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(abridged) We present the IRAM-30m observations of multiple-J CO and CI lineemission in a sample of redshift ~2-4 Herschel-ATLAS SMGs. A non-negligibleeffect of differential lensing is found for the CO emission lines, which couldhave caused significant underestimations of the linewidths, hence of thedynamical masses. The CO SLEDs are found to be similar to those of the localstarburst-dominated ULIRGs and of the previously studied SMGs. After correctingfor lensing amplification, we derived the global properties of the bulk ofmolecular gas in the SMGs using non-LTE radiative transfer modelling. The gasthermal pressure is found to be correlated with star formation efficiency.Further decomposing the CO SLEDs into two excitation components, we find alow-excitation component, which is less correlated with star formation, and ahigh-excitation one which is tightly related to the on-going star-formingactivity. Additionally, tight linear correlations between the FIR and CO lineluminosities have been confirmed for the $J \ge 5$ CO lines, implying thatthese CO lines are good tracers of star formation. The [CI](2-1) lines followthe tight linear correlation between the luminosities of the [CI](2-1) and theCO(1-0) line found in local starbursts, indicating that CI lines could serve asgood total molecular gas mass tracers for high-redshift SMGs. The total mass ofthe molecular gas reservoir, $(1-30) \times 10^{10} M_\odot$, suggests atypical molecular gas depletion time ~20-100 Myr and a gas to dust mass ratio${\delta}_{\rm GDR}$~30-100. The ratio between CO line luminosity and the dustmass appears to be slowly increasing with redshift for the SMGs, which need tobe further confirmed. Finally, through comparing the linewidth of CO and H2Olines, we find that they agree well in almost all our SMGs, confirming that theemitting regions are co-spatially located.

Molecular gas in the Herschel-selected strongly lensed submillimeter galaxies at z~2-4 as probed by multi-J CO lines