Keywords

1 Introduction

The natural frequency is essential to understand the system behaviour and able to draw the deformation profile of the structure. For a single degree of freedom (SDOF) system (where the structure is able to deform in any one direction either x or y-axis) with one mass and equivalent stiffness of the column members, identifying the natural frequency is straightforward. But in the case of multi-degree of freedom (MDOF) system, calculation of equivalent stiffness is challenging. Hence, a popular idealization-shear building concept is introduced which is used to determine the natural frequencies of a multi-degree of freedom system. A shear building may be defined as a structure in which there is no rotation at the floor level. The structure behaves like a cantilever beam, and it undergoes deflection only due to shear force. The major assumptions of shear building concepts are: (1) the total mass is located at the floor level (i.e. it transfers the system of infinite number of degrees of freedom into as many degrees as the system masses lumped at floor levels), (2) the beams are rigid when compared to columns (it means, no rotation at the joint of column and beam), (3) the deformation of the structure is independent of the axial forces of the column (which indicates that the beams and floors remain horizontal during vibration). Classical methods and approximate methods are considered under the discrete system approach (lumped masses at floor levels) and are used to calculate the natural frequency of MDOF systems. Another methodology, i.e. continuous system approach at where the mass and stiffness are uniformly distributed throughout the length of the structure, and those systems are also known as “distributed parameter systems”. The discrete system consists of a finite number of degrees of freedom (DOF), whereas in a continuous system infinite number has infinite DOF because the displacement of every point in the elastic body is specified by an infinite number of coordinates. The displacement is dependent on two variables Ф(x) and y(t). As a result, the motion of the continuous system is governed by partial differential equations to be satisfied over the entire domain of the system subjected to boundary conditions and initial conditions. In the case of a building, the structure is considered as cantilever system (i.e. one end is fixed at ground level and another end is free at the top floor).

Late 80’s the computer algorithms were developed to analyse indeterminate structures using numerical methods. Computational efforts were greatly reduced, but all these algorithms are based on flexibility and stiffness matrix methods. Kamgar and Reza [1] developed an approximate method for estimating the natural frequency of tube framed structure using Timoshenko’s beam model in which flexural and shear deformations are considered. The natural frequencies and mode shapes are calculated based on the flexural and shear rigidities along with the effects of rotational inertia. Hamdan and Jubran [2] used base beam equation to calculate the mode shape functions which satisfy all the geometric and natural boundary conditions. The mode shape functions are used in conjunction with Galerkin’s method to obtain the response of the cantilever. Malekinejad and Reza [3] presented an approximate formula for dynamic response of tubular tall building structures on the basis of D’Alembert’s principle. Using the principle and applying the compatibility conditions on the deformation of the tubes, the governing dynamic equation of the tubular structure’s motion is derived. Then, natural boundary conditions of the parallel cantilevered flexural–shear beams are derived, and by using Rayleigh–Ritz method, the value problem is solved, and trivial and nontrivial solutions are derived, which can be used for calculating natural frequencies and mode shapes of tubular structures. Roh et al. [4] developed a model, based on eigenvalues and eigenvectors which named as “frequency adaptive lumped mass stick model” has only a small number of stick elements and nodes to provide the same natural frequencies of the structure and is applied to a nuclear containment building. Cunha and Sampaio [5] studied the influence of a discrete element in the nonlinear dynamics of a continuous mechanical system subject to randomness in the model parameters. The mechanical system consists of an elastic bar that is fixed at one end (right side), and other the end is attached to a couple of springs (one linear and another nonlinear) and lumped mass. In this system, I was subjected to Gaussian white-noise distributed external force. Malekinejad et al. [6] proposed a discrete–continuous approach for free vibration analysis of the combined system of framed tube, shear core, and outer rigger belt truss. The structure is discretized at each floor of the building as a series of lumped masses placed at the centre of shear core. Sohani and Eipakchi [7] worked on beam theories proposed by Euler–Bernoulli and Timoshenko for the analysis of free vibrations. Using perturbation technique, the behaviour of beams with arbitrary varying cross sections is studied. The beam equations are solved using Wentzel, Kramers, Brillouin approximation. Liu [8] verified variational iteration technique with the domain decomposition method to solve free vibration equations of Euler–Bernoulli beam. The tested method is found to be efficient in solving uniform Euler–Bernoulli beam problems and solution converges rapidly.

A minimum study is addressed in the literature regarding the iteration procedures involved in classical methods to estimate the fundamental frequency of the structures. The present study introduces an alternative iterative procedure by using natural frequency obtained from any approach and with the mode shapes corresponding to the natural frequency obtained from the mode superposition method. The proposed methodology is limited to flexural modes especially the fundamental mode and corresponding mode shape.

2 Methodology

The proposed methodology is carried out in three levels. Initially, solving the eigenvalue problem from the modal superposition method using mass and stiffness matrix for identifying the angular frequencies and mode shapes. Next, estimate the mode shapes of the structure with the idealization of a vertical cantilever beam using the continuous approach. Consider Rayleigh quotient method to calculate the fundamental frequency. At this level, the fundamental frequencies from the two approaches must be matched well. In case of a discrepancy with frequencies, the iterative procedure must be considered. In the iteration procedure, the frequency obtained from the continuous approach is substituted in the mode superposition method to identify the mode shape corresponding fundamental frequency. Evaluate natural frequency using mode shapes by applying the Rayleigh quotient method. Continue the procedure till the natural frequency converges. The mode shape related to the converged natural frequency is considered as the exact shape profile of the frequency. A case study of the G + 9 structure is presented in the study for the identification of fundamental frequency from the conventional approaches.

2.1 Description of the Structure

A 30 m height of structure having length and width as 12 m (3bays) and 8 m (2bays) with a floor height of 3 m having column sizes 600 mm × 600 mm for bottom five storey and 400 × 400 mm for remaining storeys are considered. Beam dimensions are 300 mm × 400 mm, and the thickness of the slab is 150 mm. The shear building assumed the G + 9 structure with each level idealization is shown in Fig. 1.

Fig. 1
figure 1

Idealized G + 9 building

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2.2 Preliminary Data Collection

  1. (a)

    Calculation of mass matrix: Lumped masses are considered at the individual floor levels by calculating the dead loads of half of the column above and below the floor level including self-weight of beams and slab. The global mass matrix is framed from the lumped mass located at storey levels is shown in Table 1.

    Table 1 Global mass matrix of G + 9 structure
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  1. (b)

    Calculation of stiffness matrix: Stiffness at each floor is calculated with the assumptions of the floor is rigidly connected (in a way of fixed) for one case and floor rotation is allowed for the other case. In one case, the shear building concept is adopted that is rotation is not allowed at the floor level. The stiffness of each storey is calculated by assuming each storey columns as a cantilever bar (one end fixed and another end is free). The global stiffness matrix is shown in Table 2.

    Table 2 Global stiffness matrix of G + 9 structure (Rotation is not allowed)
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In another case, the rotation is allowed at the floor level, i.e. stiffness is calculated by assuming ground floor columns as the cantilever (K1 = 12EI/L3) and other floor columns as simply supported (K2, K3….K10 = 3 EI/L3). The stiffness matrix is framed as shown in Table 3.

Table 3 Global stiffness matrix of G + 9 structure (Rotation is allowed)
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The discrete approach is utilized in the classical method and approximate method of estimating natural frequencies and mode shapes. Approximate methods have proven to be efficient in estimating the fundamental natural frequency of the structure.

2.3 Conventional Approach (Eigenvalue Problem)

The structure is idealized as spring mass system connected in series (Fig. 1). D Alembert’s principle is adopted to write the equilibrium equation of motion in free vibration condition. The undamped equation of motion represented globally is shown in “(1)”. Damping characteristics are neglected in the study due to the basic definition of natural frequency as a function of equivalent stiffness and equivalent mass.

$$[{\text{M}}]{{\ddot{{u}}}}({{t}}) + [{\text{K}}]{{u}}({{t}}) = 0$$
(1)

where [M] is the global mass matrix, [K] is the global stiffness matrix, and u(t) is the shape vector or displacement vector of response.

Equation (1) is written in form of the matrix

$$[{\text{M}}] \times \{ {{\ddot{{u}}}}\} + [{\text{K}}] \times \{ {{u}}\} = \{ 0\}$$
(2)

The characteristic equation or frequency equation is

$$\left| {\left[ {\text{K}} \right] - {\omega }_{{{j}}} ^{2} \left[ {\text{M}} \right]} \right| = 0$$
(3)

The eigenvalues are estimated by using the algorithms of Matlab. The fundamental frequencies (ω2) of G + 9 structure in both the stiffness cases of rotation allowed and rotation not allowed are shown in Table 4. The natural frequency of 8.2 rad/s is considered in the study. The normalized mode shape values corresponding graph (height of storey is on the x-axis and corresponding mode shape values on y-axis) are shown in Fig. 2.

Table 4 Fundamental frequencies
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Fig. 2
figure 2

The first mode shape in eigenvalue problem

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2.4 Continuous Approach (Euler Bernoulli’s)

In the continuous approach (Paz [9]), the mass and stiffness are uniformly distributed throughout the length of the structure. By using the Euler–Bernoulli theory (i.e. plane of the section remains plane during flexure) an approximate equation is derived. From the solution of this approximate equation, the mode shapes are calculated.

$$EI\left( {\partial ^{4} y/\partial x^{4} } \right) + m\left( {\partial ^{2} y/\partial t^{2} } \right) = 0$$
(4)

The deflections due to shear force and inertia force caused by the rotation of column cross sections are neglected. The solution to the above equation is found by the variable separable method. The approximate solution is given in function of position Ф(x) and function of time f(t).

$${\text{i}}.{\text{e}}.\,{{y}}\left( {{{x}},{{t}}} \right) = {\rm{\Phi }}\left( {{x}} \right) \times {{f}}\left( {{t}} \right).$$
$${\rm{\Phi }}^{{{\text{IV}}}} ({{x}}) - {{a}}^{4} {\rm{\Phi }}({{x}}) = 0$$
(5)

Let, Ф(x) = A sin(ax) + B cos(ax) + C sinh(ax) + D cosh(ax)

Where A, B, C, and D stands for the constants of integration, these four define the shape and amplitude of the beam in free vibration. The constants are evaluated by considering the boundary conditions at the end of the beam. The frequency equation is shown in “(6)”

$$\cos \left( {a_{{{n}}} L} \right) \times \cosh \left( {a_{{{n}}} L} \right) + 1 = 0$$
(6)

Roots of the equation corresponds to the successive natural frequencies. The first natural frequency is given by,

$$\omega _{n} = \frac{{\left( {a_{n} L} \right)^{2} }}{{\sqrt {\frac{{EI}}{{mL^{4} }}} }}{\text{ and normal mode is }}{\rm{\Phi }}({{x}}).$$
$${\rm{\Phi }}({{x}}) = \left( {\cosh a_{n} x - \cos a_{n} x} \right) - \frac{{\left( {\cos a_{n} L + \cosh a_{n} L} \right)}}{{\left( {\sin a_{n} L + \sinh a_{n} L} \right)}}\left( {\sinh a_{n} x - \sin a_{n} x} \right)$$
(7)

where x is the level height from the base, \(a_{n} = \frac{{\sqrt {3516} }}{L}\), L is length of the structure.

The first mode shape corresponding the storey level using “(7)”, i.e. \(X = 0,{\text{3}},{\text{6}}, \ldots 0.{\text{3}}0\) is shown in Fig. 3.

Fig. 3
figure 3

Mode shape of first natural frequency

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2.5 Rayleigh Quotient Method

Using the mode shape values of first natural frequency, equivalent mass and the equivalent stiffness matrices are calculated. Rayleigh quotient method is adopted to identify the fundamental frequency of the structure. The equivalent mass and stiffness matrices are shown in “(8)” and “(9)”.

$$[{\text{M}}]_{{1 \times 1}} = \{ \rm{\Phi }\} _{{1 \times 10}}^{{\text{T}}} \times [{\text{m}}]_{{10 \times 10}} \times \{ \rm{\Phi }\} _{{10 \times 1}}$$
(8)
$$[{\text{K}}]_{{1 \times 1}} = \{ \rm{\Phi }\} _{{1 \times 10}}^{{\text{T}}} \times [{\text{k}}]_{{10 \times 10}} \times \{ \rm{\Phi }\} _{{10 \times 1}}$$
(9)

The natural frequency is calculated from the below formula comes out to be 9.32 rad/s.

$${\upomega } = \sqrt ( [{\text{K}}]/[{\text{M}}])$$
(10)

2.6 Numerical Analysis

G + 9 structure is modelled in the ETABS, and modal analysis is performed. The plan and isometric view are shown in Fig. 4. The characteristic strength given as material properties of concrete is 30 N/mm2. Sectional properties are assigned, for bottom five storey columns as 600 × 600 mm, top five storey columns as 400 × 400 mm, beam size is 300 × 400 mm, and the thickness of slab as 150 mm. Slab is modelled as diaphragm using shell 8-noded elements.

Fig. 4
figure 4

Plan and isometric view of G + 9 building

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First natural frequency is identified at 6.32 and 7.53 rad/s at first two natural frequencies. First and second modes corresponding frequencies are shown in Fig. 5.

Fig. 5
figure 5

First and second natural frequencies of the building

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2.7 Approximate Methods

Modified Stodola and Holzar’s methods are considered for the study of natural modes. In modified Stodola, mode shape is assumed at the initial stage and later it is adjusted iteratively until an approximate of the mode shape is achieved.

$$\left\{ {{u}} \right\} = {\upomega }_{{\text{j}}} ^{{\text{2}}} \left[ {\text{A}} \right]\left\{ {{u}} \right\}$$
(11)

where [A] = [f] [M] is called dynamic equation, [f] is the flexibility matrix, i.e. inverse of stiffness matrix. Table

Table 5 The converged mode shape evaluation using Stodola method
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5 describes the iterations for the mode shape convergence. The natural frequency is calculated at the 6th iteration.

The value of natural frequency is calculated as,

$$\begin{aligned} & {\upomega }^{{\text{2}}} = {\text{1}}/0.0{\text{147}} = {\text{68}}.0{\text{272}} \\ & {\upomega } = \surd {\text{68}}.0{\text{272}} = {\text{8}}.{\text{24}}\,{\text{rad}}/{\text{s}} \\ \end{aligned}$$

The natural frequency is identified as 8.24 rad/s. The converged mode shape is shown in Fig. 6.

Fig. 6
figure 6

Mode shape of the building using Stodola method

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Holzar’s method is an iterative procedure with the assumed natural frequency at a minimum value. The first natural frequency of 8.2 rad/s is estimated at converged mode shape values (shown in Table 6). The mode shapes derived from Holzar’s method are shown in Fig. 7.

Table 6 Mode shape values for assumed frequency
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Fig. 7
figure 7

First natural frequency using Holzar’s method

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The frequencies extracted from conventional method, approximation method, discrete continuous approach, numerical method and proposed iterative approach are shown in Table 7. For the superposition method of discrete approach, fundamental frequency in the case of rotation not allowed is presented.

Table 7 Fundamental frequencies from various methods
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2.8 Iterative Procedure

An iterative procedure is considered to calculate the approximate frequency from the discrete approach and continuous approach. The iteration is followed in two steps. Step 1: the natural frequency obtained from the continuous approach, and numerical analysis is substituted separately in the eigenvalue problem to retrieve mode shapes. Step 2: using the mode shapes, Rayleigh quotient frequency is determined. The Rayleigh quotient frequency is substituted in the eigenvalue problem to get the modified mode shapes. Repeat the steps till the natural frequency converges. At the 3rd iteration, the value converged to 8.24 rad/s. The normalized mode shapes are shown in Fig. 8.

Fig. 8
figure 8

Mode shape of the structure at the converged first frequency

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3 Summary and Discussions

  • In the discrete approach, the conventional method with the assumption of allowed rotation, the natural frequency is obtained as 8.24 rad/s. But in the case of restricted rotation, the natural frequency of the structure is observed to be higher than 8.24 rad/s. This is due to the provision of considering the stiffness in the system by not allowing the rotation. Hence, the natural frequency of 23 rad/s is evident. In the first case, stiffness for all the columns is taken as 12EI/L3 for all the storeys but in other case, the stiffness is taken as 12EI/L3 for the fixed base and 3EI/L3 for other storeys.

  • From the approximate methods, the structural natural frequency of 8.2 rad/s is observed in the Stadola method and Holzer’s method.

  • In the continuous system, Euler–Bernoulli’s bending theory is adopted in the discrete–continuous approach. The structure is idealized as a cantilever structure with uniform distributed mass and stiffness. The mode shape of the first mode is identified using shear building idealization. First natural frequency is determined by adopting Rayleigh quotient method. The natural frequency value obtained is 9.32 rad/s which is almost 10% greater than the value calculated from the conventional approach (i.e. eigenvalue problem second case). Because the structure is very stiff with the basic Euler’s assumption of the vertical cantilever. Hence, high stiffness is drawn to the high natural frequency. One other limitation of the approach is structured with varying sections cannot be modelled.

  • In ETABS, the natural frequency obtained from this analysis is 6.32 rad/s which is 23% lesser than the value obtained from the conventional approach.

  • The iterative procedure adopted in the study from the values derived from the discrete, continuous, and numerical approaches, the natural frequency of 8.2 rad/s is converged at 3rd iteration.

4 Conclusions

  • With the shear building assumption of the structure having lumped mass at intermediate levels and restrained floor rotations, the natural frequency of the structure is identified as 23 rad/s, which is very high for the structure. This is due to the assumption by allowing the structure to behave rigidly with the base. Since the structure has ten degrees of freedom, each degree of freedom is designed to move rigidly and relatively along with the below floor level, starting from the first mass connected at the first-floor level near the base of the structure to the top floor mass. The above idealization may work well with the SDOF system.

  • With the assumption of lumped mass and allowed floor rotations, by means of making the structure flexible at the floor levels except the floor connected to the base of the structure. Therefore, the natural frequency is reduced to 8.2 rad/s.

  • Approximate methods are in good pace with the structure assumed of lumped mass and with allowed floor rotations. It is maintained 99% accuracy.

  • The fundamental frequency obtained from the continuous approach is 94% accurate with the complete discrete method with the assumption of allowed floor rotations. This is due to the consideration of change in column sections from 6th storey of the building which is contradicting the Euler-Bernoulli’s approach considered for the study. The profile of the mode shape has not accommodated the changes in mass and stiffness. Hence, due to the limitation, this method is not suitable to be considered in the iteration procedure of identifying the first natural frequency. However, the natural frequency from the discrete–continuous approach is used in the proposed iterative procedure, but the value converged at 8.2 rad/s. Timoshenko beam theory can be considered to validate the approximate methods for the structures having varying cross sectional elements with allowed rotations, and also the Timoshenko beam theory possibly addresses the torsional modes and mode shapes in terms are shear effects.

  • The iteration procedure is efficient in estimating the natural frequency obtained from numerical method, Rayleigh quotient method, and eigenvalue problem.

  • The resonant frequency obtained from iteration procedure is highly accurate to the frequency calculated from approximate methods.

  • The proposed approach results accurate fundamental frequency for symmetric structures. For non-symmetric structures, the approach can be successful provided appropriate stiffness at each floor level is assumed.