We presents a computational framework for predicting the mechanical behavior of systems comprising slender, one-dimensional elements (fibers or beams) interacting through frictional contact. Utilizing an integral penalty-based approach, the model effectively captures conforming contact scenarios over finite-sized regions. The framework's versatility is enhanced by its applicability to various material descriptions and contact conditions. Model validation is achieved through comparisons with fully resolved continuum finite element simulations, demonstrating accuracy and robustness in addressing frictional contact. Key aspects of the presented framework include: (I) two symmetric formulations for frictionless contact: a two-full-pass and a two-half-pass approach. The two-half-pass formulation provides enhanced robustness with reduced computational cost. (II) A frictional extension that accommodates nonlinear contact laws, employing a generalized standard materials framework in conjunction with Coulomb's friction law. (III) A smoothing technique based on Bézier curves to ensure stability and applicability across different beam kinematic descriptions. The model demonstrates applicability to complex engineering problems. Examples include simulating the effective response of 3D periodic intertwined architected materials, where contact induces nonlinear stiffening phenomena. This highlights the potential for exploring the design space of materials such as woven fabrics, entangled fiber networks, and various knot topologies. The presented framework offers a robust approach for resolving conforming frictional contact between beams across a range of engineering applications, including exploring the design space of architected materials and incorporating advanced beam models to account for inelasticity and multi-scale effects.