Loads. Lecture 12: PRISMATIC BEAMS

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Ella McGee
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1 Loads After composing the joint stiffness matrix the next step is composing load vectors. reviously 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 vector of actions used directly in computations. Member loads must be converted into equivalent fixed end joint loads. Consider the following beam once again

2 The joint numbering system is the same as the previous section of notes. Joint loads would fill up the matrix [A] as follows L [ A] The remaining loads on the structure act directly on the members and are shown action on the two beam segments as follows

3 These fixed end actions may be assembled in a rectangular matrix [A ML ] where each row contains the end actions for a given member, i.e., 1 1 L 1 1 L [ ] A ML L L Given the load values [ ] A ML L L L L

4 When the fixed end reactions in [A ML ] are reversed, they constitute equivalent joint loads shown in the following figure 1 L L L L L These equivalent joint loads can be assembled as a vector [A E ] shown at the right. The equivalent joint loads are [ ] 1 L L A E g q j assembled in the vector corresponding to the joint numbering system in the previous section of notes. [ ] L L E

5 Actual joint loads [A] are then added to equivalent joint loads [A E ] to produce a matrix of composite loads [A C ] as follows p C [ ] [ ] [ ] + A A A E C 9 L L L L L If the signs on the elements of + L L If the signs on the elements of [A C ] are reversed then the matrix is equivalent to [A RL ]. 3 3

6 In summary the vector [A C ] contains information in the following manner where A D [ AC ] ARL 9L L [ A D ] [ ] A RL L 3 The formation of vectors [A D ] and [A RL ] 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 [A DL ] is the null vector. Hence 1 [ D] [ S] [ ] A D

7 with (derive for homework) then 1 1 L [ S ] 1EI 1 [ D ] [ S ] 1 [ A ] D 9L L 1 1 EI 1 L L 17 11EI 5 The reactions AR are found by substitution the matrices ARL, ARD and D from above into [A R ] [A RL ] + [A RD ][D]

8 which results in [ A R ] [ A RL ] + [ A RD ][ D ] [ A ] + [ S ][ D] RL RD 6EI L L EI L L 3 + 6EI 11EI L 6 EI 6 EI L L 17 31L

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.

9 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 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 segregates matrix elements associated with degrees of freedom from those associated with support constraints.

10 The S J matrix for the arbitrary numbering system is the 6 by 6 matrix shown below.

In order for this S J matrix to be useful the

If the fourth and sixth rows and columns

11 In order for this S 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 and columns switched to the first and second rows, while all others move downward, we obtain the following matrix:

Next the fourth and sixth column are moved to the first and

i.e., Software algorithms must have the capability to track

12 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 S J matrix we had previously, i.e., Software algorithms must have the capability to track degrees of freedom and perform the necessary matrix manipulation in order to identify pertinent information.

Example Lecture 12: The Stiffness Method Prismatic Beams. Consider again the two span beam previously discussed and determine

Example Lecture 12: The Stiffness Method Prismatic Beams. Consider again the two span beam previously discussed and determine Example 1.1 Consider again the two span beam previously discussed and determine The shearing force M1 at end B of member B. The bending moment M at end B of member B. The shearing force M3 at end B of