The near-wall behaviour of particles is important in terms of predicting both the particle-deposition and the near-wall particle-concentration. In this paper, we study the near-wall behaviour of elastic-bouncing small heavy spheric particles in fully developed turbulent pipe flows without gravity, using direct numerical simulations (DNS) with a one-way point-particle approach. The particle-concentration is assumed to be small enough such that the influence of the particles on the fluid and interparticle interactions can be neglected. The focus of the paper is on: (i) the understanding of the differences between elastic-bouncing and absorbing walls, and (ii) the evaluation of simple "local-equilibrium" models. Our results show that the near-wall behaviour of elastic-bouncing walls is very different from absorbing walls. The absence of a mean radial particle-velocity leads to a much higher particle-concentration near the wall than in the case of absorbing walls. This can be explained by the absence of a "mean drag force": the "turbophoretic effect", due to the gradient in the particle-velocity fluctuation in the radial direction, is balanced only by the "drift-velocity", due to a gradient in the particle-concentration. Our results indicate that, from a pragmatic perspective, simple "local-equilibrium" models for the "turbophoretic effect", assuming a proportionality between the particle and fluid "Reynolds-stresses", are adequate, except very close to the wall, where the reduction in the radial fluid-velocity fluctuation is not accompanied by an equivalent reduction in the radial particle-velocity fluctuation.