Elastic instability

Elastic instability is a form of instability occurring in elastic systems, such as buckling of beams and plates subject to large compressive loads.

ingle degree of freedom-systems

Consider as a simple example a rigid beam of length "L", hinged in one end and free in the other, and having an angular spring attached to the hinged end. The beam is loaded in the free end by a force "F" acting in the compressive axial direction of the beam, see the figure to the right.

Moment equilibrium condition

Assuming a clockwise angular deflection $$heta, the clockwise moment exerted by the force becomes $$M_F = F L sin heta. The moment equilibrium equation is given by

$$F L sin heta = k_ heta heta

where $$k_ heta is the spring constant of the angular spring (Nm/radian). Assuming $$heta is small enough, implementing the taylor expansion of the sine function and keeping the two first terms yields

$$F L Bigg( heta - frac{1}{6} heta^3Bigg) approx k_ heta heta

which has three solutions, the trivial $$heta = 0, and

$$heta approx pm sqrt{6 Bigg( 1 - frac{k_ heta}{F L} Bigg)}

which is imaginary (i.e. not physical) for $$F L < k_ heta and real otherwise. This implies that for small compressive forces, the only equilibrium state is given by $$heta = 0, while if the force exceeds the value $$k_ heta/L there is suddenly another mode of deformation possible.

Energy method

The same result can be obtained by considering energy relations. The energy stored in the angular spring is

$$E_mathrm{spring} = int k_ heta heta mathrm{d} heta = frac{1}{2} k_ heta heta^2

and the work done by the force is simply the force multiplied by the distance, which is $$L (1 - cos heta). Thus,

$$E_mathrm{force} = int{F mathrm{d} x = F L (1 - cos heta )}

The energy equilibrium condition $$E_mathrm{spring} = E_mathrm{force} now yields $$F = k_ heta / L as before (besides from the trivial $$heta = 0).

tability of the solutions

Any solution $$heta is stable iff a small change in the deformation angle $$Delta heta results in a reaction moment trying to restore the original angle of deformation. The net clockwise moment acting on the beam is

$$M( heta) = F L sin heta - k_ heta heta

An infinitesimal clockwise change of the deformation angle $$heta results in a moment

$$M( heta + Delta heta) = M + Delta M = F L (sin heta + Delta heta cos heta ) - k_ heta ( heta + Delta heta)

which can be rewritten as

$$Delta M = Delta heta (F L cos heta - k_ heta)

since $$F L sin heta = k_ heta heta due to the moment equilibrium condition. Now, a solution $$heta is stable iff a clockwise change $$Delta heta > 0 results in a negative change of moment $$Delta M < 0 and vice versa. Thus, the condition for stability becomes

$$frac{Delta M}{Delta heta} = frac{mathrm{d} M}{mathrm{d} heta} = FL cos heta - k_ heta < 0

The solution $$heta = 0 is stable only for $$FL < k_ heta, which is expected. By expanding the cosine term in the equation, we obtain the approximate stability condition

$$
heta| > sqrt{2Bigg( 1 - frac{k_ heta}{F L} Bigg)}

for $$FL > k_ heta, which the two other solutions satisfy. Hence, these solutions are stable.

Multiple degrees of freedom-systems

By attaching another rigid beam to the original system by means of an angular spring a two degrees of freedom-system is obtained. Assume for simplicity that the beam lengths and angular springs are equal. The equilibrium conditions become

$$F L ( sin heta_1 + sin heta_2 ) = k_ heta heta_1

$$F L sin heta_2 = k_ heta ( heta_2 - heta_1 )

where $$heta_1 and $$heta_2 are the angles of the two beams. Linearizing by assuming these angles are small yields

$$egin{pmatrix}F L - k_ heta & F L \k_ heta & F L - k_ hetaend{pmatrix}egin{pmatrix} heta_1 \ heta_2end{pmatrix} = egin{pmatrix}0 \0end{pmatrix}

The non-trivial solutions to the system is obtained by finding the roots of the determinant of the system matrix, i.e. for

$$frac{F L}{k_ heta} = frac{3}{2} mp frac{sqrt{5{2} approx left{egin{matrix} 0.382\2.618 end{matrix} ight.

Thus, for the two degrees of freedom-system there are two critical values for the applied force "F". These correspond to two different modes of deformation which can be computed from the nullspace of the system matrix. Dividing the equations by $$heta_1 yields

$$frac{ heta_2}{ heta_1} Big|_{ heta_1 e 0} = frac{k_ heta}{F L} - 1 approx left{egin{matrix} 1.618 & ext{for } F L/k_ heta approx 0.382\ -0.618 & ext{for } F L/k_ heta approx 2.618 end{matrix} ight.

For the lower critical force the ratio is positive and the two beams deflect in the same direction while for the higher force they form a "banana" shape. These two states of deformation represent the buckling mode shapes of the system.

ee also

* Buckling

Further reading

*"Theory of elastic stability", S. Timoshenko and J. Gere