Accompanying this are files named “ACI8321M.xsf” and “ACI8321I.xsf” that are in metric units and inch-pound units respectively..

This example is a cross-section of a prestressed concrete box girder bridge.

It was taken from example 1 in a paper:

Troels Brondum-Nielsen, “Serviceability limit state analysis of cracked, polygonal concrete sections under biaxial or symmetric bending“, ACI journal, title no. 83-21 March-April 1986.

This application does not uses the algorithms described in the paper although it produces results consistent with those in the paper.

The stress-strain diagram for concrete was a form of ACI 318-89 clause 10.2.7 modified to represent linear elastic behavior at small compressive strains. At small compressive strains E_c was as given in the paper.

The properties of the mild steel conformed to ACI 318-89 clause 10.2.4.

The stress strain table for the prestressing steel was derived from the information in the paper; the curved parts being approximated by a sequence of straight lines.

Four components were described: two shape components of Concrete called “Box” and “Deck”, a point component of Steel called “Bars”, and a point component of Prestressing steel called “Tendons”.

The concrete shape was described in two parts; the bottom and the two sides being included in the component called “Box”, and the deck in the component called “Deck”.

The top right corner of the deck was adopted as the origin. This was consistent with the way the information was presented in the paper. However, as a consequence of this and the sign conventions used in this application, all of the co-ordinates were negative. (For sign conventions see the User’s Guide – “Loading” and “Distortion”.)

There were no axial loads on the cross-section. Thus the choice of the location of the reference point did not affect the meaning of the bending moment values.

The concrete shrinkage of 0.3 x10^-3 was taken into account. This appears as -0.3 x10^-3 in the other distortion of the concrete components.

The prestress “epsilon p o” value of 4 x10^-3 was expressed as a stage distortion in the tendons table.

The paper gives the bending moments as M_x = 3 MNm, and M_y = -2 MNm. This is equivalent to one bending moment of magnitude 3.6055 MNm and orientation -33.69 degrees.

This discussion is in metric units only. ( 1 newton.metre = 8.850746 pound.inches: 1 MNm = 8.850746 x10³ k.in )

A load case called “Service at 146.31 d” should be present in the accompanying XSF file.

The application’s sign convention requires that 180 degrees is added to give a reference angle of 146.31 degrees. The Load case “Service at 146.31 d” investigates the behavior of the cross-section about this axis.

The “Curvature about other axis restrained” checkbox was not checked.

The “Given bending moment” method was used.

On the “Compute …” dialog the objective axial load is set to zero and the objective bending moment to 3.6055 MNm.

The computations from the “Compute” button take about 400 iterations. This ignores the given first estimates: default values are used. The computations from the “Compute from given estimates” button can take as few as 14 iterations.

The computation shows that at 0, 0 the “load case” strain is -2.343 x10^-3 . The “other” strain in the concrete of the deck being the shrinkage is -0.300 x10^-3 and there is no “stage” strain. Thus the “stress” strain is -2.043 x10^-3 . From the stress-strain table for the concrete, interpolation between lines 2 and 3 shows the stress is -2.043 / -2.500 x -25.0 = -20.43 MPa which is close to the 20.3 MPa in the paper.

A further step shows that the distortion is consistent with the “a” and “h” given in the paper.

1. Copy the load case. This should produce a new load case called “Service at 146.31 1”.
2. Change the reference angle to 180 degrees and the computation method to “Utility:- Curvature and one strain”
3. Press the compute button to obtain the “Compute” dialog.
4. Press the “Set input to current case” button. This will set the three distortion variables to represent the same distortion as obtained in the previous computation but expressed in terms of the axes at 180 degrees.
5. Press the “Compute loading” button.

The resulting bending moments 2.9990 MNm and -2.0014 MNm are close to 3 MNm and -2 MNm respectively as given in the paper.

The “load-case” strain at the reference point is -2.343 x10^-3. The “load-case” curvatures were 3.5548 x10^-3 /m and -0.3629 x10^-3 /m.

There are two components of concrete and they have the same distortions. In the following the word “concrete” means both these components.

As pointed out previously, relative to the concrete the “stress” strain at 0, 0 is -2.043 x10^-3. There is no “stage” or “other” curvature in the concrete. Thus the “stress” curvature in the concrete is the same as the “load case” curvature; 3.5548 x10^-3 /m and -0.3629 x10^-3 /m.

“a” is obtained from (-2.043 x10^-3 )/(-0.3629 x10^-3) = 5.63 m.

“h” is obtained from (-2.043 x10^-3 )/(3.5548 x10^-3) = -0.575 m.