The general assumptions in the linear beam theory are as follows: a.

The beam reference (longitudinal) axis is at any arbitrary location of the cross section.

As seen in Fig. 1, the displacement vector up can be expressed at any cross section point P as up=upx,y,zvpx,y,zwpx,y,z=ux-yφzx+zφyxvx-zφxxwx+yφxx The three components of the strains occurring in the beam can be expressed as the derivatives of the displacement functions up, vp and wp. The axial strain εxx and the two shear strain components γxz and γxy are given by the following: εxx=∂up∂x=u′-yφz′+zφy′γxz=∂up∂z+∂wp∂x=φy+w′︸γy+yφx′γxy=∂up∂y+∂vp∂x=-φz+v′︸γy-zφx′, where γz and γy are the constant shear strains that are not neglected in Timoshenko beam theory. γy=-φz+v′γz=φy+w′

The virtual work components for internal forces corresponding to normal stresses and shear stresses are given by the following: -δWi=∫xδu′N+δφz′Mz+δφy′Mydx+∫xδγzQzdx+∫xδγyQydx+∫xδφ′xMTPdx, where N is the axial force, Mz and My are bending internal moments, Qy and Qz are the corresponding shear forces and MTP is the St. Venant torsional moment.

With the introduction of the constitutive relations of Hooke's material law for the stresses corresponding to the internal forces in Eq. (6) and expressing the strains using Eqs. (4a)–(4c) and (5a)–(5b), the virtual work relationship is reformulated as -δWi=∫x{E∫AdA︸A⋅u′-∫AydA︸Ay⋅φz′+∫AzdA︸Az⋅φy′δu′-E∫AydA︸Ay⋅u′-∫Ay2dA︸Ayy⋅φz′+∫AyzdA︸Ayz⋅φy′δφz′+E∫AzdA︸Az⋅u′-∫AyzdA︸Ayz⋅φz′+∫Az2dA︸Azz⋅φy′δφy′+GAszw′+φyδw′+Asz(w′+φy)δφy+Asyv′-φzδv′-Asy(v′-φz)δφz+G∫A(z2+y2)dA︸ITφx′δφx′}dx Here, Asz=αsz⋅A and Asy=αsy⋅A are the shear areas in the z and y directions respectively, αsz, αsy are the corresponding shear correction coefficients, A is the cross-sectional area, Ay is the static moment with respect to the z axis, Az is the static moment with respect to the y axis, Ayy is the moment of inertia with respect to the z direction, Azz is the moment of inertia with respect to the y direction, Ayz is the deviation moment of inertia and IT is the torsional moment of inertia.

After the coefficient comparison in Eqs. (6) and (7) the internal forces corresponding to the normal stresses can be expressed by introducing the cross sectional stiffness matrix EA: N-MzMy=EAEAyEAzEAyEAyyEAyzEAzEAyzEAzz︸EA⋅u′-φz′φy′⇒u′-φz′φy′=(EA)-1N-MzMy The shear stress components in Eqs. (6) and (7) lead to the following relations: Mx=MTP=GITφx′ Qz=GAszw′+φyQy=GAsyv′-φz The relation described in Eq. (10a) and (10b) implies that the chosen reference axis coincides with the shear centre due to neglected shear–torsion coupling terms in Eq. (7).

For the special case where the cross section coordinate system coincides with the principal axes, the deformation relationship in Eq. (8) reduces to N-MzMy=EA000EAYY000EAzz⋅u′-φz′φy′⇒u′-φz′φy′=(EA)-1N-MzMy

The virtual work relation in Eq. (7) is rewritten for the case where the beam coordinate system coincides with the principal axis of the cross section – see Eq. (11). -δWi=∫x{EAu′δu′+EAyyφz′δφz′+EAzzφy′δφy′+(GITφx′)δφx′+GAszw′+φy(δw′+δφy)+GAsyv′-φzδv′-δφz}dx After the partial integration of Eq. (12) the cross section deformation relations and differential equilibrium conditions for the Timoshenko beam element are compiled in a first-order differential equation system (see Eqs. 13, 14). For the Timoshenko beam with an arbitrary beam reference axis at any point on the cross section (see Eq. 8), the system of differential equations can be expressed in the following form: ddxuvwφxφyφzNQyQzMxMyMz=000000S11000S13-S1200000101GAsy00000000-10001GAsz0000000000001GIT00000000S31000S33-S32000000-S21000-S23S220000000000000000000000000000000000000000000000000000000010000000000-10000⋅uvwφxφyφzNQyQzMxMyMz+000000-px-py-pz-mT-my-mz The differential equation system for the Timoshenko beam with the beam reference axis on principal axes can be represented in the following matrix form: ddxuvwφxφyφzNQyQzMxMyMz=0000001EA0000000000101GAsy00000000-10001GAsz0000000000001GIT0000000000001EAzz0000000000001EAzz0000000000000000000000000000000000000000000000000000000010000000000-10000⋅uvwφxφyφzNQyQzMxMyMz+000000-px-py-pz-mT-my-mz The entries Sij in Eq. (13) are determined by the inversion of the cross-sectional stiffness matrix in Eq. (8): u′-φz′φy′=EAEAyEAzEAyEAyyEAyzEAzEAyzEAzz-1︸S⋅N-MzMy=S11S12S13S21S22S23S31S32S33︸S⋅N-MzMy

The differential equations of motion for the differential beam element (see Fig. 2) can be written in a matrix form according to Stanoev (2007) and Müller (2012) as follows: z1,xz2,x=A11A12A21A22︸A⋅z1z2+b1b2+0m⋅z¨1 In the above-mentioned equation, matrix A is the coefficient matrix (see Eqs. 13, 14) and vector z=z1z2 is the state vector, where z1=uxvxwxφxxφyxφzxTis the vector of the displacement functions, andz2=NxQyxQzxMxxMyxMzxTis the vector of internal (section) force functions. The vector b=b1b2 contains the known excitation forces; however, for an eigenvalue problem b=0. The coefficient matrix A and the state vector z in addition to excitation force b constitute the static part of the motion equation. The kinetic part of the motion equation can be expressed using a matrix of the interpolation functions and nodal acceleration vector as follows: 0m⋅z¨1=0mΦ1(x)Φ2(x)︸bm⋅V¨(a)V¨(b)︸V¨R(e,t), where z¨1=u¨v¨w¨φ¨xφ¨yφ¨zT is the vector of accelerations, Φ1xΦ2xϵR(6×12) is the matrix of interpolation functions (see Sect. 3.3), V¨a,V¨bϵR(6×1) is the vector with nodal accelerations and mϵR(6×6) is the inertia matrix of the differential beam element (see Sect. 3.2).

The inertia matrix in Eq. (18) implies that the distributed mass μx (kg m-1) is applied eccentrically at any location (y,z) in the cross section. The inertia matrix is expressed according to Stanoev (2013): m=μ⋅1000z-y010-z00001y000-zy(y2+z2+Θpμ)00z000(z2+Θyμ)-yz-y000-yz(y2+Θzμ), where Θp, Θy and Θz in (kg m) are the mass moments of inertia for the cross section. Θy=μ⋅IyA=μ⋅AzzA,Θz=μ⋅IzA=μ⋅AyyA,Θp=Θy+Θz

The acceleration term z¨1 in Eq. (17) is expressed using the product of Hermite interpolating polynomials and the nodal acceleration vectors V¨a and V¨b.

Shape functions for axial and torsional deformations uξ and φxξ, respectively, are derived using first-order polynomial as follows: uξ=a1+a2ξ=1ξ︸Nua1a2︸a=Nu⋅a,ξ=xL To express the coefficients aj in terms of the nodal displacements uξ=0=ua, uξ=1=ub or in terms of the torsion φxξ=0=φxa,φxξ=1=φxb the following relations are applied to Eq. (20): uaub︸u=1011︸Sa1a2→a1a2=S-1⋅u=10-11uaub Substituting Eq. (21) into Eq. (20) results in the shape function for axial deformation uξ=1ξ10-11︸Guuaub︸vu=Nu⋅Gu⋅vu=Hu1︸1-ξua+Hu2︸ξub and in the following function for torsional deformation φx: φxξ=Hu1φxa+Hu2φxb

The relationships in Eq. (10a), Eq. (11) and the corresponding part of Eq. (14) are a starting point to derive approximation functions for bending deformation in the xz plane: Qz=GAszw′+φy=My′=EAzzφy′′ Using the above-mentioned relation the expression for w′ is given by w′=-φy+EAzzGAszφy′′=-φy+ηyL212φy′′︸γy←ηy=12EAzzGAszL2 The translational deformation function wξis approximated by a cubic polynomial function: wξ=c0+c1ξ+c2ξ2+c3ξ3=1ξξ2ξ3︸Nwc0c1c2c3︸c=Nw⋅c Using the constant shear strain relation in Eqs. (5b) and (26) the polynomial expression for constant shear strain can be deduced as follows: γz=ηyL212φy′′, whereφy′′=-w′′′=-6c3L3 By including Eqs. (27) and (26) in Eq. (25) the polynomial expression for φyξ results in φyξ=1L0-1-2ξ-ηy2-3ξ2c0c1c2c3=Nφy⋅c To determine the coefficients cj the following boundary conditions are applied: vw=waφyawbφyb=w(ξ=0)φy(ξ=0)w(ξ=1)φy(ξ=1)=10000-1L0-ηy2L11110-1L-2L-3+ηy2L︸Hw⋅c0c1c2c3=Hw⋅c The inversion of Eq. (29) yields c=Hw-1⋅vw=11+ηy⋅ηy+1000-ηy-L(ηy+2)2ηyηy2L-3L(ηy+4)23-L(ηy-2)22-L-2-L︸Gwwaφyawbφyb=Gw⋅vw The interpolation functions for wx,y,z and φyx,y,z, Eqs. (26) and (28), can be expressed by employing Eq. (30): wξ=1ξξ2ξ3︸Nw1ηy+1ηy+1000-ηy-L(ηy+2)2ηyηy2L-3L(ηy+4)23-L(ηy-2)22-L-2-L︸Gwwaφyawbφyb︸vw=Hw1wa+Hw2φya+Hw3wb+Hw4φyb, where the products of both matrices Nw and Gw are introduced as functions Hwj(j=1,…,4). φyξ=1L0-1-2ξ-ηy2-3ξ2︸Nφy1ηy+1ηy+1000-ηy-L(ηy+2)2ηyηy2L-3L(ηy+4)23-L(ηy-2)22-L-2-L⋅waφyawbφyb=Hφy1wa+Hφy2φya+Hφy3wb+Hφy4φyb In Eq. (32), the functions Hφyj (j=1,…,4) are introduced in an analogous manner.

A similar method is used for determining the approximation functions vξ and φzξ for bending deformation in the xy plane.

Starting with Eqs. (10b), (11) and (14), the following is obtained: Qy=GAsyv′-φz=-Mz′=-EAyyφz′′v′=φz-EAyyGAsyφz′′=φz+ηzL212φz′′←ηz=12EAyyGAsyL2 The approximations analogous to Eqs. (31) and (32) can be derived as follows: vξ=Hv1va+Hv2φza+Hv3vb+Hv4φzbφzξ=Hφz1va+Hφz2φza+Hφz3vb+Hφz4φzb The H∗j functions developed in Eqs. (31), (32), (35) and (36) are “static” shape functions for the Timoshenko beam. Supposing dependence on time alone for the nodal displacement vectors Va and Vb, the matrix of the interpolation functions Φ1xΦ2x in the inertia term Eq. (17) can be developed using Eqs. (31), (32), (35) and (36) (see Kusuma Chandrashekhara, 2018): Φ1(x)=Hu1000000Hv1000Hv200Hw10Hw20000Hu10000Hφy10Hφy200Hφz1000Hφz2,Φ2(x)=Hu2000000Hvs000Hv400Hws0Hw40000Hu20000Hφys0Hφy400Hφzs000Hφz4,

The special form of the numerical Runge–Kutta fourth-order integration method applied here is described in detail in Müller and Wolf (1975) and Schenk (2012). The integration operator is applied to the equations of motion in Eq. (16), i.e. z1,xz2,x=A⋅z1z2+b1b2+0mΦ1(x)Φ2(x)︸bm⋅V¨Re,t In order to gain sufficient numerical precision the beam axis needs to be divided into at least 20 integration intervals. The integration operator transfers the known state vector at beginning of the integration interval to the end of the interval. The integration procedure starts with the state vector at the first node a, i.e. at location x=0: z1(x=0)z2(x=0)︸z(a)=1⋯1za The integration operator is subsequently applied to the evaluated coefficient matrix A at each interval by excluding the initial state vector za. The result are static field matrices Fx,a, multiplicatively linked to za see Eq. (40). Therefore, each Fx,a matrix “transfers” the state vector at location x=0 to the end x of the integration interval considered (transfer matrix method). In the frame of the integration procedure the actual field matrix Fx,a serves column wise as an initial vector for the next interval, and the components of the state vector za remain excluded. The beam “load” vector b1b2, Eq. (38), evaluated in the actual interval, yields the β1β21 column in the Fx,a matrix after integration – Eq. (40).

The numerical integration of the inertia term bm in Eq. (38) is carried out column wise analogously to the “load” vector, by excluding the nodal accelerations V¨Re,t – the results are kinetic field matrices Hx,a at the end of each integration interval (at location x) (see Eq. 40): z1(x)z2(x)1︸=α11α12β1α21α22β2001︸⋅z1(a)z2(a)1z(x)=F(x,a)⋅z(a)+H11H120H21H220001︸⋅V¨(a)V¨(b)0+H(x,a)⋅V¨R(e) This type of numerical integration allows (slightly) varying values of the coefficients of the A matrix, the bm inertia term and the b vector along the beam axis – i.e. all stiffness, mass and external load values of the beam element may vary. After the last integration step at the second node b, at location x=L, static Le and kinetic He field matrices are obtained: z1(b)z2(b)1=L(e)⋅z1(a)z2(a)1+H(e)⋅V¨(a)V¨(b)0 where L(e)=L11L12f1L21L22f2001,H(e)=H11H120H21H220001 According to Eqs. (13) and (14) the state variable z1 represents six-component displacement vectors for Va and Vb, respectively, and the state variable z2 represents six-component internal forces vectors Sa and Sb, at locations x=0 and x=L respectively.

By solving the matrix equation in Eq. (41a) for Sa and Sb, and accounting for the definitions of internal forces in finite element beam formulation (see Fig. 2), Fa=-Sa,Fb=Sb, one can derive the element stiffness matrix Ke, the element mass matrix Me and the element forces and moments F0 using simple matrix operations as shown in Eq. (43). Then the beam element relationships for the internal forces can be formulated as F(a)F(b)=F0(a)F0(b)+KaaKabKbaKbb︸K(e)⋅V(a)V(b)+MaaMabMbaMbb︸M(e)⋅V¨(a)V¨(b)=L12-1⋅f1f2-L22⋅L12-1⋅f1︸F0+L12-1⋅L11-L12-1L21-L22⋅L12-1⋅L11L22⋅L12-1︸K(e)⋅V(a)V(b)+L12-1⋅H11-L12-1⋅H12H21-L22⋅L12-1⋅H11H22-L22⋅L12-1⋅H12︸M(e)⋅V¨(a)V¨(b)

The numerical integration according to Runge–Kutta, described in Sec. 3.4, offers the possibility of including single load or mass quantities within a beam element. Single eccentric masses can be taken into account at the integration interval boundaries. In the local coordinate system of the beam element, at a general position vector xAE=yEzET, the eccentric mass and the vector representation of dynamic equilibrium, Eq. (44a)–(44b), is as shown in Fig. 4. The beam reference axis is at point A, and vector V¨E represents the acceleration vector at the point of application x,xAE. With the help of the dynamic equilibrium conditions Eq. (44a)–(44b), additional inertia forces and moments due to eccentric mass can be determined (see Li, 2016): -NL+NR=mEV¨EML+MR=∑i=13(θEiφ¨iei)+xAE×mEV¨E, where NL and NR are internal force vectors on the left and right in differential proximity to the point at location x, ML and MR are internal moment vectors on the left and right in differential proximity to the point at location x (see Fig. 4), mE is the eccentric single mass, ΘEi are the mass moments of inertia of the single mass and φ¨i are the angular accelerations at location x.