We are interested in the design of robust (or resilient) capacitated rooted Steiner networks in the case of terminals with uniform demands. Formally, we are given a graph, capacity, and cost functions on the edges, a root, a subset of vertices called terminals, and a bound k on the number of possible edge failures. We first study the problem where k=1 and the network that we want to design must be a tree covering the root and the terminals: we give complexity results and propose models to optimize both the cost of the tree and the number of terminals disconnected from the root in the worst case of an edge failure, while respecting the capacity constraints on the edges. Secondly, we consider the problem of computing a minimum-cost survivable network, that is, a network that covers the root and terminals even after the removal of any k edges, while still respecting the capacity constraints on the edges. We also consider the possibility of protecting a given number of edges. We propose three different formulations: a bilevel formulation (with an attacker and a defender), a cutset-based formulation and a flow-based one. We compare the formulations from a theoretical point of view, and we propose algorithms to solve them and compare their efficiency in practice.