## Abstract

A wave-based model that incorporates the effects of shear deformation, rotary inertia and elastic coupling due to structural anisotropy, is developed to analyze the free vibrations of elastically restrained laminated planar frames. In this work, a generalized frame structure is represented as an assemblage of laminated beam segments that act as one-dimensional waveguides. The segments are assumed to undergo only in-plane motion, which upon applying Hamilton's principle, is described by a system of coupled differential equations. Dispersion analysis is conducted, and the nature of the wavefields associated with the propagation matrix is discussed. Generally restrained boundaries and internal joints are considered, and the associated reflection and transmission matrices are derived. Using the principle of wave-train closure, the characteristic equation is obtained by systematically assembling the propagation, reflection, and transmission matrices. The wave-based model is inherently deterministic, and solving the characteristic equation offers the advantage of determining the exact natural frequencies using conventional root finding algorithms. Application of the proposed model is demonstrated by analyzing an elastically restrained inclined laminated portal frame. Extensive computational analysis is conducted to illustrate the influence of stacking sequence, frame angle, relative frame length, orthotropicity ratios, and spring stiffness on the exact natural frequencies (and in certain cases the mode shapes) of the frame. Independent finite element simulations conducted in ansys^{®} APDL are consistently used to verify the validity of the analytical results.

## 1 Introduction

The study of flexible planar frames is an important subject in the field of applied mechanics because of its use in a wide range of engineering applications. In the aerospace, automotive, and robotics industries, frames are increasingly being constructed from fiber-reinforced composites (FRCs) [1,2]. Compared to conventional metals, FRCs are advantageous due to their high strength-to-weight and stiffness-to-weight ratios. More so, the ability to manipulate fiber orientation and stacking sequence offers the possibility of tailoring a laminate to suit a specified loading condition [3,4]. The directional nature of laminated composites renders them anisotropic in nature and can often lead to elastically coupled deformations. Consequently, laminated planar frames can exhibit both material coupling due to structural anisotropy, in addition to coupling of the displacement fields at the internal joints. It is also well established that because FRCs have a high ratio of extensional modulus to transverse shearing modulus, the effects of shear deformation are pronounced and necessary to include in dynamic models [5]. It follows that the vibration analysis of FRC frames is generally more complex compared to conventional metallic frames.

Some published studies have dealt with the dynamic analysis of laminated planar frame structures. Mahapatra et al. [6] proposed a spectral element model to analyze the transient response of asymmetrically laminated multiply-connected beam segments. The study was based on the Euler–Bernoulli beam theory (EBT), and therefore, the formulation is limited to slender frames. Miao et al. [7] studied the transient response of a laminated frame system by developing a reverberation-ray matrix method based on a first-order shear deformation theory (FSDT). Minghini et al. [8] performed free vibration analysis of transversely isotropic fiber-reinforced pultruded portal frames using a locking-free Hermitian finite element formulation. Bachoo [9] recently applied a wave-based model to investigate the free vibration characteristics of symmetrically and asymmetrically laminated planar frame structures. Exact natural frequencies were presented for an extensive range of planar frame configurations, layups, and orthotropicity ratios.

From both an experimental and theoretical perspective, laminated frames have not been as exhaustively investigated compared to their isotropic metallic counterparts. As a result, there are certain scientific and technical problems that are unresolved. The vibration analysis of elastically restrained laminated planar frames is one such problem. In practical applications, the boundaries and joints of frame structures are often treated using classical restraints (e.g., fixed, hinged, free, or sliding). Such assumptions are only valid if the stiffness at the boundaries and joints are so large that they can be considered rigid, or so small that their effects are negligible. In cases where these conditions cannot be satisfied, for example, in elastically restrained piping networks [10], it becomes necessary that the elasticity of the restraints be included in the analysis, as their influence can be significant.

In this work, a wave-based model that includes the effects of shear deformation, rotary inertia, and elastic couplings due to structural anisotropy is developed to analyze the free vibrations of a generalized laminated planar frame structure with elastic restraints. Continuity and/or equilibrium conditions are used to derive the propagation, reflection, and transmission matrices at the internal joints and boundaries of the frame. The matrices are systematically assembled according to the principle of wave-train closure and subsequently solved to determine the exact natural frequencies and corresponding mode shapes.

Knowledge of the exact natural frequencies for common structural configurations such as laminated portal frames is beneficial to designers and analysts, in terms of both benchmarking and direct practical application. To the best of the authors’ knowledge, no such results have been published for elastically restrained laminated planar frames. Consequently, the model developed herein is applied to investigate the free vibration characteristics of laminated portal frames that are elastically restrained by linear springs. Exact natural frequencies are determined for various frame angles, frame lengths, stacking sequence, and orthotropicity ratios. The results are consistently verified using the finite element (FE) package ansys^{®} APDL, and key features of the dynamic behavior are identified and discussed.

## 2 Description of Restrained Frame Structure

Consider the generalized laminated planar frame structure having two boundaries and *m* internal joints as shown in Fig. 1. The angle associated with each internal joint is represented as *α*_{i}, where *i* = 1, 2,.., *m*. The network of laminated beams can be partitioned into *m* + 1 segments having lengths *L*_{1}, *L*_{2},…, *L*_{m}, *L*_{m+1}.

The *X* − *Z* coordinate axes are chosen as the global reference for the structure. Elastic restraints in the horizontal, vertical, and rotational directions are applied at the boundaries as well as at the internal joints of the frame. The parameter *S* and its accompanying subscripts are used to represent the individual spring constants. Specifically, the first subscript identifies the location of the spring; for example, LB and RB indicate the left and right boundaries, respectively, whilst *i* = 1, 2,…, *m* represents a specific internal joint. The second subscript identifies the direction of the spring restraint, with *X*, *Z*_{,} and Ψ representing the horizontal, vertical, and rotational restraints, respectively. To maintain clarity in Fig. 1, the three spring constants are placed in brackets next to the associated boundary or joint. To complete the global description of the frame, the orientation of the horizontal spring attached to an internal joint *i* is denoted by the acute angle *θ*_{i}.

The frame vibrations can be analyzed in terms of the in-plane longitudinal, transverse and rotational displacements of the individual segments. Figure 2 shows an arbitrarily selected segment (*i*) having width (*β*), thickness (*h*), and length (*L _{i}*). A right-handed coordinate system is used as the local frame of reference with the coordinates

*x*

_{i},

*y*

_{i}, and

*z*

_{i}representing the longitudinal, lateral, and transverse directions, respectively. The axial displacement along the longitudinal reference line is given by

*u*

_{i}, whilst

*w*

_{i}and

*ψ*

_{i}are the transverse and rotational displacements, respectively. To illustrate the local coordinate system within the global

*X*−

*Z*reference, the in-plane local axes are highlighted for segment

*i*= 3 in Fig. 1.

## 3 Mathematical Model

### 3.1 Equations of Motion.

*i*) are given by [11]

*a*)

*b*)

*t*is time. The corresponding strains are

*a*)

*b*)

*ɛ*

_{x,i}and

*γ*

_{xz,i}are the normal and shear strains, respectively. The segment's constitutive equations can be written as

*σ*

_{x,i}is the longitudinal stress,

*τ*

_{xz,i}is the shear stress, and $Q\xaf11$, $Q\xaf55$ are the transformed elastic stiffness coefficients. Expressions for $Q\xaf11$ and $Q\xaf55$ in terms of the elastic constants of a lamina (

*E*

_{1},

*E*

_{2},

*G*

_{12},

*G*

_{13},

*G*

_{23}, and

*ν*

_{12}) can be readily obtained from Ref. [12].

*i*) are

*a*)

*b*)

*c*)

*m*

_{x,i}is the bending moment,

*n*

_{x,i}is the axial force,

*q*

_{xz,i}is the shear force,

*κ*is the shear correction factor, and

*A*is the cross-sectional area. The positive sign conventions for the internal forces and moment are shown in Fig. 2.

*a*)

*b*)

*c*)

*a*)

*b*)

In Eqs. (6*a*) and (6*b*), the parameters *A*_{11}, *A*_{55}, *D*_{11}, and *B*_{11} are the longitudinal, shear, bending, and bending–longitudinal coupling rigidities, respectively.

*a*)

*b*)

*c*)

In this work, it is assumed that the cross-sectional dimensions, material properties, and stacking sequence of each laminated composite segment in Fig. 1 are identical. This results in the rigidity and inertia terms (*A*_{11}, *B*_{11}, *D*_{11}, *A*_{55}, *I*_{1}, *I*_{2}, and *I*_{3}) being constant for all segments of the frame. Although these assumptions are not compulsory and can easily be relaxed, they do significantly simplify the symbolic and mathematical formalism that follows. Furthermore, in construction, it is preferred that planar frames have uniform material properties and cross-sectional dimensions to minimize the effects of localized stress intensification.

### 3.2 In-Plane Wave Motion of Laminated Segments.

*a*)–(7

*c*) are assumed to be time-harmonic and of the form

*a*)

*b*)

*c*)

*d*

_{o,i}are constants,

*k*is the wavenumber, and

*ω*is the angular frequency of vibration. At this point, it is beneficial to introduce several non-dimensional parameters. Generally, analysts prefer non-dimensionalization since it facilitates numerical simulations that can be scaled easily.

*a*)

*b*)

*a*)

*b*)

*c*)

*d*)

*e*)

*a*)–(11

*e*),

*L*is the total length of the frame,

*ξ*

_{i}is the non-dimensional position along segment

*i*,

*b*

_{o,i}, and

*c*

_{o,i}are non-dimensional wave amplitudes, and

*s*is the non-dimensional wavenumber. The rotational displacement, which by nature is a non-dimensional parameter, can be rewritten as

*a*)

*b*)

*c*)

*d*)

*e*)

*f*)

*p*is the non-dimensional frequency,

*C*

_{11},

*C*

_{22}, and

*C*

_{33}are the non-dimensional rigidities,

*η*is a measure of the dynamic coupling, and

*r*is the non-dimensional radius of gyration.

*a*)

*b*)

*c*)

*d*)

*e*)

*a*)

*b*)

*c*)

*a*)

*b*)

*c*)

*d*)

When angular frequencies are less than the cutoff frequency, $x\xaf2$ is negative and therefore indicates a pair of decaying waves. Conversely, for frequencies above the cutoff frequency, a wave-mode transition occurs and $x\xaf2$ becomes positive, indicating that the waves are now of the propagating type. For practical beams, it is generally the case that $A55I1\u226bI1I3\u2212I22$, which leads to the cutoff frequency being greater than the audible threshold frequency of 40,000*π* rad/s (20 kHz). In general, because noise and vibration problems are restricted to frequency ranges below the audible threshold frequency of 20 kHz, we limit this study to the case where *ω* < *ω*_{cutoff}.

*a*)

*b*)

*c*)

*i*) can be represented as

*a*)

*b*)

*c*)

*b*

_{i,1}−

*b*

_{i,6},

*c*

_{i,1}−

*c*

_{i,6}, and

*d*

_{i,1}−

*d*

_{i,6}are the corresponding amplitudes of each wave component. Furthermore, by reconsidering Eq. (14), the relationship between the longitudinal, transverse, and rotational wave amplitudes in Eqs. (21

*a*)–(21

*c*) are given by

*a*)

*b*)

*c*)

*d*)

*a*)

*b*)

*c*)

*d*)

*e*)

*f*)

*g*)

*h*)

### 3.3 Wave Propagation Matrix.

*a*)–(21

*c*), the displacement of each laminated segment consists of six separate wave components. The wave amplitude vectors $bi+$, $ci+$, and $di+$ are the components of the waves traveling in the positive direction whilst the vectors $bi\u2212$, $ci\u2212$, and $di\u2212$ refer to the wave components traveling in the negative direction. The amplitudes of the propagating waves at two positions

*ξ*

_{i,o}and

*ξ*

_{i,o}+

*ξ*

_{i}along the length of a uniform beam segment (

*i*) are related by the expressions

*a*)

*b*)

*a*) and (24

*b*), which are derived from Eq. (21

*c*), will also hold if the vectors $[di+di\u2212]$ are replaced with either $[bi+bi\u2212]$ or $[ci+ci\u2212]$.

### 3.4 Wave Reflection and Transmission.

Waves traveling along the frame will encounter discontinuities in the form of elastically restrained boundaries and internal joints. At a boundary, an incident wavefield is completely reflected whereas at an internal joint both reflection and transmission will occur. In the following subsections, wave amplitude relations are derived in terms of the reflection and transmission matrices. For a restrained boundary, only the equilibrium conditions are required to determine the reflection matrix. At an internal joint, however, both equilibrium and continuity of displacements must be considered to obtain the reflection and transmission matrices.

#### 3.4.1 Reflection Matrix at a Boundary.

*ξ*

_{1}= 0. The incident wavefield $(d1\u2212)$ gives rise to the reflected wavefield $(d1+)$, which is related by the expression

*a*)–(5

*c*) can be non-dimensionalized using the expressions introduced in Eqs. (10)–(13) to give

*a*)

*b*)

*c*)

*a*)

*b*)

*c*)

*a*)

*b*)

*c*)

In Eqs. (28*a*)–(28*c*), the subscript LB has been dropped from the spring constant to simplify the nomenclature.

^{®}or Mathematica

^{®}. To facilitate numerical simulations in this work, a program is written in Mathematica 12.2 to perform the required calculus and algebraic operations needed to determine the reflection matrix. As an example, the coefficient

*b*

_{31}as obtained from Mathematica 12.2 is

It is worthwhile to note that the reflection matrices for common classical boundary conditions can be obtained by manipulating the values of the spring constants to either zero or infinity. For example, the clamped boundary condition is obtained when $(S\xafX,S\xafZ,S\xaf\Psi )\u2192\u221e$, the free boundary occurs when $(S\xafX,S\xafZ,S\xaf\Psi )=0$ and the immovable simple support of a flat beam requires $(S\xafX,S\xafZ)\u2192\u221e$ and $S\xaf\Psi =0$.

#### 3.4.2 Reflection and Transmission Matrices at a Joint.

*i*and

*i*+ 1 be joined at the point

*ξ*

_{i}=

*ξ*

_{i+1}= 0 with associated angles

*α*

_{i}and

*θ*

_{i}. When a set of positively traveling waves originating from segment

*i*is incident upon the junction, both reflected and transmitted waves are generated such that

*a*)

*b*)

*i*+ 1 may be defined as

*a*)

*b*)

*a*)

*b*)

*c*)

*a*)

*b*)

*c*)

*e*

^{iωt}, the incident and reflected displacement wavefields within the region

*ξ*

_{i}≤ 0 are given by

*a*)

*b*)

*c*)

*ξ*

_{i+1}≥ 0 are

*a*)

*b*)

*c*)

*a*)–(35

*c*)) and using Eqs. (22

*a*)–(22

*d*) and Eqs. (23

*a*)–(23

*h*), we obtain the expression

*a*)–(36

*c*) and performing some algebraic manipulation for simplification leads to

Similar to the case of the reflection matrix, the coefficients associated with matrices $V1\u2212V6$ are generally long, and it is not feasible to produce a complete representation in this work. Symbolic computing tools once again offer a reasonably efficient approach to determine the coefficients required for conducting numerical simulations.

An analogous approach can also be used to determine the second pair of reflection and transmission matrices ($ri+1,i+1$, $ti+1,i$) when an incident wavefield travels from segment (*i* + 1). In this case, the incident and reflected displacement wavefields are confined to the region *ξ*_{i+1} ≥ 0, whilst the transmitted displacement wavefield is in the region *ξ*_{i} ≤ 0.

### 3.5 Calculation of Natural Frequencies and Mode Shapes.

The propagation, reflection, and transmission matrices derived in the preceding sections can be assembled and used to analyze the vibrations of an elastically restrained laminated planar frame structure. In Fig. 1 a generalized planar frame structure was illustrated. The circuit through which the wavefields travel across this structure when vibrating freely is shown in Fig. 4. The notation for the wavefield vector at a point is such that the first subscript indicates the particular segment being considered whilst the second subscript, either 0 or *L*_{i}, represents the start and end positions of the segment, respectively. The superscript identifies whether the wave is traveling in a positive (+) or negative (−) direction.

The wavefields can now be related using the following expressions:

- Wave reflection at Boundary LB and Boundary RB:(43$d1,0+=rLB.d1,0\u2212$
*a*)where $rLB$ and $rRB$ are the reflection matrices associated with the right and left boundaries, respectively.(43$dm+1,Lm+1\u2212=rRB.dm+1,Lm+1+$*b*) - Wave propagation across the length of each individual continuous segment:(44$di,Li+=f(LiL).di,0+$
*a*)where(44$di,0\u2212=f(LiL).di,Li\u2212$*b*)*i*= 1, 2,..,*m*+ 1,*L*_{i}/*L*is the fractional length of segment (*i*) and(45)$f(LiL)=[e\u2212is1LiL000e\u2212s2LiL000e\u2212is3LiL]$ - Wave reflection and transmission at each joint:(46$di+1,0+=ti,i+1di,Li++ri+1,i+1di+1,0\u2212$
*a*)where(46$di,Li\u2212=ti+1,idi+1,0\u2212+ri,idi,Li+$*b*)*i*= 1, 2,..,*m*.

*m*+ 1) and $d$ is the amplitude vector of length 12(

*m*+ 1). The non-dimensional natural frequencies (

*p*) are found when

This can be determined by using a conventional root searching computer program such as the Wittrick–Williams algorithm [16]. If required; upon evaluating the non-dimensional natural frequency (*p*), Eq. (13(*a*)) can then be used to determine the angular frequency in radians per second. The corresponding mode shapes for the frame can be obtained by deleting one row from matrix $M$ and solving for the remaining components in terms of one chosen wave component.

The proposed model can be summarized as an FSDT for unsymmetrically laminated planar frame structures. As shown in Table 1, by setting certain parameters to zero, the model can also be used to analyze symmetrically laminated planar frames based on either an FSDT or a classical EBT. The EBT model neglects the effects of shear deformation and rotary inertia and therefore *C*_{33} = *r* = 0. For isotropic metallic materials and symmetrically laminated composites, there is no bending–longitudinal elastic coupling and consequently *C*_{22} = 0.

Set | |||
---|---|---|---|

Frame model | C_{22} | C_{33} | r |

Isotropic EBT | 0 | 0 | 0 |

Isotropic FSDT | 0 | ≠0 | ≠0 |

Symmetrically laminated EBT | 0 | 0 | 0 |

Symmetrically laminated FSDT | 0 | ≠0 | ≠0 |

Unsymmetrically laminated EBT | ≠0 | 0 | 0 |

Unsymmetrically laminated FSDT | ≠0 | ≠0 | ≠0 |

Set | |||
---|---|---|---|

Frame model | C_{22} | C_{33} | r |

Isotropic EBT | 0 | 0 | 0 |

Isotropic FSDT | 0 | ≠0 | ≠0 |

Symmetrically laminated EBT | 0 | 0 | 0 |

Symmetrically laminated FSDT | 0 | ≠0 | ≠0 |

Unsymmetrically laminated EBT | ≠0 | 0 | 0 |

Unsymmetrically laminated FSDT | ≠0 | ≠0 | ≠0 |

## 4 Numerical Examples and Analysis

Numerical results are presented for an elastically restrained inclined portal frame having three identical segments, *L*_{1} = *L*_{2} = *L*_{3} as shown in Fig. 5. The first set of computations is carried out for the special case of a rectangular portal frame $(\theta =90deg)$ constructed from isotropic segments. For an isotropic material having elastic modulus *E* and Poisson's ratio *ν*, the notation for the rigidity terms is commonly represented as *A*_{11} = *EA*, *D*_{11} = *EI*, and *A*_{55} = *κAG*, where *A* is the cross-sectional area, *I* is the second moment of area, *G* = *E*/2(1 + *ν*) is the shear modulus, and *κ* = 10(1 + *ν*)/(12 + 11*ν*) is the shear correction factor [17].

To facilitate comparison with Ref. [18], the length-to-thickness ratio (*L*/*h*) of the rectangular frame is set to a value such that the slenderness ratio of each segment is 100 and Poisson's ratio is taken as *ν* = 0.29. The two boundaries of the frame are clamped whilst the inner joints are unrestrained. It follows that under these special limiting conditions, the spring constants are given by $(S\xafLB,X,S\xafLB,Z,S\xafLB,\Psi ,S\xafRB,X,S\xafRB,Z,S\xafRB,\Psi )\u2192\u221e$ and $(S\xaf1,X,S\xaf1,Z,S\xaf1,\Psi ,S\xaf2,X,S\xaf2,Z,S\xaf2,\Psi )\u21920$.

The roots of Eq. (48) are determined using a self-written computer code in Mathematica 12.2. The determinant in Eq. (48) is generally a complex number and the roots are obtained when both the real and imaginary values are simultaneously zero. The in-house computer code records the frequencies where there are simultaneous changes in sign and/or local minima with respect to both the real and imaginary functions. Although the root searching program allows for the step size of the frequency to be changed, a step size of 0.0005 was chosen as it gives well-defined asymptotic solutions and is evaluated within a reasonable amount of time. Solutions obtained for different step sizes ranging between 0.05 and 0.0005 showed only minor differences when compared against each other.

Table 2 compares the natural frequencies of the isotropic frame. The first four modes calculated using the FSDT have a percentage difference of less than $1%$ when compared with the corresponding results from Ref. [18]. Above the fourth mode, the percentage difference consistently increases. This is anticipated, since Ref. [18] based their model on the EBT. Applying the conditions stated in Table 1, the natural frequencies are recalculated using the EBT and the results are also given in Table 2. The agreement between the recalculated values and those in Ref. [18] is almost identical, with the percentage difference being less than $0.03%$ across all modes.

Non-dimensional natural frequency (p) | Non-dimensional parameter, K^{a} | |||||
---|---|---|---|---|---|---|

Present theory | Present theory | |||||

Mode | EBT | FSDT | Ref. [18] | EBT | FSDT | Ref. [18] |

1 | 28.833 | 28.782 | 28.837 | 1.790 | 1.788 | 1.790 |

2 | 113.587 | 113.193 | 113.614 | 3.553 | 3.546 | 3.553 |

3 | 185.610 | 184.476 | 185.586 | 4.541 | 4.527 | 4.541 |

4 | 200.481 | 198.885 | 200.506^{b} | 4.720 | 4.701 | 4.720^{b} |

5 | 403.153 | 398.964 | 403.166 | 6.693 | 6.658 | 6.693 |

6 | 494.570 | 487.248 | 494.573 | 7.413 | 7.358 | 7.413 |

7 | 569.687 | 559.572 | 569.681 | 7.956 | 7.885 | 7.956 |

8 | 849.530 | 832.998 | 849.606 | 9.716 | 9.621 | 9.716 |

9 | 1027.269 | 999.434 | 1027.331 | 10.684 | 10.538 | 10.684 |

Non-dimensional natural frequency (p) | Non-dimensional parameter, K^{a} | |||||
---|---|---|---|---|---|---|

Present theory | Present theory | |||||

Mode | EBT | FSDT | Ref. [18] | EBT | FSDT | Ref. [18] |

1 | 28.833 | 28.782 | 28.837 | 1.790 | 1.788 | 1.790 |

2 | 113.587 | 113.193 | 113.614 | 3.553 | 3.546 | 3.553 |

3 | 185.610 | 184.476 | 185.586 | 4.541 | 4.527 | 4.541 |

4 | 200.481 | 198.885 | 200.506^{b} | 4.720 | 4.701 | 4.720^{b} |

5 | 403.153 | 398.964 | 403.166 | 6.693 | 6.658 | 6.693 |

6 | 494.570 | 487.248 | 494.573 | 7.413 | 7.358 | 7.413 |

7 | 569.687 | 559.572 | 569.681 | 7.956 | 7.885 | 7.956 |

8 | 849.530 | 832.998 | 849.606 | 9.716 | 9.621 | 9.716 |

9 | 1027.269 | 999.434 | 1027.331 | 10.684 | 10.538 | 10.684 |

The second set of analyses deals with elastically restrained laminated composite frames. The following orthotropic material properties for AS/3501-6 graphite-epoxy are used in the numerical analysis [11]: *E*_{1} = 145 × 10^{9} Nm^{−2}, *E*_{2} = *E*_{3} = 9.6 × 10^{9} Nm^{−2}, *G*_{12} = *G*_{13} = 4.1 × 10^{9} Nm^{−2}, *G*_{23} = 3.4 × 10^{9} Nm^{−2}, and *ν*_{12} = *ν*_{13} = *ν*_{23} = 0.3. In this work, the shear correction factor (*κ*) for the laminated frames is always taken as 5/6. This value for the shear correction factor has been widely applied to AS/3501-6 graphite-epoxy in literature [5,11]. Still, the reader is cautioned that this selection is not compulsory, and depending on the material or structure being analyzed, more accurate values of the shear correction factor may be obtained experimentally. In such cases, empirically obtained values can be incorporated into the proposed model for better prediction of the dynamic characteristics.

Exact solutions for the first four natural frequencies of symmetric ([0/90/90/0]) and asymmetrically ([90/0]) laminated frames are calculated for angle *θ* = 90 deg and presented in Tables 3 and 4, respectively. The following two separate sets of dimensionless restraints are considered:

Parameter set A: $S\xafLB,X=S\xafLB,Z=S\xafRB,X=S\xafRB,Z=500$, $S\xafLB,\Psi =S\xafRB,\Psi =1$, $S\xaf1,X=S\xaf1,Z=S\xaf1,\Psi =S\xaf2,X=S\xaf2,Z=S\xaf2,\Psi =0$,

Parameter set B: $S\xafLB,X=S\xafLB,Z=S\xafRB,X=S\xafRB,Z=500$, $S\xafLB,\Psi =S\xafRB,\Psi =1$, $S\xaf1,X=S\xaf1,Z=S\xaf2,X=S\xaf2,Z=50$, $S\xaf1,\Psi =S\xaf2,\Psi =0.1$.

Parameter set A | Parameter set B | ||||||
---|---|---|---|---|---|---|---|

Present theory | Present theory | ||||||

L/h | Mode | EBT | FSDT | FE Model | EBT | FSDT | FE Model |

15 | 1 | 11.878 | 9.941 | 9.940 | 17.368 | 16.191 | 16.191 |

2 | 30.557 | 29.690 | 29.689 | 32.137 | 31.113 | 31.112 | |

3 | 39.948 | 37.940 | 37.939 | 41.439 | 39.526 | 39.525 | |

4 | 61.546 | 48.885 | 48.882 | 61.663 | 48.951 | 48.948 | |

50 | 1 | 11.998 | 11.764 | 11.764 | 17.436 | 17.290 | 17.290 |

2 | 31.232 | 31.156 | 31.156 | 32.731 | 32.642 | 32.642 | |

3 | 40.340 | 40.089 | 40.089 | 41.794 | 41.556 | 41.556 | |

4 | 62.154 | 60.651 | 60.651 | 62.275 | 60.765 | 60.765 | |

75 | 1 | 12.005 | 11.899 | 11.899 | 17.440 | 17.374 | 17.373 |

2 | 31.270 | 31.236 | 31.236 | 32.765 | 32.725 | 32.725 | |

3 | 40.362 | 40.248 | 40.248 | 41.814 | 41.706 | 41.706 | |

4 | 62.187 | 61.508 | 61.508 | 62.308 | 61.625 | 61.626 |

Parameter set A | Parameter set B | ||||||
---|---|---|---|---|---|---|---|

Present theory | Present theory | ||||||

L/h | Mode | EBT | FSDT | FE Model | EBT | FSDT | FE Model |

15 | 1 | 11.878 | 9.941 | 9.940 | 17.368 | 16.191 | 16.191 |

2 | 30.557 | 29.690 | 29.689 | 32.137 | 31.113 | 31.112 | |

3 | 39.948 | 37.940 | 37.939 | 41.439 | 39.526 | 39.525 | |

4 | 61.546 | 48.885 | 48.882 | 61.663 | 48.951 | 48.948 | |

50 | 1 | 11.998 | 11.764 | 11.764 | 17.436 | 17.290 | 17.290 |

2 | 31.232 | 31.156 | 31.156 | 32.731 | 32.642 | 32.642 | |

3 | 40.340 | 40.089 | 40.089 | 41.794 | 41.556 | 41.556 | |

4 | 62.154 | 60.651 | 60.651 | 62.275 | 60.765 | 60.765 | |

75 | 1 | 12.005 | 11.899 | 11.899 | 17.440 | 17.374 | 17.373 |

2 | 31.270 | 31.236 | 31.236 | 32.765 | 32.725 | 32.725 | |

3 | 40.362 | 40.248 | 40.248 | 41.814 | 41.706 | 41.706 | |

4 | 62.187 | 61.508 | 61.508 | 62.308 | 61.625 | 61.626 |

Parameter set A | Parameter set B | ||||||
---|---|---|---|---|---|---|---|

Present theory | Present theory | ||||||

L/h | Mode | EBT | FSDT | FE Model | EBT | FSDT | FE Model |

15 | 1 | 10.159 | 9.358 | 9.357 | 16.298 | 15.833 | 15.833 |

2 | 30.567 | 30.026 | 30.026 | 32.072 | 31.432 | 31.432 | |

3 | 35.778 | 35.262 | 35.261 | 37.636 | 37.102 | 37.101 | |

4 | 53.255 | 47.180 | 47.180 | 53.503 | 47.372 | 47.372 | |

50 | 1 | 9.922 | 9.849 | 9.848 | 16.119 | 16.076 | 16.075 |

2 | 30.941 | 30.890 | 30.891 | 32.360 | 32.300 | 32.300 | |

3 | 37.208 | 37.156 | 37.156 | 38.883 | 38.831 | 38.831 | |

4 | 51.942 | 51.354 | 51.355 | 52.209 | 51.616 | 51.617 | |

75 | 1 | 9.886 | 9.853 | 9.852 | 16.092 | 16.073 | 16.073 |

2 | 30.920 | 30.897 | 30.897 | 32.333 | 32.307 | 32.307 | |

3 | 37.360 | 37.337 | 37.337 | 39.016 | 38.993 | 38.993 | |

4 | 51.748 | 51.488 | 51.489 | 52.017 | 51.755 | 51.756 |

Parameter set A | Parameter set B | ||||||
---|---|---|---|---|---|---|---|

Present theory | Present theory | ||||||

L/h | Mode | EBT | FSDT | FE Model | EBT | FSDT | FE Model |

15 | 1 | 10.159 | 9.358 | 9.357 | 16.298 | 15.833 | 15.833 |

2 | 30.567 | 30.026 | 30.026 | 32.072 | 31.432 | 31.432 | |

3 | 35.778 | 35.262 | 35.261 | 37.636 | 37.102 | 37.101 | |

4 | 53.255 | 47.180 | 47.180 | 53.503 | 47.372 | 47.372 | |

50 | 1 | 9.922 | 9.849 | 9.848 | 16.119 | 16.076 | 16.075 |

2 | 30.941 | 30.890 | 30.891 | 32.360 | 32.300 | 32.300 | |

3 | 37.208 | 37.156 | 37.156 | 38.883 | 38.831 | 38.831 | |

4 | 51.942 | 51.354 | 51.355 | 52.209 | 51.616 | 51.617 | |

75 | 1 | 9.886 | 9.853 | 9.852 | 16.092 | 16.073 | 16.073 |

2 | 30.920 | 30.897 | 30.897 | 32.333 | 32.307 | 32.307 | |

3 | 37.360 | 37.337 | 37.337 | 39.016 | 38.993 | 38.993 | |

4 | 51.748 | 51.488 | 51.489 | 52.017 | 51.755 | 51.756 |

The results are calculated based on both the FSDT and EBT for multiple frame lengths. As expected, for both the symmetric and asymmetric layups, the percentage difference between the EBT and FSDT models $((|pEBT\u2212pFSDT|/pFSDT)\xd7100%)$ increases significantly as the relative frame length gets shorter.

In this work, the finite element package ansys^{®} Academic Research Mechanical APDL, Release 21.2 is used to verify the exact natural frequencies calculated from the proposed models. To carry out the finite element simulations in ansys^{®} APDL, specific values of the material density, thickness, width, and length are needed. The density, thickness, and width are assumed to be *ρ* = 1520 kgm^{−3}, *h* = 0.0254 m, and *β* = 2*h*, respectively, for all finite element simulations. The length of each segment is varied to meet the required ratio *L*/*h* specified in Tables 3 and 4. The element type chosen to model each segment is SHELL181, which is a four-node element with six-degrees-of-freedom at each node. The springs at the boundaries and internal joints of the frame are modeled using the Matrix 27 element. Matrix 27 relates the stiffness restraints between two nodes. Each node has the following six-degrees-of-freedom: translations in the nodal *x*, *y*, and *z* directions and rotational displacements about the nodal *x*, *y*, and *z* axes. Additional information regarding the coordinate system and other technical properties associated with SHELL181 and MATRIX27 are available in Refs. [20,21].

The size of the SHELL181 element (and hence the mesh density) is controlled by setting the element edge length using the AESIZE command in ansys^{®}. A mesh convergence study is done for each frame length to ensure that the solutions are asymptotic. Table 5 shows the results of the convergence study done for the [90/0] laminated frame with *θ* = 45 deg and restrained by Parameter Set B. The solution converges fairly rapidly to the exact values predicted by the proposed FSDT model. Based on the convergence studies conducted for the different layups and elastic restraints, the number of SHELL181 elements chosen for the frame lengths, *L*/*h* = 15, *L*/*h* = 50, and *L*/*h* = 75 are 19,431, 16,536, and 10,812, respectively.

Number of elements | Present theory (FSDT) | |||||||
---|---|---|---|---|---|---|---|---|

Mode | 24 | 540 | 1113 | 7242 | 16,536 | 18,060 | 22,500 | |

1 | 24.063 | 22.982 | 22.981 | 22.982 | 22.983 | 22.983 | 22.983 | 22.983 |

2 | 26.557 | 26.518 | 26.518 | 26.518 | 26.518 | 26.518 | 26.518 | 26.518 |

3 | 36.203 | 36.137 | 36.135 | 36.135 | 36.135 | 36.135 | 36.135 | 36.135 |

4 | 50.306 | 49.722 | 49.707 | 49.698 | 49.697 | 49.697 | 49.697 | 49.697 |

Number of elements | Present theory (FSDT) | |||||||
---|---|---|---|---|---|---|---|---|

Mode | 24 | 540 | 1113 | 7242 | 16,536 | 18,060 | 22,500 | |

1 | 24.063 | 22.982 | 22.981 | 22.982 | 22.983 | 22.983 | 22.983 | 22.983 |

2 | 26.557 | 26.518 | 26.518 | 26.518 | 26.518 | 26.518 | 26.518 | 26.518 |

3 | 36.203 | 36.137 | 36.135 | 36.135 | 36.135 | 36.135 | 36.135 | 36.135 |

4 | 50.306 | 49.722 | 49.707 | 49.698 | 49.697 | 49.697 | 49.697 | 49.697 |

Comparing the results in Tables 3 and 4, it can be observed that the percentage difference between the natural frequencies obtained from the present FSDT model and the ansys^{®} model $(|pANSYS\u2212pFSDT|/pFSDT)\xd7100%$ is always well within $0.05%$ for both symmetric and asymmetric layups, regardless of the beam length being considered. The agreement between the theoretical and finite element approaches is therefore in good agreement. Additionally, since the application of the EBT is limited to determining the frequencies of only the lowest order modes; from this point onward, only the FSDT model is used for further analysis.

As only two angles (*θ* = 45 deg and *θ* = 90 deg) were considered in the previous analysis, it is instructive to investigate the influence the subtended angle (*θ*) has on the natural frequencies. Figure 6 shows the variation of the first four natural frequencies of the asymmetric [90/0] laminated frame with angle (*θ*). The length of the frame is *L*/*h* = 50, the restraints are in accordance with Parameter Set A, and the natural frequencies are calculated in 1 deg intervals.

The symmetry associated with both the restraints and geometry results in either symmetrical or anti-symmetrical mode shapes. In Fig. 6, the terms A1 and A2 are the first and second anti-symmetrical modes, respectively, whilst S1 and S2 are the first and second symmetrical modes, respectively. Depending on the angle, the pattern or sequence of the symmetric and anti-symmetric modes can change. For example, at *θ* = 25 deg, the first four modes are symmetric, anti-symmetric, anti-symmetric, and symmetric, respectively. These mode shape patterns, which are illustrated in Figs. 7(a)–7(d), have also been verified numerically using ansys^{®}. It is also observed in Fig. 6 that at *θ* = 40 deg a cross-over point exists and the mode pattern changes. Therefore, at the angle *θ* = 75 deg to the right of the cross-over point, the mode shape pattern occurs in the sequence: anti-symmetric, symmetric, anti-symmetric, and symmetric.

*L*/

*h*= 50, and the angle used is

*θ*= 45 deg. Increasing the number of layers whilst keeping the laminate thickness constant leads to an increase in the natural frequencies of the frame. The rigidity expressions in Eqs. (6

*a*) and (6

*b*) can be used to explain the reason for the increase in natural frequencies. For an asymmetric cross-ply laminate of the form [90/0]

_{n}having layers of uniform thickness, Eqs. (6

*a*) and (6

*b*) reduces to

*a*)

*b*)

*c*)

*d*)

[90/0]_{1} | [90/0]_{2} | [90/0]_{3} | [90/0]_{4} | |||||
---|---|---|---|---|---|---|---|---|

Mode number | Present theory | FE Model | Present theory | FE Model | Present theory | FE Model | Present theory | FE Model |

1 | 19.771 | 19.771 | 23.391 | 23.391 | 23.748 | 23.748 | 23.852 | 23.852 |

2 | 24.301 | 24.302 | 24.800 | 24.800 | 24.889 | 24.889 | 24.923 | 24.924 |

3 | 34.579 | 34.579 | 37.226 | 37.226 | 37.719 | 37.720 | 37.896 | 37.896 |

4 | 49.493 | 49.494 | 58.787 | 58.787 | 59.819 | 59.820 | 60.141 | 60.141 |

[90/0]_{1} | [90/0]_{2} | [90/0]_{3} | [90/0]_{4} | |||||
---|---|---|---|---|---|---|---|---|

Mode number | Present theory | FE Model | Present theory | FE Model | Present theory | FE Model | Present theory | FE Model |

1 | 19.771 | 19.771 | 23.391 | 23.391 | 23.748 | 23.748 | 23.852 | 23.852 |

2 | 24.301 | 24.302 | 24.800 | 24.800 | 24.889 | 24.889 | 24.923 | 24.924 |

3 | 34.579 | 34.579 | 37.226 | 37.226 | 37.719 | 37.720 | 37.896 | 37.896 |

4 | 49.493 | 49.494 | 58.787 | 58.787 | 59.819 | 59.820 | 60.141 | 60.141 |

The rigidities *A*_{11}, *D*_{11}, and *A*_{55} are independent of the number of layers (2*n*) for a constant segment thickness *h*, whereas the coupling rigidity *B*_{11} decreases as the number of layers increases. The decrease in the bending–longitudinal coupling rigidity results in a stiffer laminate and hence leads to an increase in the natural frequencies.

Numerical analysis is also undertaken to investigate the changes in the non-dimensional natural frequencies with varying orthotropicity ratios (*E*_{1}/*E*_{2}) for the [0/90/90/0] and [90/0] laminates. The results are shown in Table 7 for a subtended angle *θ* = 90 deg, a frame length of *L*/*h* = 50, and Parameter set A restraints. As shown in Table 7, increasing the orthotropicity ratio results in a decrease in the non-dimensional natural frequencies for the symmetric and asymmetric layups. Finite element simulations with ansys^{®} are also used to verify the results. With the exception of *E*_{1}, the input material properties for the finite element model are the same as those given for AS/3501-6 graphite-epoxy. The value of *E*_{1} is chosen to satisfy the specified ratio in Table 7. The finite element model agrees well with theoretically obtained natural frequencies as the percentage difference between both sets of results never exceeds $0.05%$.

[0/90/90/0] | [90/0] | ||||
---|---|---|---|---|---|

E_{1}/E_{2} | Mode number | Present theory | FE Model | Present theory | FE Model |

5 | 1 | 11.917 | 11.917 | 11.007 | 11.006 |

2 | 31.212 | 31.213 | 31.148 | 31.148 | |

3 | 40.248 | 40.248 | 38.566 | 38.566 | |

4 | 61.619 | 61.619 | 57.456 | 57.456 | |

25 | 1 | 11.622 | 11.621 | 9.425 | 9.424 |

2 | 31.104 | 31.104 | 30.738 | 30.738 | |

3 | 39.945 | 39.945 | 36.740 | 36.740 | |

4 | 59.739 | 59.739 | 48.932 | 48.932 | |

50 | 1 | 11.284 | 11.283 | 8.979 | 8.977 |

2 | 30.976 | 30.976 | 30.508 | 30.507 | |

3 | 39.615 | 39.615 | 36.355 | 36.355 | |

4 | 57.571 | 57.569 | 46.243 | 46.242 |

[0/90/90/0] | [90/0] | ||||
---|---|---|---|---|---|

E_{1}/E_{2} | Mode number | Present theory | FE Model | Present theory | FE Model |

5 | 1 | 11.917 | 11.917 | 11.007 | 11.006 |

2 | 31.212 | 31.213 | 31.148 | 31.148 | |

3 | 40.248 | 40.248 | 38.566 | 38.566 | |

4 | 61.619 | 61.619 | 57.456 | 57.456 | |

25 | 1 | 11.622 | 11.621 | 9.425 | 9.424 |

2 | 31.104 | 31.104 | 30.738 | 30.738 | |

3 | 39.945 | 39.945 | 36.740 | 36.740 | |

4 | 59.739 | 59.739 | 48.932 | 48.932 | |

50 | 1 | 11.284 | 11.283 | 8.979 | 8.977 |

2 | 30.976 | 30.976 | 30.508 | 30.507 | |

3 | 39.615 | 39.615 | 36.355 | 36.355 | |

4 | 57.571 | 57.569 | 46.243 | 46.242 |

As illustrated through the various numerical simulations, both the proposed theory and the finite element model require computational resources to determine the natural frequencies of the laminated frame structure. It is therefore instructive to discuss and compare the computation time required by both methods. In this work, all simulations were undertaken on a laptop computer operating on a Microsoft Windows 10 64-bit system, with an Intel Core i7-7700HQ 2.80 GHz processor and 16 GB of random access memory (RAM).

Table 8 shows the time taken by ansys^{®} to calculate the first four natural frequencies of the [90/0] laminated beam with length *L* = 50*h* subjected to Parameter Set B restraints for an increasing number of elements. The block Lanczos solver is selected in ansys^{®}, and only the time spent computing the solution is considered. Other durations such as the time spent on modeling, pre-processing, and post-processing are therefore omitted as they are generally of less importance. It is observed that as the number of elements increases, the computing time increases significantly.

Number of elements | Present | |||||||
---|---|---|---|---|---|---|---|---|

2805 | 4134 | 7242 | 16,536 | 28,866 | 64,872 | |||

Time taken (s) | 1.28 | 1.72 | 2.25 | 4.06 | 5.98 | 12.45 | 7.19 |

Number of elements | Present | |||||||
---|---|---|---|---|---|---|---|---|

2805 | 4134 | 7242 | 16,536 | 28,866 | 64,872 | |||

Time taken (s) | 1.28 | 1.72 | 2.25 | 4.06 | 5.98 | 12.45 | 7.19 |

Note: Parameters: [90/0] laminated frame with Parameter set B restraints, *θ* = 90 deg, and *L*/*h* = 50.

The time taken by Mathematica 12.2 to compute the exact natural frequencies of the frame depends on the frequency step size used as well as on the range of the frequency band. The self-written root searching algorithm developed in Mathematica has the ability to perform multiple iterative passes that gradually reduce the frequency bandwidth whilst simultaneously increasing the step size for the required precision. As shown in Table 8, the time taken to compute the first four natural frequencies within a range Δ*p* = 55 is approximately 7.19 s. As discussed earlier in this work, 16,536 elements were used to model all beams of length *L* = 50*h*. Therefore, comparing the computing times required by ansys^{®} and Mathematica 12.2, it is observed that the latter takes approximately twice as long. One must recall, however, that it is the exact natural frequencies (in this case obtained from Mathematica) that finite element models are often benchmarked against.

## 5 Conclusions

A wave-based model that permits the free vibration analysis of elastically restrained laminated planar frame structures has been developed. The model, which is based on a first-order shear deformation theory, relies on the derivation of the propagation, reflection, and transmission matrices. The matrices are systematically assembled to provide a complete description of the wave train as it traverses the entire closed-circuit of the frame. Unlike the finite element approach, the proposed model has the advantage of determining the exact natural frequencies of the frame. The model is also shown to be capable of analyzing a range of simplified, but practically relevant laminated frames when certain parameters are set to zero (Table 1).

The proposed wave-based model is well-suited for implementation using conventional computational software, and its application is demonstrated by analyzing the vibrations of an elastically restrained inclined laminated three-member portal frame. The results of the model are consistently verified with independent finite element simulations, as the percentage difference between them never exceeds $0.05%$. Numerical simulations are also conducted to show the influence of stacking sequences on the natural frequencies of the frame. It is observed that increasing the number of cross-ply layers (*n*) in the asymmetric [90/0]* _{n}* frame whilst keeping the thickness constant increases the natural frequency. The increase is shown to be due to a corresponding decrease in the bending–extensional coupling rigidity. Additionally, numerical analysis is used to investigate the effects of frame angle, frame lengths, and orthotropicity ratios on the free vibration response. The results of the study indicate that stacking sequence can be used alongside other factors such as external restraints and frame geometry to tailor the dynamic response of a laminated frame. This feature can be beneficial to engineers and designers involved in the analysis and design of such structures.

## Conflict of Interest

There are no conflicts of interest.

## Data Availability Statement

The data sets generated and supporting the findings of this article are obtainable from the corresponding author upon reasonable request.