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D ISTORTIONAL B UCKLING O F C A ND Z M EMBERS I N B ENDING. Progress Report to AISI. Cheng Yu, Benjamin W. Schafer The Johns Hopkins University August 2004. Overview. Test Summary Finite Element Modeling Extended FE Analysis Stress Gradient Effects Conclusions. Local buckling tests.

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d istortional b uckling o f c a nd z m embers i n b ending
DISTORTIONAL BUCKLING OFCAND ZMEMBERSIN BENDING

Progress Report to AISI

Cheng Yu, Benjamin W. Schafer

The Johns Hopkins University

August 2004

overview
Overview
  • Test Summary
  • Finite Element Modeling
  • Extended FE Analysis
  • Stress Gradient Effects
  • Conclusions
test summary
Test Summary
  • 5 more tests were performed since last report in February 2004
  • Total 24 distortional buckling tests have been done. All available
  • geometry of sections in the lab have been tested.

Comparison with design methods

slide6

Test Summary -

Performance of Direct Strength Method

Test results vs. Direct Strength predictions

finite element modeling
Finite Element Modeling
  • Shell element S4R for purlins, panel and tubes, solid element C3D8
  • for transfer beam.
  • Geometric imperfection is introduced by the superposition of local
  • and distortional buckling mode scaled to 25% or 75% CDF.
  • Residual stress is not considered.
  • Stress-strain based on average of 3 tensile tests from the flats of every specimen
  • Modified Riks method and auto Stabilization method in ABAQUS were considered for the postbuckling analysis. The latter has better results and less convergence problems therefore the auto Stabilization is used.
  • The FE model was verified by the real tests.
slide8

Finite Element Modeling -

Comparison with test results

On average:

FEM-to-test ratio= 106% for 25% CDF; 93% for 75% CDF --- local buckling tests

FEM-to-test ratio= 109% for 25% CDF; 94% for 75% CDF --- distortional buckling tests

slide9

Extended Finite Element Analysis

FEA results vs. Direct Strength predictions

slide10

p

Moment diagram

Stress Gradient Effect on Thin Plate -

Moment gradient on beams

compression

Stress diagram of top flange

Stress diagram of bottom flange

tension

slide11

Stress Gradient Effect on Thin Plate -

Plate buckling

Hat section

C section

Buckling of uniformly compressed rectangular plates

slide12

Stress Gradient Effect on Thin Plate –

Analytical model

Stiffened element

Stress distribution:

Deflection function:

(by Libove 1949)

slide13

Stress Gradient Effect on Thin Plate –

Analytical model

Unstiffened element

Stress distribution:

slide14

Stress Gradient Effect on Thin Plate –

Analytical model

Finite element analysis by ABAQUS is used to verify these 3 deflection functions.

Bucking shape by FEA

Bucking shape by analytical model

Average analytical result-to-FEA ratios are

Deflection function 1: 102.4%

selected

Deflection function 2: 99.7%

Deflection function 3: 99.6%

slide15

Stress Gradient Effect on Thin Plate –

Stiffened Element Results

kmax vs. plate aspect ratio (β) for ss-ss stiffened element

(recalculation of Libove’s equations 1949 )

slide16

Stress Gradient Effect on Thin Plate –

Unstiffened Element Results

kmax vs. plate aspect ratio (β) for ss-free unstiffened element

slide17

Stress Gradient Effect on Thin Plate –

(r=0) Results

0

Comparison of stiffened and unstiffened elements subject to stress gradient r=0

kmax= buckling coefficient at the maximum stress edge

k0= buckling coefficient for plates under uniform compression stress

slide18

Stress Gradient Effect on Thin Plate –

Ultimate strength

Winter curve ---

ABAQUS r=1 --- plate under uniform compression stress

ABAQUS r=0 --- plate under stress gradient, stress is only applied at one end

conclusions
Conclusions
  • Tests that separate local and distortional buckling are necessary for understanding bending strength.
  • Current North American Specifications are adequate only for local buckling limit states.
  • The Direct Strength expressions work well for strength in local and distortional buckling.
  • Nonlinear finite element analysis with proper imperfections provides a good simulation.
  • Extended finite element analysis shows that DSM provides reasonable predictions for strengths in local and distortional buckling.
slide20

Conclusions - continued

  • An analytical method for calculating the elastic buckling of thin plate under stress gradient is derived and verified by the finite element analysis.
  • Plate will buckle at higher stress when stress gradient exists. The stress gradient has more influence on the unstiffened element than stiffened element.
  • Study on the ultimate strength of plate under stress gradient has been initialized. Up-to-date results show Winter’s curve works well for stiffened element under stress gradient.
  • More work on restraint and influence of moment gradients will be carried out by the aid of the verified finite element model.
acknowledgments
Acknowledgments
  • Sponsors
    • MBMA and AISI
    • VP Buildings, Dietrich Design Group andClark Steel
  • People
    • Sam Phillips – undergraduate RA
    • Tim Ruth – undergraduate RA
    • Jack Spangler – technician
    • James Kelley – technician
    • Sandor Adany – visiting scholar
slide22

Stress Gradient Effect on Thin Plate –

Energy method

Total potential energy:

+

( term for the elastic restraint if exists)

When buckling happens:

  • Need two assumptions to solve the elastic buckling stress:
  • the stress distribution in plate:
  • the deflection function:
slide23

Stress Gradient Effect on Thin Plate –

Analytical model

3 deflection functions are considered for the unstiffened element:

1.

2.

3.

slide24

Stress Gradient Effect on Thin Plate –

Analytical model

The coefficients in the assumed deflection function are determined by applying to

the 6 boundary conditions:

1.

2.

3.

(no deflection)

(no moment)

4.

5.

6.

(no shear force)