Chapter 2 Geometric Stability of Structures
If a system without deforming its material can /not hold its shape and position, it is referred to as a geometrically stable as simply as a stable /unstable system. without deformation .Hold its shape and position.
Unstable system can change into stable system adding more members. Because the elastic deformations of material are ignored, a beam ,link, or stable system are all served as rigid body (刚体)。
Rigid body in a plane is called rigid piece (刚片)。
?1. Degrees of freedom of planar system
The degree of freedom of a system is the number of independent coordinates which is required to locate the system fully.
Degree of freedom of a system is the number of independent coordinates which is required to change position of the system.
Degree of freedom is denoted by . ，
Restraint is the number of degree of freedom which is reduced.
? link :
link 1 restrain. ；
A rigid piece is connected to foundation by a link. The position of piece can be
located by ,so its degree of freedom is 2. That is to say, a link is equivalent ？？！12
to a restraint.
n links are equivalent to n restraints.
? Simple hinge
A hinge which just connects two rigid pieces is called simple hinge. A hinge which connects three or more pieces is called multiple hinges. Fig 2-5
There are 6 degrees of freedom when two rigid pieces are separated. After being connected by a simple hinge. The new body’s position can be located by (x, y), ( ). I.e. its degree of freedom is 4.its restraints equal to 6-4=2, ？？！12
So, a simple hinge is equivalent to 2 restraints.
A multiple hinge connecting n bars is equivalent to (n-1) simple hinge or 2(n-1) restraints.
To say standing or restraint, a simple hinge is equivalent to two links,(2-6)
In a planar system, suppose B rigid pieces .L links; H hinges. The degree of freedom of the system is (2-1) ，，，32BHL;;
a) The multiple hinges must transfer into simple hinge, then calculate. ，
b) The restraints connecting the system to the foundation are included in above
Three possible solutions may be obtained from Eq.(2-1)
restraints are insufficient, unstable system . ，0；
restraints are sufficient , ，，0；
restraints are redundant, ，0；
,it is unclear whether the system is stable or unstable , because the stability ，，0
also the arrangement of restraints.
So , is not sufficient condition but necessary condition for stability .We ，，0
must analyses the geometric construction of the system. For example:
Fig.2-7 (a) Fig.2-7(b) For Fig.2-7(a)
But, it is unstable.
For a truss structure, degree of freedom of the structure can be calculated in alternative way.
In this way, hinges are considered as bodies, while members as links .If hinges are H, links (including support links) are L, and we have
So , ，；(;；26120
?2. Geometric construction analysis of planar systems
As shown above ,systems with a sufficient number of restraints and those with redundant restraints may be stable or unstable , the conclusion depends on the arrangement of the restraints.
It is known that a hinged triangle is the simplest (Fig.2-8)stable form of framed system.
Rule 1 (Fig.2-9):
Rigid piece I is connected to hinge C by two links 1,2 ,provided that two links don’t lie on the same straight line .
If the two links lie on the same straight line (Fig.2-10),the joint C can have the
！infinitesimal displacement. Joint C will move to ,the system is termed an C
！instantaneously unstable system because is new position which two links do not C
lie on the same straight line.
This rule is also termed two components (二元体)。Two components rule is meant
by two links without in the same straight form new joint.
Adding or subtracting an unit of two components to (from) a system. Stability of the system do not change (2-11).
i.e. A stable system I ,adding units 1,2,3,4,5,6,7, the new system is still stable . Rule 2 (two pieces rule )
Two rigid pieces I,II (Fig.2-12) connected by one C hinge and one link(AB) form a stable system ,with no redundant restraint provided that the link do not cross though the hinge.
Two pieces connected by three links form a stable system with no redundant restraint provided that the three links are not either parallel to each other or cross the same point.
Rule 3(three pieces rule)
Three rigid pieces (I,II,III)joined pair wise by a hinge form a stable system provided that the three hinges (A,B,C) do not lie on the same straight line .
Three pieces joined pair wise by two links form a stable system provided that the 6 links do not cross the same point .
Assumed to be rigid piece I, foundation and rigid AB connected by hinge A and link 3 form stable system . The stable system may be regarded as a larger rigid piece .This piece connected to hinge C by link BC, 4.
So ,the system is stable.
Because three links (1,2,3) connect to foundation , this will not affect the internal stability of the system .
We may disregard the existence of the 3 support links and analyze the internal stability of the system only .
Two pieces (AE,BE) connected by hinge E and link CD ,and link CD do not cross hinge E. So the system is stable .
To analyze the internal stability separately ,2 rigid pieces (AE,BF) are connected by links EF,CD, the internal stability is unstable because two links intersect at G.
Noticing the structure is connected to foundation by 4 links ,this will affect the internal stability of the system.
In fact , three pieces are connected by three hinges (A,B and G), and three hinged do not lie on the same line .
So the system is stable.(Fig.2-19).
When 3 links connect to foundation, we can disregard the foundation .when 4 or more links connect to foundation ,the foundation must be regard as a rigid piece . Ex .4:
We substract two unit two components (1,2) ,the remained system is stable .
Two pieces are connected by hinge c and link a.
2-2, 2-7, 2-8, 2-17