1.0 INTRODUCTION
Before any structure can be designed, structural analysis must be performed to determine the effect of actions on the structure. However, before structural analysis can be successfully executed to solve for internal stresses and deformations, it is pertinent that the support condition of the structure is such that the structure is statically stable.
2.0 How do we know a structure is stable?
When using one of the numerous structural analysis software, the engineer needs not get himself worked up to determine whether the structure is stable due to its support conditions. The software will readily display error warnings to call the attention of the engineer to the inherent instability. It is left for the engineer to interpret the warning note and investigate the point of concern and then proffer necessary solution to ensure stability.
A structure is said to be stable if it assumes back the position of equilibrium after a disturbance or an excitation. This means enough reactions are summoned from the support(s) of the structure to resist actions on it without causing excessive deformation or outright displacement. To achieve this in the context of linear static analysis, one of the criteria is that the number of reactions acting on the structure must be at least equal to the number of equations of static equilibrium.
For a planar structure or element (i.e.: a 2D member like a beam), the total number of reactions must be at least three for stability. If the total number of reactions is less than three, then the structure is unstable and will collapse even under its own weight. Hence, a structure which has its number of reactions to be less than the number of equations of static equilibrium is said to be unstable.
2.1 Supports and Reaction Component
It has been learnt from the above paragraphs that to verify the stability of a structure due to support conditions the numbers of equation of static equilibrium have to be compared against the number of reactions acting on the structure. It is therefore necessary to be able to identify different types of supports and their corresponding reactions so that the stability of a structure can be correctly examined. Below are the types of supports and respective reactions.
- Roller supports: A roller support provides one degree of restraint such that it resists displacement only in the vertical direction. The structure can displace horizontally or rotate.
fig (1) Roller support and its reaction component - Hinge/pinned Supports: A hinge support has two degrees of restraints such that it resists displacement in both vertical and horizontal direction. It however allows rotation.
fig (2) Hinge Support and its reaction components - Fixed supports: A fixed support provides three degrees of restraint as it does not allow translational displacement in both vertical and horizontal direction and neither does it allow rotation. This type of support is also called rigid support.
fig (3) Fixed Support and its reaction components 2.2 Worked Examples 1: Evaluating the support conditions of member on stability.
Determine whether the support reactions of the members below are enough to achieve stability.
1)
The beam has two roller supports. The reaction at each support is one which makes the total reactions on the beam to be 2.
Number of equilibrium equation = 3
Number of reactions = 2
Since Number of equations > Number of reactions, then the beam is unstable.
2)
The beam in example (2) has one roller support and one hinge support. The total reactions on the beam are three (one reaction from the roller support plus two reactions from the hinge support).
Number of equilibrium equation = 3
Number of reactions = 3
Since Number of equations = Number of reactions, then the structure is stable
3)
The beam in example (3) has one fixed support. The total reactions on the beam are three.
Number of equilibrium equation = 3
Number of reactions = 3
Since Number of equations = Number of reactions, then the beam is stable
2.3 Other Conditions for Stability
Besides the condition in the preceding paragraphs that the number of reactions on a structure must be at least equal to the number of equations of equilibrium, other conditions that has to be met are:
- The reactions must not be all parallel
- The reactions must not be all concurrent such that they all pass through a common point.
3.0 Determinacy and Indeterminacy
3.1 Determinate Structure
A determinate structure is a structure that can be analyzed using only the equations of static equilibrium. For a determinate structure, the support reactions and internal forces – such as shear forces and bending moments – can be computed using the equilibrium equations alone.
A planar structure is determinate if the number of reactions is equal to three which is the same as the number of equilibrium equations (Fx=0, Fy = 0, Mz= 0).
It can also be said that a structure is determinate when its degree of indeterminacy is zero. The degree of indeterminacy is equals to the number of surplus reaction(s) obtained after subtracting the number of equilibrium equations from the number of support reactions acting on the structure. The surplus reactions are also called redundant.
Examples of members that are determinate are simply supported beam, cantilever beam, etc.
3.2 Indeterminate Structure
An indeterminate structure is a structure that cannot be analyzed solely by using the equations of static equilibrium. The structure can only be analyzed by combining equilibrium equations and compatibility equations.
For a planar structure if the number of reactions acting on the structure is more than three then the structure is indeterminate.
It can also be said that a structure is indeterminate if its degree of indeterminacy is greater than zero.
Structures that fall under this are continuous beam, propped cantilever, etc.
3.3 Worked Examples 2: Evaluating the determinacy and degree of indeterminacy of members.
Determine whether the beams shown below are determinate or indeterminate, and if they are indeterminate, determine their degree of indeterminacy.
1)
The beam is fixed at both ends.
A fixed ends has three reactions. Since both ends are fixed, the beam is subjected to six reactions
Determinacy Check:
6 reactions > 3 equilibrium equations
The beam is indeterminate.
Degree of indeterminacy:
The degree of indeterminacy is obtained by subtracting the number of reactions from the number of equilibrium equations.
6 – 3 = 3
The degree of indeterminacy of the beam is 3.
2)
The beam has hinge support on the left end and roller support on the right end.
Hinge supports has two reactions
Roller support has one reaction
Total reactions on the beam = 3
Determinacy Check:
3 reactions = 3 equilibrium equations
The beam is determinate
Degree of indeterminacy:
The degree of indeterminacy is obtained by subtracting the number of reactions from the number of equilibrium equations.
3 – 3 = 0
The degree of indeterminacy of the beam is 0
3.4 Determinacy and Indeterminacy of a truss system
A truss system is an assemblage of members fastened together by pin to form a triangulated structure that is inherently stable.
The basic number of members in a truss is three (see fig(i)) so as to form a triangle. When there is need to extend the truss; two members, and one joint are added one at a time so that the new extension also forms another triangle thereby maintaining the structure’s internal stability (see fig(ii) and fig(iii)).
Since the basic number of members in a truss is three and additional two members per each subsequent joint can be added to extend the truss to as many members as desire, the total number of members in any given truss can be expressed as follows:
M = (No of basic members) + 2 x (No of subsequent joints)
M = 3 + 2 x (n– 3)
M = 3 + 2(n – 3)
M = 2n -3
Where: M = number of members in the truss
n = No of joints in the truss
NB: If a given truss has the number of its actual members to be greater than the result obtained from the formular: M = 2n – 3; then the surplus members are redundant members.
To get the overall determinacy or otherwise of a truss, the ideal number of members and number of equilibrium equations are added together thus:
Number of members + Number of equilibrium equations
M +r
Where: M = 2n-3.
r = 3
M + r = 2n – 3 + 3
M + r = 2n
2n is compared against the actual sum of the number of members in the truss, and numbers of actual reactions based on the type of supports. If the actual sum is greater than 2n, the truss is indeterminate, however if the actual sum equals to 2n, the truss is determinate.
3.4.1 Worked Examples 3: Evaluating the indeterminacy of trusses.
Determine whether the below trusses are determinate or indeterminate.
1)
Using the formular
The number of joints in the truss is 4, using M + r = 2n
M + r = 2 x 4 = 8 … (1)
Inspecting the truss
On inspection the truss has 5 members and 3 reactions
M + r = 5 + 3 = 8 … (2)
Since (1) = (2), then the truss is determinate.
2)
Using the formular
The number of joints in the truss is 8, using M + r = 2n
M + r = 2 x 8 = 16 … (1)
Inspecting the truss
On inspection the truss has 14 members and 3 reactions
M + r = 14 + 3 = 17 … (2)
Since (1) < (2), then the truss is indeterminate to one degree.
