To model the behavior of continua, often discrete lattice models with nodes joined by 2-point spring elements (which resist tensile forces) and 3-point beam elements (which resist bending moments) are employed.

Consider a 1D structure of length $L$ is divided into $N$ elements of length $\Delta x$. The continuum structure has an elastic modulus $E$. In the discretized model, neighboring nodes are joined by springs. The spring stiffness in the continuum is $$ k_s = \frac{EA}{L} \,.$$

By equating the potential energy $E_s$ of the continuum model and discrete model, we may find an equivalent discrete spring stiffness $k_s^\Delta$:

$$ E_s = \int_0^L k_s (X - L) \mathrm d X = \int_0^L N k_s \left(\frac{X}{N} - \frac{L}{N} \right) \mathrm d X = N^2 \int_0^{\Delta x} k_s (x - \Delta x) \mathrm d x = N E_s^\Delta \,,$$

where $\Delta x = L/N$ and the substitution $x = X / N$ has been made. Thus,

$$ E_s^{\Delta} = \int_0^{\Delta x} N k_s (x - \Delta x) \mathrm d x \equiv \int_0^{\Delta x} k_s^{\Delta} (x - \Delta x) \mathrm d x \,, $$

and we see that $k_s^{\Delta} = N k_s = k_s (L / \Delta x) = E A / \Delta x$.

Another way of seeing this is that numerically, the strain energy between nodes $i$ and $i+1$ is $$E_{s,i}^\Delta = \frac{1}{2}k_s^\Delta \left( || \mathbf x_{i} - \mathbf x_{i+1} || - \Delta x \right)^2 $$ and the total strain energy is $$E_s^\Delta = \sum_i E_{s,i}^\Delta = N E_{s,i}^\Delta = E_s\,.$$

Now, I want to determine a similar equivalent stiffness coefficient for a discrete model that also has 3-point beam elements. The potential energy of a beam is:

$$ E_b = \int_0^L k_b \left( \kappa(X) \right)^2 \mathrm dX = \int_0^{\Delta x} Nk_b \left( \kappa(Nx) \right)^2 \mathrm dx = (N-1) E_b^{\Delta} \,, $$ where $k_b=EI$ and $I$ is the area moment of inertia of the cross-section. This part is really confusing me. Also, the form of the discrete bending energy employed in this case is: $$E_{b,i}^\Delta = \frac{1}{2} k_b^\Delta \left[ \mathbf{\hat z} \cdot (\mathbf x_{i+1} - \mathbf x_i) \times (\mathbf x_i - \mathbf x_{i-1}) \right]^2 $$

For this problem, the structure should ideally be incompressible. Thus, I obtain

$$E_{b,i}^\Delta = \frac{1}{2} k_b^\Delta \left[ \Delta x^2 \sin\theta_i \right]^2$$

where $\theta_i$ is the angle made by the vectors $\mathbf x_{i+1}-\mathbf x_i$ and $\mathbf x_i-\mathbf x_{i-1}$.

I'm not sure how to proceed from here. I thought that a good first step would be to linearize the expression. If the mesh is very fine, than the angle formed by two consecutive elements is approximately $\pi$, but I couldn't get very far with this either. I would appreciate it if someone could help me figure this out.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.