The built-in composite bar BC of length L=2 m is composed of two materials with equal cross sectional area A = 100 mm2. The first material has elastic modulus E=1 GPa. The second material is twice as stiff, with a modulus of 2E or 2 GPa. The bar is subjected to an unknown distributed load fx(x), and to an unknown concentrated load F at an unknown position along the bar. As a result of these loading conditions, the displacement field of the bar is:
To determine the displacement field of the composite bar under the given loading conditions, we need to apply the principles of mechanics and structural analysis.
1. Start by considering the equilibrium equation for the composite bar. Since the bar is subjected to internal distributed loads and a concentrated load, the equilibrium equation can be written as:
∑Fx = F - ∫f(x) dx = 0
where F is the concentrated load and f(x) is the distributed load along the bar.
2. Apply the compatibility condition, which states that the displacement of the composite bar must be continuous across its entire length. This condition leads to the following equation:
∫(ε(x) dx) = 0
where ε(x) is the strain field of the bar.
3. Use the constitutive equation of linear elasticity to relate the strain and stress in each material. The constitutive equation can be written as:
σ(x) = Eε(x)
where σ(x) is the stress at position x and E is the elastic modulus of the respective material.
4. Divide the composite bar into two sections, corresponding to the two materials. Let's call their lengths L1 and L2, such that L1 + L2 = L (the total length of the bar).
5. In the first material with elastic modulus E, the strain can be expressed as ε1(x) = -(∫(f(x) dx))/E.
6. In the second material with elastic modulus 2E, the strain can be expressed as ε2(x) = -(∫(f(x) dx))/(2E).
7. Use the beam theory assumptions, such as small deformations and plane sections remaining plane, to relate the strain and displacement fields. The equation can be written as:
ε(x) = d^2u(x)/dx^2
where u(x) is the displacement at position x.
8. Integrate the strain expressions from steps 5 and 6 twice to obtain the displacement fields in each material:
u1(x) = -(∫(∫(f(x) dx) dx))/(2E)
u2(x) = - (∫(∫(f(x) dx) dx))/(4E)
9. Apply the compatibility condition from step 2 to find the unknown concentrated load F and its position along the bar. The compatibility condition equation can be written as:
∫(u(x) dx) = 0
where u(x) is the total displacement field of the composite bar.
10. Substitute the displacement fields u1(x) and u2(x) into the compatibility condition equation and solve for the unknowns F and its position.
By following these steps, you should be able to determine the displacement field of the composite bar under the given loading conditions.