stability analysis and finite element stress analysis of a · pdf file ·...

11500 Metric Boulevard • Suite 190

Austin, TX 78758 • (512) 219-8501

Stability Analysis and Finite Element Stress Analysis

of a

SolarWing©1 Carport Structure and Solar Panel Array

By

Stephen M. Manifold

Concurrent Design, Inc.

17 January 2014


Austin, TX 78758 • (512) 219-8501

I. Introduction

This report documents a stability and stress analysis of a SolarWing© carport installation

consisting of a welded steel column, the carport roof and the reinforced concrete slab. The

SolarWing© carport roof incorporates solar panels for energizing an electric vehicle charging

station. A solid model of the SolarWing© carport and concrete slab is shown in figures 1 and 2.

Figure 1. Solid model of SolarWing© carport and concrete base.

Figure 2. Underside of the SolarWing© carport and base, showing the solar panel supporting

framework and the trunnion support.


Austin, TX 78758 • (512) 219-8501

Motivation for the analysis is concern for the tipping stability of the carport about an edge of the

concrete slab under high wind conditions and/or ice and snow buildup on the solar panels. This

report describes expected loading conditions, stability calculations using closed form solutions

and results from a finite element stress and deflection analysis.

It should be noted that the solid model used in the analysis was still in a somewhat

conceptualized state at the time this analysis was done, and did not include certain small

connecting structures such as angled iron brackets and connecting hardware that are included in

the finished structure. Furthermore, the solid model did not include wiring and electrical

conduits used to electrically connect the solar panels to the charging station. Since the high

stress areas are in the base of the support column and the round support plate, the lack of fidelity

in the upper structure is of minor consequence.

II. Loads

External loads acting on the SolarWing© carport are shown in figure 3. Expected external loads

are ice and snow buildup on the solar panels, and wind loading. The wind loading was separated

into an x-direction (south wind) load that acts to tip the structure about an east-west (E-W) edge

of the slab, and a y-direction (east wind) load that acts to tip the structure about the north-south

(N-S) edge. The ice and snow load vector acts vertically downward and also results in a tipping

moment about the N-S edge only.

The structure weight produces stabilizing moments that are proportional to the distance from the

structure center of mass to the tipping edges of the slab.


Austin, TX 78758 • (512) 219-8501

Figure 3. External loads acting on the SolarWing© carport structure.

III. Stability Analysis

Determining the stability of the SolarWing© carport structure under wind loading and frozen

precipitation accumulation was done by calculating tipping moments about an edge of the

concrete slab.

For stability, the vector product of the structure weight and the location of the center of gravity

relative to the slab edge must be greater than the vector product of the destabilizing force and the

load application centroid relative to the slab edge.

That is, for stability:


Austin, TX 78758 • (512) 219-8501

W x rstr > F x rload (1)

Where

W = weight vector of the SolarWing© carport structure including the slab

rstr = position vector of the structure center of gravity relative to the slab edge

F = external load vector

rload = position vector of the load centroid relative to the slab edge

and x denotes the vector product operator.

This is shown graphically in figure 4 for the case of an east wind and a vertical load due to ice

and snow buildup on the solar panels. Figure 5 shows the case of a south wind and ice and snow

buildup. Note that in the case of tipping about the E-W edge, ice and snow buildup produces a

moment in the same direction as the weight moment, and thus acts to stabilize the structure.

Figure 4. An east wind and ice and snow loading act to cause tipping about the N-S edge of the

SolarWing© carport slab.


Austin, TX 78758 • (512) 219-8501

Figure 5. A south wind loading acts to cause tipping about an E-W edge of the SolarWing©

carport slab.

Structure weight and center of mass location

The total weight of the SolarWing© carport structure is 16200 lb, as calculated by Solidworks

for the entire solid model. This includes the steel structure, the solar panels and the steel-

reinforced concrete slab. Component weights are given in table 1. As stated earlier in this

report, the solid model did not include wiring and electrical conduits used to electrically connect

the solar panels to the charging station.

Component Weight density Weight

Slab-concrete .084 lb/in3 12422 lb

Slab-steel rebar .283 lb/in3 430 lb

Steel .283 lb/in3 2642 lb

Solar panels 47 lbs per panel 706 lb

Total 16200

Table 1. SolarWing© carport structure component weights.


Austin, TX 78758 • (512) 219-8501

The solar panel weight is manufacturer supplied data, and includes the aluminum mounting

frame. The slab weight includes both the concrete and the reinforcing steel, and was calculated

by first calculating the total volume of the reinforcing steel and subtracting that volume from the

overall volume of the slab to find the weight of the concrete portion of the slab. The concrete

density used in the analysis is published data for class A concrete. Figure 6 shows the

reinforcing steel in the slab prior to pouring the concrete, and figure 7 shows the finished slab.

Figure 6. Steel rebar in the slab.


Austin, TX 78758 • (512) 219-8501

Figure 7. The finished slab.

The center of mass of the carport structure as calculated by Solidworks is located at a distance of

34.7 in from the N-S tipping edge of the slab, and a distance of 48 in from the E-W tipping edge.

Calculation of wind loading

Wind loading acting on the SolarWing© carport structure is given by equation 2:

F = CdA(1/2v2) (2)

Where:

Cd = drag coefficient of the structure

A = frontal area acted on by the wind

= mass density of air

v = wind velocity

In this analysis, a conservative value of 2 was used for the drag coefficient, and air density was

assumed to be 1.21x10-7

lb-sec2/in

4, which corresponds to the air density at a temperature of 0 C.

Note that the density used is the mass density and not the weight density.

The frontal areas for the south and east wind loadings, and their centroid locations, were

provided by the solid model. For the east wind loading, the frontal area is 2510 in2 and its


Austin, TX 78758 • (512) 219-8501

centroid is at a height of 117 in above the tipping point. The south wind loading frontal area

depends on the tilt angle of the roof, which is 10 degrees, giving an area of 10527 in2. The

centroid of the south wind loading is at a height of 128 in.

Wind pressure and wind loads for the east and south directions are presented in figure 8 as a

function of wind speed.

Figure 8. Wind loads vs wind speed. The x direction corresponds to a south wind, and the y

direction corresponds to an east wind.

The stability analysis assumes a wind speed of 115 mph, resulting in a wind pressure of

approximately .50 lb/in2 (72 psf) and a total south wind loading of 5240 lbs and a total east wind

loading of 1250 lbs.

Calculation of ice and snow loading

Stability calculations assumed an ice buildup of 1 inch, and a snow buildup of 3 inches on top of

the ice.

Values of ice and snow density used in the analysis are .0332 lb/in3 and .029 lb/in

3, respectively.

The density of snow used in the analysis corresponds to a value for wet snow.

0

2

4

6

8

10

12

14

0 50 100 150 200

Wind speed (mph)

Wind pressure (psi)

x-direction wind load (kips)

y-direction wind load (kips)


Austin, TX 78758 • (512) 219-8501

The total cross-sectional area of the solar panels is 39052 in2, giving the downward acting loads

shown in table 2.

Ice Snow

Weight density .033 lb/in3 .029 lb/in

3

Thickness 1in 3in

Load 1300 lb 3400 lb

Total load on carport

roof 4700 lb

Table 2. Densities and load calculations for ice and snow buildup, based on a total roof surface

area of 39052 in2.

The center of mass of the ice and snow is at a distance of 31.2 in from the N-S tipping edge of

the slab, and a distance of 48 in from the E-W tipping edge.

Stability calculations

Stability calculations are done by considering the forces and moments shown in table 3 for

tipping about the N-S edge of the slab, and table 4 for tipping about the E-W edge of the slab.

Note than in tables 3 and 4, moment arms for stabilizing loads are given a positive sign, whereas

moment arms for destabilizing loads are given a negative sign. The resulting sum of the

moments is positive, indicating that the stabilizing moment is sufficient to prevent tipping.

In table 3, the net moment is 289 x103 lb-in, indicating that the weight of the structure is more

than sufficient to offset the tipping loads produced by wind and precipitation accumulation.

Similarly, table 4 shows the same result for tipping about the E-W edge of the slab. Note that in

the case of tipping about the E-W edge, frozen precipitation accumulation actually improves

stability, so the net moment was calculated for a worst case scenario of wind loading only.

Load Magnitude Moment arm Moment

Structure weight 16200 lb 34.7 in 562 x103 lb-in

Wind 1248 lb -117 in -146 x103 lb-in

Ice and snow 4700 lb -31.2 in -147 x103 lb-in

Sum total 269 x103 lb-in

Table 3. Moments calculated for loads acting about the N-S edge of the slab.


Austin, TX 78758 • (512) 219-8501

Load Magnitude Moment arm Moment

Structure weight 16200 lb 48 in 778x103 lb-in

Wind 5236 lb -112 in -670 x103 lb-in

Ice and snow 0 lb 48 in 0 lb-in

Sum total 107 x103 lb-in

Table 4. Moments calculated for loads acting about the E-W edge of the slab.

IV. Finite Element Stress Analysis

A finite element stress analysis was also done to determine stresses and deflections in the

SolarWing© carport structure due to wind and frozen precipitation loading. The focus of the

analysis was stress at the base of the support column, since that is where internal bending

moments will be highest. Calculation of stress in the solar panels is beyond the scope of this

analysis due to insufficient detail in the solar panel solid models.

The analysis was done using Solidworks Simulation and was conducted within the framework of

linear infinitesimal elasticity theory. A nonlinear, large deflection analysis was not done due to

software limitations. Although the resulting deflections under some loading conditions were

fairly large due to the significant flexibility of the structure, error introduced into the analysis due

to the assumption of infinitesimal deflection is not expected to be great.

Finite element mesh

A finite element mesh was created for the structure including the slab and is shown in figures 9

and 10. The mesh was constructed predominantly of second order tetrahedral elements with

reduced integration, and incorporated local refinement for areas where high strain gradients

occur. Some parts of the upper solar panel support structure were modeled as shell elements in

order to minimize the total number of degrees of freedom in the model by taking advantage of

their thin-walled nature. This is shown in figure 11, where the closed box beams that support the

solar panels are modeled as shell elements. In figure 11 the shell elements are colored orange for

visualization.

Material properties

Constitutive properties of the materials used in the analysis are given in table 4.


Austin, TX 78758 • (512) 219-8501

Material Modulus Poisson ratio

AISI 1045 steel 30 msi .3

Concrete 3.4 msi .3

Solar panel 10 msi .3

Table 4. Materials used in the finite element stress analysis.

Figure 9. Finite element mesh of the SolarWing© carport structure.


Austin, TX 78758 • (512) 219-8501

Figure 10. Detail of the finite element mesh, showing localized mesh refinement.

Figure 11. Detail of the finite element mesh, showing channels modeled as shell elements

(colored orange).


Austin, TX 78758 • (512) 219-8501

Boundary conditions

The bottom of the slab was fixed in all directions to prevent any rigid body motion of the model.

Contact surfaces

All contact surfaces were modeled as bonded. This is a valid assumption as long as the bolt

preload at component connections is not exceeded by forces acting to separate the components,

and greatly expedites analysis turn-around time. Several components are welded together, such

as the main structural support column and the circular bottom plate, and a bonded contact surface

definition is appropriate. The actual welds were not modeled.

Loads and load cases

Loads described in the section on stability analysis were used in the stress analysis. Several

cases were run and are described in table 5.

Case 115 mph south wind 115 mph east wind ice and snow

1 (baseline) no no no

2 no no yes

3 yes no no

4 yes no yes

5 no yes no

6 no yes yes

Table 5. Load cases analyzed in the finite element stress analysis.

Finite element results

As expected, increasing loads result in increased deflection. Table 6 presents the maximum

deflection values for the six load cases. Figure 12 shows the deflection for the baseline case, and

figure 13 shows the worst case deflection of case 4.


Austin, TX 78758 • (512) 219-8501

Case Vertical

deflection

X-direction

deflection

Y-direction

deflection

Resultant

deflection

1 -.8 in .1 in -.3 in .9 in

2 -2.4 in .4 in -.8 2.5 in

3 -2.1 in -5.1 in -2.1 in 5.7 in

4 -2.9 in -5.2 in -2.6 in 6.1 in

5 -1.8 in -.4 in -1.0 in 2.1 in

6 -3.4 in -.7 in -1.5 in 3.8 in

Table 6. Maximum SolarWing© carport structure deflections for each load case.

Figure 12. Deflection of the SolarWing© carport for the baseline case (case 1).


Austin, TX 78758 • (512) 219-8501

Figure 13. Deflection of the SolarWing© carport for the worst case loading (case 4).

Stress in the column for the baseline case is shown in figures 14 and 15. Stress concentrations

exist at the corners of cutouts in the sheet metal fasciae at each side of the support column.

These fasciae are non-structural and are not relied on to carry load.

Figure 16 shows Von Mises stress for the baseline case in the column with the fasciae removed

for clarity. Stress concentrations exist at the corners of the structural column where it is welded

to the bottom plate. Welds are not modeled in this analysis and therefore the value of the

calculated stress is suspected to be greater than actual, but undoubtedly there is a stress

concentration in that area. The area of high stress is transmitted through the welds to the circular

bottom plate that is bolted to the studs rising out of the concrete base. All cases analyzed for this

report show the same pattern of high stress at the base of the column where it is joined to the

bottom plate, and in the bottom plate itself.


Austin, TX 78758 • (512) 219-8501

Figure 14. Von Mises stress for the baseline case.

Figure 15. Von Mises stress for the baseline case.


Austin, TX 78758 • (512) 219-8501

Figure 16. Von Mises stress for the baseline case, fasciae removed for clarity.

Figure 17 shows the Von Mises stress at the base of the column for the worst case loading

condition (case 4, 115 mph south wind and ice and snow accumulation). In figure 17 the range

of colors depicting stress is selected so that red is above the material yield strength of 45 ksi. As

can be seen, the support column and the bottom plate will experience some yielding. Since this

is a linear analysis in which the material stress-strain curve is not modeled beyond the elastic

limit, the maximum stress value, the maximum strain and the extent of yielding is unknown.

Von Mises stresses in the support column base and circular support plate for the six load cases

analyzed are given in table 7. Cases in which the stress was above the material yield strength of

45 ksi were simply noted as above yield.


Austin, TX 78758 • (512) 219-8501

Case Von Mises stress Safety factor

1 20 ksi 2.2

2 40 ksi 1.1

3 > 45 ksi < 1

4 > 45 ksi < 1

5 > 45 ksi < 1

6 40 ksi 1.1

Table 7. Von Mises stress in the support column base, and the safety factor relative to yield.

Figure 17. Von Mises stress for case 4, 115 mph south wind and ice and snow accumulation.

Stresses in the upper portion of the carport are negligible for the baseline case and for case 2 (ice

and snow loading only), and are greater in the cases involving high wind loading and

accumulation of ice and snow, although the analysis predicts no yielding even for the worst

loading case. Figure 18 shows the worst case stresses in the upper portion of the carport, with

stresses approaching 30 ksi. Table 8 presents the maximum Von Mises stress in the upper

portion of the carport for the six load cases.


Austin, TX 78758 • (512) 219-8501

Case Von Mises stress Safety factor

1 8 ksi 5

2 20 ksi 2.2

3 22 ksi 2

4 30 ksi 1.5

5 10 ksi 4

6 18 ksi 2.5

Table 8. Von Mises stress in the upper portion of the SolarWing© carport structure, and the

safety factor relative to yield.

Figure 18. Von Mises stresses in the upper portion of the SolarWing© carport structure for case

4 (115 mph south wind, ice and snow accumulation.


Austin, TX 78758 • (512) 219-8501

V. Summary and Conclusions

The stress and stability analyses were based on conservative estimates of air mass density and

drag coefficients, and a conservative value for the weight density of snow.

The stability analysis shows that the SolarWing© carport and concrete base structure is

sufficiently heavy with a low enough center of gravity to resist tipping moments produced by a

combined loading of 115 mph wind and a four inch accumulation of frozen precipitation.

The stress analysis shows that although the SolarWing© carport structure is quite flexible and

deflects significantly under severe loading conditions, the structure is robust and can withstand

loading cases of 115 mph winds and frozen precipitation accumulation. The structure will

experience some localized yielding at the base of the support column, the circular support plate

and the cut-outs in the curved fasciae under worst case loading conditions. The column material

(AISI 1045 steel), is a work-hardening material with excellent ductility (elongation of 16% and

reduction of area of 40% at its strain limit). A nonlinear analysis with the AISI 1045 stress-strain

curve as input would have to be done to verify, but it is conjectured that for the loads considered

in this analysis, material yielding in the column would result in work hardening and

redistribution of internal stress, and would be limited to localized areas around stress

concentrations. It is not expected that a catastrophic collapse of the structure would occur for the

loads investigated in this analysis.

Credits:

1 –Copyright © SolarWing 2013. All rights reserved. Patents Pending.


Austin, TX 78758 • (512) 219-8501

APPENDICES


Austin, TX 78758 • (512) 219-8501

Table of Contents

Appendix A

Figure A1 – Deflection for case 1.






Appendix B

Figure B1. Von Mises stress in lower support structure, case 1.






Appendix C

Figure C1. Von Mises stress in upper support structure, case 1.







Austin, TX 78758 • (512) 219-8501

Figure A1. Deflection for case 1.



Austin, TX 78758 • (512) 219-8501




Austin, TX 78758 • (512) 219-8501


Figure A6. Deflection for case 6


Austin, TX 78758 • (512) 219-8501




Austin, TX 78758 • (512) 219-8501

Figure C3. Von Mises stress in lower support structure, case 3.



Austin, TX 78758 • (512) 219-8501



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