Models for structural design
10.10 Some parametric dependency devices
Selecting a suitable slab depth
The following extract is taken from the parameter table for sc080.pro. Given a design bending moment M, the effective depth d has to ensure that the strength of the concrete is not exceeded and that the amount of reinforcement is within a practical percentage e.g. from 0.4% to 2%. The extract achieves both requirements, an explanation follows the extract.
Table 10.7 Extract from a parameter table for a reinforced concrete slab.
PARAMETER Start End Type Dependency conditions No. name zst() zen() zty() and notes.
...
2 ans0 1 5 5 Location of section.
3 sdr 1 5 126 20 7 20 23 26 Span:depth.
4 M 5 400 0 Ultimate moment of resistance.
5 fcu 30 60 4 Char. concrete strength.
6 fy 250 460 3 Characteristic steel strength.
...
10 dconc 75 375 1 =SQR(M*1E6/(0.1*1000*fcu)) 11 d 1E20 1E20 1 =dconc > <
12 dia 3 8 100 =INT(d/12) Tens bar diameter.
...
21 span 2 20 0 =0.7*d*sdr/1000 Effective span ...
23 expos 1 5 5 >1 <INT(fcu/10)-1 Exp. condn.
...
(SQR(M*1E3/(0.75*fy*0.02))) ┐ These two expressions relate to (SQR(M*1E3/(0.75*fy*0.004))) ┘ the > & < for the 11'th parameter.
├──────1 m──────┤ M kNm known, lever arm =0.75*d.
┌───────────────┐ ─┬─ Let steel fraction of area =p, │ │ │ then Ast mm² = d*1000*p and
│ │ d Ast*fy*0.75*d d*p*fy*0.75*d │ │ │ M = ───────────── = ─────────────
│ o o o o o │ ─┴─ 10^6 10^3 ┌ M*10^3 ┐
Rearranging d= SQR│ ───────── │ └ 0.75*fy*p ┘
Obviously an increase in steel fractional area p causes a reduction in d and vice versa.
Taking a minimum fractional area of steel reinforcement of 0.004 i.e. 0.4% and a maximum fractional area of 0.02 i.e. 2.0%, to give a maximum & minimum slab thickness respectively, then:
d >= SQR(M*1E3/(0.75*fy*0.02)) and d <= SQR(M*1E3/(0.75*fy*0.004)) as used in the PARAMETER table for sc080.pro. Expressions which follow the parameter table are associated with any isolated > = < found in the parameter table in order; this association is necessary when there is insufficient space to include the dependency condition/s on the line in the parameter table to which they refer.
For a minimum slab depth based on the concrete alone, supply dummy parameter, the 10th, dconc=SQR(M*1E6/(0.1*1000*fcu)) assuming a maximum of 30%
redistribution, BS8110-1:1997 clause 3.4.4.4. Then d, the slab depth and 11th parameter, is set to dconc, the 1E20 for the start & end values means dynamically set the start and end values according to the > and < expressions which follow, as mentioned previously these dependency conditions follow the table because of insufficient space on the 11th line for their inclusion. When these conditions are applied, the assignment of d=dconc ensures that the concrete capacity is sufficient to support the bending moment M and the two dependencies ensure that the steel reinforcement is within normal percentages.
Selecting an exposure condition
Selecting an exposure condition is straightforward as the maximum exposure condition is dependent on just the concrete cube strength. Inspection of the parameter expos, the 23'rd parameter in the above extract from the parameter table for sc080.pro, shows that when fcu=30, expos is an integer number varying between 1 and INT(30/10)-1=2; when fcu=60, expos is an integer number varying between 1 and INT(60/10)-1=5. Thus expos is constrained by the concrete strength as appropriate.
Selecting a practical bar diameter
Selecting a practical bar diameter has to meet two requirements:
• the bar must be selected from the set: 6 8 10 12 16 20 25 40 50
• the bar must be a practical size for the section depth e.g. if section depth =100 mm, then a bar diameter of 50 mm would be wrong, if section depth =400 mm, then a bar diameter of 6 mm would be wrong, unless it was used to control surface cracking.
Specifying the parameter Type in the range 100 to 125 i.e. zty()=100 to zty()=125 accesses data stored in vectors za() to zz() respectively. Currently reinforcing bar diameters are stored in za() thus to access the bar diameters, Type=100 i.e. in the above extract from the parameter table for sc080.pro, zty(12)=100.
┌3'rd ┌8'th za(1)=VEC(6,8,10,12,16,20,25,40,50) └start value └end value
The line from the parameter table below, tells the program to pick a bar diameter from the range 10 mm to 40 mm. If we divide section depth by 12 say and use as a trial diameter, for section depth =100, then bar diameter of 100/12 =8.33 mm
=400, then bar diameter of 400/12 =33.33 mm.
Obviously we need to use the trial diameter to select an available size from the set za(1)=VEC(6,8,10,12,16,20,25,40,50). This is done automatically when a dependency condition is provided e.g.
zva(3)=10mm┐ ┌zva(8)=40mm
12 dia 3 8 100 =INT(d/12) Tens bar diameter.
which is interpreted as: evaluate d/12 and find the nearest bar diameter within the range za(3) to za(8).
Selecting a suitable span
Selecting a sensible span has to meet two requirements:
• the span must give a sensible span:depth ratio for the continuity specified e.g. fixed, pinned ends or cantilever
• the span:depth ratio must be found from the location reference given in response to the prompt for the parameter ans0 with values 1 to 5 corresponding to five location descriptions for the section.
From Table 10.7 it can be seen that parameter 2, ans0, offers the user options 1 to 5 corresponding to beam span:depth ratios respectively: 20 7 20 23 & 26, given in parameter 3, to accord with the order required by proforma sc080.pro. When Type=126, subsequent numbers are stored, in this case parameter 3 stores five span:depth ratios. When ans0=1 sdr=20, when ans0=2 sdr=7, when ans0=3 sdr=20, when ans0=4 sdr=23, when ans0=5 sdr=26. For Type=126, the pattern i.e. increasing or decreasing from zst to zen is always made the same as the previous parameter, which in this case is that of parameter 2. Parameter 3 is a dummy parameter i.e. it is not used in the model, but it is used in the expression given in parameter 21 to compute a sensible span for the current set of data which is being built.
Subscripted parameters
Occasionally it is advantageous to use subscripted parameters e.g. for a continuous beam having 10 spans stored in s(1) to s(10), then assuming s(1) was the 4th parameter, the follow section from a parameter table would suffice and describe parameters s(1) to s(10) in one line of the parameter table instead of 10 lines.
PARAMETER Start End Type Dependency conditions No. name zst() zen() zty() and notes.
...
4 13 s 1.5 6 0 Spans s(1) to s(10).
14 ...
Optionally the 13, corresponding to s(10) may be omitted from the table as it may be computed to be one less than 14, the parameter number at the start of the next line.
When a subscripted parameter is the last in the parameter table, both parameter numbers must be shown.
Restricting distributed loading to be within varying spans
An extract from a parameter table follows, which in turn is followed by a description.
PARAMETER Start End Type Dependency conditions No. name zst() zen() zty() and notes.
...
2 ns 7 7 0 Number of spans.
3 sp 1 6 0 =RAN(29)*5+1 Spans l. to r.
10 uj 1 7 1 =INT(RAN(47)*ns+0.5) Span No.
17 23 splu 0 20 0 =+sn=zva(zp'-7),ztm=sp(sn) Parameter 2 sets the number of spans ns to 7. Parameter 3 sets seven spans i.e. sp(1) to sp(7) to random spans in the range 1 to 6 m; the number in brackets following the RAN function is a seed for the random number generator. As mentioned previously, it is permissible to include the end parameter number on the line as in:
3 9 sp 1 6 0 =RAN(29)*5+1 Spans l. to r.
where parameter 3 refers to sp(1) and parameter 9 refers to sp(7), or omit the 9 as in the extract. When the 9 is omitted, it is deduced by the program from the first parameter number on the next line. Parameters 10 to 16 select a random set of seven span numbers, where parameter 10 refers to uj(1) and parameter 16 refers to uj(7). The problem is to find the span lengths corresponding to the random span numbers uj(1) to
uj(7). This requires two stages:
• selecting the span numbers stored in uj(1) to uj(7) by sn=zva(zp'-7), where zp' is always the current parameter number
• assigning the current value of the parameter, which is always ztm, to the current span number sp(sn).
The two (or more) assignments are concatenated, with comma/s as separator/s. A plus sign follows the first equals sign to let the program know that multiple assignments follow.
It has been found that with a little bit of thought, it is possible to engineer dependency conditions for most situations which arise, only having to resort to writing procedures for tables of steel section properties.