Analysis by the General Method

The concepts of design and analysis can be thought of as opposite circumstances. In the design case the geometry of a member is selected to make it strong enough which means the engineer solves for many of the terms on the right side of Equation (1). For analysis, the engineer solves for the one term on the left side of Equation (1) instead. The general method equations have been intentionally arranged so that they produce design results, as above, with no iteration. That is, the equations have been selected so that they make design easy, but analysis slightly more difficult. This was a philosophical decision as the design code is intended for the construction of new structures.

To solve for the strength of a member by the CSA general method, two options are available. Firstly, for members without stirrups Equations (6), (4), and (1) may be solved simultaneously to produce a closed form solution for the shear strength of a member without stirrups. This derivation is reasonably easy and requires the solution of a quadratic equation.

Alternatively, an iterative process, suitable for spreadsheet calculations can be easily implemen­ted which works for all types of applied loading and all types of members. The process can be explained as follows. The spreadsheet will allow the determination of the shear strength of one beam/slab/column per row of the spreadsheet. The left side of the spreadsheet would include the various material, geometric, and loading properties, again, one beam per row. On the right side of the spreadsheet would be the iterative determination of the shear strength.

Considering the small sample spreadsheet below, the iteration begins with an initial estimate of the strain at the mid-depth of the member, say sx-1 = 1 x 10-3. For convenience, it is helpful to determine the values of strains in the units of parts per thousand to make mistakes easier to visually identify. Directly to the right of this value, Equations (1), (4), and (5) would be evaluated within a single spreadsheet cell to determine the shear strength for that assumed value of ex, listed as Vr_i below. This is listed as Equation type B and as the value ex is known, it is a direct substitution of values from the left side of the spreadsheet. To the right of this value, Equation (6) would then be evaluated to determine the new estimate of ex from the given applied loading, shown as ex-2. For the sake of numerical stability, it is appropriate to take the new estimate of ex as the average of the previous estimate and the newly calculated value, or ex-2 = (ex-1 + Equation (6))/2. To avoid any manual iteration actions by the user, the cells which contains the estimates of the shear strength and ex terms (Equations B, C) can be copied perhaps 10 times to the right to provide 10 iterations which is usually more than sufficient to converge. The final spreadsheet would look something like this:

bw d rho etc.

Єх-1

Vr-1

fix-2

<N

1

fix-3

Vr-3

300 925 0.83 etc.

1

200

1.2

183

1.24

177

Equation type

A

B

C

B

C

B

where Equation A is the constant initial guess of 1.0 part per thousand; Equation B is the direct substitution into Equations (1), (4), and (5); and Equation C is Equation (6) in this paper. The final value of the shear strength, perhaps Vr-10, can be copied back near the left side of the spreadsheet for easy access. Using this technique, the spreadsheet is iterative but is fully “live” in that no macros or special routines need to be called. When a value is changed on the constitutive properties on the left of the spreadsheet, the iterative equations will automatically update the predicted shear strength. That the spreadsheet requires no macros is curiously powerful as well in that nothing is hidden from the engineer with this spreadsheet: no “black-box” calculations or interpolation routes are present. From experience, engineers quickly become much more confident with the new general method partly due to this transparency.

The method above is specified in such a way that the equations (A, B, C, etc.) can be copied down over multiple rows of the spreadsheet to determine the strength of multiple members in a building, multiple experimental results, or to determine sensitivity plots by systematically varying the various beam properties on the left side of the spreadsheet.

Discussion is provided below concerning the predictions of the general method compared to experimental test results.