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.

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 (l_PA-A; l_PB-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