The Displacement Variational Method For Laminated Beams

In engineering and practical life,laminated beams have a very wide range of applications.For a long time,many scholars have used a variety of research and calculation methods to continuously strengthen and improve the research of laminated beams.The key to the calculation of laminated beams is to determine the size and distribution of contact pressures between layers.In this paper,the principle of minimum potential energy is used to solve the indirect contact force of laminated beams so as to overcoming some conclusions of traditional analytical solutions that are not in line with reality.This paper focuses on two calculation models of simply supported laminated beams and cantilever laminated beams.The upper and lower beams are studied separately.The contact force between laminated beams is regarded as external force to the lower beam.The interlayer contact pressure is discrete and loaded sequentially into the lower beam.The elastic potential energy function of the lower beam is derived by using superposition theorem.Using the analytical formula of material mechanics,the deformation potential energy formula of the upper beam and the external potential energy formula are obtained.The potential energy functions are given for the whole system of the laminated beams.On this basis,the displacement of the upper beam is calculated with the principle of minimum potential energy,then the interlayer contact pressure of laminated beams is obtained.Using the superposition principle in material mechanics,the deflection formula of the lower beam is derived.Interlayer contact analysis was performed based on the deflection difference between the upper and lower beams and the interlayer contact force obtained.The calculation process of this method is standard and easy to program.The results show that when a certain load or load combination is applied,the contact forces between the layers of the laminated beam are not all pressure,and the layers are in incomplete contact.In this case,for the composite laminated beams with complete interlayer bonding,there will be tensile force between the layers;When the full beam area is subjected to uniformly distributed loads,the contact forces between the layers of the simply supported laminated beam and the cantilever laminated beam are all pressure,and they are in full contact;When a cantilever laminated beam is subjected to a concentrated load at the free end,the layers are in point contact.The main research result of this paper is to derive the elastic potential energy formula of laminated beams,and propose a new method for studying the contact between laminated beams.This method can be used as the theoretical basis of the contact principle between laminated beams.The new algorithm presented in this paper is simple and can be widely used in other engineering calculation.