► We continue the development of Collocation methods for IGA. ► We focus on linear elasticity problems in any dimension, and develop a variational approach to IGA collocation. ► We treat NURBS multi-patch configurations, including various boundary and interface conditions. ► We investigate time dependent problems and explicit dynamic analysis. We extend the development of collocation methods within the framework of Isogeometric Analysis (IGA) to multi-patch NURBS configurations, various boundary and patch interface conditions, and explicit dynamic analysis. The methods developed are higher-order accurate, stable with no hourglass modes, and efficient in that they require a minimum number of quadrature evaluations. The combination of these attributes has not been obtained previously within standard finite element analysis.