EULER BERNOULLI BEAM THEORY: FIRST-ORDER ANALYSIS, …

EULERBERNOULLI BEAM THEORY USING THE FINITE DIFFERENCE METHOD

Article

EULERBERNOULLI BEAM THEORY: FIRST-ORDER ANALYSIS, SECOND-ORDER ANALYSIS, STABILITY, AND

VIBRATION ANALYSIS USING THE FINITE DIFFERENCE METHOD

Valentin Fogang

Abstract: This paper presents an approach to the EulerBernoulli beam theory (EBBT) using the finite difference

method (FDM). The EBBT covers the case of small deflections, and shear deformations are not considered.

The FDM is an approximate method for solving problems described with differential equations (or partial differential

equations). The FDM does not involve solving differential equations; equations are formulated with values at selected

points of the structure. The model developed in this paper consists of formulating partial differential equations with

finite differences and introducing new points (additional points or imaginary points) at boundaries and positions of

discontinuity (concentrated loads or moments, supports, hinges, springs, brutal change of stiffness, etc.). The

introduction of additional points permits us to satisfy boundary conditions and continuity conditions. First-order

analysis, second-order analysis, and vibration analysis of structures were conducted with this model. Efforts,

displacements, stiffness matrices, buckling loads, and vibration frequencies were determined. Tapered beams were

analyzed (e.g., element stiffness matrix, second-order analysis). Finally, a direct time integration method (DTIM) was

presented. The FDM-based DTIM enabled the analysis of forced vibration of structures, the damping being

considered. The efforts and displacements could be determined at any time.

Keywords: EulerBernoulli beam; finite difference method; additional points; element stiffness matrix; tapered beam;

second-order analysis; vibration analysis; boundary value problem; direct time integration method

Corresponding author: Valentin Fogang

Civil Engineer

C/o BUNS Sarl P.O Box 1130 Yaounde Cameroon

E-mail: [email protected]

ORCID iD https://orcid.org/0000-0003-1256-9862

1. Introduction

The EulerBernoulli beam has been widely analyzed in the relevant literature. Several methods have been developed,

e.g., the force method, the slope deflection method, and the direct stiffness method, etc. Anley et al. [1] considered a

numerical difference approximation for solving two-dimensional Riesz space fractional convection-diffusion problem

with source term over a finite domain. Kindelan et al. [2] presented a method to obtain optimal finite difference

formulas which maximize their frequency range of validity. Both conventional and staggered equispaced stencils for

first and second derivatives were considered. Torabi et al. [3] presented an exact closed-form solution for free vibration

Preprints (www.preprints.org) | NOT PEER-REVIEWED | Posted: 29 March 2021 doi:10.20944/preprints202102.0559.v2

© 2021 by the author(s). Distributed under a Creative Commons CC BY license.

https://doi.org/10.20944/preprints202102.0559.v2

http://creativecommons.org/licenses/by/4.0/

ana

con

me

mu

app

bou

fre

sol

num

sup

ana

exp

the

ela

FD

the

ma

pap

equ

im

spr

con

we

2.

Th

Sp

EU

alysis of Eul

ncentrated m

ethod for the

ulti-degree-o

plied the Fin

undary cond

ee vibration o

lution was ba

merical solu

pported foun

alysis of the

pressed via m

e differential

astic Winkler

DM, points ou

e governing e

aking the FD

per, a model

uations (stati

maginary poin

rings, and br

nditions and

ere conducted

Mater

he sign conve

F

ecifically, M

distribut

ULERBER

lerBernoull

masses were m

solution of t

of-freedom sy

nite Differenc

ditions and re

of a new type

ased on a sem

utions for stat

ndations usin

EulerBerno

means of diff

equations m

r foundation,

utside the be

equations. Th

DM less intere

based on fin

ics and mate

nts) at bound

rutal change o

continuity c

d with the m

rials and m

2.1

entions adopt

Figure 1.

M(x) is the be

ted load in th

RNOULLI

i conical and

modeled by D

the equations

ystems. The m

ce Method (F

esting on vari

e of tapered b

mi-analytical

tic and dynam

ng Finite Diff

oulli beam co

ferential equ

may be difficu

, a Pasternak

eam were not

he non-appli

esting in com

nite differenc

rial relation)

daries and at p

of stiffness,

conditions. Fi

model.

methods

First-or

ted for loads

Sign con

ending mome

he positive do

BEAM TH

d tapered bea

Dirac’s delta

s of motion d

method appl

FDM) to eva

iable one or

beam, with e

l technique, t

mic stability

ference Meth

onsists of so

ations, and c

ult in the pre

k foundation,

t considered.

ication of the

mparison to o

ce method w

) with finite d

positions of d

etc.). The int

irst-order ana

der analys

s, bending mo

nvention for l

ent in the sec

ownward dir

HEORY USI

ams carrying

a functions. K

describing th

lied also to e

aluate natural

two paramet

exponentially

the differenti

parameters o

hod where C

lving the gov

considering t

esence of an

or damping

. The bound

e governing e

other numeri

was presented

differences a

discontinuity

troduction of

alysis, secon

sis

oments, shea

loads, bendin

ction, V(x) is

rection.

ING THE F

any desired

Katsikadelis

he dynamic r

quations wit

l frequencies

ter elastic fou

y varying thi

ial transform

of an axially

entral Differ

verning equa

the boundary

axial force (o

(by vibratio

dary condition

equations at

cal methods

d. This model

and introduci

y (concentrat

f additional p

nd-order anal

ar forces, and

ng moments,

s the shear fo

FINITE DIF

number of a

[4] presented

esponse of st

h variable co

s of non-prism

undations. B

ckness, restin

m method. Mw

loaded unifo

rence Scheme

ations (i.e., st

y and transitio

or external d

n analysis). B

ns were appl

the beam’s e

such as the f

l consisted o

ing new poin

ted loads or m

points permit

lysis, and vib

d displaceme

shear forces

orce, w(x) is

FFERENCE

attached mas

d a direct tim

tructural line

oefficients. S

matic beams

Boreyri et al.

ing on a linea

wabora et al

form beam re

e was develo

tatics and ma

on condition

distributed ax

By the tradit

lied at the be

ends led to in

finite elemen

of formulating

nts (additiona

moments, su

tted us to sat

bration analy

ents are illust

s, and displac

the deflectio

E METHOD

ses. The

me integration

ear and nonli

Soltani et al.

, with differe

[6] analyzed

ar foundation

. [7] conside

esting on a si

oped. The cla

aterial) that a

ns. However,

xial forces), a

tional analysi

eam’s ends an

naccurate res

nt method. In

g the differen

al points or

upports, hing

tisfy boundar

ysis of structu

trated in Figu

cements.

on, and q(x) i

D

n

inear

[5]

ent

d the

n. The

ered

mply

assical

are

solving

an

is with

nd not

sults,

n this

ntial

es,

ry

ures

ure 1.

is the



Ac

fol

wh

Th

Fig

Th

app

Th

Th

Th

der

EU

ccording to th

llows:

here EI is the

he bending m

gure 2 shows

he governing

proximated a

hus, the defle

he shape func

hus, a five-po

rivatives at i

( )w x

ULERBER

2.1.1

he EulerBe

e flexural rig

moment, shea

s a segment o

equation (Eq

around the po

ection curve c

ctions fj(x) (j

oint stencil is

are expresse

2iw f

RNOULLI

1 Uniform

2.1.1.1

rnoulli beam

gidity and k(

ar force, and

2.1.1.2

of a beam ha

quation (1))

oint of intere

can be descri

j = i-2; i-1; i;

s used to writ

ed with value

4dEI

( )

( )

( )

M x

dMV x

dwx

d

2( )i if x w

BEAM TH

m beam wit

Statics

m theory (EBB

(x) is the stif

slope (x) a

Fundamen

aving equidis

Figure 2. B

has a fourth

est i as a four

ibed with the

; i+1; i+2) ca

te finite diffe

es of deflecti

4

4

( )w xk

dx

2

2

( )

( )

( )

d w xEI

dx

M xV

dxw x

dx

1 1( )i if x

(jf x

HEORY USI

hin segmen

BT), the gov

ffness of the

are related to

ntals of FD

stant points w

eam with eq

order deriva

rth degree po

e values of d

an be express

erence appro

ion at points

( ) ( )k x w x

)

( )V x EI

) (i iw f

2

2

)i

k ik j

x

ING THE F

nts

verning equat

,

elastic Wink

o the deflecti

DM for a un

with grid spa

quidistant poi

ative, the def

olynomial.

deflections at

sed using the

oximations to

i-2; i-1; i; i+

( )q x

3

3

( )d w x

dx

1( ) ix w

k

j k

x x

x x

FINITE DIF

tion of a unif

kler foundati

ion as follow

niform beam

cing h.

ints.

flection curve

equidistant

e Lagrange p

o derivatives

+1; i+2.

1( )if x w

FFERENCE

form beam l

ion.

ws:

m

e is conseque

grid points:

polynomials:

at grid point

2 2(i iw f

E METHOD

oaded with q

(1

(2

(2

(2

ently

(3a

(3b

ts. Therefore

( )x

D

q(x) is as

1)

2a)

2b)

2c)

a)

b)

e, the




(4a)

(4b)

(4c)

(4d)

2.1.1.3 FDM Formulation of equations, efforts, and deformations

Let us consider a segment k of the beam having a flexural stiffness EIk and equidistant grid points with spacing hk.

We introduce a reference flexural rigidity EIr as follows

EIk = k EIr (5)

We set

W(x) = EIr w(x) (5a)

Substituting Equations (4a), (5), and (5a) into Equation (1) yields

(6)

At point i, the bending moment, shear force, and slope are formulated with finite differences using Equations (2ac),

(4bd), (5), and (5a).

(7a)

(7b)

(7c)

2.1.1.4 FDM Formulation of loadings

Let us determine here the FDM value qi (Equation (6)) in the case of a varying distributed load q(x). Without

considering the elastic Winkler foundation the distributed load q(x) is related to the shear force V(x) as follows:

(8a)

42 1 1 2

4 4

32 1 1 2

3 3

22 1 1 2

2 2

2 1 1 2

4 6 4

2 2

2

16 30 16

12

8 8

12

i i i i i

i

i i i i

i

i i i i i

i

i i i i

i

w w w w wd w

dx h

w w w wd w

dx h

w w w w wd w

dx h

w w w wdw

dx h

2 1 1 22

2 1 1 23

2 1 1 2

16 30 16

12

2 2

2

8 8

12

i i i i ii k

k

i i i ii k

k

i i i ir i

k

W W W W WM

h

W W W WV

h

W W W WEI

h

44

2 1 1 2

14 6 4i k

i i i i i i kk r k

k hW W W W W q h

EI

( )( )

dV xq x

dx




Considering here a three-point stencil, the following FDM formulations of the first derivative can be made. The

position i is considered as the left beam’s end, an interior point on the beam, or the right beam’s end, respectively:

(8b)

(8c)

(8d)

The balance of vertical forces applied to a free body diagrams yields the following:

(8e)

(8f)

The combination of Equations (8af) yields the FDM value qi for the position i being the left beam’s end, an interior

point on the beam, or the right beam’s end.

(8g)

(8h)

, (8i)

The application of Equations (8g8i) shows that in the case of a linearly distributed load, qi is equal q(xi).

At point i, the stiffness of the elastic Winkler foundation ki is calculated similarly to Equations (8g8i).

2.1.1.5 Analysis at positions of discontinuity

Positions of discontinuity are positions of application of concentrated external loads (force or moment), supports,

hinges, springs, abrupt change of cross section, and change of grid spacing.

2.1.1.5.1 Change of grid spacing

The discretization of the beam may lead to uniform-grid segments, but the grid spacings being different from one

segment to another, as represented in Figure 3.

1 2

1

1

1

1

2 1

13 ( ) ( )

21

( )21

( ) 3 ( )2

i i

i i i

i

i i

i i

i i i

q q x dx q x dxh

q q x dxh

q q x dx q x dxh

1 2

1 1

2 1

3 4( )

2

( )

2

4 3( )

2

i i i

i

i i

i

i i i

i

V V VdV x

dx h

V VdV x

dx h

V V VdV x

dx h

1 1

1

1

( )

( )

i

i i i

i

i i i

V V q x dx

V V q x dx



Th

and

Le

spa

Th

po

Fig

EU

he governing

d slope (Equ

t us consider

acing hk and

he model dev

ints (fictive p

gure 5a,b. O

ULERBER

equation (Eq

uations (7a-c)

r segments k

d hp. Concen

veloped in thi

points ia, ib,

Opening of th

RNOULLI

Fig

quation (6)),

)) at position

2.1.1.5.2

k and p of the

ntrated loads

Figure 4

is paper cons

, ic, and id) i

he beam and

BEAM TH

gure 3. Bea

and the equ

n i are formu

2 Equatio

e beam havin

(force P and

. Beam with

sists of realiz

in the openin

introduction

HEORY USI

am with diffe

uations for th

ulated under c

ns of contin

ng flexural st

d moment M

h change in g

zing an open

ng, as represe

n of addition

ING THE F

erent grid spa

e determinat

consideration

nuity

tiffness EIk a

M*) are applie

grid spacing

ning of the be

ented in Figu

al points on t

FINITE DIF

acings.

tion of the be

n of different

and EIp, and

d at point i, a

and stiffness

eam at point

ure 5a,b.

the left side

FFERENCE

ending mome

t grid spacin

d equidistant

as represente

s

i and introdu

(5a) and righ

E METHOD

ent, shear for

gs h1 and h2

t grid points w

ed in Figure

ucing additio

ht side (5b).

(5a)

(5b)

D

rce,

2.

with

4.

onal



Th

Th

Th

usi

Th

equ

An

sup

At

at t

Th

i

k

il

w

EI

M

V

EU

he governing

hus, the gove

he FDM form

ing Equation

he following

uilibrium of

n adjustment

pport (Wil =

the beam’s e

the beam’s e

he grid may b

2 16

l ir

r il r

il ir

ik

l ir

w

I EI

M M

W

V P

ULERBER

equation (Eq

rning equatio

mulations qil

ns (8g-i).

continuity eq

the bending

of the contin

Wir = 0, no e

ends, additio

ends, as well

be such that e

*

16 30

12

il ir

ir ir

i

ik

W W

W

M

W

h

WP

W

W

RNOULLI

quation (6))

ons at positio

and qir of di

quations exp

moment (Eq

nuity equatio

equation (10d

onal points ar

as the bound

2.1.1.5.3

every node h

F

2 1

2

2 1

8

12

16

2

2

i i

il

k

i

W W

W W

h

W

h

2 4

4

i i

ic id

W W

W W

BEAM TH

is applied at

ons il and ir y

istributed loa

press the cont

quation (7a))

ons is made i

d)), or a sprin

re introduced

dary conditio

3 Non

has a non-con

Figure 6. Be

3

8

2

2

ia

k

ia ib

ia

k

W W

h

W W

W W

h

1 6

6

W

k

HEORY USI

t any point of

yield:

ading and ki

tinuity of the

) and shear fo

in case of a h

ng.

d (as shown i

ons.

n-uniform g

nstant distan

eam with a n

1

ib ic

icp

ibp

W W

W

W W

4

4

il k

k r

ir pir

p r

k hW

EI

k hW

EI

ING THE F

f the beam, i

l and kir of e

e deflection a

orce (Equatio

hinge (no con

in Figure 5a,

grid

ce from anot

non-uniform

8 8

12

16 30

1

id

id

ic

W

h

W

W

2 idW

1

4

4

il ia

r i

W W

W

FINITE DIF

.e., i-2; i-1; i

elastic Winkl

and slope (Eq

on (7b)):

ntinuity of th

,b) and so go

ther, as repre

grid.

1

2

8

0 16

2

i i

p

ir

p

W W

h

W W

h

13

2

2i

p

W

h

2

ib

i

W

W

FFERENCE

il; ir; i+1; i+2

ler foundatio

quation (7c))

he slope, Mil

overning equ

esented in Fig

2

1 2i iW W

2iWP

4

4

1

1

il kk

ir pp

q h

q h

E METHOD

2, etc.

(9

(9

on are calcula

), and the

(

= Mir = 0), a

uations are ap

gure 6.

*2 M

P

D

9a)

9b)

ated

(10a)

(10b)

(11)

(12)

a

pplied



He

com

be

Le

An

sta

Th

A u

po

A r

Co

M

M

EU

ere, the Lagra

mplicated, an

analyzed as

t us analyze

n elastic Win

atic equilibriu

he slope is de

uniform grid

int stencil is

reference fle

onsidering Eq

12

2i i

k

i i

M M

h

M EI

ULERBER

ange interpo

nd conseque

a discontinu

2.1.2

here the case

nkler foundat

um and mate

etermined usi

d with spacin

considered f

exural rigidity

quations (5-5

1

12

2

i i

i i

k

M

w w

h

RNOULLI

lation polyno

ntly the non-

uity position.

2 First-

2.1.2.1

e of a tapere

tion with stif

erial relation

ing Equation

2.1.2.2

ng hk is consi

for the follow

y EIr is intro

5a) and (14a)

1

i i

i i

k w q

w

BEAM TH

omial (Equat

-uniform grid

order anal

Statics

d beam, as sh

Figur

ffness k is co

are formulat

n (2c), and th

FDM form

idered. Equa

wing derivati

oduced (Equ

), the FDM f

2d

M

2

2

i k

k i

q h M

h M

2

(

d S

dx

dS x

dx

HEORY USI

tion (3b)) is

d is not furth

lysis of a ta

hown in Figu

re 7. Taper

onsidered. Th

ted as follow

he shear force

mulations o

ations (13a-b

ives (S(x) rep

uations (5-5a)

formulations

2

2

( )(

( )

M xk

dx

M x EI

21

1

2

2

i k

i i

M h

W

2

( )

)

i

i

i

i

Sx

x

Sx

x

ING THE F

used for FDM

her analyzed

pered beam

ure 7.

red beam.

he varying fl

ws:

e using (2b).

of equation

) have a seco

presenting M

)). Here the p

of Equation

2

2

( ) ( )

( )( )

x w x

d w xx

dx

2 2

2

i k

i i i

M h M

W

12

1 1

2

2

i

k

i

k

S

h

S

h

FINITE DIF

M formulatio

in this paper

m

exural rigidi

s, efforts, a

ond order der

M(x) or w(x))

parameter

s (13a,b) yie

( )

)

q x

1

1 0

ii

i i

k hM

EI

W

1iS

FFERENCE

on. The resu

r. In fact, thi

ity is EI(x).

and deform

rivative; con

):

is defined at

eld

4k

ir

hW

I

E METHOD

lting equatio

s case should

The equatio

(13a)

(13b)

mations

nsequently, a

(14a)

(14b)

t any position

(

(

4i kq h

D

ons are

d not

ons of

three-

n i.

(15a)

(15b)



At

app

Co

ass

(po

Fig

Th

Th

ben

At

EU

any point on

plication of E

oncentrated lo

sumed at this

oints ia and i

gure 8a,b. O

he additional

he continuity

nding mome

a point i wit

ULERBER

n the grid, Eq

Equations (2

2.1.2

oads (force P

s position. A

id) in the ope

Opening of th

points are ia

equations ex

ent and shear

th a hinge the

il

r il

il

il ir

w w

EI

M M

V V

RNOULLI

quations (15

2b), (2c), and

2.3 Analys

P and momen

As described i

ening, as rep

he beam and

a and id, and

xpress the co

r force (Equa

e slopes are n

ir il

r ir

ir

r

w W

EI

M M

P

(

i

i

dMV

d

dw

dx

BEAM TH

a-b) are appl

d (14b) yields

sis of a tap

nt M*) are ap

in section 2.1

presented in F

introduction

d the unknow

ontinuity of th

ation (15c)) a

not more equ

*

1

2

2

ir

i

i

k

W

W

M

h

( )

( )i

i

M x

dx

wx

x

HEORY USI

lied. The unk

s the shear fo

ered beam

pplied at poin

1.1.5, an ope

Figure 8a,b.

n of addition

wns are wia,

the deflection

as follows:

ual, and the m

1

2ia

k

ia

W

h

M

1

1 1

2

2

i

k

i i

k

M M

h

w w

h

ING THE F

knowns are t

orce and slop

at position

nt i (see Figu

ening of the b

al points at th

Mia, wid, a

n and slope (

moments M

2

2

id

p

id

p

W

h

M M

h

1iM

FINITE DIF

he deflection

pe:

ns of discon

ure 4). A cha

beam at poin

he left side (

and Mid.

(Equation (15

il and Mir a

1

1

i

p

i

W

MP

FFERENCE

n and bendin

ntinuity

ange in grid s

nt i introduce

(8a) and righ

5d)), and the

are zero.

(8a)

(8b)

E METHOD

ng moment. T

(1

(1

spacing is als

ed additional

ht side (8b).

e equilibrium

(16

(16

(16

(16

D

The

15c)

15d)

so

points

m of the

6a)

6b)

6c)

6d)



Th

ele

Le

Th

Th

Le

Eq

Co

mo

bou

EU

he sign conve

ement stiffne

Figu

t us define fo

he 44 eleme

he vectors ab

t us divide th

F

quations (15a

onsidering th

oments and s

undary cond

ULERBER

2.1.3

entions for be

ess matrix in

ure 9. Sign c

ollowing vec

ent stiffness m

ove are relat

he beam in n

Figure 10. F

a) with qi =0

he sign conve

shear forces i

ditions in com

i

i

V

M M

RNOULLI

3 First-o

2.1.3.1

ending mom

local coordin

onventions f

ctors:

matrix in loc

ted together w

n parts of equ

inite differen

0 and (15b) ar

entions adopt

in the elemen

mbination wi

1

1

dMV

M M

BEAM TH

rder eleme

44 elem

ments, shear fo

nates is illust

for moments,

al coordinate

with the elem

ual length h (

nce method (

re applied at

ted for bendi

nt stiffness m

th Equations

red

red

S

V

redS K

1

1

( )

0i

M x

dx

M M

HEORY USI

ent stiffness

ment stiffne

forces, displa

trated in Fig

, shear force

es of the tape

ment stiffnes

(l =nh) as sh

(FDM) discre

t any point on

ing moments

matrix (see F

s (2b) and (1

;

; ;

i i

i i

V M

w

44 redK V

2

2

0

M M

h

ING THE F

s matrix of

ess matrix

acements, and

gure 9.

s, displacem

ered beam is

s matrix K44

hown in Fig

etization for

n the grid (po

s and shear fo

igure 9), we

4b):

; ;

; ;

i k k

k k

V M

w

d

0 2M

hV

FINITE DIF

f a tapered

d slope adop

ents, and slo

denoted by

4 as follows:

gure 10.

4x4 element

ositions 1; 2;

orces in gene

can set follo

T

k

T

2iV M

FFERENCE

beam

pted for use in

opes for stiffn

K44.

t stiffness ma

; …n+1 of F

eral (see Figu

owing static c

0 0M

E METHOD

n determinin

ness matrix.

(1

(1

(1

atrix.

igure 10).

ure 1) and fo

compatibility

(1

(1

D

ng the

17a)

17b)

18)

or bending

y

19a)

19b)




(19c)

(19d)

Considering the sign conventions adopted for the displacements and slope in general (see Figure 1) and for displacements

and slope in the element stiffness matrix (see Figure 9), we can set following geometric compatibility boundary

conditions in combination with Equations (2c) and (14b):

(20a)

(20b)

(20c)

(20d)

The number of equations is 2(n+1) + 4 +4 = 2n + 10. The number of unknowns is 2(n+3) + 4 = 2n + 10,

especially 2(n+3) unknowns (M; W) at points on the beam and additional points at beam’s ends, and 4 efforts at

beam’s ends (Vi; Mi; Vk; Mk). Let us define following vector

(21)

The combination of Equations (15a,b) applied at any point on the grid, Equations (19ad), and Equations (20ad) can

be expressed with matrix notation as follows, the geometric compatibility boundary conditions (Equations (20ad))

being at the bottom.

(22)

The matrix T has 2n+10 rows and 2n+10 columns. The zero vector above has 2n+6 rows.

(23)

The matrix Taa has 2n+6 rows and 2n+6 columns, the matrix Tab has 2n+6 rows and 4 columns, the matrix Tba has 4

rows and 2n+6 columns, and the matrix Tbb has 4 rows and 4 columns.

The combination of Equations (18), (22), and (23) yields the element stiffness matrix of the beam.

(24a)

1 1

0 2 0 21

1

1 1

2 21

1

( )

2 2

( )

2 2

i r i

i r i

n k n r k

n n n nn k r k

n

w w W EI w

w w W Wdw xEI

dx h h

w w W EI w

w w W Wdw xEI

dx h h

1 0 0 1 1 2 2; ; ; ....... ;T

n nS M W M W M W

1 1 10 0

red redred redr r

S ST T

EI V EI VS S

1 aa ab

ba bb

T TT

T T

44 r bbK EI T

21 2

1

1 1

( )2 0

2

0

n nk n k n n

n

k n k n

M MdM xV V hV M M

dx h

M M M M



A g

In

As

and

Th

Th

Th

K4

the

Th

ma

EU

general matr

Equation (24

ssuming the p

d slope is illu

Figu

he 33 eleme

he vectors of

he matrix K3

44 has 4 row

e matrix Kba

he combinatio

atrix K33 as

ULERBER

rix formulatio

4b), 0 is a ze

presence of a

ustrated in F

ure 11. Sign

ent stiffness m

Equations (1

33 can be form

s and 4 colum

a has 1 row a

on of Equatio

follows:

red

red

red

S

V

S

44K

RNOULLI

on of K44 is

ero matrix w

2.1.3.2

a hinge at the

igure 11.

conventions

matrix in loc

17a), (17b), a

mulated with

mns. The ma

and 3 column

on (18) with

33

;

;

d i

d i

d

V M

w

K

aa

ba

K K

K K

33K K

44K

BEAM TH

as follows:

ith four rows

33 elem

e right end, th

for moment

al coordinate

and (18) beco

h the values o

atrix Kaa has

ns, and the m

the presence

;

;

T

i k

T

i k

red

M V

w

V

ab

bb

K

K

aa abK K

4 rEI

HEORY USI

s and 2n+6 c

ment stiffne

the sign conv

ts, shear forc

es of the tape

ome

of the matrix

s 3 rows and

matrix Kbb h

e of a hinge

1

bb

KK

0 I T

ING THE F

columns, I is

ess matrix

vention for be

es, displacem

ered beam is

x K44 (see Eq

d 3 columns,

as 1 row and

at position k

baK

1 0T

TI

FINITE DIF

the 4 4 ide

ending mom

ments, and sl

denoted by

quations (24a

the matrix K

d 1 column (a

(Mk = 0), an

FFERENCE

entity matrix

ments, shear f

lope for stiffn

K33.

ab)).

Kab has 3 row

a single valu

nd Equation

E METHOD

(2

x.

forces, displa

fness matrix.

(25a)

(25b)

(25c)

(26)

ws and 1 colu

ue).

(25c) yields

(27)

D

24b)

acements,

umn,

the



Th

Eq

Th

Th

EU

he beam is di

quation (6) w

he static comp

he geometric

12

2

i

i

k

k

V

M

V

h

M

12

1

n

ULERBER

2.1.4

vided in n pa

with qi = 0 is

patibility bou

c compatibili

1

1

2

1

3

1

2

k

k

ik

n k

k nk

n

V

M

hM

V

hV W

M

22k

k

hM

1

1

1

( )

8

(

8

n

n

dw x

dx

W W

dw x

dx

W W

RNOULLI

4 First-or

arts of equal

Figure 12. F

applied at an

undary cond

ity boundary

1

1

1

1

1

2

1

16

2

2

k

n

n n

k

W

W

W W

W

W W

W

1 16nW

1

1

0 2

1

8

12)

8

12

n

n n

w

W W

hwx

W W

h

BEAM TH

rder elemen

length h (l =

FDM discret

ny point on t

ditions (Equa

conditions (

0 23

0

0 1

3

2

1

2 2

2

6 30

12

30

2 2

2

2

n n

n

n

W W

h

W W

h

W W

W W

h

W

W

16

30

n

n

W

W W

0

3

1

2 3

8 8

12

8

1n n

n

w w

h

WEI

w w

W

HEORY USI

nt stiffness

=nh) as show

tization for 4

the grid (pos

ations (19ad

(Equations (2

2 3

1 22

2

2 3

3

16

16

0

n n

n

W W

W W

h

W

W W

W

12

1

30

12

16

n n

n

W W

h

W W

2 3

28

12

r i

n

r k

w w

I

w

h

EI

ING THE F

matrix of

wn in Figur

4x4 element s

itions 1; 2; …

d)) become

20b,d)) becom

3

2 3

3

2

0

ik

hV W

W

W

1 22

2 3

16 n

n n

W

W W

3

i

n

k

w

FINITE DIF

a uniform b

re 12.

stiffness mat

…n+1 of Figu

me

1 02W W

2 3

0

nW

k

FFERENCE

beam

trix.

ure 12).

0 22W

E METHOD

(2

(2

3 0W

D

(28a)

(28b)

(28c)

(28d)

29a)

29b)




Equations (20a) and (20c) stay unchanged.

Thus, the number of equations is n+9. The number of unknowns is n+5 + 4 = n + 9, especially n+5 unknowns W

(at points on the beam and additional points at beam’s ends) and 4 efforts at beam’s ends (Vi; Mi; Vk; Mk).

The vector becomes,

(29c)

The use of Equations (22) to (24b) yields the element stiffness matrix of the uniform beam.

2.2 Second-order analysis

The equation of static equilibrium can be expressed as follows:

(30)

The axial force (positive in tension) is denoted by N(x), and the stiffness of the elastic Winkler foundation by k(x).

Let us also consider an external distributed axial load n(x) positive along the + x axis

(31)

Combining Equations (30) and (31) yields

(32)

2.2.1 Constant stiffness EI within segments of the beam

A beam with constant stiffness in segments was considered. Substituting Equation (2a) into Equation (32) yields

(33)

Substituting Equations (4a), (4c), (4d), (5), and (5a) into Equation (33) yields the following governing equation,

(34)

Equation (34) is applied at any point on the grid with spacing h. At point i, the external distributed axial load ni is

calculated similarly to Equations (8gi). The transverse force T(x) is related to the shear force V(x) as follows:

(35)

4 2

4 2

( ) ( ) ( )( ) ( ) ( ) ( ) ( )

d w x d w x dw xEI N x n x k x w x q x

dx dx dx

2 3 2 3 2 4

2 1

2 3 2 34

1 2

4 2 5( ) ( 4 ) (6 )

12 12 3 3 2

4 2( 4 ) ( )

3 3 12 12

i i i i i ii i i i i i

r r r r r r

i i i ii i i i i

r r r r

N h n h N h n h N h k hW W W

EI EI EI EI EI EI

N h n h N h n hW W q h

EI EI EI EI

2

2

( ) ( )( ) ( ) ( ) ( )

d M x d dw xN x k x w x q x

dx dx dx

( )( )

dN xn x

dx

2 2

2 2

( ) ( ) ( )( ) ( ) ( ) ( ) ( )

d M x d w x dw xN x n x k x w x q x

dx dx dx

( )( ) ( ) ( )

dw xT x V x N x

dx

1 1 0 1 1 2 3; ; ....... ; ;T

n n nS W W W W W W 1S



Su

Th

Th

for

Ap

Eq

Th

Th

Th

for

Th

the

Fig

T

EU

ubstituting Eq

he bending m

he analysis at

rce is replace

pplying Equa

quations (37)

he combinatio

he slope is c

he analysis at

rce is replace

he sign conve

e element stif

gure 13. Sig

32

i i

i

T EI

h T

2h M

N

ULERBER

quations (2b)

moment and th

t positions of

ed by the tran

2.2.2

ations (5), (5

and (15b) ar

on of Equatio

calculated si

t positions of

ed by the tran

2.2.3

entions for be

ffness matrix

gn convention

3

3

(6

i

i

d wN

dx

Nh

E

21

2

2

2

i

i i

r

M h M

N h n h

EI E

32 ih T

RNOULLI

), (4b), (4d),

he slope are

f discontinuit

nsverse force

2 Second-

a), (14a-b), i

re applied at

ons (35), (15

imilarly to E

f discontinuit

nsverse force

3 Second-

ending mom

x in local coo

ns for mome

2

2)

ii

ir

dwN

dx

hW

EI

2

3

1

i

ir

M h M

hW

EI

21ih M

BEAM TH

(5), and (5a)

formulated u

ty is conduct

e.

-order anal

in Equation (

any point on

5c), and (15d

Equation (1

ty is conduct

e.

-order elem

ments, transve

ordinates is i

nts, transver

4( 2

3i

1

4

ii

r

i

N hM

EI

q h

21ih M

HEORY USI

) into (35) yi

using Equati

ted similarly

lysis of a ta

(32) yields T

n the grid.

d) yields the F

5d).

ted similarly

ment stiffne

ersal forces, d

illustrated in

rsal forces, di

2

1

4)

3 ir

NhW

EI

2 3

2i

r r

h n h

EI

2

i

r

N hW

EI

ING THE F

ields the FDM

ons (7a) and

y to that of th

apered beam

The FDM for

FDM formul

y to that of th

ss matrix o

displacemen

Figure 13.

isplacements

4(2

3i

1

2iW

1 1i iW W

FINITE DIF

M formulatio

d (7c), respec

he first-order

m

mulation of E

lation of the

he first-order

of a tapered

ts, and slope

s, and slopes

2

1

4)

3 ir

NhW

EI

22 i i

r

N h k

EI E

FFERENCE

on of the tran

ctively.

analysis; ho

Equation (32

transverse fo

analysis; ho

d beam

es adopted fo

for stiffness

1 ( i

4

ir

hW

EI

E METHOD

nsverse force

wever the sh

2) as follows

(3

orce

(3

wever the sh

or use in dete

s matrix.

2

)6 i

r

NhW

EI

(36

D

e

hear

s:

37)

38)

hear

ermining

2

6)




The FDM discretization is the same as Figure 10. Equations (17a) becomes,

(39)

Equations (37) and (15b) are applied at any point on the grid (the distributed load qi being zero).

The static compatibility boundary conditions are expressed similarly to Equations (19ad); however in Equations (19a)

and (19c), the shear forces are replaced by the transverse forces (Equation (38)). The analysis continues similar to the

first-order element stiffness matrix (Equations (20a24b)).

2.3 Vibration analysis of the beam

2.3.1 Free vibration analysis

Our focus here is to determine the eigenfrequencies of the beams. Damping is not considered. A second-order analysis

is conducted; the first-order analysis can easily be deduced.

2.3.1.1 Beam with constant stiffness EI within segments

The governing equation is as follows:

(40)

where is the beam’s mass per unit volume, A is the cross-sectional area, N(x) is the axial force (positive in tension),

n(x) is the external distributed axial load positive along + x axis, and k is the stiffness of the elastic Winkler

foundation. A harmonic vibration being assumed, w*(x,t) can be expressed as follows:

(41)

Here, is the circular frequency of the beam. Substituting Equation (41) into Equation (40) yields

(42)

A uniform grid with spacing hk is assumed in the segment k.

Substituting Equations (4a), (4c-d), (5), and (5a) into Equation (42) yields the following governing equation:

4 * 2 * * 2 **

4 2 2

( , ) ( , ) ( , ) ( , )( ) ( ) ( ) ( , ) 0

w x t w x t w x t w x tEI N x n x k x w x t A

x x x t

4 22

4 2

( ) ( ) ( )( ) ( ) ( ) ( ) ( ) 0

d w x d w x dw xEI N x n x k x w x A w x

dx dx dx

; ; ;T

red i i k kS T M T M

*( , ) ( ) sin( )w x t w x t

2 3 2 3 2 4 2 4

2 1

2 3 2 3

1 2

4 2 5( ) ( 4 ) (6 )

12 12 3 3 2

4 2( 4 ) ( ) 0

3 3 12 12

i k i k i k i k i k i k k kk i k i k i

r r r r r r r

i k i k i k i kk i k i

r r r r

N h n h N h n h N h k h A hW W W

EI EI EI EI EI EI EI

N h n h N h n hW W

EI EI EI EI

(43a)



Eq

det

Le

A c

Fo

Ef

We

Fig

Th

Th

res

Ap

Th

EU

quation (43a)

termined usi

t us define a

change in gr

r the special

ffect of a co

e analyzed th

gure 14.

he stiffness o

he continuity

spectively. E

pplying Equa

he transverse

k

k

h

A

ULERBER

) is applied at

ng Equation

a reference le

rid spacing ca

case of a un

oncentrated

he dynamic b

Figu

f the spring i

equations fo

Equation (11)

ations (43d) a

forces Til an

2 4iW W

il ir

il ir

T T

T T

K

M

lk r

Ak r

l

A

RNOULLI

t any point o

s (7c), (7a), a

ength lr, a ref

(43b)

(43c)

an be modele

niform beam

d mass, or a

behavior of a

ure 14. Vib

is Kp, and th

or deflection,

) is applied w

and (44a-b),

nd Tir are ca

1 (6iW

2p

r

pi

r

M

EI

KW

EI

p p

p p

K k E

M m

BEAM TH

on the grid. T

and (36), res

ference cross

ed by means

without an a

a spring

a beam carry

bration of bea

he concentrat

, slope, and b

with M*= 0. T

the balance

alculated usin

4 2 )lk W

0

0

il

il

W

W T

3/r r

r r

EI l

A l

HEORY USI

The slope, the

spectively.

s-sectional ar

of the refere

axial force or

ying a concen

am having a

ted mass is M

bending mom

The reference

of vertical fo

ng Equation

4i iW W

il ir

il ir

T T

kT T

ING THE F

e bending mo

rea Ar and th

ence length a

r a Winkler f

ntrated mass

concentrated

Mp.

ment are defi

e length of th

orces in case

(36).

1 2iW

23

30

p

r

pil

r

mW

l

kW

l

4r

r r

EI

A l

FINITE DIF

oment, and th

he vibration

and the param

foundation, E

or having a s

d mass and a

ined in Equat

he beam is lr

of a concent

0

0

0

ilW

2k

r

A h

EI

FFERENCE

the transverse

coefficient

meters lk.

Equation (43

spring, as rep

a spring.

ations (10a), (

r ((Equation

ntrated mass o

4k

Ak

h

E METHOD

e force are

as follows

(43

a) becomes

(4

presented in

(4

(4

(10b), and (1

(43b)).

or a spring y

(4

(4

4 2lk

D

3d)

43e)

44a)

44b)

11),

yields

45a)

45b)



Ef

rep

Ap

A h

A u

Th

Su

equ

EU

ffect of a sp

presented in F

pplying Equa

The governi

harmonic vib

uniform grid

he grid at the

ubstituting Eq

uation:

2

M

2d

M

2h M

N

E

ULERBER

pringmass

Figure 15. T

ations (43d) a

ing equation

bration being

d with spacin

beam’s ends

quations (5a)

*

2

*

( , )

( , )

M x t

x

M x t E

2

2

( )

( )

M xN

dx

M x E

21

2

2

2

i

i i

r

M h M

N h n h

EI EI

2

il ir

p

r

T T

MW

EI

RNOULLI

s system: W

The deflection

Figure 15.

and (44a-b),

2.3.1.2

ns are as follo

g assumed, M

ng h is assum

s and at posit

), (14a-b), an

2 *

2 *

( )

( )

wN x

wEI x

x

2

2

( )

( )

d wN x

dx

d wEI x

dx

2

3

1

i i

ir

M h M

hW

I

2p

r

piM

r

MW

EI

KW

EI

BEAM TH

We analyzed t

n of the mass

Vibration of

the balance

Tapered b

ows:

M*(x,t) can b

med.

tions of disco

nd (43c) into

*

2

*

2

( , )(

( , )

x tn x

x

x t

x

2

2

( )(

( )

w xn x

x

w x

x

2

1

0

i

r

N h

EI

0

(

iM

iM

W

W

HEORY USI

the dynamic

s is denoted b

f a beam carr

of vertical fo

beam

e expressed

ontinuity is t

Equations (4

*( , ))

w x tx

x

( ))

dw xx

dx

2 3

2i

r

n h

EI

)

il ir

ir

T T

W m

ING THE F

behavior of

by wiM.

rying a sprin

orces yields

similarly to E

the same as r

47b) and (47

*)( ) (k x w

( ) (k x w x

1

2iW

23

2

p

r

p iM

mW

l

m W

FINITE DIF

a beam carry

gmass syst

Equation (41

represented i

7a) yields Equ

( , ) (x t A

) (x A x

22 i i

r

N h k

EI E

0

(

iM

p iM

W

k W

FFERENCE

ying a spring

tem.

1). Equations

in Figure 8a,

quation (15b)

2 *

2

(( )

w xx

t

2) ( )x w x

4

Air

h

EI

)irW

E METHOD

gmass syste

(

s (46a-b) bec

(

b.

and the follo

, )0

t

0

2 4r

r

A h

EI

D

em, as

(45c)

(45d)

(46a)

(46b)

come

(47a)

(47b)

owing

iW

(48a)




The application of Equations (43b, d) yields

(48b)

For the special case of a tapered beam without an axial force or a Winkler foundation, Equation (48a) becomes

(48c)

Equations (15b) and (48a) are applied at any point on the grid. The slope and the transverse force are determined using

Equations (15d) and (38), respectively.

Effect of a concentrated mass, a spring, or a springmass system

The analysis is conducted similarly to the section above; the transverse forces Til and Tir in Equations (45a-c) are

calculated using Equation (38).

2.3.2 Direct time integration method

The direct time integration method developed here describes the dynamic response of the beam as multi-degree-of-

freedom system. The viscosity and an external loading p(x,t) are considered.

Uniform beam : The governing equation is applied at any point on the beam as follows:

(49)

The derivatives with respect to x are formulated with Equations (4a), (4c), and (4d); those with respect to t (the time

increment is t) are formulated considering a three-point stencil with Equations (50a-c),

(50a)

At the initial time t = 0, we apply a three-point forward difference approximation (Equation (8b))

(50b)

At the final time t =T, we apply a three-point backward difference approximation (Equation (8d))

(50c)

* **, ,

,

( , )

2i t t i t t

i t

w ww x t

t t

* * *2 *, , ,

2 2

,

2( , ) i t t i t i t t

i t

w w ww x t

t t

* * *2 *,0 , ,2

2 2

,0

2i i t i t

i

w w ww

t t

* * **,0 , ,2

,0

3 4

2i i t i t

i

w w ww

t t

* * *2 *, 2 , ,

2 2

,

2i T t i T t i T

i T

w w ww

t t

* * **, 2 , ,

,

4 3

2i T t i T t i T

i T

w w ww

t t

4 * 2 * **

4 2

2 * *

2

( , ) ( , ) ( , )( ) ( ) ( ) ( , )

( , ) ( , )( , )

w x t w x t w x tEI N x n x k x w x t

x x x

w x t w x tp x t A

t t

2 2 2 4 21 12 0i i i Ai l ih M h M h M W

2 44 2

4r r

lr r r

EI A h

A l EI



Th

equ

to

Th

mo

Ta

Th

Eq

Th

po

con

EU

he governing

uation is app

satisfy the bo

hus, the beam

oments M*(x

apered beam

he derivatives

quations (50a

he FDM form

int stencil an

ntinuity cond

2 *

( ,

M

x

p x

ULERBER

equation (Eq

plied at any p

oundary and

m deflection w

x,t) and the sh

Figure 16.

m: a similar a

s with respec

a-c).

mulation of E

nd a three-po

ditions. The b

2

( , )(

) (

x tN

x

t A x

RNOULLI

quation (49)

point of the b

d continuity c

w*(x,t) can b

hear force V*

Model for

analysis can b

ct to x are for

Equations (51

oint stencil, re

boundary co

2 *

2

2 *

2

(( )

( ,)

w xx

x

w x tx

t

BEAM TH

) can be form

beam at any t

conditions. T

be determined

*(x,t) are calc

the calculati

be conducted

rmulated wit

1) and (46b) a

espectively.

onditions are

*

, )( )

)

x tn x

t w

HEORY USI

mulated with

time t using a

The boundary

d with the Ca

culated using

ion of time-d

d. Thus Equa

th Equations

are applied a

Additional p

satisfied usi

*( , ))

( , )

w x t

x

x t

t

ING THE F

h FDM for x

a seven-poin

y conditions

artesian mod

g Equations

dependent vib

ation (49) be

(14a-b); tho

at any point o

points are int

ng a three-po

*( ) (k x w

FINITE DIF

= i at time t.

nt stencil. Ad

are satisfied

del represente

(7a,b).

bration of a u

comes

ose with respe

on the beam

roduced to s

oint stencil. T

( , )x t

FFERENCE

The FDM fo

dditional poin

using a five-

ed in Figure

uniform beam

(

ect to t are fo

and at any ti

satisfy the bo

Thus, the ben

E METHOD

ormulation o

nts are introd

-point stenci

16. The ben

m.

(51)

ormulated w

ime t using a

oundary and

nding mome

D

of this

duced

l.

ding

with

a five-

ent



M*

tra

Wi

har

Th

num

res

can

We

hyp

EU

*(x,t) and bea

ansverse forc

ith this mode

rmonic vibra

2

he analysis w

mber of grid

sult. Howeve

n match the e

e assume tha

perbolic curv

ULERBER

am deflection

e T*(x,t) is c

Figure 17.

el, the assum

ation (Equati

2.4 Extr

with the FDM

d points. This

er, in first-ord

exact results

at the relation

ve with the c

RNOULLI

n w*(x,t) can

calculated usi

. Model for

mptions previ

ion (41)).

rapolation

M is an approx

s means that

der analysis,

.

nship betwee

constants A,

AN

RN

BEAM TH

n be determin

ing Equation

r the calculati

ously made c

n to approx

ximation. Ge

when the nu

the exact res

en the results

B, and C, as

N B

C

HEORY USI

ned with the

n (38).

ion of time-d

can be verifi

ximate the

enerally, the

umber of poin

sult is rapidl

s R and the n

follows:

ING THE F

Cartesian m

dependent vi

ied, namely t

e exact res

accuracy of

nts is infinite

ly obtained s

number of gri

FINITE DIF

odel represen

bration of a t

the separation

ult

the results in

ely high, the

ince the FDM

id points on t

FFERENCE

ented in Figur

tapered beam

n of variable

ncreases by i

results tend

M polynomia

the beam N

E METHOD

re 17. The

m.

es and the

increasing th

toward the e

al approxima

follows a

(52)

D

he

exact

ation



Th

yie

Th

3

We

Th

usi

De

un

the

(14

gri

EU

hree couples

elds A, B, an

he exact resul

Result

3

e analyzed a

he governing

ing Equation

etails of the a

iformly distr

e present stud

4b)) and ben

id points n =

ULERBER

of values (N

nd C.

lt Re is appro

ts and dis

3.1 First-

3.1.1

uniform fixe

Figure 18.

equation (Eq

ns (7a) and (7

analysis and

ributed load”

dy. Furtherm

ding momen

4, 3, and 2 a

A

A

A

RNOULLI

Ni ; Ri) are th

oximated wh

cussions

-order ana

1 Beam su

edpinned b

. Uniform

quation (6))

7c).

results are pr

”. Table 1 list

more, the resu

nt (Equation

are presented

1

2

3

AN B

AN B

AN B

eR

BEAM TH

hen necessar

hen the numb

alysis

ubjected to

eam subjecte

m fixedpinn

is applied at

resented in th

ts the results

ults are prese

(14a)) when

d.

1

2

3

B R C

B R C

B R C

limN

AN

N

HEORY USI

ry to determi

ber of grid po

a uniform

ed to a unifo

ned beam sub

t grid points

he suppleme

s obtained wi

ented for a th

satisfying th

1 1

2 2

3 3

R N

R N

R N

BA

C

ING THE F

ne A, B, and

oints N

ly distribut

rmly distribu

bjected to a u

1, 2, 3, 4, an

entary materi

ith the classi

hree-point ste

he boundary

FINITE DIF

d C. Solving t

:

ted load

uted load, as

uniformly dis

d 5. The bou

al “fixedpin

cal beam the

encil (TPS) c

conditions. F

FFERENCE

the followin

shown in Fi

stributed loa

undary condi

nned beam s

eory (CBT) a

considered fo

Finally the re

E METHOD

g equation sy

(53a)

(53b)

(53c)

(54)

igure 18.

d.

tions are sati

subjected to a

and those obt

or the slope (

esults for a n

D

ystem

isfied

a

tained in

(Equation

number of




Table 1. Bending moments (kNm) in the beam for various number of grid points: classical beam theory (CBT),

present study, present study (three-point stencil (TPS))

Five-point grid

4 2.0m

Four-point grid

3 2.67m

Position

X(m)

CBT

(exact results)

Present study

Present study

(TPS)

Position

X(m)

CBT

Present study

0.0 -80.00 -80.00 -72.73 0.00 -80.00 -80.00

2.0 0.00 0.00 5.45 2.67 17.78 17.78

4.0 40.00 40.00 43.64 5.33 44.44 44.44

6.0 40.00 40.00 41.82 8.00 0.00 0.00

8.0 0.00 0.00 0.00

Three-point grid

2 4.0m

Two-point grid

1 8.0m

Position

X(m)

CBT

(exact results)

Present study

Position

X(m)

CBT

Present study

0.00 -80.00 -80.00 0.00 -80.00 -80.00

4.00 40.00 40.00 8.00 0.00 0.00

8.00 0.00 0.00

The results of the present study are exact for a uniformly distributed load whatever the discretization, since the exact

solution for the deflection curve is here a fourth-order polynomial which corresponds to the FDM approximation.

It is noted that the use of a three-point stencil for the bending moment and slope yields less accurate results since a

five-point stencil is used for the governing equations.



We

Th

Th

con

De

con

pre

Ta

EU

e analyzed a

he model sho

he governing

ntinuity cond

etails of the a

ncentrated lo

esent study (F

able 2. Bend

Position

X(m)

0.00

1.25

2.50

3.75

5.00

6.00

7.00

8.00

ULERBER

3.1.2

uniform fixe

Figu

owing the gri

equation (Eq

ditions are sa

analysis and

oad”. Table 2

FDM). The r

ding moment

4 C

(exac

-12

-6.

0.

7.

13

9.

4.

0.

RNOULLI

2 Beam su

edpinned b

ure 19. Unif

d points, as r

quation (6))

atisfied using

results are pr

2 lists the res

results are al

ts (kNm) in t

Eight-point

1.25m + 3 CBT

ct results)

2.89

.19

51

21

.92

28

64

00

BEAM TH

ubjected to

eam subjecte

form fixedp

represented i

is applied at

g Equations (

resented in th

sults obtained

lso presented

the beam: CB

grid

1.0m

-1

-6

0

7

1

9

4

0

HEORY USI

a concentr

ed to a conce

pinned beam

in Figure 5a,

t grid points

(7a) and (7c)

he suppleme

d with the cl

d for a three-p

BT, FDM

FDM

12.89

6.19

0.51

7.21

3.92

9.28

4.64

0.00

ING THE F

rated load

entrated load

subjected to

b, is conside

1, 2, 3, 4, 5l,

), and Equati

entary materi

assical beam

point discret

FINITE DIF

d, as shown in

a concentrat

ered.

5r, 6, 7, and

ions (10a-12)

al “fixedpin

m theory (CBT

ization of the

Thr

5.

Position

X(m)

0.00

5.00

8.00

FFERENCE

n Figure 19.

ated load

d 8. The boun

), respectivel

nned beam s

T) and those

he beam.

ree-point grid

.0m + 3.0m

E METHOD

ndary and

ly.

subjected to a

e obtained in

d

FDM

-12.89

13.92

0.00

D

a

the



Th

for

FD

We

Th

De

dis

the

Ta

Pos

X(m

0.0

2.0

4.0

6.0

8.0

Th

exa

fou

EU

he results of t

r the deflecti

DM approxim

e analyzed h

he beam is ca

etails of the a

stributed load

e present stud

able 3. Bend

sition

m)

C

(exact

00 -144

00 4.5

00 68.

00 61.

00 0.0

he results of t

act solution o

urth-order po

ULERBER

the present st

on curve is h

mation.

3.1.3

here a uniform

Figure 2

alculated at o

analysis and

d”. Table 3 li

dy (FDM).

ding moment

CBT

t results) F

4.00

50

00

50

00

the present st

of w(x) for a

olynomial. H

RNOULLI

tudy are exac

here a third-o

3 Beam su

m fixedpinn

0. Unifor

one hand with

results are pr

ists the resul

ts (kNm) in t

FDM

Five-point gr

-144.38

4.22

67.81

61.41

0.00

tudy have a h

a linearly dist

However the a

BEAM TH

ct for a conc

order polynom

ubjected to

ned beam sub

rm fixedpin

h a five-poin

resented in th

lts obtained w

the beam: CB

rid

Diffe

0.26

-6.22

-0.28

-0.15

high accurac

tributed load

accuracy inc

HEORY USI

centrated load

mial which i

a linearly

bjected to a l

nned beam su

nt grid, and a

he suppleme

with the clas

BT, FDM

erence

Po

X

6 0

2

8 3

5 4

6

8

cy. It is noted

ding is a fifth

creases with i

ING THE F

d whatever th

is exactly des

distributed

linearly distr

ubjected to a

at another han

entary materi

sical beam th

osition

X(m) (ex

0.00 -1

1.60 -

3.20 5

4.80 7

6.40 5

8.00

d that the exa

h-order polyn

increasing nu

FINITE DIF

he discretizat

scribed with

d load

ributed load,

a linearly dist

nd with a six

al “fixedpin

heory (the ex

CBT

act results)

144.00

17.92

51.84

72.96

53.12

0.00

act results ca

nomial and th

umber of grid

FFERENCE

ation, since th

the fourth-o

as shown in

tributed load

x-point grid.

nned beam s

xact results)

FDM

Six-point g

-144.15

-18.04

51.75

72.90

53.09

0.00

annot be achi

he FDM app

d points.

E METHOD

he exact solu

order polynom

Figure 20.

d

subjected to a

and those ob

grid

Differ

0.10

0.67

-0.17

-0.08

-0.06

ieved since th

roximation i

D

ution

mial

a linearly

btained in

rence

0

7

7

8

6

he

s a



We

At

I1 i

Fir

con

gri

pin

bea

Ta

Th

EU

e analyzed a

a position x

is the second

L = 8.0m

rst, the beam

nducted with

id points. De

nned fixed b

am theory (th

able 4. Bend

Position

X(m)

0.00

2.00

4.00

6.00

8.00

he results of t

ULERBER

3.1.4

tapered pinn

Figure 21

1 of the beam

d moment of

m and L0 = 2.

m is calculated

h the FDM u

etails of the a

beam subjec

he exact resu

ding moment

CBT

(exact re

0.00

17.36

-5.29

-67.93

-170.58

the present st

RNOULLI

4 Tapered

nedfixed be

. Tapered

m the second

area at x1 =

.0m

d with the fo

sing n = 9, 1

analysis and r

ted to a unifo

ults) and thos

ts (kNm) in t

T

esults)

3

8

tudy have a h

BEAM TH

d pinnedfi

eam subjecte

d pinnedfix

moment of

L1.

orce method o

3, and 17 gri

results are pr

formly distrib

se obtained i

the tapered b

FDM

Nine-point g

0.00

17.70

-4.61

-66.91

-169.22

high accurac

HEORY USI

ixed beam

ed to a unifor

xed beam sub

area I(x1) is

of the classic

id points. Th

resented in A

buted load”.

in the present

beam: CBT, F

grid Thir

-

-

cy.

1( )I x

ING THE F

subjected t

rmly distribu

bjected to a u

defined as f

cal beam the

he results are

Appendix A a

Table 4 lists

t study (FDM

FDM

FDM

rteen-point g

0.00

17.45

-5.11

-67.66

170.22

1 1 /I x L

FINITE DIF

to a uniform

ted load, as s

uniformly dis

follows,

,

ory (exact re

extrapolated

and in the su

the results o

M).

grid

F

Sevente

0.0

17

-5.

-67

-170

4

1L

FFERENCE

mly distribu

shown in Fig

stributed load

esults). Then

d to obtain th

upplementary

obtained with

FDM

een-point gri

00

.39

.22

7.83

0.44

E METHOD

uted load

gure 21.

d.

(55)

, the calculat

hose for infin

y material “ta

h the classica

id

FDM

n=

17.30

-5.40

-68.10

-170.80

D

tion is

nite

apered

al

M



We

in

Th

De

dis

ext

bea

Ta

Posi

X(m

0.00

2.00

4.00

6.00

8.00

Th

res

EU

3

e analyzed a

Figure 22.

Figu

he governing

etails of the a

stributed load

trapolated to

am theory (C

able 5. Bend

tion

m) (ex

0 -

0 -

0 -

0 -

0

he results of t

sults.

ULERBER

3.2 Secon

3.2.1

uniform fixe

ure 22. Fi

equation (Eq

analysis and

d and compre

o obtain those

CBT) and tho

ding moment

CBT

xact results)

-618.05

-451.63

-282.90

-127.54

0.00

the present st

N

E

RNOULLI

nd-order a

1 Beam su

edfree beam

ixedfree bea

-1.50

quation (6))

results are pr

essive force”

e for infinite

ose obtained

ts (kNm) in t

F

Nine-

-62

-45

-28

-12

0.

tudy have a h

2Nl

EI

BEAM TH

analysis

ubjected to

m subjected t

am subjected

0

is applied at

resented in th

”. The analys

grid points (

in the presen

the fixedfre

FDM

point grid

5.45

7.83

7.31

9.79

00

high accurac

HEORY USI

a uniform

to a uniforml

d to a uniform

t grid points.

he suppleme

sis is conduc

(Equation (5

nt study (FD

ee beam: CB

FD

Thirteen-

-621.3

-454.3

-284.8

-128.5

0.00

cy. The extra

ING THE F

ly distribut

ly distributed

mly distribut

entary materi

cted with n =

4)). Table 5

DM).

T, FDM

DM

-point grid

31

36

84

53

0

apolation tow

FINITE DIF

ted load an

d load and a

ted load and

al “fixedfre

9, 13, and 1

lists the resu

FDM

Seventeen

-619.8

-453.1

-283.9

-128.0

0.00

wards the exa

FFERENCE

nd a compr

compressive

a compressiv

ee beam subj

7 grid points

ults obtained

M

n-point grid

88

6

99

09

act results (n=

E METHOD

ressive forc

e force, as sh

ve force.

jected to a un

s. The results

with the cla

FDM

n=

-616.93

-450.70

-282.23

-127.20

0.00

= ) delivers

D

e

hown

niformly

s are then

ssical

s good



We

Th

inf

fix

Va

Ta

(e

Th

We

Th

inf

tap

Th

EU

e determined

he analysis is

finite grid po

xedpinned b

alues of the b

able 6. Buck

CBT

exact results)

0.699

he results of t

e determined

he analysis is

finite grid po

pered beam”.

he buckling f

ULERBER

3.2.2

d the bucklin

conducted w

oints. Details

beam”. The b

buckling fact

kling factors

) Ni

the present st

3.2.3

d here the bu

conducted w

oints. Details

. The bucklin

factors are

RNOULLI

2 Bucklin

ng load of a f

Figure

with n = 9, 13

of the analy

buckling load

tor are liste

of the beam

FDM

ine-point gri

0.7176

tudy have a h

3 Bucklin

uckling loads

with n = 9, 13

of the analy

ng load Ncr i

listed in Tab

crN

crN

BEAM TH

g load of a

fixedpinned

23. Buck

3, and 17 gri

ysis and the re

d Ncr is defin

ed in Table 6

: CBT, prese

id Th

high accurac

g load of a

of tapered b

3, and 17 gri

ysis and the re

s defined as

ble 7 for 0 =

2 /(EI

2EI

HEORY USI

fixedpinn

d beam, as sh

kling load of

id points. Th

results are lis

ned as follow

6.

ent study

FDM

hirteen-point

0.7073

cy.

tapered be

beams with v

id points. Th

results are lis

follows:

= L0/L1 = 0

2)l

21 /( )EI l

ING THE F

ned beam

hown in Figu

f a fixedpin

he results are

sted in the su

ws:

grid Sev

eam

various suppo

he results are

sted in the su

.25

FINITE DIF

ure 23.

nned beam.

then extrapo

pplementary

FDM

venteen-poin

0.7038

ort condition

then extrapo

pplementary

FFERENCE

olated to obta

y material “st

nt grid

ns, as shown

olated to obta

y material “st

(

E METHOD

ain those for

tability of a

(56)

FDM

n=

0.6963

in Figure 21

ain those for

tability of a

(57)

D

r

.

r



Ta

We

Th

inf

fix

Th Ta

Th

We

EU

able 7. Buck

Pinnedp

Pinned

Free

Fixed

e determined

he analysis is

finite grid po

xedfixed bea

he coefficient

able 8. Coef

CB

(exact

22.4

he results of t

e determined

ULERBER

kling factors

pinned

dfixed

fixed

fixed

3.3.1

d here the vib

conducted w

oints. The det

am”.

ts (Equatio

fficients of

BT

results)

40

the present st

3.3.2

d the vibratio

RNOULLI

of tapered

Nine

4.0

2.8

5.

2.4

1 Free vib

bration frequ

with n = 9, 13

tails of the an

on (43c)) are

f natural freq

FD

Nine-po

22.00

tudy have a h

2 Free vib

on frequencie

Figure 24.

BEAM TH

d beams with

FDM

-point grid

0255

8403

1245

4584

bration ana

uencies of a f

3, and 17 gri

nalysis and r

listed in Tab

uencies (firs

DM

oint grid

0

high accurac

bration ana

es of the tape

Vibration

HEORY USI

h 0 = L0/L1

F

Thirtee

4.02

2.84

5.12

2.21

alysis of a fi

fixedfixed b

id points. Th

results are lis

ble 8 below.

st mode) of a

FDM

Thirteen-p

22.21

cy.

alysis of a ta

ered beam re

n analysis of

ING THE F

1 = 0.25

FDM

en-point grid

249

411

284

153

ixedfixed

beam.

he results are

sted in the su

fixedfixed

M

point grid

apered free

epresented in

f a tapered fre

FINITE DIF

FDM

Seventeen

4.016

2.826

5.131

2.101

beam

then extrapo

upplementary

beam

FDM

Seventeen-p

22.28

efixed bea

Figure 24.

eefixed bea

FFERENCE

M

n-point grid

67

68

0

7

olated to obta

y file “vibrat

M

point grid

am

am.

E METHOD

FDM

n=

4.0263

2.8396

5.1452

1.7884

ain those for

ion analysis

FDM

n=

22.43

D

r

of a




At a position x1 of the beam, the second moment of area I(x1) is defined in Equation (55). The cross-sectional area

A(x1) is defined as follows:

, (58)

A1 being the cross-sectional area at x1 = L1. The analysis is conducted with n = 9, 13, and 17 grid points. The results

are then extrapolated to obtain those for infinite grid points. The details of the analysis and results are listed in the

supplementary file “vibration analysis of a tapered freefixed beam”.

The vibration frequency is defined as follows (definition adopted from Torabi [3]).

.

. (59) Table 9 lists the results obtained by Torabi [3] and those obtained in the present study. Table 9. Coefficients of natural frequencies (first mode) of a tapered beam

Torabi [3]

FDM

Nine-point grid

FDM

Thirteen-point grid

FDM

Seventeen-point grid

Present study

n=

0 = 0.10 2.6842 2.7100 2.6957 2.6906 2.6798

0 = 0.30 2.3471 2.3548 2.3506 2.3491 2.3459

0 = 0.50 2.1504 2.1493 2.1500 2.1503 2.1511

0 = 0.70 2.0165 2.0101 2.0137 2.0135 2.0133

0 = 0.90 1.9166 1.9062 1.9120 1.9157 1.9324

The results of the present study are identical to those presented by Torabi [3].

4 Conclusions The FDM-based model developed in this paper enables, with relative easiness, first-order analysis, second-order analysis,

and vibration analysis of EulerBernoulli beams. The results showed that the calculations conducted as described in this

paper yielded accurate results. First- and second-order element stiffness matrices (the axial force being tensile or

compressive) in local coordinates were determined. The analysis of tapered beams was also conducted.

The following aspects were not addressed in this study but could be analyzed with the model in future research:

Analysis of Timoshenko beams.

Analysis of linear structures, such as frames, through the transformation of element stiffness matrices from local

coordinates in the global coordinates.

2 14

1

EI

Al

2

1 1 1 1( ) /A x A x L




Second-order analysis of frames free to sidesway, the P- effect being examined.

EulerBernoulli beams resting on Pasternak foundations.

Elastically connected multiple-beam system.

Warping torsion of beams.

Lateral torsional buckling of beams.

Classical plate theory by considering additional points at boundaries.

Boundary value problem.

Initial value problem.

Linear ordinary differential equation with constants or variable coefficients.

Supplementary Materials: The following files are uploaded during submission:

“fixedpinned beam subjected to a uniformly distributed load”

“fixedpinned beam subjected to a concentrated load”

“fixedpinned beam subjected to a linearly distributed load”

“tapered pinnedfixed beam subjected to a uniformly distributed load”

“fixedfree beam subjected to a uniformly distributed load and compressive force”

“stability of a fixedpinned beam”

“stability of a tapered beam”

“vibration analysis of a fixedfixed beam”

“vibration analysis of a tapered freefixed beam”

Author Contributions:

Funding:

Acknowledgments:

Conflicts of Interest: The author declares no conflict of interest.




Appendix A: Tapered pinnedfixed beam subjected to a uniformly distributed load

The tapered beam (Figure 21) subjected to a uniformly distributed load was analyzed here with the force method of the

classical beam theory. The bending moment at the fixed-end was the redundant effort.

In the associated statically determinate system, M0(x) and m(x) are the bending moment due to the distributed load and

to the virtual unit moment at the fixed-end, respectively.

Let us introduce the dimensionless ordinate = x/l and 0 = L0/L1.

M0(x), m(x), and I(x) can be expressed as follows

(A1)

The bending moment M1 at the fixed end is the solution of the following equations:

(A2)

(A3)

(A4)

Equations (A2) and (A3) are solved numerically.

The combination of Equations (A1) and (A4) yields the bending moment at any position x, as follows:

(A5)

Details of the results are presented in the supplementary file “tapered pinnedfixed beam subjected to a uniformly

distributed load”.

References

[1] E. F. Anley, Z. Zheng. Finite Difference Method for Two-Sided Two Dimensional Space Fractional Convection-

Diffusion Problem with Source Term. Mathematics 2020, 8, 1878; https://doi.org/10.3390/math8111878

[2] M. Kindelan, M. Moscoso, P. Gonzalez-Rodriguez. Optimized Finite Difference Formulas for Accurate High

Frequency Components. Mathematical Problems in Engineering. Volume 2016, Article ID 7860618, 15 pages.

http://dx.doi.org/10.1155/2016/7860618

13 20

10 410 0 0 0

1 2

11 410 0 0 0

101

11

( ) ( ) (1 )

( ) 2 (1 )

( ) ( )

( ) (1 )

l

l

M x m x pldx d

EI x EI

m x m x ldx d

EI x EI

M

20

44

1 1 1 1 0 0

( ) ( ) / 2 (1 ) / 2

( ) /

( ) / (1 )

M x px l x pl

m x x l

I x I x L I

0 1( ) ( ) ( )M x M x M m x




[3] K. Torabi, H. Afshari, M. Sadeghi, H. Toghian. Exact Closed-Form Solution for Vibration Analysis of

Truncated Conical and Tapered Beams Carrying Multiple Concentrated Masses. Journal of Solid Mechanics

Vol. 9, No. 4 (2017) pp. 760-782

[4] J.T. Katsikadelis. A New Direct Time Integration Method for the Equations of Motion in Structural

Dynamics. Conference Paper · July 2011. Third Serbian (28th Yu) Congress on Theoretical and Applied

Mechanics. https://www.researchgate.net/publication/280154205

[5] M. Soltani, A. Sistani, B. Asgarian. Free Vibration Analysis of Beams with Variable Flexural Rigidity

Resting on one or two Parameter Elastic Foundations using Finite Difference Method. Conference Paper. The

2016 World Congress on The 2016 Structures Congress (Structures16)

[6] S. Boreyri, P. Mohtat, M. J. Ketabdari, A.Moosavi. Vibration analysis of a tapered beam with exponentially

varying thickness resting on Winkler foundation using the differential transform method. International Journal

of Physical Research, 2 (1) (2014) 10-15. doi: 10.14419/ijpr.v2i1.2152

[7] K. O. Mwabora, J. K. Sigey, J. A. Okelo, K. Giterere. A numerical study on transverse vibration of

EulerBernoulli beam. IJESIT, Volume 8, Issue 3, May 2019



Documents

EULER BERNOULLI BEAM THEORY: FIRST-ORDER ANALYSIS, …