Fundamental concepts 35
Next consider that matrix H of a certain frame is neither square
(3j ≠ n + r) nor full rank (r H < 3j or r H < n + r), then let
,
(2.41a)
.
(2.41b)
If s > 0, then the columns of H are linearly dependent, and the equilibrium
Eq. (2.39) will have non- zero solutions for t even if f = 0. The assembly can
have a total of s sets of linearly independent internal forces without the
external loading. It is therefore regarded as being statically indeterminate
because the equations of equilibrium alone are insufficient to uniquely determine the member forces. The forces are known as the states of self- stress.
When m > 0 the columns of C are linearly dependent, and similarly the
compatibility equation can have non- zero solutions for d even though e = 0.
The assembly can have m sets of linearly independent displacements. It is
known as being kinematically indeterminate since the displacements of the
joints cannot be uniquely determined by the lengths of the members.
The static and kinematic characteristics of an assembly can be given by
the pair s and m, both of which can either be greater than or equal to zero.
The possibilities can be grouped into a total of four categories.
a the assembly is both statically and kinematically determinate. It has
neither state of self- stress nor mechanism (s = 0, m = 0);
b the assembly is statically determinate and kinematically indeterminate
frame. It has no state of self- stress but is a mechanism with mobility m
(s = 0, m > 0);
c the assembly is statically indeterminate and kinematically determinate.
It has states of self- stress and is stiff (s > 0, m = 0); and
d the assembly is both statically and kinematically indeterminate. It has
states of self- stress but at the same time it is a mechanism with m
mobilities (s > 0, m > 0).
The same classification is applicable to assemblies other than trusses. The
common linkages surveyed in previous sections belong to (b) whereas the
overconstrained linkages belong to (d) which become statically indeterminate structures once their motion is locked. The existence of self- stress can
be used to detect the mobility in overconstrained linkages.
Readers should be aware that the above linear algebraic analysis, set up
only for the initial geometrical configuration, has its own limitations
despite taking both geometry and topology of the assembly into account.
The displacements may be infinitesimal, i.e. the assembly will tighten up
after a small displacement, instead of full cycle mobility (or being truly
mobile). The advanced materials on the topic can be found in Calladine
(1978), Pellegrino and Calladine (1986) and Tarnai (1984, 2001).
Next consider that matrix H of a certain frame is neither square
(3j ≠ n + r) nor full rank (r H < 3j or r H < n + r), then let
,
(2.41a)
.
(2.41b)
If s > 0, then the columns of H are linearly dependent, and the equilibrium
Eq. (2.39) will have non- zero solutions for t even if f = 0. The assembly can
have a total of s sets of linearly independent internal forces without the
external loading. It is therefore regarded as being statically indeterminate
because the equations of equilibrium alone are insufficient to uniquely determine the member forces. The forces are known as the states of self- stress.
When m > 0 the columns of C are linearly dependent, and similarly the
compatibility equation can have non- zero solutions for d even though e = 0.
The assembly can have m sets of linearly independent displacements. It is
known as being kinematically indeterminate since the displacements of the
joints cannot be uniquely determined by the lengths of the members.
The static and kinematic characteristics of an assembly can be given by
the pair s and m, both of which can either be greater than or equal to zero.
The possibilities can be grouped into a total of four categories.
a the assembly is both statically and kinematically determinate. It has
neither state of self- stress nor mechanism (s = 0, m = 0);
b the assembly is statically determinate and kinematically indeterminate
frame. It has no state of self- stress but is a mechanism with mobility m
(s = 0, m > 0);
c the assembly is statically indeterminate and kinematically determinate.
It has states of self- stress and is stiff (s > 0, m = 0); and
d the assembly is both statically and kinematically indeterminate. It has
states of self- stress but at the same time it is a mechanism with m
mobilities (s > 0, m > 0).
The same classification is applicable to assemblies other than trusses. The
common linkages surveyed in previous sections belong to (b) whereas the
overconstrained linkages belong to (d) which become statically indeterminate structures once their motion is locked. The existence of self- stress can
be used to detect the mobility in overconstrained linkages.
Readers should be aware that the above linear algebraic analysis, set up
only for the initial geometrical configuration, has its own limitations
despite taking both geometry and topology of the assembly into account.
The displacements may be infinitesimal, i.e. the assembly will tighten up
after a small displacement, instead of full cycle mobility (or being truly
mobile). The advanced materials on the topic can be found in Calladine
(1978), Pellegrino and Calladine (1986) and Tarnai (1984, 2001).