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:
To find the expressions requested, we can use the principles of mechanics and the equations governing the behavior of elastic materials. Let's go step by step:
1. The axial force resultant along the bar, N(x), can be found by integrating the distributed load p(x) along the length of the bar. The distributed load increases linearly from B to C, so we have:
N(x) = ∫ p(x) dx
= ∫ (bx) dx
= (b/2) x^2 + C1
Here, C1 is the constant of integration which represents the axial force at the fixed end B (x=0). The unknown redundant reaction R_xB will contribute to this constant.
2. The axial strain in the bar, ϵa(x), can be found by dividing the axial displacement by the original length. Since the bar is assumed to be in one-dimensional axial loading, ϵa(x) = du(x) / dx. Therefore, we need to find the displacement field u(x) first.
3. The redundant reaction R_xB can be found by imposing the deformation compatibility condition. Assuming no axial deformation at x=0, we can equate the displacement field u(x) to zero at x=0:
u(0) = 0
Solving for R_xB, we can find its expression in terms of b and L.
4. The axial strain in the bar, ϵa(x), can be found using the equation ϵa(x) = du(x) / dx. We can differentiate the displacement field expression from step 6 to find ϵa(x) in terms of x, b, E, L, and A.
5. To find the axial strain at the two ends of the bar and the position x0 where the axial strain goes to zero, we need to substitute the corresponding values of x into the expression found in step 4.
6. The displacement field, u(x), can be found by integrating the axial strain ϵa(x) over the length of the bar. Since the bar is assumed to have a constant cross-sectional area A, we can use the equation u(x) = ∫ ϵa(x) dx. Integrate the expression for ϵa(x) from step 4 to find u(x) in terms of x, b, E, L, and A.
7. To find the position where the magnitude of the displacement ux is maximum, we need to differentiate the displacement field expression from step 6 with respect to x and set it equal to zero. Solve for x to find the position x_max.
8. The maximum magnitude of displacement along the bar, umax=∣ux∣max, can be found by substituting the value of x_max into the displacement field expression from step 6.