#### This page has been superseded by Load Cases

**Load Cases**

The main computation facilities are provided in an entity called a “load case”. Each cross-section can have several load cases associated with it. Each load case is intended to represent a particular situation in the life of the structural member that is of interest. Such points of interest might be the ultimate strength, or the distortion under a known loading.

Each load case has an introductory dialog that has provision for:

**A name:** to identify the load case.

**Reference axes**: Associated with each load case is a set of reference axes that are part of the description of what is of interest. The orthogonal axes in the plane of the cross-section used to describe the cross-section are referred to as “the x, y axes”. The axis of interest in a particular analysis may be anywhere on the plane of the cross-section. To accommodate this a second set of orthogonal axes is used. The axis of interest is referred to as the “**reference axis**” and the axis at right angles to that axis as the “**other axis**“.

The dialog has provision for the x and y of the origin and the orientation angle of the references axis all expressed in terms of the global x, y axes.

**Check box – “Account for concrete displaced by bars**”: It is intended that this will normally be checked. This can be uncheck to obtain results for comparison with results from other computation facilities that do not account for the concrete displaced by the reinforcing.

**Check box – “Curvature about other axis restrained**”: This relates to the circumstances of the structural member. It can be important in computations involving trial and adjustment. Unchecking this box adds a degree of freedom to the trial and adjustment process. Usually the expression “biaxial analysis” refers to an unrestrained case.

**Maximum iterations: **A number edit box labeled “Maximum iterations =”. Although this is on a specific load case dialog it applies to all load-cases and all methods that use trail and adjustment. It is intended that this should normally not affect the computation. It is intended it should come into effect only when something is wrong. It is a limit on the total number of iterations as represented by the iteration count numbers in the left most column on the computation progress dialog. When this number is reached the computations stop and the close button on the computation progress dialog is enabled.

There is a separate iteration count limit that applies only to the trial and adjustment on the curvature. This is in the inner most loop of the nested trial and adjustment routines used for two and three degrees of freedom. The count is reset to one each time that inner-most loop is invoked. The limit is set at 150 and cannot be changed by the user. Experience so far is that whenever this limit is reached something is wrong.

A suggested value for the overall maximum iterations represented on the Load-case dialog is 2000. However this may not be appropriate in all situations.

**Computation method =** : leads to a drop-down list of “methods”. Several different computation facilities are provided each of which is called a “method”.

Only one method can be current in a load case. However the current method can be changed at any time by a simple selection from the drop-down list so that a combination of the methods can be used in an investigation of the situation the load case represents.

Note: at this stage in the development the “Given bending moment” method has been implemented only for the unrestrained case.

**Compute – button: **

The Compute – button leads to a computation dialog although it also invokes a check on the geometry of the cross-section. If a problem is found a dialog with a description of the problem appears instead.

There is a different computation dialog for each method. Generally there is a panel for the input information near the top. Under that is a compute button and near the bottom is a panel for the results.

**Display result – button:**

The Display result – button leads to a dialog where the result is displayed as text and there are facilities for writing that text to a file.

The result is first displayed as values for the six variables in terms of the reference axes. These distortion values represent only the Load-case distortion.

This is followed by the bending moment expressed in polar co-ordinates. This is the bending moment with the axial load at the origin of the references axes although the orientation angle is in terms of the global X, Y axes.

Then, if there is an axial load the cartesian co-ordinates of the axial load in terms of the global X, Y axes.

This is followed by an account for the axial force in the various kinds of materials. Here the expression “mild steel” means “steel that yields”.

Finally there is a display of the extreme strains in each component. These strains are the “stress strains”. (see “Distortion” – “Stage, Other, stress and Load-case distortion defined”.)

**The explicit algorithm**

All the different methods use the same explicit algorithm. This works with six variables;

three representing the loading; being:

- The axial load at the origin of the reference axes.
- The bending moment about the reference axis
- The bending moment about the other axis

and three representing the distortion; being:

- The axial strain at the origin of the reference axes.
- The curvature about the reference axis
- The curvature about the other axis.

This explicit algorithm takes as input the three distortion variables and outputs the three loading variables. For a description of this algorithm see *Thompson, P. J. Discussion of “Ultimate Strength Domain of Reinforced Concrete Sections under Biaxial Bending and Axial Load”, ACI Structural Journal, Nov-Dec 2013 pp 1109-1112.*

**Only Load-Case distortions displayed on the computation dialogs**

The distortion variables that appear on the computation dialogs, and the computation progress dialogs represent only the “load case” distortions.

The “stage” distortion and “other” distortions of each component are automatically taken into account so that the explicit algorithm works with the Stress Distortions.

In contrast the loading values output by the explicit algorithm and as displayed on these dialogs represent the total loading on the cross-section.

Note that the load case distortion obeys plane-sections-remain-plane over the whole cross-section.

**Methods that do not provide automatic trial and adjustment.**

Two of the methods simply invoke the explicit algorithm but require manual trial and adjustment to assess the affect of a given loading. With one the input is simply values for the three distortion variable. With the other the input is the location of three points and values for the strain at those three points. These are intended for detailed studies of a specific situation.

**Utility: Curvatures and axial strain** is a method that simply invokes the explicit algorithm: the input is values for the three distortion variables and the output is the three loading variables.

This method can be used with manual trial and adjustment of the distortion variables to find the distortion cause by a given loading.

The dialog for this method includes a button labeled “**Set input to current case**” that has a purpose when other methods are used in combination. The methods that provide automatic trial and adjustment have some of the loading variables in the input and some of the distortion variables are in the output. To accommodate this memory locations for two sets of the six variables are used; one set of six for the input and another set of six for the output. This button copies the three distortion variables from the output set to the input set.

**Utility: Strain at three points** is another method that simply invokes the explicit algorithm. The input is the location of three points and values for the strain at those three points.

This method may be seen as more convenient for manual trial and adjustment.

It is required that the three points are not in a straight line. The application automatically checks for this and can give an error message. An example of three points that are not in a straight line are the two concrete corners on one face of a rectangular column and the centre of a reinforcing bar that is in the opposite face of the column.

It is not necessary each point is a corner of the concrete or the centre of a bar. Nevertheless there is a facility for choosing a point if it is to be a corner of the concrete or the centre of a bar. The “**choose**” button opens a dialog where a component can be chosen from a drop-down list and then a specific location chosen from the table description of that component.

If the strain at a particular point is to be related to the stress-strain relationship of the material then the “other” distortions and the “stage” distortion need to be taken into account. Examples to illustrate this requirement are:

- where the concrete is to be at the maximum usable strain
- where a steel bar is to be at the first yield strain.

There are facilities on the “choose” dialog for this computation. If the material is of type “concrete” or of type “steel that yields” then the recorded maximum usable strain or first yield strain can be used. There is a check box to determine this.

If the material is not “concrete” and not “steel that yields” or the check box is not checked a number input box is provided for the strain caused by stress.

Note: At this stage in the development the application does not record the details of this computation in the XSF file. Only the resulting point location and load case strain at that point are recorded.

**Methods that provide automatic trial and adjustment.**

Generally the explicit algorithm will always work and generally the output is correct in that in accordance with the cross-section and material descriptions and the model they represent the output loading would cause the input distortion. An exception to this is the limiting strain problem. (See limiting strain problem.)

However, an automatic trial and adjustment procedure to assess the affect of a given loading that is guaranteed to work in every case is not provided. The methods provided should work in most usual cases and facilities are provided to make the procedures useful even when they do not work perfectly.

The Ultimate Bending Moment method is an adaptation of a well established computation procedure described in most standards texts. Trial and adjustment on the depth of the neutral axis described in the texts is tantamount to the trial and adjustment on the curvature about the reference axis.

**Curvature deviation**

The facilities that cope with the unrestrained case have a further degree of freedom in the trail and adjustment being the orientation of the curvature: the neutral axis is not always parallel with the reference axis. Expressed in rectangular co-ordinates in terms of the reference axes the curvature about the “other” axis is not always zero.

The trial and adjustment that deals with this works with the variables expressed in polar co-ordinates. The six variables in polar co-ordinates are:

three representing the loading; being:

- The axial load at the origin of the reference axes.
- The magnitude of the bending moment
- The orientation of the bending moment expressed as the angle from the reference axis.

and three representing the distortion; being:

- The axial strain at the origin of the reference axes.
- The magnitude of the curvature
- The orientation of the curvature expressed as the angle from the reference axis.

These distortion variables represent the “load case” distortions as distinct from the “stage” distortion and “other” distortions of each component.

A variable called “**curvature deviation”** is defined as the angle between the orientation of the bending moment and the orientation of the curvature.

Curvature deviation is made the subject of trial and adjustment. Generally it is between minus 90 degrees and plus 90 degrees and usually zero is a satisfactory first estimate.

In the unrestrained case the orientation of the objective bending moment is aligned with the reference axis so that the final value of the deviation angle is the orientation of the curvature relative to the reference axis. The curvature deviation angle is seen as a distortion variable.

This trial and adjustment process is not suitable where the bending moment is small. It has proved satisfactory for ultimate strength, first yield strength and most service load situations where there is a bending moment. Where the bending moment is very small the method “Utility: curvatures and axial load” is suggested with manual trial and adjustment on the curvatures.

**The computation dialog**

There is a different computation dialog for each method although some features are common to them all.

**Input Information**: In the panel for the **input information** near the top there is provision for objective values for three of the variables (expressed in rectangular co-ordinates in terms of the reference axes). Associated with each loading variable is a tolerance.

The unrestrained case implies an objective that the bending moment about the other axis is zero. There is no provision for a tolerance on this. However the minimum increment of the curvature deviation angle is 0.01 degrees or 36 seconds. This implies the tolerance is 0.017 % of the bending moment about the reference axis.

**First Estimates**: Under that panel is a panel for **first estimates** of the distortion variables. Associated with each variable is an increment. These are the increments used at the start of the trial and adjustment process. During the trial and adjustment process the increments are made smaller and smaller until the tolerances on the objectives are achieved.

After each successful computation these first estimates are automatically set to the corresponding values in the result. (The increment values however are not changed.)

**Compute Buttons**: Under the first estimates panel is a **compute button, **and with some methods two compute buttons. The two buttons are intended to produce the same result. The difference is that the upper of the two leads to procedures that are robust intended to have a good result if at all possible, whereas the lower leads to procedures that are less robust but quicker. .

**Auto start, Auto Close check boxes**: To the right of the computation buttons are check boxes labeled “Auto start computations” and “Auto close progress dialog”. Normally both these should be checked. Unchecking of these allows the user more control of the automatic trial and adjustment. This should be necessary only in a difficult situation where the automatic process does not work properly.

**Result panel**: The result panel near the bottom of the dialog has provision for all six variable and they are expressed in terms of the reference axes. Generally the explicit algorithm will always produce a correct output in that in accordance with the cross-section and material descriptions and the model they represent the output describes a possible state of the structural member.

Generally, even when the trial and adjustment routine fails the result from the last cycle of the computations is displayed. When the trial and adjustment routine fails a short message appears near the top of the results panel to indicate the computation failure and why. Although the result does not answer the direct question it is possibly of some use.

Sometimes (if the final cycle was in phase 2) two results are displayed. In these cases radio buttons appear below the results panel allowing the user to choose which of the results is possibly useful.

**The “Limiting strain problem”**

Some times the note “limiting strain problem” appears in the results panel and no result is displayed.

This problem occurs in the explicit algorithm rather than the trial and adjustment process. It can occur where there is a discontinuity in the stress-strain relationship of the concrete. There are such discontinuities with the rectangular stress block at the strains corresponding to the sides of the block.

It occurs when the stress distortion of the concrete component has no curvature and the strain corresponds to a discontinuity in the stress-strain relationship.

The application includes an automatic check for this condition. If it occurs during automatic trail and adjustment automatically the result of the particular cycle is ignored, a distortion variable is changed slightly and the automatic trial and adjustment is continued. This usually works and the final result is not affected.

If, however, the computation is for a final output result no result is displayed and the error message “Limiting strain problem” is displayed.

In this situation it is suggested the user make a small, possibly trivial change in the objective variables and repeat the computations.

**Computation progress dialog**

Accompanying the computations is a progress dialog that appears when a compute button is pressed. If the “Auto start computation” box is checked the computations proceed without further action by the user. If the computations are successful in that a solution is found, and the “Auto close progress dialog” box is checked the progress dialog will close and the focus will return to the computation dialog.

The main part of the progress dialog is a table that has a line to display the results of each use of the explicit algorithm.

In a panel near the top are five buttons:

“**Stop**” – interrupts the automatic process.

“**Step**” – causes one cycle of the computations, that is the incrementing of the various distortion variables and one used of the explicit algorithm.

“**Go**” – causes the automatic trial and adjustment to proceed.

“**Escape**” – the computations are abandoned and the “close” button is enabled.

“**Close**” – the progress dialog is closed and the focus is returned to the computation dialog.

The table has eleven columns. The six right hand most columns display the result values for the six variables expressed in terms of the references axes.

The first column on the left contains an iteration count.

The second, third and fourth columns contain the values of the variables that are the subject of the adjustments.

The second column contains the subject strain value. With some methods the subject location is given in the panel at the top of the dialog otherwise it is at the origin of the reference axes.

The third column contains the curvature deviation angle expressed in degrees.

The fourth column contains the curvature magnitude value. It is expressed as per inch or per metre depending on the unit system in use.

The label “Phase 1” or “Phase 2” may appear at the top of each of these three columns. The search routine for each degree of freedom has two phases:

**Phase 1** is a search for two values of the distortion variable that produce errors of opposite sign in the corresponding loading variable.

**Phase 2** is interpolation between the two values found in Phase 1.

The correspondence between the various distortion variables and the various loading variables varies between the various methods. However the Ultimate Bending Moment, Yield Bending Moment and Given Bending Moment methods use:

Curvature magnitude corresponds with axial load

Curvature deviation corresponds with bending moment orientation

Axial strain corresponds with bending moment magnitude

The correspondence between curvature deviation and bending moment orientation does not work when the bending moment magnitude is small. The problem is that small changes in the curvature deviation cause large changes in the bending moment orientation. For this situation the method Utility: Curvatures and Axial load is provided. This uses

Axial strain corresponds with axial load

It is intended this method is used in combination with manual trail and adjustment on the curvatures.

**A note on steel that has a strain limit**

It has been common to assume that reinforcing steel sustains a load regardless of how large the strain. Many standard codes of practice included this assumption. Examples are ACI318:14 clause 20.2.2.1 and Eurocode 2: (BS EN 1992-1-1:2004) clause 3.2.7 (2) b).

A steel stress – strain relationship that has a strain limit and stipulates that beyond that limit there is no stress reflects reality. However, this can cause a problem with the Ultimate Bending Moment method particularly if that breaking strain limits the strength of the cross-section. Eurocode 2: (BS EN 1992-1-1:2004) clause 3.2.7 (2) a) describes such as stress – strain relationship.

In general the existence of such a strain limit can be assessed on the material edit dialog. The “Display as text” button on that dialog reveals a stress – strain table. If such a strain limit exists t1he bottom line of that table has a zero stress value. The strain limit is the strain value on that same bottom line.

In such cases the Yield Bending Moments “method” might help.

Use the Yield Bending Moments “method” and increased the strain in the critical bar from the yield strain to a strain slightly less than the strain limit. This can be done on the “choose a bar” dialog. Uncheck the check box indicating that the yield tensile strain is to be used and a text edit box will appear where a strain can be input.