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This study presents a novel isogeometric Euler-Bernoulli beam formulation for in-plane dynamic analysis of multi-patch beam structures. The kinematic descriptions involve only displacements of the beam axis, which are approximated by non-uniform rational B-spline (NURBS) curves. Translational displacements of the control points are here considered as control variables. The motivation of this work is to propose a penalty-free method to handle in-plane dynamic analysis of multi-patch beam structures. A simple relation between cross-sectional rotations at the ends of the beams and control variables is derived, allowing the incorporation of the end rotations as degrees of freedom. This improved setting can straightforwardly tackle beam structures with many rigid multi-patch connections, a major challenging issue when using existing isogeometric Euler-Bernoulli beam formulations. Additionally, rotational boundary conditions are conveniently prescribed. Numerical examples with complicated beam structures such as circular arches and frames with kinks are considered to show the accuracy and performance of the developed formulation. The computed results are verified with those derived from the conventional finite element method, and the superior convergence properties of the proposed formulation are illustrated. We additionally discuss about the possible extension of the present approach to spatial beam structures.

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Neochloris minuta and Neochloris alveolaris grown in nitrogen-rich (+ N) and nitrogen-depleted (-N) media were tested for their heavy metal maximum biosorption capacities (qmax) and adsorption percent efficiencies (R%). By removing nitrogen from the growth media, both algal species showed an increase in their lipid content and a decrease in their protein content. Langmuir and Freundlich adsorption isotherms were used to determine the qmax and adsorption efficiencies of the + N and −N algae in the recovery of Pb2+, Cd2+, Zn2+, Cu2+, and Ni2+. When comparing the two types of algae, N. alveolaris showed the highest adsorption capacities for all five metals either in + N or -N media. The maximum adsorption efficiency percentage of the lowest concentration metal ions for N. alveolaris was 87.10% for Pb2+, 64.98% for Cd2+, 59.50% for Zn2+, 60.08% for Cu2+, and 50.61% for Ni2+. In both algae, nitrogen depletion (-N) caused an increase in the qmax values for Zn2+ and Cu2+. Additionally, the qmax of N. minuta for Cd2+, Zn2+, Cu2+ and Ni2+ increased by the nitrogen depletion demonstrating that the treatment can be applied to improve the biosorption capacity of a particular alga for multiple heavy metals. The biosorption capacity for these algae for heavy metals was also discussed in terms of their biomass compositions and the type of hard or soft metal acid based on the Pearson theory of Hard and Soft, Acid and Bases (HSAB).