Sensitivity Analysis and its Application for Optimal Steel Beam Design per IS 800:2007

Definition - sensitivity analysis:

Sensitivity analysis is the technique used to determine how independent variables will impact a particular dependent variable under a given set of assumptions (https://www.edupristine.com/blog/all-about-sensitivity-analysis).

Problem definition:

From IS 800:2007 Section 8.2.2, we have the following formula for determining the moment of resistance of a laterally unsupported beam.





Where b_b= 1.0 for plastic and compact sections 
                  = Z_e/Z_p for semi-compact sections

Z_p= plastic section modulus

Z_e= elastic section modulus

f_bd= design bending compressive stress

The parameter f_bd  in turn depends on f_cr,b, defined by IS 800:2007 as "extreme fiber bending compressive stress". The relation between f_bd  and f_cr,b is given in the form of formulae in Section 8.2.2 of IS 800:2007 and the values of f_bd for a given f_cr,b are tabulated in Table 14 of the standard for standard rolled I-sections and welded doubly symmetric I sections.

IS 800:2007 has the following simplified equation for f_cr,b:

From the above, there are the following three parameters that influence f_cr,b:

• E = Modulus of elasticity
• Slenderness ratio L_LT/r_y where L_LT is the effective length of the beam for lateral torsional buckling and r_y is the radius of gyration about minor axis.
• h_f/t_f where h_f is the center-to-center distance between flanges and t_f is the flange thickness.

Now let us determine the input parameter that influences the output most.

Discussion on the Input Parameters:

Out of the above parameters, the value of E is something that has very minor variation and a specific value of 2.05x10⁵ is prescribed by IS 800:2007 (Section 2.2.4.1) for design purpose. Also from the equation for f_cr,b, we understand that the relationship between f_cr,b and E is linear, which means the variation in f_cr,b will be very minor even if we play with the value of E which is more or less constant.

The remaining two parameters i.e. L_LT/r_y and h_f/t_f can vary significantly.

Hence we consider L_LT/r_y and h_f/t_f for our sensitivity studies.

Example:

ISMB400 beam is considered as a bench mark section. To start with, an L_LT of 2.0 m is considered.

The below are the properties of ISMB400 with an initial L_LT of 2.0 m and corresponding L_LT/r_y ratio.

Section	Unit weight (kg/m)	Grade	Yield strength fy (Mpa)	LLT(m)	ry (mm)	Depth (mm)	tf (mm)	hf (mm)	LLT/ry	hf/tf	fcr,b (MPa)
ISMB400	61.6	E250	250	2	28.2	400	16	384	70.9	24	530.3424

Analysis Approach:

The value of f_cr,b is calculated for the following variations:
1. Variations in L_LT/r_y keeping h_f/t_f  constant
2. Variations in h_f/t_f  keeping L_LT/r_y constant.

The L_LT is varied from 0.5 times the initial L_LT of 2m to twice that, i.e. between 1m and 4m. The L_LT/r_y value also varies in the same proportion.

The h_f/t_f value is varied from 0.5 times the initial value of 24 which is for the ISMB400 to twice that value, i.e. between 12 and 48.

Assumptions:

• When the h_f/t_f value is varied, it is assumed that the beam section is re-proportioned in such a way that the r_y value and hence the L_LT/r_y value remains the same.
• When the r_y value is varied, it is assumed that the beam section is re-proportioned in such a way that the value of h_f/t_f  remains the same.
• Above two assumptions mean assumption of independence between L_LT/r_y and h_f/t_f .

Results:

Tabulation of results:

		fcr,b (MPa)					Row mean (MPa)
		hf/tf
		12	18	24	36	48
LLT/ry	35.5	2121	1934	1864	1812	1794	1905
	53.2	1108	943	878	828	810	913
	70.9	733	590	530	484	466	561
	106.4	437	326	277	236	219	299
	141.8	313	224	183	147	133	200
Column mean (MPa)		942	803	746	701	684

Data visualization:

Scatter plots:

Box plots:

For better visualization of data, below box plots have been plotted.

h_f/t_f  =12 and varying L_LT/r_y                                   h_f/t_f  =18 and varying L_LT/r_y              

      

h_f/t_f  =24 and varying L_LT/r_y                                              h_f/t_f  =36 and varying L_LT/r_y 

                                    

 h_f/t_f  =48 and varying L_LT/r_y 

     

 L_LT/r_y =35.5 and varying h_f/t_f                                        L_LT/r_y =53.2 and varying h_f/t_f                                   

L_LT/r_y =70.9 and varying h_f/t_f                                    L_LT/r_y =106.4 and varying h_f/t_f

                           

                 

   L_LT/r_y =141.8 and varying h_f/t_f

Discussions, Data Analytics and Results Interpretation:

Data Analytics:

The above data tells us that there is not much variation in the f_cr,b  value across the columns (h_f/t_f) compared to the variation across the rows (L_LT/r_y).

Also we find that the means of the columns are close by while there is a lot of variation in the means of the rows.

From the scatter plots, we see negative correlation between each of the independent variables i.e. L_LT/r_y and h_f/t_f,  and the dependent variable i.e. f_cr, . This is as expected from the equation for f_cr,b.

ANOVA and discussion of its suitability:

To analyze the variation further, let us perform ANOVA. One may say both L_LT/r_y and h_f/t_f are continuous and hence the applicability of ANOVA is not appropriate.

Now the structural engineer in us has to wake up! Even though L_LT/r_y and h_f/t_f are continuous variables, we have collected data against discrete data points. This is similar to how a large structural model is discretized into distinct elements in finite element method.

When the range of the data is sufficiently large, we can study if there is any variation. If the data points are centered around a particular value, then it makes no sense to perform an ANOVA.

Also the number of data points has to be sufficiently large for accurate inference. However, in our case, we have limited to 5x5 matrix of L_LT/r_y and h_f/t_f values. Plotting the data over a larger number of data points conveyed to me that the basic shape of the graph doesn't change much in this particular case.

Single factor ANOVA is preferred as we are interested in which particular parameter between the two viz. L_LT/r_y and h_f/t_f influences the design most.

By performing single factor ANOVA across the rows and across the columns in Excel, below are the results obtained.

Anova Case 1:  Single Factor - hf/tf
SUMMARY
Groups	Count	Sum	Average	Variance
hf/tf = 12	5	4711.4	942.3	528355.7
hf/tf = 18	5	4016.6	803.3	476734.1
hf/tf = 24	5	3732.4	746.5	462615.0
hf/tf = 36	5	3507.4	701.5	455372.0
hf/tf = 48	5	3422.3	684.5	453788.0
ANOVA
Source of Variation	SS	df	MS	F	P-value	F crit
Between Groups	215994.5	4	53998.63	0.113592	0.976246	2.866081
Within Groups	9507459	20	475372.9
Total	9723453	24

Anova Case 2:  Single Factor LLT/ry
SUMMARY
Groups	Count	Sum	Average	Variance
LLT/ry= 35.5	5	9525.526	1905.105	17570.41
LLT/ry= 53.2	5	4567.121	913.4242	14410.53
LLT/ry= 70.9	5	2802.828	560.5657	11619.54
LLT/ry= 106.4	5	1494.602	298.9204	7620.852
LLT/ry= 141.8	5	1000.051	200.0102	5211.85
ANOVA
Source of Variation	SS	df	MS	F	P -value	F crit
Between Groups	9497721	4	2374430	210.3754	4.9E-16	2.866081
Within Groups	225732.7	20	11286.64
Total	9723453	24

We accept the null hypothesis for case 1 and reject it for case 2.

Result interpretation:

1. For a four times variation in L_LT/r_y, the value of maximum f_cr,b is 6.78 times higher than the minimum f_cr,b when h_f/t_f = 12. This further increases to 13.53 times when h_f/t_f = 48.
2. For a four times variation in h_f/t_f, the value of maximum f_cr,b is 1.18 times higher than the minimum f_cr,b when L_LT/r_y  = 35.5. This further increases to 2.36 times when L_LT/r_y = 141.8.
3. From the ANOVA, the difference in the means of  f_cr,b  across varying values of h_f/t_f is statistically insignificant with a very high p-value of 0.976, while it is very significant across varying values of L_LT/r_y with a nearly zero value of p.

Conclusions:

1. From the above, it can be established that the parameter L_LT/r_y influences the f_cr,b value much more than the parameter h_f/t_f.
2. From the above conclusion, it can be inferred that by increasing the r_y value by using a wider flange beam, the beam design can be optimized.
3. The same set of conclusions can be easily established by taking up similar studies for beams of other standard/non-standard sizes, both rolled and welded.