Cdi

Figure 2.26 Schematic of pinned end condition.

the beam. There are four such states of practical interest:

3. bending moment = M(x. t) = EI^x. t) = EIX"(x)Y(t),

4. shear = V(jt, t) = -£70(jt, r) = -EIX"'(x)Y(t).

It should be noted when relating these beam states that the positive convention for deflection and slope is the same at both ends of the beam. In contrast, the shear and bending moment sign conventions differ at opposite beam ends as illustrated by the free-body differential beam element used to obtain the equation of motion shown in Fig. 2.25.

The most common conditions that can occur at the beam ends involve vanishing pairs of individual states. Typical of such conditions are the following classical configurations:

• Simply-supported, hinged or pinned end, which indicates zero deflection and bending moment, is denoted by the triangular symbol in Fig. 2.26 and has v(£, t) — M(£, t) = 0 so that X(i) = X"(£) = 0.

• Cantilever or built-in end, which implies zero deflection and slope, is illustrated in Fig. 2.13 and has v(t, t) = ff (t, t) = 0 so that X(t) = X'(£) = 0.

• Free end, which corresponds to zero bending moment and shear, is illustrated in Fig. 2.14 and has M(£, t) = V(£, t) = 0 so that X"(£) = X"'(£) = 0.

• Sliding end, which corresponds to zero shear and slope, is illustrated in Fig. 2.27 and has ff (£, t ) = V(£, t ) = 0 so that X'(£) = X"'(£) = 0.

All of these conditions can occur in the same form at x = 0.

In addition to these zero-state conditions, the boundary conditions can correspond to linear constraint reactions associated with elastic and inertial elements. These types of conditions were previously observed for the torsional dynamics of beams. They may occur alone or in conjunction with others. The boundary conditions associated with these constraint reactions are of four basic types:

1. translational elastic constraint,

2. rotational elastic constraint,

3. translational inertia constraint, and

4. rotational inertia constraint.

Figure 2.27 Schematic of sliding end condition.

Figure 2.28 Schematic of translational spring end conditions. Translational Elastic Constraint

A translational elastic constraint corresponds to a spring reaction force that is equated to the shear force, as shown in Fig. 2.28. At the left end

£7^-4(0, /) = -kv(0,t) EIX"'(0) = -kX(0). (2.222)

At the right end

As previously observed, it is seen here that the constraint relations differ in sign at opposite beam ends. It should also be noted that the above conditions must be augmented by one additional condition at each end, since two are required. As illustrated in Fig. 2.28, in this instance the second condition would be zero bending moment.

Rotational Elastic Constraint

The rotational elastic constraint corresponds to a spring reaction moment that is equated to the bending moment. This is illustrated in Fig. 2.29. At the left end

£7^—^(0, t) — £ — (0, t) EIX"(0) = kX'(0). (2.224)

At the right end

As in the previous case the signs differ and one more condition is required at each end. Translational Inertia Constraint

The translational inertia constraint corresponds to the inertial reaction force associated with the translational acceleration of a rigid body or particle of mass m, attached to the Figure 2.30 Schematic of translational mass end conditions.

end of the beam, as illustrated in Fig. 2.30. This force is equated to the shear. At the left end d3V d2V

From the previously given general solution d2V

— = X(x)Y(t) = -a>2X(x)Y(t) = -co2 v = -oW (2.227) 31-

and so the boundary condition can now be written as

EI—r(0, 0 = mca4a4v(0, t) EI X"'(0) = mca4u4X(0). (2.228)

At the right end the sign is changed as the result of the shear convention to yield

The above conditions must be augmented by one additional condition at each end, since two are required. As illustrated in Fig. 2.30, in this instance the second condition would be zero bending moment.

Rotational Inertia Constraint

The rotational inertia constraint corresponds to the inertial reaction moment associated with the rotational acceleration of a rigid body attached to the end of the beam, as shown in Fig. 2.31. This moment is equated to the bending moment. At the left end d2V d3V a adv

0 0