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Iterative lateral analysis of eccentrically loaded building frames
Computers
Pergamon
0045-7!M9(!M)EOO29-2
ITERATIVE
Vol.
52, No. 5, pp. 939-9%.
1994
Copyright Q 1994 Elsevier Science Ltd Printed in Great Britain. All rightsreserved 0045-7949/94 s7.00+ o.cm
LATERAL ANALYSIS OF ECCENTRICALLY LOADED BUILDING FRAMES C. J. You&
JDepartment
& Structures
and D. E.
PANAYOTOUNAKOS$
tTechnica1 University of Crete, Hania, Crete, Greece, of Engineering Science, Section of Mechanics, National Technical University of Athens, Zographou, GR-157 73, Athens, Greece (Received 17 February 1993)
Abstract-An iterative procedure is proposed, to simplify the stress analysis of rectangular building space frames, under horizontal action. The procedure involves the solution of two associated equivalent plane frames, along with the use of equations from Rousopoulos, Theory of Elastic Complexes (Elsevier (1965)). The equivalent plane frames contain extensible connecting rods, with axial stiffnesses to be determined, which connect the individual plane subframes that form the given building space frame. The iterative procedure comprises evaluation of the column shear stiffnesses, by solving the equivalent plane frames. The column shear stiffnesses with the equations of the Theory of Elustic Complexes, furnish values of the axial deformations of the connecting rods, which in their turn indicate new values of the axial stiffnesses of the connecting rods. With these new values of the axial stiffnesses, the equivalent plane frames are solved in the process of the next iteration to furnish new values of the column shear stiffnesses, and so on. The iterative procedure converges to certain limits in the stresses of the equivalent plane frames, which limits are equal to the stresses developed in the given building space frame, under the given loading. The convergence of the iterative procedure is discussed.
NOTATION
4 A, C.E.R. 4,~ &, 4: E
Ekx1Eky
Ekz F
&k*l 4 %,y
ks
k 1x91, Mlk Mk N N
Qkx9Q, &k I(Ok U x: Xik
Y: Yik AL
areas of the connecting rods centre of elastic rotation column shear stiffnesses column torsional stiffness modulus of elasticity overall storey shear stiffnesses overall storey rotational stiffness axial force defined in Fig. 5(a) vector of axial forces defined in Fig. 10(b) N-vector defined in eqn (39) components of the horizontal loads applied on the floor slab j the 1 x N unit vector axial stiffnesses of the connecting rods stiffness of the frame structure matrix of the axial stiffnesses of the connecting rods matrix of the stiffnesses of the frame structure matrix defined in eqn (25) lengths of the connecting rods column torsional moment moment of the floorslab resultant force with respect to the C.E.R. axial force of the connecting rod vector of the axial forces of the connecting rods column shear forces column end x-displacement C.E.R. x-displacement x-displacement of the point (x, y) x-coordinate of the C.E.R. column x-coordinate y-coordinate of the C.E.R. column y-coordinate vector of the length changes in the connecting rods
A*L Au “ik uOk jk
diagonal matrix of the length changes in the connecting rods axial deformation of the connecting rod column end y-displacement C.E.R. y-displacement y-displacement of the point (x, y) upper and lower floor slab relative rotation matrix function defined in eqn (36)
INTRODUCTION
The bearing structure of a multistorey building can be approximated with a satisfactory accuracy by a space frame the members of which represent the beams and the columns of the building. It is assumed that no shear deformation is developed in the planes of the floor slabs of the different stories; in other words the floor slabs behave as rigid plane diaphragms. The analysis of a space frame requires intensive computational use, large memory size along with frequent memory operations, which often are not feasible. The analysts usually take refuge in simplifying but accurate enough assumptions which help them to espace the difficulties of the space frame analysis. There are many such methods which are more or less effective for the approximate analysis of building bearing structures under horizontal actions. In [l-7] some of the best known among them are mentioned. In (81a theory is developed for the estimation of the relative displacement of the upper floor slab of any storey of a multistorey building with respect to the 939
940
Lateral analysis of building frames
lower one, based upon the shear and torsional stiffnesses of the columns between the floor slabs. It is assumed that all the floor slabs behave as rigid plane diaphragms. The multistorey building space frames treated in the present paper, will be assumed of rectangular plan containing columns with principal directions of shear stiffness [8] parallel to the sides of the plan of the building. Hence in every storey, the principal axes of the elastic rotation [8] will be parallel to the sides of the plan. Also it will be assumed that in every building frame, an orthogonal coordinate system Oxyz with horizontal axes Ox and Oy is attached, parallel to the sides of the building plan. According to [8] to the upper floor slab of any storey, e.g., the kth storey, has a corresponding characteristic point, which plays the main role in the above mentioned relative displacement of the floor slabs; it is the Centre of Elastic Rotation (C.E.R.), defined as the centre of mass of the column shear stiffnesses of the kth storey, whose coordinates therefore are:
xi’ =
(i,xlk&)/($ %)
Y: =
(i,YtkDJk.y)/(i, Dike)-(lb)
(14
In eqns (la) and (lb) xik and y,, are the coordinates of the ith column and D,, and D,, are the shear stiffnesses of the column i, defined by
(2b) In eqns (2a) and (2b) Q,kX, Q+ are the x and y components of the shear force in column i, and u,~, u& are the x and y components of the developed horizontal relative displacement of the upper end of column i, with respect to its lower end. The overall storey shear stiffnesses of the k th storey including n columns, are defined by
the torsional stiffness of the ith column defined by
In eqn (5) Mik is the torsional moment developed in the ith column and & is the relative rotation of the upper floor slab of the kth storey with respect to its lower slabs. In the following discussion the quantity Z;=, D,,: will be considered small compared to the rest of the right hand side of eqn (4), so that this equation from now on will assume the form
E/z= i [(xi/ci=l
x:)‘D,, + (Y, - Y: )'&I.
(6)
The relative displacement of the upper floor slabs of the k th storey with respect to the lower one, is caused by the horizontal loads applied on all the Boor slabs j (j = 1,2, . , k) lying above the mentioned kth storey. Denote by H,, H,,j = 1,2,. . , k, the components of the horizontal loads, and by x,,,, y),,, j = 1,2,. . , k, the coordinates of their points of application. Then the components of the resultant horizontal force acting on the upper floor slab of the k th storey are Z;=, H, and X,“;= , H,y and the moment of this resultant force with respect to the C.E.R. of the floor slab under consideration is
Mk= i [H,~(X,-X:)-H,,~(Y,-Y:)I.
(7)
i=l
According to the above introduced quantities, the components of the relative displacement of the C.E.R. of the upper floor slab of the kth storey with respect to the lower one, are given by
vOk =
/;,
H,?lEk? )
WI
and their relative rotation by 4k
W) and the overall storey rotational stiffness of the kth storey with respect to the C.E.R. of its upper floor slab, is defined by -& = i
Dikz+ i
,=I
,=I
[tx,k -
x:)2D,k, + (yjk - y$)‘Dik,]
(4)
where x: and y: are given by (la), (lb) and Dik2is
=
Mkl’%;r
(9)
as well. Based on eqns (8a), (8b) and (9) the relative displacements (u, u) of any point (x. y) of the upper floorslab can be evaluated as follows: u = UOk-
+k(Y
u=
&(X -x:).
UOk +
-Y:)t
WW
(lob)
Equations (lOa) and (lob) indicate that points lying on lines y = const.,
or x = const.,
(lla)
C. J. YOUNISand
D. E. PANAYOTOUNAK~S
exhibit identical relative displacements u = const.,
or u = const.,
(lib)
respectively. This remark forms the basis of the idea of the two ‘associated equivalent plane frames’ corresponding to the directions x-x and y-y. Specifically, the associated equivalent plane frame corresponding to x-x is composed of all Ox plane subframes parallel to the axis, which form the given space frame, one next to the other and connected at the levels of the horizontal beams. The beams correspond to the levels of the different floor slabs, by connecting rods of area A and moment of inertia J = 0. Accordingly, the associated equivalent plane frame corresponding to y-y, is composed of all the plane subframes parallel to the axis Oy, comprising the given space frame. The zero moment of inertia (J = 0) of the connecting rods, prevents the transfer of bending moment between adjacent subframes. The two associated equivalent plane frames x-x and y-y, are loaded by the horizontal loads H,x, H, respectively,j=l,2,... , N, applied at the levels i of the horizontal beams, which correspond to the different floor slabs j, j = 1, , N. The horizontal beams in the equivalent plane frames are considered axially incompressible, since the floor slabs which they correspond to, were supposed to behave as rigid plane diaphragms. As an example, in Fig. 1 a rectangular space frame is depicted, loaded by the horizontal loads H,,, H,,, Hzx and Hz,,. In Fig. l(a) the associated equivalent plane frame x-x is shown, loaded by the horizontal loads H,, and H,,, while in
CAS w--G
941
Fig. l(b) the associated equivalent plane frame y-y is shown loaded by H,, and Hxy. All the horizontal beams A,B,, B,C,, F,E,, E,D,, F,A,, EiB,, PC,, E,B,, BG,. . . , EzB2 and D,C2 are incompressible. There is a correspondence between the developed stresses and deformations in the space frame and in the two associated equivalent plane frames. In fact, for certain values, to be determined, of the axial stiffnesses of the connecting rods of the associated equivalent plane frames, the stresses (shear forces and bending moments) and deformations of the equivalent plane frame X-X are equal to the stresses and deformations developed in the corresponding plane subframes, lying in the given space frame and parallel to the direction OX. Similarly, for these certain values of the stiffnesses of the connecting rods, the stresses and deformations in the equivalent plane frame y-y, are equal to the stresses and deformations of the 0 Y-parallel plane subframes forming the given space frame. Because of the relative rotations of the floor slabs due to the moments Mk, eqn (7), of the resultant horizontal forces with respect to the C.E.R.s, the different horizontal beams of the plane subframes forming the given space frame exhibit different horizontal displacements, as is shown in eqns (1 la) and (11b). The role of the axially deformable connecting rods, is that inside the two associated equivalent plane frames, they insure the above mentioned different horizontal displacements of the horizontal inextensible beams corresponding to different plane subframes. It is easy for one to see that in the special case where in all the floor slabs the resultant horizontal
942
Lateral analysis of building frames f Connecting rods
(4
l,+L,t
tL,t-L,-
L2-
Fig. l(a). The associated equivalent plane frame x-x.
(b) H
lY
t hl
t h2
I
1 -
L3 4
lyt
J-3 --+-
ly +
L,W
Fig. l(b). The associated equivalent plane frame y-y.
forces pass through the C.E.R.s (Mk = 0; bk = 0, k = 1,. . , N), all the floor slabs exhibit translatory displacements. This means that in the two associated equivalent plane frames, all the connecting rods have to be axially undeformable A = co, since the different horizontal beams lying in a floor slab exhibit identical horizontal displacements. The aim of the present paper is to determine the values of the axial stiffnesses of the connecting rods, so that the stresses and deformations in the two associated equivalent plane frames are equal to those of the corresponding plane subframes lying in the given space frame. The equations which have to be satisfied are those of the Theory of Elastic Complexes [8], in which the shear stiffnesses D,, , D,, eqns (2a) and (2b), will be evaluated by the solution of the two associated equivalent plane frames. An iterative procedure is proposed. Since initially the axial stiffnesses of the connecting rods are unknown, they are given arbitrary values. Following the proposed iterative procedure, it leads to a convergence of the values of the axial stiffnesses, as well as of the stresses and deformations of the two equivalent plane frames, to specific values. These specific values are the required stresses and deformations developed in the members of the given space frame for the given
loading. Additionally, in a later section of the present paper, the conditions for the convergence of the proposed iterative procedure are studied and discussed. EQUATIONS AND ANALYSIS FOR ONESTOREY STRUCTURAL SPACE FRAME. THE ITERATIVE PROCEDURE
In Fig. 2 a simple one-storey space frame is depicted loaded by horizontal loads as well as vertical member loads. In Figs 2(a) and 2(b) the associated equivalent plane frames are shown, loaded accordingly. Initially the axial stiffnesses k, = E-&IL
(12a)
k, = EAJ!,.,
WW
of the connecting rods BD and AC are unknown. In eqns (12a) and (12b) E is the modulus of elasticity, A,, A, denote the area and l,,, I, the length of each connecting rod. Those values of k, and k, are required for which the stresses and deformations developed in the associated equivalent plane frames of Figs 2(a) and 2(b), are equal to the stresses and
C.
deformations developed in the given space frame at the different corresponding subframes, lying in it. The stresses and deformations of the space frame satisfy the following equations of the Theory of Elastic Complexes [8]:
Y ,,,.+" 12t
t
.9 1. *
i=l
Mk
=
i
[txik
j=l
-
X/F
j2Diky
+
lo-’
71 I, -0.3xX
1r’
II - 0.3%
lo-’
1; = 0.266x lo-’ 1 zxi
L /5-W
X
<
0
Fig. 2. Example of a one-storey building space frame. (YIJC -
Yf)2DikxI,(14C)
131
15t
(4 -Xk*)-Hjx(Y~-Y:)l,
[f&(X,
\
B\/Uk”
1 -- 1.067x lo-’
;r Q\
(1W
i=l
51
3x lo-’
ok
Y
Eky= i Diky,
I:‘%
~‘4715X
*
LL 2 (144
lk
4; -
Ii 1
i= I
l4t
l
D lt
I, - 1.23x lo-’ I, - 1.25x lo-’
Ekx= i Dikrr
c
-3
4.om
Ekz= i
943
J. YOUNLTand D. E. PANAYOTOUNAKOS
A
(15)
I
B
C
st-
t uOk
=
;, (
Ekxr
H, >i
Eky,
(16b)
+-(164
5.0 --+l.O~
(1%
(17b)
already referred to in the Introduction. In the above equations, the column shear stiffness Dikx, D, are coupled in a complex manner with the unknown stresses and deformations, and for their evaluation the analysis of the whole space frame is more or less required. Instead of that however, one can obtain the values of the column shear stiffnesses, from the solution of the two associated equivalent plane frames, where at present the axial stiffnesses of the connecting rods are still undetermined. To overcome this deficiency, initially the axial stiffnesses are given arbitrary values, and the iterative procedure described below is followed. Step 1
The solution of the two associated equivalent plane frames yields the values of the column shear stiffnesses D,, D,, as well as the axial forces N, in the connecting rods.
5.0 p-t
Fig. 2(a). The associated equivalent plane frame x-x. 121
141
@I D
A
C
B 71
tiGzl4.0 H
Ii mrmr
t-3.5
+1.ot_3.5+
L
G
P 7?V
Fig. 2(b). The associated equivalent plane frame y-y.
result, which along with the resultant actions ZH,, Z Hjy, Mk, by the use of eqns (16a), (16b) and (161~) furnish the C.E.R. relative displacements Uok,uok, &. Then, from eqns (17a) and (17b) the horizontal displacements of the plane subframes lying in the space frame are evaluated, which coincide with the horizontal displacement of the two equivalent plane frames.
Step 2
Step 3
Substituting these values into eqns (14a), (14b) and (14c) the overall storey stiffnesses E,,, Eky and Ekr
The above evaluated displacements of the horizontal beams of the equivalent plane frames give the axial
944
z 2 ML SL”
Lateral analysis of building frames 200
1.3
160
1.2
120
s
80
1.1 $
40
1.0
0
0
2
4
6
8
10
12
14
0.9
(4
Fig. 3. Variations and convergence of the axial stiffnesses and a shear force, versus the number n of iterations.
deformations
of the connecting rods Au = ui + I,k
-
%.k 9
(18)
which substituted in k = N/Au
(19)
furnish new values for the axial stiffnesses of the connecting rods, to be used in Step 1 of the next
iteration. In eqn (19) N is the developed axial force in the connecting rod, evaluated in Step 1 of the previous iteration. The above iterative procedure is repeated until the stresses (and deformations) of the associated equivalent plane frames converge to certain limiting values. Obviously these limiting stresses (and deformations) satisfying eqns (13), (14) (15) (16) and (17) of the Theory of Elastic Complexes, are equal to the corresponding stresses and deformations of the given space frame. Indeed, for the numerical data given in Figs 2, 2(a) and 2(b), the iterative procedure provides the results shown in Fig. 3, where the values of the axial stiffnesses k, and ky of the connecting rods BD and AC, as well as the shear force QAEXof the column GE, are drawn, versus the number n of iterations. One can see that after five iterations, the values practically converge to certain constant limits. Additionally, in Table 1, a comparison is shown between the resulting limiting stresses of the associated equivalent plane frames and the corresponding stresses evaluated from the direct solution of the given space frame. One can see that the limiting stresses of the equivalent plane frames
Table 1. One-storey frame (Fig. 2). Comparison between the results of the space frame analysis and the equivalent plane frame analyses Moments and shears (1) MAEI
M AE) M EAI M EAY MBF~ MBFF
M.FBT
M,s, M CGX M CGJ M Gcr M GCY M D”.T M DH> M “Dr M HD,
MA6 MBA MBC
&ll &D MDC MDA
::
#:K=g QCBI $::WY
Results from the solution of the space frame of Fig. 2
Results from the solutions of the equivalent plane frames X-X
(2)
(3)
1.26 1.97 2.02 2.50 4.28 4.69 4.66 4.20 3.96 3.45 3.78 2.57 4.32 -0.62 3.78 1.17 1.97 4.69 4.28 3.96 -3.45 0.62 -4.32 -1.26 1.12
1.97 2.50 4.70 4.21 3.47 2.59 -0.59 1.20 1.97 4.70 -3.47 0.60 -1.12
0.82 2.24 2.22 1.51 0.14 2.03 1.93
-2.23 1.52 -0.15
Moments (M) in (t m) and shear forces (Q) in (t).
Y-Y (4) 1.26 2.02 4.27 4.66 3.95 3.78 4.32 3.79
4.27 3.95 -4.32 -1.26
2.23 0.82 2.03 1.93
C. J. YOUNISand D. E. PANAYOTOUNAKOS
Step 3: C,D, , C#$
CsD,,....
K, O/m) Fig. 4. Convergence of the iterative procedure (case with member loads).
constitute a very good approximation to the exact stresses provided by the analysis of the space frame. Figure 4 presents another point of view of the iterative procedure’s convergence. The curve shown in it represents the function N = N(k,), where N is the axial load developed in the connecting rod BD of the composite plane frame x-x, during its deformation caused by the applied loads, and k, is the axial stiffness of the connecting rod. For simplicity, only the equivalent plane frame x-x will be discussed here. According to the theorem of superposition[9], the function N = N(k,) is given by
t
L,+
1,t
L,+
Fig. S(a). Definition of the axial force F.
(b)
t
t
L,+l,t
1,+ Al, -+
945
where F is the axial force applied to the points B and D, required to keep the distance BD (Fig. Sa) constant during the deformation of the equivalent plane frame x-x, caused by the applied loads, and k,,, is the axial stiffness of the distance between the points B and D due exclusively to the structure of the equivalent plane frame x-x, the connecting rod BD being absent (Fig. 5b). The abscissae of the points c,,cz,. . . , C,, on the curve (N, k,) represent successive values of the axial stiffness k,, as they were evaluated by the iterative procedure. The same values are also shown in Fig. 3. The abscissae of the points Di = i = 1, . . . , 13, on the other hand, represent the values of the axial stiffness k, as they were determined from the axial force corresponding to the points Ci, by use of eqn (19) (Step 3). Additionally the ordinates of the points C,, i = 1,. , 14, represent the axial forces developed in the connecting rod whose axial stiffness is the abscissa of the points Di_ , . The axial forces of the points Ci are determined by the solution of the equivalent plane frame x-x (Step 1). One can see that the points Di, i = 1,2,3, . . . , lie close to the line OD,,, the tangent of the angle of which with the k, axis, represents the axial deformation AU,, of the connecting rod BD after the 14th iteration (Step 2). Indeed, the tangents of the angles of the lines ODi, i=l,2,..., with the k,-axis represent the axial deformation Aui of the connecting rod BD, as they are evaluated after each iteration by eqns (13)-(17) of the Theory of Elastic Complexes [8] (Steps 2 and 3). Actually, there is a small variation in the axial deformation Au,, from iteration to iteration, owing to a small variation in the column shear stiffnesses Dikx, Dikgas they are determined from the solution of the equivalent plane frames (Step 1) after each iteration. This variation of the D,, and D,,. in this case of a one-storey space frame, is due to the presence of member loads which contribute to the column shear forces and the column-end relative displacements independently, and additionally to the acting horizontal joint loads. Of course the contribution of the member loads to D,,_, Dlkv is small compared to that of the horizontal joint-loads, so that the variation of D,, and D,.is small. In Figs 6(a) and 6(b) of the associated equivalent plane frames x-x and y-y are shown, corresponding (a)
L,_
Fig. S(b). Definition of the stiffness K,,,.
Fig. 6(a). The associated equivalent plane frame X-X (case without member loads).
946
Lateral analysis of building frames D
A
C
B 7t 2.0
t 4.0 1.5
t-3.5
E mrmr -_t1.0+
G
F 7m
2 2
1.0
3.5 -+
Fig. 6(b). The associated equivalent plane frame y-y (case without member loads).
0.S
tan 6 I AI+ i I l.?.... I
II
to the space frame of Fig. 2 loaded exclusively by the horizontal joint loads H, and H,,, without any member load. Figure 7 presents the variations of the connecting rod axial stiffness k,, resulting from the presented iterative solution applied to the equivalent plane frames of Figs 6(a) and 6(b). The absence of member loads in this case implies constant values of Dikx, D, and hence constant Aui, for all the iterations. As shown in Fig. 7 the points , D, lie on the straight line OD, correD,,Dz,... sponding to the constant value of Aui. In this case, in the absence of member loads, the path D,C,D,C,D,Cj.. . is a zigzag line between the curve N = N(k,) and the line OD,. Also, in Figs 4 and 7 it is easy for one to check that the fast convergence of k, towards its limiting value, during the iterative procedure, is due to the horizontally asymptotic character of the curve N = N(k,) for large k,, as imposed by eqn (20), in combination with the comparatively larger and constant inclination of lines OD,,i=1,2 ,.... Except in the presence of member loads, in the general case of multistorey space frames, another source of the small variation in the values of Dikx, D,, from iteration to iteration, is the inducement of additional column shear force and column-end
Fig. 7. Convergence of the iterative procedure (case without member loads).
relative displacement by the extensions of each column into the adjacent (upper and lower) storeys. ANALYSIS OF A MULTISTOREY SPACE FRAME
Figure 8 shows an example of a two-storey space frame loaded by horizontal loads as well as by vertical member loads, while Figs 8(a) and 8(b) show the two corresponding associated equivalent plane frames, loaded accordingly. After giving initially arbitrary values to the axial stiffnesses of the connecting rods BD, FH, AC and EG, the iterative procedure described above is followed, which after five iterations practically converges to the values discussed below. In Fig. 9 the successive values of the axial stiffnesses of the above four connecting rods are shown along with the successive values of the bending moments MEc and MEH at the point E of the bars EI and EH respectively, as they arise after each iteration,
IA& - 1.067x n+ = 1.267x ‘A% I& - 1.333x = 0.521x I% IBF -0.326x
STRUCTURAL
lo-’ w3 10-3 lo-’
IRF* = 0.326x 10-3 I,,,: - 0.563x 1O-3 = 0.391x 10-S h Ico, - 0.233x lo-’ =0.715x 10-3 ‘=J, IOK, = 0.267x lo-’ IOK, = 1.067x10-3 IDa. = 1.251x 10-3 = 1.251x 10-3 ‘Da, 1% - 1.251x w3 = 1.2Slx 10-g ‘m,
Fig. 8. Example of a two-storey building space frame.
C. J. YOUNIS and D. E. PANAYOTQUNAK~~
A
B
t
C
D
Y?
16t
3.0
t--_
947
5.0 -+1.0-t
5.0 -+
Fig. 8(a). The associated equivalent plane frame x-x. 17t D
A
1
C .
t3.0
B st 1st 51
H
E”
“G
F
t 4.0
-_tl.o~-+--3.5+
t3.s
Fig. 8(b). The associated equivalent plane frame y-y. versus the number n of iterations. Obviously five iterations are sufficient to provide practically constant values for the required developed stresses and deformations in the given space frame. Additionally one can check that the variations in the axial stiffnesses of the connecting rods as they arise after each iteration with respect to their limiting values are relatively small. This can be expressed by the inequality 0.85
< 1.15
(21)
where kli,,,is the limiting axial stiffness of a connecting
tat?
160
a
52 120 a0
46
rod, and k is the value of its axial stiffness after four iterations. As mentioned in the previous section, the reasons for the variability in the connecting rod axial stiffnesses are the presence of the member loads as well as the inducement of additional shear force by the extensions of the columns into the adjacent storeys. It should be mentioned here that, as is reasonable, the column shear forces arising from the above two sources are small compared to those arising from the horizontal loads. Table 2 shows a comparison between the developed stresses at different points of the structure of Fig. 8 as they were evaluated by the iterative procedure and the corresponding stresses as they arise from an exact analysis of the given space frame. One can check the very good agreement between the results of the two methods of analysis. CONVERGENCE
5
40
2
bL
0
0 -2
Fig. 9. Variations and convergence of the axial stiffnesses and moments versus the member n of iterations.
Figure 4 and 7 show the successive values of the axial stiffness of the connecting rod, as they arise from the application of the iterative procedure on the structures described by Fig. 2 as well as Figs 6(a) and 6(b), respectively. When dealing with a one-storey structural space frame it was mentioned that in this case without member loads the values of the shear stiffnesses DikX,D, of the columns resulting from Step 1 of any iteration are identical. This fact implies that the points D, , D,, D,, . . denoting the
Lateral analysis of building frames
948
Table 2. Two-storey frame (Fig. 8). Comparison between the results of the space frame analysis and the equivalent plane frame analyses Moments and shears (1)
MA&
Results from the solution of the space frame of Fig. 8 (2)
Results from the solutions of the equivalent plane frames x-x (3) 0.97 3.95 2.38 1.18 6.81
Y-l’ (4) -0.91 2.57
-0.92 0.98 2.60 3.96 2.58 2.38 3.78 1.18 3.48 6.83 8.06 5.13 5.07 6.01 2.78 1.28 0.98 2.60 -2.37 -3.78 8.22 10.01 -9.41
5-12 6.01 1.28 0.97 -2.38 8.20 -9.41
Z Per, eQ::
-4.87 0.79 2.35 -0.04 1.92
-0.79 2.34 -0.04 1.93
z:
-4.26 -1.78
-
-4.25 - 1.78
Q GKY Q HLY
-2.49 -1.49
-
-2.48 -1.49
M
AEv
MBF,
A4sr,
M CGr M cc+ M DHr M DHY
MslV ME,, M FJr
Mr,y MGKI M GKY
M HL.Y M HLg
MAB MB,
MC, MDA MEF
Mro
M
GH
2.56 3.79 3.48 8.04 5.06 2.78 2.57 -3.79 9.97 -4.88 -
Moments (M) in (t m) and shear forces (Q) in (t).
t
h, t ha t h3
+ h4
t % -c
Fig. IO(a). Equivalent plane frame of a multistorey building.
C. J. YOUNISand D. E. PA~AYO~~AK~
Fig. 10(b). Definition of the axial forces vector
949
F.
Fig. IO(c). Definition of the elements of the stiffness matrix k,.
successive values of the connecting rod axial stiffness resulting from the successive iterations, all lie on the straight line OD, * . - D,D, , Fig. 7. On the other hand, when loads act on the space frame member (Fig. 2), the column shear stiffnesses D,, D, resulting from Step 1 exhibit small variations from iteration to iteration. As mentioned before, the above small variations are due to the presence of member loads as well as to the influence of the adjacent storeys in the case of multistorey buildings. These variations in the column shear stiffnesses imply that the angles D,OK,, 1,2,. . . , Fig. 4, where the points D, denote the successive values of the connecting rod axial stiffness, exhibit small variations, such that all the points D, lie inside the small crosshatched angle shown in Fig. 4. A measure of the small variations of the angles D,OK, is given by the inequality (21).
For the case of the one-storey space frames without member loads, the relative location of the curve N = N(k,) and the line OD, in Fig 7 makes it obvious that for any initial value of the axial stiffness k,, the convergence of the zigzag line * . +to the point C,, is certain. To this C,D,GD,C,D3 point one has to have in mind the ‘Fixed Point Iteration Theorem’ from Numerical Analysis [IO]. For the same reason, in the cases of multistorey buildings or the existence of member loads, the relative location of the curve N = N(k,) and the small crosshatched angle in Fig. 4 has as a consequence the convergence of the zigzag line C, D,CzDz ’ . * to a limiting point, lying on that portion of the curve N = N(k,), which is included in the small
crosshatched line. Of course in the above latter cases it should be pointed out that as the connecting rod
950
Lateral analysis of building frames
axial stiffness approaches its limiting value, the variations of the column shear stiffnesses DlkX,D,, diminish and the points Dj, for large i, lie on a limiting straight line passing through the origin 0. Obviously the abscissa of the section of this limiting straight line and the curve N = N(k,) is the limiting value of the axial stiffness of the connecting rod. To come to the study of the convergence of the iterative procedure in the general case of multistorey buildings, consider Fig. 10(a) where an equivalent plane frame is depicted including N connecting rods. Figure 10(b) presents the same as in Fig. 10(a) equivalent plane frame, where at the ends of the connecting rods couples of opposite axial forces Fi, -F,, i = 1,2,. . . , N are introduced, such that during the loading of the equivalent plane frame by the applied joint and member loads all the distances ii, i=l,2,..., N, remain unaltered. Denote by F=[F,,F*,..
I
kr,z
...
km
ks21 ...
km . *.
...
km ..:
...
If upon the applied couples of axial forces k;, -F,, Fig. IO(b), equal and opposite couples -F,, Fi are superposed, according to the principle of superposition 193,the connecting rods will obtain the axial deformations that are due to the loading of the equivalent plane frame, and hence the relation &+k,).AL=F
(27)
will hold, from which it follows that
kw
I
N=k;AL
(29a)
which in view of eqn (28) becomes N=k;(k,+k,)-‘.F.
(29b)
From the above formulated equations, Step 3 of the kth iteration of the iterative procedure presented before comprises the following algebraic manipulations. From eqn (29b), written as Nk=kd:.(k,+k,J’.F,
(30)
where kti is the stiffness matrix of the connecting rods, known at the beginning of the k th iteration, and F is the vector of the axial forces defined by (22), the new values Nk of the axial forces N in the connecting rods, arise. Equation (29a) is now written as
Nk=k++,,.AL kc, ...
.*. k cn
:
(31a)
0t
k c2
...
.
(24)
k c(k + k cck +
1
1j.1
ALi
il.2 AL2 . . .
Based upon (23) and (24), obviously the stiffness matrix k, of the positions of the connecting rods in the full equivalent plane frame, including the connecting rods, as shown in Fig. 10(a), is given by k=k,+k,. Denoting
the change
in length
(28)
At the same time during the superposition of the new equal and opposite couples -Fi, Fi, inside the connecting rods, axial forces N are developed which are given by
(23)
If the axial stiffness of the ith connecting rod is denoted by kcir then the stiffness matrix of the connecting rods can be introduced as the diagonal matrix
k, =
(26)
(22)
k sll
k SNI k,,
AL= [AL,,AL,, . . , ,AL,IT.
AL = (k,+ k,)-’ ’ F.
.,FN]T
the vector of the above axial forces; it is obvious that F is constant for any equivalent plane frame, depending exclusively upon its loading and the geometry of that part of its structure shown in Fig. 10(b), and hence it is independent of the axial stiffnesses of the connecting rods. Now denote by ks, the measure of each force of a couple of equal and opposite axial forces, which is developed at the position of the ith connecting rod, when a unit change of length of the distance between the ends A and B at the position of the jth connecting rod takes place, as shown in Fig. 10(c), is introduced as the matrix
k=
between the ends of the connecting rod i, i=l,2,..., N, during the loading of the equivalent plane frame of Fig. 10(a) by A&, by the applied joint and member loads, then the vector AL is defined by
(25) of the distance
kc(k
+
I).NAL
1
(31b)
from the solution of the equations from which the elements of the diagonal stiffness matrix ktk + it of the connecting rods are obtained. The matrix ku + ,) will be used in Step 1 and Step 3 of the next (k + I)th iteration.
951
C. J. Youm and D. E. PANAY~UNAK~
To study the convergence of the elements of the matrix k, evaluated from eqn (31 b), it is convenient to consider first the variations in the vector AL, included in eqn (31a) above, throughout the iterative procedure. It was referred to in Sec. 2 and at the beginning of the present section, that in the cases of multistorey buildings or the existence of member loads, there is a variability in the values of the column shear stiffnesses Dikx and D,, which implies a small variability in the values of the extensions ALi of the connecting rods as they are evaluated at the beginning of Step 3, from the results of the equations of the Theory of Elastic Complexes, obtained in Step 2 of each iteration. Also, in Fig. 4, referring to the case of one-storey buildings with member loads, it is shown that all the points Di, i = 1,2,. . . , lie inside the narrow crosshatched angle, indicating that the angles D,OK,, which represent the extension AL of the connecting rod, evaluated at the beginning of Step 3, exhibit variations that can be considered as very small. Additionally, it was referred to previously in the present section, that as the points Di of Fig. 4 tend to their limiting position, the variability of AL diminishes and the points Di, for large i, lie on a limiting straight line passing through the origin 0. In view of the above, it can be considered that after a few initial iterations the variations in the extensions AL,, i = 1,2, . , N, of the connecting rods become negligible and for the purposes of the present discussion, they may be allowed their limiting values, denoted by the diagonal matrix
A~*=[*)_
AL’
...
,,:i
(32)
from the beginning of the iterative procedure. Using the matrix AL* instead of the vector AL,, eqn (31a) can be written as
of eqn (30) by AL*-‘, it yields AL*-‘.Nk=AL*-‘.%.Ik,f%]-‘.F,
(36)
which in view of eqns (35) and (36), yields k%+,,= AL*-‘.cp(k~).F,+6~)]-‘.F.
(37)
Equation (37) constitutes a simplified and brief notation of the algebraic manipulations carried out in Step 3 of the iterative procedure, which were described by eqns (30) and (31 b) above. According to eqn (37), given k&, a new k&+ ,) arises to be used in the next iteration. Using the right hand side of eqn (37), the following function is introduced ~~)=AL*-‘,#~),~~+~~)]-‘,F,
(38)
where G(k$) is a N-vector, whose elements are functions of the N-vector c. In view of (38), eqn (37) can now be written as k&+,,=G(%),
(39)
which generates an iterative process corresponding to the fixed point problem [lo]. G(k,*)=c.
(40)
A theorem that gives conditions for the iteration process to converge is referred to in [lo], according to which the partial derivatives must satisfy the following inequalities ]aG,(k:)/%c,] c l/N, i, j = 1,2,, . . , N
(41)
for e o D, D c RN. But according to eqns (38), (36) and (29b) the above inequalities (41) assume the form ll/ALFl ]aN,(k:)/dk,]
< l/N,
i, j = 1,2,. . . , N,
(42) Nk=AL*.~~+,,+I
(33)
where I is the 1 x N unit vector. Multiplying both sides of eqn (33) from the left by AL*-‘, it follows that k&+,)=AL*-‘~Nk,
(34)
where k& + ,, is a N-vector whose elements are the diagonal elements of the matrix k+ + ,). Obviously the diagonal matrix k, is a function of the vector e k,=#W)
(35)
according to which the diagonal elements of k, are the elements of kr . Multiplying from the left both sides
where Ni(k,*) is the developed axial force in the ith connecting rod during the loading of the equivalent plane frame, and k, is the axial stiffness of the jth connecting rod. To obtain a convenient form of the partial derivatives ~Ni~~)/~k~, consider Fig. 11 where the equivalent plane frame of Fig. lo(a) is shown, including N - 2 connecting rods, since the ith and jth connecting rods are subtracted. Denote by K,,, m, n = i,j, the axial forces developed at the ends of the position of the mth connecting rod due to a unit shortening of the distance between the ends of the nth connecting rod of Fig. 10(a). Obviously kwm=km. Also denote by kci, k, the axial stiffnesses of the ith and jth connecting rods, respectively. Suppose that ALi and ALj are the shortenings in length of the ith and jth connecting rods respectively,
Lateral analysis of building frames
952
t
h t h2
t h3
+ h4
t hs 4
tIq+
1*+4+I*+L3+~3+L4+
Fig. 11. Form of an equivalent plane frame to obtain relations between any two connecting rods, by means of a condensed form of the ~~ilibrium equations.
due to the loading of the equivalent plane frame of Fig. 10(a) and E; and 4. are the required axial forces applied at the ends of the ith and ith connecting rods, such that during the foading of the equivalent plane frame of Fig. IO(a), the lengths of the above connecting rods remain unaltered. Then the following relation holds
On the other hand, since AL: in (42) is unknown, based on eqns (32) and (28) one can choose the smallest lAL,l in the set D, denoted by lAL,l,,,, when the condition (42) obtains the more conservative form
lalv,tk,*lak,I< IA-%,,,/N, i,j = 1,2,.
, N, (46)
where lALil,i,, is given by eqn (28). DISCWSION
which obviously is a condensed form of eqn (27). Solving (43) for ALi, AL,, one obtains AL<= [Fi(k, + JGj)- Ir;k~~l/[(&,,+ k,i) x
fkjj
+
k,
I-
k$l, VW
Al;i= [4(&i + 4i) - Fik~ll[(kii+ ki) x @,,+ &,I - kid. (Mb) Along with the above changes in fength of the ith and jth connecting rods, the following axial forces are developed in them N, = kci AL,,
(4W
Nj = k, AL, .
(45b)
Differentiating (4Sa) and (45b) with respect to k,: and expressions for the partial derivatives $%,dk ) (aN.jak ) (aN-/ak ) and (aN,/ak,) are obtainef;,‘to bt: us% in ihe &equalities (42).
In the present paper the replacement of the solution of a building space frame by the solution of two plane frames is proposed. Each of the plane frames contains the same number of joints as the building space frame, which results in a considerable saving in computer memory size. Of course it is a disadvantage that the plane frames have to be solved several times, since an iteration process is followed. The evaluation of the column shear stiffnesses D,,,D, from the solution of the equivalent plane frames, contains the approximation that the column torsional stiffnesses have not been taken into account. This can be considered as a negiigible simplification, given that the contribution of the column torsional stiffnesses is small compared with that of the column shear stiffnesses. In the special case where in all the floor slabs the resultant horizontal forces pass through the C.E.R.s, the displacements of the storeys are pure translations. In this case the axial stiffnesses of all the connecting rods assume infinite values and for the evaluation of the forces only one solution of the equivalent plane frames suffices.
C. J. YOUNISand D. E. PANAYOTOUNAKOS
Only one solution of the equivalent plane frames is also sufficient for one-storey space frames without member loads since in this case right after the first solution of the equivalent plane frames, the exact values of the column shear stiffnesses DikX, D, result and hence the straightforwardly be determined forces can from the equations of the Theory of Elastic Complexes. In the cases of multistorey buildings and the existence of member loads the Dlkx and D, vary from iteration to iteration and are stabilized when the axial stiffnesses of the connecting rods tend to their limiting values. The equations governing the iterative procedure comprise a nonlinear system which can assume the form of an equivalent Fixed Point Problem associated with the iteration process under consideration. Under this approach the discussion for the conditions of existence of solution and convergence of the iterative procedure can be based on a theorem relevant with Fixed Point iteration processes.
953
REFERENCES 1. P. Lustgarden, Iterative method in frame analysis. 1. Struct. Div., ASCE 89, 75-94 (1963). 2. R. W. Clough, I. P. King and E. L. Wilson, Structural analysis of multistorey buildings. J. Struct. Div., ASCE 90, (ST3), 19-34 (1964). 3. W. J. Weaver and M. F. Nelson, Three-dimensional analysis of tier buildings. J. Struct. Div., ASCE 92, (ST6), 385-404
(1966).
4. V. Cervenka and K. H. Gerstle, Approximate lateral analysis of building frames. Report to AC1 Committee 422 (1969). 5. J. Gluck, Lateral-load analysis of asymmetric multistorey structures. Proc. ASCE, ST2 (Feb. 1970). 6. AC1 Committee 442, Response of buildings to lateral forces. ACI J. (Feb., 1971). 7. C. L. Kan and A. K. Chopra, Simple model for earthquake response studies of torsionally coupled buildings. J. Engng Mech. Div., ASCE 107, (EM5), 935-951 (1981). 8. A. Rousopoulos, Theory of Elastic Complexes. Elsevier, Amsterdam, London, New York (1965). 9. R. K. Livesley, Matrix Methods of Structural Analysis, p. 9. Pergamon Press, Oxford (1969). 10. k. L. Burden, J. D. Faires and A. C. Reynolds, Numerical Analysis, 2nd Edn, pp. 26 and 445. Prindle, Weber and Schmidt (1981).
Pergamon
0045-7!M9(!M)EOO29-2
ITERATIVE
Vol.
52, No. 5, pp. 939-9%.
1994
Copyright Q 1994 Elsevier Science Ltd Printed in Great Britain. All rightsreserved 0045-7949/94 s7.00+ o.cm
LATERAL ANALYSIS OF ECCENTRICALLY LOADED BUILDING FRAMES C. J. You&
JDepartment
& Structures
and D. E.
PANAYOTOUNAKOS$
tTechnica1 University of Crete, Hania, Crete, Greece, of Engineering Science, Section of Mechanics, National Technical University of Athens, Zographou, GR-157 73, Athens, Greece (Received 17 February 1993)
Abstract-An iterative procedure is proposed, to simplify the stress analysis of rectangular building space frames, under horizontal action. The procedure involves the solution of two associated equivalent plane frames, along with the use of equations from Rousopoulos, Theory of Elastic Complexes (Elsevier (1965)). The equivalent plane frames contain extensible connecting rods, with axial stiffnesses to be determined, which connect the individual plane subframes that form the given building space frame. The iterative procedure comprises evaluation of the column shear stiffnesses, by solving the equivalent plane frames. The column shear stiffnesses with the equations of the Theory of Elustic Complexes, furnish values of the axial deformations of the connecting rods, which in their turn indicate new values of the axial stiffnesses of the connecting rods. With these new values of the axial stiffnesses, the equivalent plane frames are solved in the process of the next iteration to furnish new values of the column shear stiffnesses, and so on. The iterative procedure converges to certain limits in the stresses of the equivalent plane frames, which limits are equal to the stresses developed in the given building space frame, under the given loading. The convergence of the iterative procedure is discussed.
NOTATION
4 A, C.E.R. 4,~ &, 4: E
Ekx1Eky
Ekz F
&k*l 4 %,y
ks
k 1x91, Mlk Mk N N
Qkx9Q, &k I(Ok U x: Xik
Y: Yik AL
areas of the connecting rods centre of elastic rotation column shear stiffnesses column torsional stiffness modulus of elasticity overall storey shear stiffnesses overall storey rotational stiffness axial force defined in Fig. 5(a) vector of axial forces defined in Fig. 10(b) N-vector defined in eqn (39) components of the horizontal loads applied on the floor slab j the 1 x N unit vector axial stiffnesses of the connecting rods stiffness of the frame structure matrix of the axial stiffnesses of the connecting rods matrix of the stiffnesses of the frame structure matrix defined in eqn (25) lengths of the connecting rods column torsional moment moment of the floorslab resultant force with respect to the C.E.R. axial force of the connecting rod vector of the axial forces of the connecting rods column shear forces column end x-displacement C.E.R. x-displacement x-displacement of the point (x, y) x-coordinate of the C.E.R. column x-coordinate y-coordinate of the C.E.R. column y-coordinate vector of the length changes in the connecting rods
A*L Au “ik uOk jk
diagonal matrix of the length changes in the connecting rods axial deformation of the connecting rod column end y-displacement C.E.R. y-displacement y-displacement of the point (x, y) upper and lower floor slab relative rotation matrix function defined in eqn (36)
INTRODUCTION
The bearing structure of a multistorey building can be approximated with a satisfactory accuracy by a space frame the members of which represent the beams and the columns of the building. It is assumed that no shear deformation is developed in the planes of the floor slabs of the different stories; in other words the floor slabs behave as rigid plane diaphragms. The analysis of a space frame requires intensive computational use, large memory size along with frequent memory operations, which often are not feasible. The analysts usually take refuge in simplifying but accurate enough assumptions which help them to espace the difficulties of the space frame analysis. There are many such methods which are more or less effective for the approximate analysis of building bearing structures under horizontal actions. In [l-7] some of the best known among them are mentioned. In (81a theory is developed for the estimation of the relative displacement of the upper floor slab of any storey of a multistorey building with respect to the 939
940
Lateral analysis of building frames
lower one, based upon the shear and torsional stiffnesses of the columns between the floor slabs. It is assumed that all the floor slabs behave as rigid plane diaphragms. The multistorey building space frames treated in the present paper, will be assumed of rectangular plan containing columns with principal directions of shear stiffness [8] parallel to the sides of the plan of the building. Hence in every storey, the principal axes of the elastic rotation [8] will be parallel to the sides of the plan. Also it will be assumed that in every building frame, an orthogonal coordinate system Oxyz with horizontal axes Ox and Oy is attached, parallel to the sides of the building plan. According to [8] to the upper floor slab of any storey, e.g., the kth storey, has a corresponding characteristic point, which plays the main role in the above mentioned relative displacement of the floor slabs; it is the Centre of Elastic Rotation (C.E.R.), defined as the centre of mass of the column shear stiffnesses of the kth storey, whose coordinates therefore are:
xi’ =
(i,xlk&)/($ %)
Y: =
(i,YtkDJk.y)/(i, Dike)-(lb)
(14
In eqns (la) and (lb) xik and y,, are the coordinates of the ith column and D,, and D,, are the shear stiffnesses of the column i, defined by
(2b) In eqns (2a) and (2b) Q,kX, Q+ are the x and y components of the shear force in column i, and u,~, u& are the x and y components of the developed horizontal relative displacement of the upper end of column i, with respect to its lower end. The overall storey shear stiffnesses of the k th storey including n columns, are defined by
the torsional stiffness of the ith column defined by
In eqn (5) Mik is the torsional moment developed in the ith column and & is the relative rotation of the upper floor slab of the kth storey with respect to its lower slabs. In the following discussion the quantity Z;=, D,,: will be considered small compared to the rest of the right hand side of eqn (4), so that this equation from now on will assume the form
E/z= i [(xi/ci=l
x:)‘D,, + (Y, - Y: )'&I.
(6)
The relative displacement of the upper floor slabs of the k th storey with respect to the lower one, is caused by the horizontal loads applied on all the Boor slabs j (j = 1,2, . , k) lying above the mentioned kth storey. Denote by H,, H,,j = 1,2,. . , k, the components of the horizontal loads, and by x,,,, y),,, j = 1,2,. . , k, the coordinates of their points of application. Then the components of the resultant horizontal force acting on the upper floor slab of the k th storey are Z;=, H, and X,“;= , H,y and the moment of this resultant force with respect to the C.E.R. of the floor slab under consideration is
Mk= i [H,~(X,-X:)-H,,~(Y,-Y:)I.
(7)
i=l
According to the above introduced quantities, the components of the relative displacement of the C.E.R. of the upper floor slab of the kth storey with respect to the lower one, are given by
vOk =
/;,
H,?lEk? )
WI
and their relative rotation by 4k
W) and the overall storey rotational stiffness of the kth storey with respect to the C.E.R. of its upper floor slab, is defined by -& = i
Dikz+ i
,=I
,=I
[tx,k -
x:)2D,k, + (yjk - y$)‘Dik,]
(4)
where x: and y: are given by (la), (lb) and Dik2is
=
Mkl’%;r
(9)
as well. Based on eqns (8a), (8b) and (9) the relative displacements (u, u) of any point (x. y) of the upper floorslab can be evaluated as follows: u = UOk-
+k(Y
u=
&(X -x:).
UOk +
-Y:)t
WW
(lob)
Equations (lOa) and (lob) indicate that points lying on lines y = const.,
or x = const.,
(lla)
C. J. YOUNISand
D. E. PANAYOTOUNAK~S
exhibit identical relative displacements u = const.,
or u = const.,
(lib)
respectively. This remark forms the basis of the idea of the two ‘associated equivalent plane frames’ corresponding to the directions x-x and y-y. Specifically, the associated equivalent plane frame corresponding to x-x is composed of all Ox plane subframes parallel to the axis, which form the given space frame, one next to the other and connected at the levels of the horizontal beams. The beams correspond to the levels of the different floor slabs, by connecting rods of area A and moment of inertia J = 0. Accordingly, the associated equivalent plane frame corresponding to y-y, is composed of all the plane subframes parallel to the axis Oy, comprising the given space frame. The zero moment of inertia (J = 0) of the connecting rods, prevents the transfer of bending moment between adjacent subframes. The two associated equivalent plane frames x-x and y-y, are loaded by the horizontal loads H,x, H, respectively,j=l,2,... , N, applied at the levels i of the horizontal beams, which correspond to the different floor slabs j, j = 1, , N. The horizontal beams in the equivalent plane frames are considered axially incompressible, since the floor slabs which they correspond to, were supposed to behave as rigid plane diaphragms. As an example, in Fig. 1 a rectangular space frame is depicted, loaded by the horizontal loads H,,, H,,, Hzx and Hz,,. In Fig. l(a) the associated equivalent plane frame x-x is shown, loaded by the horizontal loads H,, and H,,, while in
CAS w--G
941
Fig. l(b) the associated equivalent plane frame y-y is shown loaded by H,, and Hxy. All the horizontal beams A,B,, B,C,, F,E,, E,D,, F,A,, EiB,, PC,, E,B,, BG,. . . , EzB2 and D,C2 are incompressible. There is a correspondence between the developed stresses and deformations in the space frame and in the two associated equivalent plane frames. In fact, for certain values, to be determined, of the axial stiffnesses of the connecting rods of the associated equivalent plane frames, the stresses (shear forces and bending moments) and deformations of the equivalent plane frame X-X are equal to the stresses and deformations developed in the corresponding plane subframes, lying in the given space frame and parallel to the direction OX. Similarly, for these certain values of the stiffnesses of the connecting rods, the stresses and deformations in the equivalent plane frame y-y, are equal to the stresses and deformations of the 0 Y-parallel plane subframes forming the given space frame. Because of the relative rotations of the floor slabs due to the moments Mk, eqn (7), of the resultant horizontal forces with respect to the C.E.R.s, the different horizontal beams of the plane subframes forming the given space frame exhibit different horizontal displacements, as is shown in eqns (1 la) and (11b). The role of the axially deformable connecting rods, is that inside the two associated equivalent plane frames, they insure the above mentioned different horizontal displacements of the horizontal inextensible beams corresponding to different plane subframes. It is easy for one to see that in the special case where in all the floor slabs the resultant horizontal
942
Lateral analysis of building frames f Connecting rods
(4
l,+L,t
tL,t-L,-
L2-
Fig. l(a). The associated equivalent plane frame x-x.
(b) H
lY
t hl
t h2
I
1 -
L3 4
lyt
J-3 --+-
ly +
L,W
Fig. l(b). The associated equivalent plane frame y-y.
forces pass through the C.E.R.s (Mk = 0; bk = 0, k = 1,. . , N), all the floor slabs exhibit translatory displacements. This means that in the two associated equivalent plane frames, all the connecting rods have to be axially undeformable A = co, since the different horizontal beams lying in a floor slab exhibit identical horizontal displacements. The aim of the present paper is to determine the values of the axial stiffnesses of the connecting rods, so that the stresses and deformations in the two associated equivalent plane frames are equal to those of the corresponding plane subframes lying in the given space frame. The equations which have to be satisfied are those of the Theory of Elastic Complexes [8], in which the shear stiffnesses D,, , D,, eqns (2a) and (2b), will be evaluated by the solution of the two associated equivalent plane frames. An iterative procedure is proposed. Since initially the axial stiffnesses of the connecting rods are unknown, they are given arbitrary values. Following the proposed iterative procedure, it leads to a convergence of the values of the axial stiffnesses, as well as of the stresses and deformations of the two equivalent plane frames, to specific values. These specific values are the required stresses and deformations developed in the members of the given space frame for the given
loading. Additionally, in a later section of the present paper, the conditions for the convergence of the proposed iterative procedure are studied and discussed. EQUATIONS AND ANALYSIS FOR ONESTOREY STRUCTURAL SPACE FRAME. THE ITERATIVE PROCEDURE
In Fig. 2 a simple one-storey space frame is depicted loaded by horizontal loads as well as vertical member loads. In Figs 2(a) and 2(b) the associated equivalent plane frames are shown, loaded accordingly. Initially the axial stiffnesses k, = E-&IL
(12a)
k, = EAJ!,.,
WW
of the connecting rods BD and AC are unknown. In eqns (12a) and (12b) E is the modulus of elasticity, A,, A, denote the area and l,,, I, the length of each connecting rod. Those values of k, and k, are required for which the stresses and deformations developed in the associated equivalent plane frames of Figs 2(a) and 2(b), are equal to the stresses and
C.
deformations developed in the given space frame at the different corresponding subframes, lying in it. The stresses and deformations of the space frame satisfy the following equations of the Theory of Elastic Complexes [8]:
Y ,,,.+" 12t
t
.9 1. *
i=l
Mk
=
i
[txik
j=l
-
X/F
j2Diky
+
lo-’
71 I, -0.3xX
1r’
II - 0.3%
lo-’
1; = 0.266x lo-’ 1 zxi
L /5-W
X
<
0
Fig. 2. Example of a one-storey building space frame. (YIJC -
Yf)2DikxI,(14C)
131
15t
(4 -Xk*)-Hjx(Y~-Y:)l,
[f&(X,
\
B\/Uk”
1 -- 1.067x lo-’
;r Q\
(1W
i=l
51
3x lo-’
ok
Y
Eky= i Diky,
I:‘%
~‘4715X
*
LL 2 (144
lk
4; -
Ii 1
i= I
l4t
l
D lt
I, - 1.23x lo-’ I, - 1.25x lo-’
Ekx= i Dikrr
c
-3
4.om
Ekz= i
943
J. YOUNLTand D. E. PANAYOTOUNAKOS
A
(15)
I
B
C
st-
t uOk
=
;, (
Ekxr
H, >i
Eky,
(16b)
+-(164
5.0 --+l.O~
(1%
(17b)
already referred to in the Introduction. In the above equations, the column shear stiffness Dikx, D, are coupled in a complex manner with the unknown stresses and deformations, and for their evaluation the analysis of the whole space frame is more or less required. Instead of that however, one can obtain the values of the column shear stiffnesses, from the solution of the two associated equivalent plane frames, where at present the axial stiffnesses of the connecting rods are still undetermined. To overcome this deficiency, initially the axial stiffnesses are given arbitrary values, and the iterative procedure described below is followed. Step 1
The solution of the two associated equivalent plane frames yields the values of the column shear stiffnesses D,, D,, as well as the axial forces N, in the connecting rods.
5.0 p-t
Fig. 2(a). The associated equivalent plane frame x-x. 121
141
@I D
A
C
B 71
tiGzl4.0 H
Ii mrmr
t-3.5
+1.ot_3.5+
L
G
P 7?V
Fig. 2(b). The associated equivalent plane frame y-y.
result, which along with the resultant actions ZH,, Z Hjy, Mk, by the use of eqns (16a), (16b) and (161~) furnish the C.E.R. relative displacements Uok,uok, &. Then, from eqns (17a) and (17b) the horizontal displacements of the plane subframes lying in the space frame are evaluated, which coincide with the horizontal displacement of the two equivalent plane frames.
Step 2
Step 3
Substituting these values into eqns (14a), (14b) and (14c) the overall storey stiffnesses E,,, Eky and Ekr
The above evaluated displacements of the horizontal beams of the equivalent plane frames give the axial
944
z 2 ML SL”
Lateral analysis of building frames 200
1.3
160
1.2
120
s
80
1.1 $
40
1.0
0
0
2
4
6
8
10
12
14
0.9
(4
Fig. 3. Variations and convergence of the axial stiffnesses and a shear force, versus the number n of iterations.
deformations
of the connecting rods Au = ui + I,k
-
%.k 9
(18)
which substituted in k = N/Au
(19)
furnish new values for the axial stiffnesses of the connecting rods, to be used in Step 1 of the next
iteration. In eqn (19) N is the developed axial force in the connecting rod, evaluated in Step 1 of the previous iteration. The above iterative procedure is repeated until the stresses (and deformations) of the associated equivalent plane frames converge to certain limiting values. Obviously these limiting stresses (and deformations) satisfying eqns (13), (14) (15) (16) and (17) of the Theory of Elastic Complexes, are equal to the corresponding stresses and deformations of the given space frame. Indeed, for the numerical data given in Figs 2, 2(a) and 2(b), the iterative procedure provides the results shown in Fig. 3, where the values of the axial stiffnesses k, and ky of the connecting rods BD and AC, as well as the shear force QAEXof the column GE, are drawn, versus the number n of iterations. One can see that after five iterations, the values practically converge to certain constant limits. Additionally, in Table 1, a comparison is shown between the resulting limiting stresses of the associated equivalent plane frames and the corresponding stresses evaluated from the direct solution of the given space frame. One can see that the limiting stresses of the equivalent plane frames
Table 1. One-storey frame (Fig. 2). Comparison between the results of the space frame analysis and the equivalent plane frame analyses Moments and shears (1) MAEI
M AE) M EAI M EAY MBF~ MBFF
M.FBT
M,s, M CGX M CGJ M Gcr M GCY M D”.T M DH> M “Dr M HD,
MA6 MBA MBC
&ll &D MDC MDA
::
#:K=g QCBI $::WY
Results from the solution of the space frame of Fig. 2
Results from the solutions of the equivalent plane frames X-X
(2)
(3)
1.26 1.97 2.02 2.50 4.28 4.69 4.66 4.20 3.96 3.45 3.78 2.57 4.32 -0.62 3.78 1.17 1.97 4.69 4.28 3.96 -3.45 0.62 -4.32 -1.26 1.12
1.97 2.50 4.70 4.21 3.47 2.59 -0.59 1.20 1.97 4.70 -3.47 0.60 -1.12
0.82 2.24 2.22 1.51 0.14 2.03 1.93
-2.23 1.52 -0.15
Moments (M) in (t m) and shear forces (Q) in (t).
Y-Y (4) 1.26 2.02 4.27 4.66 3.95 3.78 4.32 3.79
4.27 3.95 -4.32 -1.26
2.23 0.82 2.03 1.93
C. J. YOUNISand D. E. PANAYOTOUNAKOS
Step 3: C,D, , C#$
CsD,,....
K, O/m) Fig. 4. Convergence of the iterative procedure (case with member loads).
constitute a very good approximation to the exact stresses provided by the analysis of the space frame. Figure 4 presents another point of view of the iterative procedure’s convergence. The curve shown in it represents the function N = N(k,), where N is the axial load developed in the connecting rod BD of the composite plane frame x-x, during its deformation caused by the applied loads, and k, is the axial stiffness of the connecting rod. For simplicity, only the equivalent plane frame x-x will be discussed here. According to the theorem of superposition[9], the function N = N(k,) is given by
t
L,+
1,t
L,+
Fig. S(a). Definition of the axial force F.
(b)
t
t
L,+l,t
1,+ Al, -+
945
where F is the axial force applied to the points B and D, required to keep the distance BD (Fig. Sa) constant during the deformation of the equivalent plane frame x-x, caused by the applied loads, and k,,, is the axial stiffness of the distance between the points B and D due exclusively to the structure of the equivalent plane frame x-x, the connecting rod BD being absent (Fig. 5b). The abscissae of the points c,,cz,. . . , C,, on the curve (N, k,) represent successive values of the axial stiffness k,, as they were evaluated by the iterative procedure. The same values are also shown in Fig. 3. The abscissae of the points Di = i = 1, . . . , 13, on the other hand, represent the values of the axial stiffness k, as they were determined from the axial force corresponding to the points Ci, by use of eqn (19) (Step 3). Additionally the ordinates of the points C,, i = 1,. , 14, represent the axial forces developed in the connecting rod whose axial stiffness is the abscissa of the points Di_ , . The axial forces of the points Ci are determined by the solution of the equivalent plane frame x-x (Step 1). One can see that the points Di, i = 1,2,3, . . . , lie close to the line OD,,, the tangent of the angle of which with the k, axis, represents the axial deformation AU,, of the connecting rod BD after the 14th iteration (Step 2). Indeed, the tangents of the angles of the lines ODi, i=l,2,..., with the k,-axis represent the axial deformation Aui of the connecting rod BD, as they are evaluated after each iteration by eqns (13)-(17) of the Theory of Elastic Complexes [8] (Steps 2 and 3). Actually, there is a small variation in the axial deformation Au,, from iteration to iteration, owing to a small variation in the column shear stiffnesses Dikx, Dikgas they are determined from the solution of the equivalent plane frames (Step 1) after each iteration. This variation of the D,, and D,,. in this case of a one-storey space frame, is due to the presence of member loads which contribute to the column shear forces and the column-end relative displacements independently, and additionally to the acting horizontal joint loads. Of course the contribution of the member loads to D,,_, Dlkv is small compared to that of the horizontal joint-loads, so that the variation of D,, and D,.is small. In Figs 6(a) and 6(b) of the associated equivalent plane frames x-x and y-y are shown, corresponding (a)
L,_
Fig. S(b). Definition of the stiffness K,,,.
Fig. 6(a). The associated equivalent plane frame X-X (case without member loads).
946
Lateral analysis of building frames D
A
C
B 7t 2.0
t 4.0 1.5
t-3.5
E mrmr -_t1.0+
G
F 7m
2 2
1.0
3.5 -+
Fig. 6(b). The associated equivalent plane frame y-y (case without member loads).
0.S
tan 6 I AI+ i I l.?.... I
II
to the space frame of Fig. 2 loaded exclusively by the horizontal joint loads H, and H,,, without any member load. Figure 7 presents the variations of the connecting rod axial stiffness k,, resulting from the presented iterative solution applied to the equivalent plane frames of Figs 6(a) and 6(b). The absence of member loads in this case implies constant values of Dikx, D, and hence constant Aui, for all the iterations. As shown in Fig. 7 the points , D, lie on the straight line OD, correD,,Dz,... sponding to the constant value of Aui. In this case, in the absence of member loads, the path D,C,D,C,D,Cj.. . is a zigzag line between the curve N = N(k,) and the line OD,. Also, in Figs 4 and 7 it is easy for one to check that the fast convergence of k, towards its limiting value, during the iterative procedure, is due to the horizontally asymptotic character of the curve N = N(k,) for large k,, as imposed by eqn (20), in combination with the comparatively larger and constant inclination of lines OD,,i=1,2 ,.... Except in the presence of member loads, in the general case of multistorey space frames, another source of the small variation in the values of Dikx, D,, from iteration to iteration, is the inducement of additional column shear force and column-end
Fig. 7. Convergence of the iterative procedure (case without member loads).
relative displacement by the extensions of each column into the adjacent (upper and lower) storeys. ANALYSIS OF A MULTISTOREY SPACE FRAME
Figure 8 shows an example of a two-storey space frame loaded by horizontal loads as well as by vertical member loads, while Figs 8(a) and 8(b) show the two corresponding associated equivalent plane frames, loaded accordingly. After giving initially arbitrary values to the axial stiffnesses of the connecting rods BD, FH, AC and EG, the iterative procedure described above is followed, which after five iterations practically converges to the values discussed below. In Fig. 9 the successive values of the axial stiffnesses of the above four connecting rods are shown along with the successive values of the bending moments MEc and MEH at the point E of the bars EI and EH respectively, as they arise after each iteration,
IA& - 1.067x n+ = 1.267x ‘A% I& - 1.333x = 0.521x I% IBF -0.326x
STRUCTURAL
lo-’ w3 10-3 lo-’
IRF* = 0.326x 10-3 I,,,: - 0.563x 1O-3 = 0.391x 10-S h Ico, - 0.233x lo-’ =0.715x 10-3 ‘=J, IOK, = 0.267x lo-’ IOK, = 1.067x10-3 IDa. = 1.251x 10-3 = 1.251x 10-3 ‘Da, 1% - 1.251x w3 = 1.2Slx 10-g ‘m,
Fig. 8. Example of a two-storey building space frame.
C. J. YOUNIS and D. E. PANAYOTQUNAK~~
A
B
t
C
D
Y?
16t
3.0
t--_
947
5.0 -+1.0-t
5.0 -+
Fig. 8(a). The associated equivalent plane frame x-x. 17t D
A
1
C .
t3.0
B st 1st 51
H
E”
“G
F
t 4.0
-_tl.o~-+--3.5+
t3.s
Fig. 8(b). The associated equivalent plane frame y-y. versus the number n of iterations. Obviously five iterations are sufficient to provide practically constant values for the required developed stresses and deformations in the given space frame. Additionally one can check that the variations in the axial stiffnesses of the connecting rods as they arise after each iteration with respect to their limiting values are relatively small. This can be expressed by the inequality 0.85
< 1.15
(21)
where kli,,,is the limiting axial stiffness of a connecting
tat?
160
a
52 120 a0
46
rod, and k is the value of its axial stiffness after four iterations. As mentioned in the previous section, the reasons for the variability in the connecting rod axial stiffnesses are the presence of the member loads as well as the inducement of additional shear force by the extensions of the columns into the adjacent storeys. It should be mentioned here that, as is reasonable, the column shear forces arising from the above two sources are small compared to those arising from the horizontal loads. Table 2 shows a comparison between the developed stresses at different points of the structure of Fig. 8 as they were evaluated by the iterative procedure and the corresponding stresses as they arise from an exact analysis of the given space frame. One can check the very good agreement between the results of the two methods of analysis. CONVERGENCE
5
40
2
bL
0
0 -2
Fig. 9. Variations and convergence of the axial stiffnesses and moments versus the member n of iterations.
Figure 4 and 7 show the successive values of the axial stiffness of the connecting rod, as they arise from the application of the iterative procedure on the structures described by Fig. 2 as well as Figs 6(a) and 6(b), respectively. When dealing with a one-storey structural space frame it was mentioned that in this case without member loads the values of the shear stiffnesses DikX,D, of the columns resulting from Step 1 of any iteration are identical. This fact implies that the points D, , D,, D,, . . denoting the
Lateral analysis of building frames
948
Table 2. Two-storey frame (Fig. 8). Comparison between the results of the space frame analysis and the equivalent plane frame analyses Moments and shears (1)
MA&
Results from the solution of the space frame of Fig. 8 (2)
Results from the solutions of the equivalent plane frames x-x (3) 0.97 3.95 2.38 1.18 6.81
Y-l’ (4) -0.91 2.57
-0.92 0.98 2.60 3.96 2.58 2.38 3.78 1.18 3.48 6.83 8.06 5.13 5.07 6.01 2.78 1.28 0.98 2.60 -2.37 -3.78 8.22 10.01 -9.41
5-12 6.01 1.28 0.97 -2.38 8.20 -9.41
Z Per, eQ::
-4.87 0.79 2.35 -0.04 1.92
-0.79 2.34 -0.04 1.93
z:
-4.26 -1.78
-
-4.25 - 1.78
Q GKY Q HLY
-2.49 -1.49
-
-2.48 -1.49
M
AEv
MBF,
A4sr,
M CGr M cc+ M DHr M DHY
MslV ME,, M FJr
Mr,y MGKI M GKY
M HL.Y M HLg
MAB MB,
MC, MDA MEF
Mro
M
GH
2.56 3.79 3.48 8.04 5.06 2.78 2.57 -3.79 9.97 -4.88 -
Moments (M) in (t m) and shear forces (Q) in (t).
t
h, t ha t h3
+ h4
t % -c
Fig. IO(a). Equivalent plane frame of a multistorey building.
C. J. YOUNISand D. E. PA~AYO~~AK~
Fig. 10(b). Definition of the axial forces vector
949
F.
Fig. IO(c). Definition of the elements of the stiffness matrix k,.
successive values of the connecting rod axial stiffness resulting from the successive iterations, all lie on the straight line OD, * . - D,D, , Fig. 7. On the other hand, when loads act on the space frame member (Fig. 2), the column shear stiffnesses D,, D, resulting from Step 1 exhibit small variations from iteration to iteration. As mentioned before, the above small variations are due to the presence of member loads as well as to the influence of the adjacent storeys in the case of multistorey buildings. These variations in the column shear stiffnesses imply that the angles D,OK,, 1,2,. . . , Fig. 4, where the points D, denote the successive values of the connecting rod axial stiffness, exhibit small variations, such that all the points D, lie inside the small crosshatched angle shown in Fig. 4. A measure of the small variations of the angles D,OK, is given by the inequality (21).
For the case of the one-storey space frames without member loads, the relative location of the curve N = N(k,) and the line OD, in Fig 7 makes it obvious that for any initial value of the axial stiffness k,, the convergence of the zigzag line * . +to the point C,, is certain. To this C,D,GD,C,D3 point one has to have in mind the ‘Fixed Point Iteration Theorem’ from Numerical Analysis [IO]. For the same reason, in the cases of multistorey buildings or the existence of member loads, the relative location of the curve N = N(k,) and the small crosshatched angle in Fig. 4 has as a consequence the convergence of the zigzag line C, D,CzDz ’ . * to a limiting point, lying on that portion of the curve N = N(k,), which is included in the small
crosshatched line. Of course in the above latter cases it should be pointed out that as the connecting rod
950
Lateral analysis of building frames
axial stiffness approaches its limiting value, the variations of the column shear stiffnesses DlkX,D,, diminish and the points Dj, for large i, lie on a limiting straight line passing through the origin 0. Obviously the abscissa of the section of this limiting straight line and the curve N = N(k,) is the limiting value of the axial stiffness of the connecting rod. To come to the study of the convergence of the iterative procedure in the general case of multistorey buildings, consider Fig. 10(a) where an equivalent plane frame is depicted including N connecting rods. Figure 10(b) presents the same as in Fig. 10(a) equivalent plane frame, where at the ends of the connecting rods couples of opposite axial forces Fi, -F,, i = 1,2,. . . , N are introduced, such that during the loading of the equivalent plane frame by the applied joint and member loads all the distances ii, i=l,2,..., N, remain unaltered. Denote by F=[F,,F*,..
I
kr,z
...
km
ks21 ...
km . *.
...
km ..:
...
If upon the applied couples of axial forces k;, -F,, Fig. IO(b), equal and opposite couples -F,, Fi are superposed, according to the principle of superposition 193,the connecting rods will obtain the axial deformations that are due to the loading of the equivalent plane frame, and hence the relation &+k,).AL=F
(27)
will hold, from which it follows that
kw
I
N=k;AL
(29a)
which in view of eqn (28) becomes N=k;(k,+k,)-‘.F.
(29b)
From the above formulated equations, Step 3 of the kth iteration of the iterative procedure presented before comprises the following algebraic manipulations. From eqn (29b), written as Nk=kd:.(k,+k,J’.F,
(30)
where kti is the stiffness matrix of the connecting rods, known at the beginning of the k th iteration, and F is the vector of the axial forces defined by (22), the new values Nk of the axial forces N in the connecting rods, arise. Equation (29a) is now written as
Nk=k++,,.AL kc, ...
.*. k cn
:
(31a)
0t
k c2
...
.
(24)
k c(k + k cck +
1
1j.1
ALi
il.2 AL2 . . .
Based upon (23) and (24), obviously the stiffness matrix k, of the positions of the connecting rods in the full equivalent plane frame, including the connecting rods, as shown in Fig. 10(a), is given by k=k,+k,. Denoting
the change
in length
(28)
At the same time during the superposition of the new equal and opposite couples -Fi, Fi, inside the connecting rods, axial forces N are developed which are given by
(23)
If the axial stiffness of the ith connecting rod is denoted by kcir then the stiffness matrix of the connecting rods can be introduced as the diagonal matrix
k, =
(26)
(22)
k sll
k SNI k,,
AL= [AL,,AL,, . . , ,AL,IT.
AL = (k,+ k,)-’ ’ F.
.,FN]T
the vector of the above axial forces; it is obvious that F is constant for any equivalent plane frame, depending exclusively upon its loading and the geometry of that part of its structure shown in Fig. 10(b), and hence it is independent of the axial stiffnesses of the connecting rods. Now denote by ks, the measure of each force of a couple of equal and opposite axial forces, which is developed at the position of the ith connecting rod, when a unit change of length of the distance between the ends A and B at the position of the jth connecting rod takes place, as shown in Fig. 10(c), is introduced as the matrix
k=
between the ends of the connecting rod i, i=l,2,..., N, during the loading of the equivalent plane frame of Fig. 10(a) by A&, by the applied joint and member loads, then the vector AL is defined by
(25) of the distance
kc(k
+
I).NAL
1
(31b)
from the solution of the equations from which the elements of the diagonal stiffness matrix ktk + it of the connecting rods are obtained. The matrix ku + ,) will be used in Step 1 and Step 3 of the next (k + I)th iteration.
951
C. J. Youm and D. E. PANAY~UNAK~
To study the convergence of the elements of the matrix k, evaluated from eqn (31 b), it is convenient to consider first the variations in the vector AL, included in eqn (31a) above, throughout the iterative procedure. It was referred to in Sec. 2 and at the beginning of the present section, that in the cases of multistorey buildings or the existence of member loads, there is a variability in the values of the column shear stiffnesses Dikx and D,, which implies a small variability in the values of the extensions ALi of the connecting rods as they are evaluated at the beginning of Step 3, from the results of the equations of the Theory of Elastic Complexes, obtained in Step 2 of each iteration. Also, in Fig. 4, referring to the case of one-storey buildings with member loads, it is shown that all the points Di, i = 1,2,. . . , lie inside the narrow crosshatched angle, indicating that the angles D,OK,, which represent the extension AL of the connecting rod, evaluated at the beginning of Step 3, exhibit variations that can be considered as very small. Additionally, it was referred to previously in the present section, that as the points Di of Fig. 4 tend to their limiting position, the variability of AL diminishes and the points Di, for large i, lie on a limiting straight line passing through the origin 0. In view of the above, it can be considered that after a few initial iterations the variations in the extensions AL,, i = 1,2, . , N, of the connecting rods become negligible and for the purposes of the present discussion, they may be allowed their limiting values, denoted by the diagonal matrix
A~*=[*)_
AL’
...
,,:i
(32)
from the beginning of the iterative procedure. Using the matrix AL* instead of the vector AL,, eqn (31a) can be written as
of eqn (30) by AL*-‘, it yields AL*-‘.Nk=AL*-‘.%.Ik,f%]-‘.F,
(36)
which in view of eqns (35) and (36), yields k%+,,= AL*-‘.cp(k~).F,+6~)]-‘.F.
(37)
Equation (37) constitutes a simplified and brief notation of the algebraic manipulations carried out in Step 3 of the iterative procedure, which were described by eqns (30) and (31 b) above. According to eqn (37), given k&, a new k&+ ,) arises to be used in the next iteration. Using the right hand side of eqn (37), the following function is introduced ~~)=AL*-‘,#~),~~+~~)]-‘,F,
(38)
where G(k$) is a N-vector, whose elements are functions of the N-vector c. In view of (38), eqn (37) can now be written as k&+,,=G(%),
(39)
which generates an iterative process corresponding to the fixed point problem [lo]. G(k,*)=c.
(40)
A theorem that gives conditions for the iteration process to converge is referred to in [lo], according to which the partial derivatives must satisfy the following inequalities ]aG,(k:)/%c,] c l/N, i, j = 1,2,, . . , N
(41)
for e o D, D c RN. But according to eqns (38), (36) and (29b) the above inequalities (41) assume the form ll/ALFl ]aN,(k:)/dk,]
< l/N,
i, j = 1,2,. . . , N,
(42) Nk=AL*.~~+,,+I
(33)
where I is the 1 x N unit vector. Multiplying both sides of eqn (33) from the left by AL*-‘, it follows that k&+,)=AL*-‘~Nk,
(34)
where k& + ,, is a N-vector whose elements are the diagonal elements of the matrix k+ + ,). Obviously the diagonal matrix k, is a function of the vector e k,=#W)
(35)
according to which the diagonal elements of k, are the elements of kr . Multiplying from the left both sides
where Ni(k,*) is the developed axial force in the ith connecting rod during the loading of the equivalent plane frame, and k, is the axial stiffness of the jth connecting rod. To obtain a convenient form of the partial derivatives ~Ni~~)/~k~, consider Fig. 11 where the equivalent plane frame of Fig. lo(a) is shown, including N - 2 connecting rods, since the ith and jth connecting rods are subtracted. Denote by K,,, m, n = i,j, the axial forces developed at the ends of the position of the mth connecting rod due to a unit shortening of the distance between the ends of the nth connecting rod of Fig. 10(a). Obviously kwm=km. Also denote by kci, k, the axial stiffnesses of the ith and jth connecting rods, respectively. Suppose that ALi and ALj are the shortenings in length of the ith and jth connecting rods respectively,
Lateral analysis of building frames
952
t
h t h2
t h3
+ h4
t hs 4
tIq+
1*+4+I*+L3+~3+L4+
Fig. 11. Form of an equivalent plane frame to obtain relations between any two connecting rods, by means of a condensed form of the ~~ilibrium equations.
due to the loading of the equivalent plane frame of Fig. 10(a) and E; and 4. are the required axial forces applied at the ends of the ith and ith connecting rods, such that during the foading of the equivalent plane frame of Fig. IO(a), the lengths of the above connecting rods remain unaltered. Then the following relation holds
On the other hand, since AL: in (42) is unknown, based on eqns (32) and (28) one can choose the smallest lAL,l in the set D, denoted by lAL,l,,,, when the condition (42) obtains the more conservative form
lalv,tk,*lak,I< IA-%,,,/N, i,j = 1,2,.
, N, (46)
where lALil,i,, is given by eqn (28). DISCWSION
which obviously is a condensed form of eqn (27). Solving (43) for ALi, AL,, one obtains AL<= [Fi(k, + JGj)- Ir;k~~l/[(&,,+ k,i) x
fkjj
+
k,
I-
k$l, VW
Al;i= [4(&i + 4i) - Fik~ll[(kii+ ki) x @,,+ &,I - kid. (Mb) Along with the above changes in fength of the ith and jth connecting rods, the following axial forces are developed in them N, = kci AL,,
(4W
Nj = k, AL, .
(45b)
Differentiating (4Sa) and (45b) with respect to k,: and expressions for the partial derivatives $%,dk ) (aN.jak ) (aN-/ak ) and (aN,/ak,) are obtainef;,‘to bt: us% in ihe &equalities (42).
In the present paper the replacement of the solution of a building space frame by the solution of two plane frames is proposed. Each of the plane frames contains the same number of joints as the building space frame, which results in a considerable saving in computer memory size. Of course it is a disadvantage that the plane frames have to be solved several times, since an iteration process is followed. The evaluation of the column shear stiffnesses D,,,D, from the solution of the equivalent plane frames, contains the approximation that the column torsional stiffnesses have not been taken into account. This can be considered as a negiigible simplification, given that the contribution of the column torsional stiffnesses is small compared with that of the column shear stiffnesses. In the special case where in all the floor slabs the resultant horizontal forces pass through the C.E.R.s, the displacements of the storeys are pure translations. In this case the axial stiffnesses of all the connecting rods assume infinite values and for the evaluation of the forces only one solution of the equivalent plane frames suffices.
C. J. YOUNISand D. E. PANAYOTOUNAKOS
Only one solution of the equivalent plane frames is also sufficient for one-storey space frames without member loads since in this case right after the first solution of the equivalent plane frames, the exact values of the column shear stiffnesses DikX, D, result and hence the straightforwardly be determined forces can from the equations of the Theory of Elastic Complexes. In the cases of multistorey buildings and the existence of member loads the Dlkx and D, vary from iteration to iteration and are stabilized when the axial stiffnesses of the connecting rods tend to their limiting values. The equations governing the iterative procedure comprise a nonlinear system which can assume the form of an equivalent Fixed Point Problem associated with the iteration process under consideration. Under this approach the discussion for the conditions of existence of solution and convergence of the iterative procedure can be based on a theorem relevant with Fixed Point iteration processes.
953
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(1966).
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