mathcad_-_truss2d1

Mathcad_-_Truss2D1.mcdxEng. Francesco Onorato

Page 1 of 9

Converted from "Using the Finite Element Method on Truss Structures" created by Franceso Onorato

TRUSS 2D

Program for solving 2D truss structures based on the Finite Element Method

author information

This work was in response to a need to design an industrial warehouse whose structure was based on steel beams. Although the design of steel truss structures is a common analysis taught in engineering courses, and the theory allows to get the solution by handmade calculations, the desire for a robust, error-checked design process that could be performed repeatedly led me to code it in software. Typically, several iterations are common practice in the design phase but they are time consuming and potential source of errors if the process is managed by handmade calculations.

The finite element method used to design this structure was done in Mathcad for several reasons:

1. simplicity in writing the algorithm,

2. simplicity of debugging the algorithm,

3. easy customisation of the input data pre-processing and result post-processing.

4. low cost of the software for distribution to other users.

Mathcad's great flexibility in managing formulas, numerical and symbolic calculations, text and graphics in a single user-friendly environment made it ideal for this task. It required only a few days to create the following program and to validate its output against similar applications.

Input data:

This data corresponds to the following truss structure:

Created with PTC Mathcad Express. See www.mathcad.com for more information.


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The following is an Excel component, which will require you to have Excel on your system for it to operate.

Outputs Input_N excel

Foglio1!C6:H16

Outputs Input_E excel

Foglio2!C5:E23

Processing input data

Plot of the structure



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0

.3

.6

.9

.2

.5

.8

.1

.4

.6

.3

.7

1.5 3 4.5 6 7.5 9 10.5 12-1.5 0 13.5

Undeformed structure

Constrained nodes

Loaded nodes

---- Structure

The following section contains the algorithm that allows to determine the tensional level of elements, the displacements and the nodal forces of the structure is inside this area. This area must not be modified by the user.

Structure solving

For each element of the structure is computed i 1 ne

the length of the rod Li

+

N

,T,i 1

xN

,T,i 2

x

2

N,T

,i 1y

N,T

,i 2y

2

the Young modulus, the rod stiffness Ei

Ey ki

EiAi

Li

and the stiffness matrixk ((i))

|||||||||||

|

c

N,T

,i 2x

N,T

,i 1x

Li

s

N,T

,i 2y

N,T

,i 1y

Li

ki

c c c s c c c sc s s s c s s sc c c s c c c sc s s s c s s s

i 1 nn j x y



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onverting into vectors the matrixes that efine the nodal constrains and loads Vv +2 (( i 1)) j v ,i j Fv +2 (( i 1)) j F ,i j

mputing the global stiffness matrix K K

|||||||||||||||||||||||||||||||||||||||

|

for |||||

i 1 2 nn

for ||

|

j 1 2 nn

K,i j

0

for ||||||||||||||||||||||||||||||||||

e 1 ne

i T,e 1

j T,e 2

K,2 i 1 2 i 1

+K,2 i 1 2 i 1

k ((e)),1 1

K,2 i 1 2 i

+K,2 i 1 2 i

k ((e)),1 2

K,2 i 2 i 1

+K,2 i 2 i 1

k ((e)),2 1

K,2 i 2 i

+K,2 i 2 i

k ((e)),2 2

K,2 i 1 2 j 1

+K,2 i 1 2 j 1

k ((e)),1 3

K,2 i 1 2 j

+K,2 i 1 2 j

k ((e)),1 4

K,2 i 2 j 1

+K,2 i 2 j 1

k ((e)),2 3

K,2 i 2 j

+K,2 i 2 j

k ((e)),2 4

K,2 j 1 2 i 1

+K,2 j 1 2 i 1

k ((e)),3 1

K,2 j 1 2 i

+K,2 j 1 2 i

k ((e)),3 2

K,2 j 2 i 1

+K,2 j 2 i 1

k ((e)),4 1

K,2 j 2 i

+K,2 j 2 i

k ((e)),4 2

K,2 j 1 2 j 1

+K,2 j 1 2 j 1

k ((e)),3 3

K,2 j 1 2 j

+K,2 j 1 2 j

k ((e)),3 4

K,2 j 2 j 1

+K,2 j 2 j 1

k ((e)),4 3

K,2 j 2 j

+K,2 j 2 j

k ((e)),4 4

K



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possible to assembly the reduced ffness matrix Kr and the reduced ad vector Fvr

Kr

||||||||||||||

|

ii 0for |

||||||||||||

i 1 2 nn

|||||||||||

if Vvi

0

ii +ii 1jj 0

for |||||

|

j 1 2 nn

|||||

if Vvj

0

jj +jj 1Kr

,ii jjK

,i j

Kr

Fvr

||||||||

|

ii 0for |

||||

|

i 1 2 nn

|||||

if Vvi

0

ii +ii 1Fvr

iiFv

i

Fvr

e inverse of the reduced stiffness matrix R R Kr1

hich allows to define the reduced ctor of nodal displacements Uvr Uvr R Fvr

Uvr lsolve (( ,Kr Fvr))

hose knowledge makes possible e calculation of the displacement ctor Uv and the matrix U of nodal splacements of the truss structure

Uv

|||||||||||

|

ii 0for |

|||||||||

i 1 2 nn

|||||||

|

if

else

Vvi

0

ii +ii 1Uvi

Uvrii

Uvi

0

Uv

U,i j

Uv+2 (( i 1)) j

t the end the load matrix F is computed ultiplying the stiffness matrix K of the full ructure times the displacement one U

F,i j

Fv+2 (( i 1)) jFv K Uv

he knowledge of the nodal displacements und of the orientation of the rod in the plane, ows to define

((i))

||||||||||

c

N,T

,i 2x

N,T

,i 1x

Li

s

N,T

,i 2y

N,T

,i 1y

Li

Tc s c s[[ ]]

u ((i))

U,T

,i 11

U,T

,i 12

U,T

,i 21

U,T

,i 22



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e axial load f and the axial stress r each rod of the truss structure. i 1 ne fi ki u

((i)) ((i)) i

fi

Ai

Norm ((F))

|||||||||

|

Fmax max (( ,max ((F)) min ((F))))for |

|||||

|

i 1 nn

for ||||

|

j x y

||||

if


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14

21

28

35

42

49

56

63

0

7

70

4 6 8 10 12 14 16 180 2 20

Axial load

Compressed elements(rafters) Streched elements(tie rods)

Absolute value

Number of element

20

30

40

50

60

70

80

90

00

0

10

10

4 6 8 10 12 14 16 180 2 20

Axial stress

Compressed elements(rafters) Streched elements(tie rods)

Absolute value

Number of element

Output this data to an Excel component



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Output this data to an Excel component.

Output data



Page 9 of 9

Standard Output

Coordinates of nodes Displacements of nodes Forces acting on nodes

=U

0 01.431 7.8491.626 11.111.088 11.4420.551 11.110.746 7.8492.176 03.03 7.6672.059 10.9280.117 10.928

0.854 7.667

mm =F[11 2]

2

=N0

cm

Ending nodesof the element

Length of the element

Area of the cross section of the element

Axial loadof the element

Axial stressin the element

=L206.2

=A8

2=f

41231

=51.5

mm

2

=T

1 22 33 44 55 66 77 88 99 10

10 1111 1

2 11


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