NONLINEAR CONTINUUM MECHANICS FOR FINITE ELEMENT ANALYSIS
µ µ µ COLUMN THE SAME BAR-SPRING SYSTEM IS NOW POSITIONED VERTICALLY (SEE FIGURE 1 2(A)), AND AGAIN MOMENT EQUILIBRIUM ABOUT THE HINGE GIVES, PL SIN ? = M OR PL K = ? SIN ? (1 4) NONLINEAR CONTINUUM MECHANICS FOR FINITE ELEMENT ANALYSIS PL/K µ M = Kµ µ (A) P P FREE BODY DIAGRAM K L P (B) LINEAR NONLINEAR BIFURCATION FIGURE 1 2 SIMPLE COLUMN THE ABOVE EQUILIBRIUM EQUATION CAN HAVE TWO SOLUTIONS: ?RSTLY IF ? = 0, THEN SIN ? = 0, M = 0, AND EQUILIBRIUM IS SATIS?ED; AND SECONDLY, IF ? ?= 0, THEN PL/K = ?/ SIN ? THESE TWO SOLUTIONS ARE SHOWN IN FIGURE 1 2(B), WHERE THE VERTICAL AXIS IS THE EQUILIBRIUM PATH FOR ? = 0 AND THE HORSESHOE-SHAPED EQUILIBRIUM PATH IS THE SECOND SOLUTION THE INTERSECTION OF THE TWO SOLUTIONS IS CALLED A BIFURCATION POINT OBSERVE THAT FOR PL/K < 1 THERE IS ONLY ONE SOLUTION, NAMELY ? = 0 BUT FOR PL/K > 1 THERE ARE THREE SOLUTIONS FOR INSTANCE, WHEN PL/K ?1 57, EITHER ? = 0 OR ±?/2 FOR VERY SMALL VALUES OF ?, SIN ? ?? AND (1 4) REDUCES TO THE LINEAR (IN ?) EQUATION, (K ?PL)? = (1 5) AGAIN THERE ARE TWO SOLUTIONS: ? = 0 OR PL/K = 1 FOR ANY VALUE OF ?, THE LATTER SOLUTION BEING THE HORIZONTAL PATH SHOWN IN FIGURE 1 2(B) EQUATION (1 5) IS A TYPICAL LINEAR STABILITY ANALYSIS WHERE P = K/L IS THE ELASTIC CRITICAL (BUCKLING) LOAD APPLIED TO A BEAM COLUMN SUCH A GEOMETRICALLY NONLINEAR ANALYSIS WOULD YIELD THE EULER BUCKLING LOAD IN A ?NITE ELEMENT CONTEXT FOR, SAY, PLATES AND SHELLS THIS WOULD RESULT IN AN EIGENVALUE ANALYSIS, THE EIGENVALUES BEING THE BUCKLING LOADS AND THE EIGENVECTORS BEING THE CORRESPONDING BUCKLING MODES OBSERVE IN THESE TWO CASES THAT IT IS ONLY BY CONSIDERING THE ?NITE DIS- PLACEMENT OF THE STRUCTURES THAT A COMPLETE NONLINEAR SOLUTION HAS BEEN ACHIEVED NONLINEAR CONTINUUM MECHANICS FOR FINITE ELEMENT ANALYSIS