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 αi, where i = 1, 2,.., m. The network of laminated beams can be partitioned into m + 1 segments having lengths L1, L2,…, Lm, Lm+1.

Fig. 1
An elastically restrained planar frame structure with m frame angles and m + 1 individual segments
Fig. 1
An elastically restrained planar frame structure with m frame angles and m + 1 individual segments
Close modal

The XZ 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 θi.

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 (Li). A right-handed coordinate system is used as the local frame of reference with the coordinates xi, yi, and zi representing the longitudinal, lateral, and transverse directions, respectively. The axial displacement along the longitudinal reference line is given by ui, whilst wi and ψi are the transverse and rotational displacements, respectively. To illustrate the local coordinate system within the global XZ reference, the in-plane local axes are highlighted for segment i = 3 in Fig. 1.

Fig. 2
Coordinate system and degrees-of-freedom for a typical laminated segment (i)
Fig. 2
Coordinate system and degrees-of-freedom for a typical laminated segment (i)
Close modal

3 Mathematical Model

3.1 Equations of Motion.

Using an FSDT, the axial and transverse displacement fields for an arbitrary laminated segment (i) are given by [11]
u¯i=ui(xi,t)+ziψi(xi,t)
(1a)
w¯i(xi,zi,t)=wi(xi,t)
(1b)
respectively, where t is time. The corresponding strains are
εx,i=uixi+ziψixi
(2a)
γxz,i=wixi+ψi
(2b)
where ɛx,i and γxz,i are the normal and shear strains, respectively. The segment's constitutive equations can be written as
{σx,iτxz,i}=[Q¯1100Q¯55]{εx,iγxz,i}
(3)
where σx,i is the longitudinal stress, τxz,i is the shear stress, and Q¯11, Q¯55 are the transformed elastic stiffness coefficients. Expressions for Q¯11 and Q¯55 in terms of the elastic constants of a lamina (E1, E2, G12, G13, G23, and ν12) can be readily obtained from Ref. [12].
The expressions for the internal moment and forces acting within segment (i) are
mx,i(xi,t)=Aσx,izidA
(4a)
nx,i(xi,t)=Aσx,idA
(4b)
qxz,i(xi,t)=κAτxz,idA
(4c)
where mx,i is the bending moment, nx,i is the axial force, qxz,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.
Using Eqs. (2)(4), the constitutive equations that relate the bending moment and internal forces to the displacements are given by the expressions
mx,i(xi,t)=B11uixi+D11ψixi
(5a)
nx,i(xi,t)=A11uixi+B11ψixi
(5b)
qxz,i(xi,t)=A55(wixi+ψi)
(5c)
where
(A11,B11,D11)=βh/2h/2Q¯11(1,zi,zi2)dzi
(6a)
A55=κβh/2h/2Q¯55dzi
(6b)

In Eqs. (6a) and (6b), the parameters A11, A55, D11, and B11 are the longitudinal, shear, bending, and bending–longitudinal coupling rigidities, respectively.

Applying Hamilton's principle, the partial differential equations of motion for each laminated segment are given by [11,13]
A112uixi2+B112ψixi2I12uit2I22ψit2=0
(7a)
A55(2wixi2+ψixi)I12wit2=0
(7b)
B112uixi2A55(wixi+ψi)+D112ψixi2I22uit2I32ψit2=0
(7c)
In Eqs. (7a)(7c), the parameters I1, I2, and I3 are inertial terms, which can be expressed as
(I1,I2,I3)=βh/2h/2ρ(1,zi,zi2)dzi
(8)
where ρ is the mass density.

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 (A11, B11, D11, A55, I1, I2, and I3) 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.

The displacements satisfying Eqs. (7a)(7c) are assumed to be time-harmonic and of the form
ui(xi,t)=b¯o,iei(ωtkxi)
(9a)
wi(xi,t)=c¯o,iei(ωtkxi)
(9b)
ψi(xi,t)=do,iei(ωtkxi)
(9c)
where b¯o,i, c¯o,i, and do,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.
The non-dimensional translational displacements are given by
Ui(ξi,t)=ui(xi,t)L=bo,iei(ωtsξi)
(10a)
Wi(ξi,t)=wi(xi,t)L=co,iei(ωtsξi)
(10b)
where
L=i=1m+1Li
(11a)
ξi=xiL
(11b)
bo,i=b¯o,i/L
(11c)
co,i=c¯o,i/L
(11d)
s=kL
(11e)
In Eqs. (11a)(11e), L is the total length of the frame, ξi is the non-dimensional position along segment i, bo,i, and co,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
ψi(ξi,t)=do,iei(ωtsξi)
(12)
The following additional dimensionless variables are also introduced:
p2=ω2I1L4D11
(13a)
C11=(A11L2D11)
(13b)
C22=(B11LD11)
(13c)
C33=(D11A55L2)
(13d)
η2=(I2I1L)
(13e)
r2=(I3I1L2)
(13f)
where p is the non-dimensional frequency, C11, C22, and C33 are the non-dimensional rigidities, η is a measure of the dynamic coupling, and r is the non-dimensional radius of gyration.
Substituting Eqs. (9a)(9c) into Eqs. (7a)(7c) and making use of the expressions in Eqs. (10)(13), leads to the equation
[p2C11s20p2η2C22s20C33p2s2isC33p2η2C22C33s2isC33r2p2C33s21]×{bo,ico,ido,i}=0
(14)
Setting the determinant of the 3 × 3 matrix in Eq. (14) to zero gives the dispersion equation
ax¯3+bx¯2+cx¯+d=0
(15)
where
x¯=s2
(16a)
a=C222C11
(16b)
b=p2(1C222C33+C11(C33+r2)2C22η2)
(16c)
c=C11(p2C33p4r2)+p4(η4r2+C33(1+2C22η2))
(16d)
d=p4(C33p2(r2η4)1)
(16e)
Recognizing that Eq. (15) is a cubic polynomial in x¯=s2, and the roots of Eq. (15) may be given as [14]
x¯1=n~+q~cos(γ~)
(17a)
x¯2=n~+q~cos(2π3+γ~)
(17b)
x¯3=n~+q~cos(2π3γ~)
(17c)
where
n~=b3a
(18a)
q~=23ab23ac
(18b)
γ~=13cos1(F~)
(18c)
F~=4aq~3(an~3+bn~2+cn~+d)
(18d)
References [11,15] have shown that all three roots are real numbers. The roots x¯1 and x¯3 are always positive and therefore represent two pairs of propagating waves. The sign of x¯2 is dependent on the cutoff frequency
ωcutoff=A55I1I1I3I22
(19)

When angular frequencies are less than the cutoff frequency, x¯2 is negative and therefore indicates a pair of decaying waves. Conversely, for frequencies above the cutoff frequency, a wave-mode transition occurs and x¯2 becomes positive, indicating that the waves are now of the propagating type. For practical beams, it is generally the case that A55I1I1I3I22, which leads to the cutoff frequency being greater than the audible threshold frequency of 40,000π 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 < ωcutoff.

It follows that upon employing the notation
s1=x¯1+
(20a)
s2=|x¯2|+
(20b)
s3=x¯3+
(20c)
the vibratory response of any beam segment (i) can be represented as
Ui(ξi,t)=(bi,1eis1ξi+bi,2es2ξi+bi,3eis3ξi+bi,4eis1ξi+bi,5es2ξi+bi,6eis3ξi)eiωt
(21a)
Wi(ξi,t)=(ci,1eis1ξi+ci,2es2ξi+ci,3eis3ξi+ci,4eis1ξi+ci,5es2ξi+ci,6eis3ξi)eiωt
(21b)
ψi(ξi,t)=(di,1eis1ξi+di,2es2ξi+di,3eis3ξi+di,4eis1ξi+di,5es2ξi+di,6eis3ξi)eiωt
(21c)
where bi,1bi,6, ci,1ci,6, and di,1di,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. (21a)(21c) are given by
bi+=Πdi+
(22a)
bi=Πdi
(22b)
ci+=Γdi+
(22c)
ci=Γdi
(22d)
where
bi+={bi,1bi,2bi,3}T
(23a)
bi={bi,4bi,5bi,6}T
(23b)
ci+={ci,1ci,2ci,3}T
(23c)
ci={ci,4ci,5ci,6}T
(23d)
di+={di,1di,2di,3}T
(23e)
di={di,4di,5di,6}T
(23f)
Π=diag[C22s12p2η2p2C11s12C22s22+p2η2p2+C11s22C22s32p2η2p2C11s32]
(23g)
Γ=diag[is1c33p2s12s2c33p2+s22is3c33p2s32]
(23h)

3.3 Wave Propagation Matrix.

The preceding analysis showed that each laminated segment of the planar frame acts as a waveguide that supports two pairs of propagating waves and a pair of decaying waves. From Eqs. (21a)(21c), 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, ci, and di refer to the wave components traveling in the negative direction. The amplitudes of the propagating waves at two positions ξi,o and ξi,o + ξi along the length of a uniform beam segment (i) are related by the expressions
di+(ξi,o+ξi)=f(ξi).di+(ξi,o)
(24a)
di(ξi,o)=f(ξi).di(ξi,o+ξi)
(24b)
where
f(ξi)=[eis1ξi000es2ξi000eis3ξi]
(25)
is the propagation matrix. Equations (24a) and (24b), which are derived from Eq. (21c), will also hold if the vectors [di+di] are replaced with either [bi+bi] or [ci+ci].

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.

The left boundary of the laminated segment in Fig. 1 is elastically restrained at ξ1 = 0. The incident wavefield (d1) gives rise to the reflected wavefield (d1+), which is related by the expression
d1+(ξ1=0)=r.d1(ξ1=0)
(26)
The reflection matrix r, which governs the amplitudes and phase of the reflected waves, is obtained by considering the equilibrium relations. It is necessary to note that the internal moment and forces given in Eqs. (5a)(5c) can be non-dimensionalized using the expressions introduced in Eqs. (10)(13) to give
Mx,i(ξi,t)=mx,iLD11=ψiξi+C22Uiξi
(27a)
Nx,i(ξi,t)=nx,iL2D11=C22ψiξi+C11Uiξi
(27b)
Qxz,i(ξi,t)=qxz,iL2D11=1C33(Wiξi+ψi)
(27c)
The equilibrium conditions at the left boundary can be expressed in non-dimensional form as
Nx,1(0,t)cos(θ1)+Qxz,1(0,t)sin(θ1)S¯X(U1(0,t)cos(θ1)+Wi(0,t)sin(θ1))=0
(28a)
Mx,1(0,t)S¯Ψ(0,t)ψ1=0
(28b)
Qxz,1(0,t)cos(θ1)Nx,1(0,t)sin(θ1)S¯Z(W1(0,t)cos(θ1)U1(0,t)sin(θ1))=0
(28c)
where S¯X, S¯Z, and S¯Ψ are non-dimensional spring constants given by
S¯X=SX(L3D11)
(29a)
S¯Z=SZ(L3D11)
(29b)
S¯Ψ=SΨ(LD11)
(29c)

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

Substituting Eqs. (21) and (27) into Eqs. (28a)(28c) and making use of the relations in Eqs. (22a)(22d) leads to the expression
[a11a12a13a21a22a23a31a32a33].d1(ξ1=0)=[b11b12b13b21b22b23b31b32b33].d1+(ξ1=0)
(30)
In general, the coefficients of the 3 × 3 matrices in Eq. (30) are lengthy and a complete presentation of all the terms is not possible within this paper. It should be noted, however, that the expressions may be easily obtained using symbolic computing packages such as matlab® 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 b31 as obtained from Mathematica 12.2 is
b31=((S¯Z+iC11s1)η2p2(ip2+S¯Zs1)C22s1)sin(θ1)p2C11s12(p2iS¯Zs1)cos(θ1)C33p2s12
(31)
Proceeding, it follows from Eq. (26) that the reflection matrix is given as
r=[b11b12b13b21b22b23b31b32b33]1·[a11a12a13a21a22a23a31a32a33]
(32)

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¯X,S¯Z,S¯Ψ), the free boundary occurs when (S¯X,S¯Z,S¯Ψ)=0 and the immovable simple support of a flat beam requires (S¯X,S¯Z) and S¯Ψ=0.

3.4.2 Reflection and Transmission Matrices at a Joint.

As shown in Fig. 3, let two laminated segments i and i + 1 be joined at the point ξi = ξi+1 = 0 with associated angles αi and θi. 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
di+1+=ti,i+1.di+
(33a)
di=ri,i.di+
(33b)
where di+1+={di+1,1di+1,2di+1,3}T, ti,i+1 is the transmission matrix, and ri,i is the reflection matrix. The notation used to define the transmission (or reflection) matrix is such that the first subscript indicates the segment that the incident wavefield originates from whilst the second subscript indicates the segment that supports the transmitted (or reflected) wavefield.
Fig. 3
Wavefields at the junction of an internally restrained joint
Fig. 3
Wavefields at the junction of an internally restrained joint
Close modal
Analogously, the transmission and reflection matrices due to an incident set of negatively traveling waves originating from segment i + 1 may be defined as
di=ti+1,i.di+1
(34a)
di+1+=ri+1,i+1.di+1
(34b)
where di+1={di+1,4di+1,5di+1,6}T.
Expressions for the reflection and transmission matrices are determined by considering the continuity in displacements and equilibrium at the restrained joint. The displacement compatibility equations are given by
Ui(0,t)cos(θi)+Wi(0,t)sin(θi)+Ui+1(0,t)cos(αi+θi)+Wi+1(0,t)sin(αi+θi)=0
(35a)
Wi(0,t)cos(θi)Ui(0,t)sin(θi)Ui+1(0,t)sin(αi+θi)+Wi+1(0,t)cos(αi+θi)=0
(35b)
ψi+1(0,t)ψi(0,t)=0
(35c)
The equilibrium conditions at the joint are
Nx,i(0,t)cos(θi)+Qxz,i(0,t)sin(θi)+Nx,i+1(0,t)cos(αi+θi)+Qxz,i+1(0,t)sin(αi+θi)+S¯i,X(Ui(0,t)cos(θi)+Wi(0,t)sin(θi))=0
(36a)
Nx,i(0,t)sin(θi)Qxz,i(0,t)cos(θi)+Nx,i+1(0,t)sin(αi+θi)Qxz,i+1(0,t)cos(αi+θi)S¯i,Z(Wi(0,t)cos(θi)Ui(0,t)sin(θi))=0
(36b)
Mx,i+1(0,t)Mx,i(0,t)S¯i,Ψψi+1(0,t)=0
(36c)
We will now focus our attention on determining the first pair of reflection and transmission matrices, ri,i and ti,i+1. Suppressing the time dependence eiωt, the incident and reflected displacement wavefields within the region ξi ≤ 0 are given by
Ui(ξi)=bi,1eis1ξi+bi,2es2ξi+bi,3eis3ξi+bi,4eis1ξi+bi,5es2ξi+bi,6eis3ξi
(37a)
Wi(ξi)=ci,1eis1ξi+ci,2es2ξi+ci,3eis3ξi+ci,4eis1ξi+ci,5es2ξi+ci,6eis3ξi
(37b)
ψi(ξi)=di,1eis1ξi+di,2es2ξi+di,3eis3ξi+di,4eis1ξi+di,5es2ξi+di,6eis3ξi
(37c)
The transmitted wavefields within the region ξi+1 ≥ 0 are
Ui+1(ξi+1)=bi+1,1eis1ξi+1+bi+1,2es2ξi+1+bi+1,3eis3ξi+1
(38a)
Wi+1(ξi+1)=ci+1,1eis1ξi+1+ci+1,2es2ξi+1+ci+1,3eis3ξi+1
(38b)
ψi+1(ξi+1)=di+1,1eis1ξi+1+di+1,2es2ξi+1+di+1,3eis3ξi+1
(38c)
Upon substituting Eqs. (37) and (38) into the displacement compatibility relations (Eqs. (35a)(35c)) and using Eqs. (22a)(22d) and Eqs. (23a)(23h), we obtain the expression
V1.di+1++V2.di=V3.di+
(39)
Likewise, substituting Eqs. (37) and (38) into equations Eqs. (36a)(36c) and performing some algebraic manipulation for simplification leads to
V4.di+1++V5.di=V6.di+
(40)

Similar to the case of the reflection matrix, the coefficients associated with matrices V1V6 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.

Upon solving Eqs. (39) and (40), the reflection and transmission matrices are given by
ri,i=((V1)1.V2(V4)1.V5)1.((V1)1.V3(V4)1.V6)
(41)
ti,i+1=((V2)1.V1(V5)1.V4)1.((V2)1.V3(V5)1.V6)
(42)

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 ξi+1 ≥ 0, whilst the transmitted displacement wavefield is in the region ξi ≤ 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 Li, represents the start and end positions of the segment, respectively. The superscript identifies whether the wave is traveling in a positive (+) or negative (−) direction.

Fig. 4
Wave-train analysis of an elastically restrained laminated planar frame
Fig. 4
Wave-train analysis of an elastically restrained laminated planar frame
Close modal

The wavefields can now be related using the following expressions:

  • Wave reflection at Boundary LB and Boundary RB:
    d1,0+=rLB.d1,0
    (43a)
    dm+1,Lm+1=rRB.dm+1,Lm+1+
    (43b)
    where rLB and rRB are the reflection matrices associated with the right and left boundaries, respectively.
  • Wave propagation across the length of each individual continuous segment:
    di,Li+=f(LiL).di,0+
    (44a)
    di,0=f(LiL).di,Li
    (44b)
    where i = 1, 2,.., m + 1, Li/L is the fractional length of segment (i) and
    f(LiL)=[eis1LiL000es2LiL000eis3LiL]
    (45)
  • Wave reflection and transmission at each joint:
    di+1,0+=ti,i+1di,Li++ri+1,i+1di+1,0
    (46a)
    di,Li=ti+1,idi+1,0+ri,idi,Li+
    (46b)
    where i = 1, 2,.., m.

For a frame having m internal joint angles and m + 1 individual segments, Eqs. (43)(46) result in a total of 12(m + 1) algebraic equations, which can be combined and rewritten as
M.d=0
(47)
In Eq. (47), M is a square matrix of order 12(m + 1) and d is the amplitude vector of length 12(m + 1). The non-dimensional natural frequencies (p) are found when
|M|=0
(48)

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 C33 = r = 0. For isotropic metallic materials and symmetrically laminated composites, there is no bending–longitudinal elastic coupling and consequently C22 = 0.

Table 1

Rules for simplification of the present model

Set
Frame modelC22C33r
Isotropic EBT000
Isotropic FSDT0≠0≠0
Symmetrically laminated EBT000
Symmetrically laminated FSDT0≠0≠0
Unsymmetrically laminated EBT≠000
Unsymmetrically laminated FSDT≠0≠0≠0
Set
Frame modelC22C33r
Isotropic EBT000
Isotropic FSDT0≠0≠0
Symmetrically laminated EBT000
Symmetrically laminated FSDT0≠0≠0
Unsymmetrically laminated EBT≠000
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, L1 = L2 = L3 as shown in Fig. 5. The first set of computations is carried out for the special case of a rectangular portal frame (θ=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 A11 = EA, D11 = EI, and A55 = 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 + 11ν) is the shear correction factor [17].

Fig. 5
Inclined portal frame with elastic restraints
Fig. 5
Inclined portal frame with elastic restraints
Close modal

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¯LB,X,S¯LB,Z,S¯LB,Ψ,S¯RB,X,S¯RB,Z,S¯RB,Ψ) and (S¯1,X,S¯1,Z,S¯1,Ψ,S¯2,X,S¯2,Z,S¯2,Ψ)0.

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.

Table 2

Non-dimensional natural frequencies of an isotropic rectangular portal frame (θ = 90 deg)

Non-dimensional natural frequency (p)Non-dimensional parameter, Ka
Present theoryPresent theory
ModeEBTFSDTRef. [18]EBTFSDTRef. [18]
128.83328.78228.8371.7901.7881.790
2113.587113.193113.6143.5533.5463.553
3185.610184.476185.5864.5414.5274.541
4200.481198.885200.506b4.7204.7014.720b
5403.153398.964403.1666.6936.6586.693
6494.570487.248494.5737.4137.3587.413
7569.687559.572569.6817.9567.8857.956
8849.530832.998849.6069.7169.6219.716
91027.269999.4341027.33110.68410.53810.684
Non-dimensional natural frequency (p)Non-dimensional parameter, Ka
Present theoryPresent theory
ModeEBTFSDTRef. [18]EBTFSDTRef. [18]
128.83328.78228.8371.7901.7881.790
2113.587113.193113.6143.5533.5463.553
3185.610184.476185.5864.5414.5274.541
4200.481198.885200.506b4.7204.7014.720b
5403.153398.964403.1666.6936.6586.693
6494.570487.248494.5737.4137.3587.413
7569.687559.572569.6817.9567.8857.956
8849.530832.998849.6069.7169.6219.716
91027.269999.4341027.33110.68410.53810.684
a

Non-dimensional frequency representation used in Ref. [18]: K=p/3.

b

The fourth mode is only included in the author's later work [19].

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]: E1 = 145 × 109 Nm−2, E2 = E3 = 9.6 × 109 Nm−2, G12 = G13 = 4.1 × 109 Nm−2, G23 = 3.4 × 109 Nm−2, and ν12 = ν13 = ν23 = 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¯LB,X=S¯LB,Z=S¯RB,X=S¯RB,Z=500, S¯LB,Ψ=S¯RB,Ψ=1, S¯1,X=S¯1,Z=S¯1,Ψ=S¯2,X=S¯2,Z=S¯2,Ψ=0,

  • Parameter set B: S¯LB,X=S¯LB,Z=S¯RB,X=S¯RB,Z=500, S¯LB,Ψ=S¯RB,Ψ=1, S¯1,X=S¯1,Z=S¯2,X=S¯2,Z=50, S¯1,Ψ=S¯2,Ψ=0.1.

Table 3

Non-dimensional natural frequencies (p) of elastically restrained [0/90/90/0] laminated portal frames with angle θ = 90 deg

Parameter set AParameter set B
Present theoryPresent theory
L/hModeEBTFSDTFE ModelEBTFSDTFE Model
15111.8789.9419.94017.36816.19116.191
230.55729.69029.68932.13731.11331.112
339.94837.94037.93941.43939.52639.525
461.54648.88548.88261.66348.95148.948
50111.99811.76411.76417.43617.29017.290
231.23231.15631.15632.73132.64232.642
340.34040.08940.08941.79441.55641.556
462.15460.65160.65162.27560.76560.765
75112.00511.89911.89917.44017.37417.373
231.27031.23631.23632.76532.72532.725
340.36240.24840.24841.81441.70641.706
462.18761.50861.50862.30861.62561.626
Parameter set AParameter set B
Present theoryPresent theory
L/hModeEBTFSDTFE ModelEBTFSDTFE Model
15111.8789.9419.94017.36816.19116.191
230.55729.69029.68932.13731.11331.112
339.94837.94037.93941.43939.52639.525
461.54648.88548.88261.66348.95148.948
50111.99811.76411.76417.43617.29017.290
231.23231.15631.15632.73132.64232.642
340.34040.08940.08941.79441.55641.556
462.15460.65160.65162.27560.76560.765
75112.00511.89911.89917.44017.37417.373
231.27031.23631.23632.76532.72532.725
340.36240.24840.24841.81441.70641.706
462.18761.50861.50862.30861.62561.626
Table 4

Non-dimensional natural frequencies (p) of elastically restrained [90/0] laminated portal frames with angle θ = 90 deg

Parameter set AParameter set B
Present theoryPresent theory
L/hModeEBTFSDTFE ModelEBTFSDTFE Model
15110.1599.3589.35716.29815.83315.833
230.56730.02630.02632.07231.43231.432
335.77835.26235.26137.63637.10237.101
453.25547.18047.18053.50347.37247.372
5019.9229.8499.84816.11916.07616.075
230.94130.89030.89132.36032.30032.300
337.20837.15637.15638.88338.83138.831
451.94251.35451.35552.20951.61651.617
7519.8869.8539.85216.09216.07316.073
230.92030.89730.89732.33332.30732.307
337.36037.33737.33739.01638.99338.993
451.74851.48851.48952.01751.75551.756
Parameter set AParameter set B
Present theoryPresent theory
L/hModeEBTFSDTFE ModelEBTFSDTFE Model
15110.1599.3589.35716.29815.83315.833
230.56730.02630.02632.07231.43231.432
335.77835.26235.26137.63637.10237.101
453.25547.18047.18053.50347.37247.372
5019.9229.8499.84816.11916.07616.075
230.94130.89030.89132.36032.30032.300
337.20837.15637.15638.88338.83138.831
451.94251.35451.35552.20951.61651.617
7519.8869.8539.85216.09216.07316.073
230.92030.89730.89732.33332.30732.307
337.36037.33737.33739.01638.99338.993
451.74851.48851.48952.01751.75551.756

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 ((|pEBTpFSDT|/pFSDT)×100%) 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 = 2h, 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.

Table 5

Frequency convergence of the ansys® model for the [90/0] laminated frame restrained by Parameter set B: θ = 45 deg and L/h = 50

Number of elementsPresent theory (FSDT)
Mode245401113724216,53618,06022,500
124.06322.98222.98122.98222.98322.98322.98322.983
226.55726.51826.51826.51826.51826.51826.51826.518
336.20336.13736.13536.13536.13536.13536.13536.135
450.30649.72249.70749.69849.69749.69749.69749.697
Number of elementsPresent theory (FSDT)
Mode245401113724216,53618,06022,500
124.06322.98222.98122.98222.98322.98322.98322.983
226.55726.51826.51826.51826.51826.51826.51826.518
336.20336.13736.13536.13536.13536.13536.13536.135
450.30649.72249.70749.69849.69749.69749.69749.697

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 (|pANSYSpFSDT|/pFSDT)×100% 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.

Fig. 6
Lowest four non-dimensional natural frequencies (p) for an inclined [90/0] frame as a function of angle (θ); frame length L/h = 50, parameter set A restraints. Legend: (······) first anti-symmetric mode (A1), (——) first symmetric mode (S1), (—·—) second anti-symmetric mode (A2), and (- - -) second symmetric mode (S2).
Fig. 6
Lowest four non-dimensional natural frequencies (p) for an inclined [90/0] frame as a function of angle (θ); frame length L/h = 50, parameter set A restraints. Legend: (······) first anti-symmetric mode (A1), (——) first symmetric mode (S1), (—·—) second anti-symmetric mode (A2), and (- - -) second symmetric mode (S2).
Close modal

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.

Fig. 7
Mode shapes of the [90/0] laminated frame restrained with Parameter Set A; (θ = 25 deg, L/h = 50: (a) Symmetric first mode (p = 16.738), (b) anti-symmetric second mode (p = 23.485), (c) anti-symmetric third mode (p = 32.759), and (d) symmetric fourth mode (p = 49.407). Legend: (——) deformed frame and (- - -) undeformed frame.
Fig. 7
Mode shapes of the [90/0] laminated frame restrained with Parameter Set A; (θ = 25 deg, L/h = 50: (a) Symmetric first mode (p = 16.738), (b) anti-symmetric second mode (p = 23.485), (c) anti-symmetric third mode (p = 32.759), and (d) symmetric fourth mode (p = 49.407). Legend: (——) deformed frame and (- - -) undeformed frame.
Close modal
In Table 6, the effect of increasing the number of layers within the same thickness is investigated for the asymmetric cross-ply laminated frame with Parameter Set A restraints. The relative length of the frame is 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. (6a) and (6b) 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. (6a) and (6b) reduces to
A11=E1(E1+E2)hβ2(E1E2ν122)
(49a)
B11=E1(E1E2)h2β8n(E1E2ν122)
(49b)
D11=E1(E1+E2)h3β24(E1E2ν122)
(49c)
A55=κhβ2(G13+G23)
(49d)
Table 6

Influence of the number of layers on non-dimensional natural frequency (p) for an asymmetric cross-ply laminated frame subjected to Parameter set A: L/h = 50 and θ = 45 deg

[90/0]1[90/0]2[90/0]3[90/0]4
Mode numberPresent theoryFE ModelPresent theoryFE ModelPresent theoryFE ModelPresent theoryFE Model
119.77119.77123.39123.39123.74823.74823.85223.852
224.30124.30224.80024.80024.88924.88924.92324.924
334.57934.57937.22637.22637.71937.72037.89637.896
449.49349.49458.78758.78759.81959.82060.14160.141
[90/0]1[90/0]2[90/0]3[90/0]4
Mode numberPresent theoryFE ModelPresent theoryFE ModelPresent theoryFE ModelPresent theoryFE Model
119.77119.77123.39123.39123.74823.74823.85223.852
224.30124.30224.80024.80024.88924.88924.92324.924
334.57934.57937.22637.22637.71937.72037.89637.896
449.49349.49458.78758.78759.81959.82060.14160.141

The rigidities A11, D11, and A55 are independent of the number of layers (2n) for a constant segment thickness h, whereas the coupling rigidity B11 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 (E1/E2) 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 E1, the input material properties for the finite element model are the same as those given for AS/3501-6 graphite-epoxy. The value of E1 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%.

Table 7

Variation of the non-dimensional natural frequencies (p) of symmetric and asymmetric cross-ply laminated frames with orthotopicity ratio (E1/E2) subjected to Parameter set A: L/h = 50 and θ = 90 deg

[0/90/90/0][90/0]
E1/E2Mode numberPresent theoryFE ModelPresent theoryFE Model
5111.91711.91711.00711.006
231.21231.21331.14831.148
340.24840.24838.56638.566
461.61961.61957.45657.456
25111.62211.6219.4259.424
231.10431.10430.73830.738
339.94539.94536.74036.740
459.73959.73948.93248.932
50111.28411.2838.9798.977
230.97630.97630.50830.507
339.61539.61536.35536.355
457.57157.56946.24346.242
[0/90/90/0][90/0]
E1/E2Mode numberPresent theoryFE ModelPresent theoryFE Model
5111.91711.91711.00711.006
231.21231.21331.14831.148
340.24840.24838.56638.566
461.61961.61957.45657.456
25111.62211.6219.4259.424
231.10431.10430.73830.738
339.94539.94536.74036.740
459.73959.73948.93248.932
50111.28411.2838.9798.977
230.97630.97630.50830.507
339.61539.61536.35536.355
457.57157.56946.24346.242

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 = 50h 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.

Table 8

Comparison of computation time between ansys® and Mathematica 12.2 for the calculation of the first four natural frequencies

Number of elementsPresent
28054134724216,53628,86664,872
Time taken (s)1.281.722.254.065.9812.457.19
Number of elementsPresent
28054134724216,53628,86664,872
Time taken (s)1.281.722.254.065.9812.457.19

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 Δp = 55 is approximately 7.19 s. As discussed earlier in this work, 16,536 elements were used to model all beams of length L = 50h. 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.

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