# Solution to Problem 442 | Relationship Between Load, Shear, and Moment

**Problem 442**

Beam carrying the uniformly varying load shown in Fig. P-442.

Without writing shear and moment equations, draw the shear and moment diagrams for the beams specified in the following problems. Give numerical values at all change of loading positions and at all points of zero shear.

**Solution 442**

$\Sigma M_{R2} = 0$

$LR_1 = \frac{1}{3}L \,(\frac{1}{2}Lw_o)$

$R_1 = \frac{1}{6}Lw_o$

$LR_1 = \frac{1}{3}L \,(\frac{1}{2}Lw_o)$

$R_1 = \frac{1}{6}Lw_o$

$\Sigma M_{R1} = 0$

$LR_2 = \frac{2}{3}L \, (\frac{1}{2}Lw_o)$

$R_2 = \frac{1}{3}Lw_o$

**To draw the Shear Diagram**

- V
_{A}= R_{1}= 1/6 Lw_{o} - V
_{B}= V_{A}+ Area in load diagram

V_{B}= 1/6 Lw_{o}- 1/2 Lw_{o}

V_{B}= -1/3 Lw_{o} **Location of zero shear C:**

By squared property of parabola:

x^{2}/ (1/6 Lw_{o}) = L^{2}/ (1/6 Lw_{o}+ 1/3 Lw_{o})

6x^{2}= 2L^{2}

x = L / √3- The shear in AB is a parabola with vertex at A, the starting point of uniformly varying load. The load in AB is 0 at A to downward wo or -w
_{o}at B, thus the slope of shear diagram is decreasing. For decreasing slope, the parabola is open downward.

**To draw the Moment Diagram**

- M
_{A}= 0 - M
_{C}= M_{A}+ Area in shear diagram

M_{C}= 0 + 2/3 (L/√3)(1/6 Lw_{o})

M_{C}= 0.06415L^{2}w_{o}= M_{max} - M
_{B}= M_{C}+ Area in shear diagram

M_{B}= M_{C}- A_{1}(see figure for solving A_{1})**For A**:_{1}

A_{1}= 1/3 L(1/6 Lw_{o}+ 1/3 Lw_{o}) - 1/3 (L/√3)(1/6 Lw_{o}) - 1/6 Lw_{o}(L - L/√3)

A_{1}= 0.16667L^{2}w_{o}- 0.03208L^{2}w_{o}- 0.07044L^{2}w_{o}

A_{1}= 0.06415L^{2}w_{o}

M_{B}= 0.06415L^{2}w_{o}- 0.06415L^{2}w_{o}= 0 - The shear diagram is second degree curve, thus the moment diagram is a third degree curve. The maximum moment (highest point) occurred at C, the location of zero shear. The value of shears in AC is positive then the moment in AC is increasing; at CB the shear is negative, then the moment in CB is decreasing.