# Statical equilibrium in 2D

Consider the following problem from Hibbeler's Engineering Mechanics, Statics. (13th edition)

and the associated free body diagram and equilibrium equations

The problem is solved by equating $$N_B$$ to zero. Let's say I was asked to determine all support reactions and the force in cable CD (as in the picture, not requiring the ramp to just start being lifted) It seems that 3 equilibrium equations are not sufficient, as there are 4 unknowns $$F\, ,N_B\,,A_x\,and\, A_y$$

What renders this statics problem unsolvable, is it statically indeterminate?

• I find it easier to imagine the floor has vanished - this is just a pinned beam (which has a moment about A linked to the horizontal distance of its COG from A) being held up by a rope with moment liked to the perpendicular distance from rope to A. That will allow you to calculate the tension T. – Jonathan R Swift Jan 4 '20 at 10:40
• Also, your question states that $N_B$ is an unknown - it's not unknown, it's Zero! – Jonathan R Swift Jan 4 '20 at 10:42
• @JonathanRSwift The question in the book says $N_B=0$ but anyway if the question asked to determine all support reactions (at A and at B) would the problem be statically indeterminate. That's what I'm asking. I understand your point, original question tells us $N_B=0$ – Ali Kıral Jan 4 '20 at 12:22

Your question is a bit confusing. If the ramp is resting on the pier at point B They usually relax the cable with zero force to allow for the ramp to move up and down a little to accommodate the gentle swell of the water.

However, theoretically, if the cable is in tension and there is a reaction at B as well this will be an indeterminate structure and one would need to consider section properties and stresses of the cable and the ramp beam as well.

Also, the entire ramp would vibrate with a forced principal mode shape passing the two supports at C and A if the cable is loaded with tension.

• In the question, the cable is in tension, but there is not a reaction at B - so it's not indeterminate – Jonathan R Swift Jan 4 '20 at 10:38

Imagine there is no rope - determinate. Imagine there is no floor - determinate.

As you gradually increase tension in the rope the reaction force at B will gradually fall as the tension rises, until it reaches zero. If the tension rises further it will start to move. So, you see the reaction force is directly driven by the tension, provided the tension is with the range between zero and when the bridge starts to lift.

So yes, it's indeterminate in this interim state, but there is not any reason to find the static equilibrium forces in this state since the worst case forces are in the two determinate states that I described above.

• Nice explanation! – Ali Kıral Jan 4 '20 at 16:31