Exact solution of Eringen's nonlocal integral model for bending of Euler–Bernoulli and Timoshenko beams

International Journal of Engineering Science 105 (2016) 80–92

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Exact solution of Eringen’s nonlocal integral model for bending of Euler–Bernoulli and Timoshenko beams Meral Tuna, Mesut Kirca∗ Istanbul Technical University, Faculty of Mechanical Engineering, Inonu Cad. No:65 34437, Gumussuyu, Beyoglu, Istanbul, Turkey

a r t i c l e

i n f o

Article history: Received 7 April 2016 Revised 2 May 2016 Accepted 2 May 2016

Keywords: Nonlocal elasticity Eringen integral model Laplace transform Beam theory Bending Exact solution

a b s t r a c t Despite its popularity, differential form of Eringen nonlocal model leads to some inconsistencies that have been demonstrated recently for the cantilever beams by showing the differences between the integral and differential forms of the nonlocal equation, which indicates the importance and necessity of using the original integral model. With this motivation, this paper aims to derive the closed-form analytical solutions of original integral model for static bending of Euler Bernoulli and Timoshenko beams, in a simple manner, for different loading and boundary conditions. For this purpose, the Fredholm type integral governing equations are transformed to Volterra integral equations of the second kind, and Laplace transformation is applied to the corresponding equations. The analytical expressions of the beam deflections which are obtained through the utilization of the proposed solution technique are validated against to those of other studies existing in literature. Furthermore, for all boundary and loading conditions, in contrast to the differential form, it is clearly established that the integral model predicts the softening effect of the nonlocal parameter as expected. In case of Timoshenko beam theory, an additional term that includes the nonlocal parameter is introduced. This extra term is related to the shear rigidity of the beam indicating that the nonlocal effect manifests itself via not only bending, but also shear deformation. © 2016 Elsevier Ltd. All rights reserved.

1. Introduction Local theory of elasticity becomes inadequate when internal (e.g. atomic or granular distance, relaxation time etc.) and external (e.g. crack length, wave length, period of load, application area of load etc.) characteristic length or time scales are comparable as appeared in several cases such as sharp crack tip propagation in fracture mechanics, wave propagation of composites under high-frequency excitations and mechanical behavior of nano and micro structures (Benvenuti & Simone, 2013; Eringen, 1974; Fernández-Sáez, Zaera, Loya, & Reddy, 2016; Pisano & Fuschi, 2003; etc.) where the nonlocal effects are much more dominant. In order to overcome this shortcoming, as well as to investigate the size effects, different theories have been developed. First attempts to address the nonlocal effects stand back to works of Cauchy and Voight (19th century) and Cosserat (20th century) (Fernández-Sáez et al., 2016). The gradient elasticity constitutive models were developed by Mindlin (1965), Toupin (1962) and Mindlin and Eshel (1968). Meanwhile, early formulations of nonlocal elastic constitutive equations which ∗

Corresponding author. E-mail addresses: [email protected] (M. Tuna), [email protected] (M. Kirca).

http://dx.doi.org/10.1016/j.ijengsci.2016.05.001 0020-7225/© 2016 Elsevier Ltd. All rights reserved.

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were introduced by Kröner (1967), Krumhansl (1968) and Kunin (1968) were improved further by Eringen (1966, 1972, 1983, 1987), Eringen and Edelen (1972). In addition to abovementioned approaches, molecular and atomistic theories (e.g. molecular dynamic simulations) have also been offered as effective methods, although huge amount of computational effort should be allocated for large systems (Ansari, Rouhi, & Mirnezdah, 2014; Nazemizadeh & Bakhtari-Nejad, 2015; etc.). Despite their significance, no further information about the advantages, capabilities, and applications of each theory will be given here since it is not the scope of the study. Some examples of other considerable works on the topics can be found in the articles of Benvenuti and Simone (2013), Pisano, Sofi, and Fuschi (2009), Li, Yao, Chen, and Li (2015), Eltaher, Khater, and Emam (2015) and Fernández-Sáez et al. (2016), etc. In the literature, one of the most widely used methods is Eringen’s nonlocal theory of linear elasticity which incorporates an internal length parameter into the constitutive equation to capture the microstructural effects. Theory exhibits a convolution format for constitutive relation where stress at each point is related to the strain of entire domain, through a kernel function that is inversely proportional to the distance between investigated and neighboring points. In addition to classical integral model (also known as fully nonlocal model), an alternative expression is proposed, which is known as twophase local/nonlocal model, where both local and nonlocal integral constitutive equations are included through a fraction coefficient that regulate their weights (Eringen, 2002). Furthermore, the spatial integrals encountered in the formulations of nonlocal theory can be converted to their equivalent differential form for specific types of kernel functions as indicated by Eringen (1983). Due to its simplicity, many studies utilize the differential form of Eringen model in order to investigate the bending, buckling, vibration and wave propogation behavior of structural elements such as; rods, tubes, beams, plates and shells (Anjomshoa, Shahidi, Hassani, & Jomehzadeh, 2014; Dansehmehr, Rajabpoor, & Hadi, 2015; Fotouhi, Firouz-Abadi, & Haddadpour, 2013; Hosseini-Hashemi et al., 2013; Hu, Liew, Wang, He, & Yakobson, 2008; Jalali, Jomehzadeh, & Pugno, 2016; Lu et al., 2007; Nejad Hadi & Rastgo, 2016; Phadikar & Pradhan, 2010; Rahmani & Pedram, 2014; Reddy, 2007; Reddy & Pang, 2008; Roque, Ferreira, & Reddy, 2011; Salehipour , Shahidi & Nahvi, 2015; Shaat, 2015; Wang & Liew, 2007; etc.). Before any further progress, it should be mentioned that; although a few pioneering studies are referenced here, there is an extensive literature about the field. In this regard, the readers are encouraged to check the articles of Arash and Wang (2012), Eltaher et al. (2015), Khodabakshi and Reddy (2015) and Fernández-Sáez et al. (2016) to access more publications. Despite the popularity of the differential form of Eringen model, in several studies focusing on the bending behavior of cantilevered beams insubstantial results are presented (Hu et al., 2008; Peddieson, Buchanan, & McNitt, 2003; Reddy & Pang, 2008; Challamel & Wang, 2008; Li et al., 2015; etc.). In this regard, it is reported that the cantilever beams subjected to concentrated forces are insensitive to nonlocal (small-scale) parameters while in the case of uniformly distributed load nonlocal effect manifests itself as a stiffening contribution which is a questionable outcome since a softening behavior is expected within the scope of the results obtained for other boundary conditions (i.e., Reddy & Pang, 2008). To overcome this deficiency, Challamel and Wang (2008) propose a new model that couples integral model and gradient model which is based on the combination of the local and nonlocal curvatures in the constitutive equation. Furthermore, Benvenuti and Simone (2013) point out that fully nonlocal model is unable to capture the nonlocal effects of a rod (e.g. constant strain field under constant tensile stress and inconsistent stress-strain relations in the case of distributed axial load). They recover the size effects by converting the two-phase integral formulation into a specific gradient elasticity formulation. In addition, there are also some other studies that do not utilize from the differential counterpart of Eringen integral equations. For instance, Polizzotto (2001) developes the Eringen nonlocal integral model by assuming an attenuation function depending on a geodetic distance, and accordingly derives the variational statements to obtain a nonlocal finite element formulation. In another study, Pisano and Fuschi (2003) convert the two-phase integral formulation into Volterra integral equations by utilizing from the symmetry property of a specific kernel function to examine the behavior of a bar under tension. Following the conversion, the exact solution of Volterra type integral equations are obtained by using the method of successive approximations by Neumann’s series. Furthermore, Khodabakhshi and Reddy (2015) provide a general finite element formulation for the local/nonlocal two-phase integral equations and investigate the behavior of Euler–Bernoulli beams under transverse loads. Although the deflection of a simply supported beam is not in good agreement with literature, the aforementioned inconsistency encountered for cantilever beams is suppressed. In their recent study, FernándezSáez et al. (2016) indicated that the solution of Eringen integral equation coincidences with the differential form of Eringen model if corresponding boundary conditions (see; Polyanin & Manzhirov, 2008) are satisfied, which is highlighted earlier by Benvenuti and Simone (2013). In the light of this information, Fernández-Sáez et al. (2016) propose a general method to solve the integral equation, and correct the paradoxical behavior encountered with cantilever beams. Results are compared with widely used differential Eringen model and it is concluded that differential form is unable to capture the nonlocal effects correctly. The present work is motivated by the fact that closed-form exact solution of Eringen integral model has not been developed so far, despite the inconsistent results obtained from differential form, so called counterpart of integral one. Therefore, the aim of this study is to derive the analytical expression of solution of Eringen integral model. For this purpose, the general three-dimensional equations are reduced to one-dimensional form to formulate the Euler Bernoulli and Timoshenko beams. Fredholm type integral equations are split into three parts that includes two Volterra integral equations of the second kind. Although the proposed method is valid for any kernel function that depends on the distance variable, it is taken similar to those of Fernández-Sáez et al. (2016), Pisano and Fuschi (2003), Reddy (2008), in order to make comparisons. The solution of integral equation is obtained by using Laplace transformation (Wazwaz, 2011). The non-dimensional analytical

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expressions of deflections are compared to those of Fernández-Sáez et al. (2016). In the present study, in addition to Euler Bernoulli beam theory, exact analytical solution of Eringen integral model is presented for Timoshenko beam structures in a very simple manner without using any constraints. The solution technique can be further applied to the other problems in the field. 2. Analysis 2.1. Eringen nonlocal theory According to Eringen nonlocal theory (Eringen, 1972, 1983), stress at a point is not only related to strain at the corresponding point, but also related to the strain of the entire domain via a kernel function, k(|x − x¯ |, κ );

σi j ( x ) =



V

k(|x − x¯ |, κ )ti j (x¯ )dV¯

(1)

where σ ij (x) is the nonlocal stress tensor at point x , and ti j (x¯ ) is the well-known classical (local) stress tensor at point x¯ . Local stress tensor is constitutively expressed in terms of elasticity tensor, and linear strain tensor as presented in Eq. 2.

ti j (x¯ ) = Ci jkl (x¯ )εkl (x¯ )

(2)

Any spatial kernel function that satisfies the mathematical and physical requirements mentioned in articles of Eringen (1983), Khodabakhshi and Reddy (2015), etc. can be used to describe the stress decaying. In this study, kernel function is taken as

k(|x − x¯ |, κ ) =

1 − |x−x¯ | e κ 2κ

(3)

where κ is a non-local parameter that depends on both internal length scale, a, and material constant, e0 , in the form κ = e0 a. 2.2. Governing equations of beam theories Governing equations of Euler–Bernoulli (EBT) and Timoshenko (TBT) beam theories are revisited. The coordinate system is chosen such that x, y and z axes correspond to length, width, and height of the beam, respectively. The displacement field components u1 , u2 , u3 are represented as functions of x and z coordinates, based on the beam theory. Regarding these information and taking into consideration only transverse loading conditions, nonzero components of the displacement field are written in terms of transverse deflection of the mid-plane, uz , and rotation angle of cross section φ y

u1 (x, z ) = zϕy (x ) u3 (x, z ) = uz (x )

(4)

Within the framework of Euler–Bernoulli beam theory, a further assumption is made, which states that straight lines normal to the mid-plane before deformation remain straight and normal to the mid-plane after deformation (Wang, Reddy, & Lee, 20 0 0) resulting in zero shear strain on the cross-section. Thus, the relation between rotation angle and transverse deflection can be simply written as follows:

ϕy ( x ) = −

d uz ( x ) dx

(5)

In TBT this normality assumption is dismissed, and a constant state of transverse shear strain with respect to z coordinate is included (Wang et al., 20 0 0). In the light of these information, kinematic relations are represented as E εxx = −z

d 2 uz ( x ) d x2

(6a)

and T εxx =z

d ϕy ( x ) dx

γxzT = ϕy (x ) +

(6b)

d uz ( x ) dx

(6c)

where superscripts E and T denotes to EBT and TBT, respectively. Internal loads on a cross section A, are given as follows



My = Fz =

A A

zσxx dA

τxz dA

(7)

where normal (σ ) and shear (τ ) stresses are written in terms of elasticity modulus, E, and shear modulus, G, respectively, based on Eqs. (1) and (2). Bending moment My , and transverse force Fz , which are determined from the equilibrium equations, vary depending on the loading and boundary conditions.

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2.3. Equations of nonlocal beam theories Considering nonlocal elasticity, stress resultants of a beam with length L can be defined as



 My =

z A

0



 Fz =

L

A



1 − |x−x¯| e κ E εxx dx¯ dA 2κ

(8a)



L

1 − |x−x¯| e κ Gγxz dx¯ dA 2κ

0

(8b)

Regarding kinematic relations, the abovementioned equations can be rewritten as

My = EI

L 0

Fz = Ks GA

1 − |x−x¯| dϕy (x¯ ) e κ dx¯ 2κ dx¯

L 0

1 − |x−x¯| e κ 2κ

(9a)





duz (x¯ ) dx¯ dx¯

ϕy (x¯ ) +

(9b)

  where the following relations; A z2 dA = I, A dA = A are used. I and Ks correspond to moment of inertia and shear correction factor, respectively. It is obvious that, for EBT, Eq. (9b) vanishes. 2.4. Proposed solution Eqs. (9a) and (9b) are known as Fredholm integral equations of the first kind. They are considered as ill-posed problems since they might not have a solution, and if a solution exists, it may not be unique (Wazwaz, 2011). Furthermore, it is mentioned that in the case of smooth kernel functions, small errors in the left side of equations induce very large changes in the solution, which is also known as another feature of ill-posed problems. In this regard, corresponding equations are transformed into Volterra integral equations in order to obtain unique solutions more easily. Primarily, the solution of Eq. (9a) is investigated. The corresponding equation is split into two parts similar to Polyanin and Manzhirov (2008):

My = EI

x 0

L

1 − (x−x¯) e κ f (x¯ )dx¯ + 2κ

1 − (x¯−x) e κ f (x¯ )dx¯ 2κ

x

(10)

where

d ϕy ( x ) dx

f (x ) =

(11)

Limit of integrals are adjusted in a way to obtain Volterra integral equations of the first kind:

My = EI

x 0

1 (x¯−x) e κ f (x¯ )dx¯ − 2κ

x 0

1 (x−x¯) e κ f (x¯ )dx¯ + 2κ

L 0

1 (x−x¯) e κ f (x¯ )dx¯ 2κ

(12)

Since kernel function used in the study depends on the difference of variables x and x¯, Laplace transform method is utilized to solve the Eq. (12) (Wazwaz, 2011). This kind of kernel function allows the use of convolution theorem as shown below



L

x e

(x¯−x ) κ

−e

(x−x¯ ) κ





f (x¯ )dx¯



x

= L e− κ L{ f (x )} − L e κ L{ f (x )} = x

0

F (s ) s + κ1

−

F (s )

(13)

s − κ1

where L{} refer to Laplace operator, and

F (s ) = L{ f (x )}

(14)

Utilizing from Leibniz rule, the Laplace transform of the last term of Eq. (12) can be expressed as





L

L

e 0

(x−x¯ ) κ

f (x¯ )dx¯

∞ = 0



e−sx



L e 0

(x−x¯ ) κ

f (x¯ )dx¯ dx =

L 0



x¯

e− κ f (x¯ )

∞ 0



x

e−sx e κ dx dx¯ =

A s − κ1

(15)

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where the following relations are used.

∞

x

x

e−sx e κ dx = L e κ

=

0

L A=

1

(16)

s − κ1

x¯

e− κ f (x¯ )dx¯

(17)

0

It is important to mention that A is a constant since the bounds of integration is from 0 to L. After the function f(x), which depends on A, is derived, it is straight forward to obtain this constant using the equation above. Considering Eqs. (13) and (15), the Laplace transform of Eq. (12) is given as below

L {M y } 1 = EI 2κ



F ( s )κ F ( s )κ − + κs + 1 κs − 1

Aκ κs − 1



(18)

By taking the inverse transform of Eq. (18), the function f(x) is derived in terms of unknown constant, A, as follows.



f (x ) = L−1 {F (s )} = L−1

(κ s + 1 ) (κ s − 1 ) 2





2 A − L {M y } (κ s − 1 ) EI

(19)

The constant A is determined by using the Eq. (17), as mentioned before. Rotation angle of the cross section, φ y (x), is obtained via the relation that is given in Eq. (11). For Euler–Bernoulli beams; Eq. (5) is utilized to derive the deflection of beam axis, uz (x). In the case of Timoshenko beam, same procedure has to be applied to Eq. (9b), by considering the expression of rotation angle that was determined previously.

 Fz 1 = Ks GA 2κ L{Fz } 1 = Ks GA 2κ

L

|x−x¯| e− κ

φy (x¯ )dx¯ +

0

 

x e 0

L

L 0

 |x−x¯| e− κ

φy (x¯ )dx¯

(x¯−x ) κ

g(x¯ )dx¯ −

x e

(x−x¯ ) κ

g(x¯ )dx¯ +

0

G ( s )κ G ( s )κ + − + κs + 1 κs − 1



L e

(x−x¯ ) κ

g(x¯ )dx¯

(20)

0

Bκ κs − 1

 (21)

where

g( x ) =

d uz ( x ) dx

G (s ) = L{g(x )} L B=

x¯

e− κ g(x¯ )dx¯

(22)

(23)

(24)

0

The function g(x), and the constant B are determined in a similar way with resolving the function f(x) and the constant A. Integration constants obtained from Eqs. (5), (11) and (22) are established from the specified boundary conditions. 2.5. Examples In this section, considering several loading and boundary conditions, analytical expressions for the deflection of the Euler–Bernoulli and Timoshenko beams are derived by using the presented solution technique. The accuracy of the proposed method is examined via comparing the results to those of Fernández-Sáez et al. (2016) which shows the discrepancy between the results of differential and integral form of Eringen equation and solves the elastostatic bending problem of Euler–Bernoulli beams using Eringen integral constitutive equation. Non-dimensional deflections are defined as uz (x)/(qL4 /EI) and uz (x)/(FL3 /EI), depending on the loading condition. It is assumed that, beam has a circular cross section with diameter d. The method is tested for different values of nonlocal parameter (κ : 0.005 L–0.05 L [m]) as well as diameter of cross-section (d: L/10–L/50 [m]).

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2.5.1. Simply supported beam subjected to uniform distributed load For a simply supported beam that is subjected to a uniformly distributed load, q, essential boundary conditions are given below:

uz (x = 0 ) = 0 and uz (x = L ) = 0 According to natural boundary conditions, bending moment and transverse force are represented as

My = −

qx2 qLx + 2 2

(25)

qL 2

(26)

Fz = −qx +

By substituting the bending moment into Eq. (19), and using the relations mentioned in Eqs. (11) and (5) respectively, deflection of the beam axis for EBT is expressed as below.

uEz





 qκ L3 qκ 2  + H (x )Lx − x2 + 2EI 24EI



δ ( 0 )x κδ (0 ) + 2H (0 )   qκ 2 H 2 (0 )(κ L − κ x ) + H (0 )H (x )(−κ L − Lx ) + 2EI κδ (0 ) + 2H (0 )    q xH (0 ) L3 3 2 + κ +κ L+ 2EI κδ (0 ) + 2H (0 ) 6   L e− κ qκ 2 H (0 )(−κ x + κ L ) + H (x )(−κ L − Lx ) + κ x + Lx + 2EI κδ (0 ) + 2H (0 )

q (x ) = EI

Lx3 x4 − 24 12



(27)

where, δ (x) and H(x) are Dirac delta, and Heaviside step functions, respectively. Utilizing the Dirac delta function, which is zero everywhere except at the origin (x = 0 ), where it is infinite, the third term in the Eq. (27) is simplified as follows:

qκ L3 24EI



δ ( 0 )x κδ (0 ) + 2H (0 )



≡

qL3 x 24EI



since limδ (0)→∞

 x δ ( 0 )x = κδ (0 ) + 2H (0 ) κ

(28)

In addition to this, exclusion of the last term does not alter the result either, due to the argument of exponential function (L/κ : 20–200). Furthermore, remaining terms including the expression κδ (0 ) + 2H (0 ) as their denominator vanish since the division of a finite number to an infinite number is equal to zero. Thus, considering these simplifications, the transverse deflection can be rewritten as

uEz (x ) =

q EI



Lx3 x4 L3 x − + 24 12 24



+

 qκ 2  2 −x + Lx 2EI

f or

x>0

(29)

Likewise, the simplified version of the deflection for TBT is given by Eq. (30).

uTz

q (x ) = EI



Lx3 x4 L3 x − + 24 12 24



+

  qκ 2  2 q  2 qκ −x + Lx + −x + Lx + ( 2κ + L ) 2EI 2GA Ks 2GA Ks

f or

x>0

(30)

It should be mentioned that, for the sake of simplicity, the expressions are given for x > 0 since the deflection is already prescribed at x = 0, and Heaviside step function equals to 1 for values greater than zero. Eq. (29) is identical with the solution of the differential form of Eringen nonlocal equation which is able to capture the softening effect of the nonlocal parameter (Reddy, 2007; Reddy & Pang, 2008; Wang & Liew, 2007; etc.). It should be noted that Fernández-Sáez et al. (2016) encountered a relatively small discrepancy between the results of integral and differential form of constitutive equation at x = L/4 (see Fig. 2 in the corresponding study), conversely our findings overlap with the differential form in the case of EBT. The results of non-dimensional deflection at mid-span of the EBT are compared with those of Fernández-Sáez et al. (2016) (see Fig. 1a.). Similar results are observed for Timoshenko beams where beam stiffness decreases with increasing the nonlocal parameter. The non-dimensional deflection at x = L/2 of Timoshenko beam with different aspect ratios can be found in Fig. 2. As seen from the figure, the results converge to EBT for increasing slenderness, as expected. For different nonlocal parameters, the numerical values of non-dimensional deflections for both EBT and TBT are given in Table 1. 2.5.2. Cantilever beam subjected to point load For a beam that is clamped at x = 0 and subjected to a point load, F, at its free end, i.e., x = L, essential boundary conditions are given as

uz (x = 0 ) = 0 and φ (x = 0 ) = 0

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M. Tuna, M. Kirca / International Journal of Engineering Science 105 (2016) 80–92 0.155

0.01335 0.0133

0.15

0.01325

u− z(L/2)

Proposed Method

0.01315

u− z(L)

0.145 0.0132

0.14

Proposed Method Fernández-Sáez et al., 2016

Fernández-Sáez et al., 2016 0.135 0.0131 0.13

0.01305

0.125

0.013 0

0.01

0.02

0.03

0.04

0

0.05

(a) Simply supported beam subjected to uniform distributed load

0.01

0.02

0.03

0.04

0.05

(c) Cantilever beam subjected to uniform distributed load 0.0068

0.39

0.0066 0.38 0.0064 0.37

0.36

Proposed Method Fernández-Sáez et al., 2016

u− z(L/2)

u− z(L)

0.0062 0.006

Proposed Method Fernández-Sáez et al., 2016

0.0058

0.35 0.0056 0.34 0.0054 0.33

0.0052 0

0.01

0.02

0.03

0.04

0.05

0

(b) Cantilever beam subjected to point load

0.01

0.02

0.03

0.04

0.05

(d) Fixed-pinned beam subjected to uniform distributed load

Fig. 1. Comparison of non-dimensional deflection for Euler–Bernoulli beam. (a) Simply supported beam subjected to uniform distributed load. (b) Cantilever beam subjected to point load. (c) Cantilever beam subjected to uniform distributed load. (d) Fixed-pinned beam subjected to uniform distributed load.

0.0136

0.0135

0.0134 EBT 0.0133

TBT L/d=10 TBT L/d=15 TBT L/d=20

0.0132

TBT L/d=50 0.0131

0.013 0

0.01

0.02

0.03

0.04

0.05

Fig. 2. Non-dimensional deflection of simply supported beam subjected to uniform distributed load.

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87

Table 1 Comparison of non-dimensional deflection of the middle section (u¯ z (L/2 ) = uz (L/2 ) x EI/qL4 ) in simply supported beams subjected to a uniformly distributed load q (E = 10 0 0 GPa, G = 420 GPa, Ks = 0.877).

κ /L

EBT

TBT (L/d = 10)

TBT (L/d = 15)

TBT (L/d = 20)

TBT (L/d = 50)

0.0 0 0 0.005 0.010 0.015 0.020 0.025 0.030 0.035 0.040 0.045 0.050

0.0130208 0.0130239 0.0130333 0.0130489 0.0130708 0.0130989 0.0131333 0.0131739 0.0132208 0.0132739 0.0133333

0.0132329 0.0132403 0.0132541 0.0132742 0.0133006 0.0133333 0.0133724 0.0134178 0.0134696 0.0135277 0.0135921

0.0131151 0.0131201 0.0131314 0.0131491 0.0131729 0.0132031 0.0132396 0.0132823 0.0133314 0.0133867 0.0134483

0.0130739 0.0130781 0.0130885 0.0131053 0.0131283 0.0131576 0.0131931 0.0132349 0.0132830 0.0133374 0.0133980

0.0130293 0.0130326 0.0130422 0.0130580 0.0130800 0.0131083 0.0131429 0.0131837 0.0132308 0.0132841 0.0133437

Based on natural boundary conditions, bending moment and transverse force are obtained as

My = F ( x − L )

(31)

Fz = F

(32)

Consequently, the deflection is derived as;



uEz

x3 F Lx2 (x ) = − + EI 6 2

+

Fκ2 F κ 2 δ (0 )H (x )(κ L + Lx ) H (x )(−L + x ) + EI EI κδ (0 ) + 2H (0 )

   κ 2 + κ L − κ x − Lx + H (0 )H (x ) −κ 2 + κ L − κ x + Lx κδ (0 ) + 2H (0 )   L F e− κ κ 2 (κ + x )(H (x ) − H (0 )) − κ xδ (0 ) + EI κδ (0 ) + 2H (0 ) Fκ + EI





H 2 (0 )



where the third term can be rewritten as; EBT and TBT become as follows



uEz

x3 F Lx2 (x ) = − + EI 6 2

uTz

x3 F Lx2 (x ) = − + EI 6 2





F κ H (x )(κ L+Lx) EI

considering Eq. (28). Regarding those simplifications, deflection of

+

Fκ (κ x + Lx) EI

+

Fκ Fx Fκ + (κ x + Lx) + EI GA Ks GA Ks



f or

(33)

x>0

(34)

f or

x>0

(35)

As it can be seen from Eqs. (34) and (35), the derived formulation is able capture the softening effect of nonlocal parameter, on the contrary to the solution of the differential form of Eringen equation, where nonlocal parameter does not contribute to deflections at all (Peddieson et al., 2003; Wang & Liew, 2007; etc.). Comparative results of non-dimensional deflection of EBT at x = L can be found in Fig. 1b. The results overlap with corresponding points given in FernándezSáez et al. (2016) (Fig. 1b). Additional graphs of non-dimensional deflections at the free end of Timoshenko beams with different length to the diameter ratios are plotted against non-dimensional nonlocal parameter at Fig. 3. Table 2 shows the numerical values of non-dimensional deflection of EBT and TBT for different values of nonlocal parameter. 2.5.3. Cantilever beam subjected to uniform distributed load For a cantilever beam that is subjected to a uniformly distributed load; expressions for the essential boundary conditions, bending moment, and transverse force variations are given below.

uz (x = 0 ) = 0 and φ (x = 0 ) = 0 My = −

qx2 qL2 + qLx − 2 2

Fz = −qx + qL

(36)

(37)

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M. Tuna, M. Kirca / International Journal of Engineering Science 105 (2016) 80–92

0.39

0.38

0.37 EBT 0.36

TBT L/d=10 TBT L/d=15 TBT L/d=20

0.35

TBT L/d=50 0.34

0.33 0

0.01

0.02

0.03

0.04

0.05

Fig. 3. Non-dimensional deflection of cantilever beam subjected to point load. Table 2 Comparison of non-dimensional deflection of end section (u¯ z (L ) = uz (L ) x EI/F L3 ) in cantilever beams subjected to a point load, F, at free end (E = 10 0 0 GPa, G = 420 GPa, Ks = 0.877).

κ /L

EBT

TBT (L/d = 10)

TBT (L/d = 15)

TBT (L/d = 20)

TBT (L/d = 50)

0.0 0 0 0.005 0.010 0.015 0.020 0.025 0.030 0.035 0.040 0.045 0.050

0.3333333 0.3383583 0.3434333 0.3485583 0.3537333 0.3589583 0.3642333 0.3695583 0.3749333 0.3803583 0.3858333

0.3350300 0.3400636 0.3451471 0.3502806 0.3554641 0.3606976 0.3659810 0.3713145 0.3766980 0.3821315 0.3876150

0.3340870 0.3391162 0.3441950 0.3493238 0.3545026 0.3597313 0.3650101 0.3703389 0.3757176 0.3811464 0.3866252

0.3337580 0.3387847 0.3438618 0.3489889 0.3541660 0.3593931 0.3646703 0.3699974 0.3753745 0.3808016 0.3862787

0.3334010 0.3384266 0.3435019 0.3486272 0.3538026 0.3590279 0.3643032 0.3696286 0.3750039 0.3804293 0.3859046

Similarly, analytical expression of deflection is obtained as

uEz (x ) =

q EI +



x4 L2 x2 Lx3 − + 24 6 4







+



⎡

2

−



qκ δ (0 )H (x ) κ L + κ L x 2EI κδ (0 ) + 2H (0 ) 2 2



qκ κ L2 H (x ) − + κ Lx EI 2

κ x2



2

 ⎤



L2 x H 2 (0 ) κ L2 κ 2L + − κ Lx − +⎥ κδ ( 0 ) + 2 H ( 0 ) 2 2 qκ ⎢ ⎢ ⎥   + EI ⎣ H (0 )H (x ) κ L2 L2 x ⎦ 2 −κ L + − κ Lx + κδ (0 ) + 2H (0 ) 2 2

(38)

L +κ L x ) By replacing the third term with the expression qH (x )(κ2EI , and neglecting the last term due to the infinite-value expression located in the denominator, the above expression is simplified to 2 2

uEz

q (x ) = EI



Lx3 x4 L2 x2 − + 24 6 4



+

2

 qκ  −κ x2 + 2κ Lx + L2 x 2EI

f or

x>0

(39)

M. Tuna, M. Kirca / International Journal of Engineering Science 105 (2016) 80–92

89

0.153

0.148

0.143 EBT 0.138

TBT L/d=10 TBT L/d=15 TBT L/d=20

0.133

TBT L/d=50 0.128

0.123 0

0.01

0.02

0.03

0.04

0.05

Fig. 4. Non-dimensional deflection of cantilever beam subjected to uniform distributed load. Table 3 Comparison of non-dimensional deflection of end section (u¯ z (L ) = uz (L ) x EI/qL4 ) in cantilever beams subjected to a uniformly distributed load (E = 10 0 0 GPa, G = 420 GPa, Ks = 0.877).

κ /L

EBT

TBT (L/d = 10)

TBT (L/d = 15)

TBT (L/d = 20)

TBT (L/d = 50)

0.0 0 0 0.005 0.010 0.015 0.020 0.025 0.030 0.035 0.040 0.045 0.050

0.1250 0 0 0 0.1275125 0.130 050 0 0.1326125 0.13520 0 0 0.1378125 0.1404500 0.1431125 0.14580 0 0 0.1485125 0.1512500

0.12584800 0.12836943 0.13091554 0.13348674 0.13608302 0.13870438 0.14135083 0.14402237 0.14671899 0.14944069 0.15218748

0.12537700 0.12789336 0.13043468 0.13300105 0.13559245 0.13820889 0.14085037 0.14351689 0.14620844 0.14892503 0.15166666

0.12521200 0.12772673 0.13026639 0.13283106 0.13542075 0.13803547 0.14067521 0.14333997 0.14602975 0.14874455 0.15148437

0.12503400 0.12754678 0.13008462 0.13264747 0.13523532 0.13784818 0.14048603 0.14314890 0.14583676 0.14854963 0.15128750

Similarly, for TBT, the corresponding expression is written as

uTz (x ) = +

q EI



Lx3 x4 L2 x2 − + 24 6 4



+

 qκ  −κ x2 + 2κ Lx + L2 x 2EI

 q  2 qκ −x + 2Lx + (κ + L ) 2GA Ks GA Ks

f or

x>0

(40)

According to Eqs. (39) and (40), the flexibility of the beam increases with nonlocal parameter, while the results obtained from the differential form of Eringen model exhibits an increase in the bending stiffness (Reddy & Pang, 2008; Wang & Liew, 2007; etc.), conversely. As it can be noticed from Fig. 1c, the non-dimensional deflections of EBT at x = L coincide with the results of FernándezSáez et al. (2016). In addition, the effect of nonlocal parameter on the deflection of Timoshenko beams is shown in Fig. 4. Readers can access to the numerical values of non-dimensional deflections of EBT and TBT from Table 3. 2.5.4. Fixed-pinned beam subjected to uniform distributed load For a fixed-pinned beam that is subjected to a uniformly distributed load; boundary conditions are given below.

uz ( x = 0 ) = 0,

ϕ (x = 0 ) = 0 and uz (x = L ) = 0

90

M. Tuna, M. Kirca / International Journal of Engineering Science 105 (2016) 80–92 Table 4 Comparison of non-dimensional deflection of middle section (u¯ z (L/2 ) = uz (L/2 ) x EI/qL4 ) in fixed-pinned beams subjected to a uniformly distributed load (E = 10 0 0 GPa, G = 420 GPa, Ks = 0.877).

κ /L

EBT

TBT (L/d = 10)

TBT (L/d = 15)

TBT (L/d = 20)

TBT (L/d = 50)

0.0 0 0 0.005 0.010 0.015 0.020 0.025 0.030 0.035 0.040 0.045 0.050

0.0052083 0.0053275 0.0054506 0.0055777 0.0057089 0.0058442 0.0059836 0.0061273 0.0062752 0.0064273 0.0065839

0.0 05460 0 0.0055816 0.0057073 0.0058369 0.0059707 0.0061087 0.0062508 0.0063973 0.0065479 0.0067030 0.0068624

0.0053202 0.0054405 0.0055647 0.0056930 0.0058253 0.0059618 0.0061024 0.0062473 0.0063964 0.0065499 0.0067077

0.0052713 0.0053911 0.0055148 0.0056426 0.0057744 0.0059103 0.0060505 0.0061948 0.0063434 0.0064963 0.0066535

0.0052184 0.0053377 0.0054609 0.0055881 0.0057194 0.0058548 0.0059943 0.0061381 0.0062861 0.0064384 0.0065950

0.0069 0.0067 0.0065 0.0063 EBT

0.0061

TBT L/d=10 0.0059

TBT L/d=15

0.0057

TBT L/d=20 TBT L/d=50

0.0055 0.0053 0.0051 0

0.01

0.02

0.03

0.04

0.05

Fig. 5. Non-dimensional deflection of fixed-pinned beam subjected to uniform distributed load.

Bending moment and transverse force variations are written in terms of constant Fz0 that will be determined from the essential boundary conditions.

My = −

qx2 qL2 + + F z0 x − F z0 L 2 2

Fz = −qx + F z0

(41)

(42)

Expression for the deflection of fixed-pinned beam is derived by the same procedure. But, for convenience, it is not presented here explicitly due to its complexity while the numerical results can be found in the Table 4, Figs. 1d and 5. As seen from Table 4, the bending stiffness of the beam decreases with increasing nonlocal parameter. The nondimensional deflections of EBT are consistent with the work of Fernández-Sáez et al. (2016) (Fig. 1d), which emphasizes that; although both formulations (results of integral and differential form) capture the softening behavior, the effect of nonlocal parameter is even much more evident in the case of integral form. The non-dimensional deflections of TBT at x = L/2 are given in Fig. 5.

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It is important to mention that, for all cases the solutions, f(x) and g(x) are verified simply by substituting the expressions into the governing equations (i.e., Eqs. (9a) and (9b)). It should be also noted that, the expressions of the deflections are identical to those of the local theory when the nonlocal parameter is set to zero. 3. Conclusion In this paper, the exact solution of the integral form of Eringen nonlocal equation is established for static bending of Euler–Bernoulli and Timoshenko beams, and analytical expressions for the deflections of the beam axis are derived for different boundary and loading conditions. As far as authors know, this is the first time that exact analytical expressions for the bending deflections of beams are obtained through the original integral Eringen constitutive equation, while there is an extensive literature about its differential counterpart. Due to differences between integral and differential Eringen models which have been shown by Fernández-Sáez et al. (2016), recently, the derivation of a closed-form solution of the integral model is thought to be very significant. Within the scope of the proposed solution technique which is valid for any kernel function that depends on the distance variable, the Fredholm type integral governing equations are transformed to Volterra integral equations of second kind by simply adjusting the limit of integrals, where a unique solution is obtained by using the Laplace transform. For the purpose of validating the proposed technique, non-dimensional deflections of Euler–Bernoulli beam under varied conditions is compared with those of Fernández-Sáez et al. (2016), which yields consistent results on the performance of predicting the increase in the flexibility of the cantilever beam with nonlocal parameters. 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