This paper compares the chemical evolution of the Large Magellanic Cloud (LMC) to that of the Milky Way (MW) and investigates the relation between the bar and the inner disc of the LMC in the context of the formation of the bar. We obtained high-resolution and mid signal-to-noise ratio spectra with FLAMES/GIRAFFE at ESO/VLT (Proposals 072.B-0293(B) and 078.B-0323(A), P.I. Vanessa Hill) and performed a detailed chemical analysis of 106 and 58 LMC field red giant stars (mostly older than 1Gyr), located in the bar and the disc of the LMC respectively. To validate our stellar parameter determinations and abundance measurement procedures, we performed thorough tests using the well-known mildly metal-poor Milky-Way thick disc giant Arcturus (HD 124897, alpha Boo). We measured elemental abundances for O, Mg, Si, Ca, Ti (alpha-elements), Na (light odd element), Sc, V, Cr, Co, Ni, Cu (iron-peak elements), Y, Zr, Ba, La and Eu (s- and r-elements). We used the automated tool DAOSPEC to measure the equivalent width (EW) and their associated error, and we used the grid of OSMARCS model atmospheres together with the spectrum synthesis code turbospectrum to convert the EW into abundances. Since our stars are giants, atmosphere models and radiative transfer were both in spherical geometry. We built the atmosphere model for a given set of stellar parameters by interpolation onto the OSMARCS grid with the interpolation routine written by T. Masseron.