Line Load and Thermal Expansion

# 1Line Load and Thermal Expansion

## 1.1Problem definition

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

Figure 1: Frame with line load

Figure 2: Rohr2 Model

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

Figure 3: Frame with thermal expansion

Figure 4: Rohr2 Model

## 1.2 References (Kleinlogel)

A. Kleinlogel, Rahmenformeln, 11th Edition, Darmstadt 1947, Ramenform 1.

The following values are given:

• Length of frame

• Height of frame

• Line load

• Module of elasticity

• Coefficient of expansion

• temperature difference

• Outside diameter

• Inside diameter

The following formulas are used:

rigidity number

bending moment at point â€œBâ€

horizontal support load at point â€œAâ€ / â€œBâ€

vertical support load at point â€œAâ€

vertical support load at point â€œBâ€

And for the thermal expansion:

moment of intertia

bending moment at point â€œBâ€

horizontal support load at point â€œAâ€ / â€œCâ€

vertical support load at point â€œAâ€ / â€œCâ€

 Variable Description Unit Used Value length of frame mm 2000 height of frame mm 1000 line load kN/m 10 Outside diameter mm 114,3 Inside diameter mm 103,1 Module of elasticity kN/mm^2 207 Coefficient of expansion Âµm/(m*K) 12,5 temperature difference K 80

Table 1: Overview of the used variables

## 1.3 Model description (ROHR2)

The system consists of one frame. The length of the frame is 2000mm, the height 1000mm. 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 a constant line load of 10kN/m is applied.

In the second load case the thermal expansion is applied. The operation temperature is 100Â°C, assembly temperature is 20Â°C. Therefore the temperature difference is considered as 80K.

## 1.4 Result comparisons

Line loads:

 Value Reference (Kleinlogel) Rohr2 Difference [%] -3.33 kNm -3.32 kNm <0,01 8.33 kN 8.34 kN <0,01 11.67 kN 11.66 kN <0,01 3.33 kN 3.32 kN <0,01

Table 2: Comparison of bending moment and support loads for line load case

Thermal expansion

 Value Reference (Kleinlogel) Rohr2 Difference [%] -1.47 kNm -1.46 kNm <0,01 -0.73 kN -0.73 kN <0,01 0.73 kN 0.73 kN <0,01 1.47 kN 1.47 kN <0,01

Table 3: Comparison of bending moment and support loads for thermal expansion 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:

R014.r2w

R2_stresses_14

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