OS-V: 0080 Buckling of Shells and Composites with Offset

A test of influence of offset on buckling solution for shells, including composite with offset Z0 and element offset ZOFFS.



Figure 1. FE Model of the Beam with Boundary Conditions and Loadcases

Benchmark Model

Here, you solve several problems to calculate the critical load on different conditions. The model is a simply supported beam of height 1mm, breadth 2mm and length 100mm with one end constrained in all DOFs and an axial load applied on the other end.

The material properties for the beam are:
MAT1
Young's Modulus
1 x 106 N/mm2
Poisson's Ratio
0.0
Density
2 kg/mm3
Thermal Expansion Coefficient
1 x 10-4 ºC-1
Reference Temperature for Thermal Loading
300ºC
The different case description of the problem are:
  1. Buckling without offset.
  2. Buckling with moment equivalent to offset.
  3. Buckling with offset created by a frame.
  4. Buckling with offset applied through ZOFFS.
  5. Buckling of composite with non-symmetrical layup.
  6. Buckling of composite with offset.
The theoretical critical buckling load is calculated using the Euler Buckling equation:(1)
f crit =π EI ( KL ) 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOzamaaBa aaleaacaWGJbGaamOCaiaadMgacaWG0baabeaakiabg2da9iabec8a WnaalaaabaGaamyraiaadMeaaeaadaqadaqaaiaadUeacaWGmbaaca GLOaGaayzkaaWaaWbaaSqabeaacaaIYaaaaaaaaaa@435B@
Where,
f crit MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOzamaaBa aaleaacaWGJbGaamOCaiaadMgacaWG0baabeaaaaa@3AD3@
Maximum or critical force
E MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyraaaa@36C0@
Modulus of elasticity
I MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamysaaaa@36C4@
Area moment of inertia (second moment of area)
L MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamitaaaa@36C7@
Unsupported length of the beam
K MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4saaaa@36C6@
Column effective length factor (for one end fixed and the other end free, K MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4saaaa@36C6@ =2)

Results



Figure 2. First Four Buckling Eigenvalues for Non-offset (z0 = -0.5)
Quantity Theoretical No-offset Normalized
λ cr(1) 4.1123 4.1208 0.997937
λ cr(2) 16.449 16.513 0.996124
λ cr(3) 37.011 37.701 0.981698
λ cr(4) 102.81 108.19 0.950273


Figure 3. First Four Buckling Eigenvalues for Non-offset + Moment . (the effect of offset is simulated by adding a moment at the end of the beam)
Quantity Theoretical No-offset + Moment Normalized
λ cr(1) 4.1123 4.1208 0.997937
λ cr(2) 16.449 16.513 0.996124
λ cr(3) 37.011 37.701 0.981698
λ cr(4) 102.81 108.19 0.950273


Figure 4. First Four Buckling Eigenvalues for C-Frame. (the effect of offset is simulated by creating a C-shaped frame)
Quantity Theoretical C-Frame Normalized
λ cr(1) 4.1123 4.1208 0.997937
λ cr(2) 16.449 16.513 0.996124
λ cr (3) 37.011 37.700 0.981724
λ cr(4) 102.81 108.19 0.950273


Figure 5. First Four Buckling Eigenvalues for z-offset (Zoffs = -0.5)
Quantity Theoretical ZOFFS Normalized
λ cr(1) 4.1123 4.1208 0.997937
λ cr(2) 16.449 16.513 0.996124
λ cr(3) 37.011 37.700 0.981724
λ cr(4) 102.81 108.19 0.950273


Figure 6. First Four Buckling Eigenvalues for Non-symmetric Layup . (since the top layer is very weak, the load is applied to the “strong” layer with an offset of 0.5)
Quantity Theoretical Non-symmetric Layup Normalized
λ cr(1) 4.1123 4.1203 0.998058
λ cr(2) 16.449 16.510 0.996305
λ cr(3) 37.011 37.663 0.982689
λ cr(4) 102.81 107.89 0.952915


Figure 7. First Four Buckling Eigenvalues for Composites with Offset (z0 = -1)
Quantity Theoretical Offset Composite Normalized
λ cr(1) 4.1123 4.1203 0.998058
λ cr(2) 16.449 16.510 0.996305
λ cr(3) 37.011 37.663 0.982689
λ cr(4) 102.81 107.89 0.952915

Model Files

Refer to Access the Model Files to download the required model file(s).

The model files used in this problem include:

s100_buckl.zip
  • s100comp_buckl.fem
  • s100compmom_buckl.fem
  • s100comp_frame_buckl.fem
  • s100comp_buckl_zoffs.fem
  • s100comp2ply_buckl.fem
  • s100compoffs_buckl.fem