Determinacy, Indeterminacy and Stability

Determinacy, Indeterminacy and Stability Determinacya statically determinate structure is one which can be completely analysed using the 3 equilibrium equations (2D)

a structure with more unknown forces than equilibrium equations is statically indeterminate

In other words: 1. draw an FBD2. find total number of unknown reactions3. compare with equilibrium equations

2D structures have 3 equilibrium equations let n = total number of parts of the structure let r = total number of reactions (forces and moments) Therefore

Degree of Indeterminacy

this refers to the number of unknown reactions (or forces in the case of trusses)

the following examples show how to calculate the degree of indeterminacy

r = 3n, statically determinate r > 3n, statically indeterminate Example Classify each of the beams shown as statically determinate or statically indeterminate. If statically indeterminate, report the number of degrees of indeterminacy. Assume the beams are subject to external loads that can act anywhere on the beams.

Solution

Compound beams are composed of pin-connected members and must be broken up into several sections and FBDs drawn. Draw unknown reactions between the members acting in equal and opposite pairs.

Pin-connected Structures

Frames

members are rigidly connected some members form closed internal loops use method of sections to cut loops apart and draw FBD

Stability

equilibrium of a structure is not only satisfied by equations of equilibrium stability must also be ensured through provision of adequate restraint at supports

Partial Constraints

sometimes a structure has less reactions than required structure is partially constrained

is not satisfied here- member unstable

Improper Constraints

This usually occurs when the support reactions are concurrent at a point or parallel

when force P is applied sum of forces in horizontal direction not equal to zero

Example

Classify each of the following structures as determinate/ indeterminate, stable/ unstable. Also report the number of degrees of indeterminacy. Assume the beams are subject to external loads that can act anywhere on the beams.

Solution

NB: Application of Equilibrium Equations

For trusses and frames whose joints can be considered to be pin-jointed, forces at joints can be determined using equilibrium equations provided they do not contain more members or supports than are necessary to prevent collapse

if, as in (a), the structure remains rigid after supports are removed then the equilibrium equations can be used on the whole structure

But if, as in (b), the structure is non-rigid then it must be completely dismembered and the equilibrium equations must be applied to the individual members

Assignment (August 2012 Supplementary Exam)Classify each of the following structures in as, statically determinate, statically indeterminate, stable or unstable. If unstable, give reason for the condition