1. In-span loads on beam elements So far we have only been able to apply loads at nodes. example How do we then tackle loads away from nodes, or continuous loads? --------------------------------------------------------------- Option 1: add a node. example Option 2: fool the structure into thinking it has in-span loads when it doesn’t. This is the (more powerful) technique we will study in detail.

2. We have to apply loads/moments at nodes which have the same effect on the structure as the in-span loads. We conduct two analyses (A)Clamp all the nodes in the structure, apply the in- span loads and work out the reactions at the clamps. (B)Release the clamps, remove the in-span loads and reverse the reactions at nodes and conduct a standard matrix stiffness method analysis of the structure.

3. Why does this work? It is the principle of superposition which relates to linear elastic structures only. It does not matter in which order you apply loads to a structure, the deflections/rotations will be the same

4. An example – a propped cantilever P Deflected shape under load ( this is what we want to find out) P Analysis A Analysis B Fixed end moments and forces

5. So that we do not ever have to carry out analysis A we use tables of “answers” from A type analyses. Example – why can we get away with this? Fixed end moments/reactions are available for a number of load cases (tables in many textbooks) one is on DUO. Can you derive them? (Yes).

7. An example – cantilever with a UDL We know that: End deflection = End slope = Vertical support reaction = Moment support reaction =

11. We can ignore axial effects as well as those d.o.f.s which are fixed

12. FE effects example 2: Portal frame Properties E,I, A 45 o 2 1 3 4

13. Portal frame example “first load” vector, i.e. those loads already at nodes 2 1 3 4 00 Nodal load vector (transposed)

14. Analysis A

17. Node 3 Node 4 Node 1 Node 2 FE effects force vector

18. Node 1 00 FE effects force vector

19. Node 2 00

20. Node 3 Node 4 00

21. Node 4 00

22. Bringing the two load vectors together “first load” vector, i.e. those loads already at nodes 00

23. So what we now have to solve is this

24. FE effects example 3: 2 span beam Error lurking

25. This is a propped cantilever In-span bending moments