System Equations

5.2.1 Transformation and Assembly

In order to arrive at the final element expressions in Chapters3 and 4, the principle of virtual displacements on total and incremental forms, as stated in Eqs.(2.44,2.51), was applied to an individual element only. By applying the principle to a structural system, and subdivide each integral as a sum over all elements that constitute this system, the incremental equilibrium equations of the structure may then be expressed on a form similar to Eq.(4.223) for an element. Thus

𝐾Δ𝑉 = 𝑃 − 𝑅 (5.49)

where all quantities now are referred to global Cartesian coordinates and Δ𝑉 is the in-cremental displacement vector of the collected system nodes. Furthermore, (𝐾,𝑃,𝑅) are the system versions of the incremental stiffness matrix, applied node-force vector and internal node-force vector, respectively, that relate to the corresponding element quantities through

𝐾 = ^∑︁

𝑛𝑒𝑙

𝐾^(𝑒) 𝑃 = ^∑︁

𝑛𝑒𝑙

𝑃^(𝑒) (5.50)

𝑅 = ^∑︁

𝑛𝑒𝑙

𝑅^(𝑒)

where𝑛_𝑒𝑙 is the number of elements in the system in question. Here (𝐾^(𝑒),𝑃^(𝑒),𝑅^(𝑒)) originate from the element quantities of the condensed 12 DOFs element as given by Eqs.(4.224,4.225,4.226) when related to the element nodes in local coordinates. Thus, the element quantities need to be transformed to global coordinates and related to the corresponding system nodes; an operation that will be described in the following.

By assuming small incremental nodal rotations, so that all DOFs can be treated vectorially, the transformation of incremental displacements of an element node ‘𝑒𝑛’

from global coordinates to local 𝐶𝑜𝑛-coordinates becomes from Eq.(5.12)

⎧

⎪⎨

⎪⎩

Δ^𝑣_𝑡 Δ^𝑣_𝑟

⎫

⎪⎬

⎪⎭

𝑒𝑛

=

⎡

⎢

⎣

𝑜𝑛𝑇 0

0 _𝑜𝑛𝑇

⎤

⎥

⎦

⎧

⎪⎨

⎪⎩

Δ𝑣_𝑡 Δ𝑣_𝑟

⎫

⎪⎬

⎪⎭

𝑒𝑛

(5.51)

where local displacements now are identified with the accent ‘_̂︀’, and subscripts

‘𝑡’ and ‘𝑟’ again signify translations and rotations, respectively. Furthermore, the transformation of global incremental displacements from the system node ‘𝑠𝑛’ to the pertaining element node can be expressed by

⎧ where 1 is the diagonal unit matrix and^𝑛𝐸 is the eccentricity matrix, given by

𝑛𝐸 = yields the final transformation of incremental displacements from a system node in global coordinates to an eccentrically attached element node in local𝐶_𝑜𝑛-coordinates

⎧ Collection of this transformation for both endnodes into one matrix equation for the element reads

Δ^𝑣 = 𝑇_𝑒Δ𝑣 (5.55)

where the DOFs-transformation matrix of the eccentrically attached element becomes

𝑇_𝑒 =

Here (^𝑛𝐸⁽¹⁾,^𝑛𝐸⁽²⁾) are the eccentricity matrices pertaining to the external element nodes 1 and 2, respectively. Now, starting from the incremental equilibrium equa-tions of an element in local coordinates (i.e. Eq.(4.223)) and in global coordinates when referred to the pertaining system nodes, and then equalizing the incremental work done, both internally and externally, in the two systems, and finally introduc-ing Eq.(5.55), the sought transformations of element quantities from local to global coordinates become

𝑘 = 𝑇^𝑇_𝑒 ^𝑘 𝑇𝑒

𝑝 = 𝑇^𝑇_𝑒 𝑝^ (5.57)

𝑟 = 𝑇^𝑇_𝑒 𝑟^

Here again local quantities are identified with the accent ‘_̂︀’. Now (𝑘,𝑝,𝑟) can be expanded to full ‘system size’ in (𝐾^(𝑒),𝑃^(𝑒),𝑅^(𝑒)), and then assembled into (𝐾,𝑃,𝑅) through Eq.(5.50).

Note that this expand-and-add procedure is good for visualizing the assembly process. In practical computer implementation, however, the entries of (𝑘,𝑝,𝑟) are assembled directly into (𝐾,𝑃,𝑅) according to a ‘pointer-indexing’ scheme.

5.2.2 Discrete Nodal Load Contribution

The preceding subsection dealt with the assembly of element quantities into system quantities that did constitute the terms of the incremental equilibrium equations of a structure in Eq.(5.49). However, discrete loads applied at the system nodes were not accounted for. With these loads included, Eq.(5.49) retains its form, but the assembled system quantities now become

𝐾 = ^∑︁

𝑛𝑒𝑙

𝐾^(𝑒) − ^∑︁

𝑛𝑠𝑛

𝐾^(𝑛)_𝑙 𝑃 = ^∑︁

𝑛_𝑒𝑙

𝑃^(𝑒) + ^∑︁

𝑛𝑠𝑛

𝑃^(𝑛) (5.58)

𝑅 = ^∑︁

𝑛𝑒𝑙

𝑅^(𝑒)

Compared to Eq.(5.50); the new parameter𝑛_𝑠𝑛is the number of system nodes in the structure, and the new contributions (𝐾^(𝑛)_𝑙 ,𝑃^(𝑛)) are the ‘expanded’ versions of the load correction stiffness matrix and the applied node-force vector, respectively, due to discrete loading at a system node. The bases for these contributions are (𝑘⁽_𝑙^𝐶𝐿),𝑝) that refer to the 6 DOFs of the system node. In the following, expressions for the latter couple will be presented, both for unidirectional and corotational loading.

The elementary case of nodal loading is assumed to consist of a discrete force 𝐹 that acts at an offset from the node given by the eccentricity vector 𝑒_𝑙. The components of the derived nodal moment𝑀 are thus given by

𝑀_𝑥 = 𝑒_𝑦𝐹_𝑧 − 𝑒_𝑧𝐹_𝑦

𝑀_𝑦 = 𝑒_𝑧𝐹_𝑥 − 𝑒_𝑥𝐹_𝑧 (5.59)

𝑀_𝑧 = 𝑒_𝑥𝐹_𝑦 − 𝑒_𝑦𝐹_𝑥

where (𝐹_𝑥, 𝐹_𝑦, 𝐹_𝑧) and (𝑒_𝑥, 𝑒_𝑦, 𝑒_𝑧) are the components of 𝐹 and 𝑒_𝑙, respectively. In-troduction of a load eccentricity matrix 𝐸_𝑙, similar to Eq.(5.53), yields the matrix form of Eq.(5.59)

𝑀 = 𝐸^𝑇_𝑙 𝐹 (5.60)

As for the element, the discrete loading is also assumed to be rigidly attached to the system node. Thus, the load eccentricity vector in global system is updated for a new configuration according to Eq.(5.9). Furthermore, the nodal loading is known in

its initial configuration, like the distributed element loading in Subsection 5.1.6. At configuration 𝐶_𝑛 the known force in initial configuration reads

𝑛𝑜𝐹 = ℎ_𝑙(𝜆_𝑠)𝐹_𝑟𝑒𝑓 (5.61)

Thus, again the load level is determined by scaling a given reference value 𝐹𝑟𝑒𝑓

with a load factor ℎ_𝑙(𝜆_𝑠) for the solution step. For unidirectional nodal loading the corresponding node-force vector of a system node then takes the form

𝑝_𝑢 =

while for corotational loading, the node-force vector becomes

𝑝_𝑐 =

where ^𝑛𝐸_𝑙 is the load eccentricity matrix for the 𝐶_𝑛-configuration with components updated according to Eq.(5.9), and ^𝑛𝑇 is the nodal rotation matrix that is defined through Eq.(5.2).

The load correction stiffness matrix relates the increments of nodal loading to the increments of nodal displacements. Thus

Δ𝑝^(𝐶𝐿) = 𝑘⁽_𝑙^𝐶𝐿)Δ𝑣 (5.64)

Here the superscript ‘(𝐶𝐿)’ implies that load corrections due to incremental trans-lations are eliminated since in the Corotated Lagrangian description of motion the nodal reference frame moves along with the node. Thus, assuming small rotations, this relationship takes the form for corotational loading

⎧

respectively. For unidirectional loading the same relationship becomes

Note that these expressions for the load correction stiffness matrices are consistent with the corresponding element expressions as given by Eqs.(4.187,4.215).

5.2.3 Prescribed Displacements

In this work prescribed displacements or displacement boundary conditions will be applied directly in the global system. At configuration 𝐶_𝑛 a set of prescribed dis-placements that pertain to a system node, are given by

𝑛𝑣ˇ = ℎ_𝑑(𝜆_𝑠)𝑣ˇ_𝑟𝑒𝑓 (5.68) Thus, the displacement level is determined by scaling a given reference value 𝑣ˇ_𝑟𝑒𝑓 with a factor ℎ_𝑑(𝜆_𝑠) for the solution step. The treatment of displacement scaling factors is covered in Subsection 5.3.2. When going from solution step ‘𝑠’ to ‘𝑠+ 1’, the increments of prescribed nodal displacements become

Δˇ𝑣 = [ℎ_𝑑(𝜆_𝑠+1) − ℎ_𝑑(𝜆_𝑠)] 𝑣ˇ_𝑟𝑒𝑓 (5.69) By collecting all increments of prescribed nodal displacements in the system into a common vector Δ𝑉ˇ₂, the incremental equilibrium equations of a structure in Eq.(5.49) may be presented on the partitioned form

⎡

where subscripts ‘1’ and ‘2’ signify quantities pertaining to unknown and prescribed displacements, respectively. Thus, only the first of the two matrix equations needs to be solved for the unknown DOFs. However, by exchanging the second matrix equation with the trivial expression 1Δ𝑉ˇ₂ = Δ𝑉ˇ₂, a new set of equations with the same size as the original unconstrained system, that yields the correct solution for all DOFs, becomes

⎡

⎢

⎣

𝐾11 0 0 1

⎤

⎥

⎦

⎧

⎪⎨

⎪⎩

Δ𝑉₁ Δ𝑉ˇ₂

⎫

⎪⎬

⎪⎭

=

⎧

⎪⎨

⎪⎩

𝑃1 − 𝑅1 − 𝐾12Δ𝑉ˇ2

Δ𝑉ˇ₂

⎫

⎪⎬

⎪⎭

(5.71) Note that the partitioned form as indicated in Eq.(5.70) is made here only to simplify the presentation. The boundary conditions are imposed according to Eq.(5.71) in the original rows and columns without any physical rearrangement of equations.

In document Nonlinear analysis of concrete structures based on a 3D shear-beam element formulation (sider 96-101)