A uniform beam of length L has a concentrated load, $w _ { 0 }$ , at $x = L / 2$ . It is embedded at the left end and free at the right end. The correct initial value problem for the vertical deflection, $y ( x )$ , at a distance x from the embedded end is
A) $E L y ^ { \prime \prime \prime \prime } = w _ { 0 } \delta ( x - L / 2 ) , y ( 0 ) = 0 , y ^ { \prime } ( 0 ) = 0 , y ^ { \prime \prime } ( L ) = 0 , y ^ { \prime \prime \prime } ( L ) = 0$
B) $y ^ { \prime \prime \prime \prime } = E I w _ { 0 } \delta ( x - L / 2 ) , y ( 0 ) = 0 , y ^ { \prime } ( 0 ) = 0 , y ^ { \prime \prime } ( L ) = 0 , y ^ { \prime \prime \prime } ( L ) = 0$
C) $E L y ^ { \prime \prime } = w _ { 0 } \delta ( x - L / 2 ) , y ( 0 ) = 0 , y ^ { \prime } ( 0 ) = 0 , y ^ { \prime \prime } ( L ) = 0 , y ^ { \prime \prime \prime } ( L ) = 0$
D) $y ^ { \prime \prime } = E I w _ { 0 } \delta ( x - L / 2 ) , y ( 0 ) = 0 , y ^ { \prime } ( 0 ) = 0 , y ^ { \prime \prime } ( L ) = 0 , y ^ { \prime \prime \prime } ( L ) = 0$
E) $E L y ^ { \prime \prime \prime \prime } = w _ { 0 } \delta ( x + L / 2 ) , y ( 0 ) = 0 , y ^ { \prime } ( 0 ) = 0 , y ^ { \prime \prime } ( L ) = 0 , y ^ { \prime \prime \prime } ( L ) = 0$

Question

Quizplus · Accepted Answer

The correct initial value problem for the vertical deflection of a beam under a concentrated load includes the fourth derivative of the deflection ($y^{''''}$) to represent the bending moment equation, a delta function ($\delta(x - L/2)$) to represent the concentrated load at the midpoint, and boundary conditions at the embedded end ($y(0) = 0, y'(0) = 0$) and at the free end ($y''(L) = 0, y'''(L) = 0$) to represent the constraints and forces/moments at those points. Option A correctly includes these elements with the correct placement of $E$ and $L$ in the equation.

A Uniform Beam of Length L Has a Concentrated Load w0w _ { 0 }w0​

A Uniform Beam of Length L Has a Concentrated Load $w _ { 0 }$