DEFORMATION OF EARTHS SURFACE CAUSED LOADING OF TALL BUILDING
Автор: P. KOLLAR, M. MOJZEŠ, M. VAĽKO
ABSTRACT
Tall buildings can cause deformations of the earth’s crust for long distances from the area of their realization. Monitoring the deformation of tall big buildings from geodetic networks realised near the building can also have deformations from this point of view, and analysis of the measurements can result in false conclusions. The loading effect of the earth crust must be excluded from the measurement parameters. The paper presents a model computation of the loading effect produced by tall buildings constructed of simple geometric forms on the earth’s crust.
1. INTRODUCTION
When studying of surface loading of the earth, we can consider two extreme cases of deformation: absolutely rigid with no deformable body or a fluid body without any resistance form external pressure. The actual response of loading on the Earth’s surface is between these extreme cases. The materials of the Earth’s surface can have common properties, i.e., elastic, viscoelastic or plastic. If the external loading is only for a time short, then the deformation is given only consists elastic properties. This means that when the external forces are removed the Earth’s surface regain its original undeformed shape. An ideal elastic material lacks memory, i.e., any stress or deformation is uniquely determined by the final values of the loading. On the other hand, if the force effect is long term, then elastic and rheological effects can be found in the reaction of the material to external loading. Those effects are modelled as viscoelastic and plastic. In both cases is there a new factor - time. The time factor directly connects with the mechanical memory of the material depending on the long-term effect and history of the loading process. In contrast to elastic deformation, viscous and plastic deformations are not reversible, i.e., after by removing the loading, the new shape of the Earth’s surface is not the same as its original undeformed shape. Those effects are not viscoelastic but plastic.
A deeper knowledge of the physical – mechanical properties of materials and modern computer equipment can allow for a very precise model of the reaction of the Earth’s surface to loading over time, which is the next step in understanding complicated phenomena of the Earth’s crust. Geodesists need to know these laws of surface deformation and their time evolution for of geodetic networks the design, analy
This paper is analyses the surface deformation of the Earth caused by symmetric loading of its boundaries. We consider that the background is an elastic body modelled as a homogenous and isotropic half-space without any time response. The results in analytic form are presented using actual examples.
2 DEFORMATION OF A BACKGROUND BY AXISYMETRIC LOADING
This paper shows how to determine deformations of the Earth’s surface by axisymmetric loading of its boundaries (i.e. plane z = 0). The solution is based on these conditions: • The basement is a homogeneous, isotropic and perfectly elastic body (half-space) of the Boussinesq type, • The material properties of the elastic body are: Young’s modulus of elasticity Ez (unit MPa) and Poisson ratio μz (dimensionless), • The boundary of the half-space (surface z = 0) is on the area of a circle with a radius R (unit m) (Fig. 1a) or on an annulus area r1 ≤ r ≤ r2 loaded by a permanent, uniform, and normal loading of a magnitude pi (i = 1, 2) (unit MPa) (Fig. 1b). The construction of a half-space boundary flexure surface νz (ρ), where ρ = r/R, by loading p1 of its boundaries is known as the first boundary problem in the theory of elasticity. In this paper we present the Green functions approach. First, we have to consider two auxiliary problems: in the first problem we try to find the function for the deformation half-space boundary by unit loading, which is uniformly distributed on a circular line with a radius (Fig. 3). In the second problem we determine the integral relations for the displacements νz (ρ), whic
The vertical displacement at B (r, ϕ) on the half-space boundary (z = 0) by the unit force P = 1 located at
Fig. 1 Schema of loading a half-space boundary (z = 0) on the area of a circle with the radius R (Fig. 1a) and an annular area r1 ≤ r ≤ r2
3 PRACTICAL EXPERIMENTS
3.1 Half-space circle loading
Inputs for circle loading: µz = 0.3,
R = 10 m,
p1 = p2 = 0.1 MPa
The experiments for this loading case are presented for 3 different basements, for example: a clay basement (Ez = 16.0 MPa), a sandy basement (Ez = 42.0 MPa) and a rocky basement (Ez = 120.0 MPa). The deformation of a half-space by a permanent circle loading with a radius R = 10 m is given by Fig. 6.
There is room for a question: How far from the structure is a measurable effect caused by the loading? In the case of the clayed basement a vertical displacement of magnitude 1 cm is approximately 57.1 m, in the case of the sandy basement, approximately 22.3 m, and in the case of the rocky basement, approximately 9.9 m from the centre of the loading. The vertical displacement of a magnitude of 1 mm is approximately 568 m in the case of the clayed basement, in the case of the sandy basement, approximately 217 m, and in the case of the rocky
3.2 Half-space annular loading
The loading of a half-space by annular loading with radii of r1 = 8 m and r2 = 16 m with the same physical inputs as in sec. 3.1, is imaged in Fig. 7. In this case of deformation, the vertical displacement of a magnitude of 1 cm is in case of the clayed basement, approximately 110 m, in the case of the sandy basement, approximately 42 m and in the case of the rocky basement, approximately 18 m from ґthe centre of the loading. The vertical displacement of magnitude of 1 mm is, in the case of the clayed basement, approximately 1010 m, in the case of the sandy basement, approximately 416 m, and in the case of the rocky basement, approximately 148 m from the centre of the
4 CONCLUSION
This study provides basic information about the deformation of the Earth’s surface by circular or annular loading caused by motion from the accumulation of a mass (tall buildings, large dams, filling or cutting near highways, the accumulation of a water mass, the accumulation of a snow mass, etc.). The modelling of loading effects is very useful in geodesy. A suitable solution means eliminating time-dependent systematic error and increasing the accuracy of geodetic networks. It can be used in the construction and realisation of geodetic reference networks, the analysis of repeat measurements, building stability, etc. It is necessary to undertake take many geodetic experiments in order to prove theoretical models or
ACKNOWLEDGEMENT
This project was supported by Grant Agency of the Slovak Republic (1/0882/08) and by Grant Agency for Science and Research of the Slovak Republic (LPP-0216-06).
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