Two Dimensional Elasticity Solution for Transient Responce of Simply Supported Beams Under Moving Loads

8/2/2019 Two Dimensional Elasticity Solution for Transient Responce of Simply Supported Beams Under Moving Loads

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Acta Mech 217, 205218 (2011)DOI 10.1007/s00707-010-0393-7

Seyyed M. Hasheminejad Ahmad Rafsanjani

Two-dimensional elasticity solution for transient responseof simply supported beams under moving loads

Received: 16 November 2009 / Revised: 4 August 2010 / Published online: 26 October 2010 Springer-Verlag 2010

Abstract A semi-analytical analysis for the transient elastodynamic response of an arbitrarily thick simplysupported beam due to the action of an arbitrary moving transverse load is presented, based on the linear theoryof elasticity. The solution of the problem is derived by means of the powerful state space technique in conjunc-tion with the Laplace transformation with respect to the time coordinate. The inversion of Laplace transformhas been carried out numerically using Durbins approach based on Fourier series expansion. Special conver-gence enhancement techniques are invoked to completely eradicate spurious oscillations and obtain uniformlyconvergent solutions. Detailed numerical results for the transient vibratory responses of concrete beams ofselected thickness parameters are obtained and compared for three types of harmonic moving concentratedloads: accelerated, decelerated and uniform. The effects of the load velocity, pulsation frequency and beamaspect ratio on the dynamic response are examined. Also, comparisons are made against solutions based onEulerBernoulli and Timoshenko beam models. Limiting cases are considered, and the validity of the modelis established by comparison with the solutions available in the existing literature as well as with the aid of a

commercial finite element package.

1 Introduction

Beams are very important elements in civil, mechanical and aeronautical engineering. The vibration problemof single-span beams is one of the original problems of structural dynamics that has been explored in detailsfor well over a century [1]. In particular, the dynamic response of beam-type structures to moving loads hasbeen well documented in hundreds of contributions during the past few decades, owing to their extensive usein various engineering applications [2]. Historically, the analysis of moving loads on beams can be traced backto the nineteenth century, when railroad construction was first initiated [3]. Many methods have been presentedfor the response prediction. Research on this subject is still in progress, especially due to the improvement ofcomputers. It is beyond the scope of this article to present an exhaustive literature review of the moving-load-induced-vibration problem. Only the most relevant works directly related to the present study will be brieflydiscussed below in order to motivate the problem statement and method of solution.

The classical solution of a simply supported finite BernoulliEuler beam resting on a continuous elasticfoundation and subject to a constant moving concentrated load was presented by Timoshenko [ 1]. Steele [4]used Laplace transformation of the time variable to obtain an asymptotic solution for the transient responseof EulerBernoulli and Timoshenko beams on elastic foundations under moving loads. Lee and Ng [5]employed Euler beam theory and the assumed mode method to analyze the dynamic response of a beam with a

S. M. Hasheminejad (B) A. RafsanjaniAcoustics Research Laboratory, Department of Mechanical Engineering,Iran University of Science and Technology, 16844 Narmak, Tehran, IranE-mail: [email protected]


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206 S. M. Hasheminejad, A. Rafsanjani

single-sided crack subject to a moving load on the opposite side. Jaiswal and Iyengar [6] studied the dynamicresponse of an infinitely long beam resting on a foundation of finite depth and under the action of a mov-ing force. The effects of various parameters such as foundation mass, velocity of the moving load, dampingand axial force on the beam were investigated. Thambiratnam and Zhuge [7] used a simple finite element

method along with the Newmark integration scheme for treating the transient analysis of beams on an elas-tic foundation, subjected to a concentrated moving point load. Felszeghy [8] used the Fourier transformmethod and numerical integration based on the method of characteristics to obtain the transient response of asimply supported semi-infinite Timoshenko beam on an elastic foundation to a moving step load. Esmail-zadeh and Ghorashi [9] used the central difference expansions to calculate the forced vibration response ofsimply supported EulerBernoulli and Timoshenko beams carrying uniformly or partially distributed mov-ing masses or forces. Abu-Hilal and Mohsen [10] studied the dynamic response of EulerBernoulli beamswith general boundary conditions subjected to a moving harmonic load traveling with accelerating, deceler-ating and constant velocity types of motion. Yang et al. [11] used a direct numerical integration procedure todevelop a semi-analytical procedure to effectively solve the problem of a single-degree-of-freedom undampedoscillator traversing a simply supported beam. Ichikawa et al. [12] employed modal expansion and the directintegration methods to investigate the dynamic behavior of the multi-span continuous EulerBernoulli beamtraversed by a moving mass at a constant velocity. Zhu and Law [13] used Hamiltons principle and Ritzmethod in conjunction with a high-precision integration method to calculate the response time histories ofa non-uniform continuous EulerBernoulli beam under a system of moving loads. Michaltsos [14] studiedthe linear dynamic response of a simply supported single-span EulerBernoulli beam subjected to a movingsingle-axle and two-axle accelerating or decelerating load of constant magnitude. Dugush and Eisenberger[15] applied modal analysis and integral transformation methods along with direct integration to determine thedynamic deflection and the internal forces of a multispan non-uniform beam under moving loads. Kargarnovinet al. [16] used LindstedtPoincare perturbation method in conjunction with complex Fourier transforma-tion and Greens function method to calculate the time response of infinite EulerBernoulli and Timoshenkobeams, supported by non-linear viscoelastic foundations and subjected to harmonic moving loads. Wang [17]presented an analytical and numerical study of the transient dynamics of a beammass system carrying multipleaccelerating/decelerating masses rolling on an initially curved EulerBernoulli beam. Martnez-Castro et al.[18] presented a semi-analytic time-domain solution involving the conventional two-noded finite elements withHermitian shape functions for the response of multi-span non-uniform BernoulliEuler beams subjected toflexural vibrations under the presence of concentrated moving forces. Kocaturk and Simsek [19] used Lagrangeequations along with the direct time integration method of Newmark to study the dynamic response of eccentri-cally prestressed viscoelastic Timoshenko beams under a moving harmonic load. Sniady [20] presented closedform infinite series solutions for the dynamic response of a finite, simply supported Timoshenko beam loadedby a force moving with a constant velocity. Kiral and Kiral [21] used a three-dimensional finite element methodin conjunction with the Newmark integration method to study the transient response of a symmetric laminatedfixed-fixed composite beam subjected to a concentrated force traveling at a constant velocity. Just recently,Simsek and Kocatrk [22] used polynomial trial functions to study free and forced vibration characteristics ofa functionally graded simply supported EulerBernoulli beam under a concentrated moving harmonic load.

Nearly all of above-mentioned investigations are based on first- or second-order beam theories whose accu-racyreduces increasinglywhen the beam becomes thicker. Comparatively, in the elasticity theory, no hypothesesabout the distribution field of deformations and stresses are adopted, and contributions of all stresses and strainsare considered by accounting for all the elastic constants. Accordingly, the two-dimensional structural dynamicanalyses of beams based on linear elasticity theory can achieve high accuracies, keeping in mind the current

availability of computers of increased speed and capacity. Such analysis inherently accounts for the effects oftransverse shear strains and rotary inertia while it assists in bringing out the physical insights, which cannototherwise be predicted by using the various existing beam theories. Several authors investigated the dynamicbehavior of beams using the state equations of elasticity. Sundara Raja Iyengar and Raman [23] employed 2Delasticity theory and the state space approach to study free vibrations of rectangular beams of arbitrary depthand found good agreements for the calculated frequency values with the Timoshenko beam theory. Kobayashiet al. [24] used the 2D theory of elastodynamics to study the frequency spectrum and free vibration modes ofdisplacement and stress for simply supported beams of arbitrary depth with narrow rectangular cross-section.Kang and Leissa [25] employed the Ritz method in conjunction with three-dimensional dynamic equationsof the theory of elasticity to determine the free vibration frequencies and mode shapes of thick, taperedrods and beams with circular cross-section. In a series of papers, Chen and coworkers [2629] combined theelasticity-based state space method with the differential quadrature technique (SS-DQM approach) to study


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Two-dimensional elasticity solution for transient response of simply supported beams 207

h

p(x, t)

l

x

z

o

Fig. 1 Problem geometry

free vibrations of arbitrarily thick laminated beams, orthotropic beams and isotropic beams resting on aPasternak elastic foundation. Ying et al. [30] also used the state space method to present exact two-dimen-sional elasticity solutions for the bending and free vibration of functionally graded orthotropic beams on aWinklerPasternak foundation. Malekzadeh and Karami [31] implemented a mixed differential quadratureand finite element formulation to study free vibrations of short beams on elastic foundations based on thetwo-dimensional theory of elasticity. Xu and Wu [32] further used the state space method to study the free

vibrations of partial interaction composite beams with interlayer slip for various end conditions. Just recently,Li and Shi [33] utilized the state spacebased differential quadrature method (SS-DQM) to study the freevibrations of a functionally graded piezoelectric material beam under different boundary conditions.

The above review clearly indicates that nearly all investigators have so far used the classical thin or thickbeam theories for investigating the dynamic response of thin or moderately thick beams under the action ofmoving loads [122]. On the other hand, all reported investigations regarding the dynamic behavior of arbi-trary thick beams based on the linear elasticity theory appear to be confined to studying the free vibrationcharacteristics [2333]. In particular, to the best of the authors knowledge, rigorous analytic or numericaltime-domain solutions in the context of full (2D) elasticity theory for the transient forced vibration analysis ofa single-span beam of arbitrary thickness seem to be non-existent. The primary purpose of the current workis to fill this gap. Particular attention is paid to the assessment of the effects of beam thickness, and loadpulsation frequency, velocity and acceleration on the transient vibratory response. The proposed model is ofboth academic and technical interest basically due to its inherent value as a canonical problem in structural

dynamics. It is of practical value for structural engineers involved in the dynamic analysis and design of thick(short) beams under the action of moving loads. The presented semi-analytic solution is highly accurate androbust, while it is easily capable of handling general time and space varying loads (e.g., non-constant movingforces). It can serve as the benchmark for comparison to solutions obtained by other approximate analyticaland numerical methods based on classical beam theories.

2 Formulation

2.1 Basic governing equations and boundary conditions

Consider a simply supported beam having the length l and depth h, as depicted in Fig. 1. The Cartesian coor-dinate system is established, so that 0 x l and 0 z h. The constitutive equations, for a homogeneous,

isotropic, linearly elastic solid in the state of plane stress, can be written as [34]

x =E

1 2

u

x+

v

z

, z =

E

1 2

v

z+

u

x

, xz =

E

2(1 + )

u

z+

v

x

, (1)

where u and v are the axial and transverse displacement components, and E, , x , z and x z are the Youngsmodulus, Poissons ratio, axial, normal and shear stress components, respectively. In the absence of bodyforces, the equations of motion are [35]

x

x+

x z

z=

2u

t2,

x z

x+

z

z=

2v

t2, (2)

where is the mass density. Also, assuming simply supported end conditions [30], one can obtain

v(0,z, t) = v(l,z, t) = 0, x(0,z, t) = x(l,z, t) = 0. (3)


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Seeking a solution for the beam described by the system of Eqs. ( 1) through (3), one can assume

Eu(x,z, t)

Ev(x,z, t)

z

(x,z, t)

xz(x,z, t)

=

n=1

EUn(z, t) cos(nx)

EVn(z, t) sin(nx)

n

(z, t)sin

(nx)

n(z, t) cos(nx)

(4.1)

(4.2)

(4.3)(4.4)

where n = (n/ l),Un, Vn, n and n are the modal state variables, and one notes that the above form ofsolution simultaneously satisfies the equations of motion (2) and the end conditions (3).

Next, following the concept of state space (or state-vector) approach [36], the present (plane stress) elasticbeam problem can advantageously be treated as follows. Substituting (1) and (4) into the system of partialdifferential equations (2), and subsequently applying Laplace transform with respect to t [37]:

(z, s) =

0

(z, t)estdt, (5)

one arrives at the following simultaneous first-order ordinary differential equations (state equation) for eachvalue ofn:

d

dzen(z, s) = An(s)en(z, s) (6)

where en = [EUn, EVn, n, n]T, and

An(s) =

0 n 0 2(1 + )

n 0 1 2 0

0 s2/E 0 n2n + (s

2/E) 0 n 0

. (7)

Here, it is clear that the original set of partial differential equations (2) is now reduced to an ordinary differen-tial state equation involving the thickness coordinate, z. The general solution to the state equation (6) can bewritten in the form [38]

en

(z, s) = Sn

(z, s)en

(0, s), (8)

where Sn(z, s) = exp(zAn) is the transfer matrix that establishes the relation of the state vectors at the lowersurface with those at an arbitrary coordinate value z. The relevant expressions for explicit determination of theelements of the matrix exponential function [38], exp(zAn), are provided in the Appendix. It is clear that, bysetting z = h in Eq. (8), the state vector at the upper surface is obtained as en(h, s) = Sn(h, s)en(0, s). Thefinal solution can be obtained by taking account of the lateral boundary conditions explained below.

The external excitation, p(x, t), is assumed to be an arbitrarily distributed transient load applied on theupper surface of the beam (see Fig. 1). Therefore, the normal stress component at the upper surface mayadvantageously be specified in the form of a Fourier sine series representation as

z(x,z = h, t) = p(x, t) =

n=1

Pn(t) sin (nx). (9)

Substitution of the above expression into the Eq. (4.3) and subsequent Laplace transformation lead to

n(h, s) = Pn(s). Similarly, vanishing of the pertinent stresses at the upper/lower boundaries of the beamimplies that

n(0, s) = 0, n(0, s) = 0, n(h, s) = 0. (10)

Finally, making use of the above boundary conditions in the state solution (8), after some manipulations, onearrives at the following matrix relation:

S11 S12 1 0S21 S22 0 1S31 S32 0 0S41 S42 0 0

Un(0, s)

Vn(0, s)

Un(h, s)

Vn(h, s)

=

00

Pn(s)/E0

, (11)

where Si j (i, j = 1, 2, 3, 4) are the elements of the transfer matrix Sn(h, s)(see Appendix).


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2.2 Numerical inversion of the Laplace transform

It is difficult to find the analytical inverse Laplace transform of the complicated solutions for the displacementand stress components in Laplace transform domain. So one has to resort to numerical computations. In this

Section, the numerical procedure to find the solution in the physical domain shall be briefly outlined. Thecomplex inversion formula for Laplace transforms is written as [37]:

(z, t) =1

2 i

+ii

(z, s)estds, (12)

where (z, s) can be any of the modal field quantities (Un, Vn, nand n), and is an arbitrary real number

larger than all the real parts of the singularities of (z, s). Taking s = + i, the above integral takes theform:

(z, t) =et

2

+i

i

(z, + i)eitd. (13)

Expanding the function et(z, t) in a Fourier series in the interval [0, 2T0], after some manipulations, oneobtains the approximate formula [39]

(z, t) = (z, t) + ED, (14)

where

(z, t) =1

2c0(z, t) +

k=1

ck(z, t), (15)

in which ck(z, t) =et/T0

Reeik t/T0 (z, + ik/T0)

, and ED is the discretization error that can be

made arbitrarily small by choosing the free parameter large enough. As the infinite series in Eq. (15) can

only be summed up to a finite number N of terms, the approximate value of (z, t) becomes [39]:

N(z, t) =1

2c0(z, t) +

Nk=1

ck(z, t), (16)

for 0 t 2T0. The values of the and T0 are selected for increased accuracy according to the criteriaoutlined in Durbin [39]. The suggested value of T0 is between 4 and 5 for sufficient accuracy [ 40]. Usingthe preceding formula to evaluate (z, t), one can introduce a truncation error ET that must be added to thediscretization error ED to produce the total approximation error [41].

The generalized Fourier series calculated for some of the response quantities of interest does not convergesatisfactorily. This is certainly to be expected in response histories that containdiscontinuities; here pronouncednon-physical oscillations appear (Gibbs phenomenon). A superposition technique that has proven effectivein reducing these oscillations is due to Cesaro [41], according to which the so-called N-th Cesaro sum maybe written explicitly as [42]

N =1

N+ 1

Nn=0

(N+ 1 n)xn. (17)

Thus, for better convergence, Eq. (17) may be finally used in (16) to obtain

N(z, t) =1

2c0(z, t) +

1

N+ 1

Nk=1

(N+ 1 n)ck(z, t). (18)

This completes the necessary background required for the analysis of the problem. Next, some numericalexamples will be considered.


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3 Numerical results

Realizing the large number of parameters involved here while keeping in view our computing hardware lim-itations, we confine our attention to a particular model. The intent of the collection of data presented here

is merely to illustrate the kinds of results to be expected from some representative and physically realisticchoices of values for the selected parameters. From these data, some trends are noted and general conclusionsmade about the relative importance of certain parameters. Accordingly, a simply supported concrete beam( = 2,778kg/m3, E = 30 GPa, = 0.3) is considered to be subjected to a concentrated pulsating load ofconstant amplitude, f0, radial frequency, , moving at an instantaneous velocity, c(t), and acceleration, a,which may be expressed in the form [10]

p(x, t) = f0 cos(t)x

ct+ at2/2

=

n=1

Pn(t) sin (nx), (19)

where is the classical Dirac delta function and Pn(t) = (2 f0/ l) cos(t) sin

nct+ at2/2

is the modal

load coefficient.A general Mathematica code is constructed for computing the transfer matrix, Sn(h, s), treating the

system of Eqs. (11) and numerically evaluating the inverse Laplace transform of state solution (8).Three distinct loading conditions are considered: uniform motion (c = c0, a = 0), accelerated motionc|t=0 = 0, a = c

20/2l

and decelerated motion

c|t=0 = c0, a = c

20/2l

, for selected speed parame-

ters (c0 = 25, 50, 100, 200, 400, 800 m/s). Also, a dimensionless time variable is defined as t =

c0t/ l (0 < t < 1) for the uniformly moving load and t = 2c0t/ l for the accelerating/decelerating load

[2]. Furthermore, the Nterm Cesaro approximate formula (18) is employed for improving convergence andreducing the total error in the Laplace inversions of the modal quantities of interest. The computations wereperformed on a network of personal computers, and the convergence of numerical solutions was secured ina simple trial and error manner, by increasing the number of terms, N, in the Cesaro summation (18), whilelooking for steadiness or stability in the numerical values of the solutions. It was found that uniform conver-gence is obtained (i.e., a sufficiently accurate inverse Laplace transform series solution may be computed) withapplication of the Cesaro summation using up to Nmax = 200 terms.

Before presenting the main numerical results, we shall establish the overall validity of the work. Accord-

ingly, we first considered a simply supported elastic beam (l = 10 m, h = 0.0594 m, = 1922.81 kg/m3

, E=207 GPa) subjected to constant amplitude and uniform velocity partially distributed load of span width [43],with the Fourier sine series representation

p(x, t) =f0

[H(x c0 t) H(x c0 t )] =

n=1

Pn(t) sin (nx), (20)

where H is the Heaviside unit function,and

Pn(t) = (2/ l)

l0

p(x, t) sin (nx) dx = (2 f0/n ) {cos[n (c0t+ )] cos (nc0 t)} . (21)

Figure 2a, which displays the calculated midspan displacement time response, v(x = l/2,z = 0, t), due tothe uniform rectangular moving pulse loads of fixed lengths ( = 0.01, 0.1, 1 m) and amplitude ( f0 = 70 g),traveling with selected constant speeds (c0 = 3.33, 25 m/s), exhibits good agreement with the data presented inFigs. 3, 4 and 5 of Esmailzadeh and Ghorashis [43] work. Subsequently, we used our code to compute the nor-malized underload displacement response, v(x = c0t,z = 0, t)/( f0l

3/48E I), for a simply supported concretebeam (l = 15 m, h = 0.9 m, = 2, 778kg/m3, E= 30 GPa) under the action of uniformly moving pulsatingloads (c0 = 20 m/s; = 20, 40 rad/s). The results, as depicted in Fig. 2b, show good agreements with the datapresentedin Figs. 10a and 11a of Kocaturk and Simseks [19] work. Also plottedin Fig. 2c is the correspondingnormalized maximum midspan displacement magnitude, |vmax(x = l/2;z = 0; 0 < t

< 1)/( f0l3/48E I)|,

as a function of load velocity for selected beam lengths (l = 10, 12.5, 15m) under the action of a uniformlymoving concentrated load. The outcome very well confirms the data presented in Figs. 4d,e,f of Kocaturk andSimseks [19] work. Lastly, we used our general code to calculate the normalized midspan displacement timeresponse, v(x = l/2,z = 0, t)/( f0l

3/48E I), due to a uniformly moving concentrated load traveling with


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(a) (b)

MidspanD

isp.

(m)

NormalizedUnderloadDisp.

t* t (sec)

(c) (d)

Max.

Normalized

MidspanDisp.

c0 (m/s)

NormalizedMidspanDisp.

t *

0 0.2 0.4 0.6 0.8 1-0.12

-0.08

-0.04

0

0.04

0.08

0.12

c0=3.33m/s, =0.01m

c0=3.33m/s, =1m

c0=25m/s, =0.1m

0 0.15 0.3 0.45 0.6 0.75-6

-3

0

3

6

9c

0=20m/s, =20rad/s

c0=20m/s, =40rad/s

0 40 80 120 160 2000.8

1

1.2

1.4

1.6

1.8

2

l=10ml=12.5m

l=15m

0 0.2 0.4 0.6 0.8 1-2

-1.5

-1

-0.5

0

0.5

1c

0=100 m/s

c0=200 m/s

c0=400 m/s

c0=800 m/s

Fig. 2 a Midspan displacement time response due to uniform rectangular moving pulse loads of fixed lengths and amplitude,traveling with selected constant speeds on a simply supported elastic beam. b Normalized underload displacement responsefor a simply supported concrete beam under the action of uniformly moving pulsating loads. c Normalized maximum midspandisplacement magnitude as a function of load velocity for simply supported concrete beams of selected lengths under the actionof a uniformly moving concentrated load. d Comparison of the calculated normalized midspan displacement time response dueto a uniformly moving concentrated load traveling with selected constant speeds on a very short simply supported steel beamwith those calculated by using the finite element package ANSYS

selected constant speeds (c0 = 100, 200, 400, 800m/s) on a thick(l = 10 m, h = 2 m) simply supported steelbeam ( = 7,800 kg/m3, E= 210 GPa). The results, as shown in Fig. 2d, demonstrate excellent agreementswith those calculated by using the Plane 182 elements in the transient analysis module of the finite elementpackage ANSYS [44], which are denoted by markers in the same figure. It is notable that basic ANSYSpackage is incapable of handling moving load problems, and the moving load should be incorporated througha macro-program developed in ANSYS Parametric Design Language (APDL).

Figure 3 compares the calculated normalized maximum beam midspan displacement magnitude, |vmax(x =l/2;z = 0; t)/( f0l

3/48E I)|, as a function of the beam thickness parameter for a non-pulsating load ( = 0)at selected velocities (c0 = 25, 50, 100, 200, 400, 800 m/s), with those obtained by using the EulerBernoulliand Timoshenko [2] beam theories. Here, the most interesting observation is the remarkable advantage of thepresent 2D elasticity model in comparison with the other two models. In particular, it is clear that the Tim-oshenko model yields accurate results predominantly for the thin and moderately thick beams (h/ l 0.2),for all load speeds, the EulerBernoulli theory barely yields acceptable results even for the moderately thick

beams under low speed loads (i.e., h/ l < 0.2, c0 = 25 m/s). As the load velocity increases, the EulerBer-noulli model completely fails, while the Timoshenko model overestimates the response, especially for verythick beams.

Figure 4 displays the normalized maximum beam midspan displacement magnitude, |vmax(x = l/2;z =0; t)/( f0l

3/48E I)|, as a function of load pulsation frequency, , for selected beam thickness to length ratios(h/ l = 0.1, 0.25, 0.5), under three distinct loading conditions: uniform motion (c = c0, a = 0), acceleratedmotion (c|t=0 = 0, a = c

20/2l) and decelerated motion (c|t=0 = c0, a = c

20/2l), for four selected speed

parameters (c0 = 25, 50, 100, 200 m/s). The clearest observation is the appearance of distinct relatively largedisplacement peaks for low to intermediate speed parameters, nearly independent of load acceleration. Asthe speed parameter increases, these distinct peaks gradually loose strength (become wider) and divide intomultiple smaller peaks (i.e., the inertia/stiffness effects seem to prevent large displacement of the beam dur-ing the short interval of action for the fast moving loads). On the other hand, increasing the beam thickness(h/ l) leads to a notable increase in the displacement amplitudes in addition to a rightward shift in the peak


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NormalizedMaximumMidspanDisplacement

Thickness Ratio,h/l Thickness Ratio,h/l

0.1 0.2 0.3 0.4 0.50

0.5

1

1.5

2

2.5

3Euler-Bernoulli

Timoshenko

Elasticity Solution

c0= 25 m/s

0.1 0.2 0.3 0.4 0.50

0.5

1

1.5

2

2.5

3

c0= 50 m/s

0.1 0.2 0.3 0.4 0.50

0.5

1

1.5

2

2.5

3

c0=100 m/s

0.1 0.2 0.3 0.4 0.50

0.5

1

1.5

2

2.5

3

c0= 200 m/s

0.1 0.2 0.3 0.4 0.5

0.5

1

1.5

2

2.5

3

3.5

c0= 400 m/s

0.1 0.2 0.3 0.4 0.5

0

0.5

1

1.5

2

2.5

3

c0= 800 m/s

Fig. 3 Comparison of the calculated normalized maximum beam midspan displacement magnitude as a function of beam thick-ness parameter, for a non-pulsating load ( = 0) at selected load velocities, with those computed by using the EulerBernoulliand Timoshenko beam theories

frequencies, p, toward higher values. In particular, for the highest beam thickness considered (h/ l = 0.5),the above-mentioned primary displacement peak is retained even at relatively high load speeds (i.e., up toc0 = 200m/s). In other words, as the beam thickness increases, the displacement response appears to becomeless sensitive with respect to the load speed. Lastly, the maximum displacement curves for the uniform, accel-erating and decelerating loads seem to generally display a similar behavior (i.e., displaying a main primarypeak) at low to intermediate load speeds, especially for beams of high thickness parameters (h/ l = 0.5).Astheload speed increases, this trend is distorted, especially for beams of small thickness parameters (h/ l = 0.1).

Furthermore, it should be noted that the maximum displacement magnitudes associated with the uniformlymoving load in the vicinity of the primary peak frequency are always lower than those of the accelerated anddecelerated loads. The most interesting observation is perhaps the fact that for the beam with the smallestthickness parameter (h/ l = 0.1), a load pulsation safe region associated with very small maximum dis-placement magnitudes appears to the right of the primary peak at high pulsation frequencies (see the first plotin the first column of Fig. 4). As the beam thickness increases, this safe region shifts to the left of the primarypeak, and it gradually widens. In particular, for the thickest beam (h/ l = 0.5) at c0 = 100 m/s, there exists arelatively wide safe load pulsation frequency range on the left side of a relatively sharp and high magnitudedisplacement peak (see the third plot in the first column of Fig. 4). As the load speed increases, the saferegion rapidly deteriorates, while the displacement magnitude peaks notably drop.

Figure 5 presents the normalized maximum beam midspan displacement magnitude, |vmax(x = l/2;z =0; t)/( f0l

3/48E I)|, as a function of load velocity, c0, for selected beam thickness to length ratios (h/ l =0.1, 0.25, 0.5), and pulsation frequencies ( = 0, 0.5p, 0.75p, p, 1.5p), under three distinct loading con-


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c0 = 25 (m/s) c0 = 50 (m/s) c0 = 100 (m/s) c0 = 200 (m/s)

h/l

= 0.1

NormalizedM

aximumMidspanDisplacement

h/l= 0.25

h/l= 0.5

(rad/s) (rad/s) (rad/s) (rad/s)

0 100 200 300 400 500

0

5

10

15

20

25accelereted

decelerated

uniform

0 100 200 300 400 500

0

2

4

6

8

10

0 100 200 300 400 500

0

1

2

3

4

5

0 100 200 300 400 500

0

0.5

1

1.5

2

2.5

0 200 400 600 800 1000

0

5

10

15

20

25

0 200 400 600 800 10000

5

10

15

20

25

0 200 400 600 800 10000

2

4

6

8

10

0 200 400 600 800 10000

1.2

2.4

3.6

4.8

6

0 600 1 200 1800 2400

0

12

24

36

48

60

0 600 1 200 1800 2400 300030000

9

18

27

36

45

0 600 1 200 1800 2400 3000

0

3

6

9

12

15

0 600 1 200 1800 2400 3000

0

3

6

9

12

15

Fig. 4 Normalized maximum beam midspan displacement magnitude as a function of load pulsation frequency for selected beamthickness to length ratios under three distinct loading conditions

ditions. The numerical values of the peak frequencies are simply read from Fig. 4 to be p = 100 rad/s(h/ l =0.1); p = 600 rad/s(h/ l = 0.25); p = 2,000 rad/s(h/ l = 0.5). The most important observations areas follows. As expected, the largest displacement magnitudes (sharpest response peaks) occur at the critical

pulsation frequency ( = p), especially for the uniformly moving load. As the load pulsation frequencyapproaches toward the critical frequency ( p), the displacement magnitude peaks shift to lower loadvelocities. Conversely, the displacement magnitude peaks appear to shift to higher load velocities, as thepulsation frequency moves away from the critical frequency. The maximum displacement magnitudes (thepeaks) associated with the uniformly moving loads occur at lower load speeds in comparison with those ofthe accelerating/decelerating loads. In particular, at relatively low load speeds, the maximum displacementmagnitudes related to the uniformly moving loads are generally higher than those of the accelerating/decel-erating loads, while the situation is reversed at relatively high load speeds. The most interesting observationis perhaps the fact that as the beam thickness increases (e.g., h/ l = 0.5), there is an overall increase in themaximum displacement magnitudes, and a rightward shift of the peaks toward higher load velocities, in addi-tion to a notable widening of the peaks. This implies that for very thick (short) beams there is a wider range ofload velocities for which relatively large displacement magnitudes occur (i.e., the very short beams are morevulnerable to load velocity), especially for the accelerating/decelerating loads. Furthermore, there are several

sharp kink points (i.e., change in the slope variation) observed on the response curves at certain values ofload velocity. These kink points may be linked to the complex phenomenon of mode shape switching and/orthe triggering of structural modal interactions [4547], as the system input parameter (e.g., the load velocity,pulsation frequency or beam thickness) varies across the kink points.

Figure 6 shows the normalized beam midspan displacement time response, v(x = l/2;z = 0; t)/( f0l

3/48E I), for selected beam thickness to length ratios (h/ l = 0.1, 0.25, 0.5) and load velocities(c0 = 25, 50, 100, 200m/s), under three distinct loading conditions at the critical load pulsation frequency( = p). Here, it is clear that the time response curves are oscillatory, and their envelopes monotonicallyincrease with time during the passage of the load. Furthermore, increasing load speed (beam thickness) leadsto a decrease (increase) in the response oscillations, especially for the accelerating/decelerating loads. Inparticular, the least oscillatory with lowest magnitude response is observed for the thinnest beam under theuniformly moving load with the highest speed (i.e., h/ l = 0.1, c0 = 200m/s), while the largest magnitudeand most oscillatory time response are seen for the thickest beam with the lowest speed decelerating load


10/14


h/l= 0.1

NormalizedM

aximumMidspanDisplacement

h/l= 0.25

h/l= 0.5

c0 (m/s)

c0 (m/s)

c0 (m/s)

c0 (m/s)

c0 (m/s)

0 200 400 600 800 10000

0.5

1

1.5

2

0 200 400 600 800 10000

0.6

1.2

1.8

2.4

0 200 400 600 800 10000

1

2

3

4

0 200 400 600 800 10000

5

10

15

20

accelereted

decelerated

uniform

0 200 400 600 800 10000

0.5

1

1.5

2

0 200 400 600 800 10000

0.5

1

1.5

2

0 200 400 600 800 10000

0.6

1.2

1.8

2.4

0 200 400 600 800 10000

1.2

2.4

3.6

4.8

0 200 400 600 800 10000

7

14

21

28

0 200 400 600 800 10000

0.5

1

1.5

2

0 200 400 600 800 10000

0.7

1.4

2.1

2.8

0 200 400 600 800 10000

0.8

1.6

2.4

3.2

0 200 400 600 800 10000

1.5

3

4.5

6

0 200 400 600 800 10000

9

18

27

36

0 200 400 600 800 10000

0.6

1.2

1.8

2.4

Fig. 5 Normalized maximum beam midspan displacement magnitude as a function of load velocity for selected beam thicknessto length ratios and pulsation frequencies under three distinct loading conditions

c0 = 25 (m/s) c0 = 50 (m/s) c0 = 100 (m/s) c0 = 200 (m/s)

h/l= 0.1

NormalizedM

idspanDispla

cement

h/l= 0.25

h/l= 0.5

t* t* t* t*

0 0.2 0.4 0.6 0.8 1-15

-10

-5

0

5

10

15

0 0.2 0.4 0.6 0.8 1-10

-5

0

5

10

15

20

accelereted

decelerated

uniform

0 0.2 0.4 0.6 0.8 1-6

-4

-2

0

2

4

6

0 0.2 0.4 0.6 0.8 1-3

-2

-1

0

1

2

3

0 0.2 0.4 0.6 0.8 1-30

-20

-10

0

10

20

30

0 0.2 0.4 0.6 0.8 1-18

-12

-6

0

6

12

18

0 0.2 0.4 0.6 0.8 1-12

-8

-4

0

4

8

12

0 0.2 0.4 0.6 0.8 1-6

-4

-2

0

2

4

6

0 0.2 0.4 0.6 0.8 1-36

-24

-12

0

12

24

36

0 0.2 0.4 0.6 0.8 1-36

-24

-12

0

12

24

36

0 0.2 0.4 0.6 0.8 1-30

-20

-10

0

10

20

30

0 0.2 0.4 0.6 0.8 1-15

-10

-5

0

5

10

15

Fig. 6 Normalized beam midspan displacement time response for selected beam thickness to length ratios and load velocitiesunder three distinct loading conditions at the critical load pulsation frequency ( = p)

(i.e., h/ l = 0.5, c0 = 25 m/s). Another interesting observation is that by doubling the load speed the numberof oscillations in the time response curves appears roughly to be cut in half.

Figure 7 displays the normalized beam midspan displacement time response, v (x = l/2;z = 0; t) /f0l

3/48E I

, for selected beam thickness to length ratios (h/ l = 0.1, 0.25, 0.5), and load velocity(c0 = 100m/s), under three distinct loading conditions at selected load pulsation frequencies ( = 0, 0.5p, p, 1.5p). Comments very similar to the preceding remarks can readily be made. The most important


11/14


h/l= 0.1

N

ormalizedM

idspanDisplacement

h/l= 0.25

h/l= 0.5

t* t* t* t*

0 0.2 0.4 0.6 0.8 1-2

-1.5

-1

-0.5

0

0.5

1

0 0.2 0.4 0.6 0.8 1-2

-1

0

1

2

3

0 0.2 0.4 0.6 0.8 1-6

-4

-2

0

2

4

6

0 0.2 0.4 0.6 0.8 1-3

-2

-1

0

1

2

3

0 0.2 0.4 0.6 0.8 1-2

-1.5

-1

-0.5

0

0.5

1

0 0.2 0.4 0.6 0.8 1-2

-1

0

1

2

3

0 0.2 0.4 0.6 0.8 1-12

-8

-4

0

4

8

12

0 0.2 0.4 0.6 0.8 1-1.5

-1

-0.5

0

0.5

1

1.5

0 0.2 0.4 0.6 0.8 1-2

-1.5

-1

-0.5

0

0.5

1

accelereted

decelerated

uniform

0 0.2 0.4 0.6 0.8 1-2

-1

0

1

2

3

0 0.2 0.4 0.6 0.8 1-30

-20

-10

0

10

20

30

0 0.2 0.4 0.6 0.8 1-1.5

-1

-0.5

0

0.5

1

1.5

Fig. 7 Normalized beam midspan displacement time response for selected beam thickness to length ratios and load velocity(c0 = 100m/s) under three distinct loading conditions at selected load pulsation frequencies

distinction is that the time response oscillations generally increase withincreasing the loadpulsation frequency,especially for the thickest beam considered. Furthermore, the overall time response magnitudes increase, asthe load pulsation frequency is increased from zero toward the critical case of = p, especially for the

thickest beam. This trend is reversed as the increasing value of pulsation frequency passed the critical fre-quency (e.g., for = 1.5p). Moreover, except for the critical case of = p, the envelopes of the timeresponse oscillations seem to exhibit a non-monotonic behavior, which is most apparent for the thickest beamconsidered (h/ l = 0.5).

4 Conclusions

A semi-analytic analysis based on the two-dimensional linear theory of elasticity for the transient dynamicresponse of a simply supported arbitrary thick elastic beam under the action of a transverse arbitrary distrib-uted moving load is presented. The solution of the problem is obtained by means of the powerful state spacetechnique in conjunction with Durbins approach for numerical inversion of Laplace transform accompaniedwith a special solution convergence enhancement technique. The most important observations are summarized

as follows:

For all load speeds considered, there is a distinct relatively large maximum displacement peak, nearlyindependent of load acceleration, occurring at a critical value of the load pulsation frequency ( = p). Asthe beam thickness increases, the latter peak considerably magnifies as it sharpens, and the displacementresponse appears to lose its sensitivity with respect to the load speed. Also, for moderately thick and verythick beams, there exists a relatively wide safe load pulsation frequency range (a safe region) on the left sideof this peak, which is associated with very low maximum displacement response magnitudes. As the loadspeed increases, the safe region rapidly deteriorates, while the displacement magnitude peaks notablydrop.

It is found that for very thick (short) beams there is a wider range of load velocities for which relativelylarge displacement magnitudes occur. In other words, short beams seem to be more vulnerable to loadvelocity variations, especially for the accelerating/decelerating loads.


12/14


Except for the critical case of = p, the envelopes of the time response oscillations exhibit a non-monotonic behavior. On the other hand, at the critical load pulsation frequency, the overall time responsemagnitudes increase considerably, and the time response curves become oscillatory as their envelopes growmonotonically with time during the passage of the load.

Lastly, the most notable observation is the superior advantage of the proposed 2D elasticity model in pre-dictingthe transient response of low aspect ratio beams under moving loads in comparison with the classicalEulerBernoulli and Timoshenko beam models. In particular, it is demonstrated that while the Timoshenkobeam model displays an acceptable accuracy for moderately thick beams, even under high speed movingloads, the EulerBernoulli theory barely yields satisfactory results even for relatively low speed loads.Asthe load velocity increases, the EulerBernoulli model completely fails, while the Timoshenko modeloverestimates the response, especially for very thick beams.

Appendix

The elements of the transfer matrix, Sn(z, s), are:

S11 = b0 + b2

2s2

(1 + )/E+ (2 + )2n

,

S12 = b1n b3ns2(1 + )(3 + )/E+ (2 + ) 2n

,

S13 = b2n(1 + )2,

S14 = 2b1(1 + ) + b3(1 + )

4s2(1 + )/E+ (3 + ) 2n

,

S21 = b1n + b3ns2

1 +

3 + 2

/E+ (1 + 2) 2n

,

S22 = b0 + b2s2

1 2

/E 2n

,

S23 = b1

1 2

+ b3

s2

1 22

/E 2(1 + ) 2n

,

S24 = b2(1 + )2n,

S31 = b2s2(1 + )n/E+

3n

, (A-1)

S32 = b1s2/E+ b3

s4

1 2

(/E)2 s22n(1 + 2)/E 4n

,

S33 = b0 + b2s2

1 2

/E 2n

,

S34 = b1n + b3ns2(1 + )(3 + )/E+ (2 + ) 2n

,

S41 = b1s2/E+ 2n

+ b3

2s4(1 + )(/E)2 + s22n [4 + (2 )] /E+ 2

4n

,

S42 = b2ns2(1 + )/E+ 2n

,

S43 = b1n b3ns2

1 +

3 + 2

/E+ (1 + 2) 2n

,

S44 = b0 + b2

2s

2

(1 + )/E+ (2 + )2

n

where the unknown coefficientsb0 throughb3 can be exactly determined by solving the following linear systemof algebraic equations:

ez1

ez2

ez3

ez4

=

1 1 21

31

1 2 22

32

1 3 23

33

1 4 24

34

b0b1b2b3

, (A-2)

in which 1,2 = 2s2(1 + )/E+ 2n, and 3,4 = s

2(1 2)/E+ 2n.


13/14


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Two Dimensional Elasticity Solution for Transient Responce of Simply Supported Beams Under Moving Loads

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