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Section 7: PRISMATIC BEAMS

Global Stiffness Matrix For BeamsThe concept of an overall joint stiffness matrix will be explained in conjunction with the two span beam shown below. Here no loads are applied on the structure.

The restrained structure and the six possible joint displacements are labeled. Keep in mind that the axial stiffness is assumed to be large relative to flexural stiffness.

Section 7: PRISMATIC BEAMS

An overall stiffness matrix, [KJ ], can be generated that contains terms for all

ibl j i di l i l dipossible joint displacements, including those restrained at the supports. This matrix involves application of unit displacements as shown in the figures to p gthe right and below.

Section 7: PRISMATIC BEAMS

Each column in the stiffness matrix below corresponds to the restraint reactions produced for each unit displacements application in the restrained structure. Each row corresponds to an applied unit displacement There are six possible applied unit displacementsapplied unit displacement. There are six possible applied unit displacements.

Note that the matrix is square and symmetric. Expansion of the determinant of this matrix would demonstrate that the matrix is singular

[K J ] =

that the matrix is singular due to the fact that certain rows and columns are linear combinations of one another.

Section 7: PRISMATIC BEAMS

The matrix on the previous page was partitioned according to whether the displacements identified in the restrained structure, i.e.,

were free to displace in the original, unrestrained structure

With partitioning the matrix can be considered to have four elements as follows

[ ]

DRKKK[ ]

=RRRD

J KKK

Section 7: PRISMATIC BEAMS

The upper left partition, [K], is a square, symmetric matrix that corresponds to the unknown displacements. The inverse of this matrix is used in the expression

{D} = [K]-1 {{AD} {ADL}}

The lower left partition, {KRD }, is rectangular matrix contains actions that correspond to the support restraints. This partition gives the reactions for the structure due to unit values of the unknown displacements. From the course Matrix Analysis of Structures the vector of support actions (moments or forces) would be obtained from

{AR} = {ARL} + {KRD}{D}

The upper right partition, {KDR }, is simply the transpose of {KRD }. The lower right hand partition, {KRR }, is square and symmetric. It contains actions corresponding to support restraints due to unit displacements in the restrained structure. The matrices {KDR } and {KRR }

ill b d l i i d k di l h iwill be used to analyze reactions associated known support displacements, zero or otherwise.

The stiffness matrix [K] is only a small portion of [KJ]. This is a consequence of the fact that this particular structure is highly restrained to begin with. In large structures having many j i d f h i [K] i l i f [K ]joints and few supports, the matrix [K] constitutes a large portion of [KJ].

Section 7: PRISMATIC BEAMS

In Class Example

Section 7: PRISMATIC BEAMS

Arbitrary Numbering Systems

In the previous section of notes the joint displacements were numbered in a convenient order, i.e., translations proceeded rotations at each joint. Also, free displacements were numbered before constrained displacements. Consider the arbitrary numbering system below. If all matrices were generated conforming to the arbitrary numbering system below we could lose some if not all of the partition definitions developed in the last section ofwe could lose some, if not all, of the partition definitions developed in the last section of notes.

What is required of finite element software algorithms is the ability to take an arbitrary numbering system like the one above and transform it to the numbering system which

i l i d i h d f f d f h i d i hsegregates matrix elements associated with degrees of freedom from those associated with support constraints.

Section 7: PRISMATIC BEAMS

The [K J ] matrix for the arbitrary numbering system is the six by six matrix shown below.

[K J ] =

Section 7: PRISMATIC BEAMS

In order for this [K J ] matrix to be useful the actual degrees of freedom and support constraints in the structure must be recognized. If the fourth and sixth rows are switched to the first and second rows while all others move downward we obtain the following matrix:the first and second rows, while all others move downward, we obtain the following matrix:

Section 7: PRISMATIC BEAMS

Next the fourth and sixth column are moved to the first and second column, while all other columns move to the right without changing order. This rearrangement produces the [K J]matrix we had previously i ematrix we had previously, i.e.,

Software algorithms must ghave the capability to track degrees of freedom and perform the necessary matrix manipulation inmatrix manipulation in order to identify pertinent information.[K j ] =

Section 7: PRISMATIC BEAMS

Analysis of Continuous Beams in General

Continuous beams considered here are prismatic, rigidly connected to each beam segment p , g y gand supported at various points along the beam. Joints are selected at points of support, at any free end, and changes in cross section (i.e., the beam is prismatic).

A continuous beam having mA continuous beam having mmembers and m+1 joints is depicted in figure (a) to the left.

Support restraints of two typesSupport restraints of two types may exist at any joint in a continuous beam. These are restraints against rotation and/or restraints against translations.

We will only consider flexural deformations. Torsion and axial di l id ddisplacements are not considered. Thus only two displacements can occur at each joint.

Section 7: PRISMATIC BEAMS

Given the numbering system in figure (b) the translation at a particular joint is numbered prior to a rotation and it follows that the number of translations is equal to the number of joints minus one, while the rotation is twice the joint number. Thus at joint j the joints minus one, while the rotation is twice the joint number. Thus at joint j the translations and rotation are number 2j-1 and 2j respectively.

It is evident that the total numberIt is evident that the total number of possible joint displacements is twice the joints (or 2nj). If the total number of support restraints against translation and rotations is denoted nr, then the actual degrees of freedom are

r

rj

nm

nnn

+=

=

22

2

Here n is the number of degrees of freedom.

Section 7: PRISMATIC BEAMS

To relate the end displacements of a particular member to the displacements of a joint, consider a typical member in figure (c) below. The member end displacements are numbered j1 j2 k1 and k3 and correspond to end displacements 1 2 3 and 4 in figure

The new notation helps facilitate computer programming The four end

numbered j1, j2, k1 and k3 and correspond to end displacements 1, 2, 3 and 4 in figure (b).

computer programming. The four end displacements correspond to the four joint displacements as follows:

kkjj 121121 ==kkjj

kkjj2222

121121==

Since j and k are equal numerically to id (i 1) hand (i+1), then:

22222121121

+==+==

ikijikij

This indexing system is necessary to construct the joint stiffness matrix [Sj ]

Section 7: PRISMATIC BEAMS

The analysis of continuous beams consists of establishing the stiffness matrix and the load matrix The mostmatrix and the load matrix. The most important matrix generated is the overall joint stiffness matrix [SJ ]. The joint stiffness matrix consists of contributions from the beam stiffness matrix [SM ].

It is convenient to assess the contributions for one typical member iand repeat the process for members 1 through m.

So the next step involves expressing the iff ffi i h i h fistiffness coefficients shown in the figure

to the left in terms of the various member stiffnesses that contribute to the joint stiffnesses.jo t st esses.

Section 7: PRISMATIC BEAMS

This next step requires that the member stiffnesses be obtained from the matrix below:

[k]

For example the contribution to the joint stiffness (Sj )j1,j1 from member i-1 is the stiffness Sm33 for that member. Similarly, the contribution to (Sj )j1,j1 from member i is th tiff S f b ithe stiffness Sm11 from member i

Section 7: PRISMATIC BEAMS

In general the contribution of one member to a particular joint stiffness will be denoted by appending the member subscript to the member stiffness itself. From this discussion one can see that the joint stiffness matrix coefficients are generated by the followingone can see that the joint stiffness matrix coefficients are generated by the following expressions:

( ) ( ) ( )( ) ( ) ( )( ) ( )

iMiMjjJ

iMiMjjJ

SSS

SSS

211431,2

111331,1

+=

+=

( ) ( )( ) ( )iMjkJ

iMjkJ

SS

SS

411,2

311,1

=

=

which represent the transfer of elements of the first column of the member stiffness matrix [k] to the appropriate location in the joint stiffness matrix [Sj][ ] pp p j [ j]

Section 7: PRISMATIC BEAMS

Expressions analogous to the previous expressions are easily obtained for a unit rotation about the z axis at joint j:

( ) ( ) ( )( ) ( ) ( )( ) ( )

iMiMjjJ

iMiMjjJ

SSS

SSS

221442,2

121342,1

+=

+=

( ) ( )( ) ( )iMjkJ

i