Furniture Analysis problem - internal moments

[arokisbobcat]

Homework Statement

Find the internal moments on the stretchers of the chair.

Relevant Equations

Sum of Moments = 0. Forces X and Y sum to zero.

In the attached document, I am attempting the problem that they go through in Figure 4-25, Force Equilibrium in side frames. A chair with 3 stretchers is considered as having a 200 lbf. person sitting on it and a 100 lbf. applied to the top edge of the chair creating an overturning moment. I am attempting this problem from some furniture design literature from Purdue engineering. I am having a couple of questions and am seeking some help.

Here is my first question. If the 'seat load' force is 200 lbf, then wouldn't the reactions be 100 lbf. on the front leg (downward) and 300 lbf. on the rear leg (upward)? Please see my attachment labeled 'Furniture Reactions'.

My second question is how the author arrives at the internal moments on each of the 3 stretchers from top to bottom of 544 / 571 in-lb, 526 / 525 in-lb, and 513 / 521 in-lb. In my attachment labeled 'Problem', I cut the member at the front leg and take moments about each of the stretchers. Unfortunately, they equal out and I can not see how they author is getting these internal moment values stated above.

Attached are my attempts and also the literature of the stated problem in Figure 4-25 beginning on page 4-20 in the Furniture Analysis attachment. Any help would be appreciated to help me see what I am missing.

 

[erobz]

Here is my first question. If the 'seat load' force is 200 lbf, then wouldn't the reactions be 100 lbf. on the front leg (downward) and 300 lbf. on the rear leg (upward)? Please see my attachment labeled 'Furniture Reactions'.

The math looks fine, but it's not realistic. Tension isn't going to be developed in the front leg unless the chair is pinned to the ground there. The rear constraint can be met by normal forces and static friction, but basically static analysis is telling me that a "functionally normal chair" under this load is tipping.

As for the rest of the stuff, I would suspect they are approximations. The structure is statically indeterminant with all the redundant bracing, so to get internal forces/moments you have to model deflections.

 

[256bits]

Homework Statement: Find the internal moments on the stretchers of the chair.
Relevant Equations: Sum of Moments = 0. Forces X and Y sum to zero.

Here is my first question. If the 'seat load' force is 200 lbf, then wouldn't the reactions be 100 lbf. on the front leg (downward) and 300 lbf. on the rear leg (upward)? Please see my attachment labeled 'Furniture Reactions'.

Is the chair glued to the floor? Most are not, so a reaction force downwards on the front leg from the floor seems to be questionable.

Really, the book text is difficult reading, and not explicit upon what forces are acting upon the chair and where. Fv and F_H were previously explained to be external force nature, but Fv seems to be thrown around willy nilly, sometimes looking as a resultant in the diagrams, which, by the way, are lacking a distinct labelling as diagram a, b, c,...
( And if it is the side frame should not the weight of the person be cut in half , maybe maybe not. )

Anyways, the person sitting down in a chair encounters the front first, so perhaps all the 200 # is to be downwards upon the front leg(s), and not in the centre of the chair. And leaning back gives the horizontal force upon the chair back.

I think you might have to experiment as to where the external forces are being applied upon the chair.

 

[Lnewqban]

Here is my first question. If the 'seat load' force is 200 lbf, then wouldn't the reactions be 100 lbf. on the front leg (downward) and 300 lbf. on the rear leg (upward)? Please see my attachment labeled 'Furniture Reactions'.

The summation of vertical forces must be zero.
If the man weights 200 lbf, the four legs are supporting certain percentage of that force, which summation must be 200 lbf.

If the load is applied at the geometrical center of the horizontal quadrilateral shape formed by the ends of the legs, each will be exerting a vertical resistive force of 50 lbf.

Note that the diagrams shown in the attached Furniture Analysis document do not represent a regular chair, which legs can be freely lifted and slid over the floor.

The ends of the legs are represented to be connected to the floor by pins, rollers and simple supports, which seems to be used for teaching purposes of internal reactions in a structure (called chairs in this case).
The ends of the legs of most regular chairs are simple supports.

What seems to be confusing in Figure 4-25 is that both supports "1" are represented as rollers, which is indicating that the ends of that pair of legs are able to resist downwards and upward forces, but no moments or horizontal forces.

Both supports "2" are represented as pins, which is indicating that the ends of that pair of legs are able to resist vertical and horizontal forces, but no moments.
Therefore, the only forces resisting the externally applied horizontal forces appear in those supports "2", and none in "1".

Please, study the difference here:
https://skyciv.com/docs/tutorials/beam-tutorials/types-of-supports-in-structural-analysis/