Bar Subjected to Self-Weight

Let us consider a bar of radius R and length L, suspended at the upper edge and stretched under the action of self-weight (see figure).

Bar Subjected to Self-Weight, the finite element model with loads and restraints

 

 

The finite element model with applied loads and restraints

Let us use the following data: length of bar L is equal to 1 m, radius of cross-section of the bar R is equal to 0.02 m.
Material characteristics: E = 2.1E+011 Pa, ν  = 0.28, ρ = 7800 kg/m3 .
Total elongation of the bar under the action of the self-weight can be determined from the formula:
ΔL = γL2/2E,
where γ – specific weight of the bar's material, that is γ = ρg, g ~ 9.80665 m/s2.
The stress in the cross-section of the bar located at a distance x from lower (unconstrained) edge can be evaluated from formula:
σ = γx
Thus, ΔL = 1.8212E-007 m; σ = 3.8246E+004 Pa at x = 0.5L.

 

After carrying out calculation with the help of AutoFEM, the following results are obtained:

Table 1.Parameters of finite element mesh

Finite Element Type

Number of nodes

Number of finite elements

quadratic tetrahedron

440

972

Table 2.Result "Displacement, magnitude"*

Numerical Solution
Displacement w*, m

Analytical Solution
Displacement w, m

Error δ = 100%* |w* - w| / |w|

1.8173E-007

1.8212E-007

0.21

Table 3. Result "Equivalent Stress"*

Numerical Solution
Stress σ*, Pa

Analytical Solution
Stress σ, Pa

Error δ = 100%* |σ* - σ| / |σ|

3.8246E+004

3.8246E+004

0.001

 

Bar Subjected to Self-Weight, the result "Dispacement, magnitude" of finite element modelling     Bar Subjected to Self-Weight, the result "Sresses" of finite element modelling

Conclusions:

The relative error of the numerical solution compared to the analytical solution is 0.21% for displacements and 0,001% for stresses when using quadratic finite elements.

*The results of numerical tests depend on the finite element mesh and may differ slightly from those given in the tables.

 

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