1 Initial assumptions

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1 Methods of evaluation of global and local ship hull deflections based on periodic geodesic measurements assuming linear model of deflections J. Niebylski Ship Research Institute, Technical University ofszczecin, Al Piastow 41, Szczecin, Poland Abstract Control measurements during assembly of a hull on the slipway are carried out in rectangular coordinate system OXYZ which is inclined at a certain angle determined by building conditions. A mathematical model of deflections and the most appropriate methods of their measurements is presented. A diagram of real deflections of the hull obtained in the way of the measurements is also presented. 1 Initial assumptions During assembly of the ship hull its components such as sections or blocks as well as the hull structure as a whole are subject to deflections changing required configuration of the object. Moreover, the hull treated as a rigid body is subject to some movements in reference to the slipway. The phenomenon of deflection and movement of the hull treated jointly is called a deformation of the object. The most important factors causing the deformation are as follows: 1. welding deflections occurring when sections and blocks are connected, 2. thermal deformations caused by local heating of structural elements, for example during straightening, 3. exposing the object to the sun, 4. structural stress due to too short or too long elements, 5. loading of the structure which mass is increased during the assembly on the slipway and increasing support reactions 6. non-uniform settlement of the supports, 7. local deformations of the slipway It is obvious that the bottom structure is subject to the largest deformation.

2 376 Marine Technology and Transportation This is why the behaviour of the outer bottom must be controlled making periodic geodesic measurements supplying with information on deformation. Methods of determination of outer bottom deformations are presented in the paper based on periodic geodesic measurements performed from the surface of the slipway which is reference configuration when examining movements of an object. Prepared methodology is based on an assumption that so called global deformation of the outer bottom as a whole is described using a linear model of deformations. 2 Periodic measurements and determination of space movements of points of object It is generally known that for examination of deformation of an object it is represented by a finite and usually not numerous set of observed points. In the analysed case these are points selected and marked on the surface of the outer bottom, situated in places easy for access for periodic measurements. The grid of the observed points should uniformly cover the whole surface of the outer bottom. As it was already mentioned, the slipway is a reference configuration. Cartesian coordinate system OXYZ is related to the slipway. The system OXYZ is a reference system in analytical investigation. There is a defined relation of the coordinate system OXYZ with the hull system Oxyz. It should be accented that the hull system Oxyz is appointed when assembling the hull - with relation to the basic system OXYZ. Knowing the coordinates X-^Y-^ Z- of the origin Oof the system OXYZ it is possible to calculate the coordinates of any point "i" of the assembled hull in the coordinate system related to the slipway. The most convenient and the fastest method of control measurement is a method of reference line, applied in various variants and discussed in literature [2]. As a result of measurements in this method values of displacements ^ and?/y in plane {x,y} are obtained. Resultant?/^ being total horizontal displacement of point "i" is treated as a displacement vector of the point in plane {x,y} which length is and directional angle is CD (2) Apart from the components of displacements the values of mean errors /% and m,, are calculated.

3 Marine Technology and Transportation 377 Observations are usually carried out along straight lines p parallel to X axis and straight lines q parallel to Y axis. In this way an orthogonal grid is created composed of two groups of reference lines p and q. The grid is a horizontal base for measurements of deformations of an outer bottom on the slipway. The grid of the lines is referred to horizontal reference system OXY and is parallel to it. As it was already mentioned measurements of the displacements are carried out with respect to reference points. Colinearity of the reference points is periodically tested. Obtained values of displacements i^ and u are treated as real movements when they are larger than doubled values of their mean errors ^>2m- (3) where u > 2m (4) ^ S mf and m[ or m and m\ - mean unit errors of measurements of segments r and s, tis - number of segments s between point Qo and given reference point Q on line q, rir - number of segments r between axis X and given point P. Otherwise (in case when there is < instead of >) the obtained values of the displacements are supposed to occur due to observation errors and very small movements not detected by measurements. Measurements of vertical displacements of the points of the object are done by the method of optical levelling [2]. The term vertical displacements of point "i" in period [to,ti] means component 11 '' of space displacement?/?', measured in the direction of Z axis, referred to the original base plane. Value of the component can be calculated as a difference between vertical coordinates: original Z and actual Z/ <'=Z/-Z (7) In the actual measurement possible vertical displacements of base points must be taken into consideration.

4 378 Marine Technology and Transportation 3 Vectorfieldof displacements and deformation tensor Space displacements observed and calculated in fixed coordinate system OXYZ should be treated as vector i// of components %,, ^, w,. Set {uj of vectors determined on the observed object is a vector field of displacements. For a given time interval Ato-i vectors of this field depend only on original coordinates X, Y, Z of these points. The dependency can be expressed as a function of the field /^,y,z;+^/^,y,z; (8) where #x, Qy, z - unit vectors (versors). Based on the above assumptions the components of vector w/ belonging to the mentioned field can be treated as scalar functions of the original components of point "i" (9) In the theory it is assumed that functions (9) are invertible, continuous and n times differentiable Taking into consideration relation between the coordinates of system Oxyz of the hull and fixed system OXYZ of the slipway the equations can be rewritten in the following form Thus, from the above specified displacements vectors (w,w w^}, the translation matrix is obtained the rotation matrix 0 = 0 0 (12) the linear strain matrix

5 Marine Technology and Transportation = (13) For the linear model of deformations the following dependency is valid 0 (14) It is worth recalling that elements of matrix e define deflections of the object as a whole and elements of vector /- and matrix <P define displacements of the object as a rigid body Deformation described by these parameters are referred to as a global deformation of the object. 4 Conclusions The presented method of measurements and analysis allows for obtaining real deflections of the hull during building and is used for current control of the manufacturing and examination of influence of various factors on behaviour of the ship structure in sequential stages of building. This examination is caused by necessity of full automation of building, especially for assembly of the hull with more strict tolerance or without margin The presented model meets requirements resulting from trend to automation and computerisation of industrial metrology. References 1. Doerffer J.: Technology of the hull bending (in Polish). Wydawnictwo Morskie, Gdynia Niebylski J : Theoretical principles and measuring technique of optical levelling method (in Polish). Geodezja i Kartografia. Rocznik XXVII, 1978, nr Niebylski J : Geodetic measurements in inclined co-ordinate systems for the use of shipbuilding (in Polish). Prace Naukowe Politechniki Szczecihskiej 229, Instytut Okr^towy 10, Szczecin Platek A.: Geodetic measurements of displacements of machine foundations (in Polish), Zeszyty Naukowe AGH, Geodezja z. 21, Krakow Platek A.: Determination of fast deformation of a structure from relative displacements of its points Proceedings of the Mining and Geodesy Commission of Polish Academy of Sciences, Geodesy 35, Krakow Timoshenko S., Goodier J.N.: Theory of elasticity. Me Grow-Hill Book Company, Inc., New York-Toronto-London, 1951.

6 380 Marine Technology and Transportation

Optimisation of the collection of geometrical data in shipbuilding. Julian Niebylski Technical University of Szczecin, Ship Research Institute, Poland

Optimisation of the collection of geometrical data in shipbuilding Julian Niebylski Technical University of Szczecin, Ship Research Institute, Poland Abstract At the present stage of ship design a strong