Part of any structural analysis is to analyze its stability first. We have to ensure that the beam we're examining is indeed stable. It is only to be in that state if it is externally and internally stable.
External Stability
Generally, external stability happens when these two criteria are satisfied, provided that the structure is a rigid body:
- There is a proper arrangement of the truss's supports, and
- The number of supports must be greater or equal to the number of equilibrium equations.
In the case of beams, we have to consider the possibility of structures with internal devices such as hinges and links. In this case, the second criterion becomes:
- \(r{\geq}3+{e_c}\) for 2D-beams
- \(r{\geq}6+{e_c}\) for 3D-beams
Variable \(r\) and \({e_c}\) refer to the number of reaction components and conditional equations, respectively. These conditions must be satisfied if a beam is to be externally stable.
Internal Stability
Internal stability deals with the proper arrangement of a structure's components.
For beams, internal instability is most likely to occur when we introduce internal devices such as hinges or links.
To demonstrate, consider the structure below.
Let's first examine its external stability:
- There is a proper arrangement of supports.
- The number of unknown components exceeds the number of known conditions with internal devices (3=3).
Based on this, the beam is externally stable.
Let's say we apply a concentrated load on the internal pin. As a result, it will deform, as shown below.
If we release this load, the beam should return to its original position (provided the beam is elastic); however, because of the internal hinge, it may not return. When this happens, we say that the beam has collapsed and is unstable.
Despite this being an externally stable beam, it is still structurally unstable. A civil structure that behaves the same way as our example above cannot serve its function.
Summary
A beam is said to be stable if it is both externally and internally stable.
External stability happens when these two criteria are satisfied: (1) supports are correctly arranged and (2) \(r{\geq}3+{e_c}\) or \(r{\geq}6+{e_c}\) is satisfied.
Internal stability happens when there is a proper arrangement of the components making up the beam.
