Geometry, Mesh, and Properties

Rectangular domain. 1 m long, 0.1 m tall. Q4 elements. 20 elements along the length x 20 elements across the height.

Material Properties

Parameter	Symbol	Value
Young's Modulus	$E$	$200GPa$
Poisson's Ratio	$\nu$	$0.25$
Damping Coefficient	$C$	$1$ $N\cdot s/m$

Plane stress conditions applied.

Simulation Properties

Parameter	Value
Start Time ($t_{0}$)	$0s$
End Time ($t_{end}$)	$10s$
$\Delta t$	$0.1s$
$\beta = 0.5$ (CN)	$0.5$
Length ($L$)	$1m$

Boundary and Initial Conditions

B.C $\rightarrow$ $u(x=0,y,t) = 0$, $u(x=0,y=0,t) = 0$

I.C $\rightarrow$ $u(x,y,t=0) = 0

$\bar{t}_{x}(x = L,y,t) = 1000sin(t)$ $KN$

Analytical Solution

The 2D FEM solution is compared to a 1D Mass-Spring-Damper system with the following exact solution:

$u(x=L,y,t) = \frac{F}{C^{2}+E^{2}}*\left(Esin(t)-Ccos(t)\left(1-e^{-Et/C}\right)\right)$

where, F = 2100 KN, C is the damping coefficient, E is the Young's Modulus.

It should be mentioned that the numerical solution compares the displacement at the midpoint in the y-direction of the beam at the free-end (i.e., node 863) over time with the analytical solution. Below are various figures illustrating the displacement in the bar at t = 5s, t = 10s, and a comparison between Backward Euler and Crank-Nicolson temporal integration schemes vs the exact solution.

Displacement at $t=5s$

/Verification/images/Transient_Test_Case_t=5.jpeg

Displacement at $t=10s$

/Verification/images/Transient_Test_Case_t=10.jpeg

Comparison between Backward Euler, Crank-Nicolson and 1D Exact Solution

/Verification/images/Transient_Axial_Displacement_vs._Length.png