MECH 321 - Solid Mechanics II

sublimefrontΠολεοδομικά Έργα

15 Νοε 2013 (πριν από 3 χρόνια και 8 μήνες)

76 εμφανίσεις

August 29, 2013Page 1
MECH 321 -Solid Mechanics II
Week 11, Lecture 3
Statically Indeterminate Beams
Method of Superposition
August 29, 2013Page 2
The Method of Superposition has been used earlier in this
course for the analysis of combined loadings.
In order to apply this method to statically indeterminate beams
and shafts we must again first determine the redundant support
reactions.
Then, by removing them from the beam we obtain what is called
the “primary beam”, which is statically determinate and stable.
If we add to the primary beam a succession of similarly
supported beams, each loaded with a separate redundant
reaction, then by the principal of superposition, we can obtain
the actual loaded beam.
Statically Indeterminate Beams
Method of Superposition
August 29, 2013Page 3
To solve for the redundant reactions we again use the
conditions of compatibility that exist at the supports where each
of the redundant reactions act.
Once the redundant reactions are obtained, the other reactions
acting on the beam can be determined from the three equations
of equilibrium.
Statically Indeterminate Beams
Method of Superposition
August 29, 2013Page 4
Statically Indeterminate Beams
Method of Superposition
In the example shown, the redundant
reaction, B
y, is ignored in the first
step.
In order to maintain compatibility the
redundant reaction, By, is then
applied at B where the displacements
vB
and v’
B
are equal and opposite
(and therefore cancel).
Compatibility equation.
BB
vv

+
=0
August 29, 2013Page 5
Statically Indeterminate Beams
Method of Superposition
From App. C in Hibbeler.
EI
PL
vB
48
5
3
−=
EI
LB
v
y
B
3
3
=

Sub and solve.
EI
LB
EI
PL
y
348
5
0
3
3
+−=
PB
y
16
5
=
August 29, 2013Page 6
Statically Indeterminate Beams
Method of Superposition
Now that we know B
y
, the
reactions at the wall can be
determined from the equilibrium
equations.
The results are…
0=
x
A
PAy
16
11
=
PLM
PLPLM
LP
L
PM
A
A
A
16
3
16
5
16
8
16
5
2
=
−=






−=
August 29, 2013Page 7
As stated previously, the choice of the redundant reactions is
arbitrary (provided that the beam remains stable).
As another example, consider the same beam and loading, but
choose the moment at A, as the redundant reaction.
Statically Indeterminate Beams
Method of Superposition
August 29, 2013Page 8
Statically Indeterminate Beams
Method of Superposition
In this example the capacity of the
beam to resist M
A
is removed, hence
the support at A becomes a pin
support.
There is now a slope at A (θ°) caused
by load P.
In order to maintain compatibility the
redundant reaction, M
A, is then applied
at A where the displacements θ
A
and
θ'A
are equal and opposite (and
cancel).
August 29, 2013Page 9
Statically Indeterminate Beams
Method of Superposition
Compatibility equation.
AA
θ
θ

+
=
0
From App. C in Hibbeler.
EI
PL
A
16
2
=
θ
EI
LM
A
A
3
=

θ
Sub and solve.
EI
LM
EI
PL
A
316
0
2
+=
PLM
A
16
3
−=
August 29, 2013Page 10
Statically Indeterminate Beams
Method of Superposition
This is the same result (bending
moment at A) as was computed
previously.
Here the negative sign for M
A
means
that the moment acts in the opposite
sense from the direction shown in the
figure.
August 29, 2013Page 11
Statically Indeterminate Beams
Method of Superposition
Here we choose the reaction forces
at the roller supports B and C as
redundant.
When the redundant reactions are
removed, the statically determinate
beam deforms as shown here.
Each redundant force deforms the
beam as shown.
In this case the beam is indeterminate
to the second degree and therefore two
compatibility equations are needed for
the solution.
August 29, 2013Page 12
Statically Indeterminate Beams
Method of Superposition
By superposition, the compatibility
equations for the displacements at
B and C are.
BBB
vvv


+

+
=
0
When the displacements (in terms
of By
and Cy
have been determined,
they can be used with the
equilibrium equations to solve for
the remaining unknowns.
CCC
vvv


+

+
=
0
August 29, 2013Page 13
Statically Indeterminate Beams
Method of Superposition
Procedure for Analysis
The Elastic Curve.
1.Specify the unknown redundant forces or moments that must be
removed from the beam in order to make it statically determinate and
stable.
2.Using the principal of superposition, draw the statically indeterminate
beam and show it equal to a sequence of corresponding statically
determinate beams.
3.The first of these beams supports the same external loads as the
statically indeterminate beam. Each of the other beams that are
“added” to the first beam, show the beam loaded with separate
redundant forces or moments.
4.Sketch the deflection curve for each beam and indicate symbolically
the displacement or slope at the point of each redundant force or
moment.
August 29, 2013Page 14
Statically Indeterminate Beams
Method of Superposition
Procedure for Analysis
Compatibility Equations.
1. Write a compatibility equation for the displacement or slope at each
point where there is a redundant force or moment.
2.Determine all the displacements or slopes using an appropriate
method.
3.Substitute the results into the compatibility equations and solve for
the unknown redundant forces or moments.
4.If a numerical value for a redundant force or moment is positive, it
has the same sense of direction as originally assumed. A negative
numerical value indicates that the redundant force or moment acts
opposite to the assumed sense of direction.
August 29, 2013Page 15
Statically Indeterminate Beams
Method of Superposition
Procedure for Analysis
Equilibrium Equations.
Once the redundant forces and/or moments have been
determined, the remaining unknown reactions can be
found from the equations of equilibrium applied to the
loadings shown on the beam’s free-body diagram.
August 29, 2013Page 16
Example 12.22
Determine the reactions on the beam. Due to the loading and poor construction, the
roller support at B settles 12 mm. Take E = 200GPa and I = 80(10
6) mm4
=
+
August 29, 2013Page 17
Next Time
Columns and Critical Loads