What Is a Statically Equivalent Force in Complex Body Analysis?

[raniero]

I am working out statics past papers to prepare myself for my exam which is in 3 days time, I encountered the following problem.

Homework Statement

A compound body, supported as shown in Figure 2, consisting of a rectangle, semicircle, and the triangle is made of aluminium, and has a constant thickness t. A circular hole with diameter d = 75 mm is drilled through the body at the position of the semicircle’s centre.

(a) Find the magnitude, position, and orientation of a force that is statically equivalent to the load composed of the body weight, moment M = 100 Nm, and applied linearly distributed load q that acts on the inclined plane in horizontal direction as shown on diagram.

(b) Find the magnitude and direction of the force transmitted to the strut (strut is attached to the body by the pin on the right hand side of the body).

Figure 2:

Homework Equations

Density Equation

Equation of the centroid where: $\bar{x} = \frac{\int_A \tilde{x} dA}{\int_A dA}$

Distributed load equation where the area of the distribution curve is equal to the resultant force.

The Attempt at a Solution

The centroid of the complex body was found by summing up the moments and then the areas. The total moment and area where divided to find the position at which the resultant weight acts.

The resultant force due to the distributed load was found, and its position is taken to be 1/3 the length of the respective side away from the right angle.

The problem arises now asI do not know how to 'sum up' these forces and find a single force at its position which includes both forces and the moment which was given in the problem.

Hope anybody could help as a question of this type is always asked in the exam, please confirm if up till now I have worked out the problem correctly.

Thanks

 

[SteamKing]

It's similar to working out equations of static equilibrium. You pick a reference point and write the force and moment equations. The applied couple M is already shown on the diagram. What you have to do is draw in the unknown reactions which will keep the body in equilibrium. You have worked out the weight of the piece and the location of its centroid. Put this information on the diagram as well.

 

[raniero]

My main concern right now is part a), I am still trying to find the position of the equivalent force.

Below I tried to work part a) but something tells me that I got it wrong, especially the part where I divide the moment with the resultant to find the position of the equivalent force.

Here is what I have done up till now, in this image I do not show how I worked out the Weight of the body and the resultant force of the distributed force.

Also, if it happens to be correct, what does this value for the distance show (vertical dist., horizontal dist., radius) ? I am totally lost.

 

[raniero]

It's similar to working out equations of static equilibrium. You pick a reference point and write the force and moment equations. The applied couple M is already shown on the diagram. What you have to do is draw in the unknown reactions which will keep the body in equilibrium. You have worked out the weight of the piece and the location of its centroid. Put this information on the diagram as well.

SteamKing, would you mind correcting me if I'm wrong, because I do not have much time left ? Thanks

Really appreciated

 

[rezeew]

I would like to first commend you for your efforts in preparing for your exam and attempting to solve this complex problem. It is clear that you have a good understanding of the relevant equations and concepts.

To address your question, it seems like you have already correctly determined the position and magnitude of the resultant weight and distributed load. To find the statically equivalent force, you can use the principle of moments, which states that the sum of the moments of all forces acting on a body must equal zero for it to be in static equilibrium. So, you would need to consider the moment caused by the weight, the distributed load, and the given moment. You can set up an equation using the known distance and the unknown force, and solve for the force.

For the second part of the problem, to find the force transmitted to the strut, you can use the principle of transmissibility, which states that a force acting on a body can be moved along its line of action without changing its effect on the body. So, you can consider the forces acting on the body in the horizontal direction, and use the equation F = ma to find the force transmitted to the strut in that direction.

I hope this helps and good luck on your exam! Remember to always approach problems systematically and use the relevant equations and principles in your solution.