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

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

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

Cases with exact solutions Second order ODE Extension Shear Fourth order ODE bending Euler Bernoulli Timoshenko Beam on elastic foundation Hans Welleman

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

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

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 Hans Welleman

Example : Solve ODE (based on ODE) equivalent nodal load differs from the notes since we use x-z instead of x-y Hans Welleman

Example : Engineering approach Based on well known expressions, on the element, only due to q: Hans Welleman

Example: Work -Shape Functions beam with rotations only Work done by Equivalent Load is equal to distributed (element) load Hans Welleman

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 Hans Welleman

Proof DIY ! Hans Welleman

Examples of element loads (bending) Hans Welleman

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.

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)

Frame example Questions: given: q = 12 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

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

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

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

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] Hans Welleman

Beam element with left-hinge use ODE method: (alternative with shape functions, see website) Hans Welleman

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

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

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 ! Hans Welleman

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. Hans Welleman

Additional assignment on beams with hinges Not for the exam !! Hans Welleman