Lattice structures have great potential for lightweight additively manufactured parts. This paper presents a bio-inspired functionally graded lattice structure optimisation approach for compliance minimisation of statically loaded structures. Topology optimisation determines the optimal relative density distribution within a design space and the associated strain fields. A lattice structure is then generated within this design space, comprised of trusses aligned with principal strain trajectories. The trusses have a rectangular cross-section, resulting in three size variable per node during the final sizing step. A minimum feature size is implemented so that the designs are suited to the particular additive manufacturing process print resolution. The presented approach is further demonstrated to efficiently solve problems with multiple load cases and produce lattice topologies that mimic those found in bone. The optimised lattice structure examples presented are on average 12% less stiff in comparison to a conventional SIMP topology optimisation approach. However, they are well suited to multifunctional applications such as heat transfer where the surface area is increased by an average of 94%. The optimised lattice structures are also shown to be able to eliminate additive manufacturing support structure requirements.