We theoretically investigate transport-current-induced magnetic-flux penetration into superconducting strip lines with slits. Even when the individual strips have no bulk pinning, geometrical barriers prevent penetration of magnetic flux into the innermost strips while flux quasistatically penetrates into the outermost slits. The critical current of strip lines with 2N slits at zero applied magnetic field is found to be enhanced by a factor of (N+1)(1/2) above that of a single strip line without slits. Under suitable conditions, a domelike flux distribution due to the geometrical barrier can appear in the individual strips even in the absence of an applied magnetic field.