0.5 Michell truss stress measure

Michelle trusses

Unidirectional curve stress

Let $C$ be a simple curve in ${\mathbb{R}}^d$ . In other words $C$ is the image of a map $r\in C^{1,1}([0,1],{\mathbb{R}}^d)$ such that $\dot{r}\left(t\right)\ne 0$ for each $t\in [0,1]$ and $r$ is injective. We will also require without loss of generality that $r\left(t\right)$ trace out the curve $C$ at a constant speed, making $|\dot{r}\left(t\right)|$ constant on $[0,1]$ . Therefore $|\dot{r}\left(t\right)|=\frac{{\int }_0^1\left|\dot{r}\left(s\right)\right|ds}{1-0}={\mathcal{H}}^1\left(C\right)$ for each $t\in [0,1]$ and the unit tangent vector to $C$ at time $t$ will be

The curvature $\kappa$ at time $t$ is then given by

We now define a measure ${\sigma }^C$ that is proportional to the stress along the curve $C$ .

Thus ${\sigma }^C$ is a rank 1 symmetric matrix measure.

We will now treat ${\sigma }^C$ as the functional on $C_c({\mathbb{R}}^d,{\mathbb{R}}^{d\times d})$ that corresponds to the measure ${\sigma }^C$ . For any $\xi \in C_c({\mathbb{R}}^d,{\mathbb{R}}^{d\times d})$ ,

To find the divergence of the measure ${\sigma }^C$ , we let an arbitrary vector-valued function $u\in C_c^1({\mathbb{R}}^d,{\mathbb{R}}^{d\times d})$ be given and we evaluate the functional $-\text{div}{\sigma }^C\left[u\right]$ .

Using the definitions of the inner product and the tensor product, we can convert the matrix product $⟨\nabla u\left(r\left(t\right)\right);\dot{r}\left(t\right)\otimes \dot{r}\left(t\right)⟩$ into a vector product:

Therefore

where $A=r\left(0\right)$ and $B=r\left(1\right)$ . Since this equation holds for any $u\in C_c^1({\mathbb{R}}^d,{\mathbb{R}}^{d\times d})$ , we say that the measure $-\text{div}{\sigma }^C$ is given by

When the curve $C$ is just equal to the line segment $[A,B]$ , $\tau =\frac{B-A}{|B-A|}$ and $\kappa =0$ . Therefore our equation becomes

Stress in a truss

The stress in a beam is given by $\frac{1}{2}\lambda {\sigma }^{[A,B]}$ for some $\lambda$ . A finite collection of such beams is a truss . Therefore the stress in a truss that consists of $l$ nodes at $\{A_1,A_2,\cdots ,A_l\}$ is given by

where we require that ${\lambda }_{m,m}=0$ for each $m$ .

Recall that the stress $\sigma$ of a structure that is in equilibrium must satisfy the restrictions $-\text{div}{\sigma }^T=F$ and $\sigma ={\sigma }^T$ . It is easy to see that $\sigma$ will be symmetric by definition. Therefore, if our truss is in equilibrium, there must be a force distribution $F$ such that

Since the operator $-\text{div}$ is linear, we get

which will be our new requirement for a truss to be balanced [link] .

Cost of a truss

If we are given a force distribution $F$ , a set of points $\mathcal{A}=\{A_1,A_2,\cdots ,A_l\}$ , and a set of weights $\Lambda =\{{\lambda }_{m,n}:m,n=1,\cdots l\}$ such that $F={\sum }_{m,n=1}^l{\lambda }_{m,n}({\delta }_{A_m}-{\delta }_{A_n})\frac{A_m-A_n}{|A_m-A_n|}$ , then the cost of the truss structure will be the "total mass" of the structure, which is given by ${\int }_{{\mathcal{R}}^d}\left|\sigma \right|$ .

Cost minimization

Given a truss $T$ , which consists of a set of points $\mathcal{A}=\{A_1,A_2,\cdots ,A_l\}$ and a set of weights $\Lambda =\{{\lambda }_{m,n}:m,n=1,\cdots l\}$ , and given a force distribution $F$ of finite support, we define a set ${\Sigma }_F^T\left(\overline{\Omega }\right)$ of all admissible stress measures for the truss T.

The requirement $\sigma ={\sum }_{m,n=1}^l-{\lambda }_{m,n}{\sigma }^{[A_m,A_n]}$ is equivalent to the restriction that $\sigma$ be rank 1 and supported on a finite collection of simple curves.

We also define the set

of all stress measures for structures that balance the force distribution $F$ . The stresses in this set do not necessarily correspond to trusses, since we have dropped the requirement that the stress be rank 1 and concentrated on curves in ${\mathbb{R}}^d$ . In fact, the stress measures in ${\Sigma }_F\left(\overline{\Omega }\right)$ may be spread out over a region of ${\mathbb{R}}^d$ . Also, we may allow the given force $F$ to be diffuse.

Our original optimization problem was to find a truss $T^0$ such that the cost of $T^0$ is equal to ${inf}_T\left\{,inf,\left\{\int \left|\sigma \right|:\sigma ,\in ,{\Sigma }_F^T,(,\overline{\Omega }\right\}\right\}$ for a given force distribution $F$ of finite support. This is equivalent to finding a stress measure ${\sigma }^0\in {\bigcup }_T\left({\Sigma }_F^T,\left(\overline{\Omega }\right)\right)$ such that

Such a ${\sigma }^0$ , however, does not always exist [link] .