The document discusses statically indeterminate structures, which have more unknown reactive stresses than equations of static equilibrium. This leads to external and internal redundancy. The degree of static indeterminacy (T) is the sum of the external indeterminacy (E), which depends on support conditions, and internal indeterminacy (I), which depends on additional members. A continuous beam has E=3 due to its redundant supports, while a pin-jointed frame can become internally indeterminate if it has more members than required for its geometry.
5. If a structureis stable, under the action of forces acting in a plane,
three condition of equilibrium must be satisfied.
When the number of unknown reaction or stress components
exceeds the number of conditions of equilibrium, the system will
be said to be statically indeterminate. The excess members are
described as redundancy.
The degree of Redundancy is the number of unknown reactive
stresses. Over the number of conditions equations available for
solution.
6.
7. Roller support :
It consists of two rockers , the upper rocker abd
the lower rocker. Both the rockers can move
together on rollers along the bearing plate.
So a reactive force acts normal to the surface of
rolling and is dIrected through the centre of the
hinged pin.
8. A hinge support:
In this support lower rocker is fixed and cannot move.
This support has two reaction components - the
horizontal and the vertical
9. Fixed support
It has zero degree of freedom. It provides three reaction
components - horizontal and vertical components and the
magnitude of the moment about the same point.
10. External Reduedancy
For any structure , supported on external supporternale total
reaction components can be easily found.
In general, three reaction components are necessary for the
external stability of plane structure.
The arrangements of the three reaction components is very
important from stability point of view
If the reaction components have parallel lines of action or
concurrent, the structure will be unstable.
11. For a plane structure, three equation of static equilibrium
are available.
A pin provided anywhere in the structure cannot transmit
moment from one part of the structure to the other part and
provides an additional condition equation.
Similarly a link transmitting moment as well as horizontal
force from one part to the other and provides two additional
condition equations.
12. Toral number of condition equations of statical equilibrium
for any structure are equal to the three equations of statical
equilibrium plus additional condition equations because of a
pin or link anywhere in the structure.
The structure is unstable if the total number of reaction
components (R) are less than the total number of condition
equations available.
13. The degree of indeterminacy is the number by which
the reaction components exceed the condition
equations and is represented by the equation.
E =. R - r
E = Degree of external Redundancy
R = Total number of reaction components
r = Total number of condition equations available
14. Statically indeterminate Beam
A continuous beam is a typical example of externally
indeterminate structure.
R = 3+1+1+1. = 6
E = R - r = 6-3 = 3
6 reaction components
3 condition equations
So, for general system of loading the beam is statically
indeterminate to third degree
15. R = 3+3+3+3 = 12
r = 3
E = R - r = 12 - 3 = 9
I = 3a = 3*1 = 3
a - number of loop
T = E + I = 9+3 = 12
So, total indeterminacy 12
For rigid joint frame structure
16. For Hybrid structure
R = 3+3+3+2 = 14
m = 19
m are the number of members of the structure
E = R- r = 14-3= 11
b,d,f,h,m are the 5 pin joints with 3 members
I,k are pinjoint with 4 members
So, 5(3-1)+2(4-1)= 16
I = 3*6 - 16 = 2
C is the cut for Hybrid structure
Here c is 6
T = E+ I = 11+2= 13
17. Summary
Statically indeterminate structures are those structures
which cannot be analysed with the help of equations of
static equilibrium alone. This is also called hyperstatic
structure.
In the case of statically indeterminate structures, the
number of unknown is greater than the number of
independent equations derived from the conditions of
static equilibrium.
18. Additional equations based on the compatibility of
deformation, must be written in order to obtain a
sufficient numbers of equations for the determination
of all the unknown.
The number of these additional equations, necessary
for the solution of the problem, is known as the
degree of static indeterminacy of the structure
19. The total degree of static indeterminacy of the structure T may be
considered as the sum of the following two types of indeterminacies -
Degree of external indeterminacy
Degree of internal indeterminacy
So, T = E + I
The external indeterminacy is related to the support system of the
structure.
For static equilibrium there are six independent equations to be satisfied
in case of space structure., and three equations for the plane structure.
20. Pin jointed frame is statically determinate internally if it
has just the minimum number of members (m) required
to preserve its geometry.
If the number of members is more , the pin jointed frame
is internally inderminInate to the extent.
