In this article, we study the fourth-order problem with the first and second derivatives in nonlinearity under nonlocal boundary value conditions{u(4)(t)=h(t)f(t,u(t),u′(t),u″(t)),t∈(0,1),u(0)=u(1)=β1[u],u″(0)+β2[u]=0,u″(1)+β3[u]=0,\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document} $$\\begin{aligned}& \\left \\{ \\textstyle\\begin{array}{l}u^{(4)}(t)=h(t)f(t,u(t),u'(t),u''(t)),\\quad t\\in(0,1),\\\\ u(0)=u(1)=\\beta_{1}[u],\\qquad u''(0)+\\beta_{2}[u]=0,\\qquad u''(1)+\\beta_{3}[u]=0, \\end{array}\\displaystyle \\right . \\end{aligned}$$ \\end{document} where f: [0,1]timesmathbb{R}_{+}timesmathbb{R}timesmathbb{R}_{-}to mathbb{R}_{+} is continuous, hin L^{1}(0,1) and beta_{i}[u] is Stieltjes integral (i=1,2,3). This equation describes the deflection of an elastic beam. Some inequality conditions on nonlinearity f are presented that guarantee the existence of positive solutions to the problem by the theory of fixed point index on a special cone in C^{2}[0,1]. Two examples are provided to support the main results under mixed boundary conditions involving multi-point with sign-changing coefficients and integral with sign-changing kernel.