## Load Cases

**At a glance: **Descriptions of load cases processes and definitions of load cases terms here are applicable to all devices. For specific instructions on the use see the volumes specific to the Mac, PC or Android phone, iPhone & iPad.

Go to the head of the General volume.

### What is a load case

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 things of interest might be the ultimate strength, or the distortion under a known loading.

### Computation methods

Several different computation facilities are provided each of which is called a “method” and each method has a name. The name indicates the purpose and are such as ‘Ultimate bending moments’, ‘Yield bending moments’, etc.

Only one method can be current in a load case. However the current method can be changed so that a combination of the methods can be used.

There is a selection facility for the various alternative methods.

### The explicit algorithm

All the various 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*.

An earlier published description was *Thompson, Phillip J. “General Cross-section Analysis”, Transactions, Institution of Professional Engineers New Zealand, v 18, 1/CE, November 1991, pp 15-18*.

### Only Load Case distortions displayed

The distortion variables that appear on the Load Case edit facilities 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 represent the total loading on the cross-section.

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

### The need for intervention in the automatic process

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, there is no automatic trial and adjustment procedure to assess the affect of a given loading that is guaranteed to work in every case. 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.

### Curvature deviation

The facilities that cope with the unrestrained case have an added degree of freedom in the trail and adjustment being the orientation of the curvature. 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.

### Limiting strain problem

A computation 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 material in a shape component. 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 a shape 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.

### The various computation methods

The various methods are:

**Ultimate bending moments.**The input is the axial load and the main output is the ultimate strength bending moment about the reference axis. In the result:- At least one corner of a concrete component is at the ultimate compressive strain nominated for the material of that component.
- No corner of a concrete component is at a strain more compressive than its ultimate compressive strain.

- Y
**ield bending moments**. The input is the axial load and the choice of a reinforcing bar of steel that yields. The main output is the bending moment about the reference axis when that bar starts to yield. In the result the chosen reinforcing bar is at the strain nominated as the yield strain in the material description. **Utility:-Curvatures and one strain**. The input is the three distortion variable values relative to the reference axes, and the output is the three loading variable values relative to the reference axes.**Utility:-Strain at three points**. The input is the choice of three locations on the cross-section and the strain at each of those locations. The output is the three loading variable values relative to the reference axes.**Given bending moment**. The input is the axial load at the reference point and the bending moment about the reference axis. The output is the three distortion variable values relative to the reference axes. This has been implemented for the unrestrained case but not yet for the restrained case. Thus in the objective result the bending moment about the other axis is zero.**Utility:-Curvatures and Axial load**. The input is the axial load at the reference point, the curvature about the reference axis and the curvature about the other axis. The output is the strain at the reference point, the bending moment about the reference axis and the bending moment about the other axis.

### At a stage

There is a facility for the adoption of a load case result as being at a stage. This sets the stage distortion of each component equal to the total distortion of that component in that load case result. That is for all components.

On desktop devices there is an ‘Adopt stage load’ item on the Load case menu. On mobile devices there is a “Adopt this load case as at a stage” button on the first load case edit display.

### Nested one-dimensional routines

The application uses one-dimensional trial and adjustment routines nested one within another to achieve two and three dimensional trial and adjustment.

**Inner-most loop**: The Ultimate bending moment method, Yield bending moment method and Given bending moment methods do adjustments of the reference axis curvature magnitude with the objective of the axial load having a stipulated value. This is the inner most loop. It 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 adjustments of the curvature magnitude with the objective of the axial load having a particular value.**Middle loop:**In the unrestrained case this inner most loop is within a routine that adjusts the curvature deviation angle with the objective of there being zero bending moment about the other axis.**Outer-most loop:**For the Given Bending Moment method this middle loop is within a routine that adjusts the strain at the subject location with the objective of the bending moment having a stipulated value.

**A fourth loop**: The facilities that automatically determine a subject location are tantamount to a fourth loop outside of all three of these loops. This is described under the heading ‘Subject Location’.

## The input information

The kind of information to be input to a load case depends on the method selected. Nevertheless some information is the same for all methods.

#### Reference axes

**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. 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**“.

**Reference point**’ means the origin of the reference axes.

The x and y of the origin and the orientation angle of the references axis are part of this common input.

#### Concrete displaced by bars?

**Check box** or **Switch **– “**Concrete displaced by bars**”: If this is checked or on – the computations take into account that part of the concrete is displace by the reinforcing. It is intended that this will normally be checked or on. To obtain results for comparison with results from other computation facilities that do not account for the concrete displaced by the reinforcing this can be unchecked or set to off.

#### Restrained about other axis?

**Check box **or **Switch** – “**Restrained about other axis**”: This relates to the circumstances of the structural member. If this is checked or on the member is restrained and the curvature about the other axis is zero. Otherwise the objective bending moment about the other axis is zero.

This can be important in computations involving trial and adjustment. Setting this to unchecked or off adds a degree of freedom to the trial and adjustment process.

Usually the expression “biaxial analysis” refers to an unrestrained case.

#### Maximum iterations

**Maximum iterations**: An integer number. Although this is on a specific load case 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 . When this number is reached the computations stop and the buttons that close the computation dialog are enabled.

There is a separate iteration count limit that applies only to the trial and adjustment on the curvature magnitude. 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 is 2000. However, this may not be appropriate in all situations.

#### Iteration wait

**Iteration wait**: An integer number. Although this is on a specific load case it applies to all load-cases and all methods that use trail and adjustment. However, it applies only when there is a progress display.

The iteration wait controls the speed of the computation. It is approximately the wait between iterations in milli seconds. To slow down the progress increase this wait time. To speed up the progress decrease this wait time. However, if the wait time is too short the Stop, Step and Escape buttons will not work. In this regard the minimum wait time varies from device to device. The default value, 150 is intended to be above that minimum on most devices.

If the fastest possible computations are wanted switch off the progress display. On a desktop device uncheck the ‘Show progress dialog’ checkbox on the Compute dialog. On a mobile device set the ‘Show progress display’ switch to off.

### Subject location

#### What is a subject location

Most computation methods have a subject location. This is usually a corner of a shape component, or at a point in a point component although it may be anywhere on the cross section. The computation progress panel displays the subject location.

With the Ultimate bending moments method and the Yield bending moments method it is the location where the strain is set at the ultimate strain of a concrete or the yield stain of a steel. With the Given bending moment method and the Utility:- Curvatures and axial load method it is the location where the strain is the subject of trial and adjustment.

Where the subject location is to be determined by the user there are input facilities for a strain value ‘at’ and an X and a Y value. That X, Y is a subject location. The strain value is a load-case strain. Where all this is under a heading ‘First estimate’ it is only the strain value that is variable in the automatic trial and adjustment: the subject location is constant.

#### The user can choose an object as the subject location

The application includes facilities for the user to determine a subject location by selecting a point in a point component or a corner of a shape component. A special selection dialog appears for this.

Some of these selection dialogs have provision for the user to input a stress-strain value. This is an exception to the rule that there are only load-case strains on the load case facilities. The stage-strain and other-strain are automatically taken into account so that when the selection dialog is closed the corresponding load-case strain appears on the main load case dialogs.

Also some of these selection dialogs have provision to use the yield strain or ultimate strain obtained from the material properties.

#### Subject location automatically determined

The ‘Ultimate bending moments’ and ‘Given bending moment’ methods have a facility that determines a subject location automatically. On a desktop device this is invoked by the various compute buttons on the compute dialog except the ‘Compute from given estimates’ button on that dialog that does not invoke it. On a mobile device it is invoked when the computations are started and the ‘Ignore first estimate’ switch is on .

These facilities have trial and adjustment on a subject location. This trial and adjustment loop is outside of, and contains all the other trial and adjustment loops. The result of an ultimate strength computation is checked to ensure that no corner of a concrete component is at a strain greater than the ultimate strain of the concrete in that component. If such a corner is found then the subject location is changed to that corner of that component and the whole process is repeated.

With the general nature of the input accepted by this application there is a potential for this loop to continue indefinitely. To guard against this there is a limit on the number of corners tried. The error message displayed is ‘Tried too many components’. This error check is relatively simple and we are confident that it works although it has never been tested.

There has been no report of this error occurring (at the time of writing) nor have we devised a hypothetical example to illustrate it. Nevertheless, we cannot rule out the possibility that it exists.

### Adopting a result from another method

The explicit algorithm has the distortion variables as input and the loading variables as output. Nevertheless some methods have some loading variables as input and some distortion variables as output. To accommodate this the application has two sets of memory locations for the six variables; one set of six for the input and another set of six for the output.

When the selected method in a load case is changed the six output memory locations are not affected. Thus a valid result before the change is still valid.

However, part of the input required by the newly selected method may have default values rather than be consistent with information inherited from the previous method.

There are facilities with some methods for the use of information from a previous method.

Several of these provide for the selection of a point in a point component or a corner of a shape component and the strain in the previous result at that object. Examples of this are:

- With the ‘Given bending moment’ method the ‘Set first estimate to current case’ button on a mobile device .
- With the ‘Yield bending moments’ method the ‘Choose a bar’ button on any device.
- With the ‘Utility:- Strain at three points’ method the ‘Choose’ button on a desktop device and the ‘Choose a point or corner’ button on a mobile device.

Each of these also determine a subject location and some have alternative ways to determine the strain at that location; not being the strain in the previous result.

The ‘Utility:- Curvatures and one strain’ method can also use information from a previous method. A ‘Set input to current case’ button on any device causes the three distortion variables to be copied from the output set to the input set.

### 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.

The existence of such a strain limit should be apparent on the graphic image of a material. Zero stress at the strain extremity of the graph indicates such a limit.

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. On a desktop device uncheck the check box next to ‘Use Yield tensile strain=’ then change the ‘Caused by stress’ strain. (Also while on that dialog note the ‘This load case’ strain of that bar.) On a mobile device set the “Use specified yield strain” switch to off then change the stress-strain.

The result would be the bending moment at a tension failure.

This result could be used as the first estimate for the Given bending moment method. Change the selected method to Given bending moment and then:-

- On a
**desktop device**set the First estimates strain to the ‘This load case’ strain in that critical bar and the X and Y to the location of that bar. Finally use the ‘Compute from given estimates’ button rather than the ‘Compute’ button. - On a
**mobile device**tap the “Set first estimate to current case” button and select that critical bar. Also set the “Ignore first estimates” switch to off before starting the computations.