- Email: [email protected]
Numerical solution of tube buildings modeled as continuum
Computers
& Strucrures
Vol.21. No. 4.
Printed in GreatBritain.
pp. 771-776.
1985
0045-794985 $3.00 + .oo C 1985 Pergamon Press Ltd.
NUMERICAL SOLUTION OF TUBE BUILDINGS MODELED AS CONTINUUM Department
of Civil Engineering,
PETER CHANG University of Maryland, College Park, MD 20742, U.S.A.
and DOUGLAS FOUTCH
Department
of Civil Engineering,
University of Illinois, Champaign-Urbana,
IL 61801, U.S.A.
(Received 8 June 1984) Abstract-The tube building is represented by a set of six differential equations which represent the flexural and shear deformation, and the shear-lag effect of the structure. The model is divided into segments of constant properties to represent changes in member sizes along the height of the building and to reduce the numerical errors accumulated by the approximate method. A detail explanation of the fourth-order numerical technique used to solve for the differential equations is included so that readers may easily construct a computer program to study the behavior of tube structures. Also included are examples of tube buildings solved by the numerical technique compared to results of finite element method. The comparisons show that the numerical solution of the continuous models yield excellent results while requiring less than one percent of the computational time and input/output efforts associated with the finite element method. 1.
INTRODUCTION
The design of a structure can be loosely grouped into three stages: (11 the planning stage, (2) the preliminary design stage and (3) the final design stage. In the planning stage, the nonstructural aspect of the structure such as aesthetics, the usage of the structure and dimensions are decided. The structural engineer usually has little or no input at this stage of the design. Once the structure is conceived, the structural engineer is in charge of providing a structural system to support the structure under the predicted loads. The structural engineer, in order to provide the most adequate and economical supporting structure, usually needs to consider several structural systems. This is done in the preliminary stage of the design. At this design stage the structural engineer considers most of the parameters that would affect the global behavior of the building. Because local effects are not a major concern at this stage of the design, a rigorous analysis is not necessary, therefore approximate methods giving the global behavior of the structure are often used. Once the structural frame is selected the structural engineer can then use a more rigorous approach which gives the local forces and stresses needed to size the members. A method for the approximate analysis of tube frames was recently developed [l, 23. This structural system was chosen for the present study because many medium to high rise buildings utilize this type of framing due to its great efficiency at resisting lateral loads. The analysis procedure was shown to be extremely efficient when compared to more conventional approaches. Thus, many different options may be considered in the preliminary design stage at relatively little design cost to the engineer. The approach adopted for this study converts the
properties of the tube frame to those of an equivalent continuous thin-walled cantilever tube. The behavior of this equivalent tube therefore represents the global behavior of the building. The complexity of the tube’s governing differential equamakes a closed form solution tion, however, nonfeasible. The purpose of this paper is to describe in detail a simple numerical technique tailored to solve the governing differential equations resulting from the modeling technique. Because of its simplicity. the method described can be implemented on most micro or personal computers, thus making the preliminary design process much faster and inexpensive. Although some aspects of this solution scheme are not new, the combination of this scheme with the concept of modeling a discrete system as a continuum provides the engineer with a powerful, but inexpensive, package that has not been suggested previously. As mentioned above, only the solution technique will be presented in this paper. The procedure for making the conversion from the discrete system to the continuum has been described extensively elsewhere [l, 21. In fact, it has also been shown that the same differential equations, in a degenerated form, can be used quite effectively to model a regular or braced frame [ 11. Thus, the approach is quite versatile and not restricted to tube frames. The solution technique is also applicable to a broad range of differential equations and, therefore, should be useful to engineers under many circumstances.
2. GOVERNINGDIFFERENTIALEQUATION
The structures being considered are tube buildings. These buildings derived their name from the 771
712
P.
CHANG and D. FOUTCH
fact that they resist lateral loads in the same manner as a hollow thin-walled cantilever tube. The frames parallel to the direction of the load act as the web of the tube and the frames transverse to the loading direction act as flanges. This differs from the more conventional system where only the frames in the direction of loading resist the forces. Thus, approximate methods employed in the preliminary design stage of conventional frames are not applicable to tube buildings. The continuous model used is based on Reissner’s model [3] which assumes a parabolic variation of longitudinal flange displacement to account for the shear-lag effect, which is present in the flanges of the tube frame. Reissner’s model was modified to include the effects of shear deformation which is also significant for tube buildings. The behavior of the building, therefore, can be described by the set of differential equations:
lf the building is fixed at the ground level, then the boundary conditions of eqns (2) are y6(0)
=
{EZ[w'(O)p'(O)] + fEZfU'(0)}' =0
ys(0)
=
[EZ(w"(O) - p'(O), + $EZfU'(0)] =0
y4(0) = U'(0)= 0
yz(L)= y,(L)
U(L)=0 = w(L)
= 0.
Some rather interesting aspects of the behavior of thin-walled cantilever tubes as indicated by the solution of these equations have been described previously [4].
3. SOLUTION
+ 5 EZfU']"= P(s), [EZ(w"- f3')
(3)
OF DIFFERENTIAL
EQUATIONS
The solution of the problem requires the integration of the set of first-ordered differential equations [eqns (2)] over a given range of independent vari- P')]' - ;% u = 0, (1) ables, x. The solution of these equations results in EZ,[(&U' + 5 (MI" determination of values y, , y2, . . . , y6 corresponding to tabulated values of x. In the numerical where method used, the values of yi at each step Xj are based on the results of yi from the previous step. E = modulus of elasticity of the material that Hence, at some point x, all the corresponding valmakes up the tube ues Of vi: y,(O), y*(O), . . . , y&O) must be known. G(x) = shear modulus This type of problem is referred to as a forwardZ(x) = moment of inertia of the cross-section integration initial-value problem. For the tube If(x) = moment of inertia of the flanges alone building three of the boundary conditions [eqns (3)l P(x) = lateral load are given at the end x = L. Hence the forwardU(x) = shear-lag stress function integration initial-value problem cannot be applied w(x) = lateral deflection to eqns (2) directly. A technique used which nl’ - B = rotation of the section due to shear “guesses” the initial values of the variables not W = flange width of the tube structure given there and corrects them until the variables K = shear correction factor. match with the given boundary conditions can overcome the problem of not having all the boundary Because of the complexity of eqn (1). a closed conditions given at one point. This technique is form solution was not attempted. Rather, the socalled the shooting method. When the system of lution is obtained through numerical approximadifferential equations is linear, these initial tion. The application of the numerical method used “guesses” may be calculated in terms of the boundrequires that the coupled sixth-ordered differential ary values given elsewhere, making the shooting equation [eqn (l)] be transformed into six first-ormethod simple and efficient. dered differential equations. These equations are The simplest form of initial-value forward integiven in eqns (2) (see Ref. [2]). gration is the Euler’s method. In Euler’s method, the increment in y corresponding to the increment y; = [EZ(w" - fi')+$EZ,U']"= -p’.AG~=f’(x), AX, for the differential equation dy/dx = f(x, y), is y; = [EZ(w" - f3') + $EZfU']' =y6, approximated by yi + Ax (y:). A term-by-term comparison of this result to the Taylor-series ex5 1 5G -_ pansion indicates that the Euler’s expression is only y; = 2 EWZY’ - izy” accurate to the order hx(y:). This accuracy is in’ 1 - 5Z5/6Z dicated symbolically by o(Ax’y’i’), which means that the error is to the order Ax’~‘/. Many modi1 y$=(w’P)’ = FzY5 - $Y4, fications to Euler’s approximation giving more accurate results are available. The refinement of inyi = U’ = y4, tegration selected depends on the accuracy needed. The solution to the governing thin-walled tube y; = w =y3 -AGK.Y~. (2) (AGKB)’ = -P(x).
773
Numerical solution of tube buildings modeled as continuum
equation [eqns (1) or (2)] employs a fourth-order Runge-Kutta approximation, which has error to the order o(h’y”“‘). The simplest algorithm with this degree of accuracy is Kutta-Simpson’s rule. Here, the new values of yi are approximated using the slope at the previous point, the slope at the new point and the slope at the mid-point between the two. yi(X + AX) = yi(X) + Q(JJi + 2qi + 214; + Vi),
(4)
where Pi =
hr*fi[~~Yl(~)~Y2(X)9~
4i=k'f;
[
ui=hx’fi
[
Ui=AX*fj
[
. .
,y6(X)l.
1
X+$,yl(X)+$,. . .,1.6(X)+?, X+$,yl(X)+T,. . .,y6(X)+t1 , X+&3Yl(X)+Ui,. 2
where fi are the functions example
.
.,.Y6(X)+ui
9
1
(5)
defined in eqns (2). For
Once all the values of yi corresponding to the value of x are known, the application of eqn (4) is straightforward. However, the boundary conditions for a building are usually not all given at a single point; they are partially given at each of the two extremities of the structure, i.e. the ends x = 0 and x = L. This type of boundary condition constitutes a two-point boundary-value problem. To get the unknown values of yi at the point x0, trial values of yi(O) are used. If the values of yi(O) are guessed correctly, the yi at the other end of the structure would match the given boundary conditions exactly. The process of choosing a set of yi(O) so that the computed yi match the given boundary conditions is called the “shooting technique.” Since eqns (2) is linear, all the values Of yi at a given point Xi are linear combinations of the yi at another point of the beam, say x = 0. Hence we can write, for x = L, Yl(L)
=
+
al2Y2(0)
+
. .
+
b,,
Yz(L) = a2lYlm
+
a22y,(O)
+
. . .
+
b2
Y6(L)
+
a6ZY2(0)
+
. . .
+
b6,
=
UllYl@)
a6lY,@)
T
(6)
where y,(O) represents the boundary conditions at one end and yi(L) represents the boundary conditions at the other end. The uij and bj are constant coefficients that can be determined in the following manner.
CM 21:4-K
(1) Select y,(O) = y?(O) = . . . = 0 and integrate the set of first-order differential equations [eqns (2)] from 0 to L. Call the resulting values yi(L,): rl , r2, Then b, = rl, bz = r2, . . , . * ‘(;; Select y,(O) = 1, y*(O) = I, y2(0) = . . . = 0, and integrate eqns (2) from 0 to L. Call the calculated values yi(L): si, s2, . . . . Then a,, = s, b,, u7, = s2 - bz, . . . , (3) Select yz(O) = 1, y,(O) = yj(O) = . . . = 0, and integrate eqns (2) from 0 to L. Call the calculated values yi(L): tl, tz. . . . . Then al2 = tl b,, az2 = tz - b2, . . (4) Repeating steps ; gnd changing the selected boundary condition by setting one variable equal to 1 progressively while all the others are set to zero. (5) Solve eqns (6) for y;(O). Once all y;(O) are found, the solution of eqns (2) can be found directly through eqn (4). In real buildings the column and girder sizes change throughout the structure. These changes translate into abrupt changes in property of the continuous tube model making it a nonprismatic member. It is therefore convenient to analyze sections of constant (or continuously varying) properties separately and then enforce the compatibility of displacement between sections. This sectioning approach has another advantage; it reduces the accumulation of error which results from approximate numerical integration, also known as the length effect. The accumulation of error due to numerical approximation, then, is limited to the order of the error accumulated for the longest section. Thus, subdivision of the building into sections can serve as a control of error accumulation as well as modeling the change of column size. To find the values
L Prototyp
Equl”.,mr
ad*,
Fig. 1. Prototype and equivalent model.
774
P.CHANG
and
of dependent variables vi at the junction of segments and the two extremities, let us consider the jth segment. If the set of differential equations governing thejth segment is linear, the boundary conditions at one end of the segment must be a linear combination of the boundary conditions at the other end. Therefore eqn (5) is valid for each segment. However, the ends x = 0 and x = L now refer to the ends of the segment rather than the extremities of the building. Now, if the boundary conditions of the “L” end of the jth segment are forced to be compatible to the “0” end of the j + 1 segment, then the linear conversion equations of all the segments may be written related to each other. The resulting equation involves all the variables vi at all the junctions between segments of an n-segment building model. These equations can be summarized into the matrix representation as eqn (7).
D. FOLJTCH tions, it is in fact a reduction of the conventional finite element approach by several orders of magnitude. For a uniform tube, only six equations must be solved. The engineer may decide how “finely” to section the tube depending on the degree of accuracy required. Furthermore, the solution of eqn (7) requires a minimal amount of computation because of the small bandwidth associated with this formulation. 4.NUMERICALEXAMPLES
To demonstrate the accuracy of the method, the structure shown in Fig. 1 was modeled as a continuous tube. The building was also analyzed using the finite element method. The finite element model is shown in Fig. 2. The structure in the example has all columns and girders with area of cross section. A, equal to 29.1
0 0 0
‘Yl
'Y2 'Y3 'Yd 'Ys 'Y6 2YI *?,2 2Y3
_ ‘b, - ‘b2 _ ‘b3 - ‘h - ‘bs - ‘bs -*b, - 'b2 - *b3 - *b4
*Y4 2 u212a222u232u242u252u26 2 a3~2u322u332u342a352u36 2 u412a422u.432u442u452u‘k5 2 u512u~22u532u542a552u~~ 2 u612a622u632uc42u652u.56
2YS 2Y6 3Yl
0
- "b5 - "be
“fly2
0 0 0
“+‘Y3 “+‘Y4 “+‘Y5_
The first three variables of the first segment and the last three variables of the last segment are the given boundary conditions of the beam. The “‘vi in eqn (7) correspond to the initial conditions of the mth segment, and numerical integration along each segment according to eqn (4) is straightforward. The resulted matrix is of order 6(n + l), where n is the number of constant cross-section segments. Although eqn (7) appears to be a large set of equa-
Table 1. Cross sectional Section (in) 0 120 180 1140 1200
.
(7)
in2, moment of inertia, I. equal to 4000 in.4, and shear coefficient, K, equal to 0.52. The modulus of elasticity is 30,000 ksi, and the building is subjected to a uniformly distributed lateral load of 1 k/in. The model is divided into five sections of constant properties as shown in Table 1. A plot of y,(x), the global lateral deflection compared to the lateral deflection obtained from a finite element analysis is shown in Fig. 3.
property
of structure
AGK, (K)
AGK~ CK)
A(& (K)
Gf (K/in’)
114500 114500 82500 138000
8000 0 0 0
237000 229000 165000 276000
870 870 870 870
shown
in Fig. 2
Isectm (in4) 746000 746000 746000 746000
Inange (in4) 644000 644000 644000 644000
Numerical solution of tube buildings modeled as continuum
775
Fig. 2. Finite element model.
Fig. 4. Plan and elevation view of 51-story tube building.
The second example depicts a 51-story tube building with realistic dimensions and member sizes (Fig. 4). The girders throughout the structure are assumed to be W36 x 135 and all columns of each
0.8
story are identical. The change of column size along the height of the structure is shown in Table 2 and Elevation
story
(in.)
NO.
7488
51
6048
41
4608
31
3168
21
0.2
0.0 0
Latera,
1
Deflection
2
( x L/12”
1
Fig. 3. Lateral deflection of thin-walled continuous model compared to results of finite element model. Finite
Element
Approximate
Table 2. Column size along the building and their crosssectional properties Elevation (in)
Member
A (in’)
I (in4)
1728
Model
Model
II
A (in*) 288
1
’ 3;i 4608 6048 74*8
TS20 xx 370 w14 20 x 3 W14 x 283 W14x211 w14 x 90
204 109 101 62 26.5
10,132 5,440 4,900 2,660 999
102 29.1 27 15.4 6.2
Lateral
Deflection
(in,)
Fig. 5. Lateral deflection of the 51-story tube building modeled as a continuous tube and finite element.
776
P. CHANGand D. FOUTCH Table 3. Properties of modtl for 51-story building Elevation (in)
E (K/in’ )
G
AGK
(K/in’)
(K)
1 (in’)
0 288 3108 4608 6048 7488
30000 30000 30000 30000 30000
11540 240 250 360 570
1802000 1590000 1457000 655400 309000
1257,000.000 1759.000,000 1630,000,000 1000.000.000 428.000.000
the properties of the approximate model are listed in Table 3 [l]. Refer to Refs. [l] and [2] for details of the modeling technique. Assuming that the building is located in San Francisco, California, the lateral loads on the structure may be due to earthquake or wind. The design earthquake loads were determined in accordance to the ACT III recommendations, while the design wind loads were obtained from the American National Standard Institute Code-ANSI. A58.1. The lateral load due to wind is the larger load and was used in the design. Table 4 lists the wind loads determined for the example structure. The approximate solution using the continuum approach is compared to a finite element solution in Fig. 5. The amount of computer time required for the Finite Element solution was 800 times greater than the proposed method on a Cyber 175 computer.
5. SUMMARYAND
CONCLUSION
The purpose of this paper was to describe a relatively inexpensive and simple method for solving a set of differential equations which, when coupled
Table 4. Calculation of lateral wind force on Sl-story building Elevation (ft) 24 144 264 384 504 624
K*
qF
Design Load from ANSI Section 6.3.4.1 (Kift)
0.20 0.55 0.85 1.05 1.30 I .50
3 9 14 17 21 24
4.03 3.53 2.65 2.35 1.51 0.50
If (in’) 1142.000.000 1498.000.000 1389.000.000 852.000,OOO 364.000,OOO
with a continuum model of a building, provide a powerful tool for engineers. The example used to illustrate this point was the class of buildings known as tube frames which can be modeled as continuous thin-walled tubes. The analysis of the continuous structures gives the global behavior of the building which is needed both in the preliminary and the final stages of the design. Because the tube building’s final designs are often governed by the drift criterion, many iterations of design with parameter changes are often necessary at the preliminary stage of the design. Since only the global structural behavior is needed at the preliminary stage, the approximate model offers a simple and inexpensive alternative to the finite element analysis. Another added advantage is that this simple approximate technique can be easily programmed in a personal computer regardless of the size of the structure being analyzed, whereas a large finite element package requires a large computer and much computer time and manpower involved in input preparation and output interpretation. The disadvantage of the method, however, is that at this time it does not give accurate local forces and stresses, and therefore, does not provide sufficient information for use in the final stage of design. REFERENCES P. Chang, An approximate dynamic analysis of tube structures. Submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Civil Engineering in the Graduate College of the University of Illinois at Urbana-Champaign (1982). P. Chang and D. Foutch. Approximate analysis of tube buildings. Paper accepted for publication by J. Stntctural Div. ASCE. E. Reissner, Least work solution of shear lag problem. J. Aeronaur. Sci. 8(7), (1941). D. Foutch and P. Chang, A shear lag anomaly. J. Stnccrural Div. ASCE, 108, ST7 (1982).
& Strucrures
Vol.21. No. 4.
Printed in GreatBritain.
pp. 771-776.
1985
0045-794985 $3.00 + .oo C 1985 Pergamon Press Ltd.
NUMERICAL SOLUTION OF TUBE BUILDINGS MODELED AS CONTINUUM Department
of Civil Engineering,
PETER CHANG University of Maryland, College Park, MD 20742, U.S.A.
and DOUGLAS FOUTCH
Department
of Civil Engineering,
University of Illinois, Champaign-Urbana,
IL 61801, U.S.A.
(Received 8 June 1984) Abstract-The tube building is represented by a set of six differential equations which represent the flexural and shear deformation, and the shear-lag effect of the structure. The model is divided into segments of constant properties to represent changes in member sizes along the height of the building and to reduce the numerical errors accumulated by the approximate method. A detail explanation of the fourth-order numerical technique used to solve for the differential equations is included so that readers may easily construct a computer program to study the behavior of tube structures. Also included are examples of tube buildings solved by the numerical technique compared to results of finite element method. The comparisons show that the numerical solution of the continuous models yield excellent results while requiring less than one percent of the computational time and input/output efforts associated with the finite element method. 1.
INTRODUCTION
The design of a structure can be loosely grouped into three stages: (11 the planning stage, (2) the preliminary design stage and (3) the final design stage. In the planning stage, the nonstructural aspect of the structure such as aesthetics, the usage of the structure and dimensions are decided. The structural engineer usually has little or no input at this stage of the design. Once the structure is conceived, the structural engineer is in charge of providing a structural system to support the structure under the predicted loads. The structural engineer, in order to provide the most adequate and economical supporting structure, usually needs to consider several structural systems. This is done in the preliminary stage of the design. At this design stage the structural engineer considers most of the parameters that would affect the global behavior of the building. Because local effects are not a major concern at this stage of the design, a rigorous analysis is not necessary, therefore approximate methods giving the global behavior of the structure are often used. Once the structural frame is selected the structural engineer can then use a more rigorous approach which gives the local forces and stresses needed to size the members. A method for the approximate analysis of tube frames was recently developed [l, 23. This structural system was chosen for the present study because many medium to high rise buildings utilize this type of framing due to its great efficiency at resisting lateral loads. The analysis procedure was shown to be extremely efficient when compared to more conventional approaches. Thus, many different options may be considered in the preliminary design stage at relatively little design cost to the engineer. The approach adopted for this study converts the
properties of the tube frame to those of an equivalent continuous thin-walled cantilever tube. The behavior of this equivalent tube therefore represents the global behavior of the building. The complexity of the tube’s governing differential equamakes a closed form solution tion, however, nonfeasible. The purpose of this paper is to describe in detail a simple numerical technique tailored to solve the governing differential equations resulting from the modeling technique. Because of its simplicity. the method described can be implemented on most micro or personal computers, thus making the preliminary design process much faster and inexpensive. Although some aspects of this solution scheme are not new, the combination of this scheme with the concept of modeling a discrete system as a continuum provides the engineer with a powerful, but inexpensive, package that has not been suggested previously. As mentioned above, only the solution technique will be presented in this paper. The procedure for making the conversion from the discrete system to the continuum has been described extensively elsewhere [l, 21. In fact, it has also been shown that the same differential equations, in a degenerated form, can be used quite effectively to model a regular or braced frame [ 11. Thus, the approach is quite versatile and not restricted to tube frames. The solution technique is also applicable to a broad range of differential equations and, therefore, should be useful to engineers under many circumstances.
2. GOVERNINGDIFFERENTIALEQUATION
The structures being considered are tube buildings. These buildings derived their name from the 771
712
P.
CHANG and D. FOUTCH
fact that they resist lateral loads in the same manner as a hollow thin-walled cantilever tube. The frames parallel to the direction of the load act as the web of the tube and the frames transverse to the loading direction act as flanges. This differs from the more conventional system where only the frames in the direction of loading resist the forces. Thus, approximate methods employed in the preliminary design stage of conventional frames are not applicable to tube buildings. The continuous model used is based on Reissner’s model [3] which assumes a parabolic variation of longitudinal flange displacement to account for the shear-lag effect, which is present in the flanges of the tube frame. Reissner’s model was modified to include the effects of shear deformation which is also significant for tube buildings. The behavior of the building, therefore, can be described by the set of differential equations:
lf the building is fixed at the ground level, then the boundary conditions of eqns (2) are y6(0)
=
{EZ[w'(O)p'(O)] + fEZfU'(0)}' =0
ys(0)
=
[EZ(w"(O) - p'(O), + $EZfU'(0)] =0
y4(0) = U'(0)= 0
yz(L)= y,(L)
U(L)=0 = w(L)
= 0.
Some rather interesting aspects of the behavior of thin-walled cantilever tubes as indicated by the solution of these equations have been described previously [4].
3. SOLUTION
+ 5 EZfU']"= P(s), [EZ(w"- f3')
(3)
OF DIFFERENTIAL
EQUATIONS
The solution of the problem requires the integration of the set of first-ordered differential equations [eqns (2)] over a given range of independent vari- P')]' - ;% u = 0, (1) ables, x. The solution of these equations results in EZ,[(&U' + 5 (MI" determination of values y, , y2, . . . , y6 corresponding to tabulated values of x. In the numerical where method used, the values of yi at each step Xj are based on the results of yi from the previous step. E = modulus of elasticity of the material that Hence, at some point x, all the corresponding valmakes up the tube ues Of vi: y,(O), y*(O), . . . , y&O) must be known. G(x) = shear modulus This type of problem is referred to as a forwardZ(x) = moment of inertia of the cross-section integration initial-value problem. For the tube If(x) = moment of inertia of the flanges alone building three of the boundary conditions [eqns (3)l P(x) = lateral load are given at the end x = L. Hence the forwardU(x) = shear-lag stress function integration initial-value problem cannot be applied w(x) = lateral deflection to eqns (2) directly. A technique used which nl’ - B = rotation of the section due to shear “guesses” the initial values of the variables not W = flange width of the tube structure given there and corrects them until the variables K = shear correction factor. match with the given boundary conditions can overcome the problem of not having all the boundary Because of the complexity of eqn (1). a closed conditions given at one point. This technique is form solution was not attempted. Rather, the socalled the shooting method. When the system of lution is obtained through numerical approximadifferential equations is linear, these initial tion. The application of the numerical method used “guesses” may be calculated in terms of the boundrequires that the coupled sixth-ordered differential ary values given elsewhere, making the shooting equation [eqn (l)] be transformed into six first-ormethod simple and efficient. dered differential equations. These equations are The simplest form of initial-value forward integiven in eqns (2) (see Ref. [2]). gration is the Euler’s method. In Euler’s method, the increment in y corresponding to the increment y; = [EZ(w" - fi')+$EZ,U']"= -p’.AG~=f’(x), AX, for the differential equation dy/dx = f(x, y), is y; = [EZ(w" - f3') + $EZfU']' =y6, approximated by yi + Ax (y:). A term-by-term comparison of this result to the Taylor-series ex5 1 5G -_ pansion indicates that the Euler’s expression is only y; = 2 EWZY’ - izy” accurate to the order hx(y:). This accuracy is in’ 1 - 5Z5/6Z dicated symbolically by o(Ax’y’i’), which means that the error is to the order Ax’~‘/. Many modi1 y$=(w’P)’ = FzY5 - $Y4, fications to Euler’s approximation giving more accurate results are available. The refinement of inyi = U’ = y4, tegration selected depends on the accuracy needed. The solution to the governing thin-walled tube y; = w =y3 -AGK.Y~. (2) (AGKB)’ = -P(x).
773
Numerical solution of tube buildings modeled as continuum
equation [eqns (1) or (2)] employs a fourth-order Runge-Kutta approximation, which has error to the order o(h’y”“‘). The simplest algorithm with this degree of accuracy is Kutta-Simpson’s rule. Here, the new values of yi are approximated using the slope at the previous point, the slope at the new point and the slope at the mid-point between the two. yi(X + AX) = yi(X) + Q(JJi + 2qi + 214; + Vi),
(4)
where Pi =
hr*fi[~~Yl(~)~Y2(X)9~
4i=k'f;
[
ui=hx’fi
[
Ui=AX*fj
[
. .
,y6(X)l.
1
X+$,yl(X)+$,. . .,1.6(X)+?, X+$,yl(X)+T,. . .,y6(X)+t1 , X+&3Yl(X)+Ui,. 2
where fi are the functions example
.
.,.Y6(X)+ui
9
1
(5)
defined in eqns (2). For
Once all the values of yi corresponding to the value of x are known, the application of eqn (4) is straightforward. However, the boundary conditions for a building are usually not all given at a single point; they are partially given at each of the two extremities of the structure, i.e. the ends x = 0 and x = L. This type of boundary condition constitutes a two-point boundary-value problem. To get the unknown values of yi at the point x0, trial values of yi(O) are used. If the values of yi(O) are guessed correctly, the yi at the other end of the structure would match the given boundary conditions exactly. The process of choosing a set of yi(O) so that the computed yi match the given boundary conditions is called the “shooting technique.” Since eqns (2) is linear, all the values Of yi at a given point Xi are linear combinations of the yi at another point of the beam, say x = 0. Hence we can write, for x = L, Yl(L)
=
+
al2Y2(0)
+
. .
+
b,,
Yz(L) = a2lYlm
+
a22y,(O)
+
. . .
+
b2
Y6(L)
+
a6ZY2(0)
+
. . .
+
b6,
=
UllYl@)
a6lY,@)
T
(6)
where y,(O) represents the boundary conditions at one end and yi(L) represents the boundary conditions at the other end. The uij and bj are constant coefficients that can be determined in the following manner.
CM 21:4-K
(1) Select y,(O) = y?(O) = . . . = 0 and integrate the set of first-order differential equations [eqns (2)] from 0 to L. Call the resulting values yi(L,): rl , r2, Then b, = rl, bz = r2, . . , . * ‘(;; Select y,(O) = 1, y*(O) = I, y2(0) = . . . = 0, and integrate eqns (2) from 0 to L. Call the calculated values yi(L): si, s2, . . . . Then a,, = s, b,, u7, = s2 - bz, . . . , (3) Select yz(O) = 1, y,(O) = yj(O) = . . . = 0, and integrate eqns (2) from 0 to L. Call the calculated values yi(L): tl, tz. . . . . Then al2 = tl b,, az2 = tz - b2, . . (4) Repeating steps ; gnd changing the selected boundary condition by setting one variable equal to 1 progressively while all the others are set to zero. (5) Solve eqns (6) for y;(O). Once all y;(O) are found, the solution of eqns (2) can be found directly through eqn (4). In real buildings the column and girder sizes change throughout the structure. These changes translate into abrupt changes in property of the continuous tube model making it a nonprismatic member. It is therefore convenient to analyze sections of constant (or continuously varying) properties separately and then enforce the compatibility of displacement between sections. This sectioning approach has another advantage; it reduces the accumulation of error which results from approximate numerical integration, also known as the length effect. The accumulation of error due to numerical approximation, then, is limited to the order of the error accumulated for the longest section. Thus, subdivision of the building into sections can serve as a control of error accumulation as well as modeling the change of column size. To find the values
L Prototyp
Equl”.,mr
ad*,
Fig. 1. Prototype and equivalent model.
774
P.CHANG
and
of dependent variables vi at the junction of segments and the two extremities, let us consider the jth segment. If the set of differential equations governing thejth segment is linear, the boundary conditions at one end of the segment must be a linear combination of the boundary conditions at the other end. Therefore eqn (5) is valid for each segment. However, the ends x = 0 and x = L now refer to the ends of the segment rather than the extremities of the building. Now, if the boundary conditions of the “L” end of the jth segment are forced to be compatible to the “0” end of the j + 1 segment, then the linear conversion equations of all the segments may be written related to each other. The resulting equation involves all the variables vi at all the junctions between segments of an n-segment building model. These equations can be summarized into the matrix representation as eqn (7).
D. FOLJTCH tions, it is in fact a reduction of the conventional finite element approach by several orders of magnitude. For a uniform tube, only six equations must be solved. The engineer may decide how “finely” to section the tube depending on the degree of accuracy required. Furthermore, the solution of eqn (7) requires a minimal amount of computation because of the small bandwidth associated with this formulation. 4.NUMERICALEXAMPLES
To demonstrate the accuracy of the method, the structure shown in Fig. 1 was modeled as a continuous tube. The building was also analyzed using the finite element method. The finite element model is shown in Fig. 2. The structure in the example has all columns and girders with area of cross section. A, equal to 29.1
0 0 0
‘Yl
'Y2 'Y3 'Yd 'Ys 'Y6 2YI *?,2 2Y3
_ ‘b, - ‘b2 _ ‘b3 - ‘h - ‘bs - ‘bs -*b, - 'b2 - *b3 - *b4
*Y4 2 u212a222u232u242u252u26 2 a3~2u322u332u342a352u36 2 u412a422u.432u442u452u‘k5 2 u512u~22u532u542a552u~~ 2 u612a622u632uc42u652u.56
2YS 2Y6 3Yl
0
- "b5 - "be
“fly2
0 0 0
“+‘Y3 “+‘Y4 “+‘Y5_
The first three variables of the first segment and the last three variables of the last segment are the given boundary conditions of the beam. The “‘vi in eqn (7) correspond to the initial conditions of the mth segment, and numerical integration along each segment according to eqn (4) is straightforward. The resulted matrix is of order 6(n + l), where n is the number of constant cross-section segments. Although eqn (7) appears to be a large set of equa-
Table 1. Cross sectional Section (in) 0 120 180 1140 1200
.
(7)
in2, moment of inertia, I. equal to 4000 in.4, and shear coefficient, K, equal to 0.52. The modulus of elasticity is 30,000 ksi, and the building is subjected to a uniformly distributed lateral load of 1 k/in. The model is divided into five sections of constant properties as shown in Table 1. A plot of y,(x), the global lateral deflection compared to the lateral deflection obtained from a finite element analysis is shown in Fig. 3.
property
of structure
AGK, (K)
AGK~ CK)
A(& (K)
Gf (K/in’)
114500 114500 82500 138000
8000 0 0 0
237000 229000 165000 276000
870 870 870 870
shown
in Fig. 2
Isectm (in4) 746000 746000 746000 746000
Inange (in4) 644000 644000 644000 644000
Numerical solution of tube buildings modeled as continuum
775
Fig. 2. Finite element model.
Fig. 4. Plan and elevation view of 51-story tube building.
The second example depicts a 51-story tube building with realistic dimensions and member sizes (Fig. 4). The girders throughout the structure are assumed to be W36 x 135 and all columns of each
0.8
story are identical. The change of column size along the height of the structure is shown in Table 2 and Elevation
story
(in.)
NO.
7488
51
6048
41
4608
31
3168
21
0.2
0.0 0
Latera,
1
Deflection
2
( x L/12”
1
Fig. 3. Lateral deflection of thin-walled continuous model compared to results of finite element model. Finite
Element
Approximate
Table 2. Column size along the building and their crosssectional properties Elevation (in)
Member
A (in’)
I (in4)
1728
Model
Model
II
A (in*) 288
1
’ 3;i 4608 6048 74*8
TS20 xx 370 w14 20 x 3 W14 x 283 W14x211 w14 x 90
204 109 101 62 26.5
10,132 5,440 4,900 2,660 999
102 29.1 27 15.4 6.2
Lateral
Deflection
(in,)
Fig. 5. Lateral deflection of the 51-story tube building modeled as a continuous tube and finite element.
776
P. CHANGand D. FOUTCH Table 3. Properties of modtl for 51-story building Elevation (in)
E (K/in’ )
G
AGK
(K/in’)
(K)
1 (in’)
0 288 3108 4608 6048 7488
30000 30000 30000 30000 30000
11540 240 250 360 570
1802000 1590000 1457000 655400 309000
1257,000.000 1759.000,000 1630,000,000 1000.000.000 428.000.000
the properties of the approximate model are listed in Table 3 [l]. Refer to Refs. [l] and [2] for details of the modeling technique. Assuming that the building is located in San Francisco, California, the lateral loads on the structure may be due to earthquake or wind. The design earthquake loads were determined in accordance to the ACT III recommendations, while the design wind loads were obtained from the American National Standard Institute Code-ANSI. A58.1. The lateral load due to wind is the larger load and was used in the design. Table 4 lists the wind loads determined for the example structure. The approximate solution using the continuum approach is compared to a finite element solution in Fig. 5. The amount of computer time required for the Finite Element solution was 800 times greater than the proposed method on a Cyber 175 computer.
5. SUMMARYAND
CONCLUSION
The purpose of this paper was to describe a relatively inexpensive and simple method for solving a set of differential equations which, when coupled
Table 4. Calculation of lateral wind force on Sl-story building Elevation (ft) 24 144 264 384 504 624
K*
qF
Design Load from ANSI Section 6.3.4.1 (Kift)
0.20 0.55 0.85 1.05 1.30 I .50
3 9 14 17 21 24
4.03 3.53 2.65 2.35 1.51 0.50
If (in’) 1142.000.000 1498.000.000 1389.000.000 852.000,OOO 364.000,OOO
with a continuum model of a building, provide a powerful tool for engineers. The example used to illustrate this point was the class of buildings known as tube frames which can be modeled as continuous thin-walled tubes. The analysis of the continuous structures gives the global behavior of the building which is needed both in the preliminary and the final stages of the design. Because the tube building’s final designs are often governed by the drift criterion, many iterations of design with parameter changes are often necessary at the preliminary stage of the design. Since only the global structural behavior is needed at the preliminary stage, the approximate model offers a simple and inexpensive alternative to the finite element analysis. Another added advantage is that this simple approximate technique can be easily programmed in a personal computer regardless of the size of the structure being analyzed, whereas a large finite element package requires a large computer and much computer time and manpower involved in input preparation and output interpretation. The disadvantage of the method, however, is that at this time it does not give accurate local forces and stresses, and therefore, does not provide sufficient information for use in the final stage of design. REFERENCES P. Chang, An approximate dynamic analysis of tube structures. Submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Civil Engineering in the Graduate College of the University of Illinois at Urbana-Champaign (1982). P. Chang and D. Foutch. Approximate analysis of tube buildings. Paper accepted for publication by J. Stntctural Div. ASCE. E. Reissner, Least work solution of shear lag problem. J. Aeronaur. Sci. 8(7), (1941). D. Foutch and P. Chang, A shear lag anomaly. J. Stnccrural Div. ASCE, 108, ST7 (1982).