## 2 Fundamental Theory of the Relation between Pre-Stress Deviation and Member Length Deviation  
In order to analyze the effects of construction errors, member length error, which is a significant construction error and will affect the mechanical performance sensitively, was chosen as the representative construction error, and the relation between the pre-stress deviation and member length error needs to be clarified first. In a cable-bar tensile structure, the following three equations can be established [18; 19]. \[\mathbf{A}_{3n\times b}\mathbf{t}_{b\times 1}=\mathbf{P}_{3n\times 1} \tag{1}\]  
\[\mathbf{t}_{b\times 1}=\mathbf{M}_{b\times b}\big{(}\mathbf{e}_{b\times 1}-\big{(}\mathbf{e}_{0} \big{)}_{b\times 1}\big{)} \tag{2}\]  
\[\mathbf{B}_{b\times 3n}\mathbf{d}_{3n\times 1}=\mathbf{e}_{b\times 1} \tag{3}\]  
In the Equations (1)-(3), \(\mathbf{A}_{3n\times b}\) and \(\mathbf{B}_{b\times 3n}\) are the equilibrium matrix and the coordinates matrix, respectively; \(\mathbf{B}=\mathbf{A}^{\top}\), \(\mathbf{t}_{b\times 1}\) is the internal force matrix; \(\mathbf{P}_{3n\times 1}\) and \(\mathbf{d}_{3n\times 1}\) are the nodal load matrix and node displacement matrix, respectively; \(\mathbf{M}_{b\times b}\) is the member stiffness matrix; \(\mathbf{M}_{kk}=\mathbf{E}_{k}\mathbf{A}_{k}/l_{k}\), \(\mathbf{E}_{k}\), \(\mathbf{A}_{k}\) and \(\mathbf{l}_{k}\) are the elastic modulus, section size, and length for member \(k\), respectively; \(\mathbf{e}_{b\times 1}\) and \(\left(\mathbf{e}_{0}\right)_{b\times 1}\) are the member length change matrix and the initial member length change matrix, respectively; and \(b\) and \(n\) are the total number of member and free node. The combinations of Equations (1)-(3) are expressed as follows:  
\[\mathbf{t} =\mathbf{t}_{P}+\mathbf{t}_{e} \tag{4}\] \[\mathbf{d} =\mathbf{d}_{p}+\mathbf{d}_{e} \tag{5}\]  
In the Equations (4) and (5), \(\mathbf{t}_{P}=\mathbf{M}\mathbf{A}^{\mathbf{\mathrm{T}}}\big{(}\mathbf{M}\mathbf{A}\mathbf{\mathrm{T}} \big{)}^{-1}\mathbf{P}\), \(t_{e}=\mathbf{M}\big{(}\mathbf{A}\mathbf{\mathrm{T}}\big{(}\mathbf{A}\mathbf{M}\mathbf{\mathrm{A}}^{ \mathbf{\mathrm{T}}}\big{)}^{-1}\mathbf{A}\mathbf{M}-\mathbf{I}\big{)}\mathbf{e}_{0}\), \(\mathbf{d}_{p}=\big{(}\mathbf{M}\mathbf{A}\mathbf{\mathrm{A}}^{\mathbf{\mathrm{T}}}\big{)}^{-1} \mathbf{P}\), and \(\mathbf{d}_{e}=\big{(}\mathbf{M}\mathbf{A}\mathbf{\mathrm{A}}^{\mathbf{\mathrm{T}}}\big{)}^{-1} \mathbf{M}\mathbf{e}_{0}\), \(\mathbf{I}\) is the unit matrix; \(\mathbf{t}_{P}\) and \(\mathbf{t}_{e}\) are the member internal force resulting from nodal load \(\mathbf{P}\) and the initial member length defect \(\mathbf{e}_{0}\), respectively; and \(\mathbf{d}_{p}\) and \(\mathbf{d}_{e}\) are the node displacement resulting from nodal load \(\mathbf{P}\) and the initial member length defect \(\mathbf{e}_{0}\), respectively. Thus, when the nodal load \(\mathbf{P}\) remains unchanged or is zero, the member pre-stress deviations and node displacement deviations are led only by the initial member length variation. Suppose the member length changes by \(\delta\mathbf{e}_{0}\), the member pre-stress deviations and node displacement deviations are  
\[\delta\mathbf{t} =\mathbf{S}_{t}\delta\mathbf{e}_{0} \tag{6}\] \[\delta\mathbf{d} =\mathbf{S}_{d}\delta\mathbf{e}_{0} \tag{7}\]  
In the Equations (6) and (7), \(\mathbf{S}_{t}=\mathbf{M}\big{(}\mathbf{A}\mathbf{\mathrm{T}}\big{(}\mathbf{A}\mathbf{M}\mathbf{\mathrm{A} }^{\mathbf{\mathrm{T}}}\big{)}^{-1}\mathbf{A}\mathbf{M}-\mathbf{I}\big{)}\) stands for the force sensitivity matrix and \(\mathbf{S}_{d}=\big{(}\mathbf{M}\mathbf{A}\mathbf{\mathrm{A}}^{\mathbf{\mathrm{T}}}\big{)}^{-1} \mathbf{M}\mathbf{M}\) stands for the shape sensitivity matrix. When the nodal load \(\mathbf{P}\) is not taken into account, the pre-stress is \(\mathbf{t}_{0}=\mathbf{S}_{t}\mathbf{e}_{0}\). According to the function of the members during the construction process, they can be sorted into two different kinds. One kind is active member that helps tension in the structure.