Recap differential equations for structures

Recap differential equations for structures#

Let’s recap how to solve a structure using differential equations.

Given is a cantilever beam with a distributed load:

../_images/bending1field.svg

Fig. 1 Cantilever beam with distributed load#

The differential equation for the Euler-Bernoulli model can be derived leading to:

  • Kinematic relations:

    • \(\varphi=-\cfrac{\mrm{d}w}{\mrm{d}x}\)

    • \(\kappa=\cfrac{\mrm{d}\varphi}{\mrm{d}x}\)

  • Constitutive relation:

    • \(M=EI\kappa\)

  • Equilibrium relations:

    • \(\cfrac{\mrm{d}V}{\mrm{d}x}=-q\)

    • \(\cfrac{\mrm{d}M}{\mrm{d}x}=V\)

These relations can be combined into one fourth order differential equation:

\[ EI\frac{\mrm{d}^4w}{\mrm{d}x^4}=q \]

This differential equation can be solved directly to solve structures.

Solving differential equation of one field by hand#

This differential equations can be solved by integrating:

  • \(V(x) = -qx + \bar C_{1}\)

  • \(M(x) = -\cfrac{qx^2}{2} + \bar C_1 x + \bar C_2\)

  • \(\kappa(x) = -\cfrac{qx^2}{2EI}+ \tilde C_1 x + \tilde C_2\)

  • \(\varphi(x) = -\cfrac{qx^3}{6EI} + \cfrac{\tilde C_1x^2}{2} + \tilde C_2 x+ \tilde C_3\)

  • \(w(x) = \cfrac{qx^4}{24EI} + \cfrac{C_1x^3}{6} + \cfrac{C_2x^2}{2}+C_3x+C_4 \)

The boundary conditions follow from the clamped side at \(x=0\) and free end at \(x=\ell\):

  • \(w(0) = 0\)

  • \(\varphi(0) = 0\)

  • \( M(\ell) = 0\)

  • \( V(\ell) = 0\)

Solving these four equations for the integration constants gives:

  • \(C_1 = -\cfrac{q\ell}{EI}\)

  • \(C_2 = \cfrac{q\ell^2}{2EI}\)

  • \(C_3 = 0\)

  • \(C_4 = 0\)

Substituting these constants, a final solution for \(w\) can be found:

\(w(x) = \cfrac{qx^4}{24EI} - \cfrac{q\ell x^3}{6EI} + \cfrac{q\ell^2x^2}{4EI}\)

Solving differential equation of one field using SymPy#

The differential equation can also be solved using SymPy:

import sympy as sym
sym.init_printing()

import sympy as sym

EI, q, x, L = sym.symbols('EI, q, x, ell')
C_1, C_2, C_3, C_4 = sym.symbols('C_1, C_2, C_3, C_4')

V = -sym.integrate(q,x) + C_1
M = sym.integrate(V,x) + C_2
kappa = M/EI
phi = sym.integrate(kappa,x) + C_3
w = -sym.integrate(phi,x) + C_4
display(w)
\[\displaystyle - \frac{C_{1} x^{3}}{6 EI} - \frac{C_{2} x^{2}}{2 EI} - C_{3} x + C_{4} + \frac{q x^{4}}{24 EI}\]
eq1 = sym.Eq(w.subs(x,0),0)
eq2 = sym.Eq(phi.subs(x,0),0)
eq3 = sym.Eq(M.subs(x,L),0)
eq4 = sym.Eq(V.subs(x,L),0)
C_sol = sym.solve([eq1, eq2, eq3, eq4 ], [C_1, C_2, C_3, C_4])
for key in C_sol:
    display(sym.Eq(key, C_sol[key]))
\[\displaystyle C_{1} = \ell q\]
\[\displaystyle C_{2} = - \frac{\ell^{2} q}{2}\]
\[\displaystyle C_{3} = 0\]
\[\displaystyle C_{4} = 0\]
w.subs(C_sol)
\[\displaystyle \frac{\ell^{2} q x^{2}}{4 EI} - \frac{\ell q x^{3}}{6 EI} + \frac{q x^{4}}{24 EI}\]

Solving differential equations of two fields#

A similar approach can be taken when solving two fields.

Let’s investigate the following structure, consisting of two fields.

../_images/extension2fields.svg

Fig. 2 Extension bar with nodal load#

As this structure is loaded along its axis, the differential equation for extension is used.

For the first field this gives:

  • \(\cA{EA_1\cfrac{\mrm{d}^2u_1}{\mrm{d}x^2}=0}\)

  • \(\cA{N_1 = C_1}\)

  • \(\cA{u_1(x) = \cfrac{C_1}{EA}x + C_2}\)

  • Boundary conditions: \(\cA{u_1(0) = 0}\)

For the second field it gives:

  • \(\cB{EA_2\cfrac{\mrm{d}^2u_2}{\mrm{d}x^2}=0}\)

  • \(\cB{N_2 = C_3}\)

  • \(\cB{u_2(x) = \cfrac{C_3}{EA}x + C_4}\)

  • Boundary conditions: \(\cB{N_2(\ell_1+\ell_2) = F}\)

The two remaining integration constants can be solved by specifying interface conditions:

  • \(\cA{u_1(\ell_1)} = \cB{u_2(\ell_1)}\)

  • \(\cA{N_1} = \cB{N_2}\)

Solving differential equations of more fields#

The same approach can be taken to tackle problems with more field, like the one below:

../_images/bigframe.svg

Fig. 3 Frame structure with many fields#

How many integration constants should be solved for here? How many boundary- and interface conditions would be needed for that? It gets annoying very quickly as each of these conditions need to be defined carefully.

Equivalence with matrix method#

While both methods segment the structure in different parts, the matrix method applies a different principle in solving the structure than when directly solving differential equations: instead of solving for integration constants, nodal displacements are solved for. This shows big potential because setting up all the boundary- and interface conditions can be tedious and is problem-specific. The matrix method applies a generic algorithmic approach to combine all unknown nodal displacements

The similarities and differences are shown in the table below.

Table 2 Equivalence solving differential equations and matrix method#

Solving differential equations

Matrix method

Segment structure in separate fields

Segment structure in mostly repetitive elements

Define all boundary- and interface conditions

Define relations in generic algorithmic manner

Solve for integration constants \(C_1, C_2, ...\)

Solve nodal displacements \(u_1, u_2, ...\)
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