Planar Line Element (1) where q = intensity of lateral loading and EI = flexural rigidity. Integrating, we have: and or
Planar Line Element In matrix form, this becomes:
Planar Line Element We now turn our attention to the nodal deflections. The nodal rotations can be found as follows. The slope at any point along the length of the element is:
Planar Line Element We can therefore construct a matrix relationship between the nodal deflections and the undetermined coefficients.
Planar Line Element * This equation can also be interpreted as an "interpolation formula", whereby intermediate values of displacements are determined from those at specific points (in this case, the nodes). *
Planar Line Element State of Strain: The state of strain for the line element can be represented by how “curved” it is, ie:
State of Stress In this problem we consider the state of stress to be defined by the bending moment at any section. Therefore we have for the line element:
Planar Line Element v1* v2* V1 1 2 V2 21 Real Actions Virtual displacements Equilibrium Equation in Element Co-ordinates virtual nodal displacements associated virtual internal state of strain
Planar Line Element Now the internal virtual work on a length dx of the element is equal to the product of the virtual strains and the real stresses We now integrate over the length of the element, noting that matrix H alone is dependent on x. Therefore, the total internal work is: Equating internal and external work, we have:
Planar Line Element Alernatively: The element stiffness matrix is given by: Now:
Planar Line Element Hence: This stiffness matrix could have been constructed directly by considering the response of the line element to individual nodal displacements via the slope-deflection equations, (see 421-307 notes ).
Planar Line Element Hence: For conformability (matrix manipulative purposes) it is desirable to include the third planar force and displacement
Equilibrium Equation in System Co-ordinates Planar Line Element where (Note: We would continue this formulation with an enhancement for axial effects – these have been neglected in the present approach) Assemble Solve: