Jtam-A4.dvi


JOURNAL OF THEORETICAL

AND APPLIED MECHANICS

53, 2, pp. 345-356, Warsaw 2015
DOI: 10.15632/jtam-pl.53.2.345

STATIC AND SENSITIVITY ANALYSIS OF NONLOCAL NANOBEAMS

SUBJECT TO LOAD AND MATERIAL UNCERTAINTIES BY

CONVEX MODELING

Isaac Sfiso Radebe

Department of Mechanical Engineering, Durban University of Technology, Durban, South Africa

e-mail: sfisor@dut.ac.za

Sarp Adali

Discipline of Mechanical Engineering, University of KwaZulu-Natal, Durban, South Africa

e-mail: adali@ukzn.ac.za

At the nano-scale, loads acting on a nanobeam and its material properties are likely to be
not known precisely, i.e., uncertain. In the present paper, the deflection of a nanobeam sub-
ject to load and material uncertainties is studied by convex modeling of the uncertainties.
The level of uncertainty is taken to be bounded and themaximumdeflection corresponding
to the worst-case of loading ormaterial properties is obtained, that is, the uncertainties are
determined so as to maximize the deflection. The sensitivity of the deflection to the uncer-
tainty in the material properties is also investigated. Numerical results are given relating
the level of uncertainty to maximum deflection.

Keywords: nanobeams, load uncertainty, material uncertainty, convexmodeling, sensitivity

1. Introduction

Mechanics of nanobeamshas been studied extensively using continuumbasedmodels in an effort
to quantify their behavior under static, buckling and dynamic loads. Several studies focused on
continuummodeling andmechanics of nano andmicro-sized beams and carbon nanotubes using
various beam theories (seeWang and Shindo, 2006; Reddy, 2007; Reddy andPang, 2008; Adali,
2008; Zhang et al., 2010; Di Paola et al., 2011;Muc, 2011; Adali, 2011; Hosseini-Ara et al., 2012;
Thai, 2012; Thai and Vo, 2012; Eltaher et al., 2013). These studies employed Euler-Bernoulli
and Timoshenko beam models coupled with the nonlocal elastic theory (see Eringen, 2002)
to formulate the variational principles and the governing equations for nanobeams undergoing
static bending, buckling and vibrations.

The bending behavior of nano-scale structures has been the subject of several studies and,
in particular, nanobeams under static transverse loadswere studied in (Wang and Shindo, 2006;
Reddy, 2007; Reddy and Pang, 2008; Zhang et al., 2010; Di Paola et al., 2011; Thai, 2012; Thai
and Vo, 2012; Eltaher et al., 2013; Challamel and Wang, 2008; Wang et al., 2008; Ansari and
Sahmani, 2011; Fang et al., 2011; Roque et al., 2011; Li et al., 2012; Khajeansari et al., 2012).
These studies took into account a number of effects such as shear deformation, surface stress,
and elastic foundation but neglected the load andmaterial uncertainties.

As such in the previous studies, the loading was taken as deterministic and the material
properties were defined taking their average values. However, under operational conditions, the
loads often have random characteristics making it difficult to predict their magnitude and di-
stribution with accuracy. Similarly, it is usually difficult to determine the elastic constants of
nano-sized beams with some certainty. The scatter in the geometric and material properties of



346 I.S. Radebe, S. Adali

carbon nanotubes is known andwas discussed byKalamkarov et al. (2006), Huang et al. (2006),
Scarpa and Adhikari (2008), Lu and Zhong (2012) and Fereidoon et al. (2014).
The main trust of the present work is to study the bending of nanobeams in a non-

deterministic setting by taking the load andmaterial variations into account.Thus themaximum
deflection of nanobeams is determined taking the transverse loading as non-deterministic and
thematerial properties as uncertain. The problem analysis is conducted using convex modeling
of uncertainties to determine the least favourable conditions to produce the highest deflection.
Convex modeling has been used extensively in the past to deal with various engineering pro-
blems containing data uncertainties (see Adali et al., 1995a,b; Pantelidis and Ganzerli, 1998;
Jiang et al., 2007; Kang andLuo, 2009; Hu andQiu, 2010; Radebe andAdali, 2013). For further
information, the reader is referred to the review articles byWang et al. (2001) and the book by
Ben-Haim and Elishakoff (1990).
In the present study, the effect of load and material uncertainties on the deflection of a

nanobeam is studied based on the nonlocal Euler-Bernoulli beam theory. The previous work on
the subject involves the studyof the effect ofmaterial uncertainties on the buckling of a nonlocal
plate by Radebe and Adali (2014). Load and material uncertainties are modeled as uncertain-
but-bounded quantities. Explicit expressions are obtained for the least favorable deflection of
a nanobeam for a given level of uncertainty. The sensitivity of the deflection to the level of
uncertainty inmaterial properties is also studied. Numerical results are given to investigate the
effect of uncertainty on the deflection and on the sensitivity to material properties.

2. Load uncertainty

The nanobeam under consideration is subject to a combination of deterministic and uncertain
transverse loads denoted by p(x) and q̃(x), respectively, as well as a compressive axial load N0
as shown in Fig. 1. The beamhas a rectangular cross-section of dimensions h×b, where h is the
height and b is the width (Fig. 1).

Fig. 1. Beam geometry and uncertain loading

The differential equation governing its deflection w(x) based on the nonlocal elastic theory
is given by (Reddy, 2007)

EIwxxxx+N0(wxx−η2wxxxx)= (p−η2pxx)+(q̃−η2q̃xx) for 0¬ x ¬ L (2.1)

where E is Young’smodulus, I is themoment of inertia and η is the small-scale parameter. The
subscript x denotes differentiation with respect to x. The compressive axial load N0 satisfies the
buckling constraint N0 < Ncr, where the buckling load Ncr is given by (see Reddy, 2007)

Ncr =
µ2EI

L2+µ2η2
(2.2)

with µ denoting a coefficient depending on the boundary conditions.



Static and sensitivity analysis of nonlocal nanobeams... 347

The uncertain load q̃(x) acting on the beam is unknown, and only limited information is
available on its coefficients. The information required on the uncertain load is that it should
have a finite norm, i.e., it should satisfy the constraint

‖q̃(x)‖2L2 =
L∫

0

[q̃(x)]2 dx ¬ ε2 (2.3)

where 0 < ε < 1 is a given constant which determines the level of uncertainty, and the sub-
script L2 denotes the L2 norm.The solution to the deflection problem is obtained by expanding
the deterministic and uncertain loads in terms of orthogonal functions ψn(x) satisfying the
boundary conditions, viz.

p(x)=
∞∑

n=1

pnψn(x) q̃(x)=
∞∑

n=1

q̃nψn(x) (2.4)

where the coefficients are given by

pn =
1

r

L∫

0

p(x)ψn(x) dx q̃n =
1

r

L∫

0

q̃(x)ψn(x) dx (2.5)

with r given by

r = ‖ψn(x)‖2L2 =
L∫

0

[ψn(x)]
2 dx (2.6)

Here, the coefficients pn are known since the deterministic load p(x) is given, however the
coefficients q̃n are unknownandhave to be determined tomaximize the deflection corresponding
to the least favourable (worst-case) loading. The solution for the deflection function w(x) is also
expanded in terms of ψn(x) and can be expressed as

w(x) =
∞∑

n=1

Wnψn(x) (2.7)

The coefficients Wn are computed by substitutingEq. (2.7) into differential equation (2.1).Next,
the worst-case uncertain loading causing the highest deflection is obtained. FromEqs. (2.3) and
(2.5), it follows that

N∑

n=1

(q̃n)
2 ¬ ε

2

r
(2.8)

where N is a large number. The highest load is obtained when
∑N
n=1(q̃n)

2 = ε2/r, i.e., the
inequality is taken as an equality. Thus the deflection w(x; q̃) is to be maximized with respect
to the uncertain load subject to the constraint

∑N
n=1(q̃n)

2 = ε2/r. For this purpose, themethod
of Lagrange multipliers is employed with the Lagrangian at a point x = x0 given by

L(x0; q̃n)=w(x0; q̃(x0))+λ

(
N∑

n=1

(q̃n)
2−

ε2

r

)
(2.9)

where λ is a Lagrange multiplier and 0 ¬ x0 ¬ L is a point which has to be determined such
that w(x0; q̃(x0)) ismaximumat x = x0. Themaximumof L(x0; q̃n)with respect to q̃n produces
the least favourable uncertain load, viz.

max
q̃n

L(x0; q̃n) (2.10)



348 I.S. Radebe, S. Adali

which can be computed by setting its derivative with respect to q̃n to zero, viz.

∂L(x0; q̃n)

∂q̃n
=0 for n =1,2, . . . ,N (2.11)

This computation gives the coefficient q̃n at a point x0 as

q̃n(x0)=−
1

2λ

∂w(x0; q̃n(x0))

∂q̃n
(2.12)

The point x0 is an unknown and has to be determined to maximize the deflection.

2.1. Simply supported beam

The method of solution outlined above is now applied to a simply supported beam subject
to the deterministic load p(x) = p0(x/L)

3 and the uncertain load q̃(x). The simply supported
boundary conditions for the nonlocal nanobeam are given by (Reddy, 2007)

w(0)= 0 (−EI +η2N0)wxx(0)+η2k0bw(0)−η2p(0)−η2q̃(0)= 0
w(L) = 0 (−EI +η2N0)wxx(L)+η2k0bw(L)−η2p(L)−η2q̃(L)= 0

(2.13)

The deterministic and uncertain loads are expanded in terms of the orthogonal functions
ψn(x)= sinαnx, where αn =(nπ)/L. Thus

p(x)=
N∑

n=1

pn sinαnx q̃(x)=
N∑

n=1

q̃n sinαnx (2.14)

here the coefficients pn are given by

pn =(−1)n+1
2p0
(nπ)3

(n2π2−6) (2.15)

The deflection w(x) satisfying boundary conditions (2.13) can be obtained by expanding it in
terms of sinαnx as

w(x) =
N∑

n=1

Wn sinαnx (2.16)

Substituting Eq. (2.16) into differential equation (2.1), the coefficients Wn are computed as

Wn =
(1+η2α2n)(pn+ q̃n)

EIα4n− (1+η2α2n)α2nN0
(2.17)

the Lagrangian L(x0; q̃n) given by Eq. (2.9) becomes

L(x0; q̃n)=
N∑

n=1

(1+η2α2n)(pn+ q̃n)

EIα4n− (1+η2α2n)α2nN0
sinαnx0+λ

(
N∑

n=1

(q̃n)
2− 2ε

2

L

)
(2.18)

The coefficients q̃n(x0) are computed from Eqs. (2.11) and (2.18) as

q̃n(x0)=−
1

2λ

An(x0)

Bn
(2.19)

where

An(x0)= (1+η
2α2n)sinαnx0 Bn = EIα

4
n− (1+η2α2n)α2nN0 (2.20)



Static and sensitivity analysis of nonlocal nanobeams... 349

Noting that the worst case loading is given by

N∑

n=1

(q̃n)
2 =
2ε2

L
(2.21)

we can compute the Lagrange multiplier from Eqs. (2.19) and (2.21) as

λ =±
√
L√
8ε

√√√√
N∑

n=1

A2n(x0)

B2n
(2.22)

where the plus andminus signs correspond to the least andmost favourable loading cases. The
coefficients q̃n can be computed by inserting the Lagrangemultiplier (2.22) into Eq. (2.19). This
computation gives

q̃n(x0)=∓
√
2ε√
L





√√√√
N∑

n=1

A2n(x0)

B2n




−1

An(x0)

Bn
(2.23)

Theuncertain loadproducing themaximumdeflection is given byEq. (2.14)with the coefficients
given by Eq. (2.23).

3. Material uncertainty

3.1. Uncertain constants

Next, the effect of uncertainty inmaterial properties on thedeflection is investigated.Young’s
modulus Ẽ and the small scale parameter η̃ are taken as uncertain material parameters, and
they are defined as

Ẽ = E0(1+ δ1) η̃ = η0(1+ δ2) (3.1)

where E0 and η0 are the nominal (deterministic) values, and δ1 and δ2 are margins of error
(uncertainty) to be determined to maximize the deflection. The unknown constants δ1 and δ2
are required to lie in an ellipse and satisfy the inequality

∑2
i=1δ

2
i ¬ γ2 which corresponds to

inequality (2.3) of the uncertain loading case. The least favourable solution is given when the
constants lie on the boundary of the ellipse, i.e., they satisfy the equality constraint

2∑

i=1

δ2i = γ
2 (3.2)

The material uncertainty is studied for a simply supported nanobeam under a sinusoidal load
p(x)= p1 sin(πx/L). For this case, themaximumdeflection occurs at themid-point and is given
by

w
(L
2
;Ẽ, η̃

)
=

(1+α21η̃
2)p1

α41IẼ − (1+α21η̃2)α21N0
(3.3)

where α1 = π/L. Substituting Eq. (3.1) into Eq. (3.3), we obtain

w
(L
2
;Ẽ, η̃

)
=

[1+α21η
2
0(1+ δ2)

2]p1
α41IE0(1+ δ1)− [1+α21η20(1+ δ2)2]α21N0

(3.4)



350 I.S. Radebe, S. Adali

which can be linearized leading to the expression

w
(L
2
;Ẽ, η̃

)
= c0+c1δ1+ c2δ2 (3.5)

where

c0 =
1+α21η

2
0

α41IE0− (1+α21η20)α21N0
p1 c1 =−

IE0(1+α
2
1η
2
0)

[α21IE0− (1+α21η20)N0]2
p1

c2 =
2α21IE0η

2
0

[α21IE0− (1+α21η20)N0]2
p1

(3.6)

To derive expression (3.5), the relation

(1± δ)c ∼=(1∓ cδ)+O(δ2) (3.7)
is employed, where the superscript c can take positive or negative values and |δ| ≪ 1. The
Lagrangian L(δ1,δ2) to compute themaximumdeflection subject to constraint (3.2) is given by

L(δ1,δ2)= c0+ c1δ1+ c2δ2+λ

(
2∑

n=1

δ2i −γ2
)

(3.8)

The constants δi are computed from Eq. (3.8) as

δi =−
ci
2λ

(3.9)

The Lagrange multiplier λ can be computed from Eqs. (3.2) and (3.9) as

λ =± 1
2γ

√√√√
2∑

i=1

c2i (3.10)

where the plus andminus signs correspond to least andmost favourable cases. The coefficients
δi can be computed by inserting Lagrange multiplier (3.10) into Eq. (3.9). This computation
gives

δi =∓γ
1

√
c21+ c

2
2

ci (3.11)

The values of δi given by Eq. (3.11) are substituted into Eq. (3.5) to compute the mid-point
deflection w(L/2;Ẽ, η̃) subject to material uncertainty.

3.2. Sensitivity analysis

The sensitivity of the deflection to the level of uncertainty in material data can be studied
by sensitivity analysis. In general, the deflection shows different sensitivities to the material
parameters Ẽ and η̃, and this can be investigated by defining relative sensitivity indices SK(δi)
given by

SK(δi)=
∣∣∣
∂w(L/2;Ẽ, η̃)

∂δi

∣∣∣
|δi|

w(L/2;E0,η0)
(3.12)

which is normalizedwith respect to the deterministicmid-point deflection w(L/2;E0,η0). InEq.
(3.12), the sensitivity SE(γ1) denotes the relative sensitivity of the mid-point deflection with
respect to uncertainty in Ẽ, and Sη(δ2) with respect to uncertainty in η̃ so that the subscript K
stands for the respectivematerial property. The sensitivities SK(δi) can be computed fromEqs.
(3.5) and (3.12) as

SK(δi)=
|ciδi|
c0

(3.13)

noting that w(L/2;E0,η0)= c0 where the values of ci are given by Eqs. (3.6).



Static and sensitivity analysis of nonlocal nanobeams... 351

4. Numerical results

The effect of uncertain loads andmaterial properties on the deflection is studied in the present
section. The cross-section of the nanobeam is taken as square, and the height and the length
of the nanobeam are specified as b = h = 1nm and L = 10nm. The material properties are
specified as E =1000GPa, 0¬ η ¬ 2nm.

4.1. Load uncertainty

L2 norms of the uncertain and the deterministic loads can be related as

‖q̃(x)‖2L2 = ε
2 = R20‖p(x)‖2L2 (4.1)

where R0 is a proportionality constant and determines the degree of uncertainty relative to the
deterministic load with R0 =0 corresponding to no uncertainty, i.e., the deterministic case. For
the present case ‖p(x)‖2L2 = p

2
0L/7, hence

ε =

√
L

7
p0R0 (4.2)

In the calculations, the load coefficient p0 is taken as p0 =1N/m. Figure 2 shows curves of
the deflection vs. x-axis for various uncertainty levels R0 with η =2nm and N0 =0. In Fig. 2
and in the subsequent figures, the curves are obtained by setting x0 = x in equation (2.7), and
consequently at every point x the deflection is the least favourable deflection.

Fig. 2. Deflection curves vs. x-axis for various uncertainty levels with η0 =2nm and N0 =0

Figure 2 shows that, compared to the deterministic case corresponding to R0 = 0, the
deflection increases as the level of load uncertainty increases. The corresponding results for a
beam subject to a compressive axial load of N0 = 0.5Ncr are given in Fig. 3 which shows the
effect of compressive axial load on the uncertain deflection. For a simply supported beam, the
coefficient µ = π in (2.2) for Ncr.
The effect of the small scale parameter η0 on the deflection of the nanobeam subject to

an uncertain load with R0 = 0.3 is shown in Fig. 4. It is observed that both the small-scale
parameter η0 and the level of deflection are factors in the increasing of the mid-point deflec-
tion. Next, the combined effect of the small-scale parameter and the axial load on the maxi-
mum deflection is studied in Fig. 5 which shows the contour plots of the maximum deflection
with respect to 0 ¬ N0 ¬ 0.6Ncr (x-axis) and 0 ¬ η0 ¬ 2nm (y-axis) for uncertainty levels
0¬ R0 ¬ 0.3. Themaximum deflection of the beam is computed by

max
0¬x¬L

w(x) = max
0¬x¬L

(
N∑

n=1

Wn sinαnx

)
(4.3)



352 I.S. Radebe, S. Adali

Fig. 3. Deflection curves vs. x-axis for various uncertainty levels with η0 =2nm and N0 =0.5Ncr

Fig. 4. Deflection curves vs. x-axis for the deterministic case (full lines) and for an uncertainty level of
R0 =0.3 (dotted line) with η0 =0,1,2nm and N0 =0

using aminimization routine inMathematica. Figure 5 shows that an increase in the parameters
η0 or N0 as well as in the level of load uncertainty leads to higher deflection.

Fig. 5. Contour plots of the maximum deflection with respect to N0 (x-axis) and η (y-axis) for:
(a) R0 =0, (b) R0 =0.3

4.2. Material uncertainty

Next, numerical results are given for the problem studied in Section 3 for a square nano-
beam of b = h = 1nm and length L = 10nm with p1 = 0.1N/m and N0 = 0. The nominal



Static and sensitivity analysis of nonlocal nanobeams... 353

(deterministic) value of Young’s modulus is taken as E0 = 1000GPa. The results in the follo-
wing figures are obtained by employing exact expression (3.4) for the mid-point deflection of
the nanobeam. In the figures, themid-point deflection is normalized by the height h by defining
w0 = w(L/2;Ẽ, η̃)/h.

Figure 6 shows the curves of mid-point deflection w0 plotted against the uncertainty level γ
for various values of the uncertain small-scale parameter η0. It is observed that the maximum
deflection increases with increasingmaterial uncertainty and the increase is given by a nonlinear
curve. The effect of the small-scale parameter η0 on themid-point deflection is shown in Fig. 7.
It is observed that the effect of uncertainty becomes more pronounced at higher values of the
small-scale parameter.

Fig. 6. Mid-point deflection vs. the uncertainty parameter γ for various values of η0

Fig. 7. Mid-point deflection vs. the small-scale parameter for various levels of uncertainty

Next, the sensitivity of the deflection to material properties is studied in Fig. 8 which
shows the contour plots of the mid-point deflection w0 with respect to the level of uncer-
tainty and the small-scale parameter. It is observed that the sensitivity of the deflection
with respect to Young’s modulus is about 5 times more than the sensitivity to the small-
-scale parameter. Moreover, the sensitivity with respect to Young’s modulus is not affected
much with respect to the small-scale parameter, but the sensitivity with respect to the small-
-scale parameter increases with growing η0.



354 I.S. Radebe, S. Adali

Fig. 8. Contour plots of sensitivities with respect to the level of uncertainties and small-scale parameter:
(a) SE, (b) Sη

5. Conclusions

Non-probabilistic analysis of the uncertaintieswhich canarise in transverse loads and inmaterial
properties of nanobeams is given using convex modeling. The variations in uncertain quantities
are takenasuncertain-but-boundedby imposingaconstraint on theL2normof theuncertainties.
Thenanobeam ismodeled as anonlocalEuler-Bernoulli beamand the effect of axial compression
is taken into account. The uncertain load is approximated by aFourier series expression and the
coefficients of the series are determined to obtain the worst-case uncertain loading. Closed form
solutions of the problems are given, and the theory is illustrated for simply supported boundary
conditions. It is observed that the increasinguncertainty asmanifestedby increasing theL2 norm
of the uncertain load leads to higher deflections. The effect of uncertainties in Young’s modulus
and the small-scale parameter is also studied and a sensitivity index is proposed to assess the
sensitivity of the deflection to these parameters.Numerical results are given to observe the effect
of various problem parameters on the deflection. The present study complements the studies in
the literature on the static deflection of nanobeams which have taken the loads acting on the
nanobeamsand its properties asdeterministic neglecting theuncertaintieswhich canoccurunder
operational conditions.

Acknowledgements

The research reported in this paperwas supportedby researchgrants fromtheUniversityofKwaZulu-

Natal (UKZN) and from National Research Foundation (NRF) of South Africa. The author gratefully

acknowledges the supports provided by UKZN andNRF.

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Manuscript received January 5, 2014; accepted for print October 20, 2014