What is the correct expression of the internal moment in segment AB? A 10-ft-long cantilever beam is fixed at end A (x = 0). A uniform distributed load of 2 kip/ft is applied between A and B (x = 6 ft). A downward 10-kip force is applied at B, and an upward 8-kip force is applied at end C (x = 10 ft). In addition, a clockwise couple moment of 40 kip-ft is exerted at the free end C. What is the correct expression of the internal moment in segment AB? A 10-ft-long cantilever beam is fixed at end A (x = 0). A uniform distributed load of 2 kip/ft is applied between A and B (x = 6 ft). A downward 10-kip force is applied at B, and an upward 8-kip force is applied at end C (x = 10 ft). In addition, a clockwise couple moment of 40 kip-ft is exerted at the free end C. −x2 (kip-ft) −x2+24 (kip-ft) −x2−56 (kip-ft) −x2+14x−56 (kip-ft) −x2+14x (kip-ft) −x2+14x+24 (kip-ft)

Option A

Question (in proper order)

What is the correct expression of the internal moment in segment AB? A 10-ft-long cantilever beam is fixed at end A (x = 0). A uniform distributed load of 2 kip/ft is applied between A and B (x = 6 ft). A downward 10-kip force is applied at B, and an upward 8-kip force is applied at end C (x = 10 ft). In addition, a clockwise couple moment of 40 kip-ft is exerted at the free end C. What is the correct expression of the internal moment in segment AB?

(First Diagram attached below)

Explanation:

(From the second diagram attached below)

Note that

Moment of a force is the product of the force and the perpendicular distance

sum of horizontal forces, ∑Fx = 0 (that is sum of forces acting towards left = sum of forces acting towards right)

that is zero in this case

Sum of vertical forces, ∑Fy = 0 ( that is sum of forces acting upward = sum of forces acting downward)

Total upward force = Total downward force

Ay + 8 = 10 + 2 x 6

where Ay is the force acting at the fixed end

Ay + 8 = 10 + 12

Ay = 22 - 8

Ay = 14 Kip ( force is measured in kip while distance is measured in ft)

Sum of clockwise moment = Sum of anticlockwise moment

let us assume a counterclockwise moment about point A = Ma

8 x 10 + Ma = 40(counterclockwise moment) + (2 x 6) x 3(total uniform force assumed to act at the middle of the 6ft length) + 10 x 6

80 + Ma = 40 + 36 + 60

Ma = 136 - 80

Ma = 56 kip-fit

(From the third diagram attached below)

Consider a section of beam AB from A to X and of distance x fit

let the counterclockwise moment about Xx section be Mx

if we take moment about the Xx section

then

summation of moment about the Xx section ∑Mxx = 0

hence,

Mx + 56 + {(2 * x)[total of the uniform force] * x/2[acting at the middle point of the x ft distance]} = Ay * x

Mx + 56 + 2x * x/2 = 14 * x

Mx + 56 + x^2 = 14x

Mx = kip-ft