Nonlinear Beam Theory for Engineers

Transcription

1 Nonlinear Beam Theory for Engineers Dewey H. Hodges, Ph.D. Professor, School of Aerospace Engineering Georgia Institute of Technology Atlanta, Georgia Nonlinear Beam Theory for Engineers Copyright c 1994 by Dr. Dewey H. Hodges 1 1

2 Introduction By now most rotorcraft dynamicists are accustomed to writing lengthy, complicated equations for analysis of blade dynamics Many extant sets of such equations contain a host of oversimplifications no initial curvature uniaxial stress field (or alternatively cross section rigid in its own plane) only torsional warping moderate deflections (or alternatively some sort of ordering scheme) isotropic or transversely isotropic material construction no shear deformation (invalidates theory for application to composite beams) Nonlinear Beam Theory for Engineers Copyright c 1994 by Dr. Dewey H. Hodges 1 2

3 Introduction (continued) Approximation concepts such as ordering schemes have been deeply ingrained in the thinking of most analysts However, some research has pointed out problems with this concept: It is virtually impossible to apply an ordering scheme in a completely consistent manner (Stephens, Hodges, Avila, and Kung 1982) Ordering schemes can lead to more lengthy equations due to expansion of transcendental functions (Crespo da Silva and Hodges 1986) An ordering scheme that works for one set of configuration parameters may not be suitable for a different set (Hinnant and Hodges 1989) Nonlinear Beam Theory for Engineers Copyright c 1994 by Dr. Dewey H. Hodges 1 3

4 Introduction (continued) In the present approach, there are several basic and exciting departures from the old school : No ordering scheme is needed or used exact kinematics for beam reference line displacement and cross-sectional rotation geometrically-exact equations of motion A compact matrix notation is used The beam constitutive law is based on a separate finite element analysis valid for anisotropic beams with nonhomogeneous cross sections The resulting mixed formulation is in the weakest form so that the requirements for the shape functions are minimal approximate element quadrature is not required Nonlinear Beam Theory for Engineers Copyright c 1994 by Dr. Dewey H. Hodges 1 4

5 Introduction (continued) Outline Kinematical Preliminaries Analysis of 3-D Beam Deformation Constitutive Equations from 2-D Finite Element Analysis 1-D Kinematics 1-D Equations of Motion 1-D Finite Element Solution Examples Nonlinear Beam Theory for Engineers Copyright c 1994 by Dr. Dewey H. Hodges 1 5

6 Vectors and Dyadics Kinematical Preliminaries Consider a rigid body B moving in a frame A (frames and rigid bodies are kinematically equivalent) Introduce a dextral unit triad Âi fixed in A (Roman subscripts vary from 1 to 3unless otherwise specified, and unit vectors are denoted by a bold Roman symbol with a hat ) Also, introduce a dextral triad ˆB i fixed in B Now ˆB i will vary in A as a function of time A vector is a first-order tensor A vector can always be expressed as a linear combination of dextral unit vectors For example, for an arbitrary vector v with v Bi = v ˆB i, it is always true that v = ˆB i v Bi (note that summation is implied over any repeated index) Nonlinear Beam Theory for Engineers Copyright c 2004 by Dr. Dewey H. Hodges 2 1

7 Kinematical Preliminaries (continued) Vectors and Dyadics (continued) Similarly, a dyadic is a second-order Cartesian tensor Dyadics are quadratic forms of dextral unit vectors For example, consider the relationship of a dyadic T and the matrix T ij of its components projected onto a mixed set of bases The transpose of T is simply T = ˆB i T ij  j T T = ÂjT ij ˆBi = ÂiT ji ˆBj For simplicity we will not carry the names of the associated base vectors in the symbol for a particular matrix of dyadic components For one type of dyadic, however the finite rotation tensor it is quite helpful to maintain basis association in the naming of the matrices Nonlinear Beam Theory for Engineers Copyright c 2004 by Dr. Dewey H. Hodges 2 2

8 Finite Rotation Kinematical Preliminaries (continued) One can characterize the rotational motion of B in A in these two ways: ˆB i = C BA Âi = Cij BA  j Here C BA is read as the finite rotation tensor of B in A and is given by C BA = ˆB i  i Both the tensor and its components depend on time Note the convention for naming the finite rotation tensor and its corresponding matrix of direction cosines Nonlinear Beam Theory for Engineers Copyright c 2004 by Dr. Dewey H. Hodges 2 3

9 Kinematical Preliminaries (continued) Finite Rotation (continued) The transpose is indicated by reversing the superscripts ( C BA) T = C AB so that  i = C AB ˆB i C BA is an orthonormal tensor so that where is the identity tensor C BA C AB = Nonlinear Beam Theory for Engineers Copyright c 2004 by Dr. Dewey H. Hodges 2 4

10 Kinematical Preliminaries (continued) Finite Rotation (continued) The matrix of direction cosines C BA is given by C BA ij = ˆB i Âj ( ) = C AB ji C BA is a matrix of the components of the transpose of the finite rotation tensor C AB so that C BA ij =Âi C AB Âj = ˆB i C AB ˆB j The matrix of direction cosines is also orthonormal C BA C AB = where is the 3 3identity matrix Nonlinear Beam Theory for Engineers Copyright c 2004 by Dr. Dewey H. Hodges 2 5

11 Kinematical Preliminaries (continued) Finite Rotation (continued) When an intermediate frame N is involved, C BA = C BN C NA C BA = C BN C NA Consider an arbitrary vector v; it is always possible to write v Zi = v Ẑi where Z is an arbitrary frame in which the dextral unit triad Ẑ i is fixed With the column matrix notation v Z = it is easily demonstrated that v Z1 v Z2 v Z3 v B = C BA v A v A = C AB v B Nonlinear Beam Theory for Engineers Copyright c 2004 by Dr. Dewey H. Hodges 2 6

12 Kinematical Preliminaries (continued) Finite Rotation (continued) For a rigid body B moving in frame A, there exist analogous vectordyadic operations the push-forward operation on v is defined by C BA v the pull-back operation on v is defined by C AB v As can be demonstrated, these operations rotate the vector by an amount commensurate with the change in orientation from A to B and from B to A respectively. Nonlinear Beam Theory for Engineers Copyright c 2004 by Dr. Dewey H. Hodges 2 7

13 Kinematical Preliminaries (continued) Finite Rotation (continued) To visualize the pull-back operation imagine the vector v frozen at some instant in time in a frame N which has a dextral triad ˆN i that is coincident with ˆB i rotate the frame N so that ˆN i lines up with Âi the rotated image of v is the result of the pull-back operation For the push-forward operation imagine the vector v frozen at some instant in time in a frame N which has a dextral triad ˆN i that is coincident with Âi rotate the frame N so that ˆN i lines up with ˆB i the rotated image of v is the result of the push-forward operation These operations are useful for describing deformation Nonlinear Beam Theory for Engineers Copyright c 2004 by Dr. Dewey H. Hodges 2 8

14 Kinematical Preliminaries (continued) Angular Velocity & Differentiation of Vectors Note that if a vector is moving in a frame then its time derivative in that frame will be nonzero However, if a vector is fixed in a frame then its time derivative in that frame will be zero Obviously, the concept of the derivative of a vector is then frame dependent Consider a vector b fixed in B where B is moving in A. Then, But A db dt B db dt 0 =0 Nonlinear Beam Theory for Engineers Copyright c 2004 by Dr. Dewey H. Hodges 2 9

15 Kinematical Preliminaries (continued) Angular Velocity & Differentiation of Vectors (continued) Thus, when the vector v is resolved along unit vectors that are fixed in the frame in which the derivative is being taken, the derivative of v is easily expressed as Z dv dt = Ẑi v Zi where Z is an arbitrary frame as before However, as will be clear from material to follow, it is not always convenient to differentiate a vector in this way Nonlinear Beam Theory for Engineers Copyright c 2004 by Dr. Dewey H. Hodges 2 10

16 Kinematical Preliminaries (continued) Angular Velocity & Differentiation of Vectors (continued) For differentiation of a vector in a different frame A dv dt = B dv dt + ω BA v where ω BA is the angular velocity of B in A given by ω BA = ˆB 1 A d ˆB 2 dt ˆB 3 + ˆB 2 A d ˆB 3 dt ˆB 1 + ˆB 3 A d ˆB 1 dt ˆB 2 = ω BA Bi ˆB i The derivatives of the unit vectors are easily expressed in terms of the direction cosines A d ˆB i dt = ĊBA ij  j = ĊBA ij Cjk AB ˆB k = e ijk ω BA Bj ˆB k = ω BA ˆB i Nonlinear Beam Theory for Engineers Copyright c 2004 by Dr. Dewey H. Hodges 2 11

17 Kinematical Preliminaries (continued) Virtual Rotation & Variation of Vectors Just as the time derivative of a vector depends on the frame in which the derivative is taken, so does the variation of a vector As Kane and others have shown, one can express the relationship between variations in two frames as A δv = B δv + δψ BA v where δψ BA is the virtual rotation of B in A given by δψ BA = ˆB 1 A δ ˆB 2 ˆB 3 + ˆB 2 A δ ˆB 3 ˆB 1 + ˆB 3 A δ ˆB 1 ˆB 2 = δψ BA Bi ˆB i and Z δv = Ẑiδv Zi Nonlinear Beam Theory for Engineers Copyright c 2004 by Dr. Dewey H. Hodges 2 12

18 Kinematical Preliminaries (continued) Virtual Rotation & Variation of Vectors (continued) The variations of the unit vectors are easily expressed in terms of the direction cosines A δ ˆB i = δc BA ij  j = δcij BA Cjk AB where δ( ) is the usual Lagrangean variation ˆB k = e ijk δψ BA Bj ˆB k = δψ BA ˆB i It is evident that the vectors ω BA and δψ BA can be regarded as operators which produce the time derivative and variation, respectively, in A of any vector fixed in B When an additional frame N is involved, Kane s addition theorem applies ω BA = ω BN + ω NA δψ BA = δψ BN + δψ NA Nonlinear Beam Theory for Engineers Copyright c 2004 by Dr. Dewey H. Hodges 2 13

19 Kinematical Preliminaries (continued) Virtual Rotation & Variation of Vectors (continued) Note that to obtain the virtual rotation vector, one need only replace the dots in the angular velocity vector with δ s, ignoring any other terms The virtual work in A of an applied torque T acting on a body B is simply δw = T δψ BA Nonlinear Beam Theory for Engineers Copyright c 2004 by Dr. Dewey H. Hodges 2 14

20 Velocity Chain Rules Kinematical Preliminaries (continued) The velocity of a point P moving in A can be determined by time differentiation in A of the position vector p P/O where O is any point fixed in A v PA = A dp P/O Often the calculation of velocity in this way is complicated, and it is helpful to step one s way from the known to the unknown using chain rules Two such rules are necessary in what follows: 2 points fixed on a rigid body (or in a frame) 1 point moving on a rigid body (or in a frame) dt Nonlinear Beam Theory for Engineers Copyright c 2004 by Dr. Dewey H. Hodges 2 15

21 Kinematical Preliminaries (continued) Velocity Chain Rules (continued) For 2 points P and Q fixed on a rigid body B having an angular velocity ω BA in A, the velocities of these points in A are related according to v PA = v QA + ω BA p P/Q For a point P moving on a rigid body B while B is moving in A, the velocity of P in A is given by v PA = v BA + v PB where v BA is the velocity of the point in B that is coincident with P at the instant under consideration (this can often be obtained by use of the other theorem) Nonlinear Beam Theory for Engineers Copyright c 2004 by Dr. Dewey H. Hodges 2 16

22 Tilde Notation Kinematical Preliminaries (continued) These relationships for the derivative and variation can be nicely expressed in matrix notation We ve already seen how an arbitrary vector v can for an arbitrary frame Z be expressed in terms of v Z1 v Z = v Z2 v Z3 The dual matrix ṽ Z ij = e ijk v Zk has the same measure numbers but arranged antisymmetrically ṽ Z = 0 v Z3 v Z2 v Z3 0 v Z1 v Z2 v Z1 0 Nonlinear Beam Theory for Engineers Copyright c 2004 by Dr. Dewey H. Hodges 2 17

23 Kinematical Preliminaries (continued) Tilde Notation (continued) With the tilde notation identities given in Hodges (1990) are helpful When Y and Z are 3 1 column matrices, it is easily shown that ( Z) T = Z ZZ =0 ỸZ = ZY Y T Z = Z T Ỹ Ỹ Z =ZY T Y T Z Ỹ Z = ZỸ + ỸZ where is the 3 3identity matrix. Nonlinear Beam Theory for Engineers Copyright c 2004 by Dr. Dewey H. Hodges 2 18

24 Kinematical Preliminaries (continued) Tilde Notation (continued) This notation also applies to vectors The tensor ṽ has components in the Z basis given by the matrix ṽ Z such that ṽ =v where is the identity dyadic =Ẑiṽ Z ij Ẑ j Note that for any vector w, v w = ṽ w Note that for any vectors v and w with measure numbers expressed in some common basis Z, ṽ Z w Z contains the measure numbers of the cross product v w in the Z basis The ( ) operator is sometimes called a cross product operator for obvious reasons Nonlinear Beam Theory for Engineers Copyright c 2004 by Dr. Dewey H. Hodges 2 19

25 Kinematical Preliminaries (continued) Tilde Notation (continued) Now with these definitions in mind we note that and ω BA B δψ BA B = ĊBA C AB = δc BA C AB In tensorial form these are ω BA = A Ċ BA C AB and δψ BA = A δc BA C AB Nonlinear Beam Theory for Engineers Copyright c 2004 by Dr. Dewey H. Hodges 2 20

26 Kinematical Preliminaries (continued) Implications of Euler s Theorem of Rotation Since C BA is orthonormal, its elements are not independent Euler s theorem of rotation stipulates that any change of orientation can be characterized as a simple rotation A motion is a simple rotation of B in A if during the motion a line L maintains its orientation in B and in A throughout the motion Euler proved that a minimum of four parameters is necessary for a mathematical description of rotation free of singularity: the three measure numbers of a unit vector e along the line L (e i = e Ai = e Bi ) and the magnitude of the rotation α With e = e 1 e 2 e 3 T, the matrix of direction cosines is C BA = cos α + ee T (1 cos α) ẽ sin α Nonlinear Beam Theory for Engineers Copyright c 2004 by Dr. Dewey H. Hodges 2 21

27 Kinematical Preliminaries (continued) Implications of Euler s Theorem of Rotation (continued) According to Kane, Likins, and Levinson (1983), the four Euler parameters denoted by ɛ 0 and are defined as ɛ = ɛ 1 ɛ 2 ɛ 3 ɛ i = e i sin(α/2) ɛ 0 = cos(α/2) They satisfy a constraint ɛ T ɛ + ɛ 2 0 =1 Nonlinear Beam Theory for Engineers Copyright c 2004 by Dr. Dewey H. Hodges 2 22

28 The matrix of direction cosines is given by while the angular velocity is C BA = ( 1 2ɛ T ɛ ) +2 ( ɛɛ T ɛ 0 ɛ ) ω BA B =2[(ɛ 0 ɛ) ɛ ɛ 0 ɛ] Nonlinear Beam Theory for Engineers Copyright c 2004 by Dr. Dewey H. Hodges 2 23

29 Kinematical Preliminaries (continued) Implications of Euler s Theorem of Rotation (continued) It is tempting to eliminate the fourth parameter via the constraint This always introduces a singularity, but where the singularity is can be controlled Introduce Rodrigues parameters θ = where θ i =2ɛ i /ɛ 0 =2e i tan(α/2) θ 1 θ 2 θ 3 Here the singularity is at ɛ 0 =0orα = 180 Nonlinear Beam Theory for Engineers Copyright c 2004 by Dr. Dewey H. Hodges 2 24

30 Kinematical Preliminaries (continued) Implications of Euler s Theorem of Rotation (continued) Then the matrix of direction cosines is ( ) 1 θt θ 4 + θθt 2 θ C BA = 1+ θt θ 4 and the angular velocity is ω BA B = ( θ 2) θ 1+ θt θ 4 Note that in the limit when α is very small, the parameters θ i are components of the infinitesimal rotation vector Nonlinear Beam Theory for Engineers Copyright c 2004 by Dr. Dewey H. Hodges 2 25

31 Kinematical Preliminaries (continued) Implications of Euler s Theorem of Rotation (continued) In some cases it is useful to characterize rotation via the so-called finite rotation vector the components of which are Φ i = e i α so that Φ 1 Φ= Φ 2 Φ 3 Then the matrix of direction cosines is C BA = exp ( ) Φ = Φ+ Φ Φ 2... Nonlinear Beam Theory for Engineers Copyright c 2004 by Dr. Dewey H. Hodges 2 26

32 and, with α 2 =Φ T Φ, the angular velocity is [ ωb BA = sin α ] (1 cos α) T (α sin α) Φ α α 2 +ΦΦ α ( 3 = Φ ) Φ Φ Nonlinear Beam Theory for Engineers Copyright c 2004 by Dr. Dewey H. Hodges 2 27

33 Epilogue Kinematical Preliminaries (continued) The material discussed so far has focused on fundamental rigid-body kinematics We must build on this foundation in the lectures following in order to understand the kinematical foundation of rigorous beam theory The next step is to develop a suitable method for determination of strain-displacement relations Nonlinear Beam Theory for Engineers Copyright c 2004 by Dr. Dewey H. Hodges 2 28

34 Undeformed State Geometry Kinematics of Deformation For describing the deformation, one needs to introduce frame a which is unaffected by beam deformation The following development is valid even if a is not an inertial frame Next one needs to specify the position vector from some arbitrary point O fixed in a to an arbitrary material point in the undeformed beam Denote this with r(x 1,x 2,x 3,t)= where x 1 is arclength along the beam reference line, x 2 and x 3 are cross-sectional coordinates, and t is time (henceforth, explicit time dependence is ignored) Now introduce a infinite set of frames b along the undeformed beam with a dextral triad ˆb i (x 1 )fixed therein The unit vector ˆb 1 (x 1 )is tangent to the undeformed beam reference line at some arbitrary value of x 1 Nonlinear Beam Theory for Engineers Copyright c 1998 by Dr. Dewey H. Hodges 3 1

35 Kinematics of Deformation (continued) Undeformed State Geometry (continued) In order to describe the geometry of the undeformed beam, we need to introduce covariant basis vectors for the undeformed state g i (x 1,x 2,x 3 )= r x i contravariant base vectors for the undeformed state g i (x 1,x 2,x 3 )= 1 2 g e ijkg j g k Note that g i g j = δ ij and ˆb i = g i (x 1, 0, 0) Since g = det(g i g j ) > 0, one can always write r(x 1,x 2,x 3 )=r(x 1 )+ξ where ξ = x 2ˆb2 + x 3ˆb3 is the position vector from the reference line to the arbitrary point within the reference cross section Nonlinear Beam Theory for Engineers Copyright c 1998 by Dr. Dewey H. Hodges 3 2

36 Kinematics of Deformation (continued) Deformed State Geometry Consider the position vector R(x 1,x 2,x 3 )from O to the same point in the deformed beam as that described by r(x 1,x 2,x 3 )in the undeformed beam The reference cross section may undergo two types of motion due to beam deformation rigid-body translations of the order of the beam length and large rigid-body rotations small deformation of the reference cross section Nonlinear Beam Theory for Engineers Copyright c 1998 by Dr. Dewey H. Hodges 3 3

37 Kinematics of Deformation (continued) Deformed State Geometry (continued) To capture this behavior mathematically, introduce a set of frames B for the deformed beam analogous to b in the undeformed beam Global rotation of the beam (from b to B)is described by C = C Bb Based on the above, we can express R in the form where R = r + u R = R + C (ξ + w) u describes the rigid-body translation the warping w describes the cross-sectional deformation Nonlinear Beam Theory for Engineers Copyright c 1998 by Dr. Dewey H. Hodges 3 4

38 Kinematics of Deformation (continued) Deformed State Geometry (continued) Consider the plane of material points that make up the reference cross section in the undeformed beam: neither the planar form nor the section shape are preserved in general if w is nonzero these points will lie very near a plane in the deformed beam, the orientation of which is determined by six constraints on w the orientation of B is determined by orientation of this plane Consider the set of material points that make up the reference line of the undeformed beam: the reference line of the deformed beam is not the same set of material points ˆB 1 is not in general tangent to the deformed beam reference line Nonlinear Beam Theory for Engineers Copyright c 1998 by Dr. Dewey H. Hodges 3 5

39 Kinematics of Deformation (continued) Deformed State Geometry (continued) Only the covariant basis vectors for deformed state are needed G i (x 1,x 2,x 3 )= R x i The deformation is most concisely described in terms of the deformation gradient tensor A = G i g i The tensor A has a meaning that is heuristically like R r Nonlinear Beam Theory for Engineers Copyright c 1998 by Dr. Dewey H. Hodges 3 6

40 Kinematics of Deformation (continued) Polar Decomposition Theorem A very useful theorem in continuum mechanics called the polar decomposition theorem (PDT)states that the deformation gradient can always be decomposed as where A = C U C is a finite rotation tensor (corresponding to a pure rotation) U is the right stretch tensor (corresponding to a pure strain) Thus, the PDT implies G i = A g i = C U g i Nonlinear Beam Theory for Engineers Copyright c 1998 by Dr. Dewey H. Hodges 3 7

41 Kinematics of Deformation (continued) Decomposition of the Rotation Tensor That is, to arrive at the coordinate lines in the deformed beam, coordinate lines in the undeformed beam (the g i s)are first strained (pre-dot-multiplication by U) then rotated (pre-dot-multiplication by C) C rotates an infinitesimal material element at a point (x 1,x 2,x 3 ) Some of this rotation is due to global rotation and some is due to local deformation (i.e., warping) Supposing that the warping is small (in some sense), let s decompose the total rotation so that C = C(x 1 ) C local (x 1,x 2,x 3 ) where C local (x 1,x 2,x 3 )is the local rotation tensor Nonlinear Beam Theory for Engineers Copyright c 1998 by Dr. Dewey H. Hodges 3 8

42 Kinematics of Deformation (continued) Decomposition of the Rotation Tensor (continued) Recall that any finite rotation tensor can be written in the form exp( φ) = + φ + φ φ Here we let C local = exp( φ) Thus the total finite rotation tensor at some point in the cross section is C = C exp( φ) Nonlinear Beam Theory for Engineers Copyright c 1998 by Dr. Dewey H. Hodges 3 9

43 Kinematics of Deformation (continued) Strain-Displacement Relations Now let s introduce the Cauchy strain tensor (also known as the engineering strain tensor, the Jaumann strain tensor, and the Biot strain tensor) Γ = U This strain definition is nice because it does not contain the host of strain squared terms that are a well known part of other strain tensors (such as the Green strain) Now, from the PDT and the strain definition, we have Γ = C T A = exp ( ) φ C T A Nonlinear Beam Theory for Engineers Copyright c 1998 by Dr. Dewey H. Hodges 3 10

44 Kinematics of Deformation (continued) Strain-Displacement Relations (continued) Γ is a Lagrangean strain tensor Thus, it is appropriate to resolve it along the undeformed beam reference triad ˆb i yielding Γ = ˆb i Γ ij ˆbj It is also appropriate to resolve the tensor φ along the undeformed beam triad ˆb i yielding φ = ˆb i φij ˆbj It now follows that the deformation gradient tensor is resolved along the mixed bases A = ˆB i A ij ˆbj or A ij =(ˆB i G k )(g k ˆb j ) Nonlinear Beam Theory for Engineers Copyright c 1998 by Dr. Dewey H. Hodges 3 11

45 Kinematics of Deformation (continued) Strain-Displacement Relations (continued) Things are now simple enough that we can finish up in matrix form Introducing φ = φ 1 φ 2 φ 3 the matrix of strain components become Γ = exp ( ) φ A In general an expression for A can be found rather easily, but φ is unknown Nonlinear Beam Theory for Engineers Copyright c 1998 by Dr. Dewey H. Hodges 3 12

46 Kinematics of Deformation (continued) Strain-Displacement Relations (continued) For the purpose of simplifying this expression for small local deformation, we let max Γ ij (x 1,x 2,x 3 ) = ɛ<<1 max φ ij (x 1,x 2,x 3 ) = ϕ<1 Then the strain becomes Γ=E φ ( E φ φe ) + O ( ϕ 4,ϕ 2 ɛ ) where E = A + AT 2 φ = A AT 2 Nonlinear Beam Theory for Engineers Copyright c 1998 by Dr. Dewey H. Hodges 3 13

47 Kinematics of Deformation (continued) Strain-Displacement Relations (continued) Assume that ϕ = O(ɛ r ) Since ɛ (the strain)is small compared to unity, two cases are of interest: Small local rotation: r 1: Γ=E Moderate local rotation: 1 2 r<1. Γ=E φ (E φ φe) Nonlinear Beam Theory for Engineers Copyright c 1998 by Dr. Dewey H. Hodges 3 14

48 Kinematics of Deformation (continued) Strain-Displacement Relations (continued) For the vast majority of engineering beam problems that are treatable with a beam theory at all, the small local rotation theory is adequate In most of the rest, incorporation of the nonlinearities due to warping that are exhibited in moderate local rotation theory should be adequate Nonlinear Beam Theory for Engineers Copyright c 1998 by Dr. Dewey H. Hodges 3 15

49 Simple Example Kinematics of Deformation (continued) The purpose of this example is to provide a simple illustration of the theory developed so far which will give us an explicit form of the strain Consider an initially straight beam that is undergoing planar deformation without warping (w = 0) For the undeformed state, the position vector r is given by r = x 1ˆb1 + x 2ˆb2 + x 3ˆb3 The covariant and contravariant base vectors are very simple g i = r x i = ˆb i = g i Nonlinear Beam Theory for Engineers Copyright c 1998 by Dr. Dewey H. Hodges 3 16

50 Kinematics of Deformation (continued) Simple Example For the deformed state, the position vector R is given by R =(x 1 + u b1 )ˆb 1 + x 2 ˆB2 + u b2ˆb2 + x 3ˆb3 Since the warping is zero, the reference cross-sectional plane in the deformed beam is made up of the same material points which make up the reference cross-sectional plane of the undeformed beam The covariant base vectors are G i = R x i such that G 1 =(1 + u b1)ˆb 1 + x 2 ( ˆB 2 ) + u b2ˆb 2 G 2 = ˆB 2 G 3 =ˆb 3 = ˆB 3 Nonlinear Beam Theory for Engineers Copyright c 1998 by Dr. Dewey H. Hodges 3 17

51 Kinematics of Deformation (continued) Simple Example (continued) The matrix of direction cosines C = C Bb can be simply represented with a single angle ζ ˆB 1 ˆB 2 ˆB = 3 where ζ is the cross section rotation cos ζ sin ζ 0 sin ζ cos ζ With this, the matrix of deformation gradient components in mixed bases can be written as A = (1 + u b1 )cos ζ + u b2 sin ζ x 2ζ 0 0 (1 + u b1 )sin ζ + u b2 cos ζ ˆb 1 ˆb 2 ˆb 3 Nonlinear Beam Theory for Engineers Copyright c 1998 by Dr. Dewey H. Hodges 3 18

52 Kinematics of Deformation (continued) Simple Example (continued) The strain for small local rotation applies here since there is no warping Γ= 1 2 (A + AT ) so that Γ 11 =(1 + u b1)cos ζ + u b2 sin ζ x 2 ζ 1 2Γ 12 =2Γ 21 = u b2 cos ζ (1 + u b1)sin ζ Notice that these strains, when linearized, reduce to those of a Timoshenko beam Nonlinear Beam Theory for Engineers Copyright c 1998 by Dr. Dewey H. Hodges 3 19

53 Realistic Example Kinematics of Deformation (continued) For the undeformed state r(x 1,x 2,x 3 )=r(x 1 )+x αˆbα (x 1 ) where r(x 1 )is the position vector to points on the reference line If the reference line is chosen as the locus of cross-sectional centroids, then r = r if and only if x α =0 where the angle brackets stand for the average value over the cross section The covariant base vectors are g 1 =r + x αˆb α g α =ˆb α It is helpful to derive some special formulae to express g 1 in a more recognizable form Nonlinear Beam Theory for Engineers Copyright c 1998 by Dr. Dewey H. Hodges 3 20

54 Kinematics of Deformation (continued) Realistic Example (continued) Letting primes denote differentiation with respect to x 1 in a r = ˆb 1 (ˆb i ) = k ˆb i = k ˆb i Note that k = k biˆbi is the curvature vector of the undeformed beam, and k is the curvature tensor of the undeformed beam defined by ( k = C ba) C Ab = k = ˆb i kij ˆbj = ˆb i e ijl k blˆbj Normally the components of k are known in the b basis: k b1 = the twist per unit length of the undeformed beam k bα = the components of curvature of the undeformed beam Nonlinear Beam Theory for Engineers Copyright c 1998 by Dr. Dewey H. Hodges 3 21

55 Kinematics of Deformation (continued) Realistic Example (continued) With these definitions, the contravariant base vectors become g 1 = ˆb 1 g g 2 = x 3k b1ˆb1 g + ˆb 2 g 3 = x 2k b1ˆb1 g + ˆb 3 where g =1 x2 k b3 + x 3 k b2 > 0 Normally, g is near unity Nonlinear Beam Theory for Engineers Copyright c 1998 by Dr. Dewey H. Hodges 3 22

56 Kinematics of Deformation (continued) Realistic Example (continued) Let us now use the more general displacement field referred to in the development R(x 1,x 2,x 3 )=R + x α ˆBα + w i ˆBi Here R = r + u, u = u biˆbi is the displacement vector of the beam, and w = w iˆbi The push-forward operation altered the bases on the last two terms We must constrain the warping in order for ˆB i and R to be well defined R = R if and only if w i =0 ˆB 1 x α R = 0 if and only if x α w 1 =0 ˆB 2 x 3 R = ˆB 3 x 2 R if and only if x 2 w 3 x 3 w 2 =0 Nonlinear Beam Theory for Engineers Copyright c 1998 by Dr. Dewey H. Hodges 3 23

57 Kinematics of Deformation (continued) Realistic Example (continued) The strain field can be conveniently expressed in terms of 1-D variables with the following definitions: R =(1 + γ 11 ) ˆB 1 +2γ 1α ˆBα ˆB i =K ˆB i The force strain components are then γ = C(e 1 + u b + k b u b ) e 1 where γ = γ 11 2γ 12 ; e 1 = 2γ The moment strains components are given by κ = K B k b so that κ = C C T + C k b C T k b Nonlinear Beam Theory for Engineers Copyright c 1998 by Dr. Dewey H. Hodges 3 24

58 Kinematics of Deformation (continued) Realistic Example (continued) In vector-dyadic form where z is an arbitrary frame γ =C zb R C zb r κ =C zb K C zb k Letting the z frame be b, γ and κ have γ and κ as measure numbers in the b basis The vector-dyadic form gives more insight as to what these expressions mean; see Hodges (1990)for this discussion For some problems it is better to pull back to the a basis and regard the measure numbers in the a basis as the generalized strains (see Bauchau and Kang 1993) Nonlinear Beam Theory for Engineers Copyright c 1998 by Dr. Dewey H. Hodges 3 25

59 Kinematics of Deformation (continued) Realistic Example (continued) Note that K is the curvature vector of the deformed beam defined by ( K = C Ba) C ab where K B1 = the twist per unit length of the deformed beam K Bα = the components of curvature of the deformed beam Nonlinear Beam Theory for Engineers Copyright c 1998 by Dr. Dewey H. Hodges 3 26

60 Kinematics of Deformation (continued) Realistic Example (continued) The strain field for small local rotation is given by the matrix E which is ge11 = γ 11 + x 3 κ 2 x 2 κ 3 + w 1 + k b1 (x 3 w 1,2 x 2 w 1,3 )+k b2 w 3 k b3 w 2 + κ 2 w 3 κ 3 w 2 2 ge 12 =2 ge 21 =2γ 12 x 3 κ 1 + w 2 + gw 1,2 + k b1 (x 3 w 2,2 x 2 w 2,3 )+k b3 w 1 k b1 w 3 + κ 3 w 1 κ 1 w 3 2 ge 13 =2 ge 31 =2γ 13 + x 2 κ 1 + w 3 + gw 1,3 + k b1 (x 3 w 3,2 x 2 w 3,3 )+k b1 w 2 k b2 w 1 + κ 1 w 2 κ 2 w 1 E 22 = w 2,2 2E 23 =2E 32 = w 2,3 + w 3,2 E 33 = w 3,3 Notice that if the underlined terms, which are O(ɛ 2 ), are neglected, then the strain field is linear in the generalized strains γ and κ Nonlinear Beam Theory for Engineers Copyright c 1998 by Dr. Dewey H. Hodges 3 27

61 Epilogue Kinematics of Deformation (continued) For the latter example γ 11 = Γ 11 but no such relation holds for the other measures A specified warping is used in the example worked out in Danielson and Hodges (1987) In general one needs to solve for the warping as in Hodges et al. (1992)for straight, untwisted beams, and in Cesnik and Hodges (1993)initially curved and twisted beams For most cases of interest to engineers, we can find the warping as a linear function of the 1-D strain measures, leading to a strain energy density of the form U = U(γ,κ) In other words, the warping disappears from the problem, affecting only the elastic constants of the beam (as in the St. Venant problem) Nonlinear Beam Theory for Engineers Copyright c 1998 by Dr. Dewey H. Hodges 3 28

62 In class we will outline a finite element approach to cross-sectional analysis and then proceed as if the section constants are known Nonlinear Beam Theory for Engineers Copyright c 1998 by Dr. Dewey H. Hodges 3 29

63 Stiffness Modeling Introduction Real rotor blades are made of composite materials These provide certain well-known advantages High strength weight Fatigue life, damage tolerance Directional nature However, they also introduce well-known complexities Nonhomogeneous Nonisotropic Nonlinear Beam Theory for Engineers Copyright c 1996 by Dr. Dewey H. Hodges 4 1

64 Introduction (continued) Stiffness Modeling (continued) Fig. 1: V-22 Blade Section Courtesy Bell Helicopters Nonlinear Beam Theory for Engineers Copyright c 1996 by Dr. Dewey H. Hodges 4 2

65 Stiffness Modeling (continued) Introduction (continued) Rotor blades are 3-D bodies and demand a 3-D approach Consider a 3-D representation of the strain energy U = 1 2 Γ T DΓ dx 2 dx 3 dx 1 where Γ = Γ(û) and û =û(x 1,x 2,x 3 ) This offers: a complete 3-D description of the problem nonhomogeneous, anisotropic, nonlinear no problem to represent, but O(10 5 ) degrees of freedom may be required! more than adequate motivation to attempt to use beam theory Nonlinear Beam Theory for Engineers Copyright c 1996 by Dr. Dewey H. Hodges 4 3

66 Introduction (continued) Stiffness Modeling (continued) Fig. 2: Schematic of discretized wing courtesy Mr. Mark Hale, Georgia Tech Nonlinear Beam Theory for Engineers Copyright c 1996 by Dr. Dewey H. Hodges 4 4

67 Introduction (continued) Stiffness Modeling (continued) To apply beam theory to a composite rotor blade, and expect an accurate answer, requires a 1-D strain energy functional for a composite beam that is somehow representative of the original 3-D functional i.e., one must capture 3-D behavior with a 1-D model! Problem: as we ve seen, there is a 3-D quantity (warping) present in the energy functional So, in summary, to achieve the objective one must start from a general 3-D representation, and solve the problem including nonhomogeneous, anisotropic materials all possible deformation in the 3-D representation determination of the warping as a function of 1-D variables Nonlinear Beam Theory for Engineers Copyright c 1996 by Dr. Dewey H. Hodges 4 5

68 Approach Stiffness Modeling (continued) A beam has one dimension that is much larger than the other two. Dimensional reduction takes the 3-D body and represents it as a 1-D body We know that this implies that small parameters must be exploited maximum magnitude of the strain ɛ<<1 h < l(h is the a typical cross-sectional diameter; l is the characteristic length of the deformation along the beam) h<r(r =1/ kb T k b) The result is the strain energy per unit length in terms of 1-D measures of strain with asymptotically exact cross-sectional elastic constants with asymptotically exact recovering relations Nonlinear Beam Theory for Engineers Copyright c 1996 by Dr. Dewey H. Hodges 4 6

69 Approach (continued) Stiffness Modeling (continued) We call the following procedure modeling Finding 2-D (sectional) elastic constants for use in 1-D (beam) theory Finding 1-D (beam) deformation parameters from loading and sectional constants Finding 3-D displacement, strain, and stress in terms of 1-D (beam) deformation parameters Analogy from elementary beam theory: Constitutive relation: M 2 = EI 22 u 3 Equilibrium equation: M 2 =0 Recovery relation: σ 11 = M 2x 3 I 22 Approach based on Cesnik and Hodges (1996) Nonlinear Beam Theory for Engineers Copyright c 1996 by Dr. Dewey H. Hodges 4 7

70 Approach (continued) Stiffness Modeling (continued) 3-D elastic constants 2-D cross-sectional analysis (linear) warping & strain recovery relations beam cross-sectional elastic constants 3-D recovery 3-D stress, strain, & displacement fields nonlinear beam analysis 1-D displacements, generalized strains, & stress resultants Fig. 3: Process of Beam Analysis Nonlinear Beam Theory for Engineers Copyright c 1996 by Dr. Dewey H. Hodges 4 8

71 Stiffness Modeling (continued) Kinematics Based on Danielson and Hodges [1987] Assumptions: Large displacements and rotations Small strain and local rotation Nonlinear Beam Theory for Engineers Copyright c 1996 by Dr. Dewey H. Hodges 4 9

72 Kinematics (continued) Stiffness Modeling (continued) ˆ b 3 Undeformed State ˆ b 2 x 1 ˆ b 1 r r u r ˆ B 3 ˆ B 2 s R R ˆ B 1 Deformed State R Fig. 4: Schematic of Beam Kinematics Nonlinear Beam Theory for Engineers Copyright c 1996 by Dr. Dewey H. Hodges 4 10

73 Stiffness Modeling (continued) Kinematics (continued) Undeformed State r(x 1,x 2,x 3 )=r(x 1 )+x αˆbα = r(x 1 )+hζ αˆbα r = ˆb 1 (ˆb i ) = k ˆb i = k ˆb i x α =0 = 1 A S dx 2 dx 3 = 1 A S gdx 2 dx 3 g 1 = ˆb 1 g g 2 = x 3k b1ˆb1 g + ˆb 2 g 3 = x 2k b1ˆb1 g + ˆb 3 g =1 x2 k b3 + x 3 k b2 > 0 Nonlinear Beam Theory for Engineers Copyright c 1996 by Dr. Dewey H. Hodges 4 11

74 Kinematics (continued) Deformed State Stiffness Modeling (continued) R(x 1,x 2,x 3 )=R(x 1 )+x α ˆBα (x 1 )+w n (x 1,ζ 2,ζ 3 ) ˆB n (x 1 ) ( ˆB i ) =K ˆB i = K ˆB i Constraints R =(1 + γ 11 ) ˆB 1 w n (x 1,ζ 2,ζ 3 ) =0 ζ 3 w 2 (x 1,ζ 2,ζ 3 ) = ζ 2 w 3 (x 1,ζ 2,ζ 3 ) Nonlinear Beam Theory for Engineers Copyright c 1996 by Dr. Dewey H. Hodges 4 12

75 Strain Field Stiffness Modeling (continued) Under the condition of small local rotation, Jaumann strain are Γ = 1 2 (χ + χt ) I Column matrix χ mn = ˆB m R x k g k ˆb n Γ= Γ 11 2Γ 12 2Γ 13 Γ 22 2Γ 23 Γ 33 T Nonlinear Beam Theory for Engineers Copyright c 1996 by Dr. Dewey H. Hodges 4 13

76 Stiffness Modeling (continued) Strain Field (continued) Linear in γ 11, κ, and warping w and its derivatives Γ= 1 h Γ h w +Γ ε ɛ +Γ R w +Γ l w ɛ = { γ11 hκ } γ 11 =e 1 T C(e 1 + u + ku) 1 0=e α T C(e 1 + u + ku) κ = C C T + C kc T k Nonlinear Beam Theory for Engineers Copyright c 1996 by Dr. Dewey H. Hodges 4 14

77 Strain Field (continued) Stiffness Modeling (continued) Γ h = ζ ζ ζ ζ 3 ζ ζ 3 [ ( Γ R = k 1 + ki ζ 3 g ζ ζ Γ ε = 1 g ) ζ 3 ] 1 0 ζ 3 ζ 2 0 ζ ζ [ ] I Γ l = 1 g 0 Nonlinear Beam Theory for Engineers Copyright c 1996 by Dr. Dewey H. Hodges 4 15

78 Strain Energy of a Beam Stiffness Modeling (continued) The strain energy density for a beam per unit length U = 1 2 Γ T D Γ The 3-D Jaumann stress Z, which is conjugate to the Jaumann strain Γ is Z = D Γ Nonlinear Beam Theory for Engineers Copyright c 1996 by Dr. Dewey H. Hodges 4 16

79 Stiffness Modeling (continued) Basic Three-Dimensional Problem The basic 3-D problem can be now represented as the following minimization problem U [ ɛ(x 1 ),w(x 1,ζ 2,ζ 3 ), w(x ] 1,ζ 2,ζ 3 ),w (x 1,ζ 2,ζ 3 ) dx 1 ζ α + potential energy of external forces min Nonlinear Beam Theory for Engineers Copyright c 1996 by Dr. Dewey H. Hodges 4 17

80 Dimensional Reduction Stiffness Modeling (continued) In constructing a 1-D beam theory from 3-D elasticity, one attempts to represent the strain energy stored in the 3-D body by finding the strain energy which would be stored in an imaginary 1-D body. The warping displacement components w n (x 1,ζ 2,ζ 3 ) must be written as functions of the 1-D functions R(x 1 ) and ˆB n (x 1 ) too complicated to do exactly due to nonlocal dependence One can and must take advantage of small parameters Nonlinear Beam Theory for Engineers Copyright c 1996 by Dr. Dewey H. Hodges 4 18

81 Small Parameters Two small parameters Stiffness Modeling (continued) The ratio h l of the maximum dimension of the cross section (h) divided by a length parameter (l) which is characteristic of the rapidity with which the deformation varies along the beam; and The maximum dimension of the cross section times the maximum magnitude of initial curvature or twist h R Since both of them have the same numerator, expansion in h l is equivalent to the expansion in h only h R and Note: the maximum magnitude of the strain, ε, was only necessary at the definition of the strain field Nonlinear Beam Theory for Engineers Copyright c 1996 by Dr. Dewey H. Hodges 4 19

82 Stiffness Modeling (continued) Variational-Asymptotical Method Variational-asymptotical methods are applied to energy functionals rather than to differential equations essentially the work of Berdichevsky and co-workers (1976, 1979, 1981, 1983, 1985, etc.) considers small parameters if µ is a typical material modulus, the strain energy is of the form ( ) ( ) h h µε [O(1) 2 + O + O l R ( ) 2 ( ) 2 ( ) ] h h h 2 + O + O + O +... l R lr Nonlinear Beam Theory for Engineers Copyright c 1996 by Dr. Dewey H. Hodges 4 20

83 Discretization Stiffness Modeling (continued) Finite element discretization of the warping Strain energy density w(x 1,ζ 2,ζ 3 )=S(ζ 2,ζ 3 )W (x 1 ) ( ) 2 1 2U = W T EW h ( ) W T (D hε ɛ + D hr W + D hl W ) h +(1)(ɛ T D εε ɛ + W T D RR W + W T Dll W +2W T D Rε ɛ + 2 W T Dlε ɛ + 2 W T D Rl W ) Nonlinear Beam Theory for Engineers Copyright c 1996 by Dr. Dewey H. Hodges 4 21

84 Discretization (continued) Stiffness Modeling (continued) The following definitions were introduced E = [Γ h S] T D [Γ h S] D hε = [Γ h S] T D [Γ ε ] D hl = [Γ h S] T D [Γ l S] D lε = [Γ l S] T D [Γ ε ] D Rl = [Γ R S] T D [Γ l S] D εε = [Γ ε ] T D [Γ ε ] D hr = [Γ h S] T D [Γ R S] D Rε = [Γ R S] T D [Γ ε ] D RR = [Γ R S] T D [Γ R S] D ll = [Γ l S] T D [Γ l S] Note: these matrices carry information about the material properties and geometry of a given cross section Nonlinear Beam Theory for Engineers Copyright c 1996 by Dr. Dewey H. Hodges 4 22

85 Classical Theory Warping field Stiffness Modeling (continued) Constraints W = hw 0 + h 2 W 1 + h 3 W 2 W T HΨ cl =0 where H = S T S EΨ cl =0 Ψ T clhψ cl = I Nonlinear Beam Theory for Engineers Copyright c 1996 by Dr. Dewey H. Hodges 4 23

86 Stiffness Modeling (continued) Classical Theory (continued) Strain energy up to order h 2 2U =(1)[ ɛ T D εε ɛ + W T 0 EW W0 T D hε ɛ ] + 2 (h)[w T 0 EW 1 + W T 1 D hε ɛ + W T 0 D hr W 0 + W T 0 D Rε ɛ] +(h) 2 [2W0 T EW 2 + W T 1 EW W0 T (D hr + DhR)W T W2 T D hε ɛ + W T 0 D RR W W1 T D Rε ɛ] Nonlinear Beam Theory for Engineers Copyright c 1996 by Dr. Dewey H. Hodges 4 24

87 Classical Theory (continued) Evaluation of W 0 Stiffness Modeling (continued) In accord with the variational-asymptotical method, one keeps only the dominant interaction terms between W and ɛ as well as the dominant quadratic term in W 2U 0 = ( 1 h ) 2 W T EW + ( 1 h ) 2 W T D hε ɛ The Euler-Lagrange equation (including constraints) is ( ) 1 EW + D hε ɛ = HΨ cl µ h The Lagrange multiplier becomes µ =Ψ T cld hε ɛ Nonlinear Beam Theory for Engineers Copyright c 1996 by Dr. Dewey H. Hodges 4 25

88 Classical Theory (continued) Stiffness Modeling (continued) Evaluation of W 0 (continued) The warping becomes ( ) 1 EW = (I HΨ cl Ψ T h cl)d hε ɛ Note the generalized inverse EE + cl = I HΨ clψ T cl E + cl E = I Ψ clψ T cl H E + cl EE+ cl = E+ cl Nonlinear Beam Theory for Engineers Copyright c 1996 by Dr. Dewey H. Hodges 4 26

89 Classical Theory (continued) Stiffness Modeling (continued) Evaluation of W 0 (continued) Prismatic Stiffness Matrix W = he + cl D hεɛ = hw 0 2U = ɛ T Aɛ A = D εε [D hε ] T E + cl [D hε] Nonlinear Beam Theory for Engineers Copyright c 1996 by Dr. Dewey H. Hodges 4 27

90 Stiffness Modeling (continued) Classical Theory for Beams with Modest Initial Twist and Curvature Evaluation of W 1 Perturbing the warping W, one obtains W = hw 0 + h 2 W 1 The perturbation of the energy then becomes 2U 1 =h 2 W T 1 EW hW T 1 (D hε ɛ + EW 0 + h 2 D hr W 1 )+ +2h 2 W T 1 (D hr + DhR)W T 0 Nonlinear Beam Theory for Engineers Copyright c 1996 by Dr. Dewey H. Hodges 4 28

91 Stiffness Modeling (continued) Classical Theory for Beams with Modest Initial Twist and Curvature (continued) Evaluation of W 1 (continued) Minimization of the perturbed energy (including constraints) yields the Euler-Lagrange equation ( ) 1 EW 1 + HΨ cl Ψ T h cld hε ɛ +(D hr + DhR)W T 0 = HΨ cl µ The Lagrange multiplier becomes ( ) 1 µ = Ψ T h cld hε ɛ +Ψ T cl(d hr + DhR)W T 0 The warping then becomes EW 1 = (I HΨ cl Ψ T cl)(d hr + D T hr)w 0 Nonlinear Beam Theory for Engineers Copyright c 1996 by Dr. Dewey H. Hodges 4 29

92 Stiffness Modeling (continued) Classical Theory for Beams with Modest Initial Twist and Curvature (continued) Evaluation of W 1 (continued) The strain energy corrected to first order in h/r is found (all influence of the perturbed warping cancels out) where 2U = ɛ T A r ɛ A r = D εε (D hε ) T (E + cl D hε)+ + h[(e + cl D hε) T (D hr + D T hr)(e + cl D hε)+ (E + cl D hε) T D Rε D Rε T (E + cl D hε)] Higher-order corrections in h/r have been developed by Cesnik and Hodges (1996) Nonlinear Beam Theory for Engineers Copyright c 1996 by Dr. Dewey H. Hodges 4 30

93 Summary Stiffness Modeling (continued) The 1-D constitutive law that follows from the energy is of the form { FB1 M B } = [ S ] { γ 11 κ } For isotropic, prismatic beams when x 2 and x 3 are principal axes F B1 M B1 M B2 M B3 = EA GJ EI EI 3 γ 11 κ 1 κ 2 κ 3 Nonlinear Beam Theory for Engineers Copyright c 1996 by Dr. Dewey H. Hodges 4 31

94 Summary (continued) Stiffness Modeling (continued) Adding modest pretwist and initial curvature, and letting D = I 2 + I 3 J) and β =1+ν, one obtains F B1 M B1 M B2 M B3 = EA EDk 1 βei 2 k 2 βei 3 k 3 EDk 1 GJ 0 0 βei 2 k 2 0 EI 2 0 βei 3 k EI 3 For generally anistropic beams, the matrix S becomes fully populated (while remaining symmetric and positive definite, of course) γ 11 κ 1 κ 2 κ 3 Nonlinear Beam Theory for Engineers Copyright c 1996 by Dr. Dewey H. Hodges 4 32

95 Epilogue Stiffness Modeling (continued) A comprehensive beam modeling scheme is presented which allows for systematic treatment of all possible types of deformation in composite beams is ideal for 2-D finite element sectional analysis gives asymptotically exact section constants leads to the geometrically exact beam equations of Hodges (1990) gives formulae needed for recovering strain and stress distributions Nonlinear Beam Theory for Engineers Copyright c 1996 by Dr. Dewey H. Hodges 4 33

96 Objectives Geometrically Exact Beam Equations Present an exact set of equations for the dynamics of beams in a moving frame suitable for rotorcraft applications Present the equations in a matrix notation so that the entire formulation can be written most concisely Present a unified framework in which other less general developments can be checked for consistency Show how differences that normally arise in beam analyses can be interpreted in light of this unified framework Show how the present unified framework can be used to develop an elegant basis for accurate, efficient, and robust computational solution techniques Nonlinear Beam Theory for Engineers Copyright c 1996 by Dr. Dewey H. Hodges 5 1

97 Geometrically Exact Beam Equations (continued) Background Published work reveals differences in the way the displacement field is represented different numbers of kinematical variables to describe motion of the cross sectional frame different variables to describe finite rotation of this frame different orthogonal base vectors for measurement of displacement Nonlinear Beam Theory for Engineers Copyright c 1996 by Dr. Dewey H. Hodges 5 2

98 Geometrically Exact Beam Equations (continued) Present Approach In the present approach the kinematical equations are exact and separate from equilibrium and constitutive law developments generalized strains are written in simple matrix notation change of displacement and orientation variables affects only the kinematics (a relatively small portion of the analysis) Reissner (1973) derived exact intrinsic equations for beam static equilibrium (limited to unrestrained warping) no displacement or orientation variables ( intrinsic ) reduce to the Kirchhoff-Clebsch-Love equations when shear deformation is set equal to zero geometrically exact all correct beam equations can be derived from these intrinsic generalized strains were derived from virtual work Nonlinear Beam Theory for Engineers Copyright c 1996 by Dr. Dewey H. Hodges 5 3

99 Geometrically Exact Beam Equations (continued) Present Approach (continued) Here, based on Hodges (1990) the exact equilibrium equations are extended to account for dynamics derived from Hamilton s weak principle (HWP) to facilitate development of a finite element method Asymptotic analyses show that for slender, closed-section beams, a linear 2-D cross-sectional analysis determines appropriate sectional elastic constants for use in a nonlinear 1-D beam analysis Thus, in the present work we presuppose that an elastic law is given as a 1-D strain energy function The present analysis is based on exact kinematics, exact kinetics and an approximate constitutive law ( ordering scheme not needed) A mixed finite element approach can be developed from HWP in which there are many computational advantages Nonlinear Beam Theory for Engineers Copyright c 1996 by Dr. Dewey H. Hodges 5 4

100 Geometrically Exact Beam Equations (continued) Displacement/Rotation Recall that the displacement field is represented by r =r + ξ = r + x 2ˆb2 + x 3ˆb3 R =R + C (ξ + w) =r + u + x 2 ˆB2 + x 3 ˆB3 + w i ˆBi with C as the global rotation tensor Nonlinear Beam Theory for Engineers Copyright c 1996 by Dr. Dewey H. Hodges 5 5

101 Geometrically Exact Beam Equations (continued) Generalized Strains Force strains γ = γ 11 ( 2γ 12 = C e 1 + u b + k ) b u b e 1 2γ 13 where u b is the column matrix whose elements are the measure numbers of the displacement along ˆb i and ( ) signifies an antisymmetric matrix Moment strains where κ = κ 1 κ 2 = K B k b κ 3 K B = C C T + C k b C T Nonlinear Beam Theory for Engineers Copyright c 1996 by Dr. Dewey H. Hodges 5 6

102 Geometrically Exact Beam Equations (continued) Generalized Speeds Introduce column matrices of known measure numbers along ˆb i for inertial velocity of the undeformed beam reference axis v b angular velocity of the undeformed beam cross sectional frame ω b Now as generalized speeds in the sense of Kane and Levinson (1985), elements of column matrices which contain measure numbers along ˆB i for inertial velocity of the deformed beam reference axis V B = C(v b + u b + ω b u b ) angular velocity of the deformed beam cross-sectional frame Ω B = ĊCT + C ω b C T Nonlinear Beam Theory for Engineers Copyright c 1996 by Dr. Dewey H. Hodges 5 7

103 Geometrically Exact Beam Equations (continued) Hamilton s Weak Principle As developed by Borri et al. (1985), HWP for the present problem is t2 t 1 l 0 [ δ(k U)+δW ] dx1 dt = δa Here t 1 and t 2 are arbitrary fixed times K and U are the kinetic and strain energy densities per unit length, respectively δa is the virtual action at the ends of the beam and at the ends of the time interval δw is the virtual work of applied loads per unit length Nonlinear Beam Theory for Engineers Copyright c 1996 by Dr. Dewey H. Hodges 5 8