There are many important factors limiting the service life of buildings. An appropriate reliability analysis needs to take into account uncertainties in the environmental conditions as well as in structural properties. Thanks to the growth of powerful computing resources and technology, recently developed procedures in the field of stochastic mechanics have become applicable to realistic engineering systems. The presented contribution is focused on the mod- elling of uncertainties in material properties and investigates the influence of such uncertainties on structural response. Particularly, the material property is described by random field which enables description of material heterogeneity. The uncertainty in material is then propagated through the numerical model within the framework of spectral stochastic finite element method and as a result we obtain the probabilistic description of the structural response. Moreover, once the structure is built new experiments should be performed to update the knowledge about the actual state of the structure. These experiments are often expensive, but the obtained data can be used to decide about an appropriate reconstruction which can significantly extend the lifetime of building and consequently save money. Therefore, we apply Bayesian inference to update our knowledge about the material property of interest. In this contribution, we examine the application of the described methodology to the case of non-linear heat transfer.