Direct Displacement Method in Structural Analysis

Assumptions of Direct Displacement Method

The effect of elongation or reduction of a component is omitted as negligible, except where it is decisive (such as thermal loads, shrinkage, or when the component is a tie-bar or tie-rod).

The difference between the length of a deflected component and the length of its chord is omitted because of the relatively very small real movement of the component nodes.

The effect of lateral and axial forces (V, N) is usually omitted.

The degree of kinematic indeterminacy is the sum of possible rotations of rigid nodes and independent movements of any nodes.

Kinematic Indeterminacy and Formulas

The degree of kinematic indeterminacy is the sum of possible rotations of rigid nodes and independent movements of any nodes.

n_k = ∑φ + ∑Δ

Where:

• ∑φ – Number of rigid nodes intersected by a minimum of two statically indeterminate bars.
• ∑Δ – Number of restraints necessary to stop all possible linear movements (in the kinematic chain of the frame).

Kinematic Chain of Structures

A kinematic chain of a structure is a structure with all nodes replaced by pins. The number of restraints necessary to provide stability is the number of independent linear movements.

The number of independent movements can be calculated by the following formula:

∑Δ = 2n – (m + r) + o – f

Where:

• n – Number of nodes in the kinematic chain.
• m – Number of members in the kinematic chain.
• r – Number of reactions (restraints) in the kinematic chain.
• o – Number of over-rigidity (two or more reactions acting in one axis connected by elements of the kinematic chain).
• f – Number of free ends (nodes at the end of the bars with the possibility of movement perpendicular to the member axis).

The dependency between displacements of the beam axis Δ_i and deformation angles ψ_i is shown by the following relation: ψ_i = Δ_i / l_i

Semi-Rigid Supports in Displacement Method

In semi-rigid supports, movement (translation or rotation) is possible in the direction of the flexible restraints.

Translation is enabled in translationally flexible supports, and rotation is enabled in rotationally flexible supports.

Since the unknowns in the Direct Displacement Method are the displacements, the translation or rotation of semi-rigid supports increases the number of unknowns.

Dependencies between displacements of a semi-rigid support and the reaction are represented by the following formulas:

Δ = R / k → R = Δ * k
φ = M / k → M = φ * k

Because the displacements are unit displacements, the reactions are equal to the support stiffness. These reactions are treated as an additional force (in the direction of translation) or moment (in the direction of rotation).