A method for determining the moments and forces of two concentric pipes within a wellbore, comprising: determining an external pipe displacement using a computer processor; determining whether the external pipe contacts the wellbore based on the external pipe displacement; determining a bending moment and a shear force of an internal pipe and the external pipe based on contact between the internal pipe and the external pipe and the external pipe displacement if the external pipe does not contact the wellbore; determining whether contact forces between the internal pipe and the external pipe and between the external pipe and the wellbore are greater than or equal to zero if the external pipe contacts the wellbore; determining the bending moment and the shear force of the internal pipe and the external pipe based on contact between the internal pipe and the external pipe and contact between the external pipe and the wellbore if the contact forces between the internal pipe and the external pipe and between the external pipe and the wellbore are greater than or equal to zero; determining a displacement solution using a contact force between the internal pipe and the external pipe equal to zero if the contact forces between the internal pipe and the external pipe and between the internal pipe and the wellbore are not greater than or equal to zero; determining whether there is another displacement solution using a contact force between the external pipe and the wellbore equal to zero if the contact forces between the internal pipe and the external pipe and between the external pipe and wellbore are not greater than or equal to zero; and determining the bending moment and the shear force of the internal pipe and the external pipe based on the displacement solution or the another displacement solution if the contact forces between the internal pipe and the external pipe and between the external pipe and the wellbore are not greater than or equal to zero. determining an external pipe displacement using a computer processor; determining whether the external pipe contacts the wellbore based on the external pipe displacement; determining a bending moment and a shear force of an internal pipe and the external pipe based on contact between the internal pipe and the external pipe and the external pipe displacement if the external pipe does not contact the wellbore; determining whether contact forces between the internal pipe and the external pipe and between the external pipe and the wellbore are greater than or equal to zero if the external pipe contacts the wellbore; determining the bending moment and the shear force of the internal pipe and the external pipe based on contact between the internal pipe and the external pipe and contact between the external pipe and the wellbore if the contact forces between the internal pipe and the external pipe and between the external pipe and the wellbore are greater than or equal to zero; determining a displacement solution using a contact force between the internal pipe and the external pipe equal to zero if the contact forces between the internal pipe and the external pipe and between the internal pipe and the wellbore are not greater than or equal to zero; determining whether there is another displacement solution using a contact force between the external pipe and the wellbore equal to zero if the contact forces between the internal pipe and the external pipe and between the external pipe and wellbore are not greater than or equal to zero; and determining the bending moment and the shear force of the internal pipe and the external pipe based on the displacement solution or the another displacement solution if the contact forces between the internal pipe and the external pipe and between the external pipe and the wellbore are not greater than or equal to zero.