# Flexural-torsional free vibration of a simply supported beam with arbitrary cross section

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## Torsion and warping of thin-walled beams

As shown in Figure 1.7, let us consider a bi-symmetric I-section beam, for this section the center of symmetry coincides with both the center of gravity and the torsional center. When, on the one hand, the line of action of transverse loads is perpendicular to the longitudinal axis of the beam, and on the other hand, the plane of loading coincides with a plane of symmetry of the cross section (Figure 1.7-a), the beam is not subjected to axial force nor torsion. The only internal forces are therefore a bending moment and a shear force. Otherwise, when the loading plan passes through the center of symmetry of the section without coinciding with one of the principal plans, a deviated simple bending of beam (Figure 1.7-b) appears. The non-coincidence of the loading plane with a plane of symmetry of the section causes an eccentricity of the transverse loads with respect to the torsional center of the section. Hence, the beam is subjected to either a simple monoaxial or biaxial bending and twisting (Figure 1.7-c and d).

### Uniform torsion (De Saint-Venant theory)

In general, solid-section beams are distinguished from thin-walled ones and, for the latter, closed sections are distinguished from open sections.
The theory of torsion for solid sections was developed by Saint-Venant in 1855. This model is the generalization of the torsion problem of beams with circular cross sections. In the model, the torsion moment applied to the shaft leads to a twist angle and its rate along the axial coordinate. The Torsion moment is equilibrated by shear in the section and no additional axial stresses occur (Figure 1.8).

Non-uniform torsion (Vlasov’s theory)

In the classical beam theory, the internal stresses in beams result from axial and shear stresses. These stresses are carried out from the reduction of the existing subjected forces and momentums to the beam centroidal axis: (N, Fy, Fz, My, Mz, Mx). These components are defined as the followings N is the normal force perpendicular to the cross section. Fy and Fz shear forces about the principle axis Y, Z. The bending moments about Y and Z are noted My, Mz. Mx is the torsion moment. Consequently, N, My and Mz causes the normal stresses and shear stresses are due to shear forces and torsion moment Fy, Fz and Mx.
Moreover, in thin-walled open sections beams the different points of the section may have, in addition to displacements governed by the uniform torsion, additional longitudinal displacements what is known as the warping effect of the beam cross-sections. Due to this, thin-walled beams cross sections are subjected to non-uniform torsion. Since the torsion of these elements are predominated with warping effect, torsional stiffness of the element has two components: the first is the classical Saint-Venant torsional stiffness, proportional to the shear modulus . The second contribution is due to the warping of the cross sections.
Consequently, when thin-walled beams are subjected to torsion with a variable rate of twist different type of stresses occurred in the section: the uniform torsion shear stresses Msv known as Saint-Venant torsion and the warping stresses composed by the normal stresses in the longitudinal direction of the cross section and shear stresses in the cross sections plans. Hence warping stresses can be significant in relation to other stresses like the bending, and so they cannot be omitted.
Because of the warping on the beam cross section a longitudinal load differently to transverse load cannot be replaced by a statically equivalent longitudinal forces system.
Moreover, the non-uniform warping of the cross sections in the longitudinal direction results moments in some locations of the beam section what is named in literature [5] bimoment B. To illustrate this problem let us consider a cantilever beam with I cross section carrying an eccentric load 4F acting at the free end of the beam on the extremity of the bottom flange and parallel to the longitudinal axis. The force eccentricity about the beam principle axis are noted ey=b/2 and ez=h/2 as shown in Figure 1.11. According to the classical beam theory, this load can be decomposed into the equivalent system involving normal force and two bending moments My, Mz at the centroid as shown in Figure 1.11. Each of these charges can be replaced by the equivalent forces applied in the four flange tip ends of the section as depicted in Figure 1.12a-c. However, the superposition of the loads presented in Figures 1.12a,b,c leads to the loads depicted in Figure 1.12d which does not correspond to the initial load presented in 1.6a. So there is additional load for thin-walled beams defined in 1.12e as the warping effect (bimoments). The warping effect is presented initially by Vlasov and defined by two equal moments 0 having equal magnitude and opposite sign acting on the flanges subjecting flexions in opposite orientation and separated by a distance equal to the height of the beam h see Figure 1.12d.

#### Time domain, frequency domain and Fourier Transform

For modal testing, either time domain and frequency domain analysis is widely adopted in signal processing. Moreover, in vibration analysis under harmonic conditions, the frequency domain is widely adopted in the study. We can cover all range of frequencies which is impossible in time domain. Vibrational systems can be characterized by their inherent frequency components. But, some signals are given in time domain like earthquakes spectrograms. So the importance to convert signals from time into frequency domain. This process becomes easy due to the modern computational power. The most familiar method to transform signal from time to frequency domain is the Fourier transformation (FT). The importance of the Fourier transformation method is that it permits to decompose a complex signal into simpler parts witch facilitate analysis. Otherwise, it helps to transform the differential, difference equations and complex operations in the time domain to algebraic operations in the frequency domain.
In this paragraph, the Fourier transformation is presented. Let’s consider a signal x(t) given in time domain, this signal can be periodic or non-periodic. This signal can be written in the following form: ( ) = 1 ∫+∞ (Ω) Ω Ω (1.30) 2 −∞ Where (Ω) is the Fourier transformation of the time signal x(t).

Harmonic oscillator with viscous damping

The classical analysis of the free vibration undamped model is deduced from the equation (1.41) as follow: ̈+ = 0 (1.45).
As the oscillation is harmonic, the displacement u can be written as: u(x,t)=u(x) eit (1.46).
The insertion of the second derivative about time of equation (1.46) into the equation (1.45) gives: ( − Ω2) = 0 (1.47).
The solution of equation (1.47) gives the angular frequency: 0= √ (1.48).
The critical damping from which the differential equation (1.41) has no oscillating solution is written as follows: ccritical =2√ = 2m = 2 (1.49).
This gives a numerical interpretation of reduced damping ratio, which is often expressed as a percentage of critical damping: = = = (1.50).

Effect of elastic and viscous springs on the dynamic behavior of structures

The elastic springs elements are used to introduce additional stiffness in some points of the structure. By adding stiffness to the structure, the natural frequencies increase. Accordingly, the dynamic behavior of the structure is improved against low vibration frequencies. The viscous springs are a power-dissipation mechanism called also dashpots. They are used to dissipate the energy of the structures subjected to a moving force. Thus, the role of these element is to reduce the displacement amplitude of the structure near to resonance. The elastic and viscous springs can be used in parallel as shown in Figure 1.21.

Modeling linear vibrations of thin-walled beams

From the literature review it is well known that the linear vibration of thin-walled beams with arbitrary cross-sections exhibits with flexural-torsional coupled modes. otherwise, it is shown in the section 1.3 that the torsion for this types of sections is predominated with the warping effect. For this reason, it is necessary in for this types of beams to investigate a linear vibration model that take in consideration of these two important points. Thus, we decided to develop a beam model based on the Galerkin’s approach, to formulate the dynamic equations of motion. Using Hamilton’s principle, the finite element approach of the model is established. For more precision in the model, the effects of rotational inertial kinematic terms are considered. In the FE procedure the efficient model with minimum cost of calculation and that gives high precision solutions is based on the beam element with seven degree of freedom. In the model the 7th degree of freedom is related to warping effect in the thin-walled beams sections that is indispensable in torsion computation for these types of sections.

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Forced vibration analysis of thin-walled beams

Carrying out the dynamic response spectra (displacements, accelerations for any point of the beam) using the modal analysis method. The response spectra are carried out in the frequency domain. Harmonic excitation force or moment or random can be applied to this beam in any chosen point. Base motion excitation is also treated using the present FEM. The maximum values of displacement ‘resonance’ are observed near the eigenfrequencies.
The same procedure used previously is applied to solve forced vibration problem:
-Analytical method: Presenting the existing problem solution formulation.
-Numerical part: the FEM is developed to solve forced vibration problem. Some examples with our code are compared to Abaqus and Adina commercial codes to validate results.
– Experimental part:
Tests on beams with different boundary conditions and a non-symmetric section. To do this, we propose to study a specimen of thin-walled beams with arbitrary cross-sections. In this part the electrodynamic shaker machine is used to carry out the acceleration and displacement response spectra of some chosen points. Acceleration sensor are used to record the spectra and to control the vibration frequencies and the load.

Effect of the elastic and viscous springs on vibration control

The elastic springs added to structures increase their natural frequency. Otherwise, the displacement amplitude of the structures near to resonance can be reduced by introducing a power-dissipation mechanism (dashpots).
Therefore, by introducing elastic springs the vibration of the structure can be controlled by eliminating the undesirable vibration modes. Moreover, by adding dashpots in some points of the beams, the displacement of the structure near to the natural frequencies is controlled.
Consequently, this procedure can be followed for thin-walled beams in presence of 3D vibration modes. When bending and torsion modes are present, lateral and torsion viscoelastic springs should be used to improve the dynamic behavior of the structures. The optimal positions of the additional viscoelastic springs and their properties should be studied in order to obtain an optimal vibration control. The effect of adding lateral elastic springs to the beam is demonstrated in Figure 1.25.
The work is followed by analytical and numerical simulations using a home-made code (implemented on Matlab) and commercial code (Abaqus, Adina) to study the effect of viscoelastic intermediate braces and to carry out the optimum solutions for different cases of sections and boundary conditions. Validation by test setup of beams with intermediate braces is also achieved.

Free vibration solution attempts for simply supported beams

A simply supported beam in bending and torsion is depicted in Figure 2.6, the spatial and time  functions obey to: g(t) =eit (2.49) ( ) = sin( ) (2.50) With: = , n=1,2…N (2.51).
In (2.49), is the angular velocity. In (2.51), n is the vibration mode and N the number of modes under study.

Simply supported beam with doubly -symmetric I cross-section

A steel beam presented in Figure 2.10 with IPE300 cross-section is considered. To study the effect of the kinematic rotational terms on the natural frequency computation, the beam is studied in free vibration context. The natural frequencies are investigated using the analytical method that take in consideration the rotational terms developed in the section 2.2.3.2. Then, the present analytical method natural frequencies are compared to the result obtained by the analytical method without kinematic rotational terms. The geometrical and material properties of the beam are the followings:
E=210 GPa,=0.30, G=80.07 GPa, =7850kg/m3, L=4.0m.

Chapter 1 Thin-walled beams, State of art. Motivation and Interest
1.1 Introduction
1.2 Thin-walled beams elements
1.3 Torsion and warping of thin-walled beams
1.3.1 Origin of the phenomenon
1.3.2 Uniform torsion (De Saint-Venant theory)
1.3.3 Non-uniform torsion (Vlasov’s theory)
1.4 Vibration analysis
1.4.1 Free vibration
1.4.2 Forced vibration
1.4.3 Time domain, frequency domain and Fourier Transform
1.4.4 Damping
1.4.5 Effect of elastic and viscous springs on the dynamic behavior of structures
1.5 Modeling linear vibrations of thin-walled beams
1.6 Assessment of subject positioning
1.7 Objectives of research topic
1.7.1 Free vibration of beams in flexural-torsional behavior
1.7.2. Forced vibration analysis of thin-walled beams
1.7.3. Effect of the elastic and viscous springs on vibration control
1.8 Conclusion
Chapter 2 Analytical method for the vibration behavior of thin-walled beams
2.1 Introduction
2.2 Free and forced vibration analyses
2.2.1 Kinematics of the model
2.2.2 Variational formulation of motion equations
2.2.3 Free vibration analysis
2.3 Validation and discussion
2.3.1 Simply supported beam with doubly-symmetric I cross-section
2.3.2 Simply supported beam with singly-symmetric C cross-section
2.3.3 Simply supported beam with mono-symmetric T cross-section
2.3.4 Cantilever beam with doubly-symmetric I cross-section
2.3.5 Cantilever beam with singly-symmetric C cross-section
2.3.6 Cantilever beam with mono-symmetric T cross-section
2.3.7 Doubly clamped beam with doubly-symmetric I cross-section
2.3.8 Doubly clamped beam with singly-symmetric C cross-section
2.3.9 Doubly clamped beam with mono-symmetric T cross-section
2.3.10 Free vibration of a singly-symmetric cross section cantilever beam
2.3.11 Flexural-torsional free vibration of a simply supported beam with arbitrary cross section
2.4 Conclusion
Chapter 3 Finite element method for vibrations of thin-walled beams
3.1 Introduction
3.2 Finite element discretization
3.3 Generic resolution of thin-walled structures
3.3.1 Finite element formulation of thin-walled structures free vibrations
3.3.2 Forced vibration of thin-walled structures in frequency domain (Steady-state dynamic analysis)
3.4 Numerical application and analysis
3.4.1 Free vibration of a singly-symmetric cross section cantilever beam
3.4.2 Flexural-torsional free vibration of a simply supported beam with arbitrary cross section
3.4.3 Hong Hu Chen benchmark numerical study
3.4.4 Double clamped cruciform cross-section beam
3.4.5 A simply supported beam under base motion load (Earth-quake, El Centro records, acc_NS)
3.4.6 Forced vibration analysis of thin-walled beams doubly-symmetrical section
3.4.7 Forced vibration analysis of thin-walled beams mono-symmetrical Tee section
3.4.8 Forced vibration analysis of thin-walled beams with mono-symmetrical channel section
3.5 Conclusion
Chapter 4 Experimental analysis on free and forced vibrations of thin-walled beams
4.1 Introduction
4.2 Experimental setup
4.2.1 Free vibration test procedure
4.2.2 Forced vibration test procedure
4.2.3 Acquisition device and Sensors characteristics
4.3 Specimens properties
4.4 Boundary conditions
4.5 Experimental tests results
4.5.1 Impact hammer tests results
4.5.2 Forced vibration test results for cantilever beam
4.6 Repeatability condition for the test results
4.7 Validation of the numerical model by comparison to experimental results
4.7.1 Free vibration
4.7.2 Forced vibration
4.8 Effect of intermediate bracings
4.9 Conclusion
Chapter 5 Vibration of braced beams
5.1 Introduction
5.2 Brace design
5.2.1 Vibration control by means of elastic bracings
5.2.2 Dynamic behavior of braced beams
5.2.3 Motion equations
5.3 Finite element formulation for braced thin-walled beams
5.4 Numerical applications
5.4.1 Torsion free vibration of braced thin-walled beams
5.4.2 Lateral free vibration of braced thin-walled beams
5.5 Conclusion
Conclusion and Perspectives
Résumé
References.

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