Context. The bar pattern speed (Ωb) is defined as the rotational frequency of the bar, and it determines the bar dynamics. Several methods have been proposed for measuring Ωb. The non-parametric method proposed by Tremaine & Weinberg (1984, ApJ, 282, L5; TW) and based on stellar kinematics is the most accurate. This method has been applied so far to 17 galaxies, most of them SB0 and SBa types. Aims: We have applied the TW method to a new sample of 15 strong and bright barred galaxies, spanning a wide range of morphological types from SB0 to SBbc. Combining our analysis with previous studies, we investigate 32 barred galaxies with their pattern speed measured by the TW method. The resulting total sample of barred galaxies allows us to study the dependence of Ωb on galaxy properties, such as the Hubble type. Methods: We measured Ωb using the TW method on the stellar velocity maps provided by the integral-field spectroscopy data from the CALIFA survey. Integral-field data solve the problems that long-slit data present when applying the TW method, resulting in the determination of more accurate Ωb. In addition, we have also derived the ratio ℛ of the corotation radius to the bar length of the galaxies. According to this parameter, bars can be classified as fast (ℛ < 1.4) and slow (ℛ > 1.4). Results: For all the galaxies, ℛ is compatible within the errors with fast bars. We cannot rule out (at 95% level) the fast bar solution for any galaxy. We have not observed any significant trend between ℛ and the galaxy morphological type. Conclusions: Our results indicate that independent of the Hubble type, bars have been formed and then evolve as fast rotators. This observational result will constrain the scenarios of formation and evolution of bars proposed by numerical simulations.