Biharmonic eigenvalue problem of the semi-infinite strip
Authors: Horvay Gabriel 
Published: December 2016

A basic problem in the evaluation of residual stresses in simple elastic structures concerns the determination of the stress and deformation state produced by self-equilibrating, but otherwise arbitrary, normal and shear tractions acting on the edge

x
=
0

x = 0

of a semi-infinite elastic strip

(

0
≤
x
≤
∞
,
−
1
≤
y
≤
1

)

\left ( {0 \le x \le \infty , - 1 \le y \le 1} \right )

which is free along the edges

y
=
±
1

y = \pm 1

. This strip is known to experience, in accordance with St. Venant’s principle, inappreciable stresses at distances

x
≳
2

x \gtrsim 2

from the loaded edge, in spite of the very large stresses it may experience in the vicinity of the edge. An earlier paper [The end problem of rectangular strips, J. Appl. Mech. (1953)] based on the variational principle, established approximate eigenfunctions (modes of response) and eigenvalues (laws of oscillation and decay) for the various possible self-equilibrating end tractions. In this paper we give a rigorous solution of the end problem. This solution is obtained in two steps. First we solve the two “mixed” end problems: the parallel edges

y
=
±
1

y = \pm 1

of the strip are free, and along the vertical edge

x
=
0

(
a
)

x = 0\left ( a \right )

the shear displacement is given, the normal stress is zero, (b) the normal displacement is given, the shear stress is zero. These two problems are solved by extending the strip to the left, to

−
∞

- \infty

, and finding the tractions that must be applied at

y
=
±
1

(

x
>
0

)

y = \pm 1\left ( {x > 0} \right )

and at

x
=
−
∞

x = - \infty

, so that one have

σ
x

=
0
,
τ
=
0

{\sigma _x} = 0,\tau = 0

, respectively, at

x
=
0

x = 0

, while the edge values of the displacements (more specifically, of

d
v

/

d
y

dv/dy

and

u
u

) are orthogonal polynomials in

y
y

(Horvay-Spiess polynomials and Legendre polynomials, respectively). The corresponding stress functions

K
n

(

x
,
y

)

,

J
n

(

x
,
y

)

{K_n}\left ( {x,y} \right ), {J_n}\left ( {x,y} \right )

are found in the form of Fourier integrals plus polynomial terms. For

x
≥
0

x \ge 0

they may be rewritten as real parts of

∑

C

K
n
k

Φ
k

,

∑

C

J
n
k

Φ
k

\sum {C_{Knk}}{\Phi _k}, \\ \sum {C_{Jnk}}{\Phi _k}

, where
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Phi Subscript k Baseline equals z Subscript k Superscript negative 2 Baseline e Superscript minus z k x Baseline left-parenthesis cosine z Subscript k Baseline y minus y cotangent z Subscript k Baseline sine z Subscript k Baseline y right-parenthesis">

Φ
k

=

z
k

−
2

e

−
z
k
x

(

cos
⁡

z
k

y
−
y
cot
⁡

z
k

sin
⁡

z
k

y

)

{\Phi _k} = z_k^{ - 2}{e^{ - zkx}}\left ( {\cos {z_k}y - y\cot {z_k}\sin {z_k}y} \right )

or

z

k

−
2

e

−
z
k
x

(

sin
⁡

z
k

y
−
y
tan

z
k

cos
⁡

z
k

y

)

z{}_k^{ - 2}{e^{ - zkx}}\left ( {\sin {z_k}y - y\tan \\ {z_k}\cos {z_k}y} \right )

, and

sin
⁡
2

z
k

±
2

z
k

=
0

\sin 2{z_k} \pm 2{z_k} = 0

. An alternate procedure for determining the coefficients

C

K
n
k
,

C

J
n
k

{C_{Knk,}}{C_{Jnk}}

, based on a formula of R. C. T. Smith, which bypasses the extension of the strip to

x
=
−
∞

x = - \infty

, is also furnished. The second phase of the solution of the “pure” end problem—along the short edge (a) the shear stress is given, normal stress is zero, (b) the normal stress is given, shear stress is zero—consists in recombining the biharmonic eigenfunctions

K

n

,

J
n

{K_{n}}, {J_n}

within each class into functions

H
n

(

x
,
y

)

,

G
n

(

x
,
y

)

{H_n}\left ( {x,y} \right ), {G_n}\left ( {x,y} \right )

, so that the

x
=
0

x = 0

values of

H

n
,
x
y

,

G

n
,
y
y

{H_{n,xy}}, {G_{n,yy}}

, constitute two complete orthonormal sets of (transcendental) functions in

y
y

into which the given boundary stresses may be expanded.