Preheat the oven to 350°F (176°C) and adjust the oven rack to the center position. Lightly oil and flour three 8- or 9-inch round cake pans.
In a large bowl, combine the flour, sugar, baking powder, salt, cinnamon, baking soda, and nutmeg..
In a separate bowl, whisk together the oil and eggs until combined. Add the bananas and pineapple (including the pineapple juice) and stir to combine.
Stir the egg mixture into the flour mixture with a spatula or wooden spoon just until combined. Stir in the vanilla and pecans. Divide the batter evenly among the pans, smoothing the tops.
Bake the cakes for 25 to 30 minutes, until a skewer inserted in the center comes out clean.
Let the cakes cool in their pans on a wire rack for about 10 minutes.
Run a slender knife around the edges of the pans and turn the cakes out onto the racks, then turn them again so they're right side up. Let cool completely before frosting.