Lecture 12: PRISMATIC BEAMS

LoadsAfter composing the joint stiffness matrix the next step is composing load vectors. Previously it was convenient to treat joint loads and member loads separately since they are manipulated in different ways. Joint loads are can be immediately placed in a ector of actions sed directl in comp tations Member loads m st be con erted intovector of actions used directly in computations. Member loads must be converted into

equivalent fixed end joint loads.

Consider the following beam once again

Lecture 12: PRISMATIC BEAMS

The joint numbering system is the same as the previous section of notes. Joint loads would fill up the matrix [A] as follows

PL

[ ]

=

PL

A000

P00

The remaining loads on the structure act directly on the members and are shown action on the two beam segments as follows

Lecture 12: PRISMATIC BEAMS

These fixed end actions may be assembled in a rectangular matrix [AML] where each row contains the end actions for a given member, i.e.,

[ ]

−

= 82821111 LPPLPP

AML[ ]

−

82822222 LPPLPPML

−

44PLPPLP

Given the load values

[ ]

−

=

8282

44PLPPLP

AML

Lecture 12: PRISMATIC BEAMS

When the fixed end reactions in [AML] are reversed, they constitute equivalent joint loads shown in the following figure

−

8882

21

PL

PL

LP

LPLP

These equivalent joint loads can be assembled as a vector [AE] shown at the right. The equivalent joint loads are [ ]

−=

−=

8

2

81

PLP

LP

P

AEg q jassembled in the vector corresponding to the joint numbering system in the previous section of notes.

[ ]

−

−

−−

−

434

22

821

1

P

PL

PP

LPE

−

− 22

222

PP

Lecture 12: PRISMATIC BEAMS

Actual joint loads [A] are then added to equivalent joint loads [AE] to produce a matrix of composite loads [AC] as followsp C

[ ] [ ] [ ]+= AAA EC

89

8PL

PL

PLPL

PL

If the signs on the elements of

−

−=

+

−

−=

8

0008

PLP

PLP

If the signs on the elements of [AC] are reversed then the matrix is equivalent to [ARL].

−

−4

34

00

434

PP

P

−

22PP

Lecture 12: PRISMATIC BEAMS

In summary the vector [AC] contains information in the following manner

[ ]

DA

A[ ]

−

=RL

C AA

where

P

[ ]

= 89

PL

PL

AD [ ]

= 34P

PL

ARL

8

PL

−

2

4P

The formation of vectors [AD] and [ARL] sets the stage for a completed analysis. Now that the effects of the member loads have converted to equivalent joint loads implies that the vector [ADL] is the null vector. Hence

[ ] [ ] [ ]DASD 1−=

Lecture 12: PRISMATIC BEAMS

with (derive for homework)

[ ]

−=−

41121 LS[ ]

− 4114EI

then

[ ] [ ] [ ]= −1 ASD[ ] [ ] [ ]

−=

=

89

4112

14 PL

PL

EIL

ASD D

=

−

517

112

84114

2PL

PLEI

−5112EI

The reactions AR are found by substitution the matrices ARL, ARD and D from above into

[AR] = [ARL] + [ARD][D]

Lecture 12: PRISMATIC BEAMS

which results in

[ ] [ ] [ ][ ]+= DAAA RDRLR[ ] [ ] [ ][ ][ ] [ ][ ]

+=

062L

EIP

DSA RDRL

RDRLR

−

+

=5

1711260

02

34 2

EIPL

EILEIL

P

PL

−−

− 66

0

2

4

22

2

LEI

LEI

LP

=69

31107

56LP

− 646956

Lecture 12: 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, the sort of numbering system an end user of RISA or STAADS might impose on the analysis If all matrices were generated conforming to the arbitrary numbering system weanalysis. If all matrices were generated conforming to the arbitrary numbering system we could lose some, if not all, of the partition definitions developed in the last section of notes.

What is required of RISA and STAADS is the ability to take an arbitrary numbering system like the one above and transform it back 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.

Lecture 12: PRISMATIC BEAMS

The SJ matrix for the arbitrary numbering system is the 6 by 6 matrix shown below.

Lecture 12: PRISMATIC BEAMS

In order for this SJ matrix to be useful the actual degrees of freedom and support constraints in the structure must be recognized. If the fourth and sixth rows and columns 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:

Lecture 12: 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 SJmatrix 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.