thinwalled · SJin24 · Mar 31, 2026
diff --git a/analysis/fcFSM/SecAnal_fcFSM.m b/analysis/fcFSM/SecAnal_fcFSM.m
@@ -17,10 +17,6 @@
 %J_D:A basis of [q] that are in equilibrium
 %	if [q] is in equilibrium, fcFSM considers the corresponding deformation as D class deformation.
 
-%Limitations of current version of fcFSM
-%1. User defined equation constraints are not supported, and DOF constraints on nodes are not supported.
-%		there will be a general fcFSM algorithm for arbitrary supporting conditions.
-
 nNd=size(node,1);%number of nodes
 xNd=node(:,2); %X- coordinates of nodes
 zNd=node(:,3); %Z- coordinates of nodes

diff --git a/analysis/fcFSM/stripmain_fcFSM.m b/analysis/fcFSM/stripmain_fcFSM.m
@@ -6,7 +6,7 @@
 %cornerStrips: strip elements belong to curved corners.
 %curveL, curveD, curveG: L, D, and G buckling curves (load factors vs. lengths)
 %shapesL, shapesD, shapesG: L, D, and G mode shapes for each length
-%Feb. 8, 2024; Jun. 7, 2024. Sheng Jin
+%Feb. 8, 2024; Jun. 7, 2024; Mar. 31, 2026. Sheng Jin
 
 %HISTORY
 %June 2010, complete update to new Boundary conditions, Z. Li, B. Schafer
@@ -120,13 +120,6 @@
 	cornerStrips=[];
 end
 cFSM_analysis=0;%switch off cFSM. This program tests fcFSM only
-%in fcFSM, user defined constraints (equation constraints and nodal DOF constraints)
-%will be handled based on a general algorithm which will be realized in the future versions.
-%Now we ignore them.
-if BCFlag~=0
-	fprintf('\nCurrently user defined constraints and internal (at node) B.C. are ignored in fcFSM.');
-	BCFlag=0;
-end
 
 curveL=cell(nlengths,1);
 shapesL=cell(nlengths,1);
@@ -349,29 +342,53 @@
     end
 
 	%% ==additional settings for this fcFSM testing, #2=====
-	%For multiple longitudinal term problems, the C_L, J_D, J_GD should be multipalized
+	%For multiple longitudinal term problems, the C_L, J_D, J_GD should be duplicated along the diagonal directions
 	if totalm==1
 		J_GD=J_GD_oneFunc;
 		J_D=J_D_oneFunc;
-		C_L=C_L_oneFunc;
+		%C_L=C_L_oneFunc;
 	else
 		J_GD=zeros(size(J_GD_oneFunc)*totalm);
 		J_D=zeros(size(J_D_oneFunc)*totalm);
-		C_L=zeros(size(C_L_oneFunc)*totalm);
+		%C_L=zeros(size(C_L_oneFunc)*totalm);
 		for i=1:totalm
 			J_GD(size(J_GD_oneFunc,1)*(i-1)+1:size(J_GD_oneFunc,1)*i,size(J_GD_oneFunc,2)*(i-1)+1:size(J_GD_oneFunc,2)*i)=J_GD_oneFunc;
 			J_D(size(J_D_oneFunc,1)*(i-1)+1:size(J_D_oneFunc,1)*i,size(J_D_oneFunc,2)*(i-1)+1:size(J_D_oneFunc,2)*i)=J_D_oneFunc;
-			C_L(size(C_L_oneFunc,1)*(i-1)+1:size(C_L_oneFunc,1)*i,size(C_L_oneFunc,2)*(i-1)+1:size(C_L_oneFunc,2)*i)=C_L_oneFunc;
+			%C_L(size(C_L_oneFunc,1)*(i-1)+1:size(C_L_oneFunc,1)*i,size(C_L_oneFunc,2)*(i-1)+1:size(C_L_oneFunc,2)*i)=C_L_oneFunc;
 		end
-		C_L=sparse(C_L);
+		%C_L=sparse(C_L);
 	end
-
-	%the construction of fcFSM D and G constraint matrices needs stiffness matrix, so they will be determined here
-	C_D=K\J_GD*J_D;% as explained in SecAnal_fcFSM.m, this is the constraint matrix of D mode class
-	%as explained in SecAnal_fcFSM.m, constraint matrix of G, [C_G], can be obtained as such:
-	J_G=null(C_D'*J_GD);
-	C_G=K\J_GD*J_G;
 
+	% %=========================================================================
+	% %if the modal definition is regarded as a problem unrelated to support conditions
+	% %the fcFSM D and G constraint matrices can be obtained as below
+	% C_D=K\J_GD*J_D;% as explained in SecAnal_fcFSM.m, this is the constraint matrix of D mode class
+	% %as explained in SecAnal_fcFSM.m, constraint matrix of G, [C_G], can be obtained as such:
+	% J_G=null(C_D'*J_GD);
+	% C_G=K\J_GD*J_G;
+	% %=========================================================================
+
+	% traditionally, supporting conditions are not considered in the modal definitions of G, D, and L
+	% but we highly recommend that they should be
+	% because the G, D, and L solutions can better represent the buckling properties of thin-walled members
+	% here lists the way introducing the supporting conditions in the fcFSM modal definitions.
+	K_B=R'*K*R; %reduced stiffness matrix
+
+	J_B_GD=R'*J_GD; 
+	J_B_D=J_B_GD*J_D;
+
+	C_B_L=null(J_B_GD');
+	C_B_GD=K_B\orth(J_B_GD);
+	C_B_D=K_B\orth(J_B_D);
+	C_GD_G=null(J_B_D'*C_B_GD);
+
+	C_D=R*C_B_D;
+	C_G=R*C_B_GD*C_GD_G;
+	C_L=R*C_B_L;
+	% modal definition job done. The operations of null() and orth() may incur some computational consumption in large models.
+	% Optimizations can be introduced into these operations, noticing the block construction form of the J_GD and J_D matrices.
+
+
 	%perform L, D, and G modal buckling analyses
 	% L buckling
 	[tempMat_eigenMode,tempMat_eigenValue]=eigs(C_L'*K*C_L,C_L'*Kg*C_L,min([size(C_L,2),neigs]),'SM');

diff --git a/examples/An introduction to fcFSM.txt b/examples/An introduction to fcFSM.txt
@@ -1,3 +1,8 @@
+20260331
+Now, all types of supporting conditions (end boundary conditions, springs, internal boundary conditions on the nodes, and user defined constraint equations) are compatible with this fcFSM.
+Traditionally, modal definitions of G, D, and L are irrelative to supporting conditions of a member.
+But in this fcFSM, supporting conditions are introduced in the modal definitions.
+The resulted G, D, and L solutions show better representations to the buckling properties of thin-walled members, please see the newly added examples "C_120X80X15X1_N5Uz=0_N10Ux=0" and "C_120X80X15X1_m=1to5_N5Uz=0_N7Ux=N15Uz".
 
 20240606
 Now multiple longitudinal terms can be considered.

diff --git a/examples/batchcufsm5_fcFSM.m b/examples/batchcufsm5_fcFSM.m
@@ -30,6 +30,8 @@
 nameExample='CircularTube_D100X1';
 nameExample='theDefaultSection';
 nameExample='C_120X80X15X1_m=1to20';
+nameExample='C_120X80X15X1_m=1to5_N5Uz=0_N7Ux=N15Uz';
+nameExample='C_120X80X15X1_N5Uz=0_N10Ux=0';
 
 %Please specify the index of the bucklig mode/curve to be plotted (1 for the lowest buckling; 2 and so on for higher buckling modes).
 modeindex=1;

diff --git a/examples/fcFSM_examples/C_120X80X15X1_N5Uz=0_N10Ux=0/modelData.m b/examples/fcFSM_examples/C_120X80X15X1_N5Uz=0_N10Ux=0/modelData.m
@@ -0,0 +1,13 @@
+function [prop,node,elem,lengths,BC,m_all,springs,constraints,neigs,cornerStrips]=modelData(pathCUFSM)
+%This model is the same to the model 'C_120X80X15X1' except that the zdof of Node #5 and the xdof of Node #10 are fixed.
+
+%So, we will read in the data from 'C_120X80X15X1' model, then fix the two DOFs.
+
+inhertModelName='C_120X80X15X1';
+curPath=pwd;
+cd([pathCUFSM,'/examples/fcFSM_examples/',inhertModelName]);
+[prop,node,elem,lengths,BC,m_all,springs,constraints,neigs,cornerStrips]=modelData();
+cd(curPath);
+
+node(5,5)=0; %UZ dof constraint at Node #5
+node(10,4)=0; %UX dof constraint at Node #10
diff --git a/examples/fcFSM_examples/C_120X80X15X1_m=1to5_N5Uz=0_N7Ux=N15Uz/modelData.m b/examples/fcFSM_examples/C_120X80X15X1_m=1to5_N5Uz=0_N7Ux=N15Uz/modelData.m
@@ -0,0 +1,22 @@
+function [prop,node,elem,lengths,BC,m_all,springs,constraints,neigs,cornerStrips]=modelData(pathCUFSM)
+%This model is the same to 'C_120X80X15X1' model except that
+%(i) multiple longitudinal terms are considered: 1,2,3,4,5,
+%(ii) the zdof of Node #5 is fixed, and
+%(iii) the Ux of Node #7 is equal to the Uz of Node #15 (u7 = w15) 
+
+%So, we will read in the data from 'C_120X80X15X1' model, then set m as 1:5, 
+%and then the internal boundary conditions (on the nodes) and user defined constraints (equation constraints).
+
+inhertModelName='C_120X80X15X1';
+curPath=pwd;
+cd([pathCUFSM,'/examples/fcFSM_examples/',inhertModelName]);
+[prop,node,elem,lengths,BC,m_all,springs,constraints,neigs,cornerStrips]=modelData();
+cd(curPath);
+
+for i=1:length(lengths)
+    m_all{i}=1:5;
+end
+
+node(5,5)=0; %UZ dof constraint at Node #5
+
+constraints = [7 1 1.000 15 2 0.000 0 0]; %u7 = w15