geometrically stable system
结构
Under the action of any loads, the system still maintain its shape and remains its location if the deformations of the members are neglected.
F
E
2 rigid bodies, connected by 3 links, which are nonparallel and nonconcurrent cross the hinge, form an internally stable system with no redundant restraints. 。
Degrees of freedom of a system are the numbers of independent movements or coordinates which are required to locate the system fully.
for a point in plane n=2
C
structure formed by Attaching of binary systems 减二元体简化分析
W=3 ×10-(2×14+3)=-1<0 W=2 ×6-13=-1<0
计算自由度 = 体系真实 的自由度 ？
W=2 ×6-12=0 W=3 ×9-(2×12+3)=0
缺少联系 几何可变
W=2 ×6-11=1 W=3 ×8-(2×10+3)=1
summary
W>0, 缺少足够联系，体系几何可变 Restraints are not enough, unstable。 W=0, 具备成为几何不变体系所要求的最少 联系数目has the minimum necessary numbers of restraints for stable system。
Three hinged arch
大地、AC、BC为刚片;A、B、C为单铰
无多余几何不变
binary system（二元体）--- 2 non-collinear links connected by a hinge
Binary system rule: The geometric construction property of a system will not change if a binary system is attached to or detached from the system。
2 rigid bodies, connected by 1 hinge and 1 link that does not cross the hinge, form an internally stable system with no redundant restraints.
二刚片规则： two-rigid-body rule:
Because the removal of any bar in the system will increase one degree of freedom, therefore all bars are necessary restraints
Restraints, removal of which doesn’t change the degrees of freedom, is named as redundant restraints .
W=2j-b
例1：Determine the numbers of degrees of freedom of the following system
AC CDB CE EF CF DF DG FG
1 3
3
1 G 2
有几个单铰？
W=3×8-(2 ×10+4)=0
例2：Determine the numbers of degrees of
图中上部四根杆 和三根支座杆都是 必要的联系。
下部正方形中任 意一根杆，除去都 不增加自由度，都 可看作多余的联系。
例3：
W=0,但 布置不当 几何可变。 上部有多 余联系， 下部缺少 联系。
W=3 ×9-(2×12+3)=0 W=2 ×6-12=0
例4
W<0,体系 是否一定 几何不变呢？
上部 具有多 余联系
restraints。
W<0， 体系具有多余联系has redundant
W> 0 W< 0
unstable
stable?
§2-3
Geometric construction rules of planar stable framed systems
two-rigid-body rule
two-rigid-body rule:
Chapter II
Geometric Construction Analysis of Plane Systems
§2-1
Introduction
Structure consists of members, joints and supports. Structure must maintain its geometric shape and positions without consideration of the deformation of materials.
link system connected by hinges – system of bars connected by hinges at the ends of the bars.
The computed degrees of freedom ： j--the numbers of hinges; b--the numbers of links including the links at the supports
刚片-rigid body
杆件，几何不变部分 均可视为刚片 members or stable parts may be looked at as rigid bodies
形状可任意替换 may be replaced by body of any shape.
§2-1 degrees of freedom of planar system（stable system
Under the action of any loads, the system will change its shape and its location if the deformations of the members are neglected.
3 bars, when the summation of the lengths of any 2 bars is greater than the length of 3-d one, can form uniquely a triangular.
triangular joined pairwise by hinges is stable.
连接n个杆的复刚结点 等于多少个单刚结点？
The computed degrees of freedom(计算自由 度)=the total numbers of degrees of freedom of rigid bodies – total numbers of restraints
W = 3m-(2h+b) m---刚片数the numbers of rigid bodies （excluding foundation不包括地基） h---单铰数the numbers of simple joints b---单链杆数（含支杆）the numbers of links
1
体系W 等于多少？ 可变吗？
3 1
W=0,体系 是否一定 几何不变呢？
W=3 ×9-(2×12+3)=0
Restraints which reduce the degrees of freedom is named as necessary restraints,必 要联系otherwise they are called redundant restraints.多余联系
If the deformation of materials is neglected, then framed systems can be classified into two categories： 几何不变体系(geometrically stable system ) 几何可变体系(geometrically unstable system )
虚铰：联结两个刚片的两根相交链杆的作用，相当于在其交 点处的一个单铰，这种铰称为虚铰（瞬铰）If 2 noncollinear links connecting 2 rigid bodies intersect at a point outside the 2 rigid bodies, then the intersection is referred to as a virtual or instantaneous hinge。
freedom of the following system
1
2
3
3
1
按刚片计算9根杆,9个刚片
有几个单铰？ 3根单链杆
2
W=3 ×9-(2×12+3)=0
Another solution
按铰结计算
6个铰结点 12根单链杆 W=2 ×6-12=0
Discussion 2 2
有 几 个 3 单 铰？
3 rigid bodies joined pair-wise by hinges, provided that the 3 hinges don’t lie in the same straight line, form an internally stable system with no redundant restraints.