4.1 Introduction

Analytical studies on time dependent vibratory behavior of thin elastic shell structures date back to Rayleigh’s studies in 1881, and they are followed by the Love’s general formulations in 1888. Long before experimental or numerical models were developed, it was possible to express the dynamic behavior of thin shells analytically. With many recent studies, experiments on laboratory specimens have verified the accuracy of these analytical solutions (Hwang 1966, Souza and Croll 1983, Mal’tsev, Konon and Adishchev 1984).

One of the objectives of this project is to investigate the vibrations of a thin hemispherical shell by means of three-dimensional equations of elasticity. For this purpose, functions satisfying equations of motion and boundary conditions for the specific structure must be developed. Then, numerical results for the free vibration frequencies–natural frequencies and associated mode shapes–are obtained and compared to those of finite element methods. The next section proceeds with the descriptions of classic shell theory and its derivation for vibrations of thin spherical shells. The information presented here is paraphrased from Kraus (1967) and Soedel (2004).

4.2 Vibrations in Classical Thin Shell Theory

The definitions of reference surface, thickness and the edges of the reference surface are necessary to identify a shell element. In case of homogenous materials, it is a common practice to establish the midsurface as the reference. The reference surface represents the overall geometry of the element.

There are three approaches for determining the state of stress in a shell surface. The first is ‘bending theory,’ which considers forces, moments and deformations including their variation within the shell thickness. The second is ‘membrane theory,’ a simplified method, owing the thinness of the shell neglects the bending and twisting moments. The third is ‘thin shell theory,’ an intermediate step, specifically developed for ‘thin’ shells.

In thin shell theory, the integration through the thickness is simplified by substitution of stress resultants and couples. Furthermore, statically equivalent reference surface forces replace the forces acting on the shell surface as well as the body forces. Although the developers of this method did not provide a clear definition of ‘thin,’ generally a shell is accepted to be thin if the thickness is less than one tenth of the radius of curvature of the reference surface.

Equations of motion as well as the strain-displacement and stress-strain relationships appear in the governing differential equations. Because of this fact, some simplifying assumptions are established through the development of the method:

• The shell is thin.
• The deflections are small.
• Geometric nonlinearity is not introduced.
• The plane through the thickness of the shell is homogenous and remains normal to the reference surface at all times.
  

Since a complete explanation of the theory is available in a large body in literature (Kraus 1967, Soedel 2004), the subsequent sections proceed from the equations of equilibrium. Within this chapter, the procedure of obtaining dynamic parameters of a spherical dome is discussed.

Equation 4-1 presents these equations of motion based on moments and in-plane shears, shown in Figure 4-2, and membrane forces and out of plane shear shown in Figure 4-3. In these equations, ρ and h are density and shell thickness respectively while R is the nominal radius and t is time. The direction of the unit vectors t1, t2 and n are also indicated in the figures.

(Equation 4-1)

By substituting the last two equations into the first three, Q₁ and Q₂ can be eliminated from discussion. For sake of simplification Equation 4-1, can also be written as:

(Equation 4-2)

Where O is the differential operator obtained after reducing shearing forces. The next step is imposing the boundary conditions, which operate on all three displacements. In Equation 4-3, n is the number of boundary conditions to be applied.

(Equation 4-3)

Assuming that the period of vibration and the phase of motion would be the same for every point along the element, the spatial and temporal variations of the shell are separated and represented in equation 4 where ω is the natural frequency.

(Equation 4-4)

When Equation 4-4 is substituted into differential equation of boundary conditions a set of homogenous linear equations are obtained. The eigenvalues derived from these governing differential equations are the desired natural frequencies. In addition, by examining the amplitude plot of the desired closed form displacement solution for that specific root, the deformation shapes, in other words mode shape corresponding to that frequency, is obtained (if u is plotted, the obtained mode shape is over r coordinate, similarly for v and w, θ and φ respectively).

The adaptation of the above procedure to practical conditions can also be performed by employing variational methods, such as the Rayleigh-Ritz method. This method requires formulating an approximate deflection shape–usually in the form of summation of a series of sine curves, which are admissible according to the boundary conditions. Then by minimization of energy, the natural frequencies are obtained. Increasing the number of terms of the initial displacement function, the solution converges to the exact result, if an admissible function with a reasonable approximation is provided.

4.3 Vibration of Spherical Shells

In this section, θ and φ are introduced as the spherical coordinates in the plan and in the vertical section of the dome respectively. The discussion will be confined on the θ- independent free vibration –in other words axisymmetric free vibration– of non-shallow spherical shells with closed apex. Such a restriction reduces the governing equations Equation 4-1 to Equation 4-5.

(Equation 4-5)

For non-shallow shells M_θ, M_φ, and Q_φ are set equal to zero and (owing the thinness of the shell) the square of the ratio of thickness to radius (h²/R²) is approximated as zero. Through a series of reductions and substitutions (Kraus 1967, pg. 333-335), the governing differential equation is obtained.

(Equation 4-6)

In Equation 4-6, E and R are Young’s modulus and radius of the sphere respectively. The general solution for Equation 4-6 is:

(Equation 4-7)

In Equation 4-7, P_λ and Q_λ are Legendre functions of first and second kind. For spherical shells with closed apex, Q_λ(cosφ) is singular and must be excluded from the solution by setting D equal to zero. The generating functions for Legendre functions and first few Legendre functions of first order are provided in the Appendix C.

The general solution is simplified as Equation 4-8.

(Equation 4-8)

The next step is forcing Equation 4-8 to satisfy the boundary conditions for hinge end condition. The displacement normal to the reference surface is equal to zero, when φ=φ_o–where φ_o is the half of the angle that shallow shell subtends.

(Equation 4-9)

Omitting the trivial solution C=0 in Equation 4-9 reveals;

(Equation 4-10)

The roots of Equation 4-10 give the desired natural frequencies. Solving the Equation 4-6 for known frequencies give the unknown coefficient C, this reveals the amplitude of the mode shape.

4.4 Procedure Employed

The procedure followed in this study is summarized below:

• Find the angle φ₀ that the dome subtends according to the geometry of the dome.
• Notice that if φ₀ is less than 30⁰ the approximations of shallow spherical shell are applicable and roots for governing differential equation are Bessel functions. For nonshallow spherical shells, the Legendre functions are the general solutions of the governing differential equations.
• Obtain the geometric and material properties of the dome: the radius, mass density, modulus of elasticity and Poisson’s ratio.
• Note that for the domes with closed apex the Q_λ(cosφ) is singular, therefore the generic solution is w = CP_λ(cosφ).
• Determine the order of the Legendre function such that P_λ(cosφ₀)=0 for pin connection.
• When the angle that the dome subtends does not yield an integer value for the Legendre polynomial, adopt the series expansion approximation to assess the Legendre function of fractional order (Abramowitz and Stegun 1972).
  

• Insert the general solution (with known λ) into the governing differential equation  and solve for r, or use the relationship 2 + r = λ (λ +1).
• By inputting r value into the following equation, solve for Ω.
  

• Find the natural frequency ω from the following equation  by inserting the value for Ω.
• In order to find the mode shape associated with the natural frequency, utilize the series expansion approximation while plotting the w function.

4.5 Application of the Procedure to Existing Masonry Domes

The above formulations of thin shell theory are adopted in order to solve for the dynamic characteristics of the structures in study. The unique geometric shapes and boundary conditions of each building of interest present peculiar challenges in the application of this method.

Both of the structures in study are pendentive domes, which include curved spandrel edges at four corners. These additional elements make the structures of study deviate from pure spherical segmental dome behavior.

4.5.1 State Education Building (SEB)

The procedure is followed according to the physical geometry of the building [Figure 4-4]. The natural frequencies are obtained through the systematic computation explained above. A MATLAB code is written in order to plot the mode shapes in the form of vertical section through the apex of the dome. The MATLAB code is presented in Appendix B, while the results obtained through this code are provided in the following paragraphs.

• The physical Properties of the Dome:
  R=6.28m
  φ₀=45^o
  ρ=1800 N sc²/m²
  E=7.6×10⁹ N/m²
  v=0.1

• Determination of the first natural frequency:

The analysis of SEB with this method proceeds on the assumption that the open and closed apex behavior does not differ considerably for the lower modes. Consequently, the results for the closed-apex dome in the geometry of SEB are obtained and then compared with the FEA and EMA results in Chapter 5.

The roots of the Legendre polynomial, that satisfy the boundary conditions, are best found by examining the charts for P _λ (cos (π/4)) =0 [Appendix C]. The shift from positive to negative between two integer orders of Legendre polynomials, (a sign change from i to i +1) is the indication of a root between the i^th and i+1^th orders. Then, by several iterations, the desired value, which satisfies the boundary conditions for a dome that has an angle of embrace of 45°, can be found.

For SEB, it will later be seen in Chapter 5 that the calibration and validation of the FE model reveals that the boundary conditions at the peripheral edges of the actual structure do not allow rotation in all three axes. However, at these edges the physical configuration of materials found to prevent the translation in the axis perpendicular to the plane of symmetry, which is the deflection restraint with the requirement of P_λ(cosφ₀)=0 Hence, the general solutions of governing differential equations –Legendre functions– are not forced to satisfy the slope restraint requirement P'_λ(cosφ₀)=0. In the following section, the procedure as applied to SEB will be illustrated.

• First natural frequency:
  λ=2.54156 substitute the λ into the following equation,
  2 + r = λ (λ +1) then, r = 7.001
  

Now substitute the value of r into:

and, Ω₁²=0.7309 and Ω₂²=9.6740,

Due to the square roots of the ‘Ω₁²’ term, the natural frequencies are arranged in two groups: upper branch and lower branch. The lower branch includes an infinite number of natural frequencies converging to the asymptote of Ω² = 1. The upper branch, which also has infinite frequencies, results in significantly larger values [Figure 4-5]. Interestingly, the first natural frequency delivered by upper branch is higher than the highest natural frequency of the lower branch. Although the physical description of the upper branch is not clear in references, in this study, it is considered a spurious result, and the further discussion is confined to the results obtained from lower branch.

By using the formula, the first natural frequency of SEB is found as 279.55 rad/sc, which is equal to 44.49 Hz. The same exercise is repeated for the higher
order modes. The natural frequencies obtained with this routine are presented in Table 4-1. The mode shapes are plotted by the help of MATLAB code, which utilizes the series expansion method, and approximates a Legendre function of non-integer order as sum of infinite Legendre function of integer order (Legendre Polynomial). Since it is not practical to add a infinite number of Legendre polynomials, it is necessary to truncate the series at an acceptable level of accuracy. The number of the Legendre polynomials included in the series expansion approximation determines the highest possible mode shape to be obtained. In this study, the first 25 orders of Legendre polynomials are included in the analysis and the first five mode shapes are illustrated in Figure 4-6.

Table 4-1: The first five axisymmetric modes estimated by thin shell theory, SEB, PA.

4.5.2 City County Building (CCB)

A similar method is applied to the City County Building. The following paragraphs discuss the solution for the hinge end boundary conditions at the peripherals.

• The Physical Properties of the Dome:
  R=6.73m
  φ₀=35^o
  ρ=1500 N sc²/m²
  E=8×10⁹ N/m²
  v=0.1

• Determination of the first natural frequency:

The physical dimensions of the CCB are extracted from the available construction drawings [Figure 4-7]. The noteworthy difference in the CCB is –as shown in Chapter 3– the peripheral boundaries of the dome exhibit rotation-based restraints along with the translational restraints. Therefore, the meridional slope constraints at boundaries (P'_λ(cosφ₀)=0) need to be considered as a criteria in the analysis of the CCB. However, the scope of this study is limited to the dynamic parameters delivered by thin shell theory for hinge end boundary conditions.

The same procedure is repeated in order to obtain the natural frequencies and corresponding mode shapes. The first five natural frequencies are presented in Table 4-2. The mode shapes obtained for this building exhibit the similar manner as presented in Figure 4-6.

Table 4-2:  The first five axisymmetric modes estimated by thin shell theory, CCB, PA.

4.6 Concluding Remarks

As the domes in this study differs from a spherical segmental dome because of their pendentives and as the real behavior of the tile and mortar assembly deviates from those of simple constitutive law, obtaining an exact correlation with the FEA and EMA is not the main goal of the study while employing the theory of thin elastic shells. Instead, it is the intention of this chapter to introduce a well-established and previously verified method for the verification of the experimental data and finite element solutions. The comparison of the results is presented in sections [[ INSERT HYPERLINKS ]] 5.4.1.5 and 5.4.2.5.