Bar BC in the figure has length L, constant cross sectional area A, and is composed of a homogeneous material with modulus E. The bar is fixed between walls at B (x=0) and C (x = L). The bar is subjected to a variable distributed load per unit length, oriented in the direction indicated in the figure. The magnitude of the distributed load, p(x)=bx, linearly increases from B to C, with a known value for the constant parameter b. Note that b has dimensions [Nm2], so that p(x) has the desired dimensions [Nm].
Find:
1: expression for the axial force resultant along the bar, N(x), in terms of x, b, and of the unknown redundant reaction R_xB
2. expression for the axial strain in the bar, ϵa(x) in terms of x, b, E, A, and of the unknown redundant reaction R_xB
3. redundant reaction, R_xB in terms of b and L
4. expression for the axial strain in the bar, ϵa(x) in terms of x, and of the known problem parameters b, E, L and A
5. expressions for the axial strain at the two ends of the bar (in terms of the known problem parameters b, E, L and A), and for the position x0 along the bar where the axial strain goes to zero (ϵ_a(x0)=0), (in terms of L):
ϵ_a(x=0)=
ϵ_a(x=L)=
ϵ_a(x_0=0) at x_0=0 =
6. expression for the displacement field, u(x), in terms of x, and of the known problem parameters b, E, L and A
7. expression for the position where the magnitude of the displacement ux is maximum (in terms of L).
8. expression for the maximum magnitude (absolute value) of displacement along the bar, umax=∣ux∣max, in terms of the known problem parameters b, E, L and A:
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