In-span loads on beam elements So far we have only been able to apply loads at nodes. example How do...
-
Upload
sandra-harrington -
Category
Documents
-
view
220 -
download
0
Transcript of In-span loads on beam elements So far we have only been able to apply loads at nodes. example How do...
![Page 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.](https://reader035.fdocuments.net/reader035/viewer/2022062423/5697c01d1a28abf838cd0618/html5/thumbnails/1.jpg)
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.
![Page 2: 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.](https://reader035.fdocuments.net/reader035/viewer/2022062423/5697c01d1a28abf838cd0618/html5/thumbnails/2.jpg)
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.
![Page 3: 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.](https://reader035.fdocuments.net/reader035/viewer/2022062423/5697c01d1a28abf838cd0618/html5/thumbnails/3.jpg)
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
![Page 4: 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.](https://reader035.fdocuments.net/reader035/viewer/2022062423/5697c01d1a28abf838cd0618/html5/thumbnails/4.jpg)
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
![Page 5: 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.](https://reader035.fdocuments.net/reader035/viewer/2022062423/5697c01d1a28abf838cd0618/html5/thumbnails/5.jpg)
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).
![Page 6: 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.](https://reader035.fdocuments.net/reader035/viewer/2022062423/5697c01d1a28abf838cd0618/html5/thumbnails/6.jpg)
![Page 7: 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.](https://reader035.fdocuments.net/reader035/viewer/2022062423/5697c01d1a28abf838cd0618/html5/thumbnails/7.jpg)
udl of w/m
L
Properties: E, I, A
An example – cantilever with a UDL
We know that:End deflection =
End slope =
Vertical support reaction =
Moment support reaction =
EI
wL
8
4
EI
wL
6
3
wL
2
2wL
![Page 8: 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.](https://reader035.fdocuments.net/reader035/viewer/2022062423/5697c01d1a28abf838cd0618/html5/thumbnails/8.jpg)
1 21
1
1
V
U
2
2
2
V
U1
1
1
M
F
F
Y
X
2
2
2
M
F
F
Y
X
![Page 9: 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.](https://reader035.fdocuments.net/reader035/viewer/2022062423/5697c01d1a28abf838cd0618/html5/thumbnails/9.jpg)
![Page 10: 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.](https://reader035.fdocuments.net/reader035/viewer/2022062423/5697c01d1a28abf838cd0618/html5/thumbnails/10.jpg)
![Page 11: 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.](https://reader035.fdocuments.net/reader035/viewer/2022062423/5697c01d1a28abf838cd0618/html5/thumbnails/11.jpg)
2
2
2
23
2 46
612
12
2V
L
EI
L
EIL
EI
L
EI
wL
wL
We can ignore axial effects as well as those d.o.f.s which are fixed
![Page 12: 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.](https://reader035.fdocuments.net/reader035/viewer/2022062423/5697c01d1a28abf838cd0618/html5/thumbnails/12.jpg)
P
udl of w/m
2L
L2L/3
A
B C
D
Node nos.
PFE effects example 2: Portal frame
Properties
E,I, A
45o
2
1
3
4
![Page 13: 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.](https://reader035.fdocuments.net/reader035/viewer/2022062423/5697c01d1a28abf838cd0618/html5/thumbnails/13.jpg)
P
udl o f w /m
2 L
L2 L /3
A
B C
D
N ode nos.
PPortal frame example
“first load” vector, i.e. those loads already at nodes
2
1
3
4
00
00 0 01XF 1YF 1M 4XF 4YF 4M2P
2P
Nodal load vector (transposed)
![Page 14: 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.](https://reader035.fdocuments.net/reader035/viewer/2022062423/5697c01d1a28abf838cd0618/html5/thumbnails/14.jpg)
Analysis A
![Page 15: 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.](https://reader035.fdocuments.net/reader035/viewer/2022062423/5697c01d1a28abf838cd0618/html5/thumbnails/15.jpg)
27
4PL
27
2PL
27
20P
27
7P
![Page 16: 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.](https://reader035.fdocuments.net/reader035/viewer/2022062423/5697c01d1a28abf838cd0618/html5/thumbnails/16.jpg)
wL wL
3
2wL
3
2wL
![Page 17: 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.](https://reader035.fdocuments.net/reader035/viewer/2022062423/5697c01d1a28abf838cd0618/html5/thumbnails/17.jpg)
wL wL
3
2wL
3
2wL
27
4PL
27
2PL
27
20P
27
7P
Node 3
Node 4Node 1
Node 2
FE effects force vector
![Page 18: 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.](https://reader035.fdocuments.net/reader035/viewer/2022062423/5697c01d1a28abf838cd0618/html5/thumbnails/18.jpg)
27
2PL 27
7PNode 1
00
FE effects force vector
27
7P0
27
2PL
![Page 19: 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.](https://reader035.fdocuments.net/reader035/viewer/2022062423/5697c01d1a28abf838cd0618/html5/thumbnails/19.jpg)
wL
3
2wL
27
4PL
27
20P
Node 2
00
27
7P0
27
2PL27
20PwL
327
4 2wLPL
![Page 20: 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.](https://reader035.fdocuments.net/reader035/viewer/2022062423/5697c01d1a28abf838cd0618/html5/thumbnails/20.jpg)
wL
3
2wL
Node 3
Node 4
00
27
7P0
27
2PL27
20PwL
327
4 2wLPL
wL3
2wL0
![Page 21: 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.](https://reader035.fdocuments.net/reader035/viewer/2022062423/5697c01d1a28abf838cd0618/html5/thumbnails/21.jpg)
Node 4
00
27
7P0
27
2PL27
20PwL
327
4 2wLPL
wL3
2wL0 0 0 0
![Page 22: 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.](https://reader035.fdocuments.net/reader035/viewer/2022062423/5697c01d1a28abf838cd0618/html5/thumbnails/22.jpg)
00
27
7P0
27
2PL27
20PwL
327
4 2wLPL
wL3
2wL0 0 0 0
Bringing the two load vectors together
“first load” vector, i.e. those loads already at nodes
00
00 0 01XF 1YF 1M 4XF 4YF 4M2P
2P
![Page 23: 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.](https://reader035.fdocuments.net/reader035/viewer/2022062423/5697c01d1a28abf838cd0618/html5/thumbnails/23.jpg)
So what we now have to solve is this
4
4
4
3
3
3
2
2
2
1
1
1
4
4
4
2
2
1
1
1
matrix Stiffness Global1212
3
2
2
327
4
2720
272
277
V
U
V
U
V
U
V
U
M
F
F
wLwLP
P
wLPLwL
P
PLM
F
PF
Y
X
Y
X
![Page 24: 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.](https://reader035.fdocuments.net/reader035/viewer/2022062423/5697c01d1a28abf838cd0618/html5/thumbnails/24.jpg)
FE effects example 3: 2 span beam
Error lurking
![Page 25: 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.](https://reader035.fdocuments.net/reader035/viewer/2022062423/5697c01d1a28abf838cd0618/html5/thumbnails/25.jpg)
udl of w/m
L
Results of stiffness matrix
analysis
"Free" structure
"Free" BMD
Final BMD
This is a propped cantilever
In-span bending moments