Finite Element Analysis

NTU ME 720-A

(UMASS MIE/CEE 605)

Spring 1998

Course Project

by

Russ Althof

Raytheon Systems Company

I certify that this work is my own.

Optimal Design of a Steel Framed House


 




Introduction. With the increase in cost of wood products since hurricane Andrew hit southern Florida in 1992, the cost of a conventionally framed house has increased. At the same time, the price of steel products has remained stable now making steel competitive with wood for residential construction. Optimizing the steel framing members for the loads expected on the house can reduce the cost and labor during construction of the house.
 
 

Scope. This analysis optimized the steel structural members of a steel framed house. By minimizing the mass of steel members, cost and labor can be minimized. The analysis considered dead and live loads, snow loads and wind loads acting on the structural members.
 
 

House Design Details. The floor plan for the house for this analysis is shown in Figure 1. The house consists of a slab foundation and two floors. All ceiling heights are eight feet. The house consists of a main structural frame constructed of structural I-beams. The I-beams are eight inches thick in order to maintain the desired insulation. Between the I-beams, C-section members are used for ceiling and floor joists. Purlins (in hat section shapes) are used between the I-beams for attaching the roof sheeting. The roof has a 6:12 slope. Additional details of the structural members will be provided in the detail analysis.
 
 

Loads. There are four loads acting on the structural framing members - dead loads (DL), live loads (LL), snow loads (SL) and wind loads (WL). All loads are given in pounds per square foot (psf). Table I indicates the specific loads acting on the individual members and the magnitude of the load.
 
 

Table I. Loads.


 
 
 
 
Dead Load
Live Load
Snow Load
Wind Load
Ceiling Joists
5 psf
     
Floor Joists
10 psf
40 psf
   
Roof Purlins
7 psf
 
16 psf
25.76 psf
Structural I-beams
Various*

 

* Loads on the structural I-beams are a combination of the above loads. The exact combination is dependent on the location of the member.
 
 

The dead and live loads were obtained from Reference 1. Also obtained from Reference 1 are the ground snow loads. For the location of this house in North Central Texas, the maximum ground snow load is 20 psf. For a slope between 3:12 and 12:12, the equivalent snow load is:
 
 

SL = .8 * 20 = 16 psf
 
 
 
 


 
 

Figure 1. House Design Details.


 
 


The pressure load due to wind is calculated using the following equation:
 
 

WL = .00256 * V^2


 
 

where V is the wind speed in miles per hour (mph). For the location of this house, it is desired to withstand tornadic winds of up to 150 mph. Therefore, the calculated wind load is:
 
 

WL = .00256 * (150)^2 = 57.6 psf


 
 

For the 6:12 slope of the roof, the component normal to the roof purlin is:
 
 

WL = 57.6 * sin(arctan(6/12)) = 25.76 psf





Finite Element Model. All structural members were modeled as beams using ProEngineer solid modeling software along with Pro/Mechanica structural analysis. For this analysis, all joints were considered to be fixed in all directions. The structural frame model of the structural I-beams is shown in Figure 2. The joists and purlins span between the members in this model. The joists and purlins were modeled individually as single, linear beams. Because the joists and purlins are not symmetric in two axes, the cross-sections cannot be modeled exactly in Pro/Mechanica. Instead, the joists and purlins were modeled using symmetric beams with equivalent sectional properties. Because the structural I-beams are symmetric, they were modeled exactly.
 
 


 
 

Figure 2. Structural Frame.


 
 

Design Criteria. There were two criteria for designing the joists and purlins. These were deflection of the beam and maximum stresses in the beam. In accordance with Reference 2, the allowable deflections are L/240 for dead loads and L/480 for total loads where L is the span of the member. Maximum allowable stress is .6 * Sy where Sy is the minimum yield strength. This provides a 1.67 factor of safety (FS) on the design. Table III gives the allowable stresses for the different members. When evaluating wind loads, beam deflections were not considered as a design criteria due to the short duration of the applied load. In this case, only allowable yield stress was used for the criteria.
 
 

Table III. Allowable Stresses for Structural Members.


 
 
Member
Yield Strength (Sy)
Allowable Stress
Joists 1
33 ksi
19.76 ksi
Purlins 1
33 ksi
19.76 ksi
I-beams 2
60 ksi
36 ksi

 

1. Cold-formed rolled steel.

2. Hot-formed structural steel.
 
 

The joist C-section cross-section is shown in Figure 3. Commercially available sections are available to the parameters shown in Table IV.
 
 


 
 

Figure 3. Joist Cross-section.
 
 
 
 
 
 

Table IV. Joist Parameters.


 
 
Height (H)
8, 10, 12, 14 inches
Width (W)
1.625, 2.00, 2.50 inches
Thickness (t)
.018, .027, .033, .043, .054, .068, .097 inches

(25, 22, 20, 18, 16, 14, 12 ga)


 
 
 

The purlin cross-section is shown in Figure 4. Commercially available sections are available to the parameters shown in Table V.
 
 


 
 

Figure 4. Purlin Cross-section.
 
 
 
 

Table V. Purlin Parameters.


 
 
Height (H)
.75, 1.00, 1.25, 1.50 inches
Width (W)
3, 4, 5, 6 inches
Thickness (t)
.018, .027, .033, .043, .054, .068, .097 inches

(25, 22, 20, 18, 16, 14, 12 ga)


 
 
 

The I-beam cross-section is shown in Figure 5. Commercially available sections were selected from standard wide flange shapes with a "W" designation in accordance with the American Iron and Steel Institute. Wall and ceiling I-beams were selected from the W8 designation which has a nominal height of 8 inches. Roof section I-beams were selected from W4 through W8 designations since the lighter roof loads could allow for a smaller I-beam. The "W" designation also has a second part which designates lbs/ft of the beam. An example of the designation is as follows:
 
 

W8 X 20 - Nominal 8 inch high wide flange beam at 20 lbs/ft






 
 

Figure 5. I-beam Cross-section.


 
 


Detail Analysis. The joist and purlin members were analyzed first to determine if additional structural frame members were needed to reduce span lengths in order to obtain solutions for optimized joists and purlins. Once these members were optimized, the structural frame was analyzed to determine the optimal size of the structural I-beam.
 
 

First Floor Ceiling Joists. The available ceiling joists cross-sections are shown in Figure 3 and Table IV. The span of the first floor ceiling joists is 186 inches. Based on this, the design allowables are given in Table VI. For a joist spacing of 24 inches on-center (OC), the pressure load of 5 psf was converted to a linearly distributed load on the joist as follows:
 
 

distributed load = (24/12) * 5 = 10 pounds per linear foot (plf)
 
 


Table VI. First Floor Ceiling Joist Design Allowables.


 
 
Dead Load Deflection
.775 in
Total Load Deflection
NA
Allowable Stress
19.76 ksi

 
 
 

As discussed earlier, the joist cross-section cannot be modeled with accurate results as a beam due to its asymmetric shape. The joist was modeled using a rectangular beam with the same section properties as the joists. Due to the short span and light load of this joist, a simple analysis was performed on the smallest available joist shown in Table IV. The results are shown in Figure 6 indicating that the smallest joist easily meets the allowables in Table VI.
 
 


 
 

Figure 6. First Floor Ceiling Joist Static Analysis.


 
 


Second Floor Ceiling Joist Analysis. The span for the second floor ceiling joists is 356 inches resulting in the design allowables of Table VII. The uniformly distributed load for this joist is the same as the first floor ceiling joists load.
 
 

Table VII. Second Floor Ceiling Joist Design Allowables.


 
 
Dead Load Deflection
1.48 in
Total Load Deflection
NA
Allowable Stress
19.76 ksi

 
 
 

Again, this joist was modeled using a rectangular cross-section beam. An initial analysis indicated the same joist for the first floor ceiling joist meets the design allowables of Table VII. The results of the analysis are shown in Figure 7.
 
 


 
 

Figure 7. Second Floor Ceiling Joist Static Analysis.





Second Floor Floor Joist Analysis. The span for the second floor floor joists is 356 inches resulting in the design allowables of Table VIII. The uniformly distributed load was determined from the total load on the beam:
 
 

distributed load = 50 psf * (24/12) = 100 plf
 
 

Table VIII. Second Floor Floor Joist Design Allowables.


 
 
Dead Load Deflection
1.48 in
Total Load Deflection
.74 in
Allowable Stress
19.76 ksi

 
 
 

This load was used for the optimization because it is much higher than the dead load and the allowable deflection is less. Again, this joist was modeled using a rectangular cross-section beam. This beam was optimized to minimize weight while maintaining the allowables of Table VIII. The optimization resulted in a beam that was 14 inches high by .136 inches wide with an area MOI of 31.091 in^4. The maximum stresses and deflection are shown in Figure 8. Microsoft Excel Solver was then used to convert the equivalent section properties to a commercially available joist cross-section. The equivalent joist has the following parameters:
 
 

H = 14 in

W = 2.000 in

t = .068 in (14 ga)
 
 



 
 

Figure 8. Second Floor Floor Joist Optimized Design Results.


 
 


First Floor Roof Purlins. The available roof purlin cross-sections are shown in Figure 4 and Table V. The span of the first floor roof purlins is 186 inches. Based on this, the design allowables are given in Table IX. The purlin design must be optimized for both total loads and wind loads. The total load is the sum of the dead load and equivalent snow load. For a purlin spacing of 24 inches OC, the total load of 23 psf was obtained from Table I and was converted to a linearly distributed load on the purlin as follows:
 
 

distributed load = (24/12) * 23 = 46 plf
 
 

For the wind load of 25.76 psf:
 
 

distributed load = (24/12) * 25.76 = 51.52 plf
 
 

For the wind load, deflection is not a criteria for optimization. Only minimum weight and maximum stress will be considered.
 
 

Table IX. First Floor Roof Purlin Design Allowables.


 
 
Dead Load Deflection
.78 in
Total Load Deflection
.39 in
Allowable Stress
19.76 ksi

 

As with the previous analysis, the purlin was modeled using a rectangular cross-section beam because the purlin is not symmetric in two axes. Due to the shape of the purlin, a hollow, rectangular beam was used. An initial analysis to the largest purlin was made to verify the capacity of the purlin for the given span. For both total loads and wind loads, the allowables in Table IX were exceeded. Therefore, an additional structural I-beam was added to the first floor roof section in order to reduce the span of the purlin. The new allowables with a span of 93 inches are shown in Table X.
 
 

Table X. Updated First Floor Purlin Design Allowables.


 
 
Dead Load Deflection
.39 in
Total Load Deflection
.19 in
Allowable Stress
19.76 ksi

 
 
 

The beam was then optimized to minimize weight while maintaining the allowables of Table X. The optimization resulted in a beam with the dimensions shown in Figure 9. The maximum stresses and deflection are shown in Figure 10 for the wind load condition. As seen in Figure 10, the maximum deflection meets the design allowables of Table X with a higher load than the total load, therefore, no further analysis to the total load was required. The equivalent joist purlin was found using Excel Solver and is shown below:
 
 

H = 1.50 in

W = 6.00 in

t = .043 in (18 ga)
 
 

This purlin has an area MOI of .0697 in^4. Since this is slightly higher than the rectangular beam with the same distance from the beam neutral axis for both beams, the stresses and deflections will be less thus making this an acceptable purlin for this use.
 
 


 
 

Figure 9. Rectangular Beam Dimensions.
 
 


 
 

Figure 10. First Floor Roof Purlin Wind Load Optimized Design Analysis.


 
 


Second Floor Roof Purlin Analysis. The available roof purlin cross-sections are shown in Figure 4 and Table V. The original span of the second floor roof purlin was 356 inches. Based on the first floor roof purlin analysis, three more structural I-beams were added to the structural roof section for the second floor. The resultant span of the second floor purlins was reduced to 89 inches. Since this span is only 4% shorter than the first floor roof purlin and is exposed to the same loads, the optimized purlin in the previous analysis was also selected for this location.
 
 

Structural Frame Analysis. The structural frame model was updated to the required changes from the joist and purlin analysis. The updated frame is shown in Figure 11. Each member is numbered in Figure 11 and its corresponding load is shown in Table XI. The loads were determined by summing the dead, live and wind loads for each section of the house and evenly distributing them about the structural members in each section.
 
 

Table XI. Structural Frame Loads.


 
 
Member No
Windward Load (plf)
Leeward Load (plf)
1
103.3 horizontal
NA
2
240.6 horizontal
NA
3
37.2 horizontal

22.6 normal to roof

22.6 normal to roof
4
140.5 horizontal

22.6 normal to roof

25.7 vertical

22.6 normal to roof

25.7 vertical

5
221.0 horizontal

377.0 vertical

377.0 vertical
6
124.1 horizontal

37.7 vertical

24.5 normal to roof

37.7 vertical

24.5 normal to roof

7
37.2 horizontal

22.6 normal to roof

22.6 normal to roof
8
40.4 horizontal

24.5 normal to roof

24.5 normal to roof
9
40.4 horizontal

24.5 normal to roof

24.5 normal to roof
10
25.7 vertical
NA
11
402.7 vertical
NA
12
37.7 vertical
NA
13
83.7 horizontal
NA

 
 
 


 
 
 
 

Figure 11. Updated Structural Frame.

Before optimizing the size of the structural I-beams, an initial static analysis was performed to the largest available I-beam in the W8 designation. This was done to identify the need for any additional beams in the frame prior to optimization. The analysis revealed that the maximum stress was well below the maximum allowable stress. The results of this analysis are shown in Figure 12.
 
 


 
 

Figure 12. Largest I-beam Stress Analysis.


 
 


An optimization analysis was then performed on the structural frame to minimize weight which will yield the smallest I-beam and therefore the lowest cost solution. The wall, floor and ceiling I-beams were optimized within the available sizes of the W8 designation. The roof I-beams were optimized within the range of the W4 through W8 designations. The optimization was limited by a maximum stress of 36 ksi. This resulted in the optimized I-beams as shown below. The stresses are shown in Figure 13.

Wall, floor and ceiling I-beam Roof I-beam
 
 

H = 9.000 H = 4.000

W = 8.227 W = 4.000

tf = .321 tf = .200

tw = .171 tw = .170

Ixx = 107.84 Ixx = 6.44

Iyy = 29.81 Iyy = 2.13

lb/ft = 23.27 lb/ft = 7.67
 
 


 
 

Figure 13. Optimized Structural Frame Analysis Results.


 
 


Based on the above optimized parameters, the standard sizes were chosen as shown below. Since the optimization was allowed over the full range of available parameters, the optimized combination of sizes was not necessarily available. Because the stresses of concern for this model were beam bending, emphasis was placed on the section properties (inertias and maximum distance from the neutral axis). Therefore, the "equivalent" beams were selected which had similar properties.

Wall, floor and ceiling I-beam Roof I-beam
 
 

H = 8.00 H = 4.16

W = 8.00 W = 4.06

tf = .433 tf = .345

tw = .288 tw = .280

Ixx = 110.00 Ixx = 11.30

Iyy = 37.00 Iyy = 3.76

lb/ft = 31.00 lb/ft = 13.00

W8 x 31 W4 x 13 (smallest available)
 
 

Because the selected I-beams are different than the optimized I-beams, an additional static analysis was performed to verify that the selected I-beams maintained the design allowables. The analysis resulted in a maximum stress of 29.9 ksi which is within the design allowables.
 
 

Conclusion. A finite element model of a steel framed house was created to perform a structural analysis of the structural members and to optimize the design for the lowest cost selection of structural members. Applicable loads were developed and applied to the members. The optimization analysis resulted in the selection of structural members as shown in Table XII.
 
 

Table XII. Final Structural Member Sizes.


 
 
 
Member
H
W
t or tf
tw
First Floor Ceiling Joist
8.00
1.625
.018 (25 ga)
NA
Second Floor Ceiling Joist
8.00
1.625
.018 (25 ga)
NA
Second Floor Floor Joist
14.00
2.000
.068 (14 ga)
NA
Roof Purlins
1.50
6.00
.043 (18 ga)
NA
Wall, Ceiling, Floor I-beam
8.00
8.00
.433
.288
Roof I-beam
4.16
4.06
.345
.280

 
 
 

References


 
 

1. "Prescriptive Method for Residential Cold-formed Steel Framing", First Edition, Department of Housing and Urban Development, May 1996.
 
 

2. . "Commentary on Prescriptive Method for Residential Cold-formed Steel Framing", First Edition, Department of Housing and Urban Development, May 1996.