The static analysis (ANTYPE,STATIC) solution method is valid for all degrees of freedom (DOFs). Inertial and damping effects are ignored, except for static acceleration fields.

The overall equilibrium equations for linear structural static analysis are:

(15–1)

or

(15–2)

where:

= total stiffness matrix

= nodal displacement vector

= number of elements

= element stiffness matrix (described in Element Library) (may include the element stress stiffness matrix (described in Stress Stiffening))

= reaction load vector

, the total applied load vector, is defined by:

(15–3)

where:

= applied nodal load vector

= acceleration load vector

= total mass matrix

= element mass matrix (described in Derivation of Structural Matrices)

= total acceleration vector (defined in Acceleration Effect)

= element thermal load vector (described in Derivation of Structural Matrices)

= element pressure load vector (described in Derivation of Structural Matrices)

To illustrate the load vectors in Equation 15–2, consider a one element column model, loaded only by its own weight, as shown in Figure 15.1: Applied and Reaction Load Vectors. Note that the lower applied gravity load is applied directly to the imposed displacement, and therefore causes no strain; nevertheless, it contributes to the reaction load vector just as much as the upper applied gravity load. Also, if the stiffness for a certain DOF is zero, any applied loads on that DOF are ignored.

Figure 15.1:  Applied and Reaction Load Vectors

Solving for Unknowns and Reactions discusses the solution of Equation 15–2 and the computation of the reaction loads. Newton-Raphson Procedure describes the global equation for a nonlinear analysis. Inertia relief is discussed in Inertia Relief.

The overall equations for linear 1st order systems are the same as for a linear structural static analysis, Equation 15–1 and Equation 15–2. , though, is the total coefficient matrix (e.g., the conductivity matrix in a thermal analysis) and is the nodal DOF values. , the total applied load vector, is defined by:

(15–4)

Table 15.1: Nomenclature relates the nomenclature used in Derivation of Heat Flow Matrices and Derivation of Electromagnetic Matrices for thermal, magnetic and electrical analyses to Equation 15–2 and Equation 15–4. See Table 10.3: Nomenclature of Coefficient Matrices for a more detailed nomenclature description.

Table 15.1:  Nomenclature

Thermal	temperature	heat flow	heat flux heat generation convection
Scalar Magnetic	scalar potential	flux	coercive force
Vector Magnetic	vector potential	current segment	current density and coercive force
Electrical	voltage	current	-

Solving for Unknowns and Reactions discusses the solution of Equation 15–2 and Newton-Raphson Procedure describes the global equation for a nonlinear analysis.