1. Slender Structures Load carrying principles
Matrix Method
Loads on elements and some last details…
v2018-1
Hans Welleman

2. Content (preliminary schedule)
Basic cases
Extension, shear, torsion, cable
Bending (Euler-Bernoulli)
Combined systems
Parallel systems
Special system – Bending (Timoshenko)
Continuously Elastic Supported (basic) Cases
Cable revisit and Arches
Matrix Method
Hans Welleman

3. Main learning objectives
Understand the method(s) of finding the equivalent loads on nodes to model loads on elements
Be able to setup the total system
Be able to handle predescribed displacements
Be able to apply the complete method on simple cases
Hans Welleman

4. Cases with exact solutions
Second order ODE
Extension
Shear
Fourth order ODE bending
Euler Bernoulli
Timoshenko
Beam on elastic foundation
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5. Model Nodes and elements Support conditions
Degree of freedoms at the nodes (discrete)
Different elements
Loads on elements
Load on a node
Prescribed degrees of freedom
MatrixMethod
Input model (nodes/elements/loads)
Assemble the system
Impose boundary conditions
Solve unknowns
Present all results
Hans Welleman

6. Topics overview Equivalent nodal loads from loads on elements
Assemble the total system
Impose boundary conditions (prescribed displacements)
Adding spring supports (demonstrated in class)
Hinged connections between (beam) elements? ( not for exam !! )
see website for Maple scripts
Hans Welleman

7. Loads on elements equivalent concentrated forces (examples only for bending)
Possible methods:
Solve ODE (see previous slides)
Direct engineering approach (based on deformation)
Formal approach based on Work with shape functions
Rotate these loads to global coordinate system
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8. Example : Solve ODE (based on ODE)
equivalent
nodal load
differs from the notes since we use x-z instead of x-y
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9. Example : Engineering approach
Based on well known expressions, on the element, only due to q:
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10. Example: Work -Shape Functions
beam with rotations only
Work done by Equivalent Load is equal to distributed (element) load
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11. Impose shape function Shape function for rotation at left node :
All other displacements at het end of the element are zero so only Ty-1 produces work
proof?
Shape function for rotation at left node :
Repeat this for rotation at the right node
complete element definition
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12. Proof
DIY !
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13. Examples of element loads (bending)
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14. Formal definition : after all rotations
NOTE : These are forces at the element in the global directions.
Next step:
assemble system based upon the equilibrium per dof at the nodes.

15. System (assemble) Nodal equilibrium (per dof)
System load vector fnodal with nodal loads and reactions
element contributions e in global coordinate system
Sum over all elements
System stiffness matrix (global)
System displacement vector (global)
System load vector (global)

16. Frame example Questions:
given:
q = kN/m
EI = 3000 kNm2
Questions:
Define a minimum element to solve the moment distribution
Find the displacements for the dof’s
Find the moment distribution
Hans Welleman

17. Some last details ..
Impose boundary conditions (prescribed displacements)
Adding spring supports (demonstrated in class)
Hinged connections between (beam) elements? ( not for exam !! )
see website for Maple scripts
Hans Welleman

18. Non-zero prescribed displacements How does it work
Non-zero prescribed displacements How does it work? Use TRUSS example from introduction lecture on Matrix Method.
40 kN C
a = 1,0 m
all elements :
EA = 6000 kN
(5)
(1)
(3) 3a B x A
(2) D
(4) 4a 4a
row-striking with move to right hand side (rhs) z
Final system to be solved
Hans Welleman
Add row-striking with zero prescribed displacements
trick for pivot and rhs to keep system complete

19. Formal procedure as presented in the notes
unknown
Free dof’s
Prescribed dof’
In case of zero prescribed displacement of dof’s, this procedure is simplified to the previously introduced row striking technique.
Hans Welleman

20. Spring support
Add a spring to dof i of the structure and with fixed end at j :
System is extended with dof j but this dof has a zero displacement (fixed end).
So ?
Add k to Ksys[i,i]
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21. Beam element with left-hinge
use ODE method:
(alternative with shape functions, see website)
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22. Assignment 1
F and T applied at node B, q applied at element (1).
Questions
- Find the displacements for all degrees of freedom
- Find the force distribution from the element results.
Check your results with a frame analysis program (e.g. MatrixFrame, Ansys, Midas (Diana)).
Hans Welleman

23. Assignment 2 Questions - Define the dof’s to be used
- Define the element(s) to be used
- Describe the flow of actions to solve this problem
- Solve the force distribution
Check your results with a frame analysis program (e.g. MatrixFrame, Ansys, Midas (Diana)).
Hans Welleman

24. Assignment 3
Question
- Find the element definition in terms of the degrees of freedom for case A and B.
- Find the expression for the displacement field in local coordinates and degrees of freedom.
- Compare both the method based upon the ODE and the shape functions for the presented
loads. Use MAPLE !
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25. Assignment 4 Beam on elastic foundation Question
- Write a MAPLE script to model this model with elements.
Use the element description from the slides.
If you half the element length, will your solution become more accurate?
Compare your results with an analytical solution.
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26. Additional assignment on beams with hinges
Not for the exam !!
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