A new finite element which adequately represents the character of the strains near the crack-tip is presented. The element is based on displacement functions, derived from, assumed simple strain functions, by integrating the strain - displacement equations. The resulting displacement functions contain terms associated with the rigid body modes. The element is 'trapezoidal' in shape with two straight and two circular boundaries. A 'triangular' crack-tip element based on assumed simple displacement functions is also presented. This element has two of its boundaries straight, meeting at the tip of the crack, and the other circular. When employed in the singular region of the crack-tip, using as few as 312 degrees of freedom, these elements give results for displacements and stresses whose accuracies are within l«57o and ~y/o , respectively, of 'exact' solutions. The elements are combined with Constant Strain Triangular elements to analyse practical geometries used in fracture toughness testing. Relatively coarse meshes are sufficient to obtain accurate estimates of stress intensity factors by a direct displacement method not requiring an extrapolation process.