Basic Equations: Differential Form - Mechanics of Structures: Variational and Computational Methods - page 46

the boundary conditions for a beam can be written as

V = V

or s = s on S p (1.122)

M = M

for static or force conditions and
w = w
θ = θ

or u = u on S u (1.123)

for displacement or kinematic conditio ns . If axial deformation and fo rce are to be included,
then supplement Eq. (1.122) with N = N and Eq. (1.123) with u = u .

1.8.5 Displacement Form of the Governing Differential Equations
It should be apparent that the governing equations for a beam can be written in a form
quite similar to the general governing equations for an elastic solid. For the elastic solid,
we have

D T σ

= Du , σ = E , + p V = 0 (1.124a)

and for the beam,

D s s = D u u , s = E , + p = 0 (1.124b)

In the case of the general elastic solid, we derived alternative forms to Eq. (1.124) for
governing differential equations. The most common forms for the beam equations are in
terms of displacements

w ,or w and θ only, or in terms of all of the state variables (a mixed

form): displacement ; moment M ; or shear V .

A familiar form of the beam equations is the Euler-Bernoulli beam in which the shear
deformation has been neglected. To derive this form, consider only the equations related
to bending, i.e., ignore the axial extension relationships. From the kinematical relations of
Eqs. (1.97) and (1.100), and the material law of Eq. (1.106b),

w ; slope θ

EI d 2

EI d

θ
dx =−

w
dx 2

M = EI κ = (1.125)

Substitution of this relation into the equilibrium conditions, Eqs. (1.112a) and (1.113) results
in

dx EI d 2

dM
dx =−

d w

dx 2

V = (1.126a)

and

dx 2 EI d 2

d 2

dV
dx =

w
dx 2 =

− p z (1.126b)

The full set of governing differential equations is then

dx 2 EI d 2

d 2 w

dx 2 =

p z (1.127a)

dx EI d 2

d w

dx 2

V =− (1.127b)

EI d 2

w
dx 2

M =− (1.127c)

dx

d

θ =− (1.127d)