DETERMINACY [AC2:7] AND KINEMATIC STABILITY [AC2:4-6]

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Nov 14, 2013 (3 years and 7 months ago)

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CEE 371

Determinacy and Kinematic Stability, p.
1



DETERMINACY [AC2:7] AND KINEMATIC STABILITY [AC2:4
-
6]


Review of Determinacy

[AC2:7 & Lecture Notes on Structural Analysis]


A statically determinate structure is one that can be analyzed using the equations of equilibrium
(statics) only.


If a structure i
s statically indeterminate, we must use not only equations of equilibrium but also
equations of compatibility (structural deformations must be "compatible," i.e., the deformed
structure must fit together with no gaps or overlaps). If
d

such additional equ
ations are needed,
the structure is said to be "statically indeterminate to the d
th

degree."


NOTE: determinacy is independent of the loading on the structure.


Basic technique:

1.

Find
F
, the number of unknown force components by removing supports and per
haps by
also cutting into simpler segments.

2.

Find
e
, the number of equilibrium equations available for the overall structure or for all
the various segments.

3.

Calculate
d = F


e



if
d = 0
, statically determinate



if
d > 0
, statically indeterminate to the
d
th

degree



if
d < 0
, structure is geometrically unstable

Other techniques are merely specialized variants of
d = F


e
:


For
trusses
, count the joints
j
, the members
m
, and reaction components
r
. For each member,
there is one unknown force, while for ea
ch 2D joint, there are 2 equations of equilibrium. [For
each 3D joint, there are 3 equations of equilibrium.]


for a 2D truss:


d = r + m


2j


for a 3D truss:


d = r + m


3j


For rigid
frames
, see the AC2:7 and the lecture notes on Structural Ana
lysis for use of a
specialization of
d = F


e

for frames. For example, for a 2D frame without hinges:


d

= 3
m

+
r



3
j


Another technique available for
beams, trusses, or frames

[not covered in AC]: divide the
structure into "trees," free
-
body diagrams
of a skeletal structure without any closed loops. For
each 2D tree there are 3 equations of equilibrium; therefore, if the number of trees is
t
,
e = 3t
.
[For each 3D tree, there are 6 equations of equilibrium; and
e = 6t
.]


for 2D beams, trusses, or fram
es:

d = F


3t


for 3D beams, trusses, or frames:

d = F


6t

Notes on the method of trees:



In making trees, we always cut at any internal hinge and we remember that
M
= 0

at these
cuts.



The cutting of a structure into trees is not unique, but one obtai
ns the same result by any
admissible division (no loops).

CEE 371

Determinacy and Kinematic Stability, p.
2



Kinematic or Geometric Stability

[AC2:4
-
6]


Kinematic (geometric) stability


small deflection behavior.

Kinematic (geometric) instability


structure or part of a structure can change shape or
e
xperience appreciable deflections with little or no applied loadings and without stressing
of the members ("rigid body motion").


As its name implies, geometric instability is primarily a function of geometry. It is independent of:





loading





stiffne
ss of members





strength of members


In these regards, it differs from
structural instability (buckling)

which, for example in the case of
elastic columns, is characterized by critical loads of the form:



P
Euler

=


2

EI/(kL)
2

which contains load (
P
) an
d stiffness (
EI
).


We need to ensure that our computational models are kinematically stable if we are to obtain
meaningful results from structural analysis by the stiffness (displacement) method.


If from
d = F


e
, if
d < 0
, the structural model is
always

unstable. This type of instability is easy to
detect.


Example
:





d = r + m


2j



r = 3; m = 8; j =6


d = 3 + 8


2(6) =

1


However, if d


0, the structural model may still be kinematically unstable. Th
ese types of
instability are more difficult to recognize or detect, but several methods are available:



Inspection



If
d = 0
, can use evaluation of determinant of equilibrium equations


zero value will
indicate instability



Build a physical model



Proceed wit
h computer analysis; if it fails, structure is unstable (singular coefficient
matrix of equilibrium equations) [many programs, such as MASTAN2, issue a warning if
kinematic instability is encountered; some also identify the unstable shape to help the
analy
st correct the model]


CEE 371

Determinacy and Kinematic Stability, p.
3



Method of Inspection


Essentially, look for structures or parts of structures that can deflect freely or must deflect as a
mechanism

when subject to a load. A “mechanism”

is a structure or assemblage of elements
with ability to move

without significant stresses (that is, experiences “rigid body motion”).

What to look for to detect a mechanism:



Parallel forces
--

allow movement perpendicular to direction (see AC Figure 2
-
23b)



Concurrent forces
--

allow rotation about a point of concur
rency (see AC Figure 2
-
23a)

If a structural component is constrained in a manner that admits only parallel or concurrent
forces, then it is unstable. For 2D structures, for example, we need to have at least 3 nonparallel,
nonconcurrent constraining forces
.


Another source of possible instability is the presence of internal hinges. To inspect
trusses
, break
them down into basic (stable) triangular elements, and then see how these are supported and
connected. To inspect frames, break them down into trees w
ithout any internal hinges. Each
such tree is stable within itself, so we need only see if it is supported by or connected to other
trees in a stable manner. (See attached examples).


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