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3D Structures
- // calculate the stiffness matrix for a regular plan frame//
author F. Paulet-Crainiceanu// version 1.0 / December
2006clearmode(0)disp('')disp('')disp('
*******************************************')disp(' *** CALCULATION
OF THE STIFFNESS MATRIX ***')disp('
*******************************************')disp('')disp('')n=0;while
n
-
disp('')disp(' ** SPAN LENGTHS:')disp('')for j=1:ml(j)=0;
while l(j)
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// stiffness from floor displacements// first floorr(1,1)=0;if
n==1 then // first floor = last floor, one floor structure for
j=1:m+1 // translation dof r(1,1)=r(1,1)+12*c(1,j)/h(1)^3; p=n+j;
// rotational dof r(p,1)=-6*c(1,j)/h(1)^2;
endelse // first floor, more than one floor
structurer(2,1)=0;for j=1:m+1
// translation dof
r(1,1)=r(1,1)+12*(c(1,j)/h(1)^3+c(2,j)/h(2)^3);
r(2,1)=r(2,1)-12*c(2,j)/h(2)^3; p=n+j; // rotational dof
r(p,1)=6*(c(2,j)/h(2)^2-c(1,j)/h(1)^2);
r(p+m+1,1)=6*c(2,j)/h(2)^2;
endend// stiffness from floor displacements// last floorif n~=1
then // last floor, more than one floor
structurer(n,n)=0;r(n-1,n)=0;for j=1:m+1
// translation dof r(n,n)=r(n,n)+12*c(n,j)/h(n)^3;
r(n-1,n)=r(n-1,n)-12*c(n,j)/h(n)^3; // rotational dof
p=n+(n-1)*(m+1)+j; r(p,n)=-6*c(n,j)/h(n)^2;
r(p-m-1,n)=-6*c(n,j)/h(n)^2;
endend // stiffness from floor displacements// current floorsfor
i=2:n-1
r(i,i)=0; r(i-1,i)=0; r(i+1,i)=0; for j=1:m+1 // translation dof
r(i,i)=r(i,i)+12*(c(i,j)/h(i)^3+c(i+1,j)/h(i+1)^3);
r(i-1,i)=r(i-1,i)-12*c(i,j)/h(i)^3;
r(i+1,i)=r(i+1,i)-12*c(i+1,j)/h(i+1)^3; p=n+(i-1)*(m+1)+j; //
rotational dof r(p,i)=6*(c(i+1,j)/h(i+1)^2-c(i,j)/h(i)^2);
r(p+m+1,i)=6*c(i+1,j)/h(i+1)^2; r(p-m-1,i)=-6*c(i,j)/h(i)^2;
end
end
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// stiffness from node rotation// first floorif n==1 then //
first floor = last floor, one floor structure// left node, j=1
// translation dof r(1,2)=-6*c(1,1)/h(1)^2; // rotational dof
r(2,2)=4*(c(1,1)/h(1)+b(1,1)/l(1)); r(3,2)=2*b(1,1)/l(1);
// right node, j=m+1 // translation dof p=m+2;
r(1,p)=-6*c(1,m+1)/h(1)^2; // rotational dof
r(p,p)=4*(c(1,m+1)/h(1)+b(1,m)/l(m)); r(p-1,p)=2*b(1,m)/l(m);
// current nodefor j=2:m
// translation dof p=n+j; r(1,p)=-6*c(1,j)/h(1)^2; // rotational
dof r(p,p)=4*(c(1,j)/h(1)+b(1,j-1)/l(j-1)+b(1,j)/l(j));
r(p-1,p)=2*b(1,j-1)/l(j-1); r(p+1,p)=2*b(1,j)/l(j);
endelse // first floor, more than one floor structure// left
node, j=1
// translation dof p=n+1;
r(1,p)=6*(-c(1,1)/h(1)^2+c(2,1)/h(2)^2); r(2,p)=-6*c(2,1)/h(2)^2;
// rotational dof r(p,p)=4*(c(1,1)/h(1)+c(2,1)/h(2)+b(1,1)/l(1));
r(p+1,p)=2*b(1,1)/l(1); r(p+m+1,p)=2*c(2,1)/h(2);
// right node, j=m+1 // translation dof p=n+m+1;
r(1,p)=6*(-c(1,m+1)/h(1)^2+c(2,m+1)/h(2)^2);
r(2,p)=-6*c(2,m+1)/h(2)^2; // rotational dof
r(p,p)=4*(c(1,m+1)/h(1)+c(2,m+1)/h(2)+b(1,m)/l(m));
r(p-1,p)=2*b(1,m)/l(m); r(p+m+1,p)=2*c(2,m+1)/h(2);
// current nodefor j=2:m
// translation dof p=n+j;
r(1,p)=6*(-c(1,j)/h(1)^2+c(2,j)/h(2)^2); r(2,p)=-6*c(2,j)/h(2)^2;
// rotational dof
r(p,p)=4*(c(1,j)/h(1)+c(2,j)/h(2)+b(1,j-1)/l(j-1)+b(1,j)/l(j));
r(p-1,p)=2*b(1,j-1)/l(j-1); r(p+1,p)=2*b(1,j)/l(j);
r(p+m+1,p)=2*c(2,j)/h(2);
end
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end// stiffness from node rotation// last floorif n~=1 then //
last floor, more than one floor structure// left node,
j=1p=n+(n-1)*(m+1)+1;
// translation dofr(n,p)= -6*c(n,1)/h(n)^2;r(n-1,p)=
6*c(n,1)/h(n)^2;
// rotational
dofr(p,p)=4*(c(n,1)/h(n)+b(n,1)/l(1));r(p+1,p)=2*b(n,1)/l(1);r(p-m-1,p)=2*c(n,1)/h(n);//
right node, j=m+1
// translation dofp=n*(m+2);r(n,p)= -6*c(n,m+1)/h(n)^2;r(n-1,p)=
6*c(n,m+1)/h(n)^2;
// rotational
dofr(p,p)=4*(c(n,m+1)/h(n)+b(n,m)/l(m));r(p-1,p)=2*b(n,m)/l(m);r(p-m-1,p)=2*c(n,m+1)/h(n);//
right node, j=m+1for j=2:m
p=n+(n-1)*(m+1)+j; // translation dof r(n,p)= -6*c(n,j)/h(n)^2;
r(n-1,p)= 6*c(n,j)/h(n)^2; // rotational dof
r(p,p)=4*(c(n,j)/h(n)+b(n,j-1)/l(j-1)+b(n,j)/l(j));
r(p-1,p)=2*b(n,j-1)/l(j-1); r(p+1,p)=2*b(n,j)/l(j);
r(p-m-1,p)=2*c(n,j)/h(n);
end end // stiffness from node rotation// current floorfor
i=2:n-1// left node, j=1p=n+(i-1)*(m+1)+1;
// lateral
dofr(i,p)=6*(-c(i,1)/h(i)^2+c(i+1,1)/h(i+1)^2);r(i+1,p)=-6*c(i+1,1)/h(i+1)^2;r(i-1,p)=6*c(i,1)/h(i)^2;
// rotational
dofr(p,p)=4*(c(i,1)/h(i)+c(i+1,1)/h(i+1)+b(i,1)/l(1));r(p+1,p)=2*b(i,1)/l(1);r(p-m-1,p)=2*c(i,1)/h(i);r(p+m+1,p)=2*c(i+1,1)/h(i+1);//
right node, j=m+1p=n+i*(m+1);
// lateral
dofr(i,p)=6*(-c(i,m+1)/h(i)^2+c(i+1,m+1)/h(i+1)^2);r(i+1,p)=-6*c(i+1,m+1)/h(i+1)^2;r(i-1,p)=6*c(i,m+1)/h(i)^2;
// rotational
dofr(p,p)=4*(c(i,m+1)/h(i)+c(i+1,m+1)/h(i+1)+b(i,m)/l(m));
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r(p-1,p)=2*b(i,m)/l(m);r(p-m-1,p)=2*c(i,m+1)/h(i);r(p+m+1,p)=2*c(i+1,m+1)/h(i+1);//current
nodefor j=2:m
p=n+(i-1)*(m+1)+j; // lateral dof
r(i,p)=6*(-c(i,j)/h(i)^2+c(i+1,j)/h(i+1)^2);
r(i+1,p)=-6*c(i+1,j)/h(i+1)^2; r(i-1,p)= 6*c(i,j)/h(i)^2; //
rotational dof
r(p,p)=4*(c(i,j)/h(i)+c(i+1,j)/h(i+1)+b(i,j-1)/l(j-1)+b(i,j)/l(j));
r(p-1,p)=2*b(i,j-1)/l(j-1); r(p+1,p)=2*b(i,j)/l(j);
r(p-m-1,p)=2*c(i,j)/h(i); r(p+m+1,p)=2*c(i+1,j)/h(i+1);
endend// verifying the
symmetryv=r-r';vm=max(max(abs(v)));disp(['maximum of the difference
between simmetrical elements is ' + string(vm)])// calculation of
the stiffness
matrixR11=r(1:n,1:n);R12=r(1:n,(n+1):(n*(m+2)));R21=r((n+1):(n*(m+2)),1:n);R22=r((n+1):(n*(m+2)),(n+1):(n*(m+2)));K=(R11-R12*inv(R22)*R21)*EI
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1. Static
(Statics, Stability and Structural Dynamics. II.1. Structural
Statics. Statically undetermined structures. Works), (in
Romanian)
2. Atanasiu, G.M.: Structural Dynamics and Stability, Editura
"G
3. Brsan, G.M.: (Dynamics and Stability of Structures),
Romanian)4. Chopra, A. K.: Dynamics of Structures. Theory and
Applications to Earthquake Engineering,
Prentice Hall International, Inc., London, 19955. Ciongradi, I.,
Schrf, F.: (Structural Dynamics
nian)6. Clough, W.R., Penzien, J.: Dynamics of Structures,
Second Edition, McGraw-Hill, Inc., New
York, 19937. Dimoiu, I., I (Earthquake Engineering)
1999 (in Romanian)8. Fujino, Y, Shoji, G.: The Great Hanshin
Earthquake - Seismic, Structural, Economic and Social
Aspects, Proceedings of the Fifth Asia Pacific Conference on
Structural Engineering and Construction, 1995
9. Fujino, Y, Shoji, G.: The Great Hanshin Earthquake - Seismic,
Structural, Economic and Social Aspects, Proceedings of the Fifth
Asia Pacific Conference on Structural Engineering and Construction,
1995
10. Honshu-Shikoku Bridge Authority: Akashi-Kaikyo Bridge.
Engineering Note, 199511. Ifrim, M., Dobrescu, Al.:
(Applications in Dynamic Analysis and Earthquake Engineering
12. Ionescu,C., - eanu, F., Dinamica structurilor (Structural
Dynamics), Editura -Teiu Bo (in Romanian)
13. Ivan, M., Botici, A., Dogaru, E., Ivan, A., Balekics, G.,
Negru, A.: (Statics, Stability, and Dynamics of Structures.
Theory and Applications1997(in Romanian)
14. Meirovitch, L.: Dynamics and Control of Structures, John
Wiley & Sons, New York, 199015. orapciu, M.:
-Romn (Illustrated English-Romanian Dictionary of Constructions
and Construction Materials
16. (Ministry of Transportation, Constructions and Tourism). Cod
de proiectare P100. Partea I - p100-1/2006 prevederi de
, (Design Code P100. Part I p100-1/2006 design provisions for
buildings). (in Romanian)
17. , Al. (coord.), A e (Applications in Earthquake
Engineering),(in Romanian)
18.(Earthquake Engineering
19. Nikkei Construction, No. 2-24 (1995) (in Japanese)20. Nikkei
Construction, No. 6-9 (1995) (in Japanese)21. - Earthquake
Engineering,
Bibliography
-
22. - Active Control Approach For Long Span Bridge Responses To
Strong Earthquakes, Doctoral Thesis, Yokohama National University,
Japan, 1997
23. , Stere, C.: hidrotehnice (The Earthquake Engineering of
Hydro-technical Constructions
24. , D., D (Structural Dynamics and Earthquake Engineering),
Editura - (in Romanian)
25. Strat, Structural Statics, Editura Gh. Asachi 199526. Tada,
K., Jin, H., Kitagawa, M., Nitta, A., Toriumi, R.: Effect of the
Southern Hyogo Earthquake
on the Akashi-Kaikyo Bridge, Structural Engineering
International, 199527. The MathWorks Inc.: MATLAB, Reference
Guide28. Yamaguchi, H.: Structural Dynamics, Asian Institute of
Technology, 199229. Zienkiewicz, O. C., Taylor, R. L.: The Finite
Element Method, 4th edition, McGraw-Hill , 1989
