The document discusses statically indeterminate and determinate structures in structural mechanics, emphasizing that indeterminate structures have more unknown forces than equations of equilibrium can solve. It explains methods for analyzing these structures, such as energy methods and the matrix stiffness method. Additionally, it highlights how frames and pin-connected structures are classified based on their stability and determinacy.

1. STRUCTURAL MECHANICS
GROUP 5
Determinacy, Indeterminacy and Stability
NYALUGWE VICTOR N02218097P
BINALI MADALITSO N02212278Q
CHIKUDO DADISO N02215402A
NYABADZA NIGEL N02217146J
CHISHANGU LILIAN N02219753V
MANYIKA TALENT N02221442D
MANGEZI PRECIOUS N02215440Y
NYONI PRIMROSE N02216021W

2. Statically Indeterminate Structures
• Statically indeterminate structures are mechanical systems that cannot be uniquely solved using the
equations of static equilibrium alone. In other words, the forces and reactions in such structures
cannot be determined solely by the conditions of equilibrium, such as the sum of forces and the sum of
moments being equal to zero. This is because there are more unknown forces or reactions than there
are equations available to solve for them.
• Statically indeterminate structures occur when there are redundant members or supports in the
system. This means that the system contains more supports or members than necessary for stability,
leading to additional unknown forces or reactions. Examples of statically indeterminate structures
include continuous beams, arches, and trusses with redundant members.
• To determine the forces and reactions in statically indeterminate structures, additional methods or
concepts are required. These methods can include using compatibility equations, energy methods
(such as the method of virtual work or the method of least work), moment distribution method, slope-
deflection method, or the matrix stiffness method, among others.
• By employing these methods, engineers and analysts can determine the unknown forces and reactions
in statically indeterminate structures, allowing for a complete analysis and design of such structures.

3. Pin Connected Structures

4. Determinacy: Pin Connected Structures
• A statically determinate structure is one that is stable and all
unknown reactive forces can be determined from the equations of
equilibrium alone.
• For statically determinate structure:
• r=3n
• where;
• n is the total parts of structure members
• r is the total number of unknown reactive force and moment
components

5. Examples of statically determinate structures

7. r=9
n=3
9=3(3) Statically determinate

8. Determinacy and Indeterminacy: Frames
• These are structural elements composed of beams or columns connected
together to form a rigid structure. They are used to support loads and transfer
them to the foundations or supporting structures. Frames can either statically
determinant, whereby external reactions and internal forces can be
determined using equations of equilibrium alone. All the unknown forces and
reactions in the structures can be directly calculated using the principles of
statics.
• In frames,
1. Members are rigidly connected
2. Some members form closed internal loops
3. Use method of sections’ to cut loops apart and draw FBD

9. Examples

11. 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

12. Partial Constrains
• sometimes a structure has less reactions than required
• structure is partially constrained
• is not satisfied here- member unstable

13. Improper Constraints
• This usually occurs when the support reactions are concurrent at a
point or parallel

14. when force P is applied sum of forces in horizontal direction not equal to zero
In general therefore a structure will be geometrically unstable i.e will move
slightly or collapse if there are fewer reactive forces than equations of
equilibrium or if there are enough reactions instability will occur if the lines of
action of the reactive forces intersect at a common point or are parallel to one
another.

15. 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.

18. 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

19. • 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