Slender Structures Load carrying principles Matrix Method part

Slender Structures Load carrying principles Matrix Method (part 2) Elements and coordinate systems v 2020 -2 Hans Welleman 1

Content (preliminary schedule) q Basic cases q q – 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 2

Learning objectives n Understand the method(s) of finding the element stiffness matrix in both local and global coordinates n Be able to combine and reduce element stiffness matrices n Be able to use elements in different orientations Hans Welleman 3

Cases with exact solutions Second order ODE n Extension n Shear Fourth order ODE bending n Euler Bernoulli n Timoshenko n Beam on elastic foundation Hans Welleman 4

Model n n n n 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 Hans Welleman Matrix. Method • Input model (nodes/elements/loads) • Assemble the system • Impose boundary conditions • Solve unknowns • Present all results 5

Topics overview ELEMENTS n n n n n Element stiffness definition and methods Truss element stiffness (direct method) Truss element (ODE) Truss element (shape functions) Euler Bernoulli beam element (ODE) Euler Bernoulli beam element (shape functions) Combining elements Timoshenko beam element (ODE) Beam on an elastic foundation (ODE) COORDINATE SYSTEMS n n n Vector transformation Matrix transformation Example Hans Welleman 6

Element stiffness definition to ode 1 n m o r node 2 x. L f Relate forces at the element ends (in the local coordinate system) to the displacements at the element ends (in the local coordinate system) n Move from local to global coordinate system (we’ll do that later) n element stiffness matrix K(e) Hans Welleman 7

Methods n Direct Engineering approach n Solve ODE with correct BC n Based on shape functions (used in FEM) n Other methods …. work, energy (not in these lecture series) Hans Welleman (see also previous truss example) 8

Truss element (extension) (direct method) element stiffness matrix K(e) in local coordinate system element stiffness in local coordinate system Hans Welleman 9

Smooth the operation … kinematic matrix constitutive matrix Hans Welleman transpose kinematic matrix 10

Truss element (extension) (based on ODE) from distributed load (see part 3) this one is easy by hand. . Hans Welleman element definition in local coordinate system 11

MAPLE (based on ODE) Hans Welleman 12

Truss element (extension) (shape functions) nd st m 1 x L fro to 2 node Shape function, value 1 at one single dof, zero for all other dof’s Hans Welleman 13

MAPLE (based on shape functions) Hans Welleman 14

Euler Bernoulli beam (based on ODE) from distributed load (see part 3) differs from the notes since we use x-z instead of x-y Hans Welleman 15

MAPLE (based on ODE) Hans Welleman 16

Euler Bernoulli beam (based on shape functions) from distributed load (see part 3) Shape function, value 1 at one single dof, zero for all other dof’s differs from the notes since we use x-z instead of x-y Hans Welleman 17

MAPLE (based on shape functions) Hans Welleman 18

Extension + Bending constant distributed loads qx and qz differs from the notes since we use x-z instead of x-y So : combine element solutions … Hans Welleman 19

Timoshenko (based on ODE) compare to slide 15 and 17, holds for constant q differs from the notes since we use x-z instead of x-y Hans Welleman 20

Beam on elastic foundation (based on ODE) Solve only with MAPLE then “smoothen” the solution by hand …. … takes a day … All-in local x-z coordinate system Hans Welleman 21

Result … Hans Welleman 22

Summary Element definition n In local coordinate system of element n Different methods to find the stiffness matrix: - ODE - shape functions - direct methods n Exact stiffness formulation for extension, shear, bending (Euler & Timoshenko). n Only ODE method for beam on elastic foundation results in exact stiffness matrix. n Combining elements Hans Welleman 23

Rotation from local to global NOTE : angle α from global to local Hans Welleman 24

Coordinate system used for vector rotation in 2 D Hans Welleman 25

Rotate element displacement vector NOTE : angle α from global to local in global coordinate system in local element coordinate system Hans Welleman 26

Element rotation Rules matrix multiplication: - Commutive NO - Distributive YES - Associative YES Element stiffness matrix K(e) rotates according to 2 nd order tensor: element stiffness matrix in global coordinate system, notation: “e” Hans Welleman element stiffness matrix in local element coordinate system, notation: “(e)” 27

Example for 2 D-Truss element de 1 m no de 2 to no x L fro Hans Welleman 28

e K for 2 D element with 6 dof’s NOTE : angle α from global to local element stiffness matrix in global coordinate system, notation: “e” Hans Welleman element stiffness matrix in local element coordinate system, notation: “(e)” 29

Result from part 2. . Local 2 D-element definition looks in general like: equivalent load vector from load on the element, see slides part 3 all elements Hans Welleman Assemble to system 30