Jtam-A4.dvi


JOURNAL OF THEORETICAL

AND APPLIED MECHANICS

52, 1, pp. 215-225, Warsaw 2014

ANALYSIS OF AXIALLY LOADED TAPERED BEAMS WITH GENERAL
END RESTRAINTS ON TWO-PARAMETER FOUNDATION

Mohamed Hassan Taha, Mohamed Nassar

Cairo University, Faculty of Engineering, Cairo, Egypt

e-mail: mtaha@alfaconsult.org; porfmnassar@gmial.com

The stability and free vibration of axially-loaded tapered beams with elastic end restraints
resting on two-parameter foundations are studied using the differential quadrature method
(DQM). The governing differential equation is discretized at sampling points, and then the
boundary conditions due to elastic end restraints are implemented and substituted into the
governing differential equation yielding a system of homogeneous algebraic equations. The
equivalent two-parameter eigenvalue problem is obtained and solved for critical loads in the
static case and for natural frequencies in the dynamic case.The obtained solutions are found
compatiblewith those obtained fromother techniques.The influences of differentparameters
on the critical loads and natural frequencies are investigated.

Key words: tapered beams, elastic end restraints, two-parameter foundation, differential
quadraturemethod

1. Introduction

Tapered beams are used in numerous engineering applications to obtain optimum designs, and
centrally tapered beam-columns are frequently used in many structures. Indeed, in practical
situations, the conventional idealized end conditions (pinned – clamped) may lead to models
thatmisrepresent the actual situationswheremovements and/or rotationsmay occur due to the
weakening of the supporting structure at the beam ends. In other situations, the end translation
and/or rotational displacement is expected due to the flexibility of the connections at the beam
ends as in the framed structures. However, the analytical treatments of differential equations
governing the behavior of these elements are intractable while the numerical techniques offer a
tractable alternative.Closed formsandanalytical solutions for simple cases of prismatic andnon-
prismatic beams may be found in literature. Ruta (1999) used the Chebychev series to obtain
solutions fornon-prismaticbeamvibration.AsymptoticperturbationwasusedbyMaccari (1999)
to analyze the nonlinear dynamics of continuous systems.Taha (2012) investigated the nonlinear
vibration of initially stressed prismatic beams resting on an elastic foundation by employing the
elliptic integrals. Taha and Abohadima (2008) studied the free vibration of non-uniform beams
resting on an elastic foundation using the Bessel functions. Sato (1980) reported the transverse
vibration of linearly tapered beams using the Ritz method. Numerical methods were used such
as the FEMbyNaidu et al. (1995), the differential transformmethods byBanerjee et al. (2006),
Seval Çatal et al. (2008) andHo andChen (1998) and the differential quadraturemethodby Shu
(2000), Bert et al. (1994) and Essam (2012) to study different configurations of such elements.

The free vibration of tapered beams with nonlinear elastic restraints was studied by Naidu
et al. (2001) using the FEM, and the effect of tapering ratio and end restraints were analyzed.
Closed form solutions for the case of axially compressed centrally tapered beams resting

on a two parameter foundation are not found in literature. Indeed, considering the foundation
and external axial compression increases the number of significant parameters of the equivalent
mathematical model which leads to an intractable model for analytical methods. Further, the



216 M.H. Taha,M. Nassar

implementation of end conditions in DQM comprises some difficulties in the numerical stabi-
lity and accuracy analysis, and many algorithms are suggested in literature to overcome these
complications in the case of conventional boundary conditions (Shu, 2000).

In the present work, the stability and vibration behavior of an axially compressed centrally
tapered beam-columnwith elastic end restraints resting on a two-parameter foundation will be
investigated. The implementation of elastic end restraints (including conventional end condi-
tions) in the application of DQM is suggested and verified. In addition, in the present work
the DQM is used to extend Naidu et al. (2001) FEM work to consider the influences of the
two-parameter foundation and external axial loading.

The governing equations are formulated in a dimensionless form and discretized over the
studied domain. The boundary conditions due to elastic end restraints are implemented and
substituted into the governing equations yielding a system of homogeneous algebraic equations.
Using the two-parameter eigenvalue analysis, the critical loads in the static case and the natural
frequencies for the dynamic case are obtained. The obtained solutions are verified and then
used to investigate the significance of different parameters related to the studied model on the
stability and natural frequencies of the beam.

2. Formulation of the problem

2.1. Vibration equation

The equation of free vibration of the centrally tapered axially-loaded beam of length L
resting on a two-parameter foundation shown in Fig. 1, is given as

∂2

∂X2

(

EI(X)
∂2Y

∂X2

)

+(Po−k2)
∂2Y

∂X2
+ρA(X)

∂2Y

∂t2
+k1Y (X)= 0 (2.1)

where X (0¬ X ¬ L) is the distance along the beam; t is time; I(X) is themoment of inertia
of the beam cross section at a distance X; ρ is the beam mass density per unit volume; E is
the beam modulus of elasticity; A(X) is the area of the beam cross section at X; Y (X,t) is
the beam lateral displacement; Po is the external axial load acting on the beam; k1 and k2 are
the foundation stiffnesses per unit length of the beam. Using dimensionless variables x = X/L
(0¬ x ¬ 1) and y = Y/L, Eq. (2.1) can be expressed in the dimensionless form as

∂2

∂x2

(EI(x)

L3
∂2y

∂x2

)

+
Po−k2)

L

∂2y

∂x2
+ρLA(x)

∂2y

∂t2
+k1Ly(x)= 0 (2.2)

The solution to Eq. (2.2) depends on the boundary conditions at the beam ends.

Fig. 1. An axially-loaded centrally tapered beam-columnwith elastic end restraints resting on a
two-parameter foundation



Analysis of axially loaded tapered beams... 217

2.2. Boundary conditions

The boundary conditions due to elastic restraints are:

— at x =0

kT0y(0, t)=−
∂

∂x

(EIo
L3

∂2y(0, t)

∂x2

)

kR0
∂y(0, t)

∂x
=
EIo
L

∂2y(0, t)

∂x2
(2.3)

— at x =1

kTLy(1, t)=
∂

∂x

(EIo
L3

∂2y(1, t)

∂x2

)

kRL
∂y(1, t)

∂x
=−

EIo
L

∂2y(1, t)

∂x2
(2.4)

where kT0 and kTL are the elastic stiffnesses against lateral displacements at x =0,1, respecti-
vely, kR0 and kRL are the elastic stiffnesses against rotation at x =0,1, respectively and Io is
the moment of inertia of the beam cross section at x =0,1.

2.3. Mode functions

The solution to Eq. (2.2) using the separation of variables method can be assumed as

y(x,t)= φ(x)ψ(t) (2.5)

where φ(x) is the mode function, ψ(t) is a function representing the variation of lateral di-
splacement with time. Substituting Eq. (2.5) into Eq. (2.2), then Eq. (2.2) can be separated
into

d2

dx2

(

I(x)
d2φ

dx2

)

+
(Po−k2)L

2

E

d2φ

dx2
+
(k1L

4−ρL4A(x)ω2

E

)

φ(x)= 0

d2ψ

dt2
+ω2ψ(t)= 0

(2.6)

where ω is the separation constant. The solution to Eq. (2.6)2 is

ψ(t) = Asin(ωt)+Bcos(ω t) (2.7)

where A and B are constants obtained fromthe initial conditions and ω is thenatural frequency
of the lateral vibration of the beam.

The general solution to Eq. (2.6)1 depends on the distribution of section geometry along
the beam. Figure 1 shows the case of a symmetric tapered beam, where the depth of the beam
increases lineally from do at x =0 to d1 at x =0.5, then decreases linearly form d1 at x =0.5
to do at x =1, while the width of the beam b is assumed constant, then

d(x) = doη(x) (2.8)

where

η(x) =

{

1+2x(α−1) for 0¬ x ¬ 0.5

1+2(1−x)(α−1) for 0.5¬ x ¬ 1.0
(2.9)

and α = d1/do is the tapering ratio.

Using the distribution of section geometry expressed in Eqs. (2.9), the distribution of the
area andmoment of inertia of the beam cross section along the beam are given as

A(x)= Aoη(x) I(x)= Ioη
3(x) (2.10)



218 M.H. Taha,M. Nassar

where Ao and Io are the area and the second moment of area of the beam cross section at
x =0, respectively.
Equation (2.6)1 may be rewritten as

d4φ

dx4
+β(x)

d3φ

dx3
+[η1(x)+η2(x)Po]

d2φ

dx2
+[ξ1(x)− ξ2(x)ω

2]φ(x) = 0 (2.11)

where

β(x)=
2

I(x)

dI(x)

dx
η1(x)=

1

I(x)

d2I(x)

dx2
−

k2L
2

EI(x)

η2(x)=
L2

EI(x)
ξ1(x)=

k1L
4

EI(x)
ξ2(x)=

ρAL4

EI(x)

(2.12)

The boundary condition in terms of the boundaryvalues of themode functionmay be expressed
at x =0 as

kT0φ(0)=−
EIo
L3
[φ′′′+6(α−1)φ′′] kR0φ

′(0)=
EIo
L
φ′′(0) (2.13)

Also, the boundary conditions at x =1 can be expressed as

kTLφ(1)=
EIo
L3
[φ′′′+6(α−1)φ′′] kRLφ

′(1)=−
EIo
L
φ′′(1) (2.14)

3. Solution of the problem

3.1. Differential quadrature method (DQM)

In the differential quadrature method (DQM), the solution domain is discretized into N
sampling points and the derivatives of a function f(xi) at any point are approximated by a
weighted linear summation of all functional values f(xj) at the other points as (Shu, 2000)

dmf(x)

dxm

∣

∣

∣

∣

Xi

≈
N
∑

j=1

C
(m)
i,j f(xj)s i =1, . . . ,N m =1, . . . ,M (3.1)

where M is the order of the highest derivative in the governing equation, f(xj) is the functional

value at the point x = xj and C
(m)
i,j is the weighting coefficient relating the functional value

at x = xj to the m-derivative of the function f(x) at x = xi. To obtain the weighting coeffi-
cients, many polynomials with different base functions are used to approximate the functional
values. Using the Lagrange interpolation formula, the functional value at the point x can be
approximated by all the functional values f(xk) (k =1, . . . ,N) as

f(x)≈
N
∑

k=1

L(x)

(x−xk)L1(xk)
f(xk) (3.2)

where

L(x)=
N
∏

j=1

(x−xj) L1(xk)=
N
∏

k=1

(xi−xk) i,k =1, . . . ,N (3.3)

Substitution of Eq. (3.2) into Eq. (3.1) yields the weighting coefficients of the first derivative as
(Shu, 2000)

C
(1)
i,j =



















L1(xi)

(xi−xj)L1(xj)
for i 6= j ∧ i,j =1,N

−
N
∑

j=1,j 6=i
C
(1)
i,j for i = j ∧ i,j =1,N

(3.4)



Analysis of axially loaded tapered beams... 219

Differentiation of Eq. (3.1) yields theweighting coefficients of the m-th order derivative in terms
of the weighting coefficients of (m−1)-th order derivative as

C
(m)
i,k
=
N
∑

k=1

C
(1)
i,k
C
(m−1)
i,k

i,k =1, . . . ,N m =1, . . . ,M (3.5)

The accuracy and stability of the obtained solution using numericalmethods asDQMandFEM
are affected by both the number and the distribution of discretization points. In boundary va-
lue problems, it is well known that irregular distributions of the sampling points with closer
dimensions near the boundaries yield stable and accurate results. One of the frequently used di-
stributions for mesh points generation is the normalized Gauss-Chebychev-Lobatto distribution
given as

xi =
1

2

[

1− cos
( i−1

N −1
π
)]

i =1, . . . ,N (3.6)

3.2. Discretization of the boundary conditions

The boundary conditions can be discretized as:
— at x =0

KT0φ1 =−
N
∑

k=1

C
(3)
1,kφk−6(α−1)

N
∑

k=1

C
(2)
1,kφk KR0

N
∑

k=1

C
(1)
1,kφk =

N
∑

k=1

C
(2)
1,kφk (3.7)

— at x =1

KTLφN =
N
∑

k=1

C
(3)
N,k

φk+6(α−1)
N
∑

k=1

C
(2)
N,k

φk KRL

N
∑

k=1

C
(1)
N,k

φk =−
N
∑

k=1

C
(2)
N,k

φk (3.8)

where N is the number of the sampling points and the elastic restraints stiffness parameters
KT0, KTL, KR0 and KRL are defined as

KT0 =
kT0L

3

EI0
KTL =

kTLL
3

EI0
KR0 =

kR0L

EI0
KRL =

kRLL

EI0
(3.9)

Using Eqs. (3.7) and (3.8) and employed the common matrices operations, expressions for the
boundary unknown functional values φ1, φ2, φN−1 and φN in terms of the other functional
values φi, (i =3, N −2) can be obtained as

φ1 =
N−2
∑

i=3

D1,iφi φ2 =
N−2
∑

i=3

D2,iφi φN−1 =
N−2
∑

i=3

DN−1,iφi φN =
N−2
∑

i=3

DN,iφi

(3.10)

where Di,j are numerical coefficients obtained fromtheDQMweighting coefficients andmatrices
operations and φi are the required unknowns.

3.3. Discretization of the governing equation

The governing equation of the beamvibration; Eq. (2.11); can bediscretized at the sampling
point xi as

N
∑

k=1

C
(4)
i,k
φk+β(xi)

N
∑

k=1

C
(3)
i,k
φk+[η1(xi)+η2(xi)Po]

N
∑

k=1

C
(2)
i,k
φk

+[ξ1(xi)− ξ2(xi)ω
2]φi =0

(3.11)



220 M.H. Taha,M. Nassar

Substituting the boundary functional values expressed in Eq. (3.10) into Eq. (3.11), one obtains

N−2
∑

k=3

[

C
(4)
i,k
+βiC

(3)
i,k
+(η1,i+η2,iPo)C

(2)
i,k
+ δi,k(ξ1,k−ξ2,kω

2)
]

φk =0 (3.12)

where δi,j is the Kronecker delta (δi,j =1 for i = j and δi,j =0 for i 6= j).

Application of Eq. (3.12) at the sampling points xi, i = 3, . . . ,N − 2 yields a system of
N − 4 homogeneous equations in N −4 unknown functional values of the mode function (φi;
i =3,N−2)with the twoparameters (Po and ω).Using theequivalent two-parameter eigenvalue
problem, the critical axial load Pcr and the natural frequencies ωn are obtained. In addition,
the functional values of the dimensionless lateral displacement at different locations along the
beam can be obtained and used to illustrate the mode functions.

3.4. Verification of the present solution

For the case of conventional end conditions (pinned-clamped), the results of both the fun-
damental stability parameter λb and the fundamental frequency parameter λ, calculated from
the present analysis, are comparedwith those obtained from closed form solution andFEMand
are found to be compatible (Essam, 2012). The fundamental stability parameter λb and the
fundamental frequency parameter λ are defined as

λ2b =
PcrL

2

EIo
λ4 =

ρAoL
4ω21

EIo
(3.13)

where Pcr is the critical load definedas the lowest value of the external axial load afterwhich the
beamloses its stability (also called bucklingorEuler’s load).For the case of elastic end restraints,
the values of the frequency parameter λ for tapered beams, obtained from the present solution,
are presented against those obtained from the FEM (Naidu et al., 2001) for different values of
the tapering ratio α in Table 1. The values shown in the table indicate close agreement between
the results obtained from the two approaches for small values of the tapering ratio α. There are
some deviations for large values of α because in the FEM results, the beam is divided into a
limited number of prismatic elements of equal length, which may yield less accurate values.

Table 1.Values of the frequency parameter λ for the tapered columns

End restraints stiffness Tapering ratio α = d1/do
Analysis

KT0 KTL KR0 KRL 1.0 1.1 1.2 1.3 1.4 1.5

1E5 1E5 0 0
3.141 3.248 3.349 3.449 3.534 3.620 FEM
3.141 3.283 3.392 3.496 3.540 3.588 Present

1E5 1E5 0.1 0.1
3.173 3.276 3.373 3.466 3.554 3.638 FEM
3.169 3.311 3.420 3.505 3.571 3.621 Present

1E5 1E5 1 1
3.399 3.479 3.557 3.634 3.708 3.780 FEM
3.373 3.517 3.634 3.729 3.807 3.872 Present

1E5 1E5 10 10
4.156 4.201 4.247 4.294 4.341 4.388 FEM
4.109 4.274 4.419 4.549 4.667 4.776 Present

4. Numerical results

Many attempts have been made, with different discretization schemes, to select the number of
sampling points that yields stable and accurate solutions. It has been found that 15 sampling
points achieve the assigned accuracy (0.1%) for the studied model (Essam, 2012).



Analysis of axially loaded tapered beams... 221

Both the stability and the natural frequency of the beam-foundation system dependmainly
on the integrated (overall) stiffness of the system. The integrated stiffness of the system is the
magnitude of the force that produces a unit displacement. It is expected that the integrated
stiffness of the system increases as the flexural rigidity of the beam (or α) increases, as the
foundation stiffness increases, as the end restrains stiffness increases and as the applied axial
load decreases.

The foundation stiffness parameters (k1 and k2) and the loading ratio γ are defined as

k1 =
k1L
4

EI
k2 =

k2L
2

π2EI
γ =

Po
Pcr

(4.1)

In the present model, the influences of 8-independent parameters (4 for elastic end restraints,
2 for foundation, tapering ratio α and external axial load Po) on two parameters (stability
parameter λb and frequency parameter λ) are to be studied.

A MATLAB code is used to study the variations of the stability and frequency parameters
for the practical range of the beam and foundation parameters. Although the present solutions
are obtained in terms of dimensionless parameter, the properties used to obtain the present
numerical results are for a reinforced concrete beamwith: E =2.1·1010Pa, L =5m, b =0.2m,
do =0.5m, ρ =2500kg/m

3.

4.1. Investigations of the stability parameter (λb)

Thevariations of the stabilityparameter λbwithdifferentparameters of thebeam-foundation
system are shown in Figs. 2a-c. As expected, the stability parameter increases with the increase
of the tapering ratio, the foundation stiffness, the stiffness of the elastic end restraints and with
the decrease of the external compression load.

Fig. 2. Influence of the support stiffness on the stability parameter for: (a) KTO = KTL =10
5,

KRO =0; (b) KTO = KTL =10
5, KRO =10

3; (c) KRO = KRL =10
3, KTO =10

5



222 M.H. Taha,M. Nassar

The effects of the transition of end conditions from the pinned-pinned (P-P) case to pinned-
clamped (P-C) case on the stability parameter λb are shown in Fig. 2a. It is clear that the value
of the stability parameter increases as the stiffness of elastic end restraints increases. It is also
observed that, to represent the conventional clamped end condition, a value of the rotational
stiffness parameter KRO = 100 is adequate. In addition, in the case of weak foundations, the
effect of variation in KRO on the stability parameter is more significant for large values of the
tapering ratio α.

The variations of the stability parameter due to the transition of the end conditions from
the pinned-clamped (P-C) case to the clamped-clamped case (C-C) are shown in Fig. 2b for
different values of the system parameters. It is observed that the effect of variation of KRO is
more significant for large values of the tapering ratio α for all types of foundations.

The variation of the stability parameter due to relaxation of the lateral translation at one
end of the beam, while preventing the lateral translation at the other end and the rotations at
both ends is shown in Fig. 2c. It is observed that to prevent the lateral movement at the beam
end, a value of the translational stiffness parameter KTL =10

5 is enough. However, it is found
that the effect of variation in KTL on the stability parameter λb is more significant than the
variation of KRO, especially for large values of the tapering ratio α. Furthermore, it is noticed
that the effect of the tapering ratio α is negligible for the case when the lateral translation is
released at one end and other elastic restraints kept rigid (C-CF).

4.2. Investigations of the frequency parameter (λ)

The variations of the frequency parameter λ with different parameters of the beam and
the underlying foundation are shown in Figs. 3-5. It is clear in all figures that the frequency
parameter increases with the increase in the tapering ratio, the foundation stiffness, the end
restraints stiffness and with the decrease of the axial compression load.

The effects of the transition from the case of (P-P) condition to the case of (C-C) condition
on the frequency parameter λ are shown in Fig. 3a. It is found that the effect of the tapering
ratio α on the frequency parameter λ decreases as the foundation stiffness increases.

Figure 3b shows the effect of the transition of the end restraints from the (C-P) case to the
(C-C) case on the frequency parameter λ. It is observed that the effect of the axial compression
load is more noticeable for the (C-P) case in the case of weak foundation. Also, it is found that
the effect of α is more noticeable for the (C-C) case while the effect of foundation stiffness is
noticeable for the (C-P) case.

Fig. 3. Influence of the support rotational stiffness on the frequency parameter for:
(a) KTO = KTL =10

5, KRO = KRO = KR, Po =0; (b) KTO = KTL =10
5, KRO =10

3

The effects of the external axial compression and the variation of the rotational elastic
stiffness at one end on the frequency parameter for prismatic beams are shown in Fig. 4a. It is
clear that as the axial compression load increases, the natural frequency of the systemdecreases.



Analysis of axially loaded tapered beams... 223

Indeed, in the deformed configuration, the lateral component of the external axial compression
is in the opposite direction of the lateral restoring force resulting from the beamand foundation
stiffness. Consequently, as the axial compression increases, the total restoring force of the system
decreases, which leads to a flexible system with a low natural frequency. At a certain (critical)
valueof the external axial compression (Pcr), the system is transformed into anaperiodic system,
and no free vibration occurs.

Figure 4b is another version of Fig. 4a for the case of the tapered beamwith a high tapering
ratio (α =1.5). It is noticed that the influence of foundation stiffness on the natural frequency
of the system is more significant for the (C-P) case than (C-C) case.

The influences of the variation of the translational stiffness at one end on the frequency
parameter are shown in Fig. 4c for different values of the tapering ratio. It is clear that the
natural frequency of the system increases as the end lateral stiffness increases.

Fig. 4. Influence of the compression axial load on the frequency parameter for:
(a) KTO = KTL =10

5, KRO =10
3, α =1; (b) KTO = KTL =10

5, KRO =10
3, α =1.5;

(c) KTO =10
5, KRO = KRL =10

3, k1 =0

Fig. 5. Influence of the foundation parameters on the frequency parameter:
(a) KTO = KTL =10

5, KRO = KRL =10
3, (b) KTO = KTL =10

5, KRO =10
3, KRL =0



224 M.H. Taha,M. Nassar

The effects of variations of both the foundation linear stiffness k1 (Winkler effect) and the
foundation shear stiffness k2 (Pasternak effect) on the frequencyparameter are shown inFigs. 5a
and 5b.
In Fig. 5a, the influence of variation of the linear foundation stiffness k1 on the frequency

parameter for the case of (C-C) end condition is shown. Further, in Fig. 5b, the influence of
variation of the linear foundation stiffness k1 on the frequency parameter for the case of (C-P)
condition is shown. It is clear that the influence of the linear foundation stiffness is stronger for
the (C-P) case. However, the influence of the foundation shear stiffness is more significant for
prismatic beams.

5. Sammary and conclusion

The stability and free vibrational behavior of axially-loaded centrally tapered beam-columns
with elastic end restraints restingon two-parameter foundationsare investigated using theDQM.
The governing differential equation with variable coefficients is derived and discretized over the
studieddomain.Theboundaryconditions implantationatbeamendsduetoelastic endrestraints
are suggested andverified.Then, the governing differential equation is transformed into a system
of N −4 homogeneous algebraic equations in N − 4 unknown functional values of the lateral
displacement in addition to the two parameters Po and ω. Using the two parameter eigenvalue
analysis, the values of the critical loads (Pcr) for the static case (ω = 0) and the natural
frequencies ωn for a prescribed value of the axial compression load Po < Pcr are obtained. It
is found that the natural frequencies and the critical loads for the tapered beams increase as
the integrated stiffness of the beam-foundation system increases. The integrated stiffness of the
system is a qualitative combination of the beam flexural rigidity, the foundation stiffness and
the elastic end restraints stiffness. Furthermore, it is found that the natural frequency of the
beam-foundation system decreases as the axial compression load increases.

References

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Manuscript received March 4, 2013; accepted for print September 6, 2013