# Galerkin Analysis of Effect of Dead Load on Natural Frequencies of Box Beam

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## Abstract

Based on Hamilton’s principle, a nonlinear differential equation considering the effect of dead loads is formulated for box beams. By using Galerkin method, the effect of dead load on the natural frequencies of box beams with various restraint conditions being simply supported, fixed at both ends, cantilevered and fixed at one end and simply supported at the other end is studied. The effects of major physical parameters, including the magnitude of dead load, sectional inertia moment and inertia radius, span length, and restraint conditions are discussed, and the natural frequencies are presented for these box beams, with the nonlinear effect of dead load taken into account. The results show that the deterrent effect of dead loads has significant effect on the increase of the natural frequencies of box beams. It is further observed that this behavior is more apparent at lower-order natural frequencies; the varying deterrent effect of dead loads relates to the stiffness of structures and is more significant for more flexible structures.

## Keywords

Effect of dead load Box beam Natural frequency Hamilton’s principle Galerkin method## Introduction

In general, engineering structures and components are mostly subjected to a combination of constant initial loads and variable additional loads [1]. The combination of these loads can be considered as live loads and is found to have significant effect on the sustainability and durability of the structures. When the structure bears these live loads, its deformation starts from the reference state of constant initial loads that mainly contain the initial stress which causes the subsequent deformation. The stress generated by constant initial load will produce deterrent effect, which will change the internal force and deformation of the structure under live loads. The deterrent effect is not only caused by the initial middle surface tension, but also induced by the initial bending. Brunelle [2] demonstrates the effect caused on structures due to the influence of initial middle surface tension. However, studies regarding the effect due to initial bending are found scarce. The stiffening effect caused by the initial bending stress of dead load is called dead load effect according to Takabatake [3].

Takabatake [3] considered the effects of dead loads on the static characteristics of beam for the first time and derived the static force differential equation of uniform beam under dead loads. He also studied the response under the dead loads of simply supported beam and clamped–clamped beam in addition. He reported the effect of dead loads on the dynamic characteristics of beams and plates [4, 5]. Zhou and Zhu [6, 7, 8, 9] analyzed the effect of dead loads on the natural frequencies of beams and plates by means of finite element method. Zhang et al. [10, 11, 12, 13, 14, 15] have studied the influence of dead loads effect on static and dynamic characteristics of arch beams. At this end, most of the studies and their findings are primarily based on solid beam. However, these studies did not involve the impact on the thin-walled box structures.

In this paper, the effect of dead loads on the thin-walled boxes structure is analyzed. A nonlinear differential equation considering the effect of dead loads is formulated for box beams, based on Hamilton’s principle. By using Galerkin method, the effect of dead load on the natural frequencies of box beams with various restraint conditions being simply supported, fixed at both ends, cantilevered and fixed at one end and simply supported at the other end. The effects of key physical parameters mainly, magnitude of dead load, sectional inertia moment and inertia radius, span length, and restraint conditions are subsequently discussed as a function of natural frequencies by taking into account the nonlinear effect of dead load.

## Establishment of Differential Equation

### Basic Assumption

*ε*

_{x}is the strain of beam;

*v*is the deflection of beam.

### Total Energy Expression

All the strain energy caused by live loads (\(\bar{p}\)) can be expressed as follows:

All kinds of strain energy caused by dead loads (\(\hat{p}\)) can be expressed as follows:

## Controlling Differential Equation

*t*

_{1}) to another location at the time (

*t*

_{2}), to all the distance the object may experience, the distance follows Newton’s law at every moment is the distance makes the value of Lagrangian function average on time become the extreme value [16, 17] ,

## The Solution of Differential Equation

Among them, the definition of *f*_{n} is defined as follows:

Among them, the parameter *k*_{0n} is defined as follows:

## Example and Analysis

The following parameters are adopted for the beam: *E* = 210 GPa; *I*_{0} = 2.5 × 10^{−4} m^{4}; *L*_{0} = 8 m, *A*_{0} = 0.01 m^{2} (the reference value of gyration radius is that: \(r_{0} = \sqrt {I_{0} /A_{0} } = 0.158\;{\text{m}}\)). The dead load intensity is assumed to be: \(\hat{p}_{0} = 6.8\;{\text{kN}}/{\text{m}}\). The variables (\(\hat{p}\), *l*, *I*, *r*) can be expressed as follows: \(\hat{p} = \alpha_{{\hat{p}}} \hat{p}_{0}\), \(l = \alpha_{l} l_{0}\), *I* = *α*_{I}*I*_{0}, *r* = *α*_{r}*r*_{0}. The numerical results change by the above parameters.

The effect of initial bending deformation on the natural frequencies of simple box beam is analyzed. So, in the following expression, *ω*_{i} represents the circular frequencies of the *i*th vibration mode effected by initial bending deformation. *ω*_{i0} represents the circular frequencies of the *i*th vibration mode without the effect of initial bending deformation. \(\hat{p}_{0}\) represents a reference dead load per unit length of the beams. \(\hat{p}\) represents the practical value of initial load per unit length of the beams. Dimensionless form of coordinate, linear coordinate and logarithmic coordinate are adopted, respectively, for the lateral axis (\(\bar{p}_{0} /\hat{p}\)), while linear coordinate is used for the vertical axis \(\left( {\Delta = \left( {\omega_{i} - \omega_{i0} } \right)/\omega_{i0} \times 100\% } \right)\).

*m*) at unit length [3].

*α*

_{I}, span length

*α*

_{L}and gyration radius

*α*

_{r,}respectively) and \(\bar{p}_{0} /\hat{p}\) (affected by different constraint conditions). Among them, the gyration radius changes under the condition that the area changes and the sectional inertia moment keep constant. Figure 7 shows comparison of effects of initial loads when the above parameters (

*α*

_{I},

*α*

_{L}and

*α*

_{r}) of simply supported beam changes. It is demonstrated that the smaller the sectional inertia moment or the gyration radius (the area changes and the sectional inertia moment keeps constant) is or the larger the span length is, the larger effect of dead loads on the natural frequencies of beam will be. Among them, the change of span length makes the largest effect.

## Conclusion

This paper analyzes the effect of dead loads on the thin-walled box structures. A nonlinear differential equation considering the effect of dead loads is taken into account for formulating box beam model, based on Hamilton’s principle. By using Galerkin method, the effect of dead load on the natural frequencies of various box beams, with various restraint conditions being simply supported, fixed at both ends, cantilevered and fixed at one end and simply supported at the other end. The effects of key physical parameters, including the magnitude of dead load, sectional inertia moment and inertia radius, span length, and restraint conditions are discussed, and the natural frequencies are presented for these box beams, with the nonlinear effect of dead load is taken into account.

- 1.
The bending deformation generated by dead loads will produce deterrent effect which causes an increase of the natural frequencies of box beam. This trend will get more obvious when the natural frequencies belong to a lower-order range or larger weight.

- 2.
The stiffness of box beam has an impact on the deterrent effect which caused by the initial bending deformation. It is concluded that more flexible structures which have a smaller sectional inertia moment, a larger span length or a lower gyration radius, the effect becomes more significant. Among them, the change of span length makes the largest effect.

## Notes

### Acknowledgements

The work described in this paper was financially supported by the National Natural Science Foundation of China (No. 50368035), the Natural Science Foundation of Gansu Province, China (No. 145RJZA210) and Gansu Provincial Department of Housing and Urban–Rural Development (No. JK2014-16), to which the authors are grateful.

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