We consider the coupled linear system {partial derivative(tt)u + partial derivative(xxxx)u+gamma partial derivative(t)u+k(u-v) + h partial derivative(t) (u-v) = 0 epsilon partial derivative(tt)v - partial derivative(xx)v - k(u-v) - h partial derivative(t)(u-v)=0 describing the vibrations of a string-beam system related to the well-known Lazer-McKenna suspension bridge model. For epsilon >0 and k >0, the decay properties of the solution semigroup are discussed in dependence of the nonnegative parameters gamma and h, which are responsible for the damping effects.