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

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1 1716 he Unified Solution for a Beam of Rectangular Cross-Section with Different Higher-Order Shear Deformation Models bstract he 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. he 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. he unified solution is finally obtained for a beam subjected to an arbitrarily distributed load. he relation with that from the conventional beam theory is established, and therefore the difference is reasonably explained. very good agreement with the elasticity theory validates the present solution. Keywords Fundamental higher-order term, Unified higher-order shear deformation model, Shear correction factor, Unified solution, Conventional beam theory.. C. Duan a L. X. Li b State Key Laboratory for Strength and Vibration of Mechanical Structures, School of erospace, Xi an Jiaotong University, Xi an, Shaanxi, , PR China. a First author. tcduan@stu.xjtu.edu.cn b Corresponding author. luxianli@mail.xjtu.edu.cn Received In revised form ccepted vailable online INRODUCION s the basic structure, beams have been extensively used in fields such as architecture, mechanics, chemistry, aerospace and ocean engineering. wo approaches are often adopted in studying beams. One is to use the elasticity theory, two dimensional or three dimensional, while the other is to use a beam theory as a one-dimensional problem. In the first approach, the solution to a problem is derived strictly according to the elasticity theory. However, as we know, this approach can only offer analytic solutions to problems with simple geometries or loadings. For the second approach, the situation is somewhat confused. o date, there have developed quite a number of beam theories. he Euler-Bernoulli theory (EB) (Dym and Shames, 1973) is the first and classic beam theory in which the cross section is assumed normal to the neutral axis before

2 .C. Duan and L.X. Li / he Unified Solution for a Beam of Rectangular Cross-Section with Different Higher-Order Shear Deformation Models 1717 and after deformation and hence the shear deformation of cross section is not taken into account. In contrast, the imoshenko beam theory (B) (imoshenko, 19, 191) permits a uniform shear deformation of cross section with a shear correction factor. o improve the accuracy, higher-order shear deformation (HSD) models were proposed. For example, Levinson (1981) suggested a third-order rectangular beam model by taking the in-plane warping of cross section into account. Based on the same idea, Murthy (1981) used another definition of rotation, leading to the shear correction factor 5/6 rather than /3 by Levinson (1981). Following the kinematics of Levinson (1981), Bickford (198) proposed a variationally consistent beam theory and Reddy (1984a, 1984b) further developed a higher-order differential governing equation for plates. he comprehensive numerical investigations on the accuracy of various HSD models were conducted by Rohwer (199) which showed that the Murthy s (1981) and Reddy s (1984a, 1984b) theories are the best choices. Inspired by Murthy (1981) and Bickford (198), Shi (011, 007) proposed an improved third-order shear deformation theory with a variationally consistent sixth-order governing equation and respective boundary conditions. Rohwer (199) indicated that the kinematics and the rotation play important roles in the HSD model. For the kinematics, besides the third-order shear deformation model (Levinson, 1981; Murthy, 1981), the trigonometric model (kgöz and Civalek, 014a, 014b; Karama et al., 1998; Mantari et al., 01; ouratier, 1991), the hyperbolic sine model (kgöz and Civalek, 015; kavci and anrikulu, 008; El Meiche et al., 011; Soldatos, 199) and the exponential model (ydogdu, 009; Karama et al., 003) have been developed. For the rotation, two definitions are usually adopted in the existing work. s the first one, the rotation variable ψ is defined as the rotation at the neutral surface (e.g. kgöz and Civalek, 015; kgöz and Civalek, 014a, 014b; ydogdu, 009; Bickford, 198; El Meiche et al., 011; Groh, 015; Karama et al., 003; Karama et al., 1998; Levison, 1981; Mantari et al., 01; Qu, et al., 013; Reddy, 1984a, 1984b; Simsek, 010; Soldatos, 199; ouratier, 1991; Viola, et al., 013) while, as the second one, the rotation variable ϕ is defined as the average rotation over the cross section in some sense (e.g. Cowper, 1966; Murthy, 1981; Reissner, 1975; Shi, 011, 007). In this context, the beam theory in terms of ψ is called the conventional one. Compared with the B, HSD models can not only reflect the warping of cross section, but give the shear correction factor in a straightforward manner. However, the B 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., 013, 010; Gruttmann and Wagner, 001; Gruttmann et al., 1999; Hutchinson, 001; Jensen, 1983; Kaneko, 1975; Pai and Schulz, 1999). In recent years, HSD models have been widely used in composite structures (ydogdu, 009; Karama et al., 003; Mantari et al., 01; Viola, et al., 013) and functionally graded (FG) structures (El Meiche et al., 011; Simsek, 010). Much attention was also paid to real world applications by using beam theories. For example, Wang et al. (008) studied the bending problems of micro- and nano-beams based on the Eringen nonlocal elasticity theory and the B, 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 (kgöz and Civalek, 015, 014a, 014b; Challamel, 013, 011; Wang et al., 014) in nano- or micro-electro mechanical systems (NEMS or MEMS).

3 1718.C. Duan and L.X. Li / he Unified Solution for a Beam of Rectangular Cross-Section with Different Higher-Order Shear Deformation Models hough 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? re there any connections among the existing beam theories, HSD models, or shear correction factors? hese are just the issues to touch on in the present work. o this end, the paper is outlined as follows. In Section, 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. he concluding remarks are made in Section 5. FUNDMENLS OF HE BEM HEORY y 1 z o L h x Figure 1: straight beam of rectangular cross-section. 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 M ( x) y xd Qx ( ) xyd 1 wx ( ) vd 1 ( x)= yud I (1) 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. 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

4 .C. Duan and L.X. Li / he Unified Solution for a Beam of Rectangular Cross-Section with Different Higher-Order Shear Deformation Models 1719 x xy 0 x y y yx 0 y x in the x direction in the y direction () aking the end of x=0 as an example, the corresponding boundary conditions are u(0, y) given or x (0, y) given v(0, y) given or xy (0, y) given in the in the x direction y direction (3) In the x-direction, integration of the first of Eqs. () and (3) weighted by y over the cross section yields and d dx xy y xd y d0 y (4) yud yu d or y d y d (5) 0 x 0 In a similar manner, in the y-direction, integrating the second of Eqs. () and (3) over the cross section, we have and d dx y yxd+ d=0 y (6) vd v d or d d (7) 0 xy 0 Considering the first two of Eqs. (1), Eqs. (4) and (6) yield M Q0 Q + q( x)=0 (8) with y qx ( ) d (9) y In this context, the prime denotes the differentiation with respect to x. Considering the definitions in Eqs. (1), Eqs. (5) and (7) yield

5 170.C. Duan and L.X. Li / he Unified Solution for a Beam of Rectangular Cross-Section with Different Higher-Order Shear Deformation Models 0 given or M 0 given w 0 given or Q 0 given in the in the x direction y direction (10) 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 Q w Free condition : M 0 given and 0 given Simply supported condition : M 0 given and w0 0 Fixed condition : 00 and 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, 011), so, from the first of Eqs. (1), in terms of ϕ, the moment can be further expressed as ogether with Eqs. (8), the governing equations read M (11) EI (1) EI q x EI Q 0 (13) 3 HE KINEMICS 3.1 he 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). o 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 v x, y w( x) (14) 1 R u x, y y f ( x) g( y) f ( x) (15) 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 ( x) f ( x) f ( x) (16) 1 R

6 .C. Duan and L.X. Li / he Unified Solution for a Beam of Rectangular Cross-Section with Different Higher-Order Shear Deformation Models I yg( y) d (17) If we take fr(x) 0, together with Eq. (16), Eq. (15) reduces to the first-order shear deformation model as u x, y y ( x) (18) For a higher-order model, upon eliminating fr(x) via Eq. (16), Eq. (15) becomes u y f ( x) g( y) x f x (19) 1 1 hus, considering Eq. (14), the shear strain over the cross section is ( ) u v f w dg y dy f (0) xy, y, x 1 1 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. (0), we have xy x, h 0 (1) f 1 c w c () where c dg dy dg dy (3) yh yh Eventually, the axial displacement in Eq. (15) becomes c w g( y) u y w c c (4) Eq. (4) is termed as the unified HSD model for a beam corresponding to the fundamental higher-order term g(y). hus far, from Eqs. (3) and considering the higher-order property, g(y) has the following properties g y dg dy y0 0 y0 0 dg dy dg dy yh yh (5) In terms of w(x) and ϕ(x), the shear force Q(x) is expressed as

7 17.C. Duan and L.X. Li / he Unified Solution for a Beam of Rectangular Cross-Section with Different Higher-Order Shear Deformation Models Qx ( ) G xyd KG P w (6) where KP, the shear correction factor originally defined by imoshenko (19,191), takes the form of 1 KP cdg dyd c (7) It can be seen from Eq. (7) 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, 001). If we assume Eq. (6) yields which implies that the shear effect cannot be not taken into account. ccordingly, Eq. (4) reduces to w 0 (8) Qx ( ) 0 (9) u x, y y w( x) (30) which is the axial displacement assumption in the EB (Dym and Shames,1973). Compared with the first-order shear deformation model in Eq. (18), the EB in Eq. (30) is obtained under the additional assumption of Eq. (8), which is a more reasonable explanation than that of an infinite shear rigidity (e.g. Challamel, 013). 3. Four Commonly Used HSD Models One direct choice of the fundamental higher-order term is to take g (y)=y 3. his model is just the commonly used third-order model and denoted as Model- in this context. Eq. (7) yields K 56 (31) In this paper, other three commonly used HSD models are studied as well. P 3..1 he Sine Model Model-B Inspired by ouratier (1991) and considering Eqs. (5), as the sine model, the fundamental higherorder term is taken to be B h y g ( y) y sin h (3) ccordingly, from Eqs. (17) and (3), we have

8 .C. Duan and L.X. Li / he Unified Solution for a Beam of Rectangular Cross-Section with Different Higher-Order Shear Deformation Models c 1 (33) So, the axial displacement in Eq. (4) yields h B u yw sin y w 4 h (34) Further, Eq. (7) yields B K P 1 (35) 3.. he Hyperbolic Sine Model Model-C Inspired by El Meiche et al. (011) and Soldatos (199) and considering Eq. (5), as the hyperbolic sine model, the fundamental higher-order term is taken to be C y g ( y) yhsinh h ccordingly, from Eqs. (17) and (3), we have 11 cosh(1 ) sinh(1 ) c 1 cosh(1 ) (36) (37) So, the axial displacement in Eq. (4) yields C ycosh(1 ) hsinh( y h) u yw w 4sinh(1 ) 11cosh(1 ) (38) Further, Eq. (7) yields C K P cosh(1 ) sinh(1 ) 4sinh(1 ) 11cosh(1 ) (39) 3..3 he Exponential Model Model-D Inspired by Karama et al. (003) and considering Eq. (5), as the exponential model, the fundamental higher-order term is taken to be ccordingly, from Eqs. (17) and (3), we have g D ( y) y yexpy h (40)

9 174.C. Duan and L.X. Li / he Unified Solution for a Beam of Rectangular Cross-Section with Different Higher-Order Shear Deformation Models 13exp( 1 ) (3 ) erf ( ) c 1 (41) where the Gauss error function is defined as x erf ( x) exp( s ) ds (4) 0 So, the axial displacement in Eq. (4) yields u Further, Eq. (7) yields exp ( ) y y h w D yw (43) 3exp( 1 ) (3 ) erf ( ) D K P exp( 1 ) (44) 3exp( 1 ) (3 ) erf ( ) 3.3 Comparison of the Four HSD Models he First-Order erm and the hird-order erm hrough aylor s Expansion It is interesting to study the difference of the four HSD models by comparing the first two leading terms through aylor s expansion. For the third-order model, the axial displacement is 1 3 y 5, u x y y y w w (45) 4 h 3 hus, if the axial displacement is expanded in form of power series 3 y 5 5 ux, y y yc1w c 3wOy h w (46) h from aylor s expansion, we have c1 14; c3 53 For Model - B 3 B 5 c1 4 1; c3 144 For Model - B C c1 cosh1 1 4sinh1 11cosh1 1 For Model - C C c sinh1 11cosh 1 D c1 1 3exp1 3 erf 1 For Model - D D c3 3exp1 3 erf (47)

10 .C. Duan and L.X. Li / he Unified Solution for a Beam of Rectangular Cross-Section with Different Higher-Order Shear Deformation Models 175 he coefficients are summarized in able 1. It is seen that c3 increases with c1. Model type c1 c3 Model Model-B Model-C Model-D able 1: Comparison of the four HSD models in c1 and c Shear Correction Factors s 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. are summarized in able. 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. Model type Model- Model-B Model-C Model-D KP c able : Comparison of the four HSD models in KP From Eqs. (31) and (35), we can see that the commonly used shear correction factors can be reasonably explained by Eq. (7) in the manner of HSD models. For example, the B with KP =5/6 (e.g. Kaneko, 1975; imoshenko, 19, 191; Murthy, 1981; Reissner, 1975; Shi, 007, 011) is in essential equivalent to the third-order model (i.e. Model-) because of Eq. (31) while the Mindlin plate theory (Mindlin, 1951) with KP =π /1 is in essential equivalent to the sine model (i.e. Model- B) because of Eq. (35). 4 HE UNIFIED SOLUION 4.1 Derivation of the Unified Solution In this section, the unified solution will be derived for the unified HSD model in Section 3. o this end, the governing equations are re-arranged as follows EI q x 0 EI KPGw It is interesting to discuss the procedure of solving Eqs. (48). straightforward procedure is to turn Eqs. (48) into the form (48)

11 176.C. Duan and L.X. Li / he Unified Solution for a Beam of Rectangular Cross-Section with Different Higher-Order Shear Deformation Models q x EI EI w KG P (49) 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. o this end, Eqs. (48) are re-expressed as x (4) qx ( ) q w EI KPG EI qx w w KG P KG P 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 (50) x x x x x x 1 1 w dx dx dx q( x) dx dx q( x) dx a x a x a x a EI K PG 0 0 x x x 1 6EIa 1 dx dx q( x) dx 3a1x ax a3 EI K PG (51) y q(x) 1 h x L Figure : cantilever beam under an arbitrarily distributed load For the problem shown in Figure, the boundary conditions are free condition at x 0 : EI(0) 0, KPG (0) w(0) 0 fixed condition at x L: ( L) 0, w( L) 0 (5)

12 .C. Duan and L.X. Li / he Unified Solution for a Beam of Rectangular Cross-Section with Different Higher-Order Shear Deformation Models 177 So, the integration constants in Eqs. (51) are finally determined as x 1 a1 q( x) dx 6EI 0 x0 x x 1 a dx q( x) dx EI 0 0 x0 x x x x x 1 L a3 dx dx q( x) dx dx q( x) dx EI EI xl 0 0 x0 x x L 1 qxdx ( ) qxdx ( ) EI KG 0 x0 0 x0 x x x x x x x 1 L a4 dx dx dx q( x) dx EI dx dx q( x) dx EI x L x x 3 x L L dx q( x) dx q( x) dx EI 3EI 0 0 x0 0 x0 x x x 1 L dx qxdx ( ) qxdx ( ) KG P KG 0 0 xl P 0 x0 xl (53) Eqs. (51) and (53) are the unified solution for the beam problem shown in Figure. he unified solution is also derived in ppendix (see Eqs. (.4) and (.6)) by using the conventional beam theory. It is seen that Eqs. (.4) will turn into Eqs. (51) if α=1 (and hence K=KP from Eq. (.13)). However, the fact is that α is actually much less than unity for all the HSD models (see able.1), leading to major difference between K and KP. his may be the reason why K was not recognized as the shear correction factor in the previous HSD models (e.g. kgöz and Civalek, 014a, 014b; El Meiche et al., 011; Huang et al., 013; Levinson, 1987, 1981; Simsek, 010; ouratier, 1991) despite of the definition already made as early as in 190s (imoshenko, 19,191). 4. 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 q q w x L x L x L 4EI KPG q 3 3 x L 6EI (54) On the other hand, applying the boundary conditions in Eqs. (11) to the problem shown in Figure with q=const, the elasticity solution to this plane stress problem is obtained as

13 178.C. Duan and L.X. Li / he Unified Solution for a Beam of Rectangular Cross-Section with Different Higher-Order Shear Deformation Models 1 ql5xl 1xL17 0G q 3 3 uref yql 1x L 6EIqLh 3x L4 40EI3qL45x L 0G qlh x L1 4EIqLhx L1 4G+ qxy 6EIqxy Gh q 4 wref vda ql x L x L EI ql h x L x L EI (55) he 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), (.7) 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. (4). he 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 K (see able and able.1), the conventional solution greatly deviates. a) Model- b) Model-B c) Model-C d) Model-D Figure 3: Comparison of deflections from the four HSD models.

14 .C. Duan and L.X. Li / he Unified Solution for a Beam of Rectangular Cross-Section with Different Higher-Order Shear Deformation Models 179 a) Model- b) Model-B c) Model-C d) Model-D Figure 4: Comparison of the warping of cross section from the four HSD models. 5 CONCLUSIONS ND FUURE 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. he shear correction factor was then derived. he 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) he definition of rotation plays an important role for the HSD models. ) he 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.

15 1730.C. Duan and L.X. Li / he Unified Solution for a Beam of Rectangular Cross-Section with Different Higher-Order Shear Deformation Models he 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. pplication to small scale problems and extension to a plate problem are also prospective. cknowledgements his work was supported by the National Natural Science Foundations of China (Grant Nos , ) References kavci, S.S., anrikulu,.h., (008). Buckling and free vibration analyses of laminated composite plates by using two new hyperbolic shear-deformation theories. Mechanics of Composite Materials 44: kgöz, B., Civalek, O., (014a). Shear deformation beam models for functionally graded microbeams with new shear correction factors. Composite Structures 11: kgöz, B., Civalek, O., (014b). new trigonometric beam model for buckling of strain gradient micro beams. International Journal of Mechanical and Sciences 81: kgöz, B., Civalek, O., (015). novel microstructure-dependent shear deformable beam models. International Journal of Mechanical and Sciences 99: ydogdu, M., (009). new shear deformation theory for laminated composite plates. Composite Structures 89: Bickford, W.B., (198). consistent higher order beam theory. In: Developments in heoretical and pplied Mechanics. vol. XI. University of labama, labama. Challamel N., (011). Higher-order shear beam theories and enriched continuum. Mechanics Research Communications 38(5): Challamel N., (013). Variational formulation of gradient or/and nonlocal higher-order shear elasticity beams. Composite Structures 105: Cowper, G.R., (1966). he shear coefficient in imoshenko s beam theory. Journal of pplied Mechanics- ransactions of the SME 33: Dong, S.B., lpdogan, C., aciroglu, E., (010). Much ado about shear correction factors in imoshenko beam theory. International Journal of Solids and Structures 47: Dong, S.B., Çarbaş, S., aciroglu, E., (013). On principal shear axes for correction factors in imoshenko beam theory. International Journal of Solids and Structures 50: Dym, C.L., Shames, I.H., (1973). Solid Mechanics: Variational pproach. McGraw-Hill (New York). El Meiche, N., ounsi,., Ziane, N., Mechab, I., Bedia, E., (011). new hyperbolic shear deformation theory for buckling and vibration of functionally graded sandwich plate. International Journal of Mechanical and Sciences 53: Ghugal, Y.M., Sharma, R., (011). refined shear deformation theory for flexure of thick beams. Latin merican Journal of Solids and Structures 8: Guttmann, F., Sauer, R., Wagner, W., (1999). Shear stresses in prismatic beams with arbitrary cross section. International Journal for Numerical Methods in Engineering 45: Guttmann, F., Wagner, W., (001). Shear correction factors in imoshenko's beam theory for arbitrary shaped crosssection. Computational Mechanics 7: Huang, Y., Wu, J.X., Li, X.F., Yang, L.E., (013). Higher-order theory for bending and vibration of beams with circular cross section. Journal of Engineering Mathematics 80:

16 .C. Duan and L.X. Li / he Unified Solution for a Beam of Rectangular Cross-Section with Different Higher-Order Shear Deformation Models 1731 Hutchinson, J.R., (001). Shear Coefficients for imoshenko Beam heory. Journal of pplied Mechanics- ransactions of the SME 68: Jensen, J.J., (1983). On shear coefficient in imoshenko's beam theory. Journal of Sound and Vibration 87: Kaneko,., (1975). imoshenko s correction for shear in vibrating beams. Journal of Physics D-pplied Physics 8, Karama, M., bou Harb, B., Mistou, S., Caperaa, S., (1998). Bending, buckling and free vibration of laminated composite with a transverse shear stress continuity model. Composites Part B: Engineering 9B, Karama, M., faq, K.S., Mistou, S., (003). Mechanical behavior of laminated composite beam by the new multilayered laminated composite structures model with transverse shear stress continuity. International Journal of Solids and Structures 40: Levinson, M., (1981). new rectangular beam theory. Journal of Sound and Vibration 74: Levinson, M., (1987). On higher order beam and plate theories. Mechanics Research Communications 14, Mantari, J.L., Oktem,.S., Guedes Soares, C., (01). new trigonometric shear deformation theory for isotropic, laminated composite and sandwich plates. International Journal of Solids and Structures 49: Mindlin, R.D., (1951). hickness-shear and flexural vibrations of crystal plates. Journal of pplied Physics : Murthy, M.V.V., (1981). n Improved ransverse Shear Deformation heory for Laminate nisotropic Plates. N- S echnical Paper No Pai, P.F., Schulz, M.J., (1999). Shear correction factors and an energy-consistent beam theory. International Journal of Solids and Structures 36: Reddy, J.N., (1984a). refined nonlinear theory of plates with transverse shear deformation. International Journal of Solids and Structures 0: Reddy, J.N., (1984b). simple higher-order theory for laminated composite plates. Journal of pplied Mechanics- ransactions of the SME 51: Reissner, E., (1975). On transverse bending of plates, including the effect of transverse shear deformation. International Journal of Solids and Structures 11: Rohwer, K., (199). pplication of higher order theories to the bending analysis of layered composite plates. International Journal of Solids and Structures 9: Shi, G.Y., (007). new simple third-order shear deformation theory of plates. International Journal of Solids and Structures 44: Shi, G.Y., (011). Sixth-Order heory of Shear Deformable Beams With Variational Consistent Boundary Conditions. Journal of pplied Mechanics-ransactions of the SME 78(01019): Simsek, M., (010). Fundamental frequency analysis of functionally graded beams by using different higher-order beam theories. Nuclear Engineering and Design 40: Soldatos, K.P., (199). transverse shear deformation theory for homogeneous monoclinic plates. cta Mechanica 94: imoshenko, S.P., (191). On the correction for shear of the differential equation for transverse vibration of prismatic bars. Philosophical Magazine 41, imoshenko, S.P., (19). On the transverse vibrations of bars of uniform cross-section. Philosophical Magazine 43: imoshenko, S.P., Goodier J.N., (004). heory of Elasticity, singhua University Press. Beijing, China. ouratier, M., (1991). n efficient standard plate theory. International Journal of Engineering Science 9: Viola E., ornabene F., Fantuzzi N., (013). General higher-order shear deformation theories for the free vibration analysis of completely doubly-curved laminated shells and panels. Composite Structures 95:

17 173.C. Duan and L.X. Li / he Unified Solution for a Beam of Rectangular Cross-Section with Different Higher-Order Shear Deformation Models Wang B.L., Liu M.C., Zhao J.F., Zhou S.J., (014). size-dependent Reddy-Levinson beam model based on a strain gradient elasticity theory. Meccanica 49(6): Wang C.M., Kitipornchai S., Lim C.W., Eisenberger M., (008). Beam bending solutions based on nonlocal imoshenko beam theory. Journal of Engineering Mechanics-ransactions of the SCE 134(6): PPENDIX : HE CONVENIONL BEM HEORY WIH HE UNIFIED HSD MODEL ND HE UNIFIED SOLUION.1 Governing Equations and Boundary Conditions In the conventional beam theory, deflection wx () and rotation ψ(x) are defined as (e.g. imoshenko and Goodier, 004) v x y0 wx ( ),0 ( x)= u y (.1) With the assumption of Eq. (15), wx () is identical to w(x). ogether with the properties in the first two of Eqs. (5) for g(y), Eq. (15) is still mathematically valid. hus, the shear strain is xy u v f1( x) w ( x) dg dy fr( x) (.) y x Considering the shear stress free condition on the top and bottom surfaces xy x h f1 x w x c fr x (, ) ( ) ( ) ( ) 0 (.3) we have 1 fr( x) f1( x) w ( x) (.4) c where c is as defined in Eq. (3) and hence the property in the third of Eqs. (5) stand also for g(y). From the definition in the second of Eqs. (.1) and considering the third of Eqs. (5), we have ( x) f ( x) (.5) 1 With Eqs. (.4) and (.5), in terms of wx () and ψ(x), the axial displacement is eventually expressed as

18 .C. Duan and L.X. Li / he Unified Solution for a Beam of Rectangular Cross-Section with Different Higher-Order Shear Deformation Models 1733 uxy (, ) y g y c w yw y g y c w (.6) s often, Eq. (.6) can also be re-expressed as uxy (, ) yw S y w (.7) with Based on Eqs. (5), S(y) has following properties S y y g y c (.8) Sy ds dy ds dy y0 0 y0 h y 1 0 (.9) From Eq. (.7), we further obtain Q K G( w ) (.10) with 1 K ds( y) dy d (.11) In addition, in terms of wx () and ψ(x) (ydogdu, 009; kavci and anrikulu, 008; El Meiche et al., 011; Karama et al., 1998, 003; Levinson, 1981; Mantari et al., 01; Soldatos, 199; ouratier, 1991), moment M can be expressed in a unified form as M EIw EI w (.1) with 1 I ys( y) d (.13)

19 1734.C. Duan and L.X. Li / he Unified Solution for a Beam of Rectangular Cross-Section with Different Higher-Order Shear Deformation Models Interestingly, we can obtain the following relationship K K (.14) P From Eqs. (.10) and (.1), in terms of wx () and ψ(x), the governing equations are 0 K G w q x EIw EI w K G w (.15) with the boundary conditions as [e.g. Levinson, 1981] Free condition : M M0 and Q Q0 Simply supported condition : M M0 and w w0 0 Fixed condition : 0 0 and w w 0 0 (.16). he Four Commonly Used HSD Models (1) he third-order model Model- (Levinson, 1981) In the third order shear deformation model, we take g (y)=y 3, and hence c=3h /4 from Eq. (3). hen, Eq. (.8) yields 4y S ( y) y1 3 h (.17) () he sine model Model-B (ouratier, 1991) In this HSD model, g B (y) takes the form of Eq. (3), and hence c=1 from Eq. (3). hen, Eq. (.8) yields ( ) h B S y sin y h (.18) (3) he hyperbolic sine model Model-C (Soldatos, 199) In this HSD model, g C (y) takes the form of Eq. (36), and hence c=1-cosh(1/) from Eq. (3). hen, Eq. (.8) yields sinh cosh 1 1 cosh 1 S C ( y) y h y h (.19) It should be noted that Eq. (.19) firstly derived according to Eqs. (.8) corrects the original form in Soldatos (199). (4) he exponential model Model-D (Karama et al., 003) In this HSD model, g D (y) takes the form of Eq. (40), and hence c=1 from Eq. (3). hen, Eq. (.8) yields

20 .C. Duan and L.X. Li / he Unified Solution for a Beam of Rectangular Cross-Section with Different Higher-Order Shear Deformation Models 1735 D S ( y) yexp y / h.3 Shear Correction Factors From Eq. (.11), for the four HSD models, it is immediate to obtain (.0) K 3 B K C cosh(1 ) sinh(1 ) K cosh 1 1 D K exp1 (.1) In addition, from Eq. (.13), we have 45 B 3 4 C 4sinh(1 ) 11cosh(1 ) cosh 1 1 D 3exp( 1 ) 3 erf (.) he values for K and α are summarized in able.1. Compared with KP in able, Eq. (.14) can also be validated by the four HSD models. Model type Model- Model-B Model-C Model-D K α able.1: Comparison of the four HSD models in K and α..4 he Unified Solution For sake of the unified solution, Eqs. (. 15) are further re-expressed as (4) q x q x w K G EI EI qx ww KG KG (.3) Considering Eq. (.14), the first of Eq. (.3) is identical to the first of Eq. (50). From Eq. (.3), it is not difficult to obtain the unified solution as

21 1736.C. Duan and L.X. Li / he Unified Solution for a Beam of Rectangular Cross-Section with Different Higher-Order Shear Deformation Models x x x x x x 1 3 w dx dx dx q( x) dx dx q( x) dx bx 1 bx b3x b4 EI K G 0 0 x x x x 1 1 6EI dx dx q( x) dx q( x) dx 3b1x bx b1 b3 EI K G K 0 G (.4) For the problem shown in Figure, from Eqs. (.16), the corresponding boundary conditions for the conventional beam theory are Free condition at x 0 : EIw(0) EI (0) w(0) 0 KG (0) w (0) 0 Fixed condition at x L: ( L) 0, w ( L) 0 (.5) hus, the integration constants in Eqs. (.4) are determined as x 1 b1 q( x) dx 6EI 0 x0 x x 1 b dx q( x) dx EI 0 0 x0 x x x x x x 1 L L b3 dx dx q( x) dx dx q( x) dx q( x) dx EI EI EI xl 0 0 x0 0 x0 x x 1 1 qxdx ( ) qxdx ( ) KG K 0 x L G 0 x0 x x x x x x x 1 L b4 dx dx dx q( x) dx dx dx q( x) dx EI EI x L xl x x 3 x L L dx q( x) dx q( x) dx EI 3EI 0 0 x0 0 x0 x x x x 1 L dx q( x) dx L q( x) dx q( x) dx KG K 0 0 x L G K 0 G 0 xl x0 (.6) For the case of q=const, the unified solution to the unified HSD model is

22 .C. Duan and L.X. Li / he Unified Solution for a Beam of Rectangular Cross-Section with Different Higher-Order Shear Deformation Models 1737 q q ql w x 4L x3l x L xl 4EI KG KG q 3 3 1q x L x L 6EI K G (.7) Int erestingly, Eqs. (.7) will turn into Eqs. (54) if α=1 (hence K=KP from Eq. (.13)).