PURE BENDING
1 PURE BENDING
1.1 Problem definition
Determine the maximum deflections and bending stress at centre of a two times simply supported beam when two downward point loads applied at the ends.
Figure 1: Model by Timoshenko
Figure 2: 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 4.23, pg. 94.
The system which consists of two boundary conditions and three unequal sections is defined by following parameters:

Length of the stretch PAA (Ã¢â€°â„¢C)

Length of the stretch PBB (Ã¢â€°â„¢C)

Length of the stretch AB

Diameter

Spring load

Modulus of Elasticity
Where:
Variable 
Description 
Unit 
Used Value 
Maximum stress 
lbs/ inchÃ‚Â² 
3575,26 

Bending moment 
ft lbs 
29250 

Section Modulus 
mmÃ‚Â³ 
98,175 

Spring load 
lbf 
26000 
Length_{PA;A} 
inch 
1,125 

Length_{PB;B} 
inch 
1,125 

Length_{AB} 
inch 
4,917 

Diameter 
inch 
10 

Radius of curvature 
inch 
41955,03 

Modulus of Elasticity 
lbs/ inchÃ‚Â² 
30000000 

Deflection 
inch 
0,01037 
Table 1: Overview of the used variables
1.3 Model description (ROHR2)
The whole system consists of a carbon steel beam with a ten inch diameter. It has three parts, where two of them are equal long (l_{PAA}; l_{PBB}= 13,5 inch). The third is with a length of 59 inch the longest one. At two points (PA, PB) the beams are supported. Both supports are vertical and transverse supports and at point A an addition axial stop and stop for moments of torsion is added. The system is submitted to two vertical downward force at both ends (PPA, PPB). In order to have the possibility of checking and comparing the deflections with higher precision, it is necessary to create a model where the the loads P where increased by the factor 1000. A load case is created where the gravitational acceleration isn't taken into account.
Figure 3: Model with moment Informations (sectional results)
Figure 4: Model with increased loads (sectional results)
1.4 Result comparisons
Value 
Length [inch] 
Reference (Timoshenko) [lbs*ft] 
Rohr2 [lbs*ft] 
Difference [%] 
86,0 
3575,26 
3575,25 
<0,01 
Table 2: Comparison of the bending stress at point O
Value 
Length [inch] 
Reference (Timoshenko) [inch] 
Rohr2 [inch] 
Difference [%] 
86,0 
0,01037 
0,01 
<0,09 
Table 3: Comparison of the max. deflection
1.5 Conclusion
The results perfectly match those given in the references.
1.6 Files
R009_inch.r2w
R009_mm.r2w
R2_stresses_9
SIGMA Ingenieurgesellschaft mbH www.rohr2.com