The plasma membrane defines the shape of the cell and plays an indispensable role in bridging intra- and extra-cellular environments. Mechanochemical interactions between plasma membrane and cytoskeleton are vital for cell biomechanics and mechanosensing. A computational model that comprehensively captures the complex, cell-scale cytoskeleton-membrane dynamics is still lacking. In this work, we introduce a triangulated membrane model that accounts for membrane's elastic properties, as well as for membrane-filament steric interactions. The corresponding force-field was incorporated into the active biological matter simulation platform, MEDYAN ("Mechanochemical Dynamics of Active Networks"). Simulations using the new model shed light on how actin filament bundling affects generation of tubular membrane protrusions. In particular, we used membrane-MEDYAN simulations to investigate protrusion initiation and dynamics while varying geometries of filament bundles, membrane rigidities and local G-Actin concentrations. We found that bundles' protrusion propensities sensitively depend on the synergy between bundle thickness and inclination angle at which the bundle approaches the membrane. The new model paves the way for simulations of biological systems involving intricate membrane-cytoskeleton interactions, such as occurring at the leading edge and the cortex, eventually helping to uncover the fundamental principles underlying the active matter organization in the vicinity of the membrane.