Odd shaped footings 2

[akastud]

Does anyone know of any programs that will give the soil pressures of a footing that is any shape other than rectangular. Specifically I am trying to calculate the maximum soil pressure on a triangular footing where the eccentricity is 'outside the kern'. The resultant soil pressure shape is very odd, and makes it difficult to find the centroid, and even harder to find the resulting pressure once the centroid has been matched with the eccentricity.

 

[RSMENGR]

Dear AkaStud:

You can try using plate on elastic medium analogy. SAFE is one such program which can be used effectively. Though other softwares may also do the same job. See Joseph Bowles book for such a modelling.

Regards

RSMENGR

 

[ishvaaag]

Certainly you can use a modulus of subgrade reaction in combination with a model of the footing to derive compression only spring constants for the component plate elements. Hence you will obtain resulting pressures in equilibrium with the forces (that you may enter with care with the usual ssumptions at the column position, i.e., through equivalent forces on nodes there or something alike).

For Isosceles triangles in straight bending the plane of its main axis you can easily as well

1) assume a plastic distribution and use the trapeze load to equilibrate your eccentric force upon succesive trials or using the equations if determinate


or

2) if you accept a linear distribution of the stress (say triangular since outside the kern) write an small program that integrates the tentative pressure stresses, then give the center of gravity of same, and then match with the point of application of the eccentrical column force in both strength and position (this kind of things programs like Mathcad can easily do).

The Mathcad approach is as well feasible for any shape of triangle and a footing with the load eccentrical in both directions, and is mathematically very tantamount to the determination of the normal elastical stresses and axis of bending for a section in equilibrium with eccentrical flexocompression.

 

[lippie]

STAAD.pro has an automatic spring support generator for exactly what you are looking for.

 

[Bbird]

There are two approaches being discussed here.

(1) Rigid foundation - The foundation is assmed rigid and soil is elastic but does not offer any tensile resistance. When a bending is applied together with an axial load the tensile zone is assumed to have no resistance contribution. The standard method of calculation is the normal stress equation.

stress=axial load/ares+-moment/I-value*neutral distance to stress point.

(2) Elastic foundation - The foundation has a finite stiffness and can bent with the soil. The latter is modeled as springs according to the subgrade reaction in a computer simulation.

I believe akastud could be interested in the first approach although most of the posters offer solutions for the latter.

My post is intended for the first approach only.

One can calculate any geomertical shape of a foundation subjected to a combination of axial load and biaxial bending. In foundation design the biaxial bending is usually computed as a resultant applied at an angle to the othorgonal axes when performed by hand calculation.

The technique is known as the Saville variation and based on the full stress equation which has the product moment of areas Ixy. In stress analysis we normally have a section symmetrical about at least one axis and so the Ixy vanishes but the actual equation is more complicated. This equation is described in the earlier versions of JE Bowles book on "Foundation analysis and design" with an iterative method given. For completion I list it out as

stress=axial load/area +or- (Mxx-Myy*Ixy/Ixx)*x/(Ixx-Ixy*Ixy/Ixx) +or- (Myy-Mxx*Ixy/Iyy)*y/(Iyy-Ixy*Ixy/Iyy)

note : The Ixx, Iyy and Ixy refer to the centroidal axes of the stressed section, i.e. with tensile area already deducted. +or- depending on whether the applied moment is in the same direction of chosen sign convention.

With computer, the full stress equation can be solved automatically by coordinate geometery. Since the stress equation is in the form of stress=A*x+B*y+C, where A, B and C are functions of section properties of the stressed area of the bearing surface (area, Ixx, Iyy, Ixy), moments (Mxx and Myy) and axial load, by equating stress=0 the stress equation becomes the a straight line equation for the neutral axis of the form y=mx+c, where m is the slope and c is the intercept with the y axis.

The algorithm is started simply by any guess of the neutral axis. The section properties are calculated and straight line equation is solved to obtain an improved neutral axis position (slope and intercept). The computation is stopped once the improvement is small enough to satisfy a specified tolerance. The procedure is inherently stable and always converges.

The difficult part of the algorithm is the computation of the section properties and the standard technique used is to arrange the irregular section as a polygon with a series of triangles. The triangular elements'section propertis are summed together globally and if needed transferred back to centroidal axes, based on which the stress equation holds true.

The section properties of a triangular element, with respect to its own centroid and having nodes 1, 2 & 3 are

area=0.5[x1(y2-y3)+x2(y3-y1)+x3(y1-y2)
xbar (mean x) =1/3(x1+x2+x3)
ybar (mean y) =1/3(y1+y2+y3)
Ixx=area/12[y1*y1+y2*y2+y3*y3]
Iyy=area/12[x1*x1+x2*x2+x3*x3]
Ixy=area/12[x1*y1+x2*y2+x3*y3]

When transferring from centroidal to global axes

Ixx(global)=Ixx(centroid)+area*ybar*ybar etc

By noding the perimeter consecutively in anti-clockwise direction and voids in clockwise direction, with imaginery cut introduced as necessary, the above described procedure works satisfactorily for any polygon and suitable for any geometrical shape foundation.

The above procedure forms the basis of elastic design of a nonhomogeneous material (i.e. compression elastic but zero tension). It is widely used in the old days (about 20 years ago!) for the elastic method design for reinforced concrete where the steel reinforcement is transformed into equivalent areas multipied by the modulus ratio.

 

[akastud]

Thank you BBird, I was looking for the rigid analysis, and I understand the approach. Wish me luck applying it!

 

[Bbird]

akastud,

Just a tip.

In my convention I use Mxx anticlockwise and Myy clockwise as positive moments.

The algorithm ususally needs about 5 to 6 iterations. I have used it to reproduce published design charts for rectangular footings. The theory can even be expanded for inelastic stress-strain curve and in limit state design for reinforced concrete (two materials concrete and steel instead of just one material (soil) for foundation.

In the early days when ACI 307 (on chimney design) was still based on elastic theory with really complex equations derived from thin shell theory (reinforcement being a thin steel shell within the concrete windshield). I used the above procedure to independently verify the computerised ACI 307 calculations. A chimney cross section can have several unsymmetrical penetrations in the shell and should surely qualify for the term "odd shape".

The beauty of the procedure is that it is theoretically exact, based on the fundamental law of equilibrium (stress equation), applicable to all sections irrespective it is a beam or a column or a foundation (can have zero in any of the axial load P, Mxx and Myy), able to cope with any number of materials and works like a treat on any complicated shape.

 

[RSMENGR]

Dear AkaStud:

There is a spreadsheet for footing design on

http://www.sefindia.org

Try to see if it works for you.

Regards

RSMENGR

 

[akastud]

rsmengr,

The spreadsheet does the same as all other programs I have encountered, it limits you to a rectangular footing, however it looks like a very good spreadsheet for what it does, and if anyone has a similar spreadsheet with english units, I would love to have it.