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3000mm

N4. N3. WALL INFO : E=2500MPa n =0.15 t=250mm. 4000mm. EA=15x10 6. N2. N1. N5. 6000mm. 3000mm. TUTORIAL #4 Displacements of a masonry wall with tie-back.

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3000mm

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  1. N4 N3 WALL INFO: E=2500MPa n=0.15 t=250mm 4000mm EA=15x106 N2 N1 N5 6000mm 3000mm TUTORIAL #4 Displacements of a masonry wall with tie-back This presentation will provide an alternate, more efficient, method for determining and ordering the entries of the global stiffness matrix Lecture#13 - October 25th 2011 CIVL 3710 Finite Element Analysis

  2. N4 N3 WALL INFO: E=2500MPa n=0.15 t=250mm There are a total of five nodes in this question, which means that a total of 10 forces (5 in the x & y direction) and 10 displacements (5 in x & y directions) the will prevail Here are the force and displacement vectors for the structure: 4000mm EA=15x106 N2 N1 N5 6000mm 3000mm : Force Vector : Displacement Vector NOTE: the forces and displacements have not yet been placed in order

  3. N4 N3 WALL INFO: E=2500MPa n=0.15 t=250mm The local stiffness matrix for the rectangular element is given by: Et/(12(1-n 2)) times 4000mm EA=15x106 N2 N1 N5 6000mm 3000mm

  4. N4 N3 WALL INFO: E=2500MPa n=0.15 t=250mm The local stiffness matrix for the rectangular element is given by: Et/(12(1-n 2)) times (including symmetrical upper triangle) 4000mm EA=15x106 N2 N1 N5 6000mm 3000mm

  5. N4 N3 WALL INFO: E=2500MPa n=0.15 t=250mm NOTE: we want to calculate unknown displacements from known forces (there are 4 known forces, namely F3x &F3y @ N3 as well as F4x & F4y @ N4) 4000mm EA=15x106 N2 N1 N5 6000mm 3000mm Bottom 4 rowsaccount for these forces F3x F3y F4x F4y

  6. N4 N3 WALL INFO: E=2500MPa n=0.15 t=250mm NOTE: we want to calculate unknown displacements from known forces (there are also 4 unknown displacements, namely u3,v3 (N3) and u4, v4 (N4)) 4000mm EA=15x106 N2 N1 N5 6000mm 3000mm Last 4 columns account for these displacements F3x F3y F4x F4y

  7. N4 N3 WALL INFO: E=2500MPa n=0.15 t=250mm NOTE: we want to calculate unknown displacements from known forces (there are also 4 unknown displacements, namely u3,v3 (N3) and u4, v4 (N4)) 4000mm EA=15x106 N2 N1 N5 6000mm 3000mm The intersection points correspond to partition K11 used in class to measure displacements F3x F3y F4x F4y

  8. N4 N3 WALL INFO: E=2500MPa n=0.15 t=250mm NOTE: we want to calculate unknown displacements from known forces (there are also 4 unknown displacements, namely u3,v3 (N3) and u4, v4 (N4)) 4000mm EA=15x106 N2 N1 N5 6000mm 3000mm The intersection points correspond to partition K11 used in class to measure displacements We can label the columns and rows of this partition as 1 through 4 1 F3x 2 F3y 3 F4x 4 F4y 3 4 1 2

  9. N4 N3 WALL INFO: E=2500MPa n=0.15 t=250mm NOTE: we want to calculate unknown displacements from known forces (we can also consider reactions F1x &F1y @ N1 as well as F2x & F2y @ N2) 4000mm EA=15x106 N2 N1 N5 6000mm 3000mm F1x F1y F2x F2y 1 F3x 2 F3y Top 4 rowsaccount for these reaction forces 3 F4x 4 F4y 3 4 1 2

  10. N4 N3 WALL INFO: E=2500MPa n=0.15 t=250mm NOTE: we want to calculate unknown displacements from known forces (we can also consider reactions F1x &F1y @ N1 as well as F2x & F2y @ N2) 4000mm EA=15x106 N2 N1 N5 6000mm 3000mm F1x F1y F2x F2y These new intersection points correspond to partition K21 used in class to measure Reaction Forces 1 F3x 2 F3y 3 F4x 4 F4y 3 4 1 2

  11. N4 N3 NOTE: So far we have only had to calculate 16 entries for measuring unknown displacements and another 16 entries for reaction forces WALL INFO: E=2500MPa n=0.15 t=250mm NOTE: we want to calculate unknown displacements from known forces (we can also consider reactions F1x &F1y @ N1 as well as F2x & F2y @ N2) 4000mm EA=15x106 N2 N1 N5 6000mm 3000mm We can label the rows of this partition as 5 to 8 5 F1x 6 F1y 7 F2x 8 F2y These new intersection points correspond to partition K21 used in class to measure Reaction Forces 1 F3x 2 F3y 3 F4x 4 F4y 3 4 1 2

  12. N4 N3 WALL INFO: E=2500MPa n=0.15 t=250mm The local stiffness matrix for the truss element is given by: 4000mm EA=15x106 N2 N1 N5 6000mm 3000mm WHERE: q=180+tan-1(4000mm/3000mm) • =233.1o L=[(4000mm)2+(3000mm)2]1/2 =5000mm

  13. N4 N3 WALL INFO: E=2500MPa n=0.15 t=250mm The local stiffness matrix for the truss element is given by: (there are 2 known forces, namely F4x & F4y @ N4) 4000mm EA=15x106 N2 N1 N5 6000mm 3000mm Top 2 rowsaccount for these forces F4x F4y

  14. N4 N3 WALL INFO: E=2500MPa n=0.15 t=250mm The local stiffness matrix for the truss element is given by: (there are also 2 unknown displacements, namely u4 & v4 @ N4) 4000mm EA=15x106 N2 N1 N5 6000mm 3000mm Top 2 rowsaccount for these forces First 2 columns account for these displacements F4x The intersection points are part of partition K11 used in class to measure displacements F4y

  15. N4 N3 WALL INFO: E=2500MPa n=0.15 t=250mm The local stiffness matrix for the truss element is given by: (there are also 2 unknown displacements, namely u4 & v4 @ N4) 4000mm EA=15x106 N2 N1 N5 6000mm 3000mm We have to label rows & columns of the partition as 3 & 4 based on the rectangular element Top 2 rowsaccount for these forces First 2 columns account for these displacements 3 F4x The intersection points are part of partition K11 used in class to measure displacements 4 F4y 3 4

  16. N4 N3 WALL INFO: E=2500MPa n=0.15 t=250mm The local stiffness matrix for the truss element is given by: (we can also consider reactions F5x & F5y @ N5) 4000mm EA=15x106 N2 N1 N5 6000mm 3000mm Bottom 2 rowsaccount for these reactions 3 F4x The intersection points are part of partition K21 used in class to measure reactions 4 F4y F5x F5y 3 4

  17. N4 N3 NOTE: We have now only calculated 20 entries for measuring unknown displacements and another 20 entries for reaction forces (out of a possible 100) WALL INFO: E=2500MPa n=0.15 t=250mm The local stiffness matrix for the truss element is given by: (we can also consider reactions F5x & F5y @ N5) 4000mm EA=15x106 N2 N1 N5 6000mm 3000mm Since labels 1 through 8 have been used, we can label these rows using 9 & 10 Bottom 2 rowsaccount for these reactions 3 F4x 4 F4y 9 F5x 10 F5y 3 4

  18. N4 N3 WALL INFO: E=2500MPa n=0.15 t=250mm We now assemble the global stiffness matrix: Note that the forces & displacements have been ordered in accordance with previous slides 4000mm EA=15x106 6000mm 3000mm N2 N1 N5 K11 from rectangular element K21 from rectangular element K21 from truss element

  19. N4 N3 WALL INFO: E=2500MPa n=0.15 t=250mm We now assemble the global stiffness matrix: Note that the forces & displacements have been ordered in accordance with previous slides 4000mm EA=15x106 6000mm 3000mm N2 N1 N5 K11 from truss element added to K11 from rectangular element Note that the contributions of the truss element were subtracted because equal & opposite forces are used for analyses with these elements Zeros added to the empty portions of K21

  20. N4 N3 WALL INFO: E=2500MPa n=0.15 t=250mm We now assemble the global stiffness matrix: Note that the forces & displacements have been ordered in accordance with previous slides 4000mm EA=15x106 6000mm 3000mm N2 N1 N5 NOTE: these #s are valid only if displacements @ the supports are all zero K11 10x6 matrix=60 entries that did not need to be calculated. This represents 60% of the 100 entries in the 10x10 global stiffness matrix (ie. Only ~40% of the time required for assemblage) K21

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