The following relationships can be used to analyze uniform beams subject to distributed loads,

Where x = distance along beam (m), y = deflection (m), u(x) = slope (m/m), E = modulus of elasticity (Pa = N/m²), I = moment of inertia (m⁴), M(x) = moment (N m), V(x) 5 shear (N), and w(x) = distributed load (N/m). For the case of a linearly increasing load (recall Fig. P8.18), the slope can be computed analytically as

Employ (a) numerical integration to compute the deflection (in m) and (b) numerical differentiation to compute the moment (in N m) and shear (in N). Base your numerical calculations on values of the slope computed with Eq. P24.19 at equally-spaced intervals of ∆x = 0.125 m along a 3-m beam. Use the following parameter values in your computation: E 5 200 GPa, I = 0.0003 m⁴, and w⁰ = 2.5 kN/cm. In addition, the deflections at the ends of the beam are set at y(0) = y(L) = 0. Be careful of units.

Transcribed Image Text:

dy de M(x) dM V(x) -w(x) = -= dx e(x) (P24.19.1) 120EIL