The purpose of this paper is to develop a new method of seismic stiffness design for a shear building with Maxwell type viscous dampers, and to disclose basic characteristics of the effect of flexibility of spring elements in Maxwell type viscous dampers on the stiffness design. Consider an f-story planar elastic shear building with Maxwell type viscous dampers (Fig.2), called here an SBMVD model, and a spring element in a Maxwell type damper is simply called a brace. A model with braces of null flexibility is called an SBVD model. Since an SBMVD model is a non-proportionally damped structure, complex eigenvalue analysis and a combination rule presented by eq.(7) for non-proportional damped structure are employed here for evaluating the maximum interstory drift of a shear building subject to the design moderate earthquake. Then, the stiffness design problem for the SBMVD model may be formulated as follows: [Problem ERCD] Given specified set {cv} of damping coefficient of dampers and set {kv} of stiffness of braces with respect to interstory drift, find the set {ks} of story stiffnesses of an SBMVD model such that a distribution of maximum interstory drifts would coincide with a specified one. The solution to problem ERCD will be found for one set of {cv} and {kv}. If one level parameter representing {cv} and/or {kv} is introduced, then a sequence of {ks}s with respect to that parameter may be conceived and may be called a stiffness design sequence. A method of generating the design sequence with respect to parameter a representing the flexibility, i.e., inverse of stiffness, of braces is developed here for specified {cv} by means of piecewise Taylor series expansion. This procedure is initiated with the stiffness design of the corresponding SBVD model. Singularity due to null flexibility at starting point is removed by appropriate variable transformation. Six stiffness design sequences for ten and twenty story SBMVD models have been generated by the represented procedure. All the flexibilities of braces have been presented by one and the same function in terms of a. Figs.5(a-f) show variations of story stiffnesses in the examples with respect to brace flexibility ratio defined by eqn.(21). The following results have been obtained in this paper: (1)Stiffness design sequence of a shear building with Maxwell type viscous dampers has been effectively generated by piecewise Taylor series expansion with respect to a level parameter representing the flexibility of spring elements in such dampers. (2)Almost all the story stiffnesses increase from those in the corresponding SBVD model as flexibility of braces increase. Distribution of story stiffnesses is approximately proportion to that in the corresponding SBVD model. (3)The effect of flexibility of braces on the stiffness design is greater in an SBMVD model with viscous dampers of larger damping coefficients.