Page 1 Module III Module III Direct stiffness method • Introduction – element stiffness matrix – rotation transformation Direct stiffness method matrix – transformation of displacement and load vectors and stiffness matrix – equivalent nodal forces and load vectors – assembly of stiffness matrix and load vector – determination of assembly of stiffness matrix and load vector determination of nodal displacement and element forces – analysis of plane truss beam and plane frame (with numerical examples) – analysis of grid space frame (without numerical examples) grid – space frame (without numerical examples) Page 2 Module III Module III Direct stiffness method • Introduction – element stiffness matrix – rotation transformation Direct stiffness method matrix – transformation of displacement and load vectors and stiffness matrix – equivalent nodal forces and load vectors – assembly of stiffness matrix and load vector – determination of assembly of stiffness matrix and load vector determination of nodal displacement and element forces – analysis of plane truss beam and plane frame (with numerical examples) – analysis of grid space frame (without numerical examples) grid – space frame (without numerical examples) Introduction • The formalised stiffness method involves evaluating the displacement transformation matrix C MJ correctly p y • Generation of matrix C MJ is not suitable for computer programming H th l ti f di t tiff thd • Hence the evolution of direct stiffness method Page 3 Module III Module III Direct stiffness method • Introduction – element stiffness matrix – rotation transformation Direct stiffness method matrix – transformation of displacement and load vectors and stiffness matrix – equivalent nodal forces and load vectors – assembly of stiffness matrix and load vector – determination of assembly of stiffness matrix and load vector determination of nodal displacement and element forces – analysis of plane truss beam and plane frame (with numerical examples) – analysis of grid space frame (without numerical examples) grid – space frame (without numerical examples) Introduction • The formalised stiffness method involves evaluating the displacement transformation matrix C MJ correctly p y • Generation of matrix C MJ is not suitable for computer programming H th l ti f di t tiff thd • Hence the evolution of direct stiffness method Direct stiffness method • We need to simplify the assembling process of S J , the J assembled structure stiffness matrix • The key to this is to use member stiffness matrices for actions The key to this is to use member stiffness matrices for actions and displacements at BOTH ends of each member If b di l d ih f • If member displacements are expressed with reference to global co-ordinates, the process of assembling S J can be made simple simple Page 4 Module III Module III Direct stiffness method • Introduction – element stiffness matrix – rotation transformation Direct stiffness method matrix – transformation of displacement and load vectors and stiffness matrix – equivalent nodal forces and load vectors – assembly of stiffness matrix and load vector – determination of assembly of stiffness matrix and load vector determination of nodal displacement and element forces – analysis of plane truss beam and plane frame (with numerical examples) – analysis of grid space frame (without numerical examples) grid – space frame (without numerical examples) Introduction • The formalised stiffness method involves evaluating the displacement transformation matrix C MJ correctly p y • Generation of matrix C MJ is not suitable for computer programming H th l ti f di t tiff thd • Hence the evolution of direct stiffness method Direct stiffness method • We need to simplify the assembling process of S J , the J assembled structure stiffness matrix • The key to this is to use member stiffness matrices for actions The key to this is to use member stiffness matrices for actions and displacements at BOTH ends of each member If b di l d ih f • If member displacements are expressed with reference to global co-ordinates, the process of assembling S J can be made simple simple Member oriented axes (local coordinates) d t t i t d ( l b l di t ) and structure oriented axes (global coordinates) L d x y L Local axes L d L L d sin L d ? L d? x Y Y Y Y cos L d? ? Global axes y Global axes Global axes ? X X X X Global axes Page 5 Module III Module III Direct stiffness method • Introduction – element stiffness matrix – rotation transformation Direct stiffness method matrix – transformation of displacement and load vectors and stiffness matrix – equivalent nodal forces and load vectors – assembly of stiffness matrix and load vector – determination of assembly of stiffness matrix and load vector determination of nodal displacement and element forces – analysis of plane truss beam and plane frame (with numerical examples) – analysis of grid space frame (without numerical examples) grid – space frame (without numerical examples) Introduction • The formalised stiffness method involves evaluating the displacement transformation matrix C MJ correctly p y • Generation of matrix C MJ is not suitable for computer programming H th l ti f di t tiff thd • Hence the evolution of direct stiffness method Direct stiffness method • We need to simplify the assembling process of S J , the J assembled structure stiffness matrix • The key to this is to use member stiffness matrices for actions The key to this is to use member stiffness matrices for actions and displacements at BOTH ends of each member If b di l d ih f • If member displacements are expressed with reference to global co-ordinates, the process of assembling S J can be made simple simple Member oriented axes (local coordinates) d t t i t d ( l b l di t ) and structure oriented axes (global coordinates) L d x y L Local axes L d L L d sin L d ? L d? x Y Y Y Y cos L d? ? Global axes y Global axes Global axes ? X X X X Global axes 1. Plane truss member Stiffness coefficients in local coordinates 1 3 2 4 y 0 0 Degrees of freedom 1 ?? ?? Unit displacement x EA L EA L ?? ?? corr. to DOF 1 00 EA EA ? ? - ?? [] 00 0000 00 M LL S EA EA ?? ? ? ? ? ? ? - ?? = Member stiffness matrix in local coordinates 0000 L L ?? ? ? ? ? ? ?Read More

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