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**Cris Moen, Virginia Tech Cheng Yu, University of North Texas**

Elastic Buckling of Thin-Walled Structural Components with Stiffened Holes Cris Moen, Virginia Tech Cheng Yu, University of North Texas 51st AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference Monday, April 12, 2010

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**Outline Where are stiffened holes used in structures?**

Intro to thin-walled component design Elastic buckling including stiffened holes: Global buckling Distortional buckling Local buckling Conclusions

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**Applications Aerospace Metal buildings**

Fascination with perforations Aerospace Metal buildings Minimize weight, provide pass through access

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**Outline Where are stiffened holes used in structures?**

Intro to thin-walled component design Elastic buckling including stiffened holes: Global buckling Distortional buckling Local buckling Conclusions

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**Elastic buckling curve**

Buckling modes (and loads) are revealed… Pcrl Fatigue, stress concentrations, and capacity at the net cross section are also important Pcrd Pcre Schafer, B. W., and Ádàny, S. "Buckling analysis of cold-formed steel members using CUFSM: conventional and constrained finite strip methods," Eighteenth International Specialty Conference on Cold-Formed Steel Structures. Orlando, FL, 2006.

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**Thin-walled buckling modes**

Mode shapes and buckled half-wavelengths are consistent with elastic buckling curve Fatigue, stress concentrations, and capacity at the net cross section are also important Moen, C. D., and Schafer, B. W. "Experiments on cold-formed steel columns with holes," Thin-Walled Structures Vol. 46, 2008, pp

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Design curves Use elastic buckling loads with design curves to predict capacity. For thin-walled members, we have design curves for local and global instability limit states. Fatigue, stress concentrations, and capacity at the net cross section are also important

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Design curves Members with holes – calculate elastic buckling properties including holes and use similar design curves Fatigue, stress concentrations, and capacity at the net cross section are also important Moen, C. D., and Schafer, B. W. "Direct Strength Design of Cold-Formed Steel Columns with Holes," 2010 Annual Technical Session and Meeting, Structural Stability Research Council. Orlando, FL, 2010.

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**Elastic buckling curve**

Shell finite element eigen-buckling analysis is slightly painful but needed for members with holes. Fatigue, stress concentrations, and capacity at the net cross section are also important

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Motivation Simplified methods were recently developed for predicting the elastic buckling properties of thin-walled structural components with unstiffened holes. Moen, C. D., and Schafer, B. W. "Elastic buckling of cold-formed steel columns and beams with holes," Engineering Structures Vol. 31, No. 12, 2009, pp Fatigue, stress concentrations, and capacity at the net cross section are also important Are these methods viable for stiffened holes?

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**Outline Where are stiffened holes used in structures?**

Intro to thin-walled component design Elastic buckling including stiffened holes: Global buckling Distortional buckling Local buckling Conclusions

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Global buckling Fatigue, stress concentrations, and capacity at the net cross section are also important Approximate the critical elastic buckling loads for columns and beams with holes using a weighted average hand calculation.

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**Flexural buckling v Rayleigh-Ritz Energy Solution:**

Assume v(x) as half sine wave For those of you who are teaching stability this semester, this may look very familiar v

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**Flexural buckling Weighted average of moment of inertia!**

The solution is a “weighted average” equation for Euler buckling. Evenly spaced holes Weighted average of moment of inertia! Holes at the midlength of a column reduce Pcre the most Arbitrary hole spacing Strain energy contributions are highest at the midlength of the column

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Global buckling Fatigue, stress concentrations, and capacity at the net cross section are also important Hole spacing is 500 mm. Column length is 2500 mm. Hole diameter is varied from 0.1 to 0.7 of H. Q=0.12hhole Yu, C. "Behavior and design of cold-formed steel joists with edge stiffened perforations," 2007 Annual Stability Conference, April 18, April 21, Structural Stability Research Council, New Orleans, LA, United States, 2007, pp

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**Flexural buckling For evenly spaced holes:**

Strain energy contributions are highest at the midlength of the column Holes at the midlength of a column reduce Pcre the most * ABAQUS results are systematically 6% lower than the Euler buckling solution. The difference is caused by the assumption of a rigid cross-section in the classical stability equations.

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**Torsional buckling YES! Need to investigate…**

The “weighted average” approximation can be extended to the classical cubic column buckling equation… Timoshenko and Gere (1961), Chajes (1974) and the equation for lateral torsional buckling of beams with holes. YES! Need to investigate… J St. Venant torsion Cw Warping torsion Cw does not follow the weighted average approximation!

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**A comment on warping torsion**

Holes prevent warping torsion resistance from developing along the length of a member. Buckling modes dominated by torsion are especially sensitive to the presence of holes.

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**Outline Where are stiffened holes used in structures?**

Intro to thin-walled component design Elastic buckling including stiffened holes: Global buckling Distortional buckling Local buckling Conclusions

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**Distortional buckling**

For evenly spaced holes: The web provides rotational restraint to the flange. How does a stiffened hole affect this restraint? Strain energy contributions are highest at the midlength of the column Holes at the midlength of a column reduce Pcre the most

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**Distortional buckling**

For evenly spaced holes: Strain energy contributions are highest at the midlength of the column Holes at the midlength of a column reduce Pcre the most Study the hole influence within one distortional buckling half-wave.

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**Distortional buckling**

For evenly spaced holes: Edge stiffener increased web bending stiffness Strain energy contributions are highest at the midlength of the column Holes at the midlength of a column reduce Pcre the most

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**Distortional buckling**

Hole spacing is 560 mm. Column length is 2794 mm. Q=0.12hhole For evenly spaced holes: Strain energy contributions are highest at the midlength of the column Holes at the midlength of a column reduce Pcre the most

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**Outline Where are stiffened holes used in structures?**

Intro to thin-walled component design Elastic buckling including stiffened holes: Global buckling Distortional buckling Local buckling Conclusions

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**Local buckling For evenly spaced holes:**

When an unstiffened hole is added, deformation concentrates at the hole. Strain energy contributions are highest at the midlength of the column Elastic buckling stress is reduced by the presence of a hole.

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**Local buckling For evenly spaced holes:**

Depending upon the hole size and plate geometry, buckling may occur away from a hole. For evenly spaced holes: Strain energy contributions are highest at the midlength of the column Elastic buckling stress is unaffected by the hole. Moen, C. D., and Schafer, B. W. "Elastic buckling of thin plates with holes in compression or bending," Thin-Walled Structures Vol. 47, No. 12, 2009, pp

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**Local buckling Pcrl=min(Pcrlnh, Pcrlh)**

Prediction method – assume buckling occurs at a hole or away from (between) holes. Pcrl=min(Pcrlnh, Pcrlh) For evenly spaced holes: Strain energy contributions are highest at the midlength of the column Buckling loads be calculated with a finite strip analysis at the gross section and the net cross section.

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**Local buckling For evenly spaced holes:**

An important benefit of stiffened holes! Strain energy contributions are highest at the midlength of the column Holes at the midlength of a column reduce Pcre the most

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**Local buckling For evenly spaced holes:**

Wavelength stiffening occurs when holes are spaced closely together. Strain energy contributions are highest at the midlength of the column

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**Conclusions For evenly spaced holes:**

Simplified elastic buckling prediction methods outlined in Moen and Schafer (2009) are viable for stiffened holes (more validation work is needed for other cross-sections and hole sizes, shapes, spacings,…) Edge stiffeners did not improve global buckling loads because I, J, and Cw all decreased due to the presence of holes Distortional buckling loads were minimally affected by the presence of stiffened holes, although new hole modes with half waves forming between holes were observed Edge stiffeners prevented local buckling at holes, and closely spaced holes boosted the critical elastic buckling load. For evenly spaced holes: Strain energy contributions are highest at the midlength of the column Holes at the midlength of a column reduce Pcre the most

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**Thank you for your attention! This presentation can be downloaded at **

Fascination with perforations Thank you for your attention! This presentation can be downloaded at

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Engineering tools Fatigue, stress concentrations, and capacity at the net cross section are also important

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**Thin-walled component design**

Thin-walled components are susceptible to failure caused by cross-section and/or global instabilities. Elastic buckling properties are useful for predicting structural capacity. How do holes influence elastic buckling? Can edge stiffeners improve performance and how can we quantify this improvement? Fatigue, stress concentrations, and capacity at the net cross section are also important

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ENCE 455 Design of Steel Structures

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