The algorithm of solving differential equations in continuous model of tall buildings subjected to concentrated loads

CMM-2003 – Computer Methods in Mechanics

June 3-6, 2003, Gliwice, Poland

The algorithm of solving differential equations

in continuous model of tall buildings subjected to concentrated loads

Jacek A. Wdowicki and ElŜbieta M. Wdowicka

Institute of Structural Engineering

Poznań University of Technology, Poznań , Poland

e− mail: jacek.wdowicki@put.poznan.pl

Abstract

The paper presents the generalization of the algorithm of solving differential equation systems in the analysis of shear wall tall buildings in the case of concentrated loads applied at an arbitrary height. The proposed algorithm was used among others in order to obtain the flexibility matrix, required for a dynamic analysis of tall buildings by means of the continuous – discrete method.

The conducted tests confirm the correctness of algorithm realization.

Keywords: differential equation systems, shear wall structures, dynamic analysis of tall buildings, continuous-discrete approach z ∈< ,

0 h >

B N ′ ( z)

− N ( z)

= f ( z),

Introduction

(1)

z ∈ ( ,

h H >

B N ′ ( z)

− N ( z)

= 0

In multi-storey buildings, the lateral loads that arise as a

where:

result of winds and earthquakes are often resisted by a system of

B - n

shear walls acting as vertical cantilevers. Such walls are usually

w x nw diagonal matrix, containing continuous connection

flexibilities,

perforated by vertical bands of openings, which are required for

A - n

doors and windows to form a system of coupled shear walls.

w x nw symmetric, positive semi-definite matrix,

dependent on a structure,

The analysis of shear wall structures may be performed by using

either discrete or continuous methods [7]. In the continuous

w - number of continuous connections, which substitute

connecting beam bands and vertical joints,

approach, the horizontal connecting beams, floor slabs and

- vector containing unknown functions of the shear

vertical joints are substituted by continuous connections. The

( z)

fact that floor slabs have infinite in-plane stiffness is the main

force intensity in continuous connections below the

assumption used in this method. Shear walls are considered as

concentrated load,

thin-walled cantilever beams. The technique may be used for

( z) - vector containing unknown functions of the shear

both plane and spatial structures, which are essentially regular

force intensity in continuous connections above the

in form throughout the height. In recent years the use of

concentrated load,

continuum models in structural analysis has received

f ( z) - vector formed on the basis of given values of

considerable attention. These models offer an attractive, low

concentrated loads P , P , M :

cost method for analysing large structures, and they represent

the useful tool for design analysis.

f ( z)

= C L( T

L K L)−1 T

T = [ P , P , M ] T ,

K ,

For the dynamic analysis it is convenient to use a hybrid

C , L, K - matrices, dependent on a structure [9],

approach, based on the analysis of an equivalent continuous

[14],

medium and a discrete lumped mass system [15], [4]. In order

to obtain the required flexibility matrix it is necessary to

h - the ordinate of point of concentrated loads application,

determine horizontal displacements of the shear wall system,

H - structure height.

subjected to concentrated loads.

The aim of the paper is to extend the algorithm of solving

The boundary conditions have the following form:

differential equations in the analysis of shear wall tall buildings

( )

= ,

using the continuous model so as to enable us to take into

(2)

account lateral concentrated loads applied at an arbitrary height.

( h)

= N ( h),

N ′ ( h)

= N′ ( h),

N ′ ( H )

= .

Governing equations

After determination of unknown functions of shear force

Equation formulations for a three-dimensional continuous

intensity in continuous connections it is possible to obtain the

model of shear wall structures stiffening tall buildings have

functions of horizontal displacements of the structure as well as

been presented, among others, in Ref. [9], [14]. In the case of

its derivatives, by analogy with Ref. [10], using the following

structures subjected to horizontal concentrated loads, applied at

equations:

an arbitrary height, the governing differential equations can be

z ∈ ( h, H ,

V (

′ z) = − V N ( z)

stated as follows:

(3)

z ∈< ,

0 h ,

V (

′ z) = ( LT K L) 1− T − V N ( z)

where:

( z) = F ( 2

z / 2 − hz),

V ( z) - vector containing the functions of the horizontal

(8)

= −

displacements of the structure above the

g ( z)

F h / 2.

concentrated load,

The form of the solutions corresponding to non-zero

V ( z) - vector containing the functions of the horizontal

eigenvalue λ is as follows:

displacements of the structure below the

concentrated load,

λ z

− λ z

g ( z) = C e i + C e

V = ( T

L K L)−1 T

L C .

(9)

λ z

− λ z

( z) = C e i + C e

− F / λ

Boundary conditions have the following form:

Integration constants are determined from linear equation

V ( )

= ,

V ′ ( )

= ,

system resulting from boundary conditions (7), in the following

V ( h)

(4)

form:

= V ( h),

V ′ ( h)

= V ′ ( h),

V ′( h)

= V ′( h),

V ′( H )

= 0 .

R C

P ,

(10)

C = [ C , C , C , C ] T ,

Method of solution

where:

In the proposed algorithm the equation systems, described

in Eqn (1), are uncoupled by orthogonal eigenvectors of







λ h

−



P = B -1/2 A B 1/2 matrix into nw sets of two second-order

 e

− e



differential equations. Integration constants are determined for

RWi =

λ h

−

 λ e

λ e

λ e i

−



each set from the system of four linear equations resulting from





boundary conditions. Also there has been taken into account the



λ H

e i

λ e i

−





case of zero eigenvalues of P matrix, allowing for the analysis

of the structure with arbitrary plan.

In order to uncouple differential equation systems, by

 F / λ 

analogy with Ref. [11], auxiliary functions g ( z), g ( z),





− F /



i 

satisfying these relations have been introduced:

PSi =









( z)

− /

= B 2Y g ( z),





(5)

( z)

− /

= B 2Y g ( z)

After computing all elements of solutions g ( z), g ( z) of

where:

Eqn (6) and retransforming according to Eqn (5), values of

Y - nw x nw matrix columns of which are eigenvectors of

unknown functions N

( z ,

) N

( z) of the shear force

symmetric matrix P.

intensity in continuous connections throughout the building

One has obtained n

height are obtained.

w sets of two second-order differential

equations in the following form:

It can be noticed that the algorithm requires a small

operational memory as the highest order of matrix equals nw ,

z ∈< ,

0 h ,

g ′ ( z)

− g ( z)

= F ,

where nw denotes the number of vertical continuous connections

(6)

in the system, being most often of the order of 20 and not

z ∈ ( ,

h H

g ′ ( z)

− g ( z)

= 0

exceeding 100.

where:

The computational cost of labour is not high, either; it

comprises a single solution of generalized eigenvalue problem

i - i-th eigenvalue of matrix P,

and for each load scheme nw –times solution of four linear

−1 /

= Y

f ( z),

equation system is included. The solutions coefficients are

registered and subsequently they are used for computation of

Yi - eigenvector corresponding to the i-th eigenvalue,

unknown functions values for given ordinates of height. For the

realization of the presented algorithm the most essential

with boundary conditions as follows:

computational task is to determine all eigenvalues and their

corresponding orthogonal eigenvectors of symmetrical matrix

( )

= ,

P. Matrix P is dense and positive semi-definite, of n = nw order.

( h)

In general, it is possible for P matrix to have multiple

= g ( h),

g′ ( h)

= g′ ( h),

(7)

eigenvalues including zero values. Having analysed the features

g′ ( H )

= .

of the algorithms available the most appropriate as considering

accuracy of the results as well as computation time has been

Matrix A is positive semi-definite, thus matrix P can also

chosen. It consists in transforming the matrix into the

have zero eigenvalues. The solutions of Eqn (6) with boundary

tridiagonal form by means of Householder method and

conditions (7) in the case of zero eigenvalues have a simple

subsequently, computing eigenvalues and eigenvectors by QL

form:

algorithm. Procedures tred2 and tql2, published originally in

„Numerische Mathematik” and then inserted in the collection

when connecting beams and vertical joints are not present in the

[16] have been used here. The total number of operations when

system. Then Eqn (12) has the simple form:

determining n pairs of eigenvalues and eigenvectors by the use

∈

′

of this algorithm is, according to Ref. [3], of the n3 order.

( ,

h H

V ( z)

(13)

According to Ref. [16] computation time for the matrix of order

z ∈< ,

0 h ,

V ′( z) = F

POM .

n>10 is several times shorter than using Jacobi method.

Eigenvectors computed by QL algorithm are always orthogonal

The corresponding solutions are the following:

almost to working accuracy even if matrix has multiple or close

eigenvalues. Each eigenvalue and its corresponding eigenvector

V ′ ( z) = ,

are always exact for some matrix P+E

║

i where ║Ei

/║P║ is of

V ′ ( z) = F

( z − h),

the order of magnitude of the machine precision.

POM

When establishing the criterion, on the basis of which

V ′ ( z) = F

(

− h / )

2 ,

POM

eigenvalues treated as zero values in the algorithm are isolated,

(14)

error estimations of P matrix eigenvalues computed by means

V ′ ( z) = F

( z / 2 − zh),

POM

of tred2 and tql2 procedures have been used. In all the cases

V ( z) = F

(

− zh / 2

+ h / )

6 ,

analysed (see Ref. [13]) the eigenvalues, which should equal to

POM

zero, have been well separated from the others so the assumed

V ( z) = F

( 3

z / 6

− z h/ )

2 .

POM

criterion is correct.

When computing the subsequent derivatives of functions of

Horizontal displacement expressions

the system horizontal displacements V ( z), V ( z)

Eqn (12)

including the solution functions g ( z ,

) g ( z) has been

The next step of computations is determining functions of

horizontal displacements of the system and their derivatives

applied. Knowing the form of these solutions the necessary

necessary to calculate internal forces and stresses. As a result of

appropriate integrals have been calculated analytically. The

integration of Eqn (3) with appropriate boundary conditions (4)

subsequent integrals of the solutions g

( z)

g ( z

and

the following solution is obtained:

corresponding to zero eigenvalues λ i of P matrix have the

following form:

V ′ ( z)

t dt

(

′ ) ,

∫ G

( z) =

g ( t)

dt = − F h ( z − H ) / ,

∫

G i

V ′ ( z)

t dt

(

′ )

′( ) ,

∫

∫ D

( z)

g ( t) dt

∫

D i

V ′ ( z)

t dt

= − F ( 3

− z / 6

+ hz / 2

+ h / 6 − H ) 2

h / ,

′( )

′ ( ) ,

∫

∫ D

(11)

( z) = g

( t) dt + g

( t) dt =

∫

G 2 i

G i

D i

V ′ ( z)

t dt

′ ( ) ,

∫ D

2 2

= −

−

F ( h z / 2

H h z / 2

h / 2 )

4 ,

V ( z)

t dt

( z) = g

( t) dt =

∫

′ ( )

′ ( ) ,

∫

∫ D

D 2 i

D i

= − F (− z / 24 + hz / 6 + ( h / 6 − H h / )

2 z),

V ( z)

t dt

′ ( ) .

∫ D

( z) = g

( t) dt + g

( t) dt =

∫

G 3 i

G 2 i

D 2 i

During the computation of the above-mentioned function

= − F ( 2 3

h z /12

2 2

− H h z / 4

+ h z / 24

− h /12 )

0 ,

values uncoupling of equation systems is used consistently as

before. Equation (3) after introducing auxiliary functions (5) has

the following form:

( z) = g

( t) dt =

∫

D 3 i

D 2 i

(15)

z ∈ ( h, H >,

V (

′ z) = − V g ( z),

(12)

= − F (− z /120 + z h / 24 + z ( h /12 − h H / )

4 ).

z ∈< ,

0 h >,

V (

′ z) = F

− V g ( z)

POM

The integrals of solutions (9) corresponding to non-zero

where:

eigenvalues λ are more developed. In order to shorten their

−

1 /

−

V = V B

= ( T

L K L) 1 T .

POM

notation such symbols as f = F / λ

s = λ

and

i have

The first step in computing solutions (11) is determination

been introduced. So using them the subsequent integrals can be

of subsequent derivatives of displacement functions in the case

written as follows:

( z)

−1

= s ( C esz − C e− sz − C esH + C e− sH ),

The same structure has been subjected to uniformly

G 1 i

distributed horizontal load, which subsequently has been

substituted by concentrated loads applied at each storey. For the

( z)

= s− ( C esz − C e− sz + ( C − C ) esh

two kinds of loads the obtained displacements differed by 0.3%.

D i

The other conducted tests have also confirmed correctness

(− C + C ) e− sh − C e sH + C e − sH ) − f ( z − h), 2

of the algorithm realization.

The presented algorithm has also been applied in obtaining

( z)

−2

= s ( C esz + C e− sz )

+ s− (− C esH + C e− s H ) z

the flexibility matrix used in dynamic analysis for the tall

G 2 i

−

building model in the form of discrete lumped mass system. The

+ s (( C

− + C ) esh + (− C + C ) e− sh )

values of the i-th three columns of elements in flexibility matrix

+ hs− (( C − C ) esh + (− C + C ) e− sh ) for the shear wall structure represent the horizontal

displacements (V

f h / 2

−2

− s ( C + C ),

x, Vy, Φ) of the system at all levels where

lumped masses are located, induced by unit generalized forces

(PX, PY, MS) applied horizontally at the location of the i-th

( z)

−2

= s ( C esz + C e− sz − C − C )

− f z / 2

lumped mass. Natural frequencies computed by the use of this

D 2 i

method correlate well with the results given in literature, both in

+ s− (( C − C ) esh + (− C + C ) e− sh

the case without connecting beams [5], and with beams [1].

+ f sh − C esH + C e− sH ) z,

The algorithm has been used, among others, in the analysis

of the impact of concentrated loads from cranes attached to the

shear wall system of 19-storey building realized in Poznań.

( z)

−3

= s ( C esz − C e− sz + ( C − C ) esh

G 3 i

+ ( C − C ) e− sh − C + C )

Conclusions

+ (− C esH + C e− sH )( 1− 2

s z / 2 + z)

− f h / 6

The paper presents the algorithm of solving differential

+ ((− C + C ) esh + ( C − C ) e− sh ) 1− 2

s h / 2

equations in the analysis of shear wall tall buildings subjected to

−2

+ s (( C − C ) esh + ( C − C ) e− sh ) h, (16)

concentrated loads applied at an arbitrary height. In the

proposed algorithm the equation systems are consistently

uncoupled by orthogonal eigenvectors. Also there has been

( z)

−3

= s ( C esz − C e− sz − C + C )

taken into account the case of zero eigenvalues, allowing for the

D 3 i

analysis of the structure with arbitrary plan. Algorithm is very

− f z / 6

−2

− s ( C + C )

z + s − (( C − C ) e sh

economical on storage. After designing and realizing software

+ (− C + C ) e− sh + f hs − C esH + C e− sH ) 2

z / 2 .

based on the proposed algorithm a series of tests has been

carried out. The presented algorithm has also been applied in

Having computed all the elements of the above-mentioned

obtaining the flexibility matrix used in dynamic analysis of

six integrals and arranging them in vectors and then pre-

shear wall tall buildings. The conducted tests have confirmed

multiplying, according to Eqn (12), by V

correctness of the algorithm realization.

P matrix as well

adding terms from Eqn (14) one obtains the subsequent

derivatives of horizontal displacement functions of the system,

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