3000mm

N1 N2

N4 N3

N5

3000mm

6000mm

4000mm

EA=15

x106

WALL INFO:E=2500MPan=0.15t=250mm

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

CIVL 3710 Finite Element Analysis Lecture#13 - October 25th 2011

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 prevailHere are the force and displacement vectors for the structure:

F= F1x F1x F2x F2x F3x F3x F4x F4x F5x F5xt

d = u1 v1 u2 v2 u3 v3 u4 v4 u5 v5

t

: Force Vector

: Displacement Vector

NOTE: the forces and displacements have not yet been placed in order

N1 N2

N4 N3

N5

4000mm

EA=15

x106


3000mm

6000mm

The local stiffness matrix for the rectangular element is given by:Et/(12(1-n 2)) timesF1x 4B+2(1-n)/B ↑ u1

F1y 3(1+n)/2 4/B+2(1-n)B ← Symmetrical → v1

F2x 2B-2(1-n)/B -3(1-3n)/2 4B+2(1-n)/B ↓ u2

F2y=

3(1-3n)/2 -4/B+(1-n)B -3(1+n)/2 4/B+2(1-n)Bx

v2

F3x -2B-(1-n)/B -3(1+n)/2 -4B+(1-n)/B -3(1-3n)/2 4B+2(1-n)/B u3

F3y -3(1+n)/2 -2/B-(1-n)B 3(1-3n)/2 2/B-2(1-n)B 3(1+n)/2 4/B+2(1-n)B v3

F4x -4B+(1-n)/B 3(1-3n)/2 -2B-(1-n)/B 3(1+n)/2 2B-2(1-n)/B -3(1-3n)/2 4B+2(1-n)/B u4

F4y -3(1-3n)/2 2/B-2(1-n)B 3(1+n)/2 -2/B-(1-n)B 3(1-3n)/2 -4/B+(1-n)B -3(1+n)/2 4/B+2(1-n)B v4

N1 N2

N4 N3

N5

4000mm

EA=15

x106


3000mm

6000mm

F1x 4B+2(1-n)/B u1

F1y 3(1+n)/2 4/B+2(1-n)B v1

F2x 2B-2(1-n)/B -3(1-3n)/2 4B+2(1-n)/B u2

F2y=

3(1-3n)/2 -4/B+(1-n)B -3(1+n)/2 4/B+2(1-n)Bx

v2

F3x -2B-(1-n)/B -3(1+n)/2 -4B+(1-n)/B -3(1-3n)/2 4B+2(1-n)/B u3




3(1+n)/2 2B-2(1-n)/B 3(1-3n)/2 -2B-(1-n)/B -3(1+n)/2 -4B+(1-n)/B -3(1-3n)/2

-3(1-3n)/2 -4/B+(1-n)B -3(1+n)/2 -2/B-(1-n)B 3(1-3n)/2 2/B-2(1-n)B

-3(1+n)/2 -4B+(1-n)/B 3(1-3n)/2 -2B-(1-n)/B 3(1+n)/2

-3(1-3n)/2 2/B-2(1-n)B 3(1+n)/2 -2/B-(1-n)B

3(1+n)/2 2B-2(1-n)/B 3(1-3n)/2

-3(1-3n)/2 -4/B+(1-n)B

-3(1+n)/2

The local stiffness matrix for the rectangular element is given by:Et/(12(1-n 2)) times (including symmetrical upper triangle)

N1 N2

N4 N3

N5

4000mm

EA=15

x106


3000mm

6000mm

F1x 4B+2(1-n)/B u1

F1y 3(1+n)/2 4/B+2(1-n)B v1

F2x 2B-2(1-n)/B -3(1-3n)/2 4B+2(1-n)/B u2

F2y=

3(1-3n)/2 -4/B+(1-n)B -3(1+n)/2 4/B+2(1-n)Bx

v2

F3x -2B-(1-n)/B -3(1+n)/2 -4B+(1-n)/B -3(1-3n)/2 4B+2(1-n)/B u3




3(1+n)/2 2B-2(1-n)/B 3(1-3n)/2 -2B-(1-n)/B -3(1+n)/2 -4B+(1-n)/B -3(1-3n)/2

-3(1-3n)/2 -4/B+(1-n)B -3(1+n)/2 -2/B-(1-n)B 3(1-3n)/2 2/B-2(1-n)B

-3(1+n)/2 -4B+(1-n)/B 3(1-3n)/2 -2B-(1-n)/B 3(1+n)/2

-3(1-3n)/2 2/B-2(1-n)B 3(1+n)/2 -2/B-(1-n)B

3(1+n)/2 2B-2(1-n)/B 3(1-3n)/2

-3(1-3n)/2 -4/B+(1-n)B

-3(1+n)/2

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)

N1 N2

N4 N3

N5

4000mm

EA=15

x106


F3x

F3y

F4x

F4y

Bottom 4 rows account for these forces

3000mm

6000mm

F1x 4B+2(1-n)/B u1

F1y 3(1+n)/2 4/B+2(1-n)B v1

F2x 2B-2(1-n)/B -3(1-3n)/2 4B+2(1-n)/B u2

F2y=

3(1-3n)/2 -4/B+(1-n)B -3(1+n)/2 4/B+2(1-n)Bx

v2

F3x -2B-(1-n)/B -3(1+n)/2 -4B+(1-n)/B -3(1-3n)/2 4B+2(1-n)/B u3




3(1+n)/2 2B-2(1-n)/B 3(1-3n)/2 -2B-(1-n)/B -3(1+n)/2 -4B+(1-n)/B -3(1-3n)/2

-3(1-3n)/2 -4/B+(1-n)B -3(1+n)/2 -2/B-(1-n)B 3(1-3n)/2 2/B-2(1-n)B

-3(1+n)/2 -4B+(1-n)/B 3(1-3n)/2 -2B-(1-n)/B 3(1+n)/2

-3(1-3n)/2 2/B-2(1-n)B 3(1+n)/2 -2/B-(1-n)B

3(1+n)/2 2B-2(1-n)/B 3(1-3n)/2

-3(1-3n)/2 -4/B+(1-n)B

-3(1+n)/2

NOTE: we want to calculate unknown displacements from known forces (there are also 4 unknown displacements, namely u3,v3 (N3) and u4, v4 (N4))

N1 N2

N4 N3

N5

4000mm

EA=15

x106


F3x

F3y

F4x

F4y

Last 4 columns account for these displacements

3000mm

6000mm

F1x 4B+2(1-n)/B u1

F1y 3(1+n)/2 4/B+2(1-n)B v1

F2x 2B-2(1-n)/B -3(1-3n)/2 4B+2(1-n)/B u2

F2y=

3(1-3n)/2 -4/B+(1-n)B -3(1+n)/2 4/B+2(1-n)Bx

v2

F3x -2B-(1-n)/B -3(1+n)/2 -4B+(1-n)/B -3(1-3n)/2 4B+2(1-n)/B u3




3(1+n)/2 2B-2(1-n)/B 3(1-3n)/2 -2B-(1-n)/B -3(1+n)/2 -4B+(1-n)/B -3(1-3n)/2

-3(1-3n)/2 -4/B+(1-n)B -3(1+n)/2 -2/B-(1-n)B 3(1-3n)/2 2/B-2(1-n)B

-3(1+n)/2 -4B+(1-n)/B 3(1-3n)/2 -2B-(1-n)/B 3(1+n)/2

-3(1-3n)/2 2/B-2(1-n)B 3(1+n)/2 -2/B-(1-n)B

3(1+n)/2 2B-2(1-n)/B 3(1-3n)/2

-3(1-3n)/2 -4/B+(1-n)B

-3(1+n)/2


N1 N2

N4 N3

N5

4000mm

EA=15

x106


F3x

F3y

F4x

F4y

The intersection points correspond to partition K11 used in class to measure

displacements

3000mm

6000mm

F1x 4B+2(1-n)/B u1

F1y 3(1+n)/2 4/B+2(1-n)B v1

F2x 2B-2(1-n)/B -3(1-3n)/2 4B+2(1-n)/B u2

F2y=

3(1-3n)/2 -4/B+(1-n)B -3(1+n)/2 4/B+2(1-n)Bx

v2

F3x -2B-(1-n)/B -3(1+n)/2 -4B+(1-n)/B -3(1-3n)/2 277955.4 u3

F3y -3(1+n)/2 -2/B-(1-n)B 3(1-3n)/2 2/B-2(1-n)B 91911.76 380079.6 v3

F4x -4B+(1-n)/B 3(1-3n)/2 -2B-(1-n)/B 3(1+n)/2 -64826.7 -43957.8 277955.4 u4

F4y -3(1-3n)/2 2/B-2(1-n)B 3(1+n)/2 -2/B-(1-n)B 43957.8 -289500 -91911.8 380079.6 v4

3(1+n)/2 2B-2(1-n)/B 3(1-3n)/2 -2B-(1-n)/B -3(1+n)/2 -4B+(1-n)/B -3(1-3n)/2

-3(1-3n)/2 -4/B+(1-n)B -3(1+n)/2 -2/B-(1-n)B 3(1-3n)/2 2/B-2(1-n)B

-3(1+n)/2 -4B+(1-n)/B 3(1-3n)/2 -2B-(1-n)/B 3(1+n)/2

-3(1-3n)/2 2/B-2(1-n)B 3(1+n)/2 -2/B-(1-n)B

91911.76 -64826.7 43957.8

-43957.8 -289500

-91911.8


N1 N2

N4 N3

N5

4000mm

EA=15

x106


F3x

F3y

F4x

F4y

We can label the columns and rows

of this partition as 1 through 4

The intersection points correspond to partition K11 used in class to measure

displacements

1234

1 2 3 4

3000mm

6000mm

F1x 4B+2(1-n)/B u1

F1y 3(1+n)/2 4/B+2(1-n)B v1

F2x 2B-2(1-n)/B -3(1-3n)/2 4B+2(1-n)/B u2

F2y=

3(1-3n)/2 -4/B+(1-n)B -3(1+n)/2 4/B+2(1-n)Bx

v2

F3x -2B-(1-n)/B -3(1+n)/2 -4B+(1-n)/B -3(1-3n)/2 277955.4 u3

F3y -3(1+n)/2 -2/B-(1-n)B 3(1-3n)/2 2/B-2(1-n)B 91911.76 380079.6 v3

F4x -4B+(1-n)/B 3(1-3n)/2 -2B-(1-n)/B 3(1+n)/2 -64826.7 -43957.8 277955.4 u4

F4y -3(1-3n)/2 2/B-2(1-n)B 3(1+n)/2 -2/B-(1-n)B 43957.8 -289500 -91911.8 380079.6 v4

3(1+n)/2 2B-2(1-n)/B 3(1-3n)/2 -2B-(1-n)/B -3(1+n)/2 -4B+(1-n)/B -3(1-3n)/2

-3(1-3n)/2 -4/B+(1-n)B -3(1+n)/2 -2/B-(1-n)B 3(1-3n)/2 2/B-2(1-n)B

-3(1+n)/2 -4B+(1-n)/B 3(1-3n)/2 -2B-(1-n)/B 3(1+n)/2

-3(1-3n)/2 2/B-2(1-n)B 3(1+n)/2 -2/B-(1-n)B

91911.76 -64826.7 43957.8

-43957.8 -289500

-91911.8

NOTE: we want to calculate unknown displacements from known forces (we can also consider reactions F1x &F1y @ N1 as well as F2x & F2y @ N2)

N1 N2

N4 N3

N5

4000mm

EA=15

x106


F3x

F3y

F4x

F4y

1234

1 2 3 4

Top 4 rows account for these reaction forces

F1x

F1y

F2x

F2y

3000mm

6000mm

F1x 4B+2(1-n)/B u1

F1y 3(1+n)/2 4/B+2(1-n)B v1

F2x 2B-2(1-n)/B -3(1-3n)/2 4B+2(1-n)/B u2

F2y=

3(1-3n)/2 -4/B+(1-n)B -3(1+n)/2 4/B+2(1-n)Bx

v2

F3x -2B-(1-n)/B -3(1+n)/2 -4B+(1-n)/B -3(1-3n)/2 277955.4 u3

F3y -3(1+n)/2 -2/B-(1-n)B 3(1-3n)/2 2/B-2(1-n)B 91911.76 380079.6 v3

F4x -4B+(1-n)/B 3(1-3n)/2 -2B-(1-n)/B 3(1+n)/2 -64826.7 -43957.8 277955.4 u4

F4y -3(1-3n)/2 2/B-2(1-n)B 3(1+n)/2 -2/B-(1-n)B 43957.8 -289500 -91911.8 380079.6 v4

3(1+n)/2 2B-2(1-n)/B 3(1-3n)/2 -2B-(1-n)/B -3(1+n)/2 -4B+(1-n)/B -3(1-3n)/2

-3(1-3n)/2 -4/B+(1-n)B -3(1+n)/2 -2/B-(1-n)B 3(1-3n)/2 2/B-2(1-n)B

-3(1+n)/2 -4B+(1-n)/B 3(1-3n)/2 -2B-(1-n)/B 3(1+n)/2

-3(1-3n)/2 2/B-2(1-n)B 3(1+n)/2 -2/B-(1-n)B

91911.76 -64826.7 43957.8

-43957.8 -289500

-91911.8


N1 N2

N4 N3

N5

4000mm

EA=15

x106


F3x

F3y

F4x

F4y

1234

1 2 3 4

F1x

F1y

F2x

F2y

These new intersection points correspond to

partition K21 used in class to measure Reaction

Forces

3000mm

6000mm

N1 N2

N4 N3

N5

4000mm

EA=15

x106


F1x 4B+2(1-n)/B u1

F1y 3(1+n)/2 4/B+2(1-n)B v1

F2x 2B-2(1-n)/B -3(1-3n)/2 4B+2(1-n)/B u2

F2y=

3(1-3n)/2 -4/B+(1-n)B -3(1+n)/2 4/B+2(1-n)Bx

v2

F3x -2B-(1-n)/B -3(1+n)/2 -4B+(1-n)/B -3(1-3n)/2 277955.4 u3

F3y -3(1+n)/2 -2/B-(1-n)B 3(1-3n)/2 2/B-2(1-n)B 91911.76 380079.6 v3

F4x -4B+(1-n)/B 3(1-3n)/2 -2B-(1-n)/B 3(1+n)/2 -64826.7 -43957.8 277955.4 u4

F4y -3(1-3n)/2 2/B-2(1-n)B 3(1+n)/2 -2/B-(1-n)B 43957.8 -289500 -91911.8 380079.6 v4

3(1+n)/2 2B-2(1-n)/B 3(1-3n)/2 -138978 -91911.8 -74151 -43957.8

-3(1-3n)/2 -4/B+(1-n)B -91911.8 -190040 43957.8 99460.07

-3(1+n)/2 -74151 43957.8 -138978 91911.76

-43957.8 99460.07 91911.76 -190040

91911.76 -64826.7 43957.8

-43957.8 -289500

-91911.8


F3x

F3y

F4x

F4y

1234

1 2 3 4

F1x

F1y

F2x

F2y

These new intersection points correspond to

partition K21 used in class to measure Reaction

Forces

We can label the rows of this

partition as 5 to 85678

NOTE: So far we have only had to calculate 16 entries for

measuring unknown displacements and another 16

entries for reaction forces

3000mm

6000mm

N1 N2

N4 N3

N5

4000mm

EA=15

x106


F4x -cos2q -cosqsinq cos2q cosqsinq u4

F4y= EA/L

-sinqcosq -sin2q sinqcosq sin2qx

v4

F5x cos2q cosqsinq -cos2q -cosqsinq u5

F5y sinqcosq sin2q sinqcosq -sin2q v5

The local stiffness matrix for the truss element is given by:

WHERE:q=180+tan-1(4000mm/3000mm) =233.1o

L=[(4000mm)2+(3000mm)2]1/2

=5000mmF4x -cos2q -cosqsinq cos2q cosqsinq u4

F4y= 15x106 -sinqcosq -sin2q sinqcosq sin2q

xv4

F5x5000 cos2q cosqsinq -cos2q -cosqsinq u5


3000mm

6000mm



xv4



N1 N2

N4 N3

N5

4000mm

EA=15

x106


The local stiffness matrix for the truss element is given by:(there are 2 known forces, namely F4x & F4y @ N4)

Top 2 rows account for these forces

F4x

F4y

3000mm

6000mm



xv4



N1 N2

N4 N3

N5

4000mm

EA=15

x106


The local stiffness matrix for the truss element is given by:(there are also 2 unknown displacements, namely u4 & v4 @ N4)


F4x

F4y

First 2 columns account for these displacements

The intersection points are part of partition K11 used

in class to measure displacements

3000mm

6000mm

F4x -0.361 -0.480 cos2q cosqsinq u4

F4y= 15x106 -0.480 -0.639 sinqcosq sin2q

xv4



N1 N2

N4 N3

N5

4000mm

EA=15

x106


The local stiffness matrix for the truss element is given by:(there are also 2 unknown displacements, namely u4 & v4 @ N4)


F4x

F4y

First 2 columns account for these displacements


in class to measure displacements

We have to label rows & columns of the partition as 3 & 4 based on the

rectangular element

34

3 4

3000mm

6000mm

Bottom 2 rows account for these reactions



xv4



N1 N2

N4 N3

N5

4000mm

EA=15

x106


The local stiffness matrix for the truss element is given by:(we can also consider reactions F5x & F5y @ N5)

F4x

F4y

34

3 4

F5x

F5y


in class to measure reactions

3000mm

6000mm

Bottom 2 rows account for these reactions



xv4

F5x5000 0.361 0.480 -cos2q -cosqsinq u5

F5y 0.480 0.639 sinqcosq -sin2q v5

N1 N2

N4 N3

N5

4000mm

EA=15

x106


The local stiffness matrix for the truss element is given by:(we can also consider reactions F5x & F5y @ N5)

F4x

F4y

34

3 4

F5x

F5y

Since labels 1 through 8 have been used, we can label these rows using

9 & 10

910

NOTE: We have now only calculated 20 entries for

measuring unknown displacements and another 20 entries for reaction forces (out

of a possible 100)

3000mm

6000mm

N1 N2

N4 N3

N5

4000mm

EA=15

x106


We now assemble the global stiffness matrix: Note that the forces & displacements have been ordered in accordance with previous slides

F3x u3

F3y v3

F4x u4

F5y= x

v4

F1x u1

F1y v1

F2x u2

F2y v2

F5x u5

F5y v5

277955 91912 -64827 43958

91912 380080 -43958 -289500

-64827 -43958 277955 -91912

43958 -289500 -91912 380080

K11 from rectangular element

-138978 -91911.8 -74151 -43957.8

-91911.8 -190040 43957.8 99460.07

-74151 43957.8 -138978 91911.76

-43957.8 99460.07 91911.76 -190040

K21 from rectangular element

-1081.51 -1440.44

-1440.44 -1918.49

K21 from truss element

3000mm

6000mm

N1 N2

N4 N3

N5

4000mm

EA=15

x106



F3x u3

F3y v3

F4x u4

F5y= x

v4

F1x u1

F1y v1

F2x u2

F2y v2

F5x u5

F5y v5

277955 91912 -64827 43958

91912 380080 -43958 -289500

-64827 -43958 279037 -90471

43958 -289500 -90471 381998

-138978 -91911.8 -74151 -43957.8

-91911.8 -190040 43957.8 99460.07

-74151 43957.8 -138978 91911.76

-43957.8 99460.07 91911.76 -190040

K11 from truss element added to K11 from

rectangular elementNote that the contributions of the

truss element were subtracted because equal & opposite forces are used for analyses with these

elements-1081.51 -1440.44

-1440.44 -1918.49

0 0

0 0

Zeros added to the empty portions of K21

3000mm

6000mm

N1 N2

N4 N3

N5

4000mm

EA=15

x106



F3x u3

F3y v3

F4x u4

F5y= x

v4

F1x u1

F1y v1

F2x u2

F2y v2

F5x u5

F5y v5

277955 91912 -64827 43958

91912 380080 -43958 -289500

-64827 -43958 279037 -90471

43958 -289500 -90471 381998

-138978 -91911.8 -74151 -43957.8

-91911.8 -190040 43957.8 99460.07

-74151 43957.8 -138978 91911.76

-43957.8 99460.07 91911.76 -190040

-1081.51 -1440.44

-1440.44 -1918.49

0 0

0 0

K11

K21

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)

NOTE: these #s are valid only if displacements @ the supports are all zero

3000mm

6000mm

3000mm

Documents

nb31 n24b

nb31 n22b

nbu1f1y31 n24b

nbv431 n22b

15x106wall info

known forces

rectangular element

unknown displacements