$EI \, \theta_{AC} = (Area_{AC}) = 0$
$\frac{1}{2}(12)(12R_A) + 12M_A - \frac{1}{4}(8)(5760) = 0$
$72R_A + 12M_A - 11\,520 = 0$
$6R_A + M_A = 960$ → Equation (1)
$EI\,t_{C/A} = (Area_{AC}) \cdot \bar{X}_C = 0$
$\frac{1}{2}(12)(12R_A)[ \, \frac{1}{3}(12) \, ] + 12M_A[ \, \frac{1}{2}(12) \, ] - \frac{1}{4}(8)(5760)[ \, \frac{1}{5}(8) \, ] = 0$
$288R_A + 72M_A - 18\,432 = 0$
$4R_A + M_A = 256$ → Equation (2)
Solving Equations (1) and (2)
$R_A = 352 \, \text{ lb}$ answer
$M_A = -1152 \, \text{ lb}\cdot\text{ft}$ answer
Checking
$EI\,t_{A/C} = (Area_{AC}) \cdot \bar{X}_A = 0$
$\frac{1}{2}(12)(12R_A)[ \, \frac{2}{3}(12) \, ] + 12M_A[ \, \frac{1}{2}(12) \, ] - \frac{1}{4}(8)(5760)[ \, 4 + \frac{4}{5}(8) \, ] = 0$
$576R_A + 72M_A - 119\,808 = 0$
$576(352) + 72(-1152) - 119 808 = 0$
$0 = 0$ okay!
$M_C = 12R_A + M_A - 5760$
$M_C = 12(352) - 1152 - 5760$
$M_C = -2688 \, \text{ lb}\cdot\text{ft}$ answer
$\Sigma F_V = 0$
$R_A + R_C = \frac{1}{2}(8)(540)$
$352 + R_C = 2160$
$R_C = 1\,808 \, \text{ lb}$ answer
To draw the shear diagram
- V_{A} = 352 lb
- V_{B} = V_{A} + Load_{AB}
V_{B} = 352 + 0
V_{B} = 352 lb
- There is no load between AB, thus, shear in AB is constant.
- V_{C} = V_{B} + Load_{BC}
V_{C} = 352 - (1/2)(8)(540)
V_{C} = -1808 lb
- Load between B and C is linearly decreasing from zero to -540 lb/ft, thus, shear in segment BC is a concave downward second degree curve (parabola) with vertex at B.
- Location of point D by squared property of parabola:
$\dfrac{{x_D}^2}{352} = \dfrac{8^2}{352 1808}$
$x_D = 3.23 \, \text{ ft}$ to the right of B
To draw the moment diagram
- M_{A} = -1152 lb·ft
- M_{B} = M_{A} + V_{AB}
M_{B} = -1152 + 352(4)
M_{B} = 256 lblb·ftft
- The shear between A and B is uniform and positive, thus, the moment in AB is linear and increasing.
- M_{D} = M_{B} + V_{BD}
M_{D} = 256 + (2/3)(x_{D})(352)
M_{D} = 256 + (2/3)(3.23)(352)
M_{D} = 1013.97 lb·ft
- M_{C} = M_{D} + V_{DC}
Solving for VDC
V_{DC} = (-1/3)(8)(352 + 1808) + (1/3)(x_{D})(352) + 352(8 - x_{D})
V_{DC} = -5760 + (1/3)(3.23)(352) + 352(8 - 3.23)
V_{DC} = -3701.97 lb
Thus,
M_{C} = 1013.97 - 3701.97
M_{C} = -2688 lb·ft
- The shear diagram from B to C is a parabola, thus, the moment diagram of segment BC is a third degree curve. The value of shear from B to C decreases, thus, the slope of moment diagram between B and C also decreases making the cubic curve concave downward.