STATICALLY INDETERMINATE PROBLEMS IN TENSION AND COMPRESSION

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 is verified.


Figure 2: Model by Timoshenko



Figure 1: Rohr2 model


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

SIGMA Ingenieurgesellschaft mbH www.rohr2.com