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Exact solutions for stochastic Bernoulli–Euler beams under deterministic loading

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Abstract

This study deals with two general solutions for a simply supported linear elastic Bernoulli–Euler beam with a stochastic bending flexibility, subjected to a deterministic loading. Two model problems are considered. The first problem is associated with a trapezoidally distributed load, whereas the second problem treats a sinusoidally distributed load. The importance of the solution for the trapezoidal load lies in its practicality. The derivation of stochastic characteristics for random beams under a sinusoidal load is useful due to the expandability to generally distributed loads by a Fourier sine series expansion. Numerical results are reported for various cases illustrating the effect of stochasticity of the beam’s properties on its flexural response.

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Acknowledgements

Oded Rabinovitch gratefully acknowledges the support of the Abel Wolman Chair in Civil Engineering. The work was initiated when Isaac Elishakoff served as a Visiting Distinguished Professor at the Technion—I.I.T., at the auspices of the Edmond Safra Foundation. We express our sincere gratitude to the above organizations.

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Authors and Affiliations

  1. Faculty of Civil and Environmental Engineering, Technion Israel Institute of Technology, 32000, Technion City, Haifa, Israel

    Nachman Malkiel

  2. Abel Wolman Chair in Civil Engineering, Faculty of Civil and Environmental Engineering, Technion Israel Institute of Technology, 32000, Technion City, Haifa, Israel

    Oded Rabinovitch

  3. Department of Ocean and Mechanical Engineering, Florida Atlantic University, Boca Raton, FL, 33431-0991, USA

    Isaac Elishakoff

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  1. Nachman Malkiel
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  2. Oded Rabinovitch
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  3. Isaac Elishakoff
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Correspondence to Nachman Malkiel.

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Appendix 1 : Expressions for exact solutions

Appendix 1 : Expressions for exact solutions

B1 Solution for the deflection’s covariance function, where an exponential form is used for the flexibility covariance function \(\text{cov}_{f}\), and the load is a general trapezoidal function:

$$\begin{aligned} \text{cov}_{w} (\xi ,\eta ) & = \frac{{L^{8} a^{2} q_{0}^{2} }}{{18\alpha^{10} }} \cdot \\ & \left\{ {\left( {\left( {3 + (\eta + 1)\beta } \right)\eta (\eta - 1)\alpha^{3} + ( - 6\beta \eta^{2} + 2\beta - 12\eta + 6)\alpha^{2} + 18(\beta \eta + 1)\alpha - 24\beta } \right)} \right. \\ & \left( {\left( {3 + (\xi + 1)\beta } \right)(\xi - 1)\xi \alpha^{3} + \left( {6\beta \xi^{2} - 2\beta + 12\xi - 6} \right)\alpha^{2} + 18(\beta \xi + 1)\alpha + 24\beta } \right)e^{\alpha (\eta - \xi )} /2 \\ & \quad - \;2\left( {\left( {3 + (\eta + 1)\beta } \right)\eta (\eta - 1)\alpha^{3} + ( - 6\beta \eta^{2} + 2\beta - 12\eta + 6)\alpha^{2}}\right. \\ & \quad \left.{+ 18(\beta \eta + 1)\alpha - 24\beta } \right)\left( {(\beta + 3/2)\alpha^{2} + 9/2(\beta + 1)\alpha + 6\beta } \right)\xi e^{\alpha (\eta - 1)} \\ & \quad - \;\left( {(2\beta + 3)\alpha^{2} + 9(\beta + 1)\alpha + 12\beta } \right)\eta \left( {\left( {3 + (\xi + 1)\beta } \right)(\xi - 1)\xi \alpha^{3}}\right.\\ & \quad \left.{+ ( - 6\beta \xi^{2} + 2\beta - 12\xi + 6)\alpha^{2} + (18\beta \xi + 18)\alpha - 24\beta } \right)e^{\alpha (\xi - 1)} \\ & \quad + \;\left( {\left( {3 + (\eta + 1)\beta } \right)\eta (\eta - 1)\alpha^{3} + (6\beta \eta^{2} - 2\beta + 12\eta - 6)\alpha^{2}}\right. \\ & \quad\left.{ + 18(\beta \eta + 1)\alpha + 24\beta } \right)(\xi - 1)\left( {(\beta + 3)\alpha^{2} + 9\alpha - 12\beta } \right)e^{ - \alpha \eta } \\ & \quad + \;\left( {\left( {3 + (\xi + 1)\beta } \right)(\xi - 1)\xi \alpha^{3} + (6\beta \xi^{2} - 2\beta + 12\xi - 6)\alpha^{2}}\right. \\ & \quad \left.{+ (18\beta \xi + 18)\alpha + 24\beta } \right)(\eta - 1)\left( {(\beta + 3)\alpha^{2} + 9\alpha - 12\beta } \right)e^{ - \alpha \xi } \\ & \quad - \;2\left( {(2\beta + 3)\alpha^{2} + 9(\beta + 1)\alpha + 12\beta } \right)\left( {(2\xi - 1)\eta - \xi } \right)\left( {(\beta + 3)\alpha^{2} + 9\alpha - 12\beta } \right)e^{ - \alpha } \\ & \quad + \;\eta (\xi - 1)/72\left\{ {\left( {\eta^{8} - 24\eta^{6} /7 + 18\eta^{4} /5 + \xi \left( - 46/35 - 2\xi^{6} /7 + 38\xi^{4} /35 - 26\xi^{5} /7\right.}\right.}\right.\\ &\left.{ \left.{\left.\quad + 164\xi^{3} /35 + \xi^{7} - 46\xi^{2} /35 - 46/35\xi\right)} \right)} \right.\beta^{2} \\ & \quad + \;\left( {54\eta^{7} /7 - 72\eta^{6} /7 - 72\eta^{5} /5 + 108\eta^{4} /5 + (54/7)\left(\xi^{6} - 5\xi^{5} /3 - 5\xi^{4} /3 + 59\xi^{3} /15 \right.}\right.\\ & \quad \left.{\left.- 11\xi^{2} /15 - 11\xi /15 - 11/15\right)\xi } \right)\beta \\ & \quad \left. { + \;108\eta^{6} /7 - 216\eta^{5} /5 + 162\eta^{4} /5 + (108/35)\left( {5\xi^{5} - 16\xi^{4} + 31\xi^{3} /2 - 2\xi^{2} - 2\xi - 2} \right)\xi } \right\}\alpha^{9} \\ & 2/7\left\{ {\left( {\eta^{6} - 49/20\eta^{4} + 7/6\eta^{2} + \xi (23/30 + 3\xi^{4} /10 + \xi^{5} - 43\xi^{3} /20 - 2\xi^{2} /5 + 23\xi /30)} \right)} \right.\beta^{2} \\ & \quad + \;\left( {63\eta^{5} /10 - 147\eta^{4} /20 + 7\eta^{2} - 21\xi^{4} /4 + 63\xi^{5} /10 - 7\eta^{3} + 7\xi /2 - 7\xi^{3} + 7\xi^{2} /2} \right)\beta \\ & \quad \left. { + \;189\eta^{4} /20 + 189\xi^{4} /20 + 21\xi /5 + 21\eta^{2} /2 - 21\eta^{3} - 84\xi^{3} /5 + 21\xi^{2} /5} \right\}\eta (\xi - 1)\alpha^{7} \\ & \quad + \;(6/5)\eta (\xi - 1)\left( { - 15 + (\xi^{4} + \eta^{4} + \xi^{3} - 7\xi^{2} /3 - 10\eta^{2} /3 - 7\xi /3 - 5/3)\beta^{2}}\right. \\ & \quad \left.{+ (5\eta^{3} + 5\xi^{3} - 10\eta^{2} - 5\xi^{2} - 5\xi - 10)\beta } \right)\alpha^{5} \\ & \quad + \;2\left( {\left( {( - 5\xi + 1)\eta + \xi - 1} \right)\beta^{2} + 6\left( {( - 3\xi + 1)\eta + \xi - 1} \right)\beta + 9\left( {( - 2\xi + 1)\eta + \xi - 1} \right)} \right)\alpha^{4} \\ & \quad + \;36\eta (\beta^{2} + 3\beta + 3)(\xi - 1)\alpha^{3} + \left( {\left( {(114\xi - 48)\eta - 48\xi + 48} \right)\beta^{2} + \left( {(324\xi - 144)\eta}\right.}\right.\\ & \quad - \left.{\left.{144\xi + 144} \right)\beta + (324\xi - 162)\eta - 162\xi + 162} \right)\alpha^{2} \\ & \quad \left. { - \;144\beta^{2} \eta (\xi - 1)\alpha - 576\beta^{2} \left( {(\xi - 1/2)\eta - \xi /2 + 1/2} \right)} \right\}\quad {\text{for}}\; \, \xi > \eta \,. \\ \end{aligned}$$
(44)

B2 Solution for the deflection’s covariance function, where a Matérn form is used for the flexibility covariance function \(\text{cov}_{f}\), and the load is a general trapezoidal function:

$$\begin{aligned} \text{cov}_{w} (\xi ,\eta ) & = \frac{{23a^{2} q_{o}^{2} L^{8} }}{18} \cdot \{ \\ & \left[ { - (\beta /46)\left( {(\xi^{3} + 6\xi^{2} + 17\xi + 22)\beta + 3\xi^{2} + 9\xi + 12} \right)\eta^{4} } \right. \\ & + \left\{ {(\xi + 3)(\xi^{3} + 14\xi^{2} + 47\xi + 104)\beta^{2} /46 }\right.\\ & \quad \left.{+ (12\xi^{2} + 48 + 36\xi )\beta /23 - 18/23 - 9\xi^{2} /46 - 27\xi /46} \right\}\eta^{3} \\ & +\left\{\; {( - 1249\xi /46 - 3\xi^{4} /23 - 89\xi^{3} /46 - 228\xi^{2} /23 - 793/23)\beta^{2} } \right. \\ & \qquad +(3/46)(\xi + 1)(\xi^{3} + 9\xi^{2} + 3\xi + 18)\beta \\ & \qquad \qquad +\left. { 225/23 + 9\xi^{3} /46 + 117\xi^{2} /46 + 171\xi /23} \right\}\eta^{2} \\ & +\left\{ {(17\xi^{4} /46 + 245\xi^{3} /46 + 1249\xi^{2} /46 + 2168/23 + 3417\xi /46)\beta^{2} } \right. \\ & \qquad \left. { + ( - 15\xi^{4} /46 - 75\xi^{3} /23 - 429\xi^{2} /46 - 633/23 - 531\xi /23)\beta}\right. \\ & \qquad \qquad \left.{- 45\xi^{3} /46 - 234\xi^{2} /23 - 1359\xi /46 - 891/23} \right\}\eta \\ & +( - 11\xi^{4} - 156\xi^{3} - 793\xi^{2} - 2168\xi - 2750)\beta^{2} /23 \\ & + \left({12\xi^{4} + 120\xi^{3}393\xi^{2} + 999\xi + 1212}\right)\beta /23 \\ & +\left. { 351\xi^{2} /23 + 36\xi^{3} /23 + 1017\xi /23 + 1332/23} \right]e^{\eta - \xi } \\ &\!\! - ((\xi - 1)/23)\left[ {(11\beta^{2} - 12\beta )\eta^{4} } \right.+ 156(\beta - 1)(\beta + 3/13)\eta^{3} \\ &\qquad + (793\beta^{2} - 393\beta - 351)\eta^{2}+ \;(2168\beta^{2} - 999\beta - 1017)\eta \\ & \qquad \qquad \qquad \qquad \left. { + 2750\beta^{2} - 1212\beta - 1332} \right]e^{ - \eta } \\ & \!\! - ((\eta - 1)/23)\left[ {(11\xi^{4} + 156\xi^{3} + 793\xi^{2} + 2168\xi + 2750)\beta^{2}}\right.\\ & \qquad+ \left( - 12\xi^{4} - 120\xi^{3} - 393\xi^{2} - 999\xi - 1212\right)\beta \\ & \qquad \qquad \qquad \qquad \left. { - 36\xi^{3} - 351\xi^{2} - 1017\xi - 1332} \right]e^{ - \xi } \\ & \!\! +\left[ {(\beta^{2} + 12\beta /23)\eta^{4}} \right. + (36 - 332\beta^{2} - 96\beta )\eta^{3} /23 \\ & \quad + ( - 459 + 1693\beta^{2} - 93\beta )\eta^{2} /23 + (1827 - 4632\beta^{2} \\ & \quad + 1461\beta )\eta /23 - 2736/23 +\left. { 5878\beta^{2} /23 - 2736\beta /23} \right]\xi e^{\eta - 1} \\ & \!\! + \eta \left[ {(5878 + 23\xi^{4} - 332\xi^{3} + 1693\xi^{2} - 4632\xi )\beta^{2} /23} \right.\\ & \quad + ( - 2736 + 12\xi^{4} - 96\xi^{3} - 93\xi^{2} + 1461\xi )\beta /23 \\ & \quad - 2736/23 - 459\xi^{2} /23 + 1827\xi /23 +\left. { 36\xi^{3} /23} \right]e^{\xi - 1} \\ & \!\!+ (11756/23)( - 1368/2939 + \beta^{2} - 1368\beta /2939)\cdot\\ & \qquad \qquad \left( {(\xi - 1/2)\eta - \xi /2} \right)e^{ - 1} \\ & \!\! +(\beta^{2} /828)(\xi - 1)\eta^{9} + 3\beta (\xi - 1)\eta^{8} /322\\ & \!\! + (22/483)(\beta^{2} - 3/11\beta + 9/22)(\xi - 1)\eta^{7} \\ & \!\!+ (34/115)(\beta - 3/17)(\xi - 1)\eta^{6} \\ & \!\! + \;(9/46)(\beta^{2} - 26\beta /15 + 13/5)(\xi - 1)\eta^{5} \\ & \!\! + (28/23)(\xi - 1)(\beta - 6/7)\eta^{4} \\ & \!\! - (68/69)(3 + \beta )(\beta - 3/17)(\xi - 1)\eta^{3} \\ & \!\! + \;\left[ {\left(11\xi^{4} /138 - 68\xi^{3} /69 + 3914/23 - 822533\xi /2898 + 1\xi^{9} /828 \right.}\right.\\ & \qquad \left.{\left.+ 9\xi^{5} /46 - 2\xi^{6} /69 + 22\xi^{7} /483 - 1\xi^{8} /644\right)\beta^{2} } \right. \\ & \qquad + (33\xi^{4} /23 - 60\xi^{3} /23 - 2004/23 + 244171\xi /1610\\ & \qquad \qquad- 12\xi^{5} /23 + 36\xi^{6} /115 - 4\xi^{7} /161 + 3\xi^{8} /322)\beta \\ & \qquad - \;63\xi^{4} /46 + 24\xi^{3} /23 + 119538\xi /805 \\ & \qquad \qquad -2088/23 + 3\xi^{7} /161 + 27\xi^{5} /46 - \left. {9\xi^{6} /115} \right]\eta \\ & \!\! + (2750/23)(\xi - 1)(\beta^{2} - 606\beta /1375 - 666/1375)\} \quad \, for \, \xi \ge \eta .\\ \end{aligned}$$
(45)

B3 Particular solution for the deflection’s covariance function, where an exponential form is used for the flexibility covariance function \(\text{cov}_{f}\), and the load is a general sinusoidal function:

$$\begin{aligned} \chi_{nk} \left( {\xi ,\eta } \right) & = \frac{{a^{2} q_{0}^{2} L^{8} }}{{\left( {\pi^{2} n^{2} + \alpha^{2} } \right)^{2} \left( {k^{2} \pi^{2} + \alpha^{2} } \right)^{2} \left( {k - n} \right)^{3} \left( {k + n} \right)^{3} \alpha^{6} n^{6} \pi^{8} k^{6} }} \cdot \\ & \left[ {n^{4} \left( {\left( {\pi^{2} \alpha^{3} n^{2} - \alpha^{5} } \right)\sin \left( {n\pi \xi } \right) - 2\cos \left( {n\pi \xi } \right)n\pi \alpha^{4} + \sin \left( {n\pi } \right)\left( {\pi^{2} n^{2} + \alpha^{2} } \right)^{2} \left( {\alpha \xi + 2} \right)} \right) \cdot } \right. \\ & \left( {k - n} \right)^{3} \left( {\left( {\pi^{2} \alpha^{3} k^{2} - \alpha^{5} } \right)\sin \left( {k\pi \eta } \right) + 2\cos \left( {k\pi \eta } \right)k\pi \alpha^{4}}\right. \\ & \quad + \left.{\sin \left( {k\pi } \right)\left( {\pi^{2} k^{2} + \alpha^{2} } \right)^{2} \left( {\alpha \eta - 2} \right)} \right)k^{4} \left( {k + n} \right)^{3} \pi^{4} e^{{\alpha \left( {\eta - \xi } \right)}} \\ & \quad + \;n^{4} k^{4} \left( {\left( {\pi^{2} \alpha^{3} k^{2} - \alpha^{5} } \right)\sin \left( {k\pi \eta } \right) - 2\cos \left( {k\pi \eta } \right)k\pi \alpha^{4} + \sin \left( {k\pi } \right)\left( {\pi^{2} k^{2} + \alpha^{2} } \right)^{2} \left( {\alpha \eta + 2} \right)} \right) \cdot \\ & \quad \left( {\sin \left( {n\pi } \right)\left( {\pi^{2} n^{2} + \alpha^{2} } \right)^{2} \left( {\alpha \xi + 2} \right) - n\pi \alpha^{3} \left( {\pi^{2} n^{2} \xi + \alpha^{2} \xi + 2\alpha } \right)} \right)\left( {k - n} \right)^{3} \left( {k + n} \right)^{3} \pi^{4} e^{ - \alpha \eta } \\ & \quad + \;n^{4} k^{4} \left( {\left( {\pi^{2} \alpha^{3} n^{2} - \alpha^{5} } \right)\sin \left( {n\pi \xi } \right) - 2\cos \left( {n\pi \xi } \right)n\pi \alpha^{4} + \sin \left( {n\pi } \right)\left( {\pi^{2} n^{2} + \alpha^{2} } \right)^{2} \left( {\alpha \xi + 2} \right)} \right) \cdot \\ & \quad \left( {\sin \left( {k\pi } \right)\left( {\pi^{2} k^{2} + \alpha^{2} } \right)^{2} \left( {\alpha \eta + 2} \right) - k\pi \alpha^{3} \left( {\pi^{2} k^{2} \eta + \alpha^{2} \eta + 2\alpha } \right)} \right)\left( {k - n} \right)^{3} \left( {k + n} \right)^{3} \pi^{4} e^{ - \alpha \xi } \\ & \quad + \;2n^{4} \left( {k^{2} \pi^{2} + \alpha^{2} } \right)^{2} \alpha^{3} \sin \left( {k\eta \pi } \right) \cdot \left\{ { - k^{4} \pi^{2} \alpha^{4} \left( {\pi^{2} n^{2} + \alpha^{2} } \right)\left( {k - n} \right)\left( {k + n} \right)\left( {k^{2} + n^{2} } \right)\left( {\xi - \eta } \right)\sin \left( {\eta n\pi } \right)} \right. \\ & \quad + \;2n\alpha^{4} k^{4} \pi \left( {\pi^{2} k^{4} - 3\left( {\pi^{2} n^{2} + \alpha^{2} } \right)k^{2} - 2\pi^{2} n^{4} - \alpha^{2} n^{2} } \right)\cos \left( {\eta n\pi } \right) \\ & \quad + \left( {\pi^{2} n^{2} + \alpha^{2} } \right)^{2} \left. {\left( {\left( { - 2 + \eta \left( {\xi - \eta } \right)\alpha^{2} } \right)\pi^{2} k^{2} + 6\alpha^{2} } \right)\left( {k - n} \right)^{3} \sin \left( {n\pi } \right)\left( {k + n} \right)^{3} } \right\} \\ & \quad + \;\left( {k/6} \right)\left\{ {24n^{4} \left( {k^{2} \pi^{2} + \alpha^{2} } \right)^{2} \alpha^{5} \pi \cos \left( {k\pi \eta } \right)} \right. \\ & \quad \cdot \left[ { - \alpha^{2} k^{4} \left( {\left( {\pi^{2} n^{2} - \alpha^{2} } \right)k^{2} - 5\pi^{2} n^{4} - 3\alpha^{2} n^{2} } \right)\sin \left( {\eta n\pi } \right)} \right. - \left( {\pi^{2} n^{2} + \alpha^{2} } \right)\left( {k - n} \right)\\ & \quad \left( {k + n} \right)k^{4} n\pi \alpha^{2} \left( {\xi - \eta } \right)\cos \left( {\eta n\pi } \right)\left. { + \left( {\pi^{2} n^{2} + \alpha^{2} } \right)^{2} \left( {k - n} \right)^{3} \left( {k + n} \right)^{3} \left( {\xi - 2\eta } \right)\sin \left( {n\pi } \right)} \right] \\ & \quad + \;\left( {k - n} \right)\left( {k + n} \right) \cdot \left\{ {12\left( {k^{2} \pi^{2} + \alpha^{2} } \right)^{2} \left( {k - n} \right)^{2} \left( {k + n} \right)^{2} \alpha^{7} k^{3} \left( {\pi^{4} \eta \left( {\xi - \eta } \right)n^{4}}\right.}\right. \\ & \quad + \left.{\left.{\pi^{2} \left( {10 + \eta \left( {\xi - \eta } \right)\alpha^{2} } \right)n^{2} + 6\alpha^{2} } \right)\sin \left( {k\pi } \right)\sin \left( {\eta n\pi } \right)} \right. \\ & \quad + \;n\pi \left\{ {24\left( {k^{2} \pi^{2} + \alpha^{2} } \right)^{2} \left( {\pi^{2} \left( {\xi - 3\eta } \right)n^{2} + \alpha^{2} \left( {\xi - 2\eta } \right)} \right)\left( {k - n} \right)^{2} \left( {k + n} \right)^{2} \alpha^{7} k^{3} \sin \left( {k\pi } \right)\cos \left( {\eta n\pi } \right)} \right. \\ & \quad + \left( {n^{2} \pi^{2} + \alpha^{2} } \right)^{2} n^{3} \left( {k - n} \right)^{2} \left( {k + n} \right)^{2} \sin \left( {n\pi } \right) \cdot \\ & \quad \left[ {\left( {k^{2} \pi^{2} + \alpha^{2} } \right)^{2} k^{3} \pi^{3} \sin \left( {k\pi } \right)\left( { - 24 + \eta^{4} \left( {\xi - 3\eta /5} \right)\alpha^{5} - 4\alpha^{3} \eta^{3} + 6\alpha^{2} \xi \eta - 12\left( {\xi - \eta } \right)\alpha } \right)} \right. \\ & \quad \left. { - \;6\alpha^{3} \left( {\pi^{6} \eta \left( {\alpha \xi - 2} \right)k^{6} + \pi^{4} \alpha \left( {\alpha \eta + 2} \right)\left( {\alpha \xi - 2} \right)k^{4} + 8\left( {\xi + \eta /2} \right)\alpha^{4} k^{2} \pi^{2} + 4\alpha^{6} \left( {\xi + \eta } \right)} \right)} \right] \\ & \quad - \,6\alpha^{3} k^{3} \left\{ {\left( {k^{2} \pi^{2} + \alpha^{2} } \right)^{2} \left( {k - n} \right)^{2} \left( {k + n} \right)^{2} \sin \left( {k\pi } \right) \cdot } \right.\left( {\pi^{6} \xi \left( {\alpha \eta - 2} \right)n^{6} + \pi^{4} \alpha \left( {\alpha \eta - 2} \right)\left( {\alpha \xi + 2} \right)n^{4}}\right. \\ & \quad + \left.{4\left( {\xi + 2\eta } \right)\alpha^{4} n^{2} \pi^{2} + 4\alpha^{6} \left( {\xi + \eta } \right)} \right) \\ & \quad - \;n^{4} \alpha^{3} k\pi \left[ {\pi^{6} \eta \left( {n^{2} \pi^{2} \xi + \alpha^{2} \xi - 2\alpha } \right)k^{6} - 2\left( {n^{2} \pi^{2} \xi + \alpha^{2} \xi - 2\alpha } \right)\pi^{4} \left( {n^{2} \pi^{2} \eta - \eta \alpha^{2} /2 - \alpha } \right)k^{4} } \right. \\ & \quad + \;\pi^{2} k^{2} \left( {n^{6} \pi^{6} \xi \eta - \pi^{4} \alpha \left( {\alpha \xi \eta - 2\eta - 4\xi } \right)n^{4} - 2\left( { - 4 + \alpha^{2} \xi \eta - 6\alpha \left( {\xi + \eta } \right)} \right)\alpha^{2} n^{2} \pi^{2} + 8\alpha^{5} \left( {\xi + \eta /2} \right)} \right) \\ & \quad \left. { + \;\alpha \left( {\pi^{6} n^{6} \xi \left( {\alpha \eta - 2} \right) + \pi^{4} n^{4} \alpha \left( {\alpha \eta - 2} \right)\left( {\alpha \xi + 2} \right) + 4\alpha^{4} \pi^{2} n^{2} \left( {\xi + 2\eta } \right) + 4\alpha^{6} \left( {\xi + \eta } \right)} \right)} \right]\left. {} \right\}\left. {} \right\}\left. {} \right\}\left. {} \right\}\left. {} \right] \quad {\text{ for}}\; \xi \ge \eta .\\ \end{aligned}$$
(46)

B4 Particular solution for the deflection’s covariance function, where a Matérn form is used for the flexibility covariance function \(\text{cov}_{f}\), and the load is a general sinusoidal function:

$$\begin{aligned}\chi_{nk} \left( {\xi ,\eta } \right) & = \frac{{q_{0}^{2} a^{2} L^{8} e^{ - \eta - \xi } }}{{\left( {\pi^{2} n^{2} + 1} \right)^{3} \left( {\pi^{2} k^{2} + 1} \right)^{3} \left( {k - n} \right)^{3} \left( {k + n} \right)^{3} k^{6} n^{6} \pi^{8} }} \cdot \\ & \left\{ {n^{4} \left\{ {e^{2\eta } k^{4} \left( {k + n} \right)^{3} \left( {k - n} \right)^{3} \left( {\left( {\pi^{4} \left( {1 + \xi - \eta } \right)n^{4} + 6\pi^{2} n^{2} - \xi + \eta - 3} \right)\pi^{4} k^{4} + \left( {6\pi^{6} n^{4} - 6\pi^{2} } \right)k^{2}}\right.}\right.}\right. \\ & \quad - \left.{\left.{\left.{\pi^{4} \left( {\xi - \eta + 3} \right)n^{4} - 6\pi^{2} n^{2} + \xi - \eta + 5} \right)\pi^{4} \sin \left( {n\pi \xi } \right)} \right.} \right. \\ & \quad - 2e^{2\eta } k^{4} \left( {k + n} \right)^{3} n\left( {k - n} \right)^{3} \left( {\left( {\pi^{2} \left( {\xi - \eta } \right)n^{2} + \xi - \eta + 4} \right)\pi^{4} k^{4} + \left( {6\pi^{4} n^{2} + 6\pi^{2} } \right)k^{2}}\right. \\ & \quad - \left.{\pi^{2} \left( {\xi - \eta + 2} \right)n^{2} - \xi + \eta - 6} \right)\pi^{5} \cos \left( {n\pi \xi } \right) \\ & \quad - 4e^{\eta + \xi } k^{4} \pi^{2} \left( {\pi^{2} n^{2} + 1} \right)\left( {\pi^{2} k^{2} + 1} \right)^{3} \left( {k - n} \right)\left( {k + n} \right)\left( {k^{2} + n^{2} } \right)\left( {\xi - \eta } \right)\sin \left( {\eta n\pi } \right) \\ & \quad+ 16k^{4} \left( {\pi^{2} k^{2} + 1} \right)^{3} n\left( {\pi^{2} k^{4} + \left( { - \pi^{2} n^{2} - 1} \right)3k^{2} /2 - 3\pi^{2} n^{4} /2 - n^{2} /2} \right)e^{\eta + \xi } \pi \cos \left( {\eta n\pi } \right) \\ & \quad + (k + n)^{3} \left\{ {\left( {\pi^{2} n^{2} + 1} \right)^{3} \left[ {k^{4} \left( {\left( {\left( { - \xi - 2} \right)\eta}\right.}\right.}\right.}\right. \\ & \quad + \left.{\left.{\left.{\left.{\xi^{2} + 5\xi + 8} \right)\pi^{4} k^{4} + 6\pi^{2} \left( {\xi + 2} \right)k^{2} + \left( {\xi + 2} \right)\eta - \xi^{2} - 7\xi - 12} \right)\pi^{4} e^{2\eta } } \right.} \right. \\ & \quad \left. { + 4\left( {\pi^{2} k^{2} + 1} \right)^{3} \left( {6 + \pi^{2} \left( { - \eta^{2} + \eta \xi - 4} \right)k^{2} } \right)e^{\eta + \xi } + k^{4} e^{\xi } \pi^{4} \left( {\left( {\left( {\xi + 2} \right)\eta + 3\xi + 8} \right)\pi^{4} k^{4} + 6\pi^{2} \left( {\xi + 2} \right)k^{2}}\right.}\right. \\ & \quad + \left.{\left.{\left( { - \xi - 2} \right)\eta - 5\xi - 12} \right)} \right]\sin \left( {n\pi } \right) \\ & \quad - \left[ {\left( {\pi^{4} \xi \left( {\eta + 1} \right)n^{4} + 2\left( {\left( {\xi + 1} \right)\eta + 2\xi } \right)\pi^{2} n^{2} + \left( {\xi + 2} \right)\eta + 3\xi + 8} \right)\pi^{4} k^{4} + 6\pi^{2} \left( {\pi^{2} n^{2} + 1} \right)\left( {\pi^{2} n^{2} \xi + \xi + 2} \right)k^{2} } \right. \\ & \quad \left. {\left. {\left. { - \pi^{4} \xi \left( {\eta + 3} \right)n^{4} - 2\left( {\left( {\xi + 1} \right)\eta + 4\xi + 2} \right)\pi^{2} n^{2} + \left( { - \xi - 2} \right)\eta - 5\xi - 12} \right]k^{4} e^{\xi } n\pi^{5} } \right\}(k - n)^{3} } \right\}\sin (k\pi \eta ) \\ & \quad + k\left\{ {2n^{4} \left\{ {e^{2\eta } k^{4} \left( {k + n} \right)^{3} \left( {k - n} \right)^{3} \left( {\pi^{2} \left( {\pi^{4} \left( {\xi - \eta } \right)n^{4} + 6\pi^{2} n^{2} - \xi + \eta - 2} \right)k^{2} + \pi^{4} \left( {\xi - \eta + 4} \right)n^{4}}\right.}\right.}\right.\\ & \quad \left.{\left.{\left.{+ 6\pi^{2} n^{2} - \xi + \eta - 6} \right)\pi^{4} \sin \left( {n\pi \xi } \right)} \right.} \right. \\ & \quad - 2\left( {\left( {\pi^{2} \left( {\xi - \eta - 1} \right)n^{2} + \xi - \eta + 3} \right)\pi^{2} k^{2} + \pi^{2} \left( {\xi - \eta + 3} \right)n^{2} + \xi - \eta + 7} \right)e^{2\eta } k^{4} \left( {k + n} \right)^{3} n\left( {k - n} \right)^{3} \pi^{5} \cos \left( {n\pi \xi } \right) \\ & \quad - 12k^{4} \left( {\pi^{2} k^{2} + 1} \right)^{3} \left( {\left( {\pi^{2} n^{2} - 1/3} \right)k^{2} - 7\pi^{2} n^{4} /3 - n^{2} } \right)e^{\eta + \xi } \sin \left( {\eta n\pi } \right) + (k + n)(k - n)\\ & \quad\left\{ { - 4e^{\eta + \xi } k^{4} n\pi \left( {\pi^{2} n^{2} + 1} \right)\left( {\pi^{2} k^{2} + 1} \right)^{3} \left( {\xi - \eta } \right)\cos \left( {\eta n\pi } \right) + \left( {k + n} \right)^{2} \left( {k - n} \right)^{2} \left\{ {(\pi^{2} n^{2} + 1)^{3} } \right. \cdot } \right. \\ & \left[ {k^{4} \left( {\left( {\left( { - \xi - 2} \right)\eta + \xi^{2} + 4\xi + 6} \right)\pi^{2} k^{2} + \left( { - \xi - 2} \right)\eta + \xi^{2} + 8\xi + 14} \right)\pi^{4} e^{2\eta } + 4\left( {\pi^{2} k^{2} + 1} \right)^{3} \left( {\xi - 2\eta } \right)e^{\eta + \xi }}\right.\\ & \quad \left.{- k^{4} e^{\xi } \left( {\left( {\left( {\xi + 2} \right)\eta + 2\xi + 6} \right)\pi^{2} k^{2} + \left( {\xi + 2} \right)\eta + 6\xi + 14} \right)\pi^{4} } \right]\sin (n\pi ) \\ & \quad + \left. {\left. {\left. {\left( {\left( {n^{4} \pi^{4} \xi \eta + 2\left( {\left( {\xi + 1} \right)\eta + \xi - 1} \right)\pi^{2} n^{2} + \left( {\xi + 2} \right)\eta + 2\xi + 6} \right)\pi^{2} k^{2} + \pi^{4} \xi \left( {\eta + 4} \right)n^{4}}\right.}\right.}\right.}\right.\\ & \quad \left.{\left.{\left.{\left.{+ 2\left( {\left( {\xi + 1} \right)\eta + 5\xi + 3} \right)\pi^{2} n^{2} + \left( {\xi + 2} \right)\eta + 6\xi + 14} \right)k^{4} e^{\xi } n\pi^{5} } \right\}} \right\}} \right\}\pi \cos (k\pi \eta ) \\ & \quad + \, (k \, + \, n)(k \, - \, n)\left\{ {k^{3} (k + n)^{2} n^{4} (k - n)^{2} } \right. \cdot \left\{ {\left( {\pi^{2} k^{2} + 1} \right)^{3} } \right.\left[ {\left( {\left( { - \eta^{2} + \left( {\xi + 5} \right)\eta - 2\xi - 8} \right)\pi^{4} n^{4} }\right.}\right.\\ & \quad + \left.{\left.{6\pi^{2} \left( {\eta - 2} \right)n^{2} + \eta^{2} + \left( { - \xi - 7} \right)\eta + 2\xi + 12} \right)} \right.e^{2\eta } \\ & \quad \left. { + \left( {\left( {\left( {\xi + 3} \right)\eta + 2\xi + 8} \right)\pi^{4} n^{4} + 6\pi^{2} \left( {\eta + 2} \right)n^{2} + \left( { - \xi - 5} \right)\eta - 2\xi - 12} \right)e^{\eta } } \right]\sin \left( {k\pi } \right) \\ & \quad - ke^{\eta } \left[ {\left( {\pi^{4} \left( {\xi + 1} \right)n^{4} + 6\pi^{2} n^{2} - \xi - 3} \right)\eta \pi^{4} k^{4} } \right. + 2\pi^{2} \left( {\left( {\left( {\xi + 2} \right)\eta + \xi } \right)\pi^{4} n^{4}}{6\pi^{2} \left( {\eta + 1} \right)n^{2} + \left( { - \xi - 4} \right)\eta - \xi - 2} \right)k^{2} \\ & \quad + \left. {\left. {\left( {\left( {\xi + 3} \right)\eta + 2\xi + 8} \right)\pi^{4} n^{4} + 6\pi^{2} \left( {\eta + 2} \right)n^{2} + \left( { - \xi - 5} \right)\eta - 2\xi - 12} \right]\pi } \right\}\pi^{4} \sin (n\pi \xi ) \\ & \quad - \, 2\left\{ {\left( {\left( {\left( { - \eta^{2} + \left( {\xi + 4} \right)\eta - 2\xi - 6} \right)\pi^{2} n^{2} - \eta^{2} + \left( {\xi + 8} \right)\eta - 2\xi - 14} \right)e^{2\eta } }\right.}\right.\\ & \quad +\left.{\left.{ \left( {\left( {\left( {\xi + 2} \right)\eta + 2\xi + 6} \right)\pi^{2} n^{2} + \left( {\xi + 6} \right)\eta + 2\xi + 14} \right)e^{\eta } } \right)} \right.\left( {\pi^{2} k^{2} + 1} \right)^{3} \sin \left( {k\pi } \right) \\ & \quad - ke^{\eta } \pi \left. {\left( {\eta \pi^{4} \left( {\pi^{2} n^{2} \xi + \xi + 4} \right)k^{4} + 2\left( {\left( {\left( {\xi + 1} \right)\eta + \xi - 1} \right)\pi^{2} n^{2} + \left( {\xi + 5} \right)\eta + \xi + 3} \right)\pi^{2} k^{2}}\right.}\right. \\ & \quad +\left.{\left.{ \left( {\left( {\xi + 2} \right)\eta + 2\xi + 6} \right)\pi^{2} n^{2} + \left( {\xi + 6} \right)\eta + 2\xi + 14} \right)} \right\}k^{3} (k + n)^{2} n^{5} (k - n)^{2} \pi^{5} \cos (n\pi \xi ) \\ & \quad + 4k^{3} \left( {\pi^{2} k^{2} + 1} \right)^{3} \left( {k + n} \right)^{2} \left( {6 + \eta \pi^{4} \left( {\xi - \eta } \right)n^{4} + \pi^{2} \left( { - \eta^{2} + \eta \xi + 14} \right)n^{2} } \right)\left( {k - n} \right)^{2} e^{\eta + \xi } \sin \left( {k\pi } \right)\sin \left( {\eta n\pi } \right) \\ & \quad + \left\{ {8k^{3} \left( {\pi^{2} \left( {\xi - 4\eta } \right)n^{2} + \xi - 2\eta } \right)} \right.\left( {\pi^{2} k^{2} + 1} \right)^{3} \left( {k + n} \right)^{2} \left( {k - n} \right)^{2} e^{\eta + \xi } \sin \left( {k\pi } \right)\cos \left( {\eta n\pi } \right) \\ & \quad + (\pi^{2} n^{2} + 1)^{3} (k + n)^{2} n^{3} (k - n)^{2} \left\{ {k^{3} \left( {\pi^{2} k^{2} + 1} \right)} \right.^{3} \left[ {\left( {\left( { - \xi - 2} \right)\eta^{2} + \left( {\xi^{2} + 9\xi + 16} \right)\eta - 2\xi^{2} }\right.}\right.\\ & \quad - \left.{\left.{16\xi - 28} \right)} \right.e^{2\eta } + \left( { - \eta^{5} /5 + \xi \eta^{4} /3 - 8\eta^{3} /3 + \left( {12 + 5\xi } \right)\eta - 12\xi - 28} \right)e^{\eta + \xi } \\ & \quad + \left( {\left( {\xi^{2} + 7\xi + 12} \right)\eta + 2\xi^{2} + 16\xi + 28} \right)e^{\eta } + \left. {\left( {\left( {\xi + 2} \right)\eta^{2} + \left( {7\xi + 16} \right)\eta + 12\xi + 28} \right)e^{\xi } } \right]\pi^{3} \sin \left( {k\pi } \right) \\ & \quad + \left( { - \left( {3\xi - 8} \right)\eta \pi^{8} k^{8} - 4\left( {\left( {2\xi - 5} \right)\eta + \xi - 3} \right)\pi^{6} k^{6} - \left( {\left( {5\xi - 12} \right)\eta + 12\xi - 28} \right)\pi^{4} k^{4}}\right. \\ & \quad - \left.{24\left( {\xi + \eta /3} \right)\pi^{2} k^{2} - 8\xi - 8\eta } \right)e^{\eta + \xi } \\ & \quad - k^{4} \left. {\left( {\eta \pi^{4} \left( {\xi^{2} + 5\xi + 8} \right)k^{4} + 2\left( {\left( {\xi^{2} + 6\xi + 10} \right)\eta + \xi^{2} + 4\xi + 6} \right)\pi^{2} k^{2} }\right.}\right.\\ & \quad +\left.{\left.{ \left( {\xi^{2} + 7\xi + 12} \right)\eta + 2\xi^{2} + 16\xi + 28} \right)e^{\eta } \pi^{4} } \right\}\sin (n\pi ) \\ & \quad - 3k^{3} \left\{ {\left( {\pi^{2} k^{2} + 1} \right)} \right.^{3} \left( {k + n} \right)^{2} \left\{ {\left( {\xi \pi^{8} \left( { - 8 + 3\eta } \right)n^{8} + 4\left( {\left( {2\xi + 1} \right)\eta - 5\xi - 3} \right)\pi^{6} n^{6} + \left( {\left( {5\xi + 12} \right)\eta }\right.}\right.}\right.\\ & \quad - \left.{\left.{\left.{12\xi - 28} \right)\pi^{4} n^{4} + 8\pi^{2} \left( {\xi + 3\eta } \right)n^{2} + 8\xi + 8\eta } \right)} \right.e^{\eta + \xi } /3 \\ & \quad + e^{\xi } n^{4} \left. {\left( {\pi^{4} \xi \left( {\eta^{2} + 5\eta + 8} \right)n^{4} + 2\left( {\left( {\xi + 1} \right)\eta^{2} + \left( {6\xi + 4} \right)\eta + 10\xi + 6} \right)\pi^{2} n^{2} + \left( {\xi + 2} \right)\eta^{2}}\right.}\right. \\ & \quad + \left.{\left.{\left( {7\xi + 16} \right)\eta + 12\xi + 28} \right)\pi^{4} /3} \right\}(k - n)^{2} \sin (k\pi ) \\ & \quad - kn^{4} \left\{ {\eta \pi^{8} \left( {\pi^{4} n^{4} \xi + 4\pi^{2} n^{2} \xi + 3\xi - 8} \right)} \right.k^{8} - 2\left( {n^{6} \pi^{6} \xi \eta + 2n^{4} \pi^{4} \xi \eta}\right.\\ & \quad -\left.{\pi^{2} \left( {3\left( {\xi + 2} \right)\eta + 2\xi + 2} \right)n^{2} + \left( { - 4\xi + 10} \right)\eta - 2\xi + 6} \right)\pi^{6} k^{6} \\ & \quad + \left( {n^{8} \pi^{8} \xi \eta - 4n^{6} \pi^{6} \xi \eta + \left( { - 18\eta \xi - 8} \right)\pi^{4} n^{4} -4\left({\left( {2\xi - 9} \right)\eta - 3\xi - 3} \right)\pi^{2} n^{2} +\left( {5\xi - 12} \right)\eta + 12\xi - 28} \right)\pi^{4} k^{4}\\ & \quad + 2\left( {2n^{8} \pi^{8} \xi \eta + \left( {\left( {3\xi + 2} \right)\eta + 6\xi + 2} \right)\pi^{6} n^{6} - 2\left( {\left( {2\xi - 3} \right)\eta - 9\xi - 3} \right)\pi^{4} n^{4} }\right.\\ & \quad-\left.{ \left( {\left( {5\xi - 24} \right)\eta - 24\xi - 28} \right)\pi^{2} n^{2} + 12\xi + 4\eta } \right)\pi^{2} k^{2} \\ & \quad \left. {\left. {\left. {\left. {\left. {\left. { + \xi \pi^{8} \left( { - 8 + 3\eta } \right)n^{8} + 4\left( {\left( {2\xi + 1} \right)\eta - 5\xi - 3} \right)\pi^{6} n^{6} + \left( {\left( {5\xi + 12} \right)\eta - 12\xi - 28} \right)\pi^{4} n^{4}}\right.}\right.}\right.}\right.}\right.}\right.\\ & \quad + \left.{\left.{\left.{\left.{\left.{\left.{8\pi^{2} \left( {\xi + 3\eta } \right)n^{2} + 8\xi + 8\eta } \right\}e^{\eta + \xi } \pi /3} \right\}} \right\}n\pi } \right\}} \right\}} \right\}\quad {\text{for}} \;\xi \ge \eta. \\ \end{aligned}$$
(47)

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Malkiel, N., Rabinovitch, O. & Elishakoff, I. Exact solutions for stochastic Bernoulli–Euler beams under deterministic loading. Acta Mech 232, 2201–2224 (2021). https://doi.org/10.1007/s00707-020-02895-1

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