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 â€œsâ€

• Dimension â€œlâ€

• Dimension â€œhâ€

• Dimension â€œaâ€

• Dimension â€œdâ€

• Dimension â€œcâ€

• Module of elasticity

• Coefficient of expansion

• Temperature difference

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 â€œBâ€ / â€œEâ€

Bending moment at point â€œCâ€ / â€œDâ€

Horizontal support load at point â€œAâ€ / â€œFâ€

For the line load:

Auxilliary value

Bending moment at point â€œBâ€ / â€œEâ€

Bending moment at point â€œCâ€ / â€œDâ€

Horizontal support load at point â€œAâ€ / â€œFâ€

Vertical support load at point â€œAâ€ / â€œFâ€

For the point load:

Auxilliary value

Bending moment at point â€œBâ€

Bending moment at point â€œEâ€

Bending moment at point â€œCâ€

Bending moment at point â€œDâ€

Vertical support load at point â€œAâ€ / â€œFâ€

Horizontal support load at point â€œFâ€

Horizontal support load at point â€œAâ€

The following variables are used:

 Variable Description Unit Used Value Outside diameter mm 114,3 Inside diameter mm 107,1 Dimension â€œsâ€ mm 1414,21 Dimension â€œlâ€ mm 3500 Dimension â€œhâ€ mm 2000 Dimension â€œaâ€ mm 1000 Dimension â€œdâ€ mm 1500 Dimension â€œcâ€ mm 1000 Module of elasticity kN/mm^2 207 Coefficient of expansion Âµm/(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 â€œTable1â€ and â€œFigure7â€. The frame is supported by two simple supports. At point â€œAâ€ all displacements are fixed, at point â€œCâ€ 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 â€œCâ€ and point â€œDâ€.

In the third load case a single load of 10kN is applied at point â€œBâ€.

The operation temperature is 100Â°C, assembly temperature is 20Â°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

 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

 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

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