Units in empirical formulas

There are many formulas in structural design codes, that are empirical and if you enter them in Calcpad “as are”, you will not obtain the results in the expected units. Moreover, there are even formulas, that are not dimensionally correct and won’t calculate. Lets take for example the shear resistance without reinforcement, according to Eurocode 2. It is calculated by the following formula:

  V_Rd,c = (C_Rd,c k (100 ρ_l   f_ck)^1/3 + k₁ σ_cp) b_w d (6.2a)

Here, f_ck and σ_cp are in MPa, so, we will end up with MPa^1/3 + MPa, which does not have physical meaning and cannot be calculated. The k factor is equal to:

  k = 1 + sqrt(200/d) ≤ 2.0

If d is in mm, we will get: 1 + mm^-0.5, which also does not make any sense.

On the other hand, the minimum resistance is:

  V_Rd,c = (v_min + k₁ σ_cp) b_w d (6.2b), where

    v_min = 0.035k^3/2 f_ck^1/2

How can we solve this problem?

Obviously, we have no choice, except to manually compensate for the missing dimensions. For example the value of “200” in the formula for k is actually in mm. So, we will have k = 1 + sqrt(200mm/d) and get a dimensionless result as expected.

If we apply the same approach for the other formulas, we will obtain the following little program for calculation of shear resistance without reinforcement to Eurocode 2:

'Section width -'b_w = 250mm
'Effective height -'d = 450mm
'Main reinforcement ratio -'ρ_l = 0.02
'Characteristic concrete strength -'f_ck = 25MPa
'Partial safety factor for concrete -'γ_c = 1.5
'Design concrete strength -'f_cd = f_ck/γ_c
'Normal stress -'σ_cp = 0MPa
'Factors:'k_1 = 0.15', 'C_Rd,c = 0.18/γ_c
k = min(1 + sqr(200mm/d); 2)
v_min = 0.035*k^(3/2)*sqr(f_ck)*MPa^0.5|MPa
'Shear resistance
'<p class="ref">(6.2a)</p>
V_Rd,c = (C_Rd,c*k*(100*ρ_l*f_ck)^(1/3)*MPa^(2/3) + k_1*σ_cp)*b_w*d
'Minimum shear resistance 
'<p class="ref">(6.2b)</p>
V_Rd,c = (v_min + k_1*σ_cp)*b_w*d

When you run the above program in Calcpad, the results will look as follows:

You should not worry about that formulas do not look exactly as in the design codes. It is far more important to give the correct results as it can be easily verified.