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.

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:

bending moment at point “B”

horizontal support load at point “A” / “B”

vertical support load at point “A”

vertical support load at point “B”

And for the thermal expansion:

moment of intertia

bending moment at point “B”

horizontal support load at point “A” / “C”

vertical support load at point “A” / “C”

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	µm/(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 “A” all displacements are fixed, at point “C” 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 100°C, assembly temperature is 20°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