NTU ME 720A
(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 Ibeams.
The Ibeams are eight inches thick in order to maintain the desired insulation.
Between the Ibeams, Csection members are used for ceiling and floor joists.
Purlins (in hat section shapes) are used between the Ibeams 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.





Ceiling Joists 


Floor Joists 



Roof Purlins 




Structural Ibeams 

* Loads on the structural Ibeams 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
Ibeams 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 crosssections cannot be modeled exactly in Pro/Mechanica. Instead,
the joists and purlins were modeled using symmetric beams with equivalent
sectional properties. Because the structural Ibeams 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.



Joists ^{1} 


Purlins ^{1} 


Ibeams ^{2} 


1. Coldformed rolled steel.
2. Hotformed structural steel.
The joist Csection crosssection is shown in Figure
3. Commercially available sections are available to the parameters shown
in Table IV.
Figure 3. Joist Crosssection.
Table IV. Joist Parameters.
Height (H) 

Width (W) 

Thickness (t) 
(25, 22, 20, 18, 16, 14, 12 ga) 
The purlin crosssection is shown in Figure 4. Commercially
available sections are available to the parameters shown in Table V.
Figure 4. Purlin Crosssection.
Table V. Purlin Parameters.
Height (H) 

Width (W) 

Thickness (t) 
(25, 22, 20, 18, 16, 14, 12 ga) 
The Ibeam crosssection 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 Ibeams were selected from the W8 designation which has
a nominal height of 8 inches. Roof section Ibeams were selected from W4
through W8 designations since the lighter roof loads could allow for a
smaller Ibeam. 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. Ibeam Crosssection.
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 Ibeam.
First Floor Ceiling Joists. The available
ceiling joists crosssections 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 oncenter
(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 

Total Load Deflection 

Allowable Stress 

As discussed earlier, the joist crosssection 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 

Total Load Deflection 

Allowable Stress 

Again, this joist was modeled using a rectangular
crosssection 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 

Total Load Deflection 

Allowable Stress 

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 crosssection 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 crosssection. 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 crosssections 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 

Total Load Deflection 

Allowable Stress 

As with the previous analysis, the purlin was modeled
using a rectangular crosssection 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 Ibeam 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 

Total Load Deflection 

Allowable Stress 

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 crosssections 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 Ibeams 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.










22.6 normal to roof 


22.6 normal to roof 25.7 vertical 
25.7 vertical 

377.0 vertical 


37.7 vertical 24.5 normal to roof 
24.5 normal to roof 

22.6 normal to roof 


24.5 normal to roof 


24.5 normal to roof 













Figure 11. Updated Structural Frame.
Before optimizing the size of the structural Ibeams,
an initial static analysis was performed to the largest available Ibeam
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 Ibeam Stress Analysis.
An optimization analysis was then performed on the structural frame to minimize weight which will yield the smallest Ibeam and therefore the lowest cost solution. The wall, floor and ceiling Ibeams were optimized within the available sizes of the W8 designation. The roof Ibeams 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 Ibeams as shown below. The stresses are shown in Figure 13.
Wall, floor and ceiling Ibeam Roof Ibeam
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 Ibeam Roof Ibeam
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 Ibeams are different than the
optimized Ibeams, an additional static analysis was performed to verify
that the selected Ibeams 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.





First Floor Ceiling Joist 




Second Floor Ceiling Joist 




Second Floor Floor Joist 




Roof Purlins 




Wall, Ceiling, Floor Ibeam 




Roof Ibeam 




References
1. "Prescriptive Method for Residential Coldformed
Steel Framing", First Edition, Department of Housing and Urban Development,
May 1996.
2. . "Commentary on Prescriptive Method for Residential Coldformed Steel Framing", First Edition, Department of Housing and Urban Development, May 1996.