1PURE BENDING 2

1.1 Problem definition

Determine the deflections and bending stresses at the middle of the carbon steel beam.

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. 95.

The thirty inch high carbon structure is stressed at both ends by a constant line load. The following values are given:

• Length of several chapter

• Total length

• Constant line load

• Moment of Inertia

• Max. fiberdistance

• Modulus of Elasticity

Where:

Variable	Description	Unit	Used Value
	Maximum stress	lbs/ inch²	9836,066
	Bending Moment	ft lbs	6000000,0
	Section Modulus	inch³/cm³	610,0/9996,10

	Moment of Inertia	inch4/cm4	9150,0/380853,25
	Max. fibre distance	inch	15,00
	Chapter length	ft	10,00
	Total length	ft	40,00
	Line load	lbs/ ft	10000,00
	Radius of curvature	inch	45750,00
	Modulus of Elasticity	lbs/ inch²	30000000
	Deflection	inch	0,01574

Table 1: Overview of the used variables

1.3 Model description (ROHR2)

The system consists of four equal long bars (l_C= 10 ft ≙ 120 inch). For this model it necessary to generate a new profile (IPB30“ see figure 4) with the given moment of inertia of 9150,0 inch⁴. The section parameters of the profile are written in the profile database (see figure 3).The beam is supported by two simple supports. At point B the torsion and axial displacement is fixed to avoid a singular system: At both sides, between end and boundary condition, constant line loads are applied. The line load were increased by a factor of 100 to increase the number of significant digits in the output. In the Lc Load1 the gravitational acceleration is not taken into account.

Figure 3: Section parameters

Figure 4: Declaration the axes of an I- profile

Figure 5: Input from the PROFDS.r2u

Figure 8: Results for the increased load

Figure 7: δ from the Load case 1

Figure 6: Sectional results (equivalent stress) a point C

1.4 Result comparisons

Value	Length [inch]	Reference (Timoshenko) [lbs/inch²]	Rohr2 [lbs/inch²]	Difference [%]
	480,0	9836,07	9836,1	<0,01

Table 2: Comparison of the stress at the middle of the system

Value	Length [inch]	Reference (Timoshenko) [inch]	Rohr2 [inch]	Difference [%]
	480,0	0,15738	0,15726	<0,08

Table 3: Comparison of the max. deflection

1.5 Conclusion

The results are perfectly matching the reference values.

1.6 Files

R010_inch.r2w

R010_mm.r2w

R2_stresses_10.xls

PROFDS.r2u