The Unified Solution for a Beam of Rectangular Cross-Section with Different Higher-Order Shear Deformation Models

The unified solution is studied for a beam of rectangular cross section. With the rotation defined in the average sense over the cross section, the kinematics with higher-order shear deformation models in axial displacement is first expressed in a unified form by using the fundamental higher-order term with some properties. The shear correction factor is then derived and discussed for the four commonly used higher-order shear deformation models including the third-order model, the sine model, the hyperbolic sine model and the exponential model. The unified solution is finally obtained for a beam subjected to an arbitrarily distributed load. The relation with that from the conventional beam theory is established, and therefore the difference is reasonably explained. A very good agreement with the elasticity theory validates the pre-sent solution.


Latin American Journal of Solids and
and after deformation and hence the shear deformation of cross section is not taken into account.In contrast, the Timoshenko beam theory (TBT) (Timoshenko, 1922(Timoshenko, , 1921) permits a uniform shear deformation of cross section with a shear correction factor.
Compared with the TBT, HSD models can not only reflect the warping of cross section, but give the shear correction factor in a straightforward manner.However, the TBT is amazing in evaluating deflection due to its sophisticated physics in the shear correction factor.In virtue of this, much attention was paid to evaluating the shear correction factor (e.g.Cowper, 1966;Dong et al., 2013Dong et al., , 2010;;Gruttmann and Wagner, 2001;Gruttmann et al., 1999;Hutchinson, 2001;Jensen, 1983;Kaneko, 1975;Pai and Schulz, 1999).
In recent years, HSD models have been widely used in composite structures (Aydogdu, 2009;Karama et al., 2003;Mantari et al., 2012;Viola, et al., 2013) and functionally graded (FG) structures (El Meiche et al., 2011;Simsek, 2010).Much attention was also paid to real world applications by using beam theories.For example, Wang et al. (2008) studied the bending problems of micro-and nano-beams based on the Eringen nonlocal elasticity theory and the TBT, and found that the shear deformation and small-scale effect are significant in these problems.In addition, HSD models have been adopted to study small-scale structures (Akgöz and Civalek, 2015, 2014a, 2014b;Challamel, 2013Challamel, , 2011;;Wang et al., 2014) in nano-or micro-electro mechanical systems (NEMS or MEMS).Structures 13 (2016) 1716-1737 Though there have been massive researches on beam problems, some confusions are still pending.For example, what are the proper quantities in characterizing governing equations and boundary conditions of a beam problem?Are there any connections among the existing beam theories, HSD models, or shear correction factors?These are just the issues to touch on in the present work.To this end, the paper is outlined as follows.In Section 2, with the rotation defined in the average sense over the cross section, the beam theory is summarized, including derivation of the governing equations and the boundary conditions.In Section 3, the HSD model is expressed in a unified form for the kinematics in axial displacement by introducing the fundamental higher-order term with some properties.Four commonly used HSD models, viz. the third-order model, the sine model, the hyperbolic sine model and the exponential model, are then studied in detail, and the shear correction factors are finally calculated.In Section 4, the unified solution is derived and discussed.The concluding remarks are made in Section 5.For a straight beam of rectangular cross-section, the coordinate system is shown in Figure 1.In the present work, according to the quantities in the plane elasticity theory, moment M, shear force Q, deflection w and rotation ϕ for a beam are respectively defined as

FUNDAMENTALS OF THE BEAM THEORY
where σx , σy and τxy are the stress components in the x-y plane.u(x, y) and v(x, y) are respectively the axial and transverse displacements in this plane.A and I are respectively the area and the moment of inertia of cross section.It is easily seen that definitions for M, Q and w are the same as often while the definition for rotation is somewhat different.
In this context, in parallel to Eqs. (1), the beam theory will be also established from the plane elasticity theory.Ignoring body forces, the governing equations in the plane elasticity theory are Taking the end of x=0 as an example, the corresponding boundary conditions are in the direction in the direction (0, ) given or (0, ) given (0, ) given or (0, ) given (3) In the x-direction, integration of the first of Eqs. ( 2) and (3) weighted by y over the cross section yields and In a similar manner, in the y-direction, integrating the second of Eqs. ( 2) and (3) over the cross section, we have and Considering the first two of Eqs.(1), Eqs. ( 4) and (6) yield In this context, the prime denotes the differentiation with respect to x.
Considering the definitions in Eqs.(1), Eqs. ( 5) and ( 7) yield Latin American Journal of Solids and Structures 13 (2016) 1716-1737 in the direction in the direction 0 given or 0 given 0 given or 0 given With Eqs. ( 10), all the practical boundary conditions in the x-direction and/or in the y-direction can be prescribed.For instance, we have Free condition : and Simply supported condition : and Fixed condition : and 0 given 0 given 0 given 0 0 It should be noted that, due to the definitions in Eqs.(1), Eqs. ( 11) can also serve as the boundary conditions in the plane elasticity theory.
For a beam structure, the fundamental assumption is that σy<<τxy <<σx (e.g.see Ghugal and Sharma, 2011), so, from the first of Eqs.(1), in terms of ϕ, the moment can be further expressed as

The Unified Higher-Order Shear Deformation Model
In the second of Eqs. ( 13), Q must be expressed in terms of w(x) and/or ϕ(x).To this end, first, as often, the transverse displacement is assumed to be independent of the thickness coordinate y, and hence we have Next, the axial displacement is assumed to be expressed by where y and g(y) are respectively called the first-order term and the fundamental higher-order term with respect to y while f1(x) and fR(x) are the corresponding coefficient functions with respect to x.
It is worth noticing that g(y) is a pure higher-order term in this context.Substituting Eq. ( 15) into the fourth of Eqs.(1) yields where Latin American Journal of Solids and Structures 13 (2016) 1716-1737   1 ( ) If we take fR(x)≡0, together with Eq. ( 16), Eq. ( 15) reduces to the first-order shear deformation model as For a higher-order model, upon eliminating fR(x) via Eq.( 16), Eq. ( 15) becomes Thus, considering Eq. ( 14), the shear strain over the cross section is ) For a beam of rectangular cross section, the shear stress free condition is often adopted on the top and bottom surfaces, which requires So, from Eq. ( 20), we have where Eventually, the axial displacement in Eq. ( 15) becomes Eq. ( 24) is termed as the unified HSD model for a beam corresponding to the fundamental higher-order term g(y).
Thus far, from Eqs. ( 23) and considering the higher-order property, g(y) has the following properties In terms of w(x) and ϕ(x), the shear force Q(x) is expressed as Latin American Journal of Solids and Structures 13 (2016) 1716-1737   ( ) where KP, the shear correction factor originally defined by Timoshenko (1922Timoshenko ( ,1921)), takes the form of It can be seen from Eq. ( 27) that the shear correction factor will be certainly determined as long as g(y) is known, rather than other procedures (e.g.Cowper, 1966;Hutchinson, 2001).
If we assume Eq. ( 26) yields which implies that the shear effect cannot be not taken into account.
Accordingly, Eq. ( 24) reduces to which is the axial displacement assumption in the EBT (Dym and Shames,1973).Compared with the first-order shear deformation model in Eq. ( 18), the EBT in Eq. ( 30) is obtained under the additional assumption of Eq. ( 28), which is a more reasonable explanation than that of an infinite shear rigidity (e.g.Challamel, 2013).

Four Commonly Used HSD Models
One direct choice of the fundamental higher-order term is to take g A (y)=y 3 .This model is just the commonly used third-order model and denoted as Model-A in this context.Eq. ( 27) yields 5 6 In this paper, other three commonly used HSD models are studied as well.

The Exponential Model-Model-D
Inspired by Karama et al. (2003) and considering Eq. ( 25), as the exponential model, the fundamental higher-order term is taken to be Accordingly, from Eqs. ( 17) and ( 23), we have Latin American Journal of Solids and Structures 13 (2016) 1716-1737 where the Gauss error function is defined as So, the axial displacement in Eq. ( 24) yields Further, Eq. ( 27) yields It is interesting to study the difference of the four HSD models by comparing the first two leading terms through Taylor's expansion.
For the third-order model, the axial displacement is Thus, if the axial displacement is expanded in form of power series from Taylor's expansion, we have The coefficients are summarized in Table 1.It is seen that c3 increases with c1.

Model type c1 c3
Model

Shear Correction Factors
As already indicated in Section 3.1, given g(y), the shear correction factor can be evaluated for the beam of rectangular cross section.Values of the shear correction factor for the four HSD models in Section 3.2 are summarized in From Eqs. ( 31) and ( 35), we can see that the commonly used shear correction factors can be reasonably explained by Eq. ( 27) in the manner of HSD models.For example, the TBT with KP =5/6 (e.g.Kaneko, 1975;Timoshenko, 1922Timoshenko, , 1921;;Murthy, 1981;Reissner, 1975;Shi, 2007Shi, , 2011) ) is in essential equivalent to the third-order model (i.e.Model-A) because of Eq. ( 31) while the Mindlin plate theory (Mindlin, 1951) with KP =π 2 /12 is in essential equivalent to the sine model (i.e.Model-B) because of Eq. ( 35).

Derivation of the Unified Solution
In this section, the unified solution will be derived for the unified HSD model in Section 3. To this end, the governing equations are re-arranged as follows It is interesting to discuss the procedure of solving Eqs.(48).A straightforward procedure is to turn Eqs.(48) into the form Latin American Journal of Solids and Structures 13 (2016) 1716-1737 From the first of Eqs. ( 49) -a third-order differential equation, the rotation ϕ can be first obtained, and then, from the second of Eqs. ( 49) -a first-order differential equation, the deflection w can be further obtained.
In this context, a more general approach is used instead.To this end, Eqs. ( 48) are re-expressed as     (4) ( ) From the first of Eqs.(50), w is firstly obtained by solving a fourth-order differential equation, and ϕ is then directly obtained from the second of Eqs.(50).Eventually, we have For the problem shown in Figure 2, the boundary conditions are

Comparison of the Results
With the unified solution, the results can be comparatively studied in detail.For this purpose, the special case of q=const is considered.From Eqs. ( 51) and (53), it is not difficult to obtain the unified solution to this case as On the other hand, applying the boundary conditions in Eqs. ( 11) to the problem shown in Figure 2 with q=const, the elasticity solution to this plane stress problem is obtained as Latin American Journal of Solids and Structures 13 (2016) 1716-1737 The deflections from different solutions are plotted in Figure 3 for the four HSD models in which "Reference", "Conventional" and "Present" denote the solution from Eqs. ( 55), (A.27) and ( 54), respectively.It is seen that the current solution can always agree better with the reference for all the four HSD models than the conventional solution.
With Eqs. ( 51), the axial displacement (i.e. the warping of the cross section) can also be obtained through Eq. ( 24).The variation with y at x=L is plotted in Figure 4.It is seen that the warping from all the four HSD models are in considerably good agreement with the reference for the present solution.However, due to the apparent difference between KP and KT (see Table 2 and  Table A.1

CONCLUSIONS AND FUTURE WORK
In this paper, with the definitions of average deflection and average rotation, the governing equations and the boundary conditions were derived from the plane elasticity theory.Based on the kinematics in axial displacement, the unified HSD model was proposed by using the fundamental higher-order term with some properties.The shear correction factor was then derived.The unified solution was finally obtained for the unified HSD model subjected to an arbitrarily distributed load, and compared with the conventional one.From the current work, following conclusions can be made.
1) The definition of rotation plays an important role for the HSD models.
2) The kinematics of HSD models in axial displacement can be expressed by using the fundamental higher-order term with some properties.3) Given a loading, the unified solution can be derived for a beam with different fundamental higher-order terms in axial displacement.4) Given the fundamental higher-order term, the shear correction factors can be reasonably obtained.
Latin American Journal of Solids and Structures 13 (2016) 1716-1737 The beam theory used in the current work is a lower-order (fourth-order) one in terms of deflection w.Future work will focus on a variationally consistent higher-order (sixth-order) beam theory and the shear correction factor of arbitrary cross section.Application to small scale problems and extension to a plate problem are also prospective.

Free condition : and
Simply supported condition : and 0 Fixed condition : 0 and 0

A.4 The Unified Solution
For sake of the unified solution, Eqs.(A. 15) are further re-expressed as dx dx q x dx dx q x dx q x dx EI EI EI L dx dx dx q x dx dx dx q x dx I E I L L dx q x dx q x dx EI EI L dx q x dx L q x dx q x dx K AG K AG K AG For the case of q=const, the unified solution to the unified HSD model is Int erestingly, Eqs.(A.27) will turn into Eqs.(54) if αT=1 (hence KT=KP from Eq. (A.13)).

Figure 1 :
Figure 1: A straight beam of rectangular cross-section.
First-Order Term and the Third-Order Term Through Taylor's Expansion

Figure 2 :
Figure 2: A cantilever beam under an arbitrarily distributed load

Figure 3 :Figure 4 :
Figure 3: Comparison of deflections from the four HSD models.
23)Considering Eq.(A.14), the first of Eq. (A.23) is identical to the first of Eq. (50).From Eq. (A.23), it is not difficult to obtain the unified solution as Latin American Journal of Solids and Structures 13 shown in Figure2, from Eqs. (A.16), the corresponding boundary conditions for the conventional beam theory are

Table 1 :
Comparison of the four HSD models in c1 and c3

Table 2 .
It is seen that they vary a bit with different HSD models.In addition, KP becomes smaller if the HSD model (i.e.Model-D with the biggest c3) is farther deviated from the first-order shear deformation model.

Table 2 :
Comparison of the four HSD models in KP

Table A .1: Comparison
Table A.1.Compared with KP in Table 2, Eq.(A.14) can also be validated by the four HSD models.
of the four HSD models in KT and αT.