Board sagging due to blocked expansion

If board expansion is blocked, the board will bend e.g. sag.
If W is the board width and ΔW the width increase which is blocked, we can setup the following model:
Legend
 
The formulas (F2) and (F4) are worked out in a small BASIC program from which S is calculated after the input of the W and W + ΔW values.
Example of a BASIC program for calculation of the board sagging:
10 DEFSNGA-Z
20 INPUT " W : ";K
30 INPUT " W + ΔW : ";L
40 G=K/L:A=ATN(1) : B=A/2
50 C=SIN(A)/A
60 IF (G C) THEN A=A-B
70 IF (G C) THEN A=A+B
80 IF (G=C) THEN PRINT"*"A"*",INT(250*K*SIN(A)/1000:END
90 B=B/2
100 PRINT A
110 GO TO 50
In Excel
Note: From the factor (ϕ / sinϕ) the value of ϕ can be calculated also if one uses the tool Goal Seek in Excel.
Method:
1. Set the value (ϕ / sinϕ) e.g. in cel A1 and set a real ‘default’ value for ϕ in cel A
2.Set in cel A3 the formula =A2/(SIN(A2*PI()/180)).2. Next select that cel A3 and activate the “Tools / Goal seek” from the menu bar.
3. Set in the “To value” box the value of cel A1 (this is ϕ / sinϕ).
4. Set in the “By changing” box the cel code A2.
5. Click at OK. Next the value of ϕ will be expressed in cel A2.