In this paper, limiting force profile for laterally loaded rigid piles in sand is differentiated from on-pile force profile, from which elastic-plastic solutions are established and presented in explicit expressions. Nonlinear responses of the piles are characterised by slip depths mobilised from mudline and the pile-tip. At the states of tip-yield and rotation point yield, expressions for some critical depths are developed, which allow the on-pile force profiles to be constructed. The solutions and the expressions are developed concerning a constant subgrade modulus (k) and a linearly increasing modulus with depth (Gibson k), respectively. Capitalised on three measurable parameters, the solutions agree well with measured data and numerical predictions. Non-dimensional responses are presented for various eccentricities at the tip-yield state and in some cases at the rotation point yield state. Nonlinear responses are obtained for typical eccentricities from elastic state right up to failure. Comments are made regarding displacement based capacity. A case study is elaborated to illustrate (i) the use of the current solutions; (ii) the impact of the k distributions; (iii) the evaluation of stresses on pile surface; and (iv) the deduction of soil modulus. The current solutions are easy to be implemented and suitable for general design.