Suspension 50 m: max tension, BM at 10 m, BMD/SFD

Data: span $L = 50$ m, dip $d = 5$ m, three-hinged stiffening girder (hinges at both ends and mid-span). Dead load $w = 8$ kN/m carried entirely by the cable. A moving load $W = 80$ kN crosses the girder. The cable is a parabola, $y = \dfrac{4d\,x(L-x)}{L^2} = 0.008\,x(50-x)$; $\tan\alpha_{support} = \dfrac{4d}{L} = 0.4$.

Horizontal pull in the cable

Dead load: $H_d = \dfrac{wL^2}{8d} = \dfrac{8\times 2500}{40} = 500$ kN.

Live load $W$ at distance $a$ from the left end (hinge at mid-span: girder moment at $C$ is zero):

$$H_L = \frac{W a}{2d} \ (a\le 25),\qquad \frac{W(L-a)}{2d}\ (a\ge 25)$$

The maximum is with $W$ at mid-span: $H_L = \dfrac{80\times 25}{10} = 200$ kN.

$$H_{max} = 500 + 200 = 700\ \text{kN}$$

Maximum tension in the cable

It occurs at the supports, where the cable is steepest:

$$T_{max} = H_{max}\sqrt{1 + \left(\frac{4d}{L}\right)^2} = 700\sqrt{1+0.16} = 753.9\ \text{kN}$$

(vertical component at the support $= 700\times 0.4 = 280$ kN).

Maximum bending moment at 10 m from the left end

Section $x_s = 10$ m, cable ordinate $y_s = 0.008\times 10\times 40 = 3.2$ m. For a unit load at $a$, $M = M_b - H\,y_s$:

• $a\le 10$: $M = 0.8a - \dfrac{a}{10}(3.2) = 0.48a$ (peak $+4.8$ m at $a=10$)
• $10\le a\le 25$: $M = 0.2(50-a) - 0.32a = 10 - 0.52a$ ($-3.0$ m at $a = 25$)
• $a\ge 25$: $M = -0.12\,(50-a)$

The ILD is $0 \to +4.8$ (at 10 m) $\to -3.0$ (at 25 m) $\to 0$.

$$M_{max}^{+} = 80\times 4.8 = 384.0\ \text{kN m (load at the section)}, \qquad M_{max}^{-} = 80\times(-3.0) = -240.0\ \text{kN m (load at mid-span)}$$

BMD and SFD (load at 10 m, giving the maximum positive moment)

With $W$ at 10 m: $R_A = 64$ kN, $R_B = 16$ kN, $H_L = \dfrac{80\times 10}{10} = 80$ kN.

$$M(x) = 64x - 80\times 0.008x(50-x) = 32x + 0.64x^2 \ (x\le 10), \qquad M(x) = 800 - 48x + 0.64x^2 \ (x\ge 10)$$ $$V(x) = V_b - H_L\tan\alpha_x,\quad \tan\alpha_x = 0.4 - 0.016x$$

 BMD: A(0) -> +384 at x=10 -> 0 at hinge C (x=25) -> -100 at x=37.5 -> 0 at B
      sagging from A to C, hogging from C to B (parabolic segments)
 SFD: +32 at A -> +44.8 at 10-  | jump of -80 |  -35.2 at 10+ -> -16 at C -> +16 at B

Answer: $T_{max} = 753.9$ kN; maximum BM at 10 m = $+384.0$ kN m (load at the section); at the same section the minimum is $-240.0$ kN m with the load at mid-span.