Thin-walled T-section Tapered Cantilever Beam

The Problem

A thin-walled T-section cantilevered beam is fixed at one end and is subjected to a vertically downward point load of 368N at the free end.

The section is of uniform thickness of 2mm.

The Young's modulus value is 200kN/mm2and Poisson’s ratio is 0.25

368N

2mm

100mm

100mm

2mm

t=2mm

N.B. the dimensions represent only the free end and gets bigger towards the fixed end. The web in middle of the beam 500mm from the free end is 150mm instead of 100mm

Finite Element Model

1 Work in units of N (force) and mm (length) and MPa (=N/mm2modulus/stress).

2 Use Plate/Shell QUAD8 element.

3 Take the global axes origin to be at the bottom left corner the fixed end- (X,Y, Z).

4 Generate the nodes and element manually for three elements model. UseTools Subdivide for generation of more elements.

Use QUAD8 elements as required

4@25

A

4@ 250mm

200mm

Q S

R

X

Z

B

P

4@ 25mm

Finite Element Output

1 Vertical deflection (DY) along the line AB.

2 Bending stress (SZZ), in Global axes, along the line AB.

3 Bending stress (SZZ), in Global axes, along the line PQ.

4 Vertical shear stress (SYZ), in Global axes, along the line PQ.

Theory

1 Use the double integration method to find the vertical deflection along the top surface (line AB).

2 Use the bending theory to find the bending stress along the top surface (line AB).

3 Use the bending theory to find the bending stress along the web (line PQ).

4 Use the bending theory to find the vertical shear stress distribution along the web (line PQ).

Results–DY-Vertical deflection (mm) along AB

Distance from fixed end (mm) Theory FE

0

150

300

450

600

750

900

1050

1200

Results–SZZ (Global axes)-Bending stress (N/mm2) along AB

Distance from fixed end (mm) Theory FE

0

150

300

450

600

750

900

1050

1200

Results–SZZ (Global axes) –Bending stress (N/mm2) along PQ

Distance from bottom(mm) Theory FE

0(P)

12.5

25

37.5

50

62.5

75

87.5

100(Q)

Results–SYZ (Global axes)-Shear stress(N/mm2) along PQ

Distance from bottom(mm) Theory FE

0(P)

12.5

25

37.5

50

62.5

75

87.5

100(Q)

Plot Graph

1 Plot on the same graph the span wise finite element vertical deflectionvalues (along AB) superimposed on the theoretical values.

2 Plot on the same graph the span wise the finite element bending stress values (along AB) superimposed on the theoretical values.

3 Plot on the same graph the finite element bending stress values (along PQ) superimposed on the theoretical values.

4 Plot on the same graph the finite element vertical shear stress values (along PQ) superimposed on the theoretical values.

The experimental values I are shown below but does not have to be same as theory and

strand7.

Load (N) Deflection (mm) Stress@ Flange 1 Stress@Flange 2 Stress @Web

0 0 0 0 0

18 0.14 0.9 0.4 0.2

28 0.22 1 0.6 0.2

38 0.3 1.4 0.6 0.1

48 0.38 1.5 1 -0.2

58 0.47 1.7 1.2 -0.3

68 0.55 1.8 1.3 -0.5

88 0.73 2.2 1.7 -0.8

108 0.89 2.5 1.9 -0.8

128 1.06 2.8 2.2 -1.1

148 1.24 3 2.7 -1.3

168 1.42 3.4 3 -1.6

188 1.64 3.7 3.2 -1.8

208 1.84 3.9 3.6 -2

228 2.07 4.3 4 -2.2

248 2.32 4.6 4.4 -2.6

268 2.65 4.9 4.7 -2.8

288 3.05 5.2 5.2 -3

308 3.39 5.3 5.5 -3.2

328 3.8 5.6 5.9 -3.4

348 4.19 6 6.3 -3.6

368 4.58 6.4 6.7 -4

The Problem

A thin-walled T-section cantilevered beam is fixed at one end and is subjected to a vertically downward point load of 368N at the free end.

The section is of uniform thickness of 2mm.

The Young's modulus value is 200kN/mm2and Poisson’s ratio is 0.25

368N

2mm

100mm

100mm

2mm

t=2mm

N.B. the dimensions represent only the free end and gets bigger towards the fixed end. The web in middle of the beam 500mm from the free end is 150mm instead of 100mm

Finite Element Model

1 Work in units of N (force) and mm (length) and MPa (=N/mm2modulus/stress).

2 Use Plate/Shell QUAD8 element.

3 Take the global axes origin to be at the bottom left corner the fixed end- (X,Y, Z).

4 Generate the nodes and element manually for three elements model. UseTools Subdivide for generation of more elements.

Use QUAD8 elements as required

4@25

A

4@ 250mm

200mm

Q S

R

X

Z

B

P

4@ 25mm

Finite Element Output

1 Vertical deflection (DY) along the line AB.

2 Bending stress (SZZ), in Global axes, along the line AB.

3 Bending stress (SZZ), in Global axes, along the line PQ.

4 Vertical shear stress (SYZ), in Global axes, along the line PQ.

Theory

1 Use the double integration method to find the vertical deflection along the top surface (line AB).

2 Use the bending theory to find the bending stress along the top surface (line AB).

3 Use the bending theory to find the bending stress along the web (line PQ).

4 Use the bending theory to find the vertical shear stress distribution along the web (line PQ).

Results–DY-Vertical deflection (mm) along AB

Distance from fixed end (mm) Theory FE

0

150

300

450

600

750

900

1050

1200

Results–SZZ (Global axes)-Bending stress (N/mm2) along AB

Distance from fixed end (mm) Theory FE

0

150

300

450

600

750

900

1050

1200

Results–SZZ (Global axes) –Bending stress (N/mm2) along PQ

Distance from bottom(mm) Theory FE

0(P)

12.5

25

37.5

50

62.5

75

87.5

100(Q)

Results–SYZ (Global axes)-Shear stress(N/mm2) along PQ

Distance from bottom(mm) Theory FE

0(P)

12.5

25

37.5

50

62.5

75

87.5

100(Q)

Plot Graph

1 Plot on the same graph the span wise finite element vertical deflectionvalues (along AB) superimposed on the theoretical values.

2 Plot on the same graph the span wise the finite element bending stress values (along AB) superimposed on the theoretical values.

3 Plot on the same graph the finite element bending stress values (along PQ) superimposed on the theoretical values.

4 Plot on the same graph the finite element vertical shear stress values (along PQ) superimposed on the theoretical values.

The experimental values I are shown below but does not have to be same as theory and

strand7.

Load (N) Deflection (mm) Stress@ Flange 1 Stress@Flange 2 Stress @Web

0 0 0 0 0

18 0.14 0.9 0.4 0.2

28 0.22 1 0.6 0.2

38 0.3 1.4 0.6 0.1

48 0.38 1.5 1 -0.2

58 0.47 1.7 1.2 -0.3

68 0.55 1.8 1.3 -0.5

88 0.73 2.2 1.7 -0.8

108 0.89 2.5 1.9 -0.8

128 1.06 2.8 2.2 -1.1

148 1.24 3 2.7 -1.3

168 1.42 3.4 3 -1.6

188 1.64 3.7 3.2 -1.8

208 1.84 3.9 3.6 -2

228 2.07 4.3 4 -2.2

248 2.32 4.6 4.4 -2.6

268 2.65 4.9 4.7 -2.8

288 3.05 5.2 5.2 -3

308 3.39 5.3 5.5 -3.2

328 3.8 5.6 5.9 -3.4

348 4.19 6 6.3 -3.6

368 4.58 6.4 6.7 -4

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