13 determinacy and kinematic stability notes sp04

CEE 371 Determinacy and Kinematic Stability, p. 1

DETERMINACY [AC2:7] AND KINEMATIC STABILITY [AC2:4-6]

Review of Determinacy [AC2:7 & Lecture Notes on Structural Analysis]

A statically determinate structure is one that can be analyzed using the equations of equilibrium (statics) only.

If a structure is statically indeterminate, we must use not only equations of equilibrium but also equations of compatibility (structural deformations must be "compatible," i.e., the deformed structure must fit together with no gaps or overlaps). If d such additional equations are needed, the structure is said to be "statically indeterminate to the dth degree."

NOTE: determinacy is independent of the loading on the structure.

Basic technique:1. Find F, the number of unknown force components by removing supports and perhaps by

also cutting into simpler segments.2. Find e, the number of equilibrium equations available for the overall structure or for all

the various segments.3. Calculate d = F – e

if d = 0, statically determinateif d > 0, statically indeterminate to the dth degreeif d < 0, structure is geometrically unstable

Other techniques are merely specialized variants of d = F – e:

For trusses, count the joints j, the members m, and reaction components r. For each member, there is one unknown force, while for each 2D joint, there are 2 equations of equilibrium. [For each 3D joint, there are 3 equations of equilibrium.]

for a 2D truss: d = r + m – 2jfor a 3D truss: d = r + m – 3j

For rigid frames, see the AC2:7 and the lecture notes on Structural Analysis for use of a specialization of d = F – e for frames. For example, for a 2D frame without hinges:

d = 3m + r – 3j

Another technique available for beams, trusses, or frames [not covered in AC]: divide the structure into "trees," free-body diagrams of a skeletal structure without any closed loops. For each 2D tree there are 3 equations of equilibrium; therefore, if the number of trees is t, e = 3t. [For each 3D tree, there are 6 equations of equilibrium; and e = 6t.]

for 2D beams, trusses, or frames: d = F – 3tfor 3D beams, trusses, or frames: d = F – 6t

Notes on the method of trees: In making trees, we always cut at any internal hinge and we remember that M = 0 at

these cuts. The cutting of a structure into trees is not unique, but one obtains the same result by any

admissible division (no loops).


Kinematic or Geometric Stability [AC2:4-6]

Kinematic (geometric) stability … small deflection behavior.Kinematic (geometric) instability … structure or part of a structure can change shape or

experience appreciable deflections with little or no applied loadings and without stressing of the members ("rigid body motion").

As its name implies, geometric instability is primarily a function of geometry. It is independent of:

• loading• stiffness of members• strength of members

In these regards, it differs from structural instability (buckling) which, for example in the case of elastic columns, is characterized by critical loads of the form:

PEuler = 2 EI/(kL)2

which contains load (P) and stiffness (EI).

We need to ensure that our computational models are kinematically stable if we are to obtain meaningful results from structural analysis by the stiffness (displacement) method.

If from d = F – e, if d < 0, the structural model is always unstable. This type of instability is easy to detect.

Example:

d = r + m – 2j r = 3; m = 8; j =6d = 3 + 8 – 2(6) = –1

However, if d 0, the structural model may still be kinematically unstable. These types of instability are more difficult to recognize or detect, but several methods are available:

Inspection If d = 0, can use evaluation of determinant of equilibrium equations – zero value will

indicate instability Build a physical model Proceed with computer analysis; if it fails, structure is unstable (singular coefficient

matrix of equilibrium equations) [many programs, such as MASTAN2, issue a warning if kinematic instability is encountered; some also identify the unstable shape to help the analyst correct the model]


Method of Inspection

Essentially, look for structures or parts of structures that can deflect freely or must deflect as a mechanism when subject to a load. A “mechanism” is a structure or assemblage of elements with ability to move without significant stresses (that is, experiences “rigid body motion”).What to look for to detect a mechanism:

Parallel forces -- allow movement perpendicular to direction (see AC Figure 2-23b) Concurrent forces -- allow rotation about a point of concurrency (see AC Figure 2-23a)

If a structural component is constrained in a manner that admits only parallel or concurrent forces, then it is unstable. For 2D structures, for example, we need to have at least 3 nonparallel, nonconcurrent constraining forces.

Another source of possible instability is the presence of internal hinges. To inspect trusses, break them down into basic (stable) triangular elements, and then see how these are supported and connected. To inspect frames, break them down into trees without any internal hinges. Each such tree is stable within itself, so we need only see if it is supported by or connected to other trees in a stable manner. (See attached examples).

13 determinacy and kinematic stability notes sp04

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