Suspension 100 m: max/min tension, SFD, BMD

Assumptions. The loads given are for the whole bridge, so each of the two girders and cables carries half: $w$ = 40/2 = 20 kN/m and $P$ = 200/2 = 100 kN (if the data are taken for a single cable, double all forces below; the method does not change). Span $L$ = 100 m, dip $d$ = 10 m.

Cable pull $H$

A three-hinged girder gives the cable a uniform load, so the cable stays parabolic.

• Dead load part: $H_d = \dfrac{wL^2}{8d} = \dfrac{20(100)^2}{8(10)} = 2500.0$ kN
• Point loads: $R_A = 1.5P = 150.0$ kN; $M_{0C} = 150.0(50) - 100(25) = 5000.0$ kN m, so $H_p = \dfrac{M_{0C}}{d} = 500.0$ kN

$$H = H_d + H_p = 2500.0 + 500.0 = 3000.0\ \text{kN}$$

Maximum and minimum tension

End slope: $\tan\theta = 4d/L = 0.40$, $\sec\theta = 1.0770$.

• Maximum tension at the supports A and B: $T_{max} = H\sec\theta = 3000.0(1.0770) = 3231.10$ kN
• Minimum tension at mid-span (crown, where the cable is horizontal): $T_{min} = H = 3000.0$ kN

Length of the cable

Using $l = L\left[1+\dfrac{8}{3}\left(\dfrac{d}{L}\right)^2-\dfrac{32}{5}\left(\dfrac{d}{L}\right)^4\right]$ with $d/L = 0.1$:

$$l = 100\left[1 + \frac{8}{3}(0.1)^2 - \frac{32}{5}(0.1)^4\right] = 102.60\ \text{m}$$

Girder shear force and bending moment (per girder)

The dead load is carried wholly by the cable, so it produces no SF or BM in the girder. For the three loads, with $y = 0.004x(100-x)$ and $H_p$ = 500 kN:

$$M = M_0 - H_p\,y,\qquad S = S_0 - H_p\,\frac{dy}{dx}$$

• $0 \le x \le 25$: $M = 2x^2 - 50x$, $S = 4x - 50$
• $25 \le x \le 50$: $M = 2x^2 - 150x + 2500$, $S = 4x - 150$
• The right half is the mirror image (SF with opposite sign).

So $M = 0$ at the load points (0, 25, 50, 75, 100 m), and $M = -312.5$ kN m at 12.5 m, 37.5 m, 62.5 m and 87.5 m (the greatest hogging value). Shear force is $-50$ to $+50$ kN in each quarter, changing sign at the mid-point of each panel (a saw-tooth).

SFD (kN), +-50 in each quarter span
            *           *           *           *
          **          **          **          **
         *           *           *           *
       **          **          **          **
------*-----------*-----------*-----------*------
    **          **          **          **
   *           *           *           *
 **          **          **          **
*
(top = 50.0, bottom = -50.0; '-' is zero line)

BMD (kN m), all hogging, zero at the loads
*-----------*-----------*-----------*-----------*

 *         * *         * *         * *         *

  *       *   *       *   *       *   *       *

   *     *     *     *     *     *     *     *
    *   *       *   *       *   *       *   *
     ***         ***         ***         ***
(top = 0.0, bottom = -312.5; '-' is zero line)

Answer: $T_{max} = 3231.10$ kN at the supports; $T_{min} = 3000.0$ kN at the crown (mid-span); cable length = 102.60 m; girder $M_{max} = -312.5$ kN m (hogging) at 12.5, 37.5, 62.5 and 87.5 m; $S_{max} = 50$ kN.