Large Deformation Analyses of Space-Frame Structures ...Large Deformation Analyses of Space-Frame Structures 339 where qˆ is the total torsion of the rod at x 1 due to the torque
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Large Deformation Analyses of Space-Frame Structures,Using Explicit Tangent Stiffness Matrices, Based on theReissner variational principle and a von Karman Type
Nonlinear Theory in Rotated Reference Frames
Yongchang Cai1,2, J.K. Paik3 and Satya N. Atluri3
Abstract: This paper presents a simple finite element method, based on assumedmoments and rotations, for geometrically nonlinear large rotation analyses of spaceframes consisting of members of arbitrary cross-section. A von Karman type non-linear theory of deformation is employed in the updated Lagrangian co-rotationalreference frame of each beam element, to account for bending, stretching, and tor-sion of each element. The Reissner variational principle is used in the updatedLagrangian co-rotational reference frame, to derive an explicit expression for the(12x12) symmetric tangent stiffness matrix of the beam element in the co-rotationalreference frame. The explicit expression for the finite rotation of the axes of the co-rotational reference frame, from the global Cartesian reference frame is derivedfrom the finite displacement vectors of the 2 nodes of each finite element. Thus,the explicit expressions for the tangent stiffness matrix of each finite element of thebeam, in the global Cartesian frame, can be seen to be derived as text-book exam-ples of nonlinear analyses. When compared to the primal (displacement) approachwherein C1 continuous trial functions (for transverse displacements) over each el-ement are neccessary, in the current approch the trial functions for the transversebending moments and rotations are very simple, and can be assumed to be linearwithin each element. The present (12×12) symmetric tangent stiffness matrices ofthe beam, based on the Reissner variational principle and the von Karman type sim-plified rod theory, are much simpler than those of many others in the literature. Thepresent approach does not involve such numerical procedures as selective reducedintegration or suppression of attendant Kinematic modes. The present methodolo-
1 Key Laboratory of Geotechnical and Underground Engineering of Ministry of Education, De-partment of Geotechnical Engineering, Tongji University, Shanghai 200092, P.R.China. E-mail:[email protected]
2 Center for Aerospace Research & Education, University of California, Irvine3 Lloyd’s Register Educational Trust (LRET) Center of Excellence, Pusan National University, Ko-
gies can be extended to study the very large deformations of plates and shells aswell. Metal plasticity may also be included, through the method of plastic hinges,etc. This paper is a tribute to the collective genius of Theodore von Karman (1881-1963) and Eric Reissner (1913-1996).
Keywords: Large deformation, Unsymmetrical cross-section, Explicit tangentstiffness, Updated Lagrangian formulation, Rod, Space frames, Reissner varia-tional principle
1 Introduction
In the past decades, large deformation analyses of space frames have attracted muchattention due to their significance in diverse engineering applications. Many differ-ent methods were developed by numerous researchers for the geometrically non-linear analyses of 3D frame structures. Bathe and Bolourchi (1979) employed thetotal Lagrangian and updated Lagrangian approaches to formulate fully nonlinear3D continuum beam elements. Punch and Atluri (1984) examined the performanceof linear and quadratic Serendipity hybrid-stress 2D and 3D beam elements. Basedon geometric considerations, Lo (1992) developed a general 3D nonlinear beamelement, which can remove the restriction of small nodal rotations between twosuccessive load increments. Kondoh, Tanaka and Atluri (1986), Kondoh and Atluri(1987), Shi and Atluri(1988) presented the derivations of explicit expressions ofthe tangent stiffness matrix, without employing either numerical or symbolic inte-gration. Zhou and Chan (2004a, 2004b) developed a precise element capable ofmodeling elastoplastic buckling of a column by using a single element per mem-ber for large deflection analysis. Izzuddin (2001) clarified some of the conceptualissues which are related to the geometrically nonlinear analysis of 3D framed struc-tures. Simo (1985), Mata, Oller and Barbat (2007, 2008), Auricchio, Carotenutoand Reali (2008) considered the nonlinear constitutive behavior in the geometri-cally nonlinear formulation for beams. Iura and Atluri (1988), Chan (1994), Xueand Meek (2001), Wu, Tsai and Lee(2009) studied the nonlinear dynamic responseof the 3D frames. Lee, Lin, Lee, Lu and Liu (2008), Lee, Lu, Liu and Huang (2008),Lee and Wu (2009) gave the exact large deflection solutions of the beams for somespecial cases. Gendy and Saleeb (1992); Atluri, Iura, and Vasudevan(2001) hadbrief discussions of arbitrary cross sections. Dinis, Jorge and Belinha (2009), Han,Rajendran and Atluri (2005), Lee and Chen (2009), Rabczuk and Areias (2006),Shaw and Roy (2007), Wen and Hon (2007) applied meshless methods to the anal-yses of nonlinear problems with large deformations or rotations. Large rotations inbeams, plates and shells, and attendant variational principles involving the rotationtensor as a direct variable, were studied extensively by Atluri and his co-workers
Large Deformation Analyses of Space-Frame Structures 337
(see, for instance, Atluri 1980, Atluri 1984, and Atluri and Cazzani 1994).
This paper presents a simple finite element method, based on assumed momentsand rotations, for geometrically nonlinear large rotation analyses of space framesconsisting of members of arbitrary cross-section. A von Karman type nonlineartheory of deformation is employed in the updated Lagrangian co-rotational refer-ence frame of each beam element, to account for bending, stretching, and torsionof each element. The Reissner variational principle (1953) [see also Atluri andReissner (1989)] is used in the updated Lagrangian co-rotational reference frame,to derive an explicit expression for the (12x12) symmetric tangent stiffness matrixof the beam element in the co-rotational reference frame. The explicit expressionfor the finite rotation of the axes of the co-rotational reference frame, from theglobal Cartesian reference frame is derived from the finite displacement vectors ofthe 2 nodes of each finite element. Thus, the explicit expressions for the tangentstiffness matrix of each finite element of the beam, in the global Cartesian frame,can be seen to be derived as text-book examples of nonlinear analyses. When com-pared to the primal (displacement) approach wherein C1 continuous trial functions(for transverse displacements) over each element are necessary, in the current ap-proach the trial functions for the transverse bending moments and rotations arevery simple, and can be assumed to be linear within each element. The present(12×12) symmetric tangent stiffness matrices of the beam, based on the Reissnervariational principle and the von Karman type simplified rod theory, are much sim-pler than those of many others in the literature, such as, Simo (1985), Bathe andBolourchi (1979), Kondon, Tanaka and Atluri (1986), Kondoh and Atluri (1987),and Shi and Atluri (1988). The present approach does not involve such numericalprocedures as selective reduced integration or suppression of attendant Kinematicmodes. The present methodologies can be extended to study the very large de-formations of plates and shells as well. Metal plasticity may also be included,through the method of plastic hinges, etc. Furthermore, Unlike in the formulationsof Simo(1985), Crisfield (1990) [and many others who followed them], which leadto the currently popular myth that the stiffness matrices of finitely rotated structuralmembers should be unsymmetric, the (12x12) stiffness matrix of the beam elementin the present paper is enormously simple, and remains symmetric throughout thefinite rotational deformation. This paper is a tribute to the collective genius ofTheodore von Karman (1881-1963) and Eric Reissner (1913-1996).
2 Von-Karman type nonlinear theory for a rod with large deformations
We consider a fixed global reference frame with axes xi (i = 1,2,3) and base vectorsei. An initially straight rod of an arbitrary cross-section and base vectors ei, inits undeformed state, with local coordinates xi (i = 1,2,3), is located arbitrarily in
space, as shown in Fig.1. The current configuration of the rod, after arbitrarily largedeformations (but small strains) is also shown in Fig.1.
The local coordinates in the reference frame in the current configuration are xi andthe base vectors are ei (i = 1,2,3). The nodes 1 and 2 of the rod (or an element ofthe rod) are supposed to undergo arbitrarily large displacements, and the rotationsbetween the ei (i = 1,2,3) and the ek (k = 1,2,3) base vectors are assumed to bearbitrarily finite. In the continuing deformation from the current configuration, thelocal displacements in the xi (ei) coordinate system are assumed to be moderate,and the local gradient (∂u10/∂x1) is assumed to be small compared to the transverserotations (∂uα0/∂x1)(α = 2,3). Thus, in essence, a von-Karman type deformationis assumed for the continued deformation from the current configuration, in the co-rotational frame of reference ei (i = 1,2,3) in the local coordinates xi (i = 1,2,3).If H is the characteristic dimension of the cross-section of the rod, the preciseassumptions governing the continued deformations from the current configurationareu10
H� 1;
HL� 1
uα0
H≈ O(1)(α = 2,3)
∂u10
∂x1� ∂uα0
∂x1(α = 2,3)
and(
∂uα0∂x1
)2(α = 2,3) are not negligible.
As shown in Fig.2, we consider the large deformations of a cylindrical rod, sub-jected to bending (in two directions), and torsion around x1. The cross-section isunsymmetrical around x2 and x3 axes, and is constant along x1.
As shown in Fig.2, the warping displacement due to the torque T around x1 axis isu1T (x2,x3) and does not depend on x1, the axial displacement at the origin (x2 =x3 = 0) is u10 (x1), and the bending displacement at x2 = x3 = 0 along the axis x1are u20 (x1) (along x2) and u30 (x1) (along x3).
We consider only loading situations when the generally 3-dimensional displace-ment state in the ei system, donated as
ui = ui (xk) i = 1,2,3; k = 1,2,3
is simplified to be of the type:
u1 = u1T (x2,x3)+u10 (x1)− x2∂u20
∂x1− x3
∂u30
∂x1
u2 = u20 (x1)− θx3
u3 = u30 (x1)+ θx2
(1)
Large Deformation Analyses of Space-Frame Structures 339
where θ is the total torsion of the rod at x1 due to the torque T .
2.1 Strain-displacement relations
Considering only von Karman type nonlinearities in the rotated reference frameei (xi), we can write the Green-Lagrange strain-displacement relations in the up-dated Lagrangian co-rotational frame ei in Fig.1 as:
x3, e3
1
2210u
220u
230u
2θ
220θ
230θ
11, ex
Von Karman nonlinear strains in rotated reference frame ei
12
11~,~ ex
Undeformed element
Initial configuration
Current configuration
10u
30u20u
x2, e2
x1, e1
22~,~ ex33
~,~ ex
22 , ex33 , ex
L
l
u1
u2
Figure 1: Kinematics of deformation of a space framed member
Figure 2: Large deformation analysis model of a cylindrical rod
ε11 =∂u1
∂x1+
12
(∂u2
∂x1
)2
+12
(∂u3
∂x1
)2
=∂u10
∂x1+
12
(∂u20
∂x1
)2
+12
(∂u30
∂x1
)2
− x2∂ 2u20
∂x21− x3
∂ 2u30
∂x21
ε12 =12
(∂u1
∂x2+
∂u2
∂x1
)=
12
(∂u1T
∂x2− ∂u20
∂x1+
∂u20
∂x1− ∂ θ
∂x1x3
)=
12
(∂u1T
∂x2−θ x3
)ε13 =
12
(∂u1
∂x3+
∂u3
∂x1
)=
12
(∂u1T
∂x3− ∂u30
∂x1+
∂u30
∂x1+θx2
)=
12
(∂u1T
∂x3+θx2
)ε22 =
∂u2
∂x2+
12
(∂u1
∂x2
)2
+12
(∂u2
∂x2
)2
+12
(∂u3
∂x2
)2
≈ 0+12
(∂u20
∂x1
)2
+0≈ 0
ε23 ≈ 0
ε33 ≈ 0
(2)
Large Deformation Analyses of Space-Frame Structures 341
where θ = dθ/dx1.
By letting
χ22 =−u20,11
χ33 =−u30,11
ε011 = u10,1 +
12
(u20,1)2 +
12
(u30,1)2 = ε
0L11 + ε
0NL11
(3)
the strain-displacement relations can be rewritten as
ε11 = ε011 + x2χ22 + x3χ33
ε12 =12
(u1T,2−θx3)
ε13 =12
(u1T,3 +θx2)
ε22 = ε33 = ε23 = 0
(4)
where , i denotes a differentiation with respect to xi.
The matrix form of the Eq.(4) is
εεε = εεεL +εεε
N (5)
where
εεεL =
εL
11εL
12εL
13
=
u10,1 + x2χ22 + x3χ33
12 (u1T,2−θx3)12 (u1T,3 +θx2)
(6)
εεεN =
εN
11εN
12εN
13
=
12 (u20,1)
2 + 12 (u30,1)
2
00
(7)
2.2 Stress-Strain relations
Taking the material to be linear elastic, we assume that the additional second Piola-Kirchhoff stress, denoted by tensor S1 in the updated Lagrangian co-rotational ref-erence frame ei of Fig.1 (in addition to the pre-existing Cauchy stress due to priordeformation, denoted by τττ0), is given by:
3 Updated Lagrangian formulation in the co-rotational reference frame ei
3.1 The use of the Reissner variational principle in the co-rotational updatedLagrangian reference frame
If τ0i j are the initial Cauchy stresses in the updated Lagrangian co-rotational frame
ei of Fig.1, S1i j are the additional (incremental) second Piola-Kirchhoff stresses in
the same updated Lagrangian co-rotational frame with axes ei, Si j = S1i j +τ0
i j are thetotal stresses, and ui are the incremental displacements in the co-rotational updated-Lagrangian reference frame, the functional of the Reissner variational principle(Reissner 1953) [see also Atluri and Reissner (1989)] for the incremental S1
i j and ui
in the co-rotational updated Lagrangian reference frame is given by [Atluri 1979,1980]
Where V is the volume in the current co-rotational reference state, Sσ is the surfacewhere tractions are prescribed, bi = b0
i +b1i are the body forces per unit volume in
the current reference state, and Ti = T 0i + T 1
i are the given boundary tractions.
The conditions of stationarity of ΠR, with respect to variations δS1i j and δui lead
to the following incremental equations in the co-rotational updated- Lagrangianreference frame.
∂B∂S1
i j=
12
[ui, j +u j,i] (20)
[S1
i j + τ0iku j,k
], j
+ρb1i =−
(τ
0i j), j−ρb0
i (21)
n j[S1
i j + τ0iku j,k
]−T 1
i =−n jτ0i j + T 0
i at Sσ (22)
In Eq.(19), the displacement boundary conditions,
ui = ui at Su (23)
are assumed to be satisfied a priori, at the external boundary, Su. Eq.(21) leads toequilibrium correction iterations.
If the variational principle embodied in Eq.(19) is applied to a group of finite ele-ments, Vm, m = 1,2, · · · ,N, which comprise the volume V , ie, V = ∑Vm, then
ΠR =
∑m
∫Vm
{−B(S1
i j)+
12
τ0i juk, juk, j +
12
Si j (ui, j +u j,i)−ρbiui
}dV −
∫Sσm
TiuidS
(24)
Let ∂Vm be the boundary of Vm, and ρm be the part of ∂Vm which is shared by theelement with its neighbouring elements. If the trial function ui and the test function∂ui in each Vm are such that the inter-element continuity condition,
u+i = u−i at ρm (25)
(where + and – refer to either side of the boundary ρm) is satisfied a priori, then itcan be shown (Atluri 1975,1984; Atluri and Murakawa 1977; Atluri, Gallagher andZienkiewicz 1983) that the conditions of stationarity of ΠR in Eq.(24) lead to:
∂B∂S1
i j=
12[ui, j +u j,i
]in Vm (26)
Large Deformation Analyses of Space-Frame Structures 345
[S1
i j + τ0iku j,k
], j
+ρb1i =−τ
0i j, j−ρb0
i in Vm (27)
[ni(S1
i j + τ0iku j,k
)]++[ni(S1
i j + τ0iku j,k
)]−=−
[niτ
0i j]+− [niτ
0i j]−
at ρm (28)
n j[S1
i j + τ0iku j,k
]− T 1
i =−n jτ0i j + T 0
i at Sσm (29)
Eq.(28) is the condition of traction reciprocity at the inter-element boundary, ρm.Eqs(27) and (28) lead to corrective iterations for equilibrium within each element,and traction reciprocity at the inter-element boundaries, respectively.
Carrying out the integration over the cross sectional area of each rod, and usingEqs.(4) and (12), Eq.(24) can be easily shown to reduce to:
ΠR = ∑elem
∫l
(−1
2σσσ
T D−1σσσ
)dl +
∫l
N011
12(u2
20,1 +u230,1)
dl
+∫l
(N11ε
0L11 + M22χ22 + M33χ33 + T θ
)dl− Qq
(30)
where D is given in Eq.(15), C = D−1, l is the length of the rod element, σσσ is givenin Eq.(14), σ0
i j =[N0
11 M022 M0
33 T 0]T is the initial element-generalized- stress
in the corotational reference coordinates ei, and σσσ =σσσ0 +σσσ =[N11 M22 M33 T
]Tis the total element generalized stresses in the corotational reference coordinates ei.Q is the nodal external generalized force vector (consisting of force as well asmoments) in the global Cartesian reference frame, and q is the incremental nodalgeneralized displacement vector (consisting of displacements as well as rotations)in the global Cartesian reference frame. It should be noted that while ΠR in Eq.(30)represents a sum over the elements, the relevant integrals are evaluated over eachelement in it’s own co-rotational updated Lagrangian reference frame.
In the functional in Eq.(30), only the squares of (u20,1) and (u30,1) occur withineach element. Thus, (u20,1) and (u30,1) are assumed directly to be linear withineach element, in terms of their respective nodal values. This will be enormouslysimple and advantageous in the case of plate and shell elements. This is in contrastto the primal (displacement) approach (Cai, Paik and Atluri 2010) wherein u20 andu30 were required to be C1 continuous over each element, and thus were assumed tobe Herimitian polynomials over each element. In this paper, however, we assume:
uθ = Nθ aθ =[
φ1 0 φ2 00 φ1 0 φ2
]1θ201θ302θ202θ30
(42)
whereφ1 = 1−ξ
φ2 = ξ
(ξ =
x1
l
)(43)
Assuming that ‘a’ represents the vector of generalized displacements of the nodesof the rod element in the updated Lagrangian co-rotational frame ei of Fig.1, thedisplacement vectors of node i are:ia =
Thus, KL is the usual linear symmetric (12×12) stiffness matrix of the beam in theco-rotational reference frame, with the geometric parameters I2, I3, I22, I33, I23 andIrr, and the current length l.
It is clear from the above procedures, that the present (12×12) symmetric tan-gent stiffness matrices of the beam in the co-rotational reference frame, based onthe Reissner variational principle and simplified rod theory, are much simpler thanthose of Kondon, Tanaka and Atluri (1986), Kondoh and Atluri (1987), and Shi andAtluri (1988). Moreover, the explicit expressions for the tangent stiffness matrix ofeach rod can be seen to be derived as text-book examples of nonlinear analyses.
3.4 Cubic trial functions of the displacements in the beam element, using theReissner variational principle
When using the Reissner functional in Eq.(30), one may directly assume the ro-tation field (u20,1) and (u30,1) as linear functions in terms only of their respectivenodal values, as in Eq.(42). Alternatively, u20 and u30 may be assumed as cubicpolynomials in terms of the four nodal values 1u20, 2u20, 1u20,1, 2u20,1 (1u30, 2u30,1u30,1, 2u30,1 for u30), and derive the element fields for u20,1 (and u30,1) from thesecubic polynomials [even though the Reissner principle does not demand it]. Thiswill be particularly advantageous for plate and shell elements which demand C1
Large Deformation Analyses of Space-Frame Structures 353
continuity while using the potential energy approach, while C1 continuity of thedisplacement field will not be demanded in the Reissner principle.
In general, we assume over each element:
u20 = α1 +α2ξ +α3ξ2 +α4ξ
3
u30 = γ1 + γ2ξ + γ3ξ2 + γ4ξ
3 (67)
By letting
u20|ξ=ξ0= 1u20, u20|ξ=ξ1
= 2u20, u20,1|ξ=ξ0= 1
θ30, u20,1|ξ=ξ1= 2
θ30
u30|ξ=ξ0= 1u30, u30|ξ=ξ1
= 2u30,−u30,1|ξ=ξ0= 1
θ20,−u30,1|ξ=ξ1= 2
θ20(68)
we can approximate the displacement function in each rod element by
By using the cubic trial functions of Eq.(69) and deriving the equations in a sameway as the section 3.3, we obtain the respective discrete equations, as follows.
4 Transformation between deformation dependent co-rotational local [ei],and the global [ei] frames of reference
As shown in Fig.1, xi (i = 1,2,3) are the global coordinates with unit basis vectorsei. xi and ei are the local coordinates for the rod element at the undeformed element.The basis vector ei are initially chosen such that (Shi and Atluri 1988, Cai, Paik andAtluri 2010)
e1 = (∆x1e1 +∆x2e2 +∆x3e3)/L
e2 = (e3× e1)/|e3× e1|e3 = e1× e2
(76)
where ∆xi = x2i − x1
i ,L =(∆x2
1 +∆x22 +∆x2
3) 1
2 .
Then ei and ei have the following relations:e1e2e3
=
∆x1/L ∆x2/L ∆x3/L−∆x2/S ∆x1/S 0
−∆x1∆x3/(SL) −∆x2∆x3/(SL) s/L
e1e2e3
(77)
where S =(∆x2
1 +∆x22) 1
2 .
Large Deformation Analyses of Space-Frame Structures 355
Thus we can define a transformation matrix λλλ 0 between ei and ei as
λλλ 0 =
∆x1/L ∆x2/L ∆x3/L−∆x2/S ∆x1/S 0
−∆x1∆x3/(SL) −∆x2∆x3/(SL) S/L
(78)
When the element is parallel to the x3 axis, S =[∆x2
1 +∆x22] 1
2 = 0 and Eq.(64) isnot valid. In this case, the local coordinates is determined by
e1 = e3, e2 = e2, e3 =−e1 (79)
Let xi and ei be the co-rotational reference coordinates for the deformed rod ele-ment. In order to continuously define the local coordinates of the same rod elementduring the whole range of large deformation, the basis vectors ei are chosen suchthat
e1 = (∆x1e1 +∆x2e2 +∆x3e3)/l = a1e1 +a2e2 +a3e3
e2 = (e3× e1)/|e3× e1|e3 = e1× e2
(80)
where ∆xi = x2i − x1
i , l =(∆x2
1 +∆x22 +∆x2
3) 1
2 .
We denote e3 in Eq.(77) as
e3 = c1e1 + c2e2 + c3e3 (81)
Then ei and ei have the following relations:e1e2e3
Thus, the transformation matrix λλλ , between the 12 generalized coordinates in theco-rotational reference frame, and the corresponding 12 coordinates in the globalCartesian reference frame, is given by
λλλ =
λλλ 0
λλλ 0λλλ 0
λλλ 0
(86)
Letting xi and ei be the reference coordinates, and repeating the above steps [Eq.(70)– Eq.(86)], the transformation matrix of each incremental step can be obtained in asame way.
Then the element matrices are transformed to the global coordinate system using
a = λλλT a (87)
K = λλλT Kλλλ (88)
F = λλλT F (89)
where a, K, F are respectively the generalized nodal displacements, element tan-gent stiffness matrix and generalized nodal forces, in the global coordinates system.
The Newton-Raphson method, modified Newton-Rapson method or the artificialtime integration method (Liu 2007a, 2007b; Liu and Atluri 2008) can be employedto solve Eqs.(59) and (74). In this implementation, the Newton-Raphson algorithmis used. In all examples, the assumptions of linear trial functions of the rotationswere employed, except where stated otherwise.
5 Numerical examples
5.1 Buckling of a beam
The (12×12) tangent stiffness matrix for a beam in space should be capable ofpredicting buckling under compressive axial loads, when such an axial load inter-acts with the transverse displacement in the beam. We consider a simply supportedbeam subject to an axial force as shown in Fig.3 and assume that EI = 1 and L = 1.
Large Deformation Analyses of Space-Frame Structures 357
The buckling loads of the beam obtained by the present method using differentnumbers of elements are shown in Tab.1. It is seen that the buckling load predictedby the present method agrees well with the analytical solution (buckling load isπ2).
Figure 3: A simply supported beam subject to an axial force
Table 1: Buckling load of the simply supported beam
Present method(Number of elements) Analytical1 2 3 4 10 solution
When the beam is fixed at x1 = 0, while at the other end it is free and under acompressive load P, the buckling load of the beam obtained by the present methodusing different number of elements is shown in Tab.2 (the analytical solution isπ2EI4L2 ).
Table 2: Buckling load of the beam fixed at x1 = 0
Present method(Number of elements) Analytical1 2 3 4 10 solution
5.2 Large deformation analysis of a cantilever beam with a symmetric crosssection
A large deflection and moderate rotation analysis of a cantilever beam subject to atransverse load at the tip, as shown in Fig. 5, is considered. The cross section ofthe beam is a square with h = 1. The Poisson’s ration is ν = 0.3. Fig.5 shows theresults obtained in the analysis of the cantilever problem. It is seen that the presentresults using 10 elements agree well with those of Bathe and Bolourchi (1979).
Figure 4: A cantilever beam subject to a transverse load at the tip
PL2 / E
I
L/δ
Figure 5: Deflections of a cantilever under a concentrated load
Large Deformation Analyses of Space-Frame Structures 359
5.3 Large rotations of a cantilever subject to an end-moment and a transverseload
An initially-straight cantilever subject to an end moment M∗ = ML2πEI (Crisfield
1990) as shown in Fig.6, is considered. The beam is divided into 10 equal ele-ments. When M∗ = 1, the beam is curled into a complete circle as shown in Fig.6.
If a non-conservative, follower-type transverse load P∗ = PL2
2πEI is applied at the tip,instead of M∗, the initial and deformed geometries of the cantilever are shown inFig.7.
5.4 Large deformation analysis of a cantilever beam with an asymmetric crosssection
We consider the large deflection of a cantilever beam with an asymmetric crosssection, as shown in Fig.8. The Poisson’s ration is ν = 0.3. The areas of thesymmetric and asymmetric cross section in Fig.8 are all equal to 1.
Fig.9 shows the comparison of the deflections in x3 direction, between the casesof symmetric and asymmetric cross sections. Fig.10 shows the deflection in x2direction for the cantilever beam with an asymmetric cross section. However, thedeflections in x2 direction are zero in the case of a symmetric cross section.
5.5 Large displacement analysis of a 45-degree space bend
The large displacement response of a 45-degree bend subject to a concentrated endload [Bathe and Bolourchi (1979)] is calculated as shown in Fig.11. The radiusof the bend is 100, the cross section area is 1 and lies in the x1− x2 plane. Theconcentrated is applied in the x3 direction.
8 equal straight elements and 140 equal load steps are used in the analysis of theproblem. Fig.12 shows the tip deflection predicted by the present method and Batheand Bolourchi (1979). It can be seen that the results of the present method agreeexcellently with the results of Bathe and Bolourchi (1979).
5.6 A framed dome
A framed dome shown in Fig.13 is considered (Shi and Atluri 1988). A concen-trated vertical load P is applied at the crown point. Each member of the dome ismodeled by 4 elements.
The linear approaches of the displacements in Eq.(42) are robust for most casesin the large deformation analysis of the space frames. However, the solution wasfound to diverge when λ > 0.59 by using the linear interpolations for rotations
Figure 6: Initial and deformed geometries for cantilever subject to an end-moment
L
EI2PLP
2*
π=
P*=0.0
P*=1.0
P*=2.0
P*=4.0
P*=6.0
Figure 7: Initial and deformed geometries for cantilever subject to a transverse load
Large Deformation Analyses of Space-Frame Structures 361
L=50
EI
P
δ
x1
x2
x3
bb
a(a=0.2, b=1.3)
P
Asymmetric cross section
x2
x3
bbP
a
Symmetric cross section
(a=0.2, b=5/6)
x2
x3
Figure 8: A cantilever beam with an asymmetric cross section
for this example. Thus, the nonlinear stiffness matrix in Eq.(75), which is derivedfrom cubic trial functions of the displacements, was used, and the converged resultsshown in Fig.14 were obtained.
6 Conclusions
Based on the Reissner variational principle and a von Karman type nonlinear theoryin a rotated reference frame, a simplified finite deformation theory of a cylindricalrod subjected to bending and torsion has been developed. The present (12×12)symmetric explicit tangent stiffness matrices of the beam are much simpler thanthose of many others based on the primal approach or potential energy approach.
Figure 9: Comparison of the deflections in x3 direction of a cantilever beam
Lu 2
Figure 10: Deflections in x2 direction for the cantilever beam with asymmetriccross section
The explicit expressions for the tangent stiffness matrix of each element can beseen to be derived as text-book examples of nonlinear analyses. The proposedmethod is capable of handling large rotation geometrically nonlinear analysis offrames with arbitrary cross sections, which haven’t been considered by a majorityof previous studies. Numerical examples demonstrate that the present method isjust as competitive as the existing methods in terms of accuracy and efficiency.
Large Deformation Analyses of Space-Frame Structures 363
Figure 11: Model of a 45-degree circular bend
0 1 2 3 4 5 6 70.0
0.1
0.2
0.3
0.4
0.5
0.6
Non
-dim
ensi
onal
tip
defle
ctio
n
Load parameter
Present method
Bathe,Bolourchi(1979)
Linear solutionu3/R
-u2/R
-u1/R
EIPRk
2
=
u3/R
Figure 12: Three-dimensional large deformation of a 45-degree circular bend
Figure 13: Framed dome (the unit of length is metre)
The present method can be extended to consider the formation of plastic hinges ineach beam of the frame; and also to consider large-rotations of plates and shells,by implementing only a von Karman type nonlinear theory in the co-rotationalreference frame of each beam/plate element. It is noted that the present approachdoes not involve any reduced integration, or suppression of Kinematic modes.
Acknowledgement: The authors gratefully acknowledge the support of the Na-ture Science Foundation of China (NSFC, 10972161). This research was alsosupported by the World Class University (WCU) program through the NationalResearch Foundation of Korea funded by the Ministry of Education, Science andTechnology (Grant no.: R33-10049). The second author is also pleased to acknowl-edge the support of the Lloyd’s Register Educational Trust (LRET) which is anindependent charity working to achieve advances in transportation, science, engi-
Large Deformation Analyses of Space-Frame Structures 365
0.0 0.5 1.0 1.5 2.0 2.50.0
0.2
0.4
0.6
0.8
Present method
Kondoh,Tanaka,Atluri(1986)
Shi,Atluri(1988)
MN8.123λP ×=
crP/Pλ =
d(m)
Figure 14: Force-displacement curve for the crown point of a framed dome
neering and technology education, training and research worldwide for the benefitof all.
References
Atluri, S.N. (1975): On ’hybrid’ finite element models in solid mechanics, Ad-vances in computer methods for partial differential equations, R. Vichrevetsky, Ed.,AICA, pp. 346-355.
Atluri, S.N. (1979): On rate principles for finite strain analysis of elastic and inelas-tic nonlinear solids, Recent research on mechanical behavior, University of TokyoPress, pp. 79-107.
Atluri, S.N. (1980): On some new general and complementary energy theoremsfor the rate problems in finite strain, classical elastoplasticity. Journal of StructuralMechanics, Vol. 8(1), pp. 61-92.
Atluri, S.N. (1984): Alternate stress and conjugate strain measures, and mixedvariational formulations involving rigid rotations, for computational analyses offinitely deformed plates and shells: part-I: thoery. Computers & Structures, Vol.18(1), pp. 93-116.
Atluri, S.N.; Cazzani, A. (1994): Rotations in computational solid mechanics,invited feature article. Archives for Computational Methods in Engg., ICNME,Barcelona, Spain, Vol 2(1), pp. 49-138.
Atluri, S. N.; Gallagher, R. H.; Zienkiewicz, O. C. (Editors). (1983): Hybrid &Mixed Finite Element Methods, J. Wiley & Sons, 600 pages.
Atluri, S.N.; Iura, M.; Vasudevan, S.(2001): A consistent theory of finite stretchesand finite rotations, in space-curved beams of arbitrary cross-section. Computa-tional Mechanics, vol.27, pp.271-281
Atluri, S.N.; Murakawa, H. (1977): On hybrid finite element models in nonlinearsolid mechanics, finite elements in nonlinear mechanics, P.G. Bergan et al, Eds.,Tapir Press, Norway, vol. 1, pp. 3-40.
Atluri, S.N.; Reissner, E. (1989): On the formulation of variational theorems in-volving volume constraints. Computational Mechanics, vol. 5, pp 337-344.
Auricchio, F.; Carotenuto, P.; Reali, A. (2008): On the geometrically exact beammodel: A consistent, effective and simple derivation from three-dimensional finite-elasticity. International Journal of Solids and Structures, vol.45, pp. 4766–4781.
Bathe, K.J.; Bolourchi, S. (1979): Large displacement analysis of three- dimen-sional beam structures. International Journal for Numerical Methods in Engineer-ing, vol.14, pp.961-986.
Cai, Y.C.; Paik, J.K.; Atluri S.N. (2010): Large deformation analyses of space-frame structures, with members of arbitrary cross-section, using explicit tangentstiffness matrices, based on a von Karman type nonlinear theory in rotated referenceframes. CMES: Computer Modeling in Engineering & Sciences, vol. 53, no. 2, pp.123-152.
Chan, S.L. (1996): Large deflection dynamic analysis of space frames. Computers& Structures, vol. 58, pp.381-387.
Crisfield, M.A. (1990): A consistent co-rotational formulation for non-linear, three-dimensional, beam-elements. Computer Methods in Applied Mechanics and Engi-neering, vol.81, pp. 131–150.
Dinis, L.W.J.S.; Jorge, R.M.N.; Belinha, J. (2009): Large deformation applica-tions with the radial natural neighbours interpolators. CMES: Computer Modelingin Engineering & Sciences, vol.44, pp. 1-34
Gendy, A.S.; Saleeb, A.F. (1992): On the finite element analysis of the spatialresponse of curved beams with arbitrary thin-walled sections. Computers & Struc-tures, vol.44, pp.639-652
Han, Z.D.; Rajendran, A.M.; Atluri, S.N. (2005): Meshless Local Petrov- Galerkin(MLPG) approaches for solving nonlinear problems with large deformations and
Large Deformation Analyses of Space-Frame Structures 367
rotations. CMES: Computer Modeling in Engineering & Sciences, vol.10, pp. 1-12
Iura, M.; Atluri, S.N. (1988): Dynamic analysis of finitely stretched and rotated3-dimensional space-curved beams. Computers & Structures, Vol. 29, pp.875-889
Izzuddin, B.A. (2001): Conceptual issues in geometrically nonlinear analysis of3D framed structures. Computer Methods in Applied Mechanics and Engineering,vol.191, pp. 1029–1053.
Kondoh, K.; Tanaka, K.; Atluri, S.N. (1986): An explict expression for thetangent-stiffness of a finitely deformed 3-D beam and its use in the analysis ofspace frames. Computers & Structures, vol.24, pp.253-271.
Kondoh, K.; Atluri, S.N. (1987): Large-deformation, elasto-plastic analysis offrames under nonconservative loading, using explicitly derived tangent stiffnessesbased on assumed stresses. Computational Mechanics, vol.2, pp.1-25.
Lee, M.H.; Chen, W.H. (2009): A three-dimensional meshless scheme with back-ground grid for electrostatic-structural analysis. CMC: Computers Materials &Continua, vol.11, pp. 59-77
Lee, S.Y.; Lin, S.M.; Lee, C.S.; Lu, S.Y.; Liu, Y.T. (2008): Exact large deflec-tion of beams with nonlinear boundary conditions.CMES: Computer Modeling inEngineering & Sciences, vol.30, pp. 27-36
Lee, S.Y.; Lu, S.Y.; Liu, Y.R.; Huang, H.C. (2008): Exact large deflection so-lutions for Timoshenko beams with nonlinear boundary conditions. CMES: Com-puter Modeling in Engineering & Sciences, vol.33, pp. 293-312
Lee, S.Y.; Wu, J.S. (2009): Exact Solutions for the Free Vibration of ExtensionalCurved Non-uniform Timoshenko Beams. CMES: Computer Modeling in Engi-neering & Sciences,vol.40, pp.133-154.
Liu, C. S. (2007a): A modified Trefftz method for two-dimensional Laplace equa-tion considering the domain’s characteristic length. CMES: Computer Modeling inEngineering & Sciences, vol. 21, pp. 53-66.
Liu, C.S. (2007b): A highly accurate solver for the mixed-boundary potential prob-lem and singular problem in arbitrary plane domain. CMES: Computer Modelingin Engineering & Sciences, vol. 20, pp. 111-122.
Liu, C.S.; Atluri, S.N. (2008): A novel time integration method for solving a largesystem of non-linear algebraic equations. CMES: Computer Modeling in Engineer-ing & Sciences, vol.31, pp. 71-83.
Lo, S.H. (1992): Geometrically nonlinear formulation of 3D finite strain beamelement with large rotations. Computers & Structures, vol.44, pp.147-157
Mata, P.; Oller, S.; Barbat A.H. (2007): Static analysis of beam structures un-der nonlinear geometric and constitutive behavior. Computer Methods in Applied
Mechanics and Engineering, vol.196, pp. 4458–4478.
Mata, P.; Oller, S.; Barbat A.H. (2008): Dynamic analysis of beam structuresconsidering geometric and constitutive nonlinearity. Computer Methods in AppliedMechanics and Engineering, vol.197, pp. 857–878.
Punch, E.F.; Atluri, S.N. (1984): Development and testing of stable, invariant,isoparametric curvilinear 2-D and 3-D hybrid-stress elements. Computer Methodsin Applied Mechanics and Engineering, Vol.47, pp.331-356
Rabczuk, T.; Areias, P. (2006): A meshfree thin shell for arbitrary evolving cracksbased on an extrinsic basis. CMES: Computer Modeling in Engineering & Sci-ences,vol.16, pp.115-130.
Reissner, E. (1953): On a variational theorem for finite elastic deformations, Jour-nal of Mathematics & Physics, vol. 32, pp. 129-135.
Shaw, A.; Roy, D. (2007): A novel form of reproducing kernel interpolationmethod with applications to nonlinear mechanics. CMES: Computer Modeling inEngineering & Sciences,vol.19, pp.69-98.
Simo, J.C. (1985): A finite strain beam formulation. The three-dimensional dy-namic problem. Part I. Computer Methods in Applied Mechanics and Engineering,vol. 49, pp.55–70.
Shi, G.; Atluri, S.N. (1988): Elasto-plastic large deformation analysis of space-frames: a plastic-hinge and strss-based explict derivation of tangent stiffnesses.International Journal for Numerical Methods in Engineering, vol.26, pp.589-615.
Wen, P.H.; Hon, Y.C. (2007): Geometrically nonlinear analysis of Reissner- Mindlinplate by meshless computation. CMES: Computer Modeling in Engineering & Sci-ences, vol.21, pp.177-191.
Wu, T.Y.; Tsai, W.G.;Lee, J.J. (2009):Dynamic elastic-plastic and large deflec-tion analyses of frame structures using motion analysis of structures. Thin-WalledStructures, vol.47, pp. 1177-1190.
Xue, Q.; Meek J.L. (2001): Dynamic response and instablity of frame structures.Computer Methods in Applied Mechanics and Engineering, vol.190, pp. 5233–5242.
Zhou, Z.H.; Chan,S.L. (2004a): Elastoplastic and Large Deflection Analysis ofSteel Frames by One Element per Member. I: One Hinge along Member. Journalof Structural Engineering, vol. 130, pp.538–544.
Zhou, Z.H.; Chan,S.L. (2004b): Elastoplastic and Large Deflection Analysis ofSteel Frames by One Element per Member. II: Three Hinges along Member. Jour-nal of Structural Engineering, vol. 130, pp.545–5553.