1. Principles of Form Synthesis II
©2002 The Ohio State University
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2. Form Synthesis Principles
1. Triangle Principle
2. Tetrahedron Principle
3. Hollow Shaft principle
4. Mating Surface Principle
5. I-Beam Principle
6. Supplementary Shape Principle
7. Anti-Buckling Principle
8. Direct Path Principle
9. Force Flow Principle
10. Metal Removal Principle
11. Redundancy Avoidance Principle
12. Leverage Principle
13. Shape Merging Principle
14. Roughly Uniform Size Principle
15. Symmetry Principle.
Based on the calculations done in the previous section, it is apparent that some geometries are more efficient than others for carrying loads. These calculations apply to the body of a part where the stress patterns can become fully developed. In general, tension, compression, uniform shear, and transverse shear stress patterns are efficient in that they provide uniform stress; but bending, torsion, and spot contact patterns are inefficient because they do not provide uniform stress if solid sections are used. These facts lead to several general principles that are useful to designers. There are fifteen principles in all, and these are listed. Each of these will be discussed in detail.

3. Triangle Principle Football stadium structures Bicycle frame
Airplane landing gear
Roof trusses
Automobile frame component
As much as possible, we should attempt to use members loaded in uniform tension and compression. This will give the most efficient utilization of material. In the plane, this means that we should use triangle structures (Fig. 2.1) as much as possible. If pinned joints (momentless) are used, the bodies of the members will be in pure tension or compression assuming the loads are directed along the centroidal axes of the members. The stresses will not be uniform at the joint elements; however, these will be considered separately.
Examples of this principle are seen in football stadium structures, bicycle frames, airplane landing gear, roof trusses, and automobile frame components.

4. Tetrahedron Principle
Antenna supports
Drilling platforms
Jack stands
Wire spoked wheels
Guyed power poles
The tetrahedron principle is a spatial analog of the triangle principle as shown in Fig Here we again use 100 percent tension/compression members. In spatial structures, this means tetrahedrons (four sided structures.
Examples of the use of this principle are seen in antenna supports, frame supports, drilling platforms, jack stands, wire spoked wheels, and guyed power poles.

5. Hollow Shaft Principle
When a shaft is loaded in torsion, the most efficient shape is a round hollow shaft. Use as large a diameter as possible and as small a thickness as possible to generate the most uniform stress field. Buckling (discussed later) must be considered if the wall becomes too thin. In general, for a given area, as the diameter increases, the stresses become more uniform and lower for a given torque. This is due in part to leverage which will also be discussed later.
Examples of the use of this principle are seen in automobile drive shafts, helicopter masts, ASV drive shafts, drill pipes, and aircraft transmission shafting.

6. Mating Surface Principle
Examples:
All lower kinematic pairs (surface contact)
Journal bearings
Scotch yoke
Piston
Roller chain
Press fit on a shaft
V-belt (instead of a gear)
Wobble plate shoe
Forces are transmitted from one part to another in compression if the two parts are not rigidly connected together by some kind of welding or bonding technique. When compression is involved, the forces are transmitted most efficiently when the surfaces in contact have similar dimensions; i.e., they fit together. This reduces the Hertzian contact stresses.
Examples where the mating surface principle is applied are all lower kinematic pairs, journal bearings, scotch yokes, pistons, roller chains, press fits on a shaft, V-belts, and wobble plate shoes.

7. Mating Surface Summary
The mating surface principle is especially important when there is sliding since relative motion and high contact pressures cause and surface fatigue. A general rule is to avoid point/line contact when sliding is involved.
Some examples are given here along with the type and severity of the stresses.

8. I-Beam Principle
When a beam is subjected to bending, the material should be placed as far away from the neutral axis as possible in a thin strip for a uniform stress pattern.

9. Examples of I-Beam Principle
Some examples of the I-beam principle include wrench handles, box beams, channel sections, sandwich construction, and formed sheets.

10. Supplementary Shape Principle
A supplementary shape often can be added to provide a load path leading to a strong stress pattern
To further increase the efficiency of the body of a part, supplementary structural shapes can often be added which maximize the percentage of the material of the part which stressed in one of the strong types of stress patterns and which minimize the percentage subjected to weak patterns. These shapes generally provide a direct load path from the load to a support point. Some examples of supplementary shapes are given in the slide:
The web of the I-beam separates the flanges so that the moment of inertia of the section is increased and most of the material is placed where the stresses are highest.
Wing spars, shown in the lower right-hand corner, provide a load path which is almost pure compression. These apply the triangle principle.
By closing a cross section with weld material, the strength of the section is increased by about a factor of 50.

11. More on Supplementary Shapes
Shear plates and struts enforce the triangle principle.
The honey-comb structure separates the top and bottom of the plate surfaces so that the effective moment of inertia of the plate is increased.
The triangle principle is applied in the crane boom.

12. Anti-Buckling Principle
As sections become thin, buckling becomes possible
Buckling requires:
Compressive load
Thin or long section
Sources of Buckling
Geometry change
Material change
In the design of high performance machinery, the body of a part must often contain thin wall or small dimension cross sections which are subjected to compressive loading. Such situations invite failure by elastic instability or buckling. Buckling is a phenomenon which is a consequence of the great strength difference between various stress patterns.
There are two different sources of buckling: buckling due to geometry change and buckling due to material change.
The material change is due to yielding or a nonlinear relationship between stress and strain. That is, as the material is stressed, it becomes less stiff. With yielding there is plastic flow of the material, and there may also be a local geometry change.
In buckling, the problem is assumed to be load controlled. This means, that the original load is applied even when the geometry or material changes. If the change causes the structure to be weaker, than the continued application of the load will make the structure collapse.
Alternatively, the structure could be displacement controlled. In this case, a displacement would be applied. The load would then reduce after buckling, and collapse would not occur. This is not the usual situation in general machine design.
Buckling-like phenomena is seen in a simple tension test if the test is load controlled (the normal test is displacement controlled). As the material is loaded and necking begins, the specimen will fail quickly if the load is not reduced. If the load is reduced, the classical stress-strain curve can be produced. The phenomena is not buckling in the sense we will consider here because the tension test is by definition associated with tensile stresses while buckling is normally associated with compressive stresses.

13. Examples of Buckling
Thin beams loaded in cantilever fashion have an initial stress pattern due to bending and a buckled stress pattern corresponds to bending and torsion of a noncircular section as shown.
Circular rings or shells under external pressure have a strong stress pattern of pure compression and the weak stress pattern of compression and bending.
Shear panels have a strong stress pattern of pure shear and a weak stress pattern of bending and shear as shown.
Circular arches have a strong stress pattern of pure compression and a weak stress pattern of bending and compression.

14. More Examples of Buckling
Hollow cylinders under torsion. Here the strong stress pattern is torsion of a thin walled section, and the weak stress pattern is torsion and bending.
Web buckling in a wide flanged beam. Here the strong stress pattern is compression, and the weak stress pattern is bending. This type of buckling is possible in any wide flanged section, not just wide-flanged beams.
Wires in torsion have a strong stress pattern of pure torsion and a weak stress pattern of torsion and bending.
Thin tubes in bending have a strong stress pattern of general bending and a weak stress pattern of local bending.

15. Buckling Stiffeners
Buckling stiffeners add a small amount of material to stiffen (not necessarily strengthen) structure
Flange stiffeners
Flat plate stiffeners
Buckling must be prevented, and this can usually be done by adding some form of anti-buckling stiffener. The idea behind buckling stiffeners is to stiffen the structure so that the deformations necessary for buckling cannot occur. Note the difference between strengtheners and stiffener. Stiffeners prevent the buckling modes from being possible. Strengthers add material to improve the load carrying capacity. Stiffeners normally do not reduce the nominal stress significantly, and they may carry little or no actual load.
In most cases, a small amount of material can make a significant difference in terms of buckling prevention. Some examples of this are shown in the following.
Flange stiffeners are made by rolling over the edges of the flanges of a wide flange beam to increase the local moment of inertia. This can be seen in this example of the plastic patio chair.
Flat plate stiffeners are formed by corrugating the plate in order to increase the stiffness in the compressive direction.
Wing stiffeners are made by bonding angle stiffeners to wing skin.
Wing stiffeners

16. More Buckling Stiffeners
Tube stiffeners for bending
Tapered column
Tube stiffeners for bending increase the local stiffness to prevent local buckling. Tapered columns increase the diameter in the region of the maximum bending moment. House floor braces prevent lateral buckling of floor joists.
House floor braces

17. More Buckling Stiffeners (cont’d)
Derrick boom lacing
Use of internal pressure
Derrick boom lacing: Note that we must check buckling of each member subjected to compression. Failure can occur due to either overall buckling or buckling of individual members.
Use of internal pressure reduces the compressive stresses and therefore lessens the possibility of buckling. This procedure will work for both torsion and bending.

18. Direct-Path Principle
Put material in a straight-line path between loads and supports
This principle states that material should be in a straight line directly between points of application of forces: material should not be placed in circuitous paths.

19. Force-Flow Principle Internal forces flow like fluids in laminar flow
Stress fields in complex shapes can be visualized with respect to magnitude and direction. For the force-flow principle, establish the shape of the part so that the stress pattern is uniform tension/compression or uniform shear. Also, take advantage of leverage for the material.
Assume that the stresses act like a fluid in a channel under laminar flow. The flow stream lines will indicate the relative magnitude of the stresses (bunched up force streamlines imply high stresses). When visualizing the force flow, recognize that the forces will take the stiffest path. Note that this is also a form of the direct path principle; i.e., gradually blend one part into the other.

20. Example of force flow
We can also use flow lines as a guide to how the part should be shaped. For example, in the bracket shown, we should makes the bracket slant toward the line so that the flow lines do not have to turn a sharp corner.

21. Flow Line examples
Here are also some real examples of the force-flow principles in design.

22. Metal Removal Principle
After selecting a strong stress pattern
After visualizing the flow of forces
Remove all material where stresses are low
Consider economics, function, and manufacturing processes
In this principle, we visualize how the force flow will be in the part, and then remove any material which is under stressed. By removing the material, we will also usually improve the stress patterns.
Metal removal steps are as follows:
After selecting a strong stress pattern and after visualizing the flow of forces, remove all material where stresses are low. Then consider economics, function, and manufacturing processes.

23. Examples of Metal Removal
Note that the I-beam and hollow-shaft principles are applications of the metal removal principle.

24. Examples of Metal Removal (cont’d)
Here are a couple more examples. Note, in the sketches, that the shaded areas represent low stress regions that may be removed. This flooring grate uses metal removal for both aesthetic and functional purpose.

25. Another Example of Metal Removal Principle
Since the web is mainly in shear and this is a strong stress pattern, material may be removed from this region. This is accomplished by making the web thin. However, due to occasional and unavoidable out-of-plane loading and the fact that cast parts have a minimum thickness, the web has a minimum thickness. More material may be removed while maintaining the necessary strength by the inclusion of the holes through the web.

26. Examples of Material Removal
Here are some more examples of material removal used in everyday applications. Note how the material removal is both economical and functional.

27. Redundancy Avoidance Principle
Identify optimum load paths, and do not put in structure at other places unless absolutely necessary
Do not use multiple load paths for the same force unless done to enhance the factor of safety. Then make all paths equally efficient. Generally, redundant pieces waste material and cause excess weight. The general idea is to figure out where you want the load to flow and then to put the material there (only).

28. Load Leverage Principle
When moments or torques must be carried, regions of force application should be separated as much as possible
When moments or torques must be carried, regions of force application should be separated as much as possible to obtain large moment arms. This will give the lowest stress for a given load.

29. Load Leverage Example
It is important to make sure that the loads are property located to achieve leverage. To do this, the use of bosses and projections are sometimes required.

30. Limitations on Leverage
Weight can increase due to extra length, and supplementary shapes needed to achieve leverage
Sections can become so thin that buckling occurs
Space may be too large to achieve good leverage
Manufacturing problems caused by the introduction of leverage may make the part too expensive
Sometimes it is not possible or practical to apply leverage. Some examples where this is the case are given in the following:
1) Weight can increase due to extra length and supplementary shapes needed to achieve leverage.
2) Sections can become so thin that buckling occurs.
3) Space may be too large to achieve good leverage.
4) Manufacturing problems caused by the introduction of leverage may make the part too expensive.

31. Shape/stiffness Merging
When a large force must be accommodated and widely distributed to a general area, use shape merging
A problem which frequently arises in the design of machine parts is the necessity to accept a highly concentrated, large magnitude force or group of forces at one location and transmit these forces to another location where the forces must be much more widely distributed. Obviously, the machine part which is acted upon by such forces must be of such a shape as to accept the concentrated forces at one location and transmit them through an efficient shape merging section to a different location where forces are widely distributed.

32. General Regions Involved in Shape Merging
Compact region where there is a highly concentrated load
Merge region
Thin walled region into which the load must be transferred
In general, there are three general regions involved in shape merging. These are:
1) Start with a compact region where there is a highly concentrated load.
2) Merge region
3) Thin walled region into which the load must be transferred.

33. Types of Merging Regions
Tapered Sections (thickness or diameter changes with length)
Fan Sections (width changes with length)
In general there are four types of merging forms. These are:
Tapered Sections (thickness or diameter changes with length)
Fan Sections (width changes with length)

34. Types of Merging Regions (cont’d)
Ribs (height and possibly thickness changes with length)
Branching Ribs
Ribs (height and possibly thickness changes with length)
Branching Ribs

35. Roughly Uniform Size Principle
Adjoining portions of a part should be roughly the same size
Adjoining portions of a part should be roughly the same size. If sound shape merging ideas are used, the stresses will tend to vary with the size of the part. Note that this is a "rule of thumb" and depends a little on experience.

36. Symmetry Principle
Unless there is a reason to the contrary, make a part symmetric
Makes parts easy to machine on a lathe
Makes assembly easier
Reduces inventory needs
In practical problems, it is often not possible to determine the exact direction of the loads. In such cases, the part should be made as symmetric as possible so that it can withstand loads from any direction. Often, symmetry will lower manufacturing costs. For example, an axisymmetric part can be machined on a lathe.
When left and right handed parts are used, symmetry can reduce inventory by making one part work in both the left and right handed modes. Also, assembly time is reduced if it is not necessary to orient the part in the left-right orientation.

37. Rules of Thumb for Part Shapes
Ranking according to order of expense
Rectangular solids
Planes with general boundaries
Circular cylinders
Cones
Axisymmetric geometries
Spheres
General geometries
Some shapes are more easily manufactured than others. The following shapes are ranked in order of expense (from least costly to manufacture to most expensive).
1) Rectangular solids
2) Planes with general boundaries
3) Circular cylinders
4) Cones
5) Axisymmetric geometries
6) Spheres
7) General geometries.

38. Limits to Form Synthesis Principles
Function limitations
Space limitations
Although the stress patterns of tension, compression, and transverse shear are more efficient than those of bending, torsion, and spot contact for a machine part body, nevertheless these latter stress patterns are frequently used in good designs – even high performance machines. This can happen for the following reasons:
There are space limitations. In this example, the goal post cannot be placed on the goal line due safety concerns.
Cost limitations can often limit the design. The sign is hung from a pole – a simple and cost effective solution.
Cost limitations

39. Limits to Form Synthesis Principles (cont’d)
Manufacturing process limitations (e.g., hollow crankshaft)
Fastening constraints; for example, the mounting brackets on engine block
Appearance
Manufacturing processes could limit choices. For example, hollow shafting for crank shafts are not practical to manufacture.
There might be fastening constraints. For example, the mounting brackets on engine block transmission.
The product could be heavily governed by appearance. Such examples would be door handles and some types of chairs.

40. Examples of Limits
Here are examples of cost and space limitations in everyday applications.

41. Summary Many different principles for Form Synthesis
Use of each principle depends on the application and the driving factors
Engineers determine the appropriate principle for the “best” design
As seen in the module, there are many different applicable principles in Form synthesis. Although each of these principles are effective, their application depends on the factors and limitations of the design. Ultimately, a good engineer will determine the appropriate form synthesis principles in order to find the “best” design.

42. Credits
This module is intended as a supplement to design classes in mechanical engineering. It was developed at The Ohio State University under the NSF sponsored Gateway Coalition (grant EEC ). Contributing members include:
Gary Kinzel ……………………………………..Primary author
Walter Starkey……………Primary source of original material
Phuong Pham and Matt Detrick ……….…….. Module revisions

43. Disclaimer
This information is provided “as is” for general educational purposes; it can change over time and should be interpreted with regards to this particular circumstance. While much effort is made to provide complete information, Ohio State University and Gateway do not guarantee the accuracy and reliability of any information contained or displayed in the presentation. We disclaim any warranty, expressed or implied, including the warranties of fitness for a particular purpose. We do not assume any legal liability or responsibility for the accuracy, completeness, reliability, timeliness or usefulness of any information, or processes disclosed. Nor will Ohio State University or Gateway be held liable for any improper or incorrect use of the information described and/or contain herein and assumes no responsibility for anyone’s use of the information. Reference to any specific commercial product, process, or service by trade name, trademark, manufacture, or otherwise does not necessarily constitute or imply its endorsement.