Download presentation

Presentation is loading. Please wait.

Published byAinsley Collinson Modified over 3 years ago

2
Finite Element Methodology Planar Line Element

3
Planar Line Element v1 1 2 v2 21 v(x) x

4
Planar Line Element q BMD SFD

5
Planar Line Element (1) where q = intensity of lateral loading and EI = flexural rigidity. Integrating, we have: and or

6
Planar Line Element In matrix form, this becomes:

7
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:

8
Planar Line Element We can therefore construct a matrix relationship between the nodal deflections and the undetermined coefficients.

9
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). *

10
Planar Line Element State of Strain: The state of strain for the line element can be represented by how “curved” it is, ie:

11
Planar Line Element

12
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:

13
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

14
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:

15
Planar Line Element Alernatively: The element stiffness matrix is given by: Now:

16
Planar Line Element So that:

17
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 ).

18
Planar Line Element Hence: For conformability (matrix manipulative purposes) it is desirable to include the third planar force and displacement

19
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:

20
We’ve only just begun ………… ‘The Carpenters’

Similar presentations

OK

CE 329 Structural Analysis Spring 2005. Objectives ― General List Course Objectives Describe Topical Coverage for Class Provide the Formula for Computing.

CE 329 Structural Analysis Spring 2005. Objectives ― General List Course Objectives Describe Topical Coverage for Class Provide the Formula for Computing.

© 2018 SlidePlayer.com Inc.

All rights reserved.

Ads by Google

Ppt on group 14 elements Free ppt on 5g wireless communication technology Ppt on high voltage engineering Ppt on poem ozymandias by percy bysshe shelley Ppt on conceptual art video Ppt on email etiquettes presentation college Maths ppt on surface area and volume Download ppt on business plan Ppt on tamper resistant labels Educational backgrounds for ppt on social media