10.2 Definition of Mechanical Properties of Biological Materials
343
10.2.4 Bending
Pure compression or tension are relatively uncommon states of stress in biological materials. By far the most common situation is that of bending. This
loading regime is experienced by many marine organisms subjected to wave or
current loading, such as branches of hard corals, sea anemones, macroalgae,
barnacles and many others.
To describe the behaviour of such biological materials under a bending regime,
we apply methods developed in structural engineering, namely beam theory.
A beam is an element designed to resist forces acting normal to its axis. In
Fig. 1O.3a, a simple cantilever beam loaded by force F at its end is shown. To
keep the beam in equilibrium the shear force, Fl , equal to force F is applied
at the fixing point O. Because of bending of the beam by the force F, the
so called bending moment, M, occurs. In general, force tends to move a
body along the line of force action, but the force also tends to rotate the body
about an axis. The ability of a force to cause a body to rotate is measured
by the moment of the force, which is a product of force and a perpendicular
distance from the moment centre to the line of the force action. Thus, at the
cross-section AB of the beam in Fig. 10.3a, the bending moment becomes:
M(x) = -Fx.
(10.14)
At point P, the moment M = 0, and it becomes maximum at point 0, i.e.
M = - F l. In this equation, the sign convention was used according to which a
positive bending moment elongates the lower part of the beam and compresses
the upper part (see Fig. 1O.3b). For a negative bending moment, the situation
is reversed. Also a positive shear force tends to deform the element by causing
the right-hand face to move downward with respect to the left-hand face (Cere
and Timoshenko, 1991).
Let us consider a cross-section AB ofthe beam (Fig. 1O.3c). The neutral axis,
x, denotes the point where there is no stress- the material here neither extends
nor contracts. At points not on the neutral axis, tension or compression stresses
in the beam are related to the bending moment, M, by a formula known as
the flexure formula (Cere and Timoshenko, 1991):
(10.15)
in which y is the distance from the neutral axis, S = I/y is the section moduli,
and I is the moment of inertia (or second momentum of area) of the crosssection of the beam with respect to the neutral axis:
1= ty2dA,
(10.16)
where A is the cross-section area. From Eq. (10.15) it follows that the maximum
stresses appear at the edges of the beam (see Fig. 10.3c).
343
10.2.4 Bending
Pure compression or tension are relatively uncommon states of stress in biological materials. By far the most common situation is that of bending. This
loading regime is experienced by many marine organisms subjected to wave or
current loading, such as branches of hard corals, sea anemones, macroalgae,
barnacles and many others.
To describe the behaviour of such biological materials under a bending regime,
we apply methods developed in structural engineering, namely beam theory.
A beam is an element designed to resist forces acting normal to its axis. In
Fig. 1O.3a, a simple cantilever beam loaded by force F at its end is shown. To
keep the beam in equilibrium the shear force, Fl , equal to force F is applied
at the fixing point O. Because of bending of the beam by the force F, the
so called bending moment, M, occurs. In general, force tends to move a
body along the line of force action, but the force also tends to rotate the body
about an axis. The ability of a force to cause a body to rotate is measured
by the moment of the force, which is a product of force and a perpendicular
distance from the moment centre to the line of the force action. Thus, at the
cross-section AB of the beam in Fig. 10.3a, the bending moment becomes:
M(x) = -Fx.
(10.14)
At point P, the moment M = 0, and it becomes maximum at point 0, i.e.
M = - F l. In this equation, the sign convention was used according to which a
positive bending moment elongates the lower part of the beam and compresses
the upper part (see Fig. 1O.3b). For a negative bending moment, the situation
is reversed. Also a positive shear force tends to deform the element by causing
the right-hand face to move downward with respect to the left-hand face (Cere
and Timoshenko, 1991).
Let us consider a cross-section AB ofthe beam (Fig. 1O.3c). The neutral axis,
x, denotes the point where there is no stress- the material here neither extends
nor contracts. At points not on the neutral axis, tension or compression stresses
in the beam are related to the bending moment, M, by a formula known as
the flexure formula (Cere and Timoshenko, 1991):
(10.15)
in which y is the distance from the neutral axis, S = I/y is the section moduli,
and I is the moment of inertia (or second momentum of area) of the crosssection of the beam with respect to the neutral axis:
1= ty2dA,
(10.16)
where A is the cross-section area. From Eq. (10.15) it follows that the maximum
stresses appear at the edges of the beam (see Fig. 10.3c).