二階梁理論及其在側向-扭轉挫屈的應用(I)

行政院國家科學委員會專題研究計畫 成果報告

二階梁理論及其在側向-扭轉挫屈的應用(I)

計畫類別： 個別型計畫 計畫編號： NSC91-2211-E-009-041-執行期間： 91 年 08 月 01 日至 92 年 07 月 31 日 執行單位： 國立交通大學機械工程學系 計畫主持人： 蕭國模 報告類型： 精簡報告 處理方式： 本計畫可公開查詢

中

華

民

國 92 年 8 月 20 日

二階梁理論及其在側向-扭轉挫屈的應用(I)

A Second Or der Beam Theor y and Its Application In

Later al-Tor sional Buckling (I)

計畫編號：NSC 91-2211-E-009-041

執行期限：91 年 08 月 1 日至 92 年 07 月 31 日

主持人：蕭國模 國立交通大學機械工程學系

計畫參與人員：陳弘虎

中文摘要 本研究提出一個二階梁理論共旋轉法以探討大旋轉但小應變之三維尤拉梁 的非線性分析。將梁分割成若干個梁元素。每一個元素座標系統是建立在元素當 前的變形位置上。梁元素的變形、平衡方程式及構成方程式，皆在元素座標系統 中建立。以虛功原理及完整非線性梁理論之二階一致線性化推導梁元素的平衡方 程式及構成方程式。平衡方程式及構成方程式保留至變形參數的二次項及部份的 三次項所以能考慮撓曲、扭曲、與軸向變形間非線性的耦合效應。本研究將以數 值例題探討提出之二階梁理論的正確性及實用性。 關鍵詞: 梁, 幾何非線性, 共旋轉法, 虛功原理 Abstr act

A consistent co-rotational total Lagrangian formulation of second order beam theory is presented for the nonlinear analysis of three-dimensional elastic Euler beam with large rotations but small strains. The beam structure is divided into several segments. A set of segment coordinate system is constructed at the current configuration of the deformed beam segment. The deformations, equilibrium equations and constitutive equations of the beam segment are defined in the segment coordinates. The principle of virtual work and the consistent linearization of the fully geometrically nonlinear beam theory is used to derive the equilibrium equations and constitutive equation of the beam segment. The nonlinear coupling among bending, twisting, and stretching deformations is considered by retaining all terms up to the second order and some terms of third order of the deformation parameters in the

equilibrium equations and constitutive equations of the second order beam theory. Numerical examples are presented to demonstrate the accuracy and effectiveness of the proposed second order beam theory.

Keywor ds: Beam, Geometrical Nonlinearity, Co-rotational Formulation, Virtual

1 INTRODUCTION

Three-dimensional beams are very important structural elements in all type of engineering systems. In many applications, these beams undergo finite rotations that require a nonlinear formulation to their structural analysis. In order to capture correctly all coupling among bending, twisting, and stretching deformations of the beam, the formulation of beam equations might be derived by the fully geometrically non-linear beam theory [1].

The beam structure is divided into several segments. A set of segment coordinate system is constructed at the current configuration of the deformed beam segment. The deformation, equilibrium equations and constitutive equations of the beam segment are defined in this segment coordinates. The deformations of the beam segment are determined by the unit extension of the centroid axis and the rotation of segment cross section coordinates, which are rigidly tied to segment cross section, relative to the segment coordinate system. Three rotation parameters proposed in [2] are used to describe the rotation of the segment cross section coordinates.

The principle of virtual work and the consistent linearization [2] of the fully geometrically nonlinear beam theory is used to derive the equilibrium equations and constitutive equation of the beam. In order to consider the nonlinear coupling among bending, twisting, and stretching deformations, the rotation parameters are retained up to the second order in the equilibrium equations and constitutive equation of the beam segment. Thus, the beam theory proposed here is called a second order beam theory. However, some third order terms, which may not be negligible for some problems, are also retained. Numerical examples are presented to demonstrate the accuracy and effectiveness of the proposed second order beam theory.

2 NONLINEAR FORMULATION

2.1 Basic assumptions

In this study, The beam structure is divided into several segments. The following assumptions are made in derivation of the beam segment behavior.

(1) The beam is prismatic and slender, and the Euler-Bernoulli hypothesis is valid. (2) The cross section of the beam is doubly symmetric.

(3) The cross section of the beam does not deform in its own plane and strains within this cross section can be neglected.

(4) The out-of-plane warping of the cross section is the product of the twist rate of the beam element and the Saint Venant warping function for a prismatic beam of the same cross section.

(5) The deformation displacements and rotations of the beam segment are small. (6) The strains of the beam are small.

In conjunction with the co-rotational formulation, assumption (5) can always be satisfied, if the segment size is properly chosen.

2.2 Coor dinate systems

In this paper, a co-rotational total Lagrangian formulation is adopted. In order to describe

the system, we define three sets of right handed rectangular Cartesian coordinate systems: (1) A fixed global set of coordinates, X_i (i = 1, 2, 3) (see Fig. 1); the nodal coordinates,

displacements, rotations, the governing equations of the system are defined in this coordinates.

P X₃ 2 X X₁ x3 2 x 1 x xc s 3 S y P x z Q xS 2 1 _x u P v w ( ,0,0l ) 2 x xS_{3 x}S₂ S 1

b

a

_b

(3) Segment coordinates, x_i (i = 1, 2, 3) (see Fig. 1); a set of segment coordinates is

associated with each segment, which is constructed at the current configuration of the beam segment.

Fig.1 Coordinate systems. Fig.2 Rotation vector.

2.3 Rotation vector and rotation par ameter s

For convenience of the later discussion, the term 'rotation vector' is used to represent a finite rotation. Figure 2 shows that a vector b which as a result of the application of a rotation vector φa is transported to the new position b . The relation between b and b may be expressed as [3]

b =cosφb+ −(1 cos )(φ a b⋅ +) sin (φ a×b) ₍₁₎ where φ is the angle of counterclockwise rotation, and a is the unit vector along the axis of rotation.

Let e_i and e_iS (i = 1, 2, 3) denote the unit vectors associated with the x_i and x_iS axes, respectively. Here, the traid e_iS in the deformed state is assumed to be achieved by the successive application of the following two rotation vectors to the traid e_i [2]:

θ_n =θ n_{n , θt} =θ t (2,3)_t where n = + + = { , ( ) , ( ) } { , , } 0 0 2 22 32 1 2 2 22 32 1 2 2 3 θ θ θ θ θ θ n n (4) t={cosθ θ θ_n, ₂, ₃} (5)

cosθ_n = −(1 θ₂2 −θ₃2 1 2) (6) θ₂ = −dw s ds ( ) , θ₃ =dv s ds ( ) (7,8)

in which n is the unit vector perpendicular to the vectors e₁ and e₁S, and t is the tangent unit vector of the deformed centroid axis of the beam segment. Note that e₁S coincides with t . θ_n is the inverse of cosθ_n . v s( ) and w s( ) are the lateral deflections of the centroid axis of the beam segment in the x₂ and x₃ directions, respectively, and s is the arc length of the deformed centroid axis.

Using Eqs. (2)-(8), the relation between the vectors e_i and e_iS (i = 1, 2, 3) in the

segment coordinate system may be obtained as

e_iS =[ ,t R₁,R₂]e_i =Re_i (9) where R is the so-called rotation matrix. The rotation matrix is determined byθ_i (i = 1, 2,

3). Thus, θ_i are called rotation parameters in this study.

Let θ={ ,θ θ θ_{1 2}, ₃} be the column matrix of rotation parameters, δθ be the variation of

θ. The traid eiS (i = 1, 2, 3) corresponding to θ may be rotated by a rotation vector

δφ={δφ δφ δφ₁, ₁, ₁} to reach their new positions corresponding to θ +δθ [2]. When θ₂ and θ₃ are much smaller than unity, the relationship between δθ and δφ may be approximated by [2] δ θ θ θ θ δ δ θ = φ φ. 1 2 2 1 0 0 1 3 2 3 2 1 − −           =T− (10)

2.4 Kinematics of beam segment

The deformations of the beam segment are described in the current segment coordinate system. From the kinematic assumptions made in this paper, the deformations of the beam segment may be determined by the displacements of the centroid axis of the beam segment, orientation of the cross section (segment cross section coordinates), and the out-of-plane warping of the cross section. Let Q (Fig. 1) be an arbitrary point in the beam segment and P

be the point corresponding to Q on the centroid axis. The position vector of point Q in the

undeformed and deformed configurations may be expressed as

r₀ =xe₁ + ye₂ +ze₃ (11) and

r =x s_c( )e₁ +v s( )e₂ +w s( )e₃ +θ_1,se₁s +ye₂s +ze₃s (12) where

x s_c( )= +x u s( ) (13) is the x₁ coordinate of point P, u s( ) is the displacement of point P in the x₁ direction,

respectively as mentioned, s is the arc length of the deformed centroid axis measured from

node 1 to point P.

In this study, the Green strains are used for the measure of strain. Using assumption 3, we only consider the strain components ε₁₁,ε₁₂ and ε₁₃. If x, y and z in Eq. (11) are

regarded as the Lagrangian coordinates, the Green strain ε₁₁,ε₁₂ and ε₁₃ are given by [4]

ε11 1 2 1 = (r r_,t_{x ,x} − ) ε12 1 2 = r r_,t_{x ,y} ε₁₃ 1 2 = r r_,t_{x ,z}. (14) Using the chain rule for differentiation, r_,x in Eq. (14) may be expressed as

r_,x =r_,s(1+ε (15)₀) ε ∂ ∂ 0 = −1 s x (16)

where ε₀ is the unit extension of the centroid axis.

F a x₂ x₃ M₂ x₁ 3 M Bθ 2 3 F 1 M s F₁ 3 F F Bθ b 1 M F 3 2 M₂ M₁

Fig.3 Free body of a portion of beam segment.

2.5 Equilibr ium equations and constitutive equations

Here the equilibrium equations and constitutive equations of the beam segment are derived by the virtual work principle in the current segment coordinate. It is assumed that no external load is applied between the end points of the beam segment. The generalized displacements are chosen to be u_i (i = 1, 2, 3)

θi (i = 1, 2, 3), the rotation parameters, and θ1,s, the twist rate of the centerline.

The generalized forces corresponding to u_i, θ_i, andθ_1,s are F_i (i = 1, 2, 3), the

forces in the x_i directions, M_iθ (i = 1, 2, 3), the generalized moments, and Bθ, the generalized bimoment. Figure 3 shows a portion of the deformed centerline of the beam segment and the generalized forces at sections a and b. From Eq. (10) and

the contragradient law [5], the relation between the conventional moment M_i (i = 1,

2, 3), the moments about the x_i axes, and the generalized moments M_iθ may be given by

M =T−tMθ (17)

where M =

{

M₁ M₂ M₃

}

, Mθ =

{

M₁θ Mθ₂ Mθ₃

}

, and T−t is the transpose of T−1 given in Eq. (10).

Consider infinitesimal generalized virtual displacements from the equilibrium configuration, with components δu_i, δθ_i (i = 1, 2, 3), andδθ_1,s as functions of the arc length of the centerline, s. The generalized forces at sections a and b are regarded

as the external forces for the free body shown in Fig. 3. The virtual work of the external forces may be expressed by

( )

δW_ext tδ θ tδ Bθδθ _s a b = + +    F u M θ _1, (18) F={F F1, 2,F3} δu={δu₁,δu₂,δu₃}={δ δ δu v w, , } δ θ={δθ δθ δθ₁, ₂, ₃} (19) where [ ]_ab denotes that the value of the term in brackets at the upper limit subtracts

The virtual work done by the internal stresses going through the virtual strains (that corresponding to the imposed virtual displacements) for the free body shown in Fig. 3 may be expressed by

(

)

δW_int σ δε σ δε σ δε dV a b =

∫

11 11+ 12 12 + 13 13 (20)

where V is the volume of the undeformed beam segment between section a and b,

dV=dAds/ (1+ε , δε₀) _1j (j = 1, 2, 3) are the variation of ε_1j with respective genearlized displacements. σ_1j (j = 1, 2, 3) are second Piola-Kirchhoff stresses.

For linear elastic material, the stress-strain relations are given by

σ11=Eε11, σ12 =2Gε12, σ13 =2Gε13 (21)

where E is the Young’s modulus and G is shear modulus.

The principle of virtual work requires that

δW_int =δW_ext (22) Substituting Eqs. (14), (18)-(20), and (21) into Eq. (22), and equating the terms on both sides of Eq. (22) corresponding to the same virtual generalized displacements and their derivatives, we may obtain

(

)

M₁θ_,s = E I_y −I_z θ θ_{2,s 3,s} (23) M₂θ_,s−F₃ =F₁ (24) M_3s F₂ F_{1 3} 1C _{1s 2 s} 2 , , , θ _{+ = θ + θ θ (25)} F_1,s =0, F_2,s =0, F_3,s =0 (26-28) M₁ B_s E ₀I_{p 1s} C 1 _{0 1s} 1 _{2s 3 2 3s} EI_{4 1}3_s 2 1 2 1 2 θ ₊ θ _{= ε θ + +ε θ + θ θ − θ θ + θ} , , [( ) , , , ] , (29)

(

)

M₂ E 1 4 ₀ I_{y 2s} I_y I_{z 1 3s} 3 _{yz 1ss 3s} 1C _{1s 3} 2 θ _{= [ + ε θ +( − )θ θ − θ θ ]+ θ θ} , , Ω , , , (30)

(

)

M₃ E 1 4 ₀ I_{z 3s} I_y I_{z 1 2s} 3 _{yz 1ss 2s} 1C _{1s 2} 2 θ _{= [ + ε θ +( − )θ θ − θ θ ]− θ θ} , , Ω , , , (31)

(

)

Bθ =C₁ 1+4ε θ_{0 1,ss} −3EΩ_yzθ θ_{2,s 3,s} (32) F₁ E A ₀ 1 5 ₀ I_{p 1}2_s I_{y 2}2_s I_{z 3}2_{s 1}2_ssA C ₁2_s 2 1 2 5 2 = [ ε ( + ε )+ θ _, + ( θ _, + θ _, +θ _{, ωω})]+ θ _, (33) where C=GJ is the torsional rigidity, and C₁= EA_ωω is the warping rigidity.

Equations (23)-(28) are equilibrium equations and Eqs. (29)-(33) are constitutive equations of the beam segment in the current segment coordinates. Note that because all the terms up to the second order of the generalized displacements are retained in Eqs. (23)-(33), the beam theory proposed here is called second order beam theory. However, some third order terms (the underlined terms in Eqs. (24), (25), and (29)), which may not be negligible for some problems, are also retained.

3. NUMERICAL EXAMPLES

Consider a cantilever beam subjected to end torsion T as shown in Fig . 4.

Because only the primary equilibrium path is considered, the ways of generating end torsion are rendered irrelevant here. It is assumed that warping is free at both end sections. Three cases of cross section are considered: (1) b=0 6. mm, h=10mm, (2) b=0 6. mm, h=30mm. The rest geometry and material properties of the beam are: L=240mm , Young's modulus E =71240N mm/ 2 , and shear modulus

G=27190N mm/ 2.

It is observed that if v s( )=w s( )= θ_1,ss( )s =0 are assumed in Eqs. (23)-(33), all force and displacement boundary conditions may be satisfied, and only Eqs. (26), (29), and (33) are not trivial.

From F1 =0 at the free end, Eqs. (26) and (33), and using the approximation

1 5 2 0 1

+ ε ≈ , one may obtains

ε₀ 1 1 θ₁3 2 = − + A C E Ip s ( ) _, (34)

8 t Sect. e-e b L 3 X W, X₂ e e V , T X₁, U 0 1 2 3 1 3 0 2 θ1,s (10 -3 rad/mm ) T /C Linear solution Eq. (44) without EI₄/2 Eq. (44) (1 0 -3 ra d /m m ) 1 3 2 4 T /C Linear solution Eq. (44) without EI₄/2 Eq. (44) (1 0 -3 ra d /m m ) T C C EI A C E I EI C s p p s =θ1 + 4 − + + θ13 1 1 2 1 1 2 , [ ( )( )] , (35)

The results for cases (1)-(2)are shown in Figs. (5)-(6). As can be seen, the discrepancy between curves A, B, and C are remarked for case (2). Thus the underline term in Eq. (29) (or.Eq. (35))] may be required for reliable solutions.

Fig. 4. Cantilever beam subjected to end torsion.

Fig. 6. Torsion-twist rate curve (case 2).

4 CONCLUSIONS

A consistent co-rotational total Lagrangian formulation of second order beam theory is presented for the nonlinear analysis of three-dimensional elastic Euler beam with large rotations but small strains.

The beam structure is divided into several segments. A set of segment coordinate system is constructed at the current configuration of the deformed beam segment. The deformation, equilibrium equations and constitutive equations of the beam segment are defined in the segment coordinates. In order to describe the orientation of the beam cross section, a set of segment cross section coordinates associated with each cross section of the beam segment is employed. Three rotation parameters are used to describe the rotation of the segment cross section coordinates. In this paper the deformations of the beam segment are determined by the unit extension of the centroid axis and rotations of segment cross section coordinate systems relative to segment coordinate system.

The principle of virtual work and the consistent linearization of the fully geometrically nonlinear beam theory is used to derive the equilibrium equations and constitutive equation of the beam segment. In order to consider the nonlinear coupling among bending, twisting, and stretching deformations, all terms up to the second order of the deformation parameters are retained in the equilibrium equations and constitutive equations of the second order beam theory. However, some third order terms, which may not be negligible for some problems, are also retained. Numerical examples are presented to demonstrate the accuracy and effectiveness of the proposed second order beam theory.

REFERENCES

[1] J.C. Simo and L. Vu-Quoc, “The role of non-linear theories in transient dynamic analysis of flexible structures”, J. of Sound and Vibration, 119, 487-508 (1987).

[2] K.M. Hsiao, “Corotational total Lagrangian formulation for three-dimensional beam element”, AIAA Journal, 30, 797-804 (1992).

[3] H. Goldstein, Classical Mechanics, Addision-Wesley, Reading, MA, (1980).

[4] T.J. Chung, Continuum Mechanics, Prentice-Hall N.J., (1988).

[5] D.J. Dawe, Matrix and Finite Element Displacement Analysis of Structures,