Problem: Direct Integration - Deflection Curve
Statement:
For the beam and loading shown, determine, as a function of $x$, the equations of the deflec-
tion curve, $y(x)$, and its slope, $\theta(x)$. Use the direct integration method. Your work
must show two functions with all constants solved for:
y(x), $\theta(x)$
To start the integration process, you will need to develop the function:
$\omega(x)$
What is the deflection at point C? Use our sign convention. Plot the deflected curve and
the slope.
Note that this a continuation of homework problem 15 so the determination of $\omega(x)$, $V(x)$,
and $M(x)$ using integration is shown in the solutions.
$\omega_A$
A
C
$x$
$L_C$
L
$\omega_B$
B
Find: The functions for the deflection and the slope in the beam. Quiz response is $y(L_C)$, in units
of inches.
Procedure:
1. Determine the equation $\omega(x)$ using the two known points for the distributed load.
2. Integrate $\omega(x)$ four times to get $V(x)$, $M(x)$, $\theta(x)$, and $y(x)$ with integration constants
$C_1$, $C_2$, $C_3$, $C_4$.
3. Determine and apply the boundary conditions to solve for the constants of integration.
4. Write the equations $\theta(x)$ and $y(x)$.
5. Calculate the value $y(L_C)$ and plot the functions in Excel (or another plotting software).
Solution:
The beam is made of a W8x35 A992 steel beam.
$E = 29000$ ksi, $I = 127$ in$^4$
