Thermal Expansion, Line Load and Point Load

1Thermal Expansion, Line Load and Point Load

1.1Problem definition

Determine bending moment and support loads in a steel frame submitted to thermal expansion.


Figure 1: Frame with thermal expansion



Figure 2: Rohr2 Model



Determine bending moment and support loads in a steel frame submitted to line load.


Figure 3: Frame with line load



Figure 4: Rohr2 Model



Determine bending moment and support loads in a steel frame submitted to a point load.


Figure 5: Frame with point load



Figure 6: Rohr2 Model




1.2 References (Kleinlogel)

A. Kleinlogel, Rahmenformeln, 11th Edition, Darmstadt 1947, Ramenform 99.


The following values are given:

  • Outside diameter

  • Inside diameter

  • Dimension “s”

  • Dimension “l”

  • Dimension “h”

  • Dimension “a”

  • Dimension “d”

  • Dimension “c”

  • Module of elasticity

  • Coefficient of expansion

  • Temperature difference

  • Line load

  • Single load


The following formulas for the fixed values are used:


Fixed value

Fixed value

Fixed value

Fixed value

Fixed value

Fixed value

Fixed value

For the thermal expansion:


Auxilliary value

Bending moment at point “B” / “E”

Bending moment at point “C” / “D”

Horizontal support load at point “A” / “F”


For the line load:


Auxilliary value

Bending moment at point “B” / “E”

Bending moment at point “C” / “D”

Horizontal support load at point “A” / “F”

Vertical support load at point “A” / “F”



For the point load:


Auxilliary value

Bending moment at point “B”

Bending moment at point “E”

Bending moment at point “C”

Bending moment at point “D”

Vertical support load at point “A” / “F”

Horizontal support load at point “F”

Horizontal support load at point “A”


The following variables are used:


Variable

Description

Unit

Used Value

Outside diameter

mm

114,3

Inside diameter

mm

107,1

Dimension “s”

mm

1414,21

Dimension “l”

mm

3500

Dimension “h”

mm

2000

Dimension “a”

mm

1000

Dimension “d”

mm

1500

Dimension “c”

mm

1000

Module of elasticity

kN/mm^2

207

Coefficient of expansion

µm/(m*K)

12,5

Temperature difference

K

80

Line load

kN/m

10

Single load

kN

10

Table 1: Overview of the used variables



1.3 Model description (ROHR2)

The needed dimensions and the geometry can be seen in “Table1” and “Figure7”. The frame is supported by two simple supports. At point “A” all displacements are fixed, at point “C” all displacements and the torsion is fixed.

In the first load case the thermal expansion is applied.

In the second load case a constant line load of 10kN/m is applied between point “C” and point “D”.

In the third load case a single load of 10kN is applied at point “B”.

The operation temperature is 100°C, assembly temperature is 20°C. Therefore the temperature difference is considered as 80K.


Figure 7: Geometry of the model



1.4 Result comparisons

Thermal expansion:

Value

Reference (Kleinlogel)

Rohr2

Difference [%]

-0.105 kNm

-0.105 kNm

<0.01

-2.10 kNm

-2.10 kNm

<0.01

0.105 kN

0.105 kN

<0.01

Table 2: Comparison of bending moment and support loads for thermal expansion case

Line load:

Value

Reference (Kleinlogel)

Rohr2

Difference [%]

-3.453 kNm

-3.450 kNm

-0,09

0.595 kNm

0.6 kNm

0,84

3.453 kN

3.450 kN

-0,09

7.5 kN

7.5 kN

<0.01

Table 3: Comparison of bending moment and support loads for line load case

Point load:

Value

Reference (Kleinlogel)

Rohr2

Difference [%]

7.019 kNm

7.020 kNm

<0.02

-2.981 kNm

-2.980 kNm

-0,03

1.181 kNm

1.183 kNm

0,17

-3.105 kNm

-3.103 kNm

-0,06

-2.857kN

-2.857kN

<0.01

2.981kN

2.980kN

0,03

-7.019kN

-7.020kN

<0.02

Table 4: Comparison of bending moment and support loads for the point load case

1.5 Conclusion

The results are closely matching the reference values. The remaining difference has it's origin in the neglected axial flexibility of the beams in the reference.

1.6 Files:

R015.r2w

R2_stresses_15.xls

SIGMA Ingenieurgesellschaft mbH www.rohr2.com