A High-Frequency Model of a Rectilinear Beam with a T-Shaped Cross Section

Abstract

This paper derives an analytical model of a straight beam with a T-shaped cross section for use in the high-frequency range, defined here as approximately 1 to 35 kHz. The web, the right part of the flange, and the left part of the flange of the T-beam are modeled independently with two-dimensional elasticity equations for the in-plane motion and Mindlin flexural plate equation for the out-of-plane motion. The differential equations are solved with unknown wave propagation coefficients multiplied by circular spatial domain functions. These algebraic equations are then solved to yield the wave propagation coefficients and thus produce a solution to the displacement field in all three directions. An example problem is formulated and compared with solutions from fully elastic finite element modeling, a previously derived analytical model, and Timoshenko beam theory. It is shown that the accurate frequency range of this new model is significantly higher than that of the analytical model and the Timoshenko beam model, and, in the frequency range up to 35 kHz, the results compare very favorably to those from finite element analysis.

1. Introduction

This paper is a direct extension of a previous effort [1,2] that modeled the dynamics of a straight T-beam. This model accurately captured the physics of the beam up to about 8 kilohertz (kHz) for the example problem studied. The previous beam model became inaccurate above 8 kHz because of the limitations of the out-of-plane (normal) component, which was a Love‒Kirchhoff [3] plate formulation. The assumptions of the Love‒Kirchhoff plate equations typically produce model results that are too stiff, especially as the frequency increases. In the new model derived here, the Love‒Kirchhoff plate equations are replaced by Mindlin plate equations [4], whose theoretical basis includes shear deformation and rotary inertia terms, resulting in a higher-frequency range of analysis. This permits the new analytical model of a straight T-shaped beam to extend the frequency range of this system compared to previous modeling efforts. It is primarily intended for use in models that have reinforced plates that need improved accuracy at higher frequencies.

Beams are structural members and are designed to resist an external force either applied directly or applied to a body that they are supporting. They provide a concentrated or distributed stiffness in a mechanical system that has various design objectives. There is an extremely large body of analytical and experimental papers that model and analyze various types of beams. The first equation for the motion of a beam was developed by Bernoulli and Euler [5], and this equation is presented in almost every text on mechanical vibrations. This theory uses the assumption that all sections rotate orthogonally to the neutral axis of the beam. Timoshenko [6] revised this equation so that the rotation angle was a function of the shear effects and polar inertia of the beam. Beam theory has been made more accurate by the inclusion of higher-order displacement functions, usually in the axial direction. Bickford [7] used a third-order polynomial through the thickness of the beam to model the in-plane displacement field. Karama et al. [8] used an exponential function to model the shear distribution in the beam. Other functional distributions [9] are possible and have also been similarly applied to plate theory. These higher-order models are lumped parameter models and do not have the ability to model higher-frequency wave motion in the normal direction of the beam.

Beam shapes have become more geometrically diverse and their cross sections have been modified from rectangular to L-, T-, I-, and H-shaped, channel-shaped, and hollow box designs. Park et al. [10] studied the longitudinal wave motion of finite coupled thin plates. Kessissoglou [11] added the flexural motion to the previous reference so that a full three-dimensional analysis of an L-shaped finite plate became possible. Du et al. [12] investigated the free vibration of coupled rectangular plates with general boundary conditions, using energy methods to derive the differential equations of motion. Chen et al. [13] analyzed vibrations in box-type plate structures. Wang [14] addressed the problem of finite coupled plates whose intersection contained a mass. T-shaped beams have been the subject of some investigations [15,16], generally applied to concrete or reinforced concrete structures where static analysis and ultimate strength are the major foci of the research. Langley and Heron [17] derived a method to calculate the wave transmission coefficients of plate and beam junctions. Keir et al. [18] investigated coupled rectangular plates, with an emphasis on the effect of active control of these systems. Mitrou et al. [19] researched wave transmission in two-dimensional structures using a mixed finite element and wave and finite element methods.

Beam-reinforced structures have been analyzed for many years. Fluid-loaded stiffened plates have been researched using flexural wave plate models by Mace [20,21] and Lin and Hayek [22], with a general emphasis on structural acoustic responses. The plate governing equations in a stiffened plate analysis has been extended to admit fully elastic wave propagation by Hull and Welch [23], and this allows much higher-frequency studies compared to previous flexural wave models. Models of these structures typically have an infinite length assumption as the energy propagates down the length of the structure and, due to the large spatial lengths and slight damping present, does not form a standing wave pattern. The model of the beam reinforcement has typically been an infinite length Bernoulli‒Euler or Timoshenko model, which has very low to low frequency range limitations.

This paper develops an analytical model of a T-shaped beam for high-frequency range analysis. In this model, the web and flange of the T-beam are modeled independently with two-dimensional elasticity equations accounting for the in-plane motion and the Timoshenko plate equations accounting for the out-of-plane motion. The method presented here combines six in-plane components and six out-of-plane components into a single model that can predict the response of a T-beam to various external loads. The web and flange are joined by 12 equilibrium and constraint equations and the free and forced edges are modeled with 12 additional force and moment equations. Modeling the beam in this manner allows the web and flange equations of motion to incorporate coupled in-plane and out-of-plane plate responses, and these equations make the model much more accurate compared to previous shear deformation models [7,8]. For the specific example presented in this paper, the three-dimensional displacement fields are studied with respect to independent loads in the three primary axes of the Cartesian coordinate system. Comparisons are made to a previous analytical model, a Timoshenko beam model, and a finite element model. It is shown that, for the specific example presented here, the accurate frequency region of the normal displacement divided by normal pressure is increased from 1 kHz of the Timoshenko beam model, or 8 kHz of the previous analytical model, to 35 kHz for the new beam model. Beam mode shapes are also calculated.

2. Methods

The system under consideration is an inverted T-beam with continuous excitation at the top of the structure. A schematic of this system with the normal load is shown in Figure 1. Usually this type of beam is symmetric, i.e., −a = b, where −a and b are the respective widths of the left and right parts of the flange, but this is not a necessary condition for the model. The narrow top portion is called the web, and the wide lower portion is called the flange. The problem is analytically modeled using the two-dimensional plane stress elastic equations for in-plane motions of the web and the flange, and Mindlin plate equations are used for the out-of-plane motion of the web and the flange. This model is an extension of a previous model wherein the classical plate equation was used to model the out-of-plane motions. The model makes the following assumptions: (1) the system has infinite spatial extent in the y-direction, (2) the excitation is at a fixed frequency and fixed wavenumber in the y-direction, (3) the angle at the intersection of the web and the flange is always a right angle, (4) the material properties of the web and flange are identical, and (5) the particle motion is linear. The model is developed by analyzing the system as three separate components: the web, the left part of the flange, and the right part of the flange. For all three components, the two-dimensional elasticity equations of motion are used for the in-plane motion and Mindlin plate equations are used for out-of-plane motion.

The equations modeling the in-plane motion of the web and flange begin with the Navier‒Cauchy [24] fully elastic equations of motion. These are reduced to two-dimensional plane stress equations of motion. The first one, in the x-direction, is written as

$$\frac{E}{1-{\upsilon }^2}\frac{{\partial }^2{u}_n\left(x_n,y,t\right)}{\partial x_n^2}+\frac{E}{2\left(1-\upsilon \right)}\frac{{\partial }^2{v}_n\left(x_n,y,t\right)}{\partial x_n\partial y}+G\frac{{\partial }^2{u}_n\left(x_n,y,t\right)}{\partial y^2}=\rho \frac{{\partial }^2{u}_n\left(x_n,y,t\right)}{\partial t^2}$$

(1)

and the second one, in the y-direction, is written as

$$\frac{E}{1-{\upsilon }^2}\frac{{\partial }^2{v}_n\left(x_n,y,t\right)}{\partial x_n^2}+\frac{E}{2\left(1-\upsilon \right)}\frac{{\partial }^2{u}_n\left(x_n,y,t\right)}{\partial x_n\partial y}+G\frac{{\partial }^2{v}_n\left(x_n,y,t\right)}{\partial y^2}=\rho \frac{{\partial }^2{v}_n\left(x_n,y,t\right)}{\partial t^2},$$

(2)

where u_n(x_n,y,t) is the displacement in the x-direction, v_n(x_n,y,t) is the displacement in the y-direction, ρ is density, E is Young’s modulus, G is the shear modulus, υ is Poisson’s ratio, and the subscript n denotes either the web (n = w), left flange (n = fl), or right flange (n = fr). The specific orientations of the three coordinate systems to the three components of the model are depicted in Figure 2. Note that they all share a common y-axis.

The equations modeling the out-of-plane motion of the components in the transverse z-direction are derived using Mindlin plate theory, and this set of equations is written as [4]

$$D_n\frac{{\partial }^2{\xi }_n\left(x_n,y,t\right)}{\partial x_n^2}+\frac{D_n\left(1-\upsilon \right)}{2}\frac{{\partial }^2{\xi }_n\left(x_n,y,t\right)}{\partial y^2}+\frac{D_n\left(1+\upsilon \right)}{2}\frac{{\partial }^2{\psi }_n\left(x_n,y,t\right)}{\partial x_n\partial y}+$$

$$\kappa G{t}_n\left(\frac{\partial w_n\left(x_n,y,t\right)}{\partial x_n}-{\xi }_n\left(x_n,y,t\right)\right)=J_n\rho \frac{{\partial }^2{\xi }_n\left(x_n,y,t\right)}{\partial t^2}$$

(3)

$$D_n\frac{{\partial }^2{\psi }_n\left(x_n,y,t\right)}{\partial y^2}+\frac{D_n\left(1-\upsilon \right)}{2}\frac{{\partial }^2{\psi }_n\left(x_n,y,t\right)}{\partial x_n^2}+\frac{D_n\left(1+\upsilon \right)}{2}\frac{{\partial }^2{\xi }_n\left(x_n,y,t\right)}{\partial x_n\partial y}+$$

$$\kappa G{t}_n\left(\frac{\partial w_n\left(x_n,y,t\right)}{\partial y}-{\psi }_n\left(x_n,y,t\right)\right)=J_n\rho \frac{{\partial }^2{\psi }_n\left(x_n,y,t\right)}{\partial t^2}$$

(4)

and

$$\kappa G{t}_n\left(\frac{{\partial }^2{w}_n\left(x_n,y,t\right)}{\partial x_n^2}+\frac{{\partial }^2{w}_n\left(x_n,y,t\right)}{\partial y^2}-\frac{\partial {\xi }_n\left(x_n,y,t\right)}{\partial x_n}-\frac{\partial {\psi }_n\left(x_n,y,t\right)}{\partial y}\right)=t_n\rho \frac{{\partial }^2{w}_n\left(x_n,y,t\right)}{\partial t^2},$$

(5)

where w_n(x_n,y,t) is the out-of-plane displacement, ξ_n(x_n,y,t) is the angle of rotation of a normal line with respect to the y-coordinate, ψ_n(x_n,y,t) is the angle of rotation of a normal line with respect to the x-coordinate, κ is the shear correction factor, t_n is the thickness, D_n is the flexural rigidity and is equal to

$$D_n=\frac{E{t}_n^3}{12\left(1-{\upsilon }^2\right)}.$$

(6)

J_n is equal to

$$J_n=\frac{t_n^3}{12},$$

(7)

and it is noted that t_w = b_w and t_fl = t_fr = h_f.

The solutions to Equations (1) and (2) for the in-plane motion are [25]

$$u_n\left(x_n,y,t\right)=U_n\left(x_n\right)\exp \left(\mathrm{i}{k}_{y}y\right)\exp \left(\mathrm{i}\omega t\right)$$

(8)

and

$$v_n\left(x_n,y,t\right)=V_n\left(x_n\right)\exp \left(\mathrm{i}{k}_{y}y\right)\exp \left(\mathrm{i}\omega t\right),$$

(9)

where

$$U_w\left(x_w\right)=C_1\alpha \cos \left(\alpha x_w\right)-C_2\alpha \sin \left(\alpha x_w\right)+C_3\mathrm{i}{k}_y\sin \left(\beta x_w\right)+C_4\mathrm{i}{k}_y\cos \left(\beta x_w\right)$$

(10)

$$U_{fl}\left(x_{fl}\right)=C_9\alpha \cos \left(\alpha x_{fl}\right)-C_{10}\alpha \sin \left(\alpha x_{fl}\right)+C_{11}\mathrm{i}{k}_y\sin \left(\beta x_{fl}\right)+C_{12}\mathrm{i}{k}_y\cos \left(\beta x_{fl}\right)$$

(11)

$$U_{fr}\left(x_{fr}\right)=C_{17}\alpha \cos \left(\alpha x_{fr}\right)-C_{18}\alpha \sin \left(\alpha x_{fr}\right)+C_{19}\mathrm{i}{k}_y\sin \left(\beta x_{fr}\right)+C_{20}\mathrm{i}{k}_y\cos \left(\beta x_{fr}\right)$$

(12)

$$V_w\left(x_w\right)=C_1\mathrm{i}{k}_y\sin \left(\alpha x_w\right)+C_2\mathrm{i}{k}_y\cos \left(\alpha x_w\right)-C_3\beta \sin \left(\beta x_w\right)+C_4\beta \cos \left(\beta x_w\right)$$

(13)

$$V_{fl}\left(x_{fl}\right)=C_9\mathrm{i}{k}_y\sin \left(\alpha x_{fl}\right)+C_{10}\mathrm{i}{k}_y\cos \left(\alpha x_{fl}\right)-C_{11}\beta \cos \left(\beta x_{fl}\right)+C_{12}\beta \sin \left(\beta x_{fl}\right)$$

(14)

$$V_{fr}\left(x_{fr}\right)=C_{17}\mathrm{i}{k}_y\sin \left(\alpha x_{fr}\right)+C_{18}\mathrm{i}{k}_y\cos \left(\alpha x_{fr}\right)-C_{19}\beta \cos \left(\beta x_{fr}\right)+C_{20}\beta \sin \left(\beta x_{fr}\right).$$

(15)

In Equations (10)–(15), α and β are modified wavenumbers and are equal to

$$\alpha =\sqrt{k_p^2-k_y^2}$$

(16)

and

$$\beta =\sqrt{k_s^2-k_y^2},$$

(17)

where k_p is the plate wavenumber, expressed as

$$k_p=\frac{\omega }{c_p}=\frac{\omega }{\sqrt{\frac{E}{\left(1-{\upsilon }^2\right)\rho }}},$$

(18)

where k_s is the shear wavenumber expressed as

$$k_s=\frac{\omega }{c_s}=\frac{\omega }{\sqrt{\frac{G}{\rho }}}.$$

(19)

The solutions to w_n(x_n,y,t) for Equations (3)–(5) for the out-of-plane motion are [25]

$$w_n\left(x_n,y,t\right)=W_n\left(x_n\right)\exp \left(\mathrm{i}{k}_{y}y\right)\exp \left(\mathrm{i}\omega t\right),$$

(20)

where

$$W_w\left(x_w\right)=C_5\exp \left({\lambda }_1{x}_w\right)+C_6\exp \left({\lambda }_2{x}_w\right)+C_7\exp \left({\lambda }_3{x}_w\right)+C_8\exp \left({\lambda }_4{x}_w\right)$$

(21)

$$W_{fl}\left(x_{fl}\right)=C_{13}\exp \left({\lambda }_5{x}_{fl}\right)+C_{14}\exp \left({\lambda }_6{x}_{fl}\right)+C_{15}\exp \left({\lambda }_7{x}_{fl}\right)+C_8\exp \left({\lambda }_8{x}_{fl}\right)$$

(22)

and

$$W_{fr}\left(x_{fr}\right)=C_{21}\exp \left({\lambda }_5{x}_{fr}\right)+C_{22}\exp \left({\lambda }_6{x}_{fr}\right)+C_{23}\exp \left({\lambda }_7{x}_{fr}\right)+C_{24}\exp \left({\lambda }_8{x}_{fr}\right).$$

(23)

In Equations (21)–(23), the values of λ_i are out-of-plane eigenvalues and for Equation (21) are equal to [26]

$${\lambda }_{1,2,3,4}=\pm \sqrt{\frac{-b\pm \sqrt{b^2-4ac}}{2a}},$$

(24)

where

$$a=D_w\kappa G$$

(25)

$$b=-2D_w\kappa G{k}_y^2+D_w\rho {\omega }^2+J_w\kappa G\rho {\omega }^2$$

(26)

$$c=D_w\kappa G{k}_y^4-D_w\rho {\omega }^2{k}_y^2+J_w\kappa G\rho {\omega }^2{k}_y^2+J_w{\rho }^2{\omega }^4-\kappa G{b}_w\rho {\omega }^2$$

(27)

and

$${\lambda }_{5,6,7,8}=\pm \sqrt{\frac{-b\pm \sqrt{b^2-4ac}}{2a}},$$

(28)

where

$$a=D_f\kappa G$$

(29)

$$b=-2D_f\kappa G{k}_y^2+D_f\rho {\omega }^2+J_f\kappa G\rho {\omega }^2$$

(30)

$$c=D_f\kappa G{k}_y^4-D_f\rho {\omega }^2{k}_y^2+J_f\kappa G\rho {\omega }^2{k}_y^2+J_f{\rho }^2{\omega }^4-\kappa G{h}_f\rho {\omega }^2,$$

(31)

with

$$D_f=D_{fl}=D_{fr}$$

(32)

and

$$J_f=J_{fl}=J_{fr}.$$

(33)

The solutions to the rotational angles are written as

$${\xi }_n\left(x_n,y,t\right)={\mathsf{\Xi }}_n\left(x_n\right)\exp \left(\mathrm{i}{k}_{y}y\right)\exp \left(\mathrm{i}\omega t\right)$$

(34)

and

$${\psi }_n\left(x_n,y,t\right)={\mathsf{\Psi }}_n\left(x_n\right)\exp \left(\mathrm{i}{k}_{y}y\right)\exp \left(\mathrm{i}\omega t\right),$$

(35)

where

$${\mathsf{\Xi }}_w\left(x_w\right)=C_5{a}_1\exp \left({\lambda }_1{x}_w\right)+C_6{a}_2\exp \left({\lambda }_2{x}_w\right)+C_7{a}_3\exp \left({\lambda }_3{x}_w\right)+C_8{a}_4\exp \left({\lambda }_4{x}_w\right)$$

(36)

$${\mathsf{\Xi }}_{fl}\left(x_{fl}\right)=C_{13}{c}_1\exp \left({\lambda }_5{x}_{fl}\right)+C_{14}{c}_2\exp \left({\lambda }_6{x}_{fl}\right)+C_{15}{c}_3\exp \left({\lambda }_7{x}_{fl}\right)+C_{16}{c}_4\exp \left({\lambda }_8{x}_{fl}\right)$$

(37)

$$\begin{array}{c}{\mathsf{\Xi }}_{fr}\left(x_{fr}\right)=C_{21}{c}_1\exp \left({\lambda }_5{x}_{fr}\right)+C_{22}{c}_2\exp \left({\lambda }_6{x}_{fr}\right)+C_{23}{c}_3\exp \left({\lambda }_7{x}_{fr}\right)\\ +C_{24}{c}_4\exp \left({\lambda }_8{x}_{fr}\right)\end{array}$$

(38)

$${\mathsf{\Psi }}_w\left(x_w\right)=C_5{b}_1\exp \left({\lambda }_1{x}_w\right)+C_6{b}_2\exp \left({\lambda }_2{x}_w\right)+C_7{b}_3\exp \left({\lambda }_3{x}_w\right)+C_8{b}_4\exp \left({\lambda }_4{x}_w\right)$$

(39)

$${\mathsf{\Psi }}_{fl}\left(x_{fl}\right)=C_{13}{d}_1\exp \left({\lambda }_5{x}_{fl}\right)+C_{14}{d}_2\exp \left({\lambda }_6{x}_{fl}\right)+C_{15}{d}_3\exp \left({\lambda }_7{x}_{fl}\right)+C_{16}{d}_4\exp \left({\lambda }_8{x}_{fl}\right)$$

(40)

and

$${\mathsf{\Psi }}_{fr}\left(x_{fr}\right)=C_{21}{d}_1\exp \left({\lambda }_5{x}_{fr}\right)+C_{22}{d}_2\exp \left({\lambda }_6{x}_{fr}\right)+C_{23}{d}_3\exp \left({\lambda }_7{x}_{fr}\right)+C_{24}{d}_4\exp \left({\lambda }_8{x}_{fr}\right),$$

(41)

where

$$a_n=\frac{\kappa G{b}_w{\lambda }_n}{D_w{k}_y^2-D_w{\lambda }_n^2-J_w\rho {\omega }^2+\kappa G{b}_w}$$

(42)

$$b_n=\frac{\kappa G{b}_w\mathrm{i}{k}_y}{D_w{k}_y^2-D_w{\lambda }_n^2-J_w\rho {\omega }^2+\kappa G{b}_w}$$

(43)

$$c_n=\frac{\kappa G{h}_f{\lambda }_{\left(n+4\right)}}{D_f{k}_y^2-D_f{\lambda }_{\left(n+4\right)}^2-J_f\rho {\omega }^2+\kappa G{h}_f}$$

(44)

and

$$d_n=\frac{\kappa G{h}_f\mathrm{i}{k}_y}{D_f{k}_y^2-D_f{\lambda }_{\left(n+4\right)}^2-J_f\rho {\omega }^2+\kappa G{h}_f}.$$

(45)

The constants C_i in Equations (10)–(15), (21)–(23), and (36)–(41) are wave propagation coefficients and are determined from the boundary conditions of the system. The solutions to the rotations are included here because they will be used to calculate the out-of-plane shear force and moment in the x-direction.

The various forces of the structure, shown in Figure 3, are now mathematically defined. These will be used in the force and moment balance equations to solve for the values of C_i. The normal in-plane forces are determined using [27]

$$N_{xx}^{\left(n\right)}\left(x_n\right)=\frac{E{t}_n}{1-{\upsilon }^2}\left[\frac{\mathrm{d}{U}_n\left(x_n\right)}{\mathrm{d}{x}_n}+\mathrm{i}{k}_y\upsilon V_n\left(x_n\right)\right]$$

(46)

and the shear in-plane forces are

$$N_{xy}^{\left(n\right)}\left(x_n\right)=\frac{E{t}_n}{2\left(1+\upsilon \right)}\left[\mathrm{i}{k}_y{U}_n\left(x_n\right)+\frac{\mathrm{d}{V}_n\left(x_n\right)}{\mathrm{d}{x}_n}\right].$$

(47)

The shear out-of-plane forces are calculated using [26]

$$V_x^{\left(n\right)}\left(x_n\right)=\kappa G{t}_n\left[\frac{\mathrm{d}{W}_n\left(x_n\right)}{\mathrm{d}{x}_n}-{\mathsf{\Xi }}_n\left(x_n\right)\right]+\frac{D_n\left(1-\upsilon \right)}{2}\left[k_y^2{\mathsf{\Xi }}_n\left(x_n\right)-\mathrm{i}{k}_y\frac{{\mathrm{d}\mathsf{\Psi }}_n\left(x_n\right)}{\mathrm{d}{x}_n}\right]$$

(48)

and the moments with respect to the x-axis are determined using

$$M_{xx}^{\left(n\right)}\left(x_n\right)=-D_n\left[\frac{{\mathrm{d}\mathsf{\Xi }}_n\left(x_n\right)}{\mathrm{d}{x}_n}+\mathrm{i}{k}_y\upsilon {\mathsf{\Psi }}_n\left(x_n\right)\right],$$

(49)

where it is noted that the exponential terms exp(iωt) and exp(ik_yy) are suppressed from Equations (46)–(49).

There are 24 boundary conditions on the beam. The boundary conditions at the top of the web (x_w = h_w) are

$$N_{xx}^{\left(n\right)}\left(h_w\right)=-b_w{P}_0$$

(50)

$$N_{xy}^{\left(n\right)}\left(h_w\right)=-b_w{F}_0$$

(51)

$$V_x^{\left(n\right)}\left(h_w\right)=-b_w{Q}_0$$

(52)

and

$$M_{xx}^{\left(n\right)}\left(h_w\right)=0,$$

(53)

where P₀ is the normal external pressure acting in the x-direction of the web, F₀ is the axial external pressure acting in the y-direction of the web, and Q₀ is the transverse external pressure acting in the z-direction of the web. Implicit in these pressure loads is the multiplication of exponential functions in y-direction wavenumber and frequency. In general, the most important loading quantity is the normal pressure. Note that these forcing functions act on the top of the web, because this model allows the beam to be loaded at a location other than the neutral axis of the beam, and this corresponds more closely to the actual physical problem than loading the beam on its neutral axis. There are three force balances at the intersection of the web and flange (x_w = x_fl = x_fr = 0). These force balances are written as

$$N_{xx}^{\left(w\right)}\left(0\right)-V_x^{\left(fl\right)}\left(0\right)+V_x^{\left(fr\right)}\left(0\right)=0$$

(54)

$$V_x^{\left(w\right)}\left(0\right)+N_{xx}^{\left(fl\right)}\left(0\right)-N_{xx}^{\left(fr\right)}\left(0\right)=0$$

(55)

$$N_{xy}^{\left(w\right)}\left(0\right)-N_{xy}^{\left(fl\right)}\left(0\right)+N_{xy}^{\left(fr\right)}\left(0\right)=0,$$

(56)

and there is a moment balance at this location written as

$$M_{xx}^{\left(w\right)}\left(0\right)-M_{xx}^{\left(fl\right)}\left(0\right)+M_{xx}^{\left(fr\right)}\left(0\right)=0.$$

(57)

There are eight continuity equations for the intersection of the web and the flange. The continuity terms are shown in Figure 4, and the displacement continuity equations are written as

$$U_w\left(0\right)=W_{fl}\left(0\right)=W_{fr}\left(0\right)$$

(58)

$$W_w\left(0\right)=-U_{fl}\left(0\right)=-U_{fr}\left(0\right)$$

(59)

$$V_w\left(0\right)=V_{fl}\left(0\right)=V_{fr}\left(0\right),$$

(60)

and the slope continuity equations are written as

$${\mathsf{\Xi }}_w\left(0\right)={\mathsf{\Xi }}_{fl}\left(0\right)={\mathsf{\Xi }}_{fr}\left(0\right).$$

(61)

The boundary conditions at the free end of the left flange (x_fl = a) are

$$N_{xx}^{\left(fl\right)}\left(a\right)=0$$

(62)

$$N_{xy}^{\left(fl\right)}\left(a\right)=0$$

(63)

$$V_x^{\left(fl\right)}\left(a\right)=0$$

(64)

and

$$M_{xx}^{\left(fl\right)}\left(a\right)=0,$$

(65)

where it is noted that a < 0. The boundary conditions at the free end of the right flange (x_fr = b) are

$$N_{xx}^{\left(fr\right)}\left(b\right)=0$$

(66)

$$N_{xy}^{\left(fr\right)}\left(b\right)=0$$

(67)

$$V_x^{\left(fr\right)}\left(b\right)=0$$

(68)

and

$$M_{xx}^{\left(fr\right)}\left(b\right)=0.$$

(69)

Inserting Equations (10)–(15), (21)–(23), and (36)–(41) into Equations (50)–(69) produces a 24-by-24 algebraic matrix equation given by

$$\left[A\right]\left\{x\right\}=\left\{b\right\},$$

(70)

where the entries of [A] are in the Appendix A as Equations (A1)–(A132), the vector {x} is

$$\left\{x\right\}={\left\{\begin{array}{ccccc}{C}_1& C_2& \dots & C_{23}& C_{24}\end{array}\right\}}^{\mathrm{T}}$$

(71)

and the {b} vector is

$$\left\{b\right\}={\left\{\begin{array}{cccccccc}-b_w{P}_0& -b_w{F}_0& -b_w{Q}_0& 0& 0& \dots & 0& 0\end{array}\right\}}^{\mathrm{T}}.$$

(72)

The solution to the wave propagation coefficients C_i in Equation (70) is found using

$$\left\{x\right\}={\left[A\right]}^{-1}\left\{b\right\}.$$

(73)

Once these coefficients are known, they can be inserted into Equations (10), (13), and (21), and the response of the web for external loading in three dimensions can be calculated. Additionally, the displacement of the flange can also be calculated, but it is typically not a quantity of interest.

To integrate this beam model into a reinforced structural model, the dynamic stiffness components of the beam are typically calculated and used. For a symmetric T-beam, there are four unique and nonzero terms. The first term is the dynamic stiffness of the normal displacement to normal pressure and is written as

$$K_{zz}=\frac{-b_w{P}_0}{U_w\left(h_w\right)};$$

(74)

the second term is the dynamic stiffness of normal displacement to axial pressure (and equal to axial displacement to normal pressure) and is written as

$$K_{zy}=K_{yz}=\frac{-b_w{F}_0}{U_w\left(h_w\right)}=\frac{-b_w{P}_0}{V_w\left(h_w\right)};$$

(75)

the third term is the dynamic stiffness of axial displacement to axial pressure and is written as

$$K_{yy}=\frac{-b_w{F}_0}{V_w\left(h_w\right)};$$

(76)

and the fourth term is the dynamic stiffness of transverse displacement to transverse pressure and is written as

$$K_{xx}=\frac{-b_w{Q}_0}{W_w\left(h_w\right)},$$

(77)

where the units of Equations (74)–(77) are stiffness per unit length, which in the metric system is (N/m)/m or alternatively N·m⁻². Finally, it is noted that, if a second flange is present on the top of the beam, i.e., an I- or an H-beam design, this dynamic contribution can be added to the model in the same method as the bottom flange equations.

3. Results

The model is now analyzed using an example problem where the beam has material and geometric properties that are consistent with an application in underwater structures. The T-beam has the following physical dimensions: height of the web h_w = 0.2436 m (9.59 in), width of the web b_w = 0.0140 m (0.550 in), height of the flange h_f = 0.0333 m (1.310 in), and width of the flange b_f = 0.1981 m (7.800 in), which results in a = −0.0991 m and b = 0.0991 m. The beam is made of steel that has the following mechanical properties: Young’s modulus E = 200 × 10⁹ N·m⁻², shear modulus G = 76.92 × 10⁹ N·m⁻², Poisson’s ratio υ = 0.30, and density ρ = 7800 kg·m⁻³. The value for the shear correction factor is κ = 0.8333. The analytical model was programmed and the results were displayed using the MATLAB programming language.

The beam is independently loaded on its top surface with three separate loads that correspond to normal (web in-plane), axial (web in-plane) and transverse (web out-of-plane) pressure. Although any location of the beam can be chosen for displacement output, the top of the web is investigated here because this location is pertinent to the analysis of reinforced structures. This allows the dynamic stiffness of the beam to be calculated and subsequently used in analysis of beams attached to plates or elastic bodies. Thus, the output of the model is normal, axial, and transverse beam displacement at the top of the web. It is noted, however, that by far the most important model output is the normal displacement divided by normal pressure as this corresponds to the main design objective of most beams. Plots of the other outputs are included for completeness. The finite element model results were produced using COMSOL finite element program using a model that consisted of 2200 quadratic serendipity hexahedral elements and a total of 49,659 degrees of freedom.

Figure 5 is a comparison of the normal displacement divided by the normal pressure versus the axial wavenumber and frequency in the decibel scale referenced to m Pa⁻¹. The analytical model results are on the left and the finite element results are on the right. Figure 6 is a plot of the normal beam stiffness K_zz versus frequency at zero axial wavenumber in the decibel scale referenced to (N/m)/m. The analytical model is the solid line, the previous analytical model [1,2] are the square markers, the Timoshenko beam model [6] are the diamond markers and the finite element results are depicted with circular markers. The Timoshenko beam stiffness in the normal direction was calculated using the following equation [6]:

$$K_{zz}\left(k_y\equiv 0\right)=\frac{AI{\rho }^2{\omega }^4-A^2\kappa G\rho {\omega }^2}{A\kappa G-I\rho {\omega }^2},$$

(78)

where A is the area, and I is the area moment of inertia of the T-beam incorporating both the web and the flange components as a single entity. These values are A = 0.01 m², and I = 6.044 × 10⁻⁵ m⁴. Figure 7 is a comparison of the axial displacement divided by the axial pressure versus the axial wavenumber and frequency. Figure 8 is a plot of the axial beam stiffness K_yy versus frequency at zero axial wavenumber. The analytical model is the solid line, the previous analytical model [1,2] are the square markers and the finite element results are depicted with circular markers. Note that the two analytical models are identical in this plot because there is no out-of-plane motion present in the structure when it is subjected to axial loading. Figure 9 is a comparison of the transverse displacement divided by the transverse pressure versus the axial wavenumber and frequency. Figure 10 is a plot of the transverse beam stiffness K_xx versus frequency at zero axial wavenumber. Figure 11 is a comparison of the axial displacement divided by the normal pressure versus the axial wavenumber and frequency. Note that Figure 11 is also identical to the normal displacement divided by the axial pressure. The beam transfer functions of normal displacement divided by axial forcing and axial displacement divided by normal forcing are both zero for zero axial wavenumber; thus, the beam stiffness terms K_zy and K_zy have no physical meaning (for k_y = 0 only), and these quantities do not have corresponding stiffness plots. Note that in Figure 5 through Figure 11 there is broad-based agreement between the new analytical model and the finite element model. Additionally, the Timoshenko beam model in Figure 6 is valid to approximately 1 kHz, where the magnitude begins to diverge from both the analytical models and the finite element model.

To ensure the finite element results have accurately converged, the finite element model was rerun at a frequency of 35 kHz and a wavenumber of 50 rad·m⁻¹ with various sized elements. The previous results were generated with a web thickness (b_w) divided by element size ratio of 10. This ratio was changed to 1, 2, 4, 8, 16, 64, 128 and 256 and a comparison of the web normal displacement at each ratio was compared to the web normal displacement at a ratio of 256. It was found that using a ratio of 10 produced results that had approximately a 0.4% difference (−8 dB) compared to a mesh size that was 10 times more dense. These convergence results are shown graphically in Figure 12 where the dashed line corresponds to the finite element results presented in this section.

The beam system presented here supports an infinite number of propagating modes that are illustrated in Figure 5, Figure 7, Figure 9 and Figure 11, Figure 7 as high-displacement regions. Because the most important response of the system is the normal displacement divided by the normal pressure, the first four dominant modes from this transfer function near zero axial wavenumber are plotted. Figure 13 is a plot of the mode shape at 202 Hz, Figure 14 is a plot of the mode shape at 3305 Hz, Figure 15 is a plot of the mode shape at 7228 Hz, and Figure 16 is a plot of the mode shape at 15,036 Hz. In Figure 13, Figure 14, Figure 15 and Figure 16, the left portion of the plots are the axial displacements and the right portion are the normal displacements. The transverse displacements for these modes are zero or very close to zero and are not shown. The mode shapes are symmetric about the web, i.e., the right flange response will be identical to the response of the left flange. In all of the plots, the axial wavenumber was set equal to 2 rad m⁻¹ as a value of zero will yield an axial displacement of zero everywhere. The first mode shape is a low-frequency flexural wave whereas the others are more complex in their displacement shapes and are based on the geometric properties of the web and flange. Note that the beam stiffness K_zz is zero at these resonant frequency locations.

4. Conclusions

A high-frequency analytical model for a T-shaped beam was derived and compared to a previous analytical model, a Timoshenko beam model, and a fully elastic finite element model. This new model was constructed with two-dimensional elastic equations for the in-plane motion and Mindlin plate equations for the out-of-plane motion. This allows for almost a total elastic response of the entire system. For the beam example problem presented, the analytical model and the finite element compared favorably up to 35 kHz. Four of the mode shapes of the beam were plotted. The application of this model to a reinforced structure is discussed.