Question

Consider the same general setup as in Problem 1, but with an additional distributed load along the beam. This distributed load is described by the function

G(x)=−(3x 2 ) j^ kN/m,x∈[0,L]
where the boldface type indicates that G is a vector. a) Draw an FBD of this system with the distributed load. b) The distributed load is measured to be−27^kN/m at the end of the beam. Find the magnitude and line of action of the equivalent resultant force due to the distributed load G(x).
c) Draw thẻ FBD of this beam using thẻ équivalēnt rẻsultant forcẻ you found in part (c). d) If the force applied to the end of the rope is F=17.5kN, solve for the vertical reaction force at the support (Ry)and the mass of the beam per-unit-meter (m x). Take the magnitude of the acceleration due to gravity to beg−9.81 m/s 2

Answer 1

Step-by-step explanation:

a) To draw the FBD of the system with the distributed load, we can represent the beam as a simple object with length L and represent the distributed load as a force acting along the length of the beam. The force acting on the beam due to the distributed load is G(x).

b) The equivalent resultant force due to the distributed load G(x) can be found by integrating the load along the length of the beam:

G_equiv = ∫ G(x) dx = ∫ -3x^2 j^ dx

= -x^3 j^ |x=L

= -L^3 j^

The magnitude of the equivalent force is |G_equiv| = L^3, and the line of action is j^ (downwards).

c) To draw the FBD of the beam using the equivalent resultant force, we can represent the beam as a simple object with length L and represent the equivalent force as a single force acting at the end of the beam.

d) The force applied to the end of the rope is F=17.5kN. The vertical reaction force at the support, Ry, can be found using the equation of static equilibrium:

ΣFy = Ry - G_equiv - F = 0

Ry = G_equiv + F = -L^3 j^ + 17.5 kN j^

The mass of the beam per-unit-meter, mx, can be found using the equation of motion:

ΣFy = mx * g

mx = (Ry - F) / g = (-L^3 + 17.5) / -9.81 = (17.5 + L^3) / 9.81 kg/m

Note that the units of L and g must be consistent.