Curvature and its affect on strength

[trueacoustics]

Dave made a comment about curvature being a variable when determining strength of a material. I was wondering how curvature does affect strength in structures/materials. I know that "the more circular an object is the stronger it will be" is common knowledge. I am looking for something more quantitative.

I appreciate it,

Tony

 

[Studiot]

What is the context for this question?

Shape does indeed affect the strength of an object, but remember that the strength of a material forming that object is (almost) independent of its shape.

However it also depends upon the nature of the loading.

A good example would be a concrete dam which is curved into the water in plan for strength (of the structure as a whole) but the concrete itself will have exactly the same strength as flat airport runway concrete.

A bad example would be a beam which is definitely weaker in circular cross section than almost any other.

 

[trueacoustics]

The context is just how you described with the damn. Similarly, it is interesting to here that circular geometry is disadvantageous with a beam. I was interested in the relationship of a structure's shape and ability to withstand a load.

 

[Studiot]

What are your thoughts on the subject?

For instance you can bridge across a motorway with an arch bridge or a bridge formed from concrete beams.
Both will carry the same load.
Geometrically and structurally they are very different.
Can you think of any reason why?

 

[trueacoustics]

For instance you can bridge across a motorway with an arch bridge or a bridge formed from concrete beams.
Both will carry the same load.
Geometrically and structurally they are very different.
Can you think of any reason why?

I am not sure what you are asking. However, my thoughts on the subject in general are as follows. I believe circular geometries to withstand greater loads, such as the damn holding back water or a pipe through deep water holding back its pressure, because the displacement is as evenly distributed as possible; at least more so then geometries with edges or sharper bends. When looking at a circle, the tangent points are as numerous and their intersections as obtuse as possible. This means that each tangent plane is supporting the minimum amount of weight. Once these planes or angles are reduce by edges or narrowing curves, the load which each must bare is increased. Moreover, the more acute the angles, and the fewer the planes, the more concentrated and strenuous the load becomes at particular points. This can lead to a loss in structural integrity.

Just my thoughts.

 

[Studiot]

I am not sure what you are asking.

You appear to like thinking around a subject - this is great, but Physics Forums doesn't spoon feed answers. So this is meant as encouragement.

Here are some sketches to aid discussion.
The first set continues the motorway bridge theme. There are five ways shown to support load L above a span S.

1) An arch
2) A beam
3) A frame
4) A triangle
5) Despite the longer length, supports for bridges above motorways are usually raked back as shown

Each of these transfer the load L to the supports at ground level by different mechanisms

What do you think about each? In particular why do you think we rake back the supports as in (5)?

A curved shape is not the only strong shape. (1) and (4) have the same strength.

The second set shows cross sections of beams.

The further you can get the bulk of the material from the middle the stronger the beam. This is why we do not use round bars as beams. The steel is just as strong but the shape is wrong. We use $I$ sections.
Similarly if the rectangles were scaffold planks, I'm sure you have experience of which way round is stronger as the rectangles show.

 

[sophiecentaur]

Dave made a comment about curvature being a variable when determining strength of a material. I was wondering how curvature does affect strength in structures/materials. I know that "the more circular an object is the stronger it will be" is common knowledge. I am looking for something more quantitative.

I appreciate it,

Tony

Gaudi, the creative genius and designer of many fascinating buildings in Barcelona did a lot of his designs of arched structures in the Sagrada Familia 'upside down'. He used models in which he hung various masses from a network of strings. Thew would settle down in a shape in which there was only tension in the strings (which he could measure and adjust). A building (the right way up) which had similar mass distribution and which had struts in the same pattern as the strings, would have only compression in the struts and have a lot of inherent strength as there were no 'twisting' force'. This method was used earlier in building design (even Robert Hooke used it, apparently).
Here is an interesting link, about the process.

It's a method that works in 3D, and makes strong design possible without the need of a computer.

 

[trueacoustics]

You appear to like thinking around a subject - this is great, but Physics Forums doesn't spoon feed answers. So this is meant as encouragement.

I do not expect spoon feeding, I simply did not understand the question. I did however give you "my thoughts on the subject."

Each of these transfer the load L to the supports at ground level by different mechanisms.

What do you think about each? In particular why do you think we rake back the supports as in (5)?

I will first look at the "mechanism by which each bridge transfers weight to the ground." Sketch 1 combines both the transfer of the weight to the ground and the suspension of the weight above the ground into one fixed structure. This differs from the rest, and is unique to the half circle. Through this design, the weight compresses and strengthens the structure as it concentrates the load to the abutments. I would imagine 4 to be similar in strength, but not necessarily equivalent. This is because of my previous explanation. I believe 2 and 3 to be identical. Each bridge uses a horizontal plane suspended by perpendicular piers. 2 may be stronger since less emphasis is being placed on the joints. In each of these, the integrity of the supporting plane will obviously be greatly influenced by the span S. 5 is similar to this design, but the "piers" have been raked as mentioned. Since this design typically entails the mounding of Earth for use as the piers, or the removal of Earth for a recessed road, the raked line pictured is just the retaining wall. The Earth itself supports the bridge. However, since the Earth is being compressed, and since lateral pressures increase when the retaining wall is made more vertical, the retaining wall is sloped back to prevent Earth pressure build up.

The further you can get the bulk of the material from the middle the stronger the beam. This is why we do not use round bars as beams.

Can you explain this further?

I do not know if it is the same for steel, but I can speak for wood. The shear strength against the grain is incredible. However, shear strength with the grain is quite poor. Therefore, when "bridging" supports are needed with wood, and $I$ effect is used.

 

[trueacoustics]

Thanks for the link sophie.