Answer :
The stresses in the timber beam and aluminum plate can be calculated using the bending stress formula and considering the different elastic moduli. The total stress is a combination of these two stresses, and the beam deflection must account for composite structure properties.
The question requires understanding stress in composite beams and deflection due to applied bending moments. For a timber beam with an aluminum plate bonded to it, we must consider the different modulus of elasticity of each material when calculating stress and deflection under bending moments.
To calculate the stress in both the timber beam (E_w = 8.75 GPa) and the aluminum plate (E_al = 70 GPa), we use the formula for bending stress, σ = M*y/I. Where M is the bending moment, y is the distance from the neutral axis to the fiber location, and I is the moment of inertia of the beam cross-section.
The stress in the timber and aluminum will be different because of the different elastic moduli.
The total stress is the combination of stresses in the timber beam and aluminum plate. The deflection of the beam can be calculated using the formula δ = M*L^2/(48*E*I), where L is the length of the beam and E is the elastic modulus of each material.
Stress in Timber Beam
Calculate the neutral axis location by considering the area moments about the bottom of the beam of both materials.
Find the moment of inertia, I, for the composite section.
Substitute the required values into the bending stress formula.
Stress in Aluminum Plate
Use the same bending moment as for timber but account for the different elastic modulus.
Calculate the stress using the distance from the neutral axis to the center of the aluminum plate.
Total Stress and Deflection
The total stress is the sum of the stresses in the timber and aluminum.
Deflection can be calculated using the same method for single material beams but must take into account the composite structure and different moduli of each material.
