m.
l!‘“{(lqbBq-i8B2 + A2)
where U,=(-l)p+‘pjAI-Q;
1,2 ,...,
By now, introducing the expressions for the Y-matrix and W”-vector given by eqns (68) and (69) into eqn (56), we tind:
in the form
b(P) = U,PP-
m.
(67)
y = L,IT = (n x n)-matrix, M2 = K2 - JTC[ &eS- ‘A, JT) + &r)]SB,
Proof From the second of the reductional we obtain the relation
formulae (61)
based on the last equation,
one may
,+(P)=p,P,=(-l)‘+‘p,A’-SPBq-’ 1,2 ,...,m;
= (m x I)-vector,
1, = coefficient = ECU,l
(j = 1,2,. . . , m; q = sum. index 1,2,. . . , j). Consequently, have:
= (1 x n)-vector, A, = k, - &a)L,
+(_])‘+‘A’-‘PI-1
(j=
+ fi(eW’J’?. (68)
j = sum. index 1,2,. . . , m. Through equation formula
#C(B)E f(a) + &#“b)
Y = ,@)a;
t = sum. index 0, 1,2,. . . , m
4 -sum.
q=sum.
index 1,2 ,...,
m.
It is necessary now to introduce proposition.
(70a) the following
Proposition 5.1.
index 1,2 ,...,
j).
On the other hand, for a different running of the indices q and j, the last expression for b(P) remains unchanged, but the q, j-indices take the values
The expansion of the expression
has the form
q = sum. index 1,2,...,m; j = sum. indexq,q+l,..., Consequently, obvious.
m.
the validity of eqns (66) and (67) is
where & are constant matrices of dimensions
coetkients (n x n).
and r/_i.,_i
are
D. E. PANAYOTQIJNAKCS et al.
866 Proof
Putting Q = JbX b = (m x n)-matrix
(71)
and using corollary 5.1, we obtain:
= UJ(U,QB’+
U,QB’ +. . .
. ..+U.QBm-‘hBO+... e. . +
+ U,QB’ +. . .
U,fi(U,QB”
. ..+U.QB”-‘)yBm-‘.
(72)
Through the substitutions U,/KJ, = A, = (m x m)-matrices,
foundation, according to the previously developed theory one may list the following steps in order to facilitate the user of digital computers: Srep 1. By the relations (70a), (45), (49), (55a), (56a), (69) and the corollary 5.1 we calculate the (1 x n)-vector M2, while through the same procedure, as well as proposition 5.1 a d formula (76), we calculate the (m x m)-matrix d . Furthermore, using eqn (75) we evaluate the (1 x n)-vector &Y Srep 2. Through Step 1 and the use of the second of eqns (69) we determine the (m x m)-matrix S-‘, while, at the same time, based on relations (69a) and the third equation of (69), we also obtain the (m x I)-vector M,. In the sequel, applying the first of eqns (69) we calculate the (m x I)-vector WX of the horizontal displacements. Step 3. From the results of Steps 1 and 2, also using the tinal formula (77), we determine the (m x n)-matrix ‘Y of the rotations of the nodes of the frame.
(i; i = 1,2, . . . , m), 6. NUMERICAL RESULTS AND DJSCUSSION
BkyB’ = E, = (m x m)-matrices, (k; I=O,l,...,
m-l),
the expansion (72) can be written as tc[Btc(Q)vI = A,/Qa,- I.,-, (i; j = sum. indices 1,2, . . . , m).
(73)
Multiplying the last equation on the left with the (1 x m)-vector J TC and taking into account the validity of relation jTCAoJ = ECU,&
= ~~ = consts,
one may have:
As an application of the proposed methodology in this paper we shall try to determine the nodal slopes and displacements of the four-storey and fourcolumn plexus frame shown in Fig. 4. The structure is subjected to uniformly distributed loadings, which vary from one span to the next, with the dimensionless ratio q2/q, = 2. Four concentrated unqual horizontal forces are also acting in the direction of the four respective horizontal beams of the frame. The flexural rigidities EJ vary from one span to the next, but they are assumed constant within each span. Also, the dimensionless ratio h/a is considered to be equal to 1. Based on relations (l), (9a), (12), (30) and (45) we formulate the following matrices of the geometrical characteristics of the frame:
Through the last formula, the vectorial eqn (70) takes the following simple form: &x= h4&,
(75)
-2
where AZ=6-i = [(l,bBq-’ - /+,bB’-‘gB’-‘)0B2 q=sum.index i,i=sum.indices
1,2 ,...,
+ AZ]-’
m
l,Z,...,m.
(76)
Finally, based on eqns (75) and (76) the exact matrix solution of eqn (68) becomes: ‘I’= #(a + &&b
+ eS-‘A4,~)6X
(77)
In conclusion, for the evaluation of the structure redundants of a plexus frame supported on an elastic
J a*
-2 2
0 l-l
0 0
Static analysis
j=l
j=2
ta
_t___
of frames on elastic foundation
867
j=4
j=3
i-lt h
i=2+ h
i-3+ h
i= 4+ h
20
+a
1
Fig. 4. Numerical example.
C = E = (4
x 4)-unit matrices.
(78)
As a first application we consider that the supports of the frame are rigid on the same level. Consequently, the &-vector becomes equal to zero, while the fundamental matrix equations (55) and (57) take, respectively, the following forms:
2/l
Ocl=Ix
2 1 0 0 1410 0141’ 0 0 1 2
1I r -1
1
r -1 where
7
0
07
-7
-11
In the beginning, according to the previously developed theory, we formulated the characteristic polynomial of the matrix El with the following five roots: po =
1329; p1= - 1248; p2 = 330; P3 =
-32;
p,= 1.
D. E. PANAYOT~UNAK~S er al.
868
In the sequel, based on eqn (65), we evaluated the matrix 8, i.e. - 18,009 -5016 8=
-30,912
-448 20,432
I
-5016 63,889
jy,=_
-5016
-448
-63,889
-20,432
--30,912 20,432
- -5016 18,0091 (82)
- 20,432 -
eqn (52) we calculate the (4 x 4) matrix % Yas follows:
“‘91 24EJ
’ X
i 0.332199 0
0.216128 0
0.147386 0
0.156280 0
Also, from relations (69) we find L, =(-0.000063,
(87)
O.OOOOO6, O.OOOOO6, -0.000063)T;
1., = -0.00012;
1, = -0.00034;
1, = -0.003264,
(83)
1, = -0.02624.
Through eqns (66), (67) and the second part of (69) one may obtain: i
q-1
’
As a second application we consider that the four-storey and four-column plexus frame shown in Fig. 4 is supported on four elastic supports with equal translational spring constants S. The dimensionless ratio a’J”/6EJ is taken equal to 0.75. In this case, the fundamental matrix equations (55), (56) and (57) are reduced to the following form:
m
~2YzJ +
2,W" =k,,
(88)
+
ih2 =K2,
(8%
JT'4'~,
--!
8,Y’ + ‘Pa, = a + a2W”JT + hba,,
PO)
-0.593066 -0.396498 -0.008605 J -0.005736 -0.036729 O.OOQ499-0.086673 0.288282 -0.067339 -0.036729 0.142863 -0.130010 -0.131741 -0.396498 o.fmO749 -0.101009
in whicithe
(4 x 4)-matrices 8,, f,, a,, a, a2 and the vector , are the same as in eqns (81), while the remaining matrices and vectors are given by:
3920 -432 3212 ’ -3:;; 32,052 3920 - -32,052 57.260 3096 3096 57,260 20,180 9232 I -9232 -20,180
r-2
2
0
07
(84) On the other hand, based on eqns (69) and selecting the dimensionless ratio aq,/P qua1 to 4, we have:
M , = &;
a i a,=4
(1 .oOOOOo,2.000000,
40
-5 1-5
1
04 i
0
0
2
2
- 3.OoOOOO, 4.OooOOO); II’“= -&(-0.332199, -0.147386,
-0.216128,
-0. 156280)T.
(85)
Finally, based on the formula (77). as well as on relations (85), the second part of qns (84) and on eqn (82), we evaluate the matrix V’ of the slopes of the nodes, i.e. p*
x
-- a’qt 8U
0.026378 -0.012976 0.620499 -0.614819 0.434861 -0:320695 0.351313 -0.508372 1.176598 - 1.532732 1.606571 -1.524232 1.912498 -2.641767 2.783818 -2.471136
’
(86)
while,
based
on
the second
of qns
(85) and
Now, from the 6fth of qns (70a) we evaluate the coefficients 1, (q = 1,2,3,4). while through the combination of the fourth and the third of qns (70a) with relations (88), (81), and the second of (84) and the corollary 5.1, we calculate the M,-vector. On the
869
Static analysis of frames on elastic foundation
other hand, from the first of eqns (70a) one may obtain the (4 x 4)-matrices /I and y, while, according to the proposition 5.1, one may also have the (4 x 4>matrices A,, Q,. So, the coefficients pY, included in the previous proposition, can be easily determined. By now, based on the above, as well as on eqn (76), we iind the matrix g-l, while, through the third of eqns (69) the vector M,. Furthermore, using eqn (75) and the first two parts of relations (69), we evaluate the #‘X and W-vectors,
i.e.
Comparing the solution results of relations (86) and (87), corresponding to a rigid soil, with the results of (93) derived from the case of an elastic soil, one may conclude that the effect of an elastic foundation on the upper structure is appreciable.
REFERENCES 1. G. Kani, Analysb of Mulriclorey Frames. Fred. Uog. Publ. Co., Ne\; Yo;k, N.Y. (19j6). 2. M. F. Rubinsteio. Multistorv. frame analvsis . bv dkital computers, Presented at the 2nd Conf. on Electronic Computation, AXE, Pittsburgh, PA (1960). 3. J. E. Soehrens, The electronic computer as a tool oo electronic computation. ASCE, Pittsburgh, PA (1960). 4. P. Lustgardeo~ Iterative method in f&e a&y&, Joumal of the Structural Division. ASCE. Vol. 89. ’ No !3T2,April, 1963, Z-94. . 5. S. S. Texcan, Discussion of “Simplified Formulation of Stiffness Matrices” by M. Wright, Journal ofthe SWUCrural Division, ASCE, Vol. 89, No ST6, December, 1963, pp. 445-450. 6. R. W. Clough, E. L. Wilson and I. P. King, Large capacity multistory frame analysis programs, Jou&l of the Struclural Division, ASCE, Vol. 89, No ST4, August, 1963, pp. 179-204. 7. R. W. Clough, I. P. King and E. L. Wilson, Structural analysis of multistory buildings, Journal of the Structural Division, ASCE, Vol. 90, No ST3, June, 1964, pp. 1e34. 8. E. D. Paoayotounakos, Static and dynamic analysis of plexus-frames based on the matrix-algebra, Technica Chronica, Vols l-2, Athens (I 966). 9. E. D. Paoayotounakos and N. Galidakis, Trfgerroste, Techica Chronica, Vols 425-426, Athens (1959). 10. R. W. Clough. Structural analysis by means of a matrix algebra program. ASCE Con/. on Electronic Computation, Kansas City, p. 109 (November 1958). 11. G. L. Rogers, Dynamics of Framed Structures. John Wiley and-Sons, inc., New-York, N.Y. (1959). 12. J. H. Argyris. Enerav Theoremsand StructuralAnalvsis. Butte&&s Publ&tioos, London, England (l&j.- ’ 13. C. H. Samson and H. W. Bergmann, Analysis of low-aspect-ratio aircraft structures, Journal of AeroSpace Sciences, Vol. 27, No 9, September, 1960, pp. 619-693. 14. S. S. Texcan, Moment equations for computer analysis of frames, Journal of the Structural Division, ASCE, Vol. 90, No ST3, June, 1964, pp. 35-53. 15. J. S. Przcmieoiecki, Theory of Matrix Structural Analyst&McGraw-Hill, N.Y. (1968). 16. Jennings, Matrix Computationfor Engineers and Scienfisrs, John Wiley and Sons, N.Y. (1978). II
8x = -&;
( - 24.294810, - 23.692039,
-23.786440, W”-- -$&
- 24.224697)
(-0.366388,
-0.192900,
-0.269922,
-0.173642).
(92)
Finally, using formulae (77), (51) and (52) we obtain the following solutions for the W, 4and W-matrices: y_
-0QI 24EJ
-0.308968 0.853872 -0.855832 0.272967 0.461652 -0.339069 1.740275 1.230110 - 1.927782 3.001748 1.945198 -3.382358
x
-0.227488 -0.246797 -0.307535 -0.453721
&&!L
24EJ
x
0.173642 0 i 0.366388 0 0.269922 0 0.192900 0
WY,0441 24E/
x
0 0 24.294810 0 0 0 0 23.692039 0 ’ 0 0 23.786440 0 24.224697 0 0 (93)