Beam on Elastic Foundation
This program provides analysis of solid beams continuously supported by an elastic material. Typical applications are for concrete beams supporting uniform and concentrated building loads, transferring the loads to the underlying soil.
This program is based upon the elastic beam formulas presented in Formulas for Stress and Strain, 5th Ed., by Raymond J. Roark and Warren C. Young (Article 7.5 and Table 7 & 8) and Beams on Elastic Foundation by M. Hetenyi, University of Michigan Press, 1946.
NOTE! THIS PROGRAM ASSUMES ELASTIC BEAM IS ALWAYS IN COMPRESSION. No provision is made when the beam has upward deflection. In this instance the beam is actually "pulling" the soil upward instead ignoring the soil/beam interaction. This is so because of the nature of the equations used.
The established formulas have been formulated into computer code that is used by this program, which can analyze beams considering:
|•||Left end free, guided, pinned, or free.|
|•||Right end free, guided, pinned, free, or infinite.|
|•||Up to 7 uniform loads, 11 concentrated loads, and 5 applied moments.|
Assumptions and Limitations
The flexibility of the beam in relation to the spring constant of the soil is limited. In the program you will see the item "Beta * Length". The value Beta is a measure of the beams flexibility and is equal to:
(Width * Subgrade Modulus / (4 * EI) ) ^1/4
According to the reference text when the value Beta * Length > 6.0 the beam is so flexible that the behavior changes. In this case the program displays a message and no results are given.
The data entry for this example is shown in the screen captures that accompany the Data Entry Tabs and Results & Graphics Tabs sections to follow.
Data Entry Tabs
This set of tabs provides entries for all input in this calculation. While you are entering data and switching between these tabs you can view the desired resulting information on the tabs on the right-hand side of the screen (calculated values, sketches, diagrams, etc.). A recalculation is performed after any entry data is changed. After each data entry you can view the results on the right-hand set of tabs.
This tab collects all the analysis information except loading.
Enter the length of the beam between end supports.
Depth & Width
Enter the beam depth and width to be used for calculation of moment of inertia.
Free : Indicates the beam end has no vertical, horizontal, or rotational restraint.
Guided : Indicates the beam end has horizontal restraint, but is not allowed to rotate or move horizontally.
Pinned : Indicates the beam end is free to rotate, but cannot translate vertically or horizontally.
Fixed : Indicates the beam end is fully restrained against vertical and horizontal translation and cannot rotate.
Elastic modulus of the beam's material. Steel is 29,000 ksi; concrete is 57,000 * sqrt(f'c)
The compressive modulus (commonly called the K-Value) of the supporting material. A Soil Engineer, based upon field testing of the soil typically supplies this value. The units for this number are pounds per square inch per inch of deflection, or just an ordinary spring constant.
Calculated using Width * Depth3 / 12 for rectangular sections
Beta * Length
Beta is a measure of the difference in flexural stiffness between the beam and foundation. beta = (Beam Width*Subgrade Modulus) / ( 4.0 * E * I)) ^ .25. See the major section "Assumptions & Limitations" for more information.
These entries define load factors to be applied to the loads entered on the next three tabs. You can use these to build ACI type factored load combinations for the analysis run. There is one load factor for Dead, Live and Short Term loads. The "Overall" factor is applied to the summation of the three.
The "Current Load Combination" selection tells the program which loads to use. In our example you can see that "Dead, Live & Short Term Loads" has been selected. This means that all loads of all types will have the factors applied to them and then the overall factor applied. In this example the final load applied to the beam is:
(1.40 * DL + 1.70 * LL * 1.55 * ST) * 0.83
Uniform Loads Tab
This load is a uniform intensity load applied from SLoc (start distance) to ELoc (end distance). The values for dead, live, and short-term loads are combined according to Load Combination. If ELoc is specified greater than Span (except for an Infinite right support), then the excess distance is ignored.
Point Loads Tab
Up to 11 point loads can be applied to the beam, with the dead, live, and short-term components combined according to Load Combination. If the ELoc distance is specified greater that Span (except for Infinite right supports), the load is ignored.
The user may apply up to 5 concentrated moments at any location on the beam. The sign convention follows the right hand rule, where a positive moment applies a torque to the beam in a counter-clockwise direction.
Results & Graphics Tabs
This set of tabs provides the calculated values resulting from your input on the "Data Entry Tabs". Because a recalculation is performed with each data entry, the information on these tabs always reflects the accurate and current results, problem sketch, or stress/deflection diagram.
Shear ( Reactions )
Maximum positive and negative shears and the locations where they occur are given by checking the span at 250th points.
Maximum positive and negative moments and the locations where they occur are given by checking the span at 250th points.
Maximum positive and negative rotations and the locations where they occur are given by checking the span at 250th points.
Maximum positive and negative deflections and the locations where they occur are given by checking the span at 250th points.
Using the deflection values given above and multiplying by the subgrade modulus gives the soil pressures. The basis of the calculation is (Spring Force * Distance) = Force. Maximum positive and negative soil pressures and the locations where they occur are given by checking the span at 250th points.
Values @ Beam Ends
|•||Reactions Ra and Rb (left and right supports) are given for ends which have Pinned and Fixed support restraints.|
|•||Rotations at Left and Right (left and right ends supports) are given for ends which have Free, Pinned, and Infinite support restraints.|
|•||Moments Ma and Mb (left and right ends supports) are given for ends which have Guided and Fixed support restraints.|
|•||Deflections Da and Db (left and right ends supports) are given for ends which have Free and Guided support restraints.|
This tab provides a sketch of the beam with loads and resulting values shown. Using the [Print Sketch] button will print the sketch in large scale on a single sheet of paper.
This displays a moment, shear, and deflection diagram for the beam with the applied loads and end conditions. Note the two tabs...."Graphic Diagram" and "Data Table". The Data Table tab provides the entire internal analysis at the 1/500th points within the beam.
This tab allows you to control which areas of the calculation to print. Checking a box will signal that the information described by the item will be printed. However, if there is no information in for a particular selection it will not be printed. So these checkboxes are best described as "If this particular area of the calculations contains data then print it".