Equilibrium, Determinacy and Instability of Structures

Before starting discussion, it is necessary to explain the term of rigid body. The term rigid body as used here implies that the structure offers significant resistance to its change of shape, whereas a nonrigid structure offers negligible resistance to its change shape when detached from the supports and would often collapse under its own weight when not supported externally. Statics is the study of rigid bodies that are stationary (at rest). When the sum of the applied loads and support reactions is zero and there is no resultant couple at any point in the structure, the structure must be at rest (statical equilibrium) and principles of statics can be applied. In mathematical terms, $x$ and $y$ direction as the mutually perpendicular direction, the condition may be written as

$\sum{{{F}_{x}}}=0$ 

$\sum{{{F}_{y}}}=0$

Fig. 1 Couple $Fa$ form even $\sum{{{F}_{x}}=0}$

However, the above equations are not sufficient to guarantee the equilibrium of a body. For example, in the Figure 1 at the left, the forces $F$ acting on rectangular plate on horizontal surface satisfy the condition $\sum{{{F}_{x}}}=0$ , but form a couple $Fa$ which will cause the plate to rotate in a clockwise. Further condition to be added for the statical equilibrium of a body acted upon by a system of coplanar forces, namely, that the sum of the moments of all the forces acting on the body about any point in their plane must be zero. Therefore, designating a moment in the $xy$ plane about the $z$ axis as ${{M}_{z}}$ (using right handed coordinate system in Figure 2), we have $\sum{{{M}_{z}}}=0$. Combining all the equations, we obtain the necessary conditions for a system of coplanar forces to be in equilibrium.

$\sum{{{F}_{x}}}=0$ 

$\sum{{{F}_{y}}}=0$ 

$\sum{{{M}_{z}}}=0$ 

Next, we will call this 3 equations as equilibrium equations.

Fig. 1 Right-Handed Coordinate System - A Right handed coordinate system consist of an ordered set of three mutually perpendicular axes ($x,y,z$) which have a common origin and whose positive directions point in the same directions as the ordered set of the thumb, forefinger, and middle finger of the right hand when positioned. However, once this decision is made then the positive directions of the $y$ and $z$ axes must be as indicated by the corresponding configuration.

Forces and couples are actions that tends to maintain or change the position of a structure and can be classified into two types, external forces and internal forces.

External forces are action of other bodies on the structure under consideration. The applied force can be due to physical contact (i.e. pushing) or close proximity (i.e. gravitational, magnetic). If unbalanced, an external force will cause motion of the body. It is usually convenient to further classify forces as applied forces and reaction forces. Applied forces usually referred to as loads (i.e. live loads and wind loads), have a tendency to move the structure. Reaction forces, or reactions, are the forces exerted by supports on the structure and have a tendency to prevent its motion and keep it in equilibrium. The reactions are usually among the unknowns to be determined by the analysis.

Internal forces are the tensile and compressive forces within parts of the body as found from the product of stress and area. Although internal forces can cause deformation of a body, motion is never caused by internal forces. Internal Forces acting inside of structure that hold parts of itself. The internal forces are also among the unknowns in the analysis and are determined by applying the equations of equilibrium to the individual members or portions of the structure.

Determinacy
When all the forces in a structure can be determined from equilibrium equations, the structure is referred to as statically determinate. Since we have 3 equilibrium equations, to be a statically determinate externally, structure must be supported by exactly three reactions. Structures having more unknown forces than available equilibrium equations are called statically indeterminate. The reactions excess of those necessary for equilibrium are called external redundants, and the number of external redundants is referred to as the degree of external indeterminacy. Thus, if a structure has $r$ reactions ($r$ > 3), then the degree of external indeterminacy (${{i}_{_{e}}}$) can be written as

${{i}_{e}}=r-3$

Structures supported by fewer than 3 reaction are not sufficient to maintain its equilibrium, therefore, referred as statically unstable externally. The conditions of static instability, determinacy and indeterminacy of plane internally stable structures can be summarized as follows:

$r$ < 3 the structure is statically unstable externally
$r$ = 3 the structure is statically determinate externally
$r$ > 3 the structure is statically indeterminate externally, where r = number of reactions.

Fig 3. Geometrically unstable structure

However, even sufficient number of reactions ($r\ge 3$) supported the structure, it may still unstable due to improper arrangement of supports. Such structures are referred to as geometrically unstable externally. Example of geometric instability in plane structure shown in Figure 3. It can be seen that although there is a sufficient number of reactions (r = 3), all of them are in vertical direction, so cannot prevent horizontal translation. 

Fig 4. Portal Frame with Internal Hinge

Consider a portal frame in Figure 4 which compose of 3 rigid member, AB, BC and CD and supported by hinge at A and D, also there is internal hinge at B. Obviously, three equilibrium equations are not sufficient to determine the four unknown support reaction for this structure. However, the presence of internal hinge at B provide additional equation that can be used to determine the four unknowns. The additional equation is based on the condition that an internal hinge cannot transmit moment. In other words, with the presence hinge at B, sum of moments about B due to loads and reactions acting on the member AB and BCD must be zero; that is, $\sum{M_{B}^{AB}=0}$ (sum of moment at B on the AB member) and $\sum{M_{B}^{BCD}=0}$ (sum of moment at B on the BCD member).

Fig 5. Beam Structure with Internal Roller

For internal connection which allow not only rotation but also translation is modeled as internal roller. Consider a structure in Figure 5 consisting of two rigid member AB and BC that are connection by internal roller at B. The structure has 5 external support reactions (2 at A and 3 at C) means with only 3 equilibrium equations are not sufficient to determine the five unknown reactions at external support. However, an internal roller can transmit neither moment nor force in the direction parallel to the supporting surface, it provides two equations of condition;

$\sum{F_{x}^{AB}}=0$ or $\sum{F_{x}^{BC}}=0$

and

$\sum{M_{B}^{AB}}=0$ or $\sum{M_{B}^{BC}}=0$

These two additional equations of condition can be used in conjunction with the three equilibrium equations to determine the five unknown external reactions. Thus, the structure of figure above is statically determinate externally.

From the foregoing discussion, we can conclude that if there are ${{e}_{c}}$ additional equations (one equation for each internal hinge and two equations for each internal roller), the degree of external indeterminacy, ${{i}_{e}}$, is expressed as.

${{i}_{e}}=r-(3+{{e}_{c}})$, then if

${{i}_{e}}$ < 0, the structure is statically unstable externally
${{i}_{e}}$ = 0, the structure is statically determinate externally
${{i}_{e}}$ > 0, the structure is statically indeterminate externally

Based on the above formula,we can classify the structure in Figure 6 below as externally unstable, statically determinate, or statically indeterminate.

Fig 6. Example of Statically Determinate Structure

Number of reactions, $r$ =  5

Since structure has 2 internal hinge, there are ${{e}_{c}}$ = 2 additional equations (1 for each internal hinge).
Degree of external indeterminacy
${{i}_{e}}=r-(3+{{e}_{c}})$ 
${{i}_{e}}=5-(3+2)=0$, the beam is statically determinate structure.