Solution to Problem 439 | Relationship Between Load, Shear, and Moment | Strength of Materials Review at MATHalino

Problem 439
A beam supported on three reactions as shown in Fig. P-439 consists of two segments joined by frictionless hinge at which the bending moment is zero.

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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 439

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$\Sigma M_H = 0$
$8R_1 = 4000(4)$
$R_1 = 2000 \, \text{lb}$

$\Sigma M_A = 0$
$8V_H = 4000(4)$
$V_H = 2000 \, \text{lb}$

$\Sigma M_D = 0$
$10R_2 = 2000(14) + 400(10)(5)$
$R_2 = 4800 \, \text{lb}$

$\Sigma M_H = 0$
$14R_3 + 4(4800) = 400(10)(9)$
$R_3 = 1200 \, \text{lb}$

To draw the Shear Diagram

V_A = 0
V_B = 2000 lb
V_B2 = 2000 - 4000 = -2000 lb
V_H = -2000 lb
V_C = -2000 lb
V_C = -2000 + 4800 = 2800 lb
V_D = 2800 - 400(10) = -1200 lb
Location of zero shear:
x / 2800 = (10 - x) / 1200
1200x = 28000 - 2800x
x = 7 ft

439-load-shear-and-moment-diagrams-continuous-beam-with-internal-hinge.gif

To draw the Moment Diagram

M_A = 0
M_B = 2000(4) = 8000 lb·ft
M_H = 8000 - 4000(2) = 0
M_C = -400(2)
M_C = -8000 lb·ft
M_x = -800 + ½ (2800)(7)
M_x = 1800 lb·ft
M_D = 1800 - ½(1200)(3)
M_D = 0
Zero M is 4 ft from R₂

Solution to Problem 439 | Relationship Between Load, Shear, and Moment

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