We introduce a simple family of barred galaxy models with flat rotation curves. They are built by convolving the axisymmetric logarithmic model with a needle density. The density contours in the bar region are highly triaxial and elongated, but become spheroidal in the outer parts. The mass distribution differs markedly from the elliptical shape assumed in other analytical models, like Ferrers or Freeman bars. The surface density profile along the bar major axis is flattish, as befits models for bars in early-type galaxies (SB0, SBa). The 2D orbital structure of the models is analysed with surfaces of section and characteristic diagrams and it reveals qualitatively new features. For some pattern speeds, additional Lagrange points can exist along the major axis and give rise to off-centred, trapped orbits. When the bar is weak, the orbital structure is very simple, comprising just prograde, aligned x1 orbits and retrograde anti-aligned x4 orbits. As the bar strength increases, the x1 family becomes unstable and vanishes, with propeller orbits dominating the characteristic diagram.