Zero-field-cooled (ZFC) and field-cooled (FC) magnetic properties of strip-like Josephson-junction (JJ) arrays with very strong demagnetizing effects are calculated from basic laws. Similar to slab-like JJ arrays without considering demagnetizing effects, a vortex state evolves to a critical state (CS) with increasing maximum JJ currents in the ZFC case, and a vortex state always remains with a negative low-field susceptibility in the FC case. However, the strong demagnetizing effects cause qualitative changes in the CS, where the overall feature of the field and current profiles turns out to be similar to that in type-II superconducting strips, but not like the ordinary Bean CS in slab-like JJ arrays, the CS current profile is never flat and the critical current is no longer a step function of the maximum JJ current as in slab-like JJ arrays. The calculated results of different types of JJ arrays indicate that although the intergranular CS in granular superconductors may have a common origin, the discovered paramagnetic Meissner effect in them is still difficult to explain.