Question

Three struts are pinned together at A, B, and C to form a triangular

implant which will be used to support two adjacent vertebrae, as shown in
the photo. The implant is attached directly to the bones at A, B, and C. The
maximum expected stresses on the implant due to the vertebrae are labeled
in the photo: V is a vertical force and H is a horizontal force. Strut AC is
vertical and thus is parallel to force V. All angles between the struts are 60
degrees each. Typically, titanium is best used for struts under compression
and aluminum is best used for struts under tension.
a. Determine FAB, FAC, and FBC .
b. Determine the material each strut should be made of, in order to
make the best implant.
c. The forces H and V are exerted by the vertebrae attached to each
respective pin. Calculate the total force exerted by the vertebra on
pin C.

Answer 1

a. Fab = 3000 N tension

Fac = 1500 N compression

Fbc = 3000 N compression

b. Aluminum for AB, titanium for AC and BC.

c. Fc = 4000 N

Step-by-step explanation:

a. Draw free body diagrams of the pins at A and B, assuming that all internal forces are in tension (pulling away from the pin).

Sum of the forces on A in the x direction:

∑F = ma

H − Fab sin 60° = 0

Fab = H / sin 60°

Fab = 2600 / sin 60°

Fab = 3000 N

Sum of the forces on A in the y direction:

∑F = ma

-Fab cos 60° − Fac = 0

Fac = -Fab cos 60°

Fac = -H / tan 60°

Fac = -2600 / tan 60°

Fac = -1500 N (it's negative, so it's actually in compression).

Sum of the forces on B in the x direction:

∑F = ma

Fab cos 30° + Fbc cos 30° = 0

Fbc = -Fab

Fbc = -H / sin 60°

Fbc = -3000 N (it's negative, so it's in compression)

b. AB is in tension, so it should be aluminum.

AC and BC are in compression, so they should be titanium.

c. Draw a free body diagram of the entire triangle ABC. Assume there is both a horizontal and vertical reaction force at C, and assume they point in the +x and +y directions.

Sum of the forces in the x direction:

∑F = ma

Fcx + H = 0

Fcx = -H

Fcx = -2600 N

Sum of the forces in the y direction:

∑F = ma

Fcy − V = 0

Fcy = V

Fcy = 3000 N

The magnitude of the net force is:

Fc² = Fcx² + Fcy²

Fc² = (-2600 N)² + (3000 N)²

Fc = 4000 N