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Abstract In this paper, we discuss the positive solutions of beam equations with the nonlinearities including the slope and bending moment under nonlocal boundary conditions involving Stieltjes integrals. We pose some inequality conditions on nonlinearities and the spectral radius conditions on associated linear operators. These conditions mean that the nonlinearities have superlinear or sublinear growth. The existence of positive solutions is obtained by fixed point index on cones in C 2 [ 0 , 1 ] $C^{2}[0,1]$ , and some examples are given for beam equations subject to mixed integral and multi-point boundary conditions with sign-changing coefficients.