Theoretical Models

The theoretical modelling of any structure attempts to incorporate simplifying as-sumptions that improve the computation efficiency, without significant detriment to the resulting natural frequency estimation. Aspects of modelling a tall building for natural frequency estimation include, but are not limited to, the distribution of mass and stiffness, the type of deflection actions, and the interaction between the soil and foundations. The assumptions adopted are directly linked to the complexity of a tall building structural system. Some simple structural systems may be accu-rately modelled as an idealised beam with simple boundary conditions, while more complex structures may require significantly more detail.

The theoretical modelling of tall building structures has two options for repre-senting the dynamic system: a lumped parameter system, or a distributed param-eter system. For lumped paramparam-eter systems, the structure is discretised into a set of points, and the motion of each point in the system is modelled with a tributary mass that is concentrated at the point. The set of lumped masses are typically

con-|2²

(a) System Representation (b) Free Body Diagram

)

Figure 2.1: Lumped prarmeter model of a two-storey building experiencing transla-tional vibration.

nected by elements that represent the stiffness and damping of the structure. Most often the models represent the stiffness by springs or equivalent beam elements. The damping is typically represented by dash-pots, or more often ignored due to its in-significance on the natural frequency for large civil engineering structures. Once the system model has been established, the natural frequencies of the system can be de-termined using modal analysis techniques [74]. The dynamic behaviour is described by second order differential equations, and the discretisation of the structure limits the results to a finite number of natural frequencies.

An example of a lumped parameter system model of a two storey building is presented in Figure 2.1. The example can be extended to include structures that include more than two storeys, and the mass and stiffness parameters can vary throughout the system. The lumped masses are readily calculated, and the stiffness parameters can be calculated based on the arrangement and specification of walls, columns, and other structural elements that resist lateral deformation. It is noted that the model presented assumes a rigid connection between the first storey and the foundations. If required, this assumption can be replaced with a more suitable alternative that models soil structure interactions.

Depending on the number of degrees-of-freedom contained within a model, lumped parameter systems can be computationally intensive to solve for the natural frequen-cies. Degrees-of-freedom are the set of independent displacements or rotations that define the deformed position of the system. For example, Figure 2.1 has two degrees-of-freedom, x₁ and x₂. A coarse discretisation of a system can aid in reducing the degrees-of-freedom, but the effect on estimation accuracy needs to be considered.

A coarse discretisation coupled with an approximation technique, such as Rayleigh’s method [74], is a common approach when using lumped parameter models for quick theoretical estimates of the fundamental natural frequency [68]. The premise of Rayleigh’s method is that the maximum kinetic energy and the maximum potential

energy must be equal in a conservative system. The estimates from this method are an upper bound for the fundamental natural frequency. Approximations of higher mode frequencies can be obtained by applying the Ritz method [171], which is an extension of the Rayleigh method.

For the distributed parameter case, the mass of the structure is considered to be a series of infinitely small elements that are distributed throughout the system.

Vibration of the structure causes each of the infinite number of elements to contin-uously move relative to each other. A continuous function of the relative position along the system is used to spacially describe the time response of the system. As for the lumped parameter system representation, the assumptions used in formulating the model are critical to the accuracy of the results.

Of particular interest to expeditious estimates of natural frequency is the mod-elling of a tall building as an idealised beam. For tall buildings with relatively con-stant properties throughout height, the discrete set of elements (beams, columns, walls, etc.) that comprise the structure are replaced by a continuous medium of equivalent properties. By assuming the incompressibility of the continuous medium, the estimation of the natural frequencies is reduced to a single linear differential equation with constant coefficients.

A number of fundamental theories exist for the analysis of the transverse vibra-tion of idealised beams [88]. The Euler-Bernoulli beam model, also known as the classical beam model, is one such theory that models the flexural action of uniform, slender beams that are composed of linear, homogeneous material. Note that the classical beam model only incorporates flexural deformations via the assumption that plane sections remain plane. Equation (2.1) presents the formula for estimat-ing the natural frequency of a fixed base uniform cantilever usestimat-ing classical beam theory;

where L is the length of the beam, E is the modulus of elasticity, I is the area moment of inertia about the neutral axis, m is the mass per unit length, and the subscript i denotes the mode of vibration.

Another potentially important translational deformation action in tall buildings is shear deformation. Flexural deformations generally dominate when considering a slender structure, however shear deformations become important with reduced slen-derness ratios and for analysing higher modes of vibration. Equation (2.2) presents

the formula for estimating the natural frequencies of a fixed base uniform cantilever

where κ is the shear coefficient, G is the shear modulus, and ρ is the mass density.

The natural frequency estimates from the shear beam model is proportional to 1/L, as opposed to 1/L² for the flexural beam model, and the natural frequencies increase linearly with the mode number.

For most tall buildings, the lateral stiffness is not constant with height, and tends to decrease from a maximum at the base to a minimum at the top. This is certainly the case for tall buildings with a core comprised of reinforced concrete shear walls.

A small percentage of the walls will extend from the foundations to the roof, while others will be terminated, or have reduced dimensions, at various levels according to the design lateral load resistance requirements. Equations for idealised beams with tapers are applicable in these cases, which take the form of Equations (2.1) or (2.2), and have adjusted values of λ_i that account for a tapering effect [115, 182]. A taper in the dimension perpendicular to the vibration motion tends to increase the natural frequency for all modes of vibration. For a taper in the plane of vibration motion, the first mode natural frequency tends to increase, while the second and third modes tend to decrease. More complex theoretical models that account for arbitrary distributions of mass and stiffness have also been developed [105].

Rotatory inertia is also ignored in classical beam theory. Rotatory inertia is the inertia associated with the local rotation of a beam cross section during flexural de-formation. Corrections to classical beam theory to account for rotatory inertia were established by Rayleigh [133]. A model proposed by Timoshenko [172] incorporates the flexural deformations, shear deformations, and rotatory inertia effects. The ef-fect of rotatory inertia on natural frequency is generally less than shear deformation, and both tend to reduce the natural frequencies of beams compared with flexural theory predictions.

Closed form solutions of models that incorporate flexural and shear deformations are generally not attainable, which leads to the application of numerical methods for solving such models to determine the natural frequencies. Alternatively, the flexural and shear deformations can be combined using Dunkerley’s formula [45] to calculate a lower bound estimate of the fundamental natural frequency [78]. Dunker-ley’s formula is not limited to combining the flexural and shear deformations. The generic form of Dunkerley’s formula, Equation (2.3), allows the input of multiple

deformation actions.

1 f² = 1

f_f² + 1

f_s² + . . . (2.3)

Where f_f and f_s are the fundamental frequencies from a flexural beam model and shear beam model, respectively.

The application of Dunkerley’s formula implies the system is modelled as isolated components. This approach has been applied to shear wall buildings [60], coupled shear wall buildings [135], and wall-frame buildings [57, 64, 147] by decomposing the deflections into the component actions. The fundamental natural frequency from each action is then combined using Dunkerley’s formula, with results found to be within 3% of estimates from Rayleigh’s method [60]. Furthermore, the soil structure interaction can be incorporated into the model by modelling the deflection actions from the rotatory and translational motion of foundations on elastically yielding soils.

Rotatory, or rocking motion, of structures becomes more pronounced as the stiffness of the building increases relative to the stiffness of the ground. Rocking motion can manifest as part of the translational mode of a building, or as a rocking mode for a very stiff building resting on soft ground. Salvadori and Heer [138] com-bined linearly varying shear and flexural rigidities with rocking and translational motion using Dunkerley’s formula. The resulting formula for the fundamental nat-ural frequency was able to match the upper and lower bounds of a large full-scale measurements database, when using inputs that bounded expected structural and soil parameters.

Another action that can influence the natural frequency estimation is the axial compression and deformation of walls and columns. In some cases this action is excluded since it is deemed insignificant relative to the desired accuracy of the estimation. When axial compression has been included in the estimation method, a decrease in the natural frequency was observed [36, 134]. The decrease was most significant for the fundamental natural frequency, and reduced with each increase in mode of vibration considered. This result can be explained by the increasing dominance of shear deformations with increasing mode order. Furthermore, the effect of axial deformation becomes important when building height and slenderness ratios increase [158], or in other words, when flexural deformations are dominant.

The preceeding discussion has focused on the estimation of translational modes of vibration. Attention is now turned to estimating the natural frequency of torsional modes of vibration. Both the lumped parameter and distributed parameter systems of modelling are applicable for estimating the torsional natural frequencies. For the case of an idealised uniform beam, exact closed form solutions are only obtainable for circular cross sections. Non-circular cross sections tend to warp during torsional deformations, however this effect is not significant for simple closed sections. The

equation for estimating the torsional natural frequency of a uniform beam is given

where J is the torsion constant, G is the shear modulus, ρ is the mass density, and I_p is the polar area moment of inertia of the cross section about the axis of torsion.

Similar to the shear beam model, the natural frequency estimates from the torsion beam model is proportional to 1/L, and the natural frequencies increase linearly with the mode number.

The theoretical techniques discussed in this section are not extensive, but form the core of available methods that are relatively quick to apply for tall building natural frequency estimation. More complex techniques are available, and may be useful for certain structure types. Those included in the discussion also present the foundation for understanding the formation of empirical estimates, which are discussed in the following section.