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 PA-A (≙C)

  • Length of the stretch PB-B (≙C)

  • Length of the stretch A-B

  • Diameter

  • Spring load

  • Modulus of Elasticity

Where:

Variable

Description

Unit

Used Value

Maximum stress

lbs/ inch²

3575,26

Bending moment

ft lbs

29250

Section Modulus

mm³

98,175

Spring load

lbf

26000

LengthPA;A

inch

1,125

LengthPB;B

inch

1,125

LengthAB

inch

4,917

Diameter

inch

10

Radius of curvature

inch

41955,03

Modulus of Elasticity

lbs/ inch²

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 (lPA-A; lPB-B= 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