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Description

Theoretical Background
Overview
For analysis of any structure, it is modeled as a set of simple, idealized
elements connected at nodes. Analysis by direct stiffness method can be
divided into following steps.
1. Formulation of element stiffness matrix in local Coordinates ( ୣ ).
2. Formation of element transformation matrix T.
3. Transformation of element stiffness matrix in global Coordinates ( ୣ ).
4. Addition of all element stiffness matrices at pertinent DOF to form a
structural stiff

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Theoretical Background
Overview
For analysis of any structure, it is modeled as a set of simple, idealized elements connected at nodes. Analysis by direct stiffness method can be divided into following steps. 1.Formulation of element stiffness matrix in local Coordinates (
).2.Formation of element transformation matrix T.3.Transformation of element stiffness matrix in global Coordinates (
).4.Addition of all element stiffness matrices at pertinent DOF to form astructural stiffness matrix (K). 5.Formation of Nodal load vector (P) in global coordinates.6.Formation of Element load Vector (
) in local coordinates for framesonly. 7.Transformation of Element load vector in global coordinates for framesonly. 8.Formation of Nodal displacement vector (U).9.Solving
to get unknown displacements at unconstrained joints.10.Making use of displacements for step 6 to get reactions at constrained joints. 11.Transformation of global displacements to local displacements tocalculate the member forces.
Coordinate System
Global: Structure Nodes are always described in global coordinates. It could be expressed by uppercase letters of X, Y and Z.
Global Coordinates
Local: Element internal forces are described in the local coordinates. It is represented by lowercase letters of x,y and z.
Local Coordinates
2D structures will be defined in X-Y plane where as 3D structures will be defined in X-Y-Z plane.
Sign Conventions
Horizontal force is positive if directed to right side, vertical force is positive upward and moment is positive in the counterclockwise direction as shown in the Fig 3.3.
Sign Conventions
Degrees Of Freedom(DOF)
It is defined as an independent displacement of a node along X, Y or Z axis.These displacements are always independent of each other.For example a hinge support can have only one displacement (rotation
.Displacement is being used in a generalized context here as it could be rotation as well as translation.Displacement in a structure depends upon structure type as there could be one, two or none. DOF in both local and global coordinate system remains same for a particular case. But in case of trusses this is not the case as there is only one axial deformation in local coordinates and two or three translations at each node in 2D and 3D trusses respectively. Degrees of freedom associated with each type of element and its numbering can be summarized as shown in the Fig (3.4)
Typical Degrees of Freedom for various types of elements
Element Stiffness Matrix
Each element stiffness properties are calculated based on the nature of element DOF at each node, these properties are grouped together to form an element stiffness matrix.
Structural Stiffness Matrix
Element stiffness matrices are then augmented into a single matrix which governs the behaviour of the entire idealized structure, known as structural stiffness matrix. This is obtained by multiplication of element stiffness matrix to transformation matrix as in (3.1a)
Load Vector
Load vector is calculated such that known forces and Unknown reactions are arranged as

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