Combined state of stress bending and torsion
CIVE 302 Solid Mechanics Laboratory
San Diego State University
Department of Civil and Environmental Engineering
Purpose of Experiment:
The combined loading experiment aims to provide deep study into the theory of plane stress. The principle of superposition is also examined and verified.
Description of Experiment:
This experiment has two parts A and B. Part A works on torsion and bending of a steel pipe. The system includes the pipe that is fixed to a support and has a plate welded to the free end and we applied the load at the end of the plate. Actually, we have to increase the load from 40lbs to maximum is 200lbs, however, we just do 2 load is 100lb and 200lb this this experiment. The pipe is hollow with inner diameter Id=1^{15/16 }in., outer diameter Od =2^{3/8}in., Area=1.477in^{2}
In addition, Polar all data about the system are: moment of Inertia J= 1.736 in^{4}, Moment of Inertia I=.868 in^{4}, Section Modulus S= .731 in^{3}, moment arm for bending d= 28in., torsion lever arm L=20in., Poisson’s ration u= .275, and Elastic Modulus of Elasticity E= 30x10^{6}psi. Strain gages are oriented at principal stresses and connected to dummy gages in a half bridge connection.
We did exactly the same in part B as part A, however, we skip the torsion and just focus on bending. The strain gages remain in place but are not at principal stresses any more since torsion is removed. Loading is the same as Part A is 100lb and 200lb.
Experimental Results:
Part A
Applied Load (lbs) 
Strain (ustrain) 

eq1 
eq2 

0 
2500 
2500 

100 
2530 
2710 

200 
2500 
2840 
Applied Load (lbs) 
Strains *10^{4} 

e1 
e2 

100 
Exprmntl. Avg. 
0.3 
2.1 

Theoretical 
2.1 
1 

200 
Exprmntl. Avg. 
0 
3.4 

Theoretical 
2.81 
95.8 
Part B
Applied Load (lbs) 
Strain (ustrain) 

eq1 
eq2 

0 
2500 
2500 

100 
2630 
2520 

200 
2750 
2520 
Applied Load (lbs) 
Strains *10^{4} 

e1 
e2 

100 
Exprmntl. Avg. 
1.3 
0.2 

Theoretical 
1.3 
0.35 

200 
Exprmntl. Avg. 
2.5 
0.2 

Theoretical 
2.5 
7.0 
Sample Calculations:
Part A: (P=200lb, maximum),
Experimental
e= P_{i}  P_{0} =(28402500)x10^{6 }in/in.= 340x10^{6 }in/in.
M = Pd= (200lb*28in)= 5600inlb.
s_{x}=Pd/S= (5600)/(.731)= 7660 psi
s_{y}= 0 psi
T= PL= (200lb.*20in.)= 4000inlb.
t=t_{xy}=t_{yx}=PLcJ= (4000*31)/(32*1.736)= 2232psi
Principal stresses:
s_{1}=(sx/2)+[(sx/2)^{2}+(t_{xy})^{2}]^{½}= 3830+[(4433)]= 8263psi
s_{2}= (3125)(3840.3)=  603psi
Principal Strains:
e_{1}=s_{1}/Eu*s_{2}/E and e_{2}=s_{2}/Eu*s_{1}/E
e_{1}= 2.81x10^{4 }in/in. (T)
e_{2}= 9.58x10^{5 }in/in. (C)
Part B: (P= 200lb maximum) Theoretical data:
*NO torsion, just bending stresses and strains.
M= 5600inlb. (same as part A)
s_{x}=Pd/S= (5600)/(.731)= 7660 psi (uniaxial stress)= s_{1}
s_{2}=0 psi (therefore principal angles are 0^{o} and 90^{o})
e_{1}= (s_{1})/E= 7660psi/30000000psi= 2.55x10^{4 }in/in.
e_{2}=(u*s_{1}/E)= .275*7660/30000000= 7.02x10^{4 }in/in.
Data Analysis:
In this experiment, from the table in the experimental result, we can see the results sometimes are the theoretical (strain 1 in part B) and sometime are significantly different than the experimental data. Therefore, these data do not represent for the relationship between these terms. Because these data jump differently, so we cannot calculate the average to find the percentage of errors. It means they can be smaller than 10% (reasonable) and they can be larger 10% (not accepted)
Sources of Errors:
This experiment had more errors than expected probably due to the many supports, welds, and loadings occurring simultaneously. Sometimes, we do not know the actual reason because we get some correct results (100% accuracy), and some results are total off. There are a lot of errors due to old equipment, many experiments in many classes. Moreover, the loads can affect the system by time. Therefore, when we unload it, the system cannot go back as new.
Discussion Questions:
1) The strains varied more than expected, the results change differently. However, their magnitude are correct (negative for compression and positive for tension). Errors could have entered in many different places causing for the difference.
2) The theory seems very applicable to the conditions that occurred. The theory will help us to predict the experiment. Moreover, with the prediction, we can avoid some wrong actions and number during the experiment.
3)A load that would be applied along the y axis could cause a stress in the s_{yy}.
Conclusion:
The torsion and bending are the fundamental elements in the real life. Therefore, they are observed that the principle of superposition is an extremely practical and convenient form of determining the complex effects generated by multiple forces acting on a body of solid materials. Through this experiment, we know the impact of load to the material. Indeed, some cases the failure of material due to over bending or over torsion.. Also, it is assumed that the material is only within the elastic range. Plastic deformation would require a more advanced analysis.
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