STATICALLY DETERMINED PLANE BAR STURCTURES (BEAMS)

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STATICALLY DETERMINED PLANE BAR STURCTURES
(BEAMS)
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Superposition principle
To determine cross-sectional forces we need to
know all external loadings including reaction of
different type of supports. This can be easily
done if we assume solely for this purpose that
the body is non-deformable as we did in
Theoretical Mechanics. Therefore, we deal with
linear systems and we can exploit superposition
principle. It says, in our case, that the effect
of sum of different loadings is equal the sum of
effects of individual loadings. This is a
principal property of so called linear systems.
As the result we will use linear algebra as
already done considering the relation of
M-Q-q. Quite often it is worthwhile to apply this
principle when we have to deal with complicated
loadings, and therefore it is strongly
recommended in practical use. It is much easer to
avoid mistakes when considering simple loadings
and summing up their effects than to deal with
very complex ones.
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Superposition principle
An example of the use of superposition principle
for determination of support reactions
1 kN/m
Y
4 MNm
2 kN
? MK0
14(224/2)
22
4
0
328RP0
(224)RP
RP - 4
? Y0
RL 6 RP
RL 6 4 2
RL
- 2
0
RP
- 14
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Superposition principle
Y
? MK0
? Y0

2 kN
RL 2,0
RR 4,0
RL126/81,5
RR10,5

0,5 kN
4 kNm
RL2-4/8 - 0,5
RP20,5

0,5 kN
0,5 kN
RL314 2/81,0
RP33,0
3,0 kN
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Beams
Typical simple beams
Pin-pointed (or simply supported) beam
Cantilever beam
Semi-cantilever beam
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Beams
Fundamental assumptions used for drawing diagrams
of cross sectional forces
1. Diagram of bending moment will appear always
on the side of a beam which is subjected to the
tension caused by this bending moment. Therefore,
no sign is necessary.
2. We will make use of q-Q-M dependence
3 . Shear and normal forces will be always marked
with their sign according to the following
convention
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Typical beams
P
q?0
Qc
Mcxb
qd
Mdx2/2exf
Qdxe

Pb/l
-
Pa/l
P
P/2
P/2
l/2
l/2
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Typical beams
qd
Qdxe
Mdx2/2exf
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M
a
b
l
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Typical beams
P(1a/l)
Pa/l
Pa
P(1a/l)
P

1(a/l)2ql/2
-
1-(a/l)2ql/2
Pa/l
q
l
a
qa2/2
l
l
a
a

-
(l/2)1-(a/l)2
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Cantilever beam
P(1a/l)
Pa/l
l
a
Pa
P(1a/l)
P

-
Pa/l
q
M
1(a/l)2ql/2
M/l
M/l
l
a
l
a
qa2/2

-
(l/2)1-(a/l)2
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Gerbers beams
The concept of multiple co-linear beams
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Gerbers beams
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Gerbers beams
A
B
Formal definition the set of aligned simple
bars, hinged together and supported in the way
which assures kinematical stability
Number of unknown reactions4 horizontal 4
vertical
Equilbrium equations
?X 0
1 equation
4 unknown horizontal reactions
Structure is statically indetermined with respect
to horizontal reactions
2 equations
4 unknown vertictal reactions
MA 0
2 equations
MB 0
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Gerbers beams
Number of unknown reactions to be found1
horizontal reaction from 1 equilibrium equation
(or we accept indeterminancy with respect to
normal cross-sectional forces)(2 n) remaining
reactions (vertical reactions, moment reactions)
from 2 equilibrium equations and n equations of
vanishing bending moment in hinges
Number of hinges is determined from the second of
the above condition structures appears to be
kinemtaically unstable if there are too many
hinges, and hyper-stiff if there are too little
hinges). However, the location of hinges even
if their number is correct one cannot be
arbitrary!
WRONG!
WRONG!
GOOD!
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Gerbers beams
Partitioning of a Gerbers beam into series of
simple beams allows for better understanding of
structures work!
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Gerbers beams
SUPERPOSITION!






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Gerbers beams
SUPERPOSITION!
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• stop

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Gerbers beams