The following relationships can be used to analyze uniform beams subject to distributed loads,
dy/dx = θ(x); dθ/dx = M(x)/EI; dM/dx = V(x); dV/dx = -w(x)
where x = distance along the beam (m), y = deflection (m), θ = slope (m/m), M = bending moment (N·m), E = modulus of elasticity (Pa), I = moment of inertia (m^4), V = shear force (N) and w = distributed load (N/m). For the case of a linearly increasing load (see Figure 1a), the moments, M at selected beam locations, x are very accurately (i.e., with no errors) recorded as shown in Table 3:
Table 3: Bending moment at designated beam locations
Location, x (m): 0, 1, 2, 3
Bending moment, M (N·m): 0, 611.1111, 388.8889, -1500
Figure 1: (a) Beam with linearly increasing load, and (b) deflection of the beam due to linearly increasing load.
For this problem, the values for the parameters are: E = 200 GPa, I = 0.0003 m^4, w0 = 2.5 kN/m and L = 3 m. Note that, for numerical differentiation, a second order accurate approximation is required. Use h = 0.5 m for both numerical differentiation and integration.
a. From data in Table 3, find an equation that describes the bending moment as a function of location, i.e., M(x).
b. What are the bending moments at x = 0.5 and 1.5 m using the equation obtained in part (a)?
c. Calculate the shear force, V at x = 1.0 m using numerical method.
d. Calculate the slope, θ at x = 1.5 m using numerical method.
e. If the analytical bending moment of this linearly increasing load case is accurately described by a third order polynomial, what are the analytical shear force, V at x = 1.0 m and slope, θ at x = 1.5 m?