A uniform beam of length L has a concentrated load, $w _ { 0 }$ , at $x = L / 2$ . It is embedded at the left end and simply supported at the right end. If $y ( x )$ is the vertical deflection, then the correct differential equation for y, is
A) $E L y ^ { \prime \prime \prime \prime } = w _ { 0 } \delta ( x - L / 2 )$
B) $y ^ { \prime \prime \prime \prime } = E l w _ { 0 } \delta ( x - L / 2 )$
C) $E L y ^ { \prime \prime } = w _ { 0 } \delta ( x - L / 2 )$
D) $y ^ { \prime \prime } = E l w _ { 0 } \delta ( x - L / 2 )$
E) $E L y ^ { \prime \prime \prime \prime } = w _ { 0 } \delta ( x + L / 2 )$

Question

A uniform beam of length L has a concentrated load, $w _ { 0 }$ , at $x = L / 2$ . It is embedded at the left end and simply supported at the right end. If $y ( x )$ is the vertical deflection, then the correct differential equation for y, is
A) $E L y ^ { \prime \prime \prime  \prime } = w _ { 0 } \delta ( x - L / 2 )$
B) $y ^ { \prime \prime \prime \prime } = E l w _ { 0 } \delta ( x - L / 2 )$
C) $E L y ^ { \prime \prime } = w _ { 0 } \delta ( x - L / 2 )$
D) $y ^ { \prime \prime } = E l w _ { 0 } \delta ( x - L / 2 )$
E) $E L y ^ { \prime \prime \prime \prime  } = w _ { 0 } \delta ( x + L / 2 )$

Quizplus · Accepted Answer

The correct differential equation for the vertical deflection $y(x)$ of a beam under a concentrated load is given by the Euler-Bernoulli beam equation, which in this case is $E I y^{''''} = w_0 \delta(x - L/2)$, where $E$ is the modulus of elasticity, $I$ is the moment of inertia of the beam's cross-section, and $\delta$ is the Dirac delta function representing the concentrated load. Option A correctly represents this relationship, albeit with a slight notation difference, assuming $L$ in the option might be a typo or a simplification for $EI$.

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

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