Question

a. The basic differential equation of the deflection of a simply supported, uniformly loaded beam is given as (deflection is represented by the curved dashed line in the Figure): $\text{EI}\frac{d^2y}{dx^2} = \frac{wLx}{2} - \frac{wx^2}{2}$ where, E = the modulus of elasticity and I = the moment of inertia. The boundary conditions are y(0) = y(L) = 0. The following parameter values apply: E = 200 GPa, I = 30,000 cm², w (uniform load) = 15 kN/m, and L = 2 m. Using the finite-difference approach ($\Delta$x = 0.5 m), calculate the deflection of the beam (y) at locations x = 1.0 m and x = 1.5 m and evaluate the percent true relative error if the analytical solution is: y = $\frac{wLx^3}{12EI} - \frac{wx^4}{24EI} - \frac{wL^3x}{24EI}$ b. Creeping oil flow underground can be modeled (in 2 dimensions) as driven by an effective pressure (p) as: $\frac{\partial^2p}{\partial x^2} + \frac{\partial^2p}{\partial y^2} = x + y$

          a. The basic differential equation of the deflection of a simply supported, uniformly loaded beam is given as (deflection is represented by the curved dashed line in the Figure):
$\text{EI}\frac{d^2y}{dx^2} = \frac{wLx}{2} - \frac{wx^2}{2}$

where, E = the modulus of elasticity and I = the moment of inertia. The boundary conditions are
y(0) = y(L) = 0. The following parameter values apply: E = 200 GPa, I = 30,000 cm², w (uniform load) = 15 kN/m, and L = 2 m. Using the finite-difference approach ($\Delta$x = 0.5 m), calculate the deflection of the beam (y) at locations x = 1.0 m and x = 1.5 m and evaluate the percent true relative error if the analytical solution is:
y = $\frac{wLx^3}{12EI} - \frac{wx^4}{24EI} - \frac{wL^3x}{24EI}$
b. Creeping oil flow underground can be modeled (in 2 dimensions) as driven by an effective pressure (p) as:
$\frac{\partial^2p}{\partial x^2} + \frac{\partial^2p}{\partial y^2} = x + y$

a. The basic differential equation of the deflection of a simply supported, uniformly loaded beam is given as (deflection is represented by the curved dashed line in the Figure):
EI(d^2y)/(dx^2) = (wLx)/(2) - (wx^2)/(2)

where, E = the modulus of elasticity and I = the moment of inertia. The boundary conditions are
y(0) = y(L) = 0. The following parameter values apply: E = 200 GPa, I = 30,000 cm², w (uniform load) = 15 kN/m, and L = 2 m. Using the finite-difference approach (Δx = 0.5 m), calculate the deflection of the beam (y) at locations x = 1.0 m and x = 1.5 m and evaluate the percent true relative error if the analytical solution is:
y = (wLx^3)/(12EI) - (wx^4)/(24EI) - (wL^3x)/(24EI)
b. Creeping oil flow underground can be modeled (in 2 dimensions) as driven by an effective pressure (p) as:
(∂^2p)/(∂ x^2) + (∂^2p)/(∂ y^2) = x + y

Added by Charles M.

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