Statically Determinate and Indeterminate Structures by Force / Energy Methods

Static and Kinematic Indeterminacy

Statically Determinate Structures

Conditions of equilibrium are sufficient to analyse the structure. Bending moment and shear force is independent of the cross-sectional area of the components and flexural rigidity of the members. No stresses are caused due to temperature change. No stresses are caused due to lack of fit or differential settlement.

Statically Indeterminate

Structures

Additional compatibility conditions are required. Bending moment and shear force depends upon the cross-sectional area and flexural rigidity of the members. Stresses are caused due to temperature variation. Stresses are caused due to lack of fit or differential settlement.

Static Indeterminacy

If a structure cannot be analyzed for external and internal reactions using static equilibrium conditions alone then such a structure is called indeterminate structure.

(i) D_S = D_Se + D_Si

Where,

D_S = Degree of staticindeterminacy

D_Se = External static indeterminacy

D_Si = Internal static indeterminacy

External static indeterminacy:

It is related with the support system of the structure and it is equal to number of external reaction components in addition to number of static equilibrium equations.

(ii) D_Se = r_e - 3 For 2D

D_Se = r_e – 6 For 3D

Where, r_e = total external reactions

Internal static indeterminacy:

It refers to the geometric stability of the structure. If after knowing the external reactions it is not possible to determine all internal forces/internal reactions using static equilibrium equations alone then the structure is said to be internally indeterminate.

For geometric stability sufficient number of members are required to preserve the shape of rigid body without excessive deformation.

(iii) D_Si = 3C - r_r …… For 2D

D_Si = 6C - r_r …… For 3D

where, C = number of closed loops.

and

r_r = released reaction

(iv) r_r = ∑(m_j - 1) …… For 2D

r_r = 3∑(m_j - 1) ……. For 3D

where m_j = number of member connecting with J number of joints.

and J = number of hybrid joint.

(v) D_s = m + r­_e – 2j ….. For 2D truss

D_Se = r_e - 3 & D_Si = m – (2j – 3)

(vi) D_S = m + r_e – 3j ….. For 3D truss

D_Se = r_e – 6 & D_Si = m – (3j - 6)

(vii) D_S = 3m + r_e – 3j - r_r ….. 2D Rigid frame

(viii) D_s = 6m + r­_e – 6j - r_r ….. 3D rigid frame

(ix) D_S = (r_e – 6) + (6C – r_r) ….. 3D rigid frame

Kinematic Indeterminacy

It the number of unknown displacement components are greater than the number of compatibility equations, for these structures additional equations based on equilibrium must be written in order to obtain sufficient number of equations for the determination of all the unknown displacement components. The number of these additional equations necessary is known as degree of kinematic indeterminacy or degree of freedom of the structure.

A fixed beam is kinematically determinate and a simply supported beam is kinematically indeterminate.

(i) Each joint of plane pin jointed frame has 2 degree of freedom.

(ii) Each joint of space pin jointed frame has 3 degree of freedom.

(iii) Each joint of plane rigid jointed frame has 3 degree of freedom.

(iv) Each joint of space rigid jointed frame has 6 degree of freedom.

Degree of kinematic indeterminacy is given by:

1. D_k = 3j - r_e ………. For 2D Rigid frame when all members are axially extensible.
2. D_k = 3j - r_e - m ………. For 2D Rigid frame if 'm' members are axially rigid / inextensible.
3. D_k = 3(j + j’) - r_e – m + r_r …… For 2D Rigid frame when J' = Number of Hybrid joints is available.
4. D_k = 6(j + j’) - r_e – m + r_r ….. For 3D Rigid frame
5. D_k = 2(j + j’) - r_e – m + r_r ….. For 2D Pin jointed truss.
6. D_k = 3(j + j’) - r_e – m + r_r …… For 3D Pin jointed truss.