Statical Determinate or Statical Indeterminate?

1. What is statical determinate and statical indeterminate?

In simple way, 

Statical Determinate = Number of Equation ≥ Number of Unknown Forces

Therefore,

Statical Indeterminate = Number of Equation < Number of Unknown Forces

In Statical Determinate Structures, the force in the system can be obtained by Static Equilibrium Equations.

Static Equilibrium Equations :

1. Σ H = 0   (The sum of all horizontal forces in structure system must be zero)

2. Σ V = 0   (The sum of all vertical forces in structure system must be zero)

3. Σ M = 0  (The sum of moment at any point in structure system must be zero)

In Statical Indeterminate Structures, the force in the system can be obtained by Static Equilibrium Equations and Deformation Compatibility Equations.

2. Type of structure system

There are two types of structure system :

1. Pin-Jointed Frame

(A)

(B)

(C)

Pin-Jointed Frame is a frame which its members are jointed with pin connection.
The member only resist the axial forces.
The pin connection only resist the member's axial forces (that is separated into 2 forces direction: X-direction and Y-direction) as shown in Figure (C).

2. Rigid Frame

(A)

(B)

Rigid Frame is a frame which its members are jointed with rigid connection.
The applied load is transferred to the support by bending moment, shear and axial reaction within the member (3 types of reaction in each member).

3. Pin-Jointed Frame

• Number of unknown forces = Member force + Support reaction (M + R)
• Number of equations = 2/joint, that is :
• Σ H = 0   (The sum of all horizontal forces in all joint must be zero)
• Σ V = 0   (The sum of all vertical forces in all joint must be zero)

• Degree of Indeterminacy (Id) : 

Id = (M + R) - 2N

          Id ≤ 0  (Statical Determinate)

          Id > 0 (Statical Indeterminate)

Example :

1.

Number of Unknown Force :

• Number of member force (M) = 10
• Number of reaction force (R) = 4

Number of Equation :

• Number of Joint (N) = 7

Degree of Indeterminacy (Id) = (10 + 4) - 2(7) = 0 (Statical Determinate)

2.

Number of Unknown Force :

• Number of member force (M) = 12
• Number of reaction force (R) = 5

Number of Equation :

• Number of Joint (N) = 7

Degree of Indeterminacy (Id) = (12 + 5) - 2(7) = 3 (Statical Indeterminate)

3.

Number of Unknown Force :

• Number of member force (M) = 9
• Number of reaction force (R) = 4

Number of Equation :

• Number of Joint (N) = 6

Degree of Indeterminacy (Id) = (9 + 4) - 2(6) = 1 (Statical Indeterminate)

4.

Number of Unknown Force :

• Number of member force (M) = 13
• Number of reaction force (R) = 4

Number of Equation :

• Number of Joint (N) = 8

Degree of Indeterminacy (Id) = (13 + 4) - 2(8) = 1 (Statical Indeterminate)

4. Rigid Frame

• Number of Unknown Force = 
• Member's response = Bending moment, shear, and axial force (3 force/member) = 3(M)
• Support's reaction = R
• Number of Equation :
• Σ H = 0   (The sum of all horizontal forces in all joint must be zero)
• Σ V = 0   (The sum of all vertical forces in all joint must be zero)
• Σ M = 0  (The sum of moment in all joint must be zero)

• Degree of Indeterminacy (Id) : 
Id = (3M + R) - 3N
Id ≤ 0  (Statical Determinate)
Id > 0 (Statical Indeterminate)

Example :
1.

Number of Unknown Force :

• Number of member force (M) = 8
• Number of reaction force (R) = 7

Number of Equation :

• Number of Joint (N) = 9

Degree of Indeterminacy (Id) = (3(8) + 7) - 3(9) - 1 (due to moment release in pin joint) = 3 (Statical Indeterminate)

2.

Number of Unknown Force :

• Number of member force (M) = 7
• Number of reaction force (R) = 6

Number of Equation :

• Number of Joint (N) = 7

Degree of Indeterminacy (Id) = (3(7) + 6) - 3(7) = 6 (Statical Indeterminate)

3.

Number of Unknown Force :

• Number of member force (M) = 8
• Number of reaction force (R) = 8

Number of Equation :

• Number of Joint (N) = 9

Degree of Indeterminacy (Id) = (3(8) + 8) - 3(9) = 5 (Statical Indeterminate)