Mgr.inz.arch. Makowska Agnieszka

Cracow University of Technology

 

 

COMPUTER AIDING FOR ESTIMATING THE CURVILINEAR ENGIEERING STRUCTURES

 

 

1.Introduction

 

This paper is the continuation of determining of the curvelinear objects parameters. 

The most of contemporary building use soft form and irregular shapes.

The base of the cost calculation is the exact calculation of length or the surface area or the volume. The new techniques and technology use very expensive materials, hence it is very important to do the precise calculation to reduce the cost of investment.

Exact calculation of the surface or the volume of straight-line forms is easy, in the case of curvilinear objects it can be controversial.

The attempt of description of curvilinear objects made with the help of cubic spline interpolation has been presented in this paper.   

 

2. Elementary theory of  cubic spline

 

We have got n+1 points in the interval  <a,b>  : a= x0 , x1 ,  . . . , xn = b, call this nodes and value function y=f(x) in this points: f(x0)=y0 , f(x1), ...,f(xn). Pair (xk, yk) we call data points.

We seek estimate values function f(x) class C2  between nodes as of polynomial of third degree  for x <xi-1, xi>. Let us mark us:

              for i=0,1,2,...,n                                                                 (1)

from definition of function f(x) to appear that f’’(x) is the continuous function in interval  <a ,b> and linear  for x <xi-1, xi>, so we have:

                                                                             (2)

 

where:

Integrating twice (2) we obtain:

          (3)

 where 

  (4)

 

Using function’s conditions of continuity and first derivative by algebraic conversion we have linear system of equation :

                                         for i=1,…,n-1                     (5)                        

where:                                                     

                        for i=1,2,..,n-1                    (6)    

System  (5)  has got n-1 equations and n+1 unknown coefficients: M0,  M1,...,  Mn .

Often we accept two additional conditions :

M0=0, Mn=0   (f”(x0)=0, f”(xn )=0: Natural cubic Spline).

Let us mark as :

                                                       for i=2,3,...,n-1                  (7)

                                                      for i=2,3,...,n-1                   (8)    

and for symmetric expression :

.                                                                                               (9)

 From the first equation of system (5)

         we have:

,  or

  by recurrence end algebraic conversion we obtain:

 ,    for k=1,...,n-2                                                        (10)

Using principle of mathematical induction easily proof the truth of this expression.

 From the last equation of system (5) ,we have:

                                                                                                 (11)

after calculation coefficients Mk  we construct function  f(x):

 

 

                                                        (12)

Fig.1 The visual graph of function  f[x]

 

where fi  is  the right side of expression  (3).

We define Heaviside’s function: 

                                                    

and then function f(x) is expressed one rule:

                                                              (13)

                                     

Now we can write in any programming language the basic procedure named splajnP, in which the input parameters are data points: (x0,y0) , (x1,y1) ,  . . . , (xn,yn) , and on the exit of that procedure we will get function  f(x) and her graph.

 

 

Fig.2 Procedure SplajnP written in Mathematica program

 

 

3. Application  and estimation of error

 

Example 1.

In publications or WEB we can often see graph of function, but we do not know value of this function. We will show in three steps, how can we find a rule of function.

1. step: we import a scanned graph of function to the Mathematica program

Fig.3  Graph of unknown function g[x]

 

2 step: we read and write coordinates of screen witch Fig.3:

 

 

Fig.4 Coordinates of screen function g[x]

We read coordinates of beginning (xB, yB) and  end (xE,yE) of graph and corresponding them coordinates of screen are: (xBS,yBS) and  (xES,yES).

Now we define a procedure using linear interpolation which will  transform  coordinates of screen on coordinates of  real:

 

 

Fig.5 ChangeCord procedure

 

After executing of this procedure with parameters DataScrCord we obtain real coordinates:

 

Fig.6 Real coordinates

 

Step 3: we evaluated procedure SplajnP [DataRealCord] and we obtain function f[x] and her graph :

 

Fig.7. f[x] Function graph

The compatibility of Fig.3 and Fig.7 is almost ideal, author knows g function:

g[x]=x Exp[-x] , therefore we can find estimate error in this method:

 

Fig 8.  Estimate error

 

Adding graphic and numeric instructions to procedure SplajnP we can obtain new procedures.

Example 2. In the procedure SplajnTheWall the first parameter are data points, and the second is height of the wall; output parameters are the graph and area of the wall surface.

Fig.9 Result of Procedure SplajnTheWall[data,4]

 

Example 3. We construct the circle with centre on curve and in normal plane  which moves  after curve.

 

Fig.10 The circle in normal plane

 

In the SplajnPipeline procedure the first parameter are data points, and the second is radius of pipeline; parameters output are graph and area of the pipeline surface.

 

 

 

Fig.11 Result of Procedure SplajnPipeline [data,0.7]

Example 4. In the SplajnVolume procedure input parameter are data points, output parameters are graph and volume of the solid formed by the revolution of the curve y=f[x] around x-axis.

 

 

Fig.12 Result of Procedure SplajnVolume[DataHeart]

 

Fig.12 represents the heart of Zygmunt’s bell.

These calculations were made without taking into account of the handle of the bell heart. The mass of the handle was estimated as 20 kg.  The density of the heart is unknown, there are some admixtures: phosphorus, sulphur, etc. According to the accessible data new heart mass is about 350 kg. Specific mass of the heart is assumpted as 7.7 [g/cm3].

Counted mass of bell: mass=47371.7[cm3]*7.7[g/cm3]/1000+20[kg]=384.762 [kg], error of calculations=(384.762 –350)/350*100% =   9.9 %

The heart of Zygmunt Bell is in the Wawel Cathedral in Cracow.

 

 

 

Conclusion

The presented above method of calculations enables the precise defining of the necessary object parameters to its optimalization from the engineer’s point of view, for example the quantity of necessary material to object construction. The presented method of calculations permits to qualify of object with engineering point of view permits his optimization necessary parameters. The method creates the possibility of the objects modelling as well as quick amendments of parameters in the project process. On the base of support of prepared procedures in Mathematica program, it permits to calculate the interesting us parameters.

 

Literature

[1].Stephen Wolfram Mathematica 4 book: Cambridge Universuty Press 1999.

[2]. Z. Fortuna D.Macukow. J.Wasowski: Metody numeryczne WNT Warszawa 1982.

 

 

 

 

 

Abstract

The natural cubic splain theory elements apllied to building engineering have been presented in the paper. Coefficients of cubic spline have been determined in recurring form.  Graphics and numerical procedures were executed in programme "Mathematica". Application of that theory for estimating of curvilinear objects has been presented on chosen examples.