Plane Frame Analyzer
Plane Frame Analysis
For all member, young’s modulus E = 210000, moment of Inertia I = 0.0002, cross sectional area = 0.01, spring stiffness = 10
Solution:
Degree of Freedom Matrix
Nodal support condition can be classified into following irrespective of its inclination in the global axis. Support inclination had taken into consideration later by redistributing it into global stiffness matrix.
1 – Free to displace or rotate,
0 – Restrained from displacement and rotation.
Spring support is taken as rigid joint criteria; however the stiffness of spring is distributed in global stiffness matrix. Later the nodal resultant force is calculated by multiplying spring stiffness and the vertical displacement at that node.
DOF Matrix
Member:1 ( 0 ---> 1 ) => [0 0 1 1 1 1]
Member:2 ( 1 ---> 2 ) => [1 1 1 1 1 1]
Member:3 ( 2 ---> 3 ) => [1 1 1 1 0 1]
Support Inclination Transformation Matrix [SI_T]
Member 1( 0 ---> 1 )
Member 2( 1 ---> 2)
Member 3( 2 --->3)
Element Stiffness Matrix
Stiffness matrix, combing the axial, shear and bending effects for the bar element in the global coordinates is given by
A – Cross-sectional area
E – Modulus of Elasticity
I – Moment of Inertia
L – Length of member
C, S – Direction cosines
Stiffness matrix in Local Coordinate system
Stiffness Matrix (Local Coordinate System) [K~]
Member 1( 0 ---> 1 )
Member 2( 1 ---> 2)
Member 3( 2 --->3)
Element Direction Cosine Matrix [DC_T]
Stiffness Matrix (Global Coordinate System) (after applying transformation from element direction cosine matrix [K^] = [DC_T]T [K~] [DC_T]
Member 1( 0 ---> 1 )
Member 2( 1 ---> 2)
Member 3( 2 --->3)
Stiffness Matrix (Global Coordinate System) (after applying transformation from support inclination
transformation matrix [K] = [SI_T]T [K^] [SI_T]
Member 1( 0 ---> 1 )
Member 2( 1 ---> 2)
Member 3( 2 --->3)
Fixed End Reaction Matrix
Fixed reaction is calculated by the numerical method shown in the figure below. The force vector in the member is distributed as force vector along the lateral and longitudinal direction of the member. Longitudinal force is Fx which taken as the axial load applied on the member.
Support settlement is redistributed as additional force in the fixed end reaction matrix. New force matrix = [F] + [K][Settlement].
Above Picture and Formula Credit: TURAN BABCAN
Fixed End Reaction
Member 1
( 0 ---> 1 )
[F~]
Element Direction
Cosine Transformation
[F^] = [DC_T] [F~]
Support Inclination Transformation
[F] = [SI_T] [F^]
[F~]
Element Direction
Cosine Transformation
[F^] = [DC_T] [F~]
Support Inclination Transformation
[F] = [SI_T] [F^]
[F~]
Element Direction
Cosine Transformation
[F^] = [DC_T] [F~]
Support Inclination Transformation
[F] = [SI_T] [F^]
Member 2
( 1 ---> 2 )
Member 3
( 2 ---> 3 )
Global Stiffness Matrix
Global stiffness matrix after curtailment (removing the elements which are restrained from displacement
Solve for displacement [U] ([K][U] = [F])
Result Matrix (after applying Gauss elimination method)
Nodal Reaction
[N] = [K][U] – [F]
Since spring support is allowed to displace and rotate in all directions we have to find the reaction under the spring support by
Reaction under spring = – spring stiffness x displacementY
Result matrix (Force at the element end nodes, Fx, Fy and Mxy)
End resultant in each member is easily calculated by back computation
[U^] = [SI_T]T [U]
[U~] = [DC_T]T [U]
[N~] = [K~][U~] – [F~}
Shear Force Plot
Bending Moment Plot
Deflection Plot
Axial Force and Reaction Force Plot
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