Using the superposition principle, the generalized displacement u eiwt of the rigid foundation can be represented as. In the past, the most commonly studied case has been that of an incident vertical S wave of unit amplitude. The amplitude of the free field motion at any point on the surface is then equal to two.
This effect becomes particularly important when the incident wave's wavelength is comparable to the characteristic dimension of the foundation. The most important unknown of the soil-foundation-superstructure interaction problem is the movement of the foundation. Resistance to this movement is provided by the soil and the inertial forces of the foundation.
Once the foundation displacement is determined, all other quantities such as the stress and strain of the superstructure can be obtained.
CHAPTER III - LONG FOUNDATIONS SUBJECTED TO SH-WAVE EXCITATION
A simple idea of the image method makes it possible to solve some cases of the two-dimensional half-space problem. For the SV and P wave motions in the plane, the second spatial derivatives of the potentials are also required. To apply the image method to the z component of the displacement, w, consider the spreading object or foundation shown in Figure 3-1.
In the following sections, several :::problems in foundation dynamics are solved precisely according to this idea. As will be shown, certain aspects of soil-structure interaction can be understood and explained using this approach.
Since a circle is symmetrical with respect to its origin, it turned out that the angle of attack has no influence on the response of the foundation. The disadvantage of the semicircular foundation model from an analysis point of view is that the embedding ratio of the foundation cannot be varied. In contrast, the semi-elliptical shape of the foundation has both the semicircular foundation and the strip foundation (Al Sb) as borderline cases.
The embedding ratio can be easily changed by changing the focal length of the ellipse. Using the above coordinates, the integral solution of the two-dimensional scalar wave equation degenerates into an infinite series of Mathieu and modified Mathieu functions. This simplification makes the longitudinal element of the impedance matrix independent of all other components.
To calculate the value of the impedance function Is, the radiating wave wR from the elliptical foundation was expanded into a generalized Fourier series of orthogonal Mathieu functions.
Massless foundation)
As the foundation is subjected to a unit excitation, the surface boundary condition for the half-space is automatically satisfied by symmetry if only Mathieu functions are used. Therefore, the forces resisting the foundation can be obtained by integrating the shear stress o-i;z over the surface. The equivalent stiffness and damping coefficients associated with the real and imaginary parts of Ks are shown in Figure 3-2.
These results show that the equivalent stiffness and damping coefficients are fairly constant for dimensionless frequency values greater than 0. At low frequencies, the stiffness coefficient increases with the embedment ratio; at high frequencies, the value of the stiffness coefficient is between the values corresponding to the strip foundation (AlSb) (h/b = 0) and the semicircular foundation (AlbS,Alb 9).
26- Umek{Al 71
In both cases, there is no mass movement of the wall relative to the foundation, and,. The presence of the canyon will change both the resistance functions and the driving forces derived in the previous section. Due to the presence of the canyon, the driving force depends on the angle of the fall.
This arrangement was made so that the canyon effect could be studied without interaction with the superstructure. Equation (3-4. 7) now satisfies all the boundary conditions of the interaction problem as is now the displacement at the foundation pth. This envelope, I 6eJ P' provides an upper bound on the response of the pth foundation if it is the only structure in .
These figures have been adjusted so that the influence of incidence angle and separation distance can be studied together. This phenomenon can be explained by the standing waves created by the incident interference and the reflected wave from the larger back wall. 85, which shows that the resonant frequency 11 of the small wall is highly dependent on the distance to the larger walls.
11, the response of the large center wall is not greatly affected by the smaller outer wall. As can be seen from the above analysis, the weight and size of the structure play an important role in the interaction process. This last condition is made for the half-space free area requirement, and can be satisfied simply by taking w?)(!:_).
They should be interesting to compare with the elliptical foundation because of the similarity in overall dimensions. 2], it was noted that the 11radiation11 attenuation is directly proportional to the area of the scattering surface. Therefore, the comparison of these two cases was performed by equating the circumference of the two cross-sections.
The size of the box was also changed to gain additional insight into the problem. The asymptotic amplitude of the Bessel function for large arguments r decays as 1/{r [Abramowitz and Stegun (1971)],. The two types of viscoelastic materials defined above will now be used for the investigation of the material's distribution of energy.
Each figure includes the ratios, a2/a1 = 5 and 10, so that the effect of the medium size can be studied. The static impedance of the enclosed medium is always greater due to the additional confinement nearby. Therefore, as the outer boundary moves further away, the order of the mode shapes in the enclosed medium becomes higher for the same a0.
The introduction of damping in the material smooths out the large amplitude oscillation and contributes to the imaginary part of the impedance. Therefore, one can conclude that the harmonic behavior of a bounded medium is not the same as that of the semi-infinite.
By using the Green's function on the surface of the half space, the displacements everywhere can be expressed in terms of the unknown traction via an integral formula. In this section, a stricter approach is followed, in which the singularity of the Green's functions is properly accounted for. In the applications presented in this section, the former approach is followed because the integral of the lower order functions, Ix - x. )n, over the area, A., can be carried out with less.
The infinite integral is convergent since·, when z ➔ co, the nonoscillatory part of the integrand for. Since the calculation of the influencing functions f_ requires considerable effort, it is therefore wise to save the results for the future. By taking advantage of all these properties, the calculation of the final results can be carried out economically.
Their unique solutions must be determined by the compatibility conditions of the three displacement components on the foundation surface. Again equation (4-1.3) is reduced to an independent integral equation for the tangential stresses in the direction of the applied load. For the calculation of the vertical flexibility, Cvv' and the horizontal flexibility, Chh, the displacements must fulfill the conditions for rigid body translation.
But for the evaluation of the rocking compliances, Cmm, and for the torsional compliance, Cu, rigid body rotation is imposed. Results for the cases in which the ratio of the sides c/b is equal to ½, 1 and 2 are plotted in Figure 4-2. 4 versus a0• In these cases, the side b is chosen to be the characteristic dimension of the rectangular foundation, and the area depends only on the ratio c/b.
Therefore, the force, Ql' and the displacement, .6.I\, are related to the size of the foundation area. 7, the torsional resistance is greater for the elongated foundations if the area is kept constant during the equation, because the moment depends on the leverage of the loads.
Both of the above analyzes used the relaxed boundary conditions of equation (4-2. 11) for the vertical. From studies of simply bonded foundations, it is known that the stress is usually concentrated at the outer edges of the. 75 shows a consistent change of 15% even for the static solution, because more than half of the area was removed.
The "fixed" boundary conditions required for calculating the driving force and torque for the incident plane SH wave are then. To illustrate the differences created by the geometric shape of the foundation, consider a square foundation with a hole in it. An application of the above integral formulation, which can be extended to flexible foundation analyses, will be made in chapter.
We now use method II, taking advantage of the simplicity of polar coordinates. Matching boundary conditions for displacements at the center point, rp_, of the ring, the discrete form of Eq. To illustrate the degree of convergence, the static compliance value for the annulus was calculated.
This rapid convergence is caused by the smoothing process during the double integration of the stress distribution over the foundation area. This is again evidence that the stress concentration occurs at the outer edge of the ring foundation. Therefore, the influence of the hole on the compliances is not significant until it is quite large.
From the results of Section [3-4], the physical phenomena of this multi-structure interaction problem vary according to the order of the structures. Method II: Observation points for this method are sampled at grid nodes of the grid.
S VIBRATION (A)
HORIZONTAL
S VIBRATION (B)
S VIBRATION t:. measured vertical motion
VERTICAL
0 HORIZONTAL
S VIBRATION 6 measured vertical motion
