<p>Alternative approach: Integrate beam ODE and solve for integration constants</p>

e4cdc970-3a6a-4b6e-9084-6b3eb7d45d34 28 4 5 decimal

&[DATE] &[TIME] - &[FILENAME]

Stiffness Matrix and Element Load Vector for a Straight Shear Stiff Beam Element

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 #

u w.I φ.I w.J φ.J 41 mat :

q.I

q.J

element DOFs

0 Q

0 M

l M

x

l, EI

I

J

w.J

l Q

z

w.I

element nodal loads:

f 0 Q - 0 M - l Q l M 41 mat :

φ.J

x l ≡

φ.I

Shape function

polynomial ansatz for the shape function (displacement interpolation)

x w c.5 x 5^* c.4 x 4^* + c.3 x 3^* + c.2 x 2^* + c.1 x * + c.0 + :

displacement

x φ x w x diff - :

slope (no shear)

x κ x φ x diff :

curvature (bending deformation)

x M EI x κ * :

bending moment

x Q x M x diff :

shear force

x q x Q x diff - :

distributed load

The ansatz is selected such that it can represent linear variations of distributed load

x q radcan Maxima 120 c.5 * x * 24 c.4 * + (EI *

Express the ansatz parameters via nodal displacements and nodal values of distributed load

L 0 w w.I ≡ 0 φ φ.I ≡ l w w.J ≡ l φ φ.J ≡ 0 q q.I ≡ l q q.J ≡ 61 sys c.0 c.1 c.2 c.3 c.4 c.5 61 sys Solve : c.0 w.I ≡ (c.1 φ.I - (≡ (c.2 120 EI * 3 w.J - w.I + (* - l φ.J 2 φ.I * + (* + (* l 4^2 q.J * 3 q.I * + (* + 120 l 2^* EI * / ≡ (c.3 120 EI * 2 w.J - w.I + (* - l φ.J φ.I + (* + (* l 4^3 q.J * 7 q.I * + (* + 120 l 3^* EI * / - (≡ (c.4 q.I 24 EI * / ≡ (c.5 q.I q.J - 120 l * EI * / - (≡ (61 sys

L L Assign :

x w x w :

Update the definition

<p>Alternative approach: Integrate beam ODE and solve for integration constants</p>

int() MaximaTakeover int() now use Maxima

x q q.I 1 x l / - (* q.J x l / * + :

linear element load function

Integrate the beam ODE

x Q Q.0 x q x 0 x int - :

x M M.0 x Q x 0 x int + :

x φ φ.0 1 EI / x M x 0 x int * + :

x w w.0 x φ x 0 x int - :

This shape function has the integration constants as parameters and can exactly represent the solution for linearly varying distributed loads.

x w w.0 x 5^q.J * 5 l * x 4^* x 5^- (q.I * + 20 l * x 3^* Q.0 * - 60 l * x 2^* M.0 * - 120 φ.0 * l * x * EI * - (120 l * EI * (/ - (-

Replace the integration constants by the element nodal degrees of freedom

0 w w.I ≡ 0 φ φ.I ≡ l w w.J ≡ l φ φ.J ≡ 41 sys w.0 φ.0 M.0 Q.0 41 mat Solve Assign w.I φ.I 260 EI * 3 w.J - w.I + (* - φ.J l * + 2 φ.I * l * + (* l 4^q.J * + (* 3 l 4^* q.I * + 60 l 2^* / - 340 EI * 2 w.J - w.I + (* - φ.J l * + φ.I l * + (* l 4^q.J * + (* 7 l 4^* q.I * + 20 l 3^* / 41 sys

x κ x w - (x 2 diff :

Theory: expressions for stiffness matrix and element load vector from the principle of virtual work

Virtual work per volume due to bending

w.i δκ M * ≡

Virtual work of the distributed load

w.e δw q * ≡

Principle of the virtual work

δW.i δκ M * x 0 l int ≡

δW.e δw q * x 0 l int ≡

δW.i δW.e ≡

Interpolate the actual and the virtual displacements from nodal values

N w u diff transpose ≡

x w x N u * ≡

Interpolation matrix

Compute the bending strain (curvature)

x κ w'' - ≡ x N'' - (u * ≡ B u * ≡

As long as B is independent of u (geometric linearity) B can be obtained as derivative of κ wrt u

B κ u diff transpose ≡

M EI κ * ≡ EI B * u * ≡

δκ B δu * ≡ δu transpose B transpose * ≡

substitute everything into the principle

δu transpose EI * B transpose B * x 0 l int * u * δu transpose N q * x 0 l int * ≡

must hold for any δu

δu transpose EI B transpose B * x 0 l int * u * N q * x 0 l int - (* 0 ≡

K EI B transpose B * x 0 l int * ≡

f.e N q * x 0 l int ≡

K u * f.e ≡

Actual computation with the given displacement function

int() diff() MaximaTakeover int() and diff() now use Maxima

Compute the nodal displacement-strain-matrix transpose(B) = dκ/du

i 14 range B 1 i el x κ u i el diff : for

B x M EI / 11 sys u mat2sys jacobian Maxima transpose :

B 6 l - 2 x * + (* l 3^/ - 22 l * 3 x * - (* l 2^/ - 6 l 2 x * - (* l 3^/ - 2 l 3 x * - (* l 2^/ - 14 mat

Sometimes mat2sys is required because Maxima's jacobian wants both arguments as lists.

Compute the element stiffness matrix

BtB B transpose B * (:

Here we have to compute the matrix product in advance, because this is not done in the preprocessing of the Maxima call and Maxima uses a different operator for non-commutative multiplication

EI l K.e BtB EI * x 0 l int :

Element load vector

Derive the virtual work of the distributed element loads wrt. the nodal DOFs

i 14 range f.e i el x q x w * x 0 l int (u i el diff : for

q.I q.J l f.e x q x w * x 0 l int 11 sys u mat2sys jacobian Maxima transpose :

q.I q.J l f.e f.e :

EI l K.e 12 EI * l 3^/ 6 EI * l 2^/ - 12 EI * l 3^/ - 6 EI * l 2^/ - 6 EI * l 2^/ - 4 EI * l / 6 EI * l 2^/ 2 EI * l / 12 EI * l 3^/ - 6 EI * l 2^/ 12 EI * l 3^/ 6 EI * l 2^/ 6 EI * l 2^/ - 2 EI * l / 6 EI * l 2^/ 4 EI * l / 44 mat

q.I q.J l f.e l 3 q.J * 7 q.I * + (* 20 / l 2^2 q.J * 3 q.I * + (* 60 / - l 7 q.J * 3 q.I * + (* 20 / l 2^3 q.J * 2 q.I * + (* 60 / 41 mat

Extremal values

Check these locations for extreme values of M, if they are within 0...l

L x Q x Solve (: x 340 EI * 2 q.J q.I - (w.J * q.I q.J - (w.I * + (* φ.J φ.I + (l * q.I * - (* l 4^q.J 2^q.I 2^+ (* + (* 4 l * l 3^q.I * 30 φ.J φ.I + (* EI * + (* q.J * + sqrt l 2^q.I * 10 sqrt * - l 10 sqrt * q.J q.I - (* / ≡ (x 340 EI * 2 q.J q.I - (w.J * q.I q.J - (w.I * + (* φ.J φ.I + (l * q.I * - (* l 4^q.J 2^q.I 2^+ (* + (* 4 l * l 3^q.I * 30 φ.J φ.I + (* EI * + (* q.J * + sqrt l 2^q.I * 10 sqrt * + l 10 sqrt * q.J q.I - (* / - (≡ (21 sys

Check this location for extreme value of Q, if it is within 0...l

L x q x Solve (: x l q.I * q.J q.I - / - (≡

Also check the values at the nodes (x=0 and x=l)

Check the displacement (w) values for Re(polyroots()) of the following coefficient vector

c.1 2 c.2 * 3 c.3 * 4 c.4 * 5 c.5 * 51 mat φ.I - 120 EI * 3 w.J - w.I + (* - l φ.J 2 φ.I * + (* + (* l 4^2 q.J * 3 q.I * + (* + 60 l 2^* EI * / 120 EI * 2 w.J - w.I + (* - l φ.J φ.I + (* + (* l 4^3 q.J * 7 q.I * + (* + 40 l 3^* EI * / - q.I 6 EI * / q.I q.J - 24 l * EI * / - 51 mat