1STATICALLY INDETERMINATE PROBLEMS IN TENSION AND COMPRESSION

1.1Problem definition

Determinate the existing compression stress, when a double shell system with two different material is compressed by two vertical loads. Also, the negative elongation (unit compression) is verified.

1.2 References (Timoshenko)

S. Timoshenko, Strength of Material, Part I, Elementary Theory and Problems, 3rd Edition, D. Van Nostrand Co., Inc., New York, NY, 1956, Chapter 1.6, pg. 21-22.

System consists of two concentric, coupled beams submitted to compressive forces. The following parameters are given:

• Modulus of Elasticity (carbonsteel, copper) ,

• Compression load

• ratio of the E- Modulus

• Diameter ,

Where:

Variable	Description	Unit	Used Value
	Stresses in the steel beam	lbs/ inch²	3060,67
	Stresses in copper beam	lbs/ inch²	1632,36

	Modulus of Elasticity (Steel)	lbs/ inch²	30000000
	Modulus of Elasticity (Copper)	lbs/ inch²	16000000
	Ratio of E- Modulus		1,88
	Outside Diameter (Copper)	inch	8,0
	Diameter (Steel)	inch	4,0
	Compressive Load	lbf	100000
	Unit Compression (Copper)	inch	0,000102
	Length	ft/ inch	4,17/ 50,04

Table 1: Overview of the variables used

1.3 Model description (ROHR2)

The ROHR2 model consists of a double jacked line with full couplings at both ends. The length of the bars is 4,17 ft. They are made of two different materials. The first is a cylinder of carbon steel, the second part consists of copper (Pipe). This material has following mechanical parameters:

• Young's Modulus= 16 x 10⁶ lbs/ inch²

• Alfa T (68 °F)= 10,8 μm/ m

• Tensile Strength= 28x 10³ lbs/ inch²

The dimensions are taken from the example by Timoshenko. The copper tube has an outside diameter of eight inch. The inside diameter of the copper tube of four inch is equal to the outside diameter of the steel beam. An anchor point fixes the two parts at the middle and coupling are connecting the inner and outer parts at both ends. With this boundary conditions, it is guaranteed that the forces (100000 lbf) are transferred to both materials. The normal force is applied at the ends (P1a, P2a) and oriented towards the anchor point in the middle. The force were scaled by a factor at both model-versions (R005_inch: 1000; R005_mm: 10) at the lower ends, to increase the number of available digits in the output. As the gravitational acceleration should not be taken into account, an additional load case (occasional loads; Load1) is defined which contains only the two end forces.

Figure 3: Tension by P=100000 lbs (outer ≙ σc; inner ≙ σs)

Figure 4: Compression displacements by increased loads (outer ≙ εc)

1.4 Result comparisons

Value	Length [inch]	Reference (Timoshenko) [lbs/ inch²]	Rohr2 [lbs/ inch²]	Difference [%]
	50.04	3060,67	3071,4	<0,35
		1632,36	1628,8	<0,22

Table 2: Comparison for the tension of both materials

Value	Length [inch]	Reference (Timoshenko) [inch]	Rohr2 [inch]	Difference [%]
	50.04	0,005105	0,005119	<0,28

Table 3: Comparison of the unit compression for copper

1.5 Conclusion

The results are very close to the reference by Timoshenko. The difference are within the precision expected, given the number of digits in the input.

1.6 Files

R005_inch.r2w

R005_mm.r2w

R2_stresses_5.ods

MATDAT.r2u