BENDING MOMENT AND SHEARING FORCE DIAGRAMS 1
1 BENDING MOMENT AND SHEARING FORCE DIAGRAMS 1
1.1 Problem definition
Determinate the support pressures of a beam, which stresses by a line load.
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 3.22, pg. 85.
A profile of carbon steel stresses under an triangle line load. The following parameters are given:

Triangleline Load

maximum of the line load

Length of the structural section
Where:
Variable 
Description 
Unit 
Used Value 
Support load _{0 R1} 
lbf 
48000,00 

Support load _{24 R2} 
lbf 
96000,00 

Total length 
ft 
24,00 
Triangleline load 
lbf 
500,00 
Line load _{24 R2} 
lbs/ ft 
12000,00 

Sum of support loads 
lbf 
144000,00 
Table 1: Overview of the used variables
1.3 Model description (ROHR2)
One 24 ft long structural section (I100) is used. A triangleline load (500 lbs/ ft) is applied. The boundary conditions at both ends are: The left side of the beam, where the line load increases from zero is named 0R1 (My, Mz not transmitted) and the point of maximum stress is named 24R2 (Mx, My, Mz not transmitted). In ROHR2 it is necessary to break down the entire triangleline load into many parts of constant loads. These loads have to be chosen so that the sum under the load surface is equal to the total line load. Seven different systems were generated for this model. It should illustrate, how the results (support loads) behave, when different discretization are used. In the Load case 1 the gravitational acceleration is not considered.
Figure 4: 8 inch model
Figure 3: 6 inch model
Figure 6: 4 inch model
Figure 5: 3 inch model
Figure 8: 2 inch model
Figure 7: 1 inch model
Figure 10: W [lbf] from the Lc Dead Weight
Figure 9: 0,2 inch model with the support loads F_{R1, FR2}
1.4 Result comparisons
Version 
Value [lbf 
Reference 0 R1 [lbf] 
Rohr2 0 R1 [lbf] 
Difference 0 R1 [%] 
Reference 24 R2 [lbf] 
Rohr2 24 R2 [lbf] 
Difference 24 R2 [%] 
_{} 
48000 
51266,7 
<6,81 
96000 
93933,4 
<2,3 

50100,0 
<4,38 
95100,1 
<0,95 

49266,7 
<2,64 
95933,4 
<0,07 

48975,0 
<2,04 
96225,1 
<0,24 

48766,7 
<1,60 
96433,4 
<0,46 

48641,7 
<1,34 
96558,4 
<0,59 

48601,7 
<1,26 
96598,4 
<0,63 
Table 2: Comparison of the support loads with the reference
Version 
Value 
Reference (Timoshenko) [lbf] 
Rohr2 [lbf] 
Difference [%] 
All seven 
144000 
145200 
<0,84 
Table 3: Comparison of the sum of support loads
1.5 Conclusion
The results become more precise with every version. This accounts for the parts, which become smaller. The results approach to an ideally triangleline load. One other condition which is given in the reference is, that the support load at the two points 0 R1 and 24 R2 reach ordained values (see table 2and table 3 ). It is to remark, that the dimensions, lengths, loads and materials are free selectable, not given by Timoshenko.
Figure 11: force curve by Timoshenko
Figure 12: force curve of 0,2 inch Model
Figure 13: moment curve by Timoshenko
Figure 14: moment curve of 0,2 inch Model
The figures also show, that the course of force and momentcurve is exactly the same as in the reference.
1.6 Files
R007_inch.r2w
R007_mm.r2w
R2_stresses_7
SIGMA Ingenieurgesellschaft mbH www.rohr2.com