Derivation of stringing equation

The stringing equation relates the tension $T_2$ at erection (stringing) to the known tension $T_1$ in the toughest condition. It uses the sag and conductor-length formulas of a parabolic span.

Symbols: span $L$; area $A$; modulus $E$; coefficient of expansion $\alpha$. Condition 1 (toughest) has $T_1, W_1, \theta_1$. Condition 2 (stringing) has $T_2, W_2, \theta_2$.

Step 1 – maximum sag and conductor length

For a parabolic span with level supports:

$$D = \frac{WL^2}{8T}, \qquad l = L + \frac{8D^2}{3L}$$

Substituting D:

$$l = L + \frac{8}{3L}\cdot\frac{W^2L^4}{64T^2} = L + \frac{W^2L^3}{24T^2}$$

Hence

$$l_1 = L + \frac{W_1^2L^3}{24T_1^2}, \qquad l_2 = L + \frac{W_2^2L^3}{24T_2^2}$$

Step 2 – change in length from physics

Between the two conditions, the conductor length changes by expansion and by elastic strain:

$$l_2 - l_1 = \alpha(\theta_2-\theta_1)L + \frac{T_2-T_1}{AE}L \qquad (l_1 \approx L)$$

Step 3 – equate and simplify

$$\frac{W_2^2L^3}{24T_2^2} - \frac{W_1^2L^3}{24T_1^2} = \alpha(\theta_2-\theta_1)L + \frac{(T_2-T_1)L}{AE}$$

Multiply by $AE/L$:

$$\frac{W_2^2L^2AE}{24T_2^2} = T_2 - T_1 + \alpha(\theta_2-\theta_1)AE + \frac{W_1^2L^2AE}{24T_1^2}$$

Multiply by $T_2^2$:

$$T_2^2\left[T_2 - T_1 + \alpha(\theta_2-\theta_1)AE + \frac{W_1^2L^2AE}{24T_1^2}\right] = \frac{W_2^2L^2AE}{24}$$

or

$$T_2^2\,(T_2 + K_2) - K_1 = 0$$

with $K_2 = -T_1 + \alpha(\theta_2-\theta_1)AE + \dfrac{W_1^2L^2AE}{24T_1^2}$ and $K_1 = \dfrac{W_2^2L^2AE}{24}$.

This is the stringing equation. With $T_1 = \text{UTS}/\text{FS}$ known, the cubic gives $T_2$, and the stringing sag is $D_2 = W_2L^2/(8T_2)$. When $\theta_2$ is the maximum temperature and $W_2$ is the bare conductor weight, the same equation gives the maximum sag in the easiest condition.