Statically Indeterminate Beams

Statically Indeterminate Beams





A beam with a larger number of reactions than can be calculated using equations of static
equilibrium is said to be
statically indeterminate
. The number of reactions is excess of the
number of equilibrium equations is called the
degree of statical indeterminacy redundancy
.





To determine the forces is a statically indeterminate beam, account must be taken of the
deflections of the beam from

which, or the basis of compatibility, supplementary equations
needed to calculate the reactions are obtained.

Types of Statically Indeterminate Beams











Continuous Beam: has more than one span which is continuous internal supports.




Note that t
he degree of statical indeterminacy can be reduced by introducing hinges on the beam.
In the general case,

DSR = 3b + r
-

3j

where b = number of members

r = number of reactions

j = number of joint in beam

Note that if the beam is subjected
to only ver
tical loads
, then DSR = 2b + 4
-

2j.




Analysis of Statically Indeterminate Beams


Essentially the same procedure as for the statically determinate beams is employed. The principal
objective is to determine the redundant reactions which then reduces the
problem to a statically
determinate one.




Method of Successive Integration


The equations needed are exactly the same as was obtained for the statically determinate beams.
There are always enough BCs to obtain the constants of integration as well as the

redundant
reactions.

Note that judicious choice of BCs will greatly simplify/minimise the computation involved.
Using the 2nd Order Differential Equation of the deflection curve also simplifies the
computation by reducing the number of constants.




Exa
mple

(Problem 8.2
-
7)

Obtain the equation of the deflection curve and the reactions R
A
, R
B

and M
A

for a propped
cantilever beam A
-
B supporting a triangular load of maximum intensity q.


Solution


Alternative Solution





Use of Castigliano's Theorem to obtain Redundant Reaction

(See Example 1: Page 538)







Consider a section
?
-
?

at distance x from support B.







According to Castigliano's Theorem, the deflection
?
B

due to the force R
B

is given by







Moment
-
Area Method for Statically Indeterminate Structures


This method involves the use of the two
-
mom
ent area theorems to obtain the supplementary
equations required to calculate the redundant reactions. These supplementary equations are based
as known slopes and deflections of the beam and their member will always equal the number of
redundant reactions.


This procedure is based on the flexibility method. The statically indeterminate beam is split into
a primary beam (released beam), which is statically determinate and is acted upon by the
externally applied loads, and a secondary beam (which is the same
as the primary beam but with
only the redundant reaction acting). The bending moment diagrams for both the primary and
secondary structures are drawn. By applying the two moment
-
area theorems and using values of
known boundary conditions, the redundant rea
ctions are computed.




Example

(Problem 8.3
-
1)

Determine the reactions R
A
, R
B

and M
A

for a propped cantilever beam AB subjected to a uniform
load of intensity q.


Solution








Example

(Problem 8.3
-
3)

Determine the reactions R
a
, R
b
, and M
a

for the propped cantilever beam AB loaded as shown in
the figure.


Also, draw the
shear
-
force and bending
-
moment diagrams for the beam, labeling all
critical ordinates.


Solution








Moment
-
Area Method


The two moment
-
area theorems are used to obtain additional equations to represent conditions as
the slope and deflection of the beam and hence obtain the redundant reactions.

The approach is very similar to the method of superposition is that the statical
ly indeterminate
beam is represented by an equivalent statically determinate primary structure acted upon by the
applied loads and a secondary structure (which is the primary structure with only the redundant
reaction acting on it).

The effect of the exte
rnal load on the primary structure and the redundant load on the secondary
structure are obtained using the moment
-
area method and compatibility conditions are enforced
to obtain the redundant reactions. Note that
deformation

corresponding to the selected
redundant
is utilized.







Example

(Problem 10.3
-
7)

The load on a fixed
-
end beam
AB

of length
L

is distributed in the form if a sine curve. The
intensity of the load is given by the equation
q

=
q
0
sin

x/L
. Beginning with the forth
-
order
differential equation of the deflection curve (the load equation), obtain the reactions of the beam
and the equation of the deflection curve.


Solution








Example

(Example 10
-
1 pg. 685)

A propped cantilever beam
AB

of length
L

supports a uniform load of intensity
q
. Analyze this
beam by solving the second
-
order differential equation of the deflection curve (the bending
-
mome
nt equation). Determine the reactions, shear forces, bending moments, slopes, and
deflections of the beam.


Solution








Example

(10
-
7 pg. 709)

The continuous beam shown has three spans of equal length
L

and constant moment of inertia
I
.
The first span is subject to a uniform load of intensity
q

and the third span supports a
concentrated load
P

of magnitude
qL
. The concentrated load acts at distan
ce 3
L
/4 from support
3. Determine the reactions of the beam using the three
-
moment equation, and then construct the
shear
-
force and bending
-
moment diagrams for the beam.


Solution








Example

(Problem 10.4
-
6)

A continuous beam
ABC

with two unequal spans, one length
L

and one length 2
L
, supports a
uniform load of intensity
q
. Determine the reactions R
A
, R
B
, and R
C

for this beam. Also, draw the
shear
-
force an
d bending
-
moment diagrams, labeling all critical ordinates.


Solution








Example


Solve the problem below using Castigliano's 2nd theorem. Obtain the support reactions.


Solution








Method of Superposition


This method involves the repl
acement of the statically indeterminate beam by a statically
determinate primary structure acted upon by the external loads and secondary structures acted
upon by the redundants.
The deformations (deflection or rotation) corresponding to the selected
redun
dants

are determined in both the primary R secondary structure and compatibility of
deformation is enforced. The method envisages the use of Tables of Load
-
Deflection in
specifying the deformations due to the external loading and redundant reactions.







Example

(Problem 9.5
-
10)

A beam
ABC

having flexural rigidity
EI

= 75 kNm
2

is loaded by a force
P

= 480 N at end
C

and
tied down at end
A

by wire having axial rigidity
EA

= 900 kN. What is the deflection at point
C

when the load
P

is applied?


Solution








Example

(Problem 9.9
-
7)

The cantilever beam
ACB

shown is subjected to a uniform load of intensity
q

acting between
points
A

and
C
. Determine the
angle of rotation

A

at the free end
A
. (Obtain the solution by
using the modified form of Castigiano's theorem.)