V. Stresses in a Tapered Cantilever
To find the maximum deflection and principal stress due to a load on the free end of a cantilever beam with tapered section.
Reference
Hand calculation using the following reference:
Harris, C.O., Introduction to Stress Analysis, The Macmillan Co., 1959. Page 114, Problem 61.
Problem
Find the maximum deflection, δ, and the principal normal stress, σ, in the beam.
| E = 30×(10)6 psi |
| Load at free end, P = 10 lb |
| Width at support, b = 3 in., Depth, d = 0.5 in. |
| Length, L = 20 in. |
Beam with a tapering cross section
Comparison
| Result | Theory | STAAD.Pro | Difference |
|---|---|---|---|
| Maximum deflection at free end (in.) | -0.04267 | -0.04265 | none |
| Principle stress (psi) | 1,600 | 1,600 | none |
STAAD Input
The file C:\Users\Public\Public Documents\STAAD.Pro CONNECT Edition\Samples\ Verification Models\01 Beams\Stresses in a Tapered Cantilever.STDis typically installed with the program.
STAAD PLANE :A CANTILEVER BEAM OF TAPERED SECTION
START JOB INFORMATION
ENGINEER DATE 18-Sep-18
END JOB INFORMATION
*
* REFERENCE: HARRIS, C.O., INTRODUCTION TO STRESS ANALYSIS,
* THE MACMILLAN CO., NEW YORK, 1956
*
* USING A TAPERED BEAM ELEMENT
*
INPUT WIDTH 72
UNIT INCHES POUND
JOINT COORDINATES
1 0 0 0; 2 2 0 0; 3 4 0 0; 4 6 0 0; 5 8 0 0; 6 10 0 0; 7 12 0 0;
8 14 0 0; 9 16 0 0; 10 18 0 0; 11 20 0 0;
MEMBER INCIDENCES
1 11 10; 2 10 9; 3 9 8; 4 8 7; 5 7 6; 6 6 5; 7 5 4; 8 4 3; 9 3 2;
10 2 1;
MEMBER PROPERTY AMERICAN
1 TAPERED 3 0.5 2.7 0.5 0.01
2 TAPERED 2.7 0.5 2.4 0.5 0.01
3 TAPERED 2.4 0.5 2.1 0.5 0.01
4 TAPERED 2.1 0.5 1.8 0.5 0.01
5 TAPERED 1.8 0.5 1.5 0.5 0.01
6 TAPERED 1.5 0.5 1.2 0.5 0.01
7 TAPERED 1.2 0.5 0.9 0.5 0.01
8 TAPERED 0.9 0.5 0.6 0.5 0.01
9 TAPERED 0.6 0.5 0.3 0.5 0.01
10 TAPERED 0.3 0.5 0.03 0.5 0.01
DEFINE MATERIAL START
ISOTROPIC MATERIAL1
E 3e+07
POISSON 0.290909
END DEFINE MATERIAL
CONSTANTS
BETA 90 ALL
MATERIAL MATERIAL1 ALL
SUPPORTS
11 FIXED
LOAD 1 POINT LOAD AT TIP
JOINT LOAD
1 FY -10
PERFORM ANALYSIS
PRINT MEMBER PROPERTIES ALL
PRINT MEMBER FORCES
PRINT SUPPORT REACTION
SECTION 0.5 ALL
PRINT MEMBER SECTION FORCES ALL
PRINT JOINT DISPLACEMENTS LIST 1
PRINT MEMBER STRESSES ALL
FINISH
STAAD Output
MEMBER PROPERTIES. UNIT - INCH ----------------- MEMB PROFILE AX/ IZ/ IY/ IX/ AY AZ SZ SY 1 TAPERED 1.42 0.97 0.03 0.12 1.42 0.01 0.68 0.12 2 TAPERED 1.28 0.70 0.03 0.11 1.28 0.01 0.54 0.11 3 TAPERED 1.12 0.48 0.02 0.09 1.12 0.01 0.42 0.09 4 TAPERED 0.97 0.31 0.02 0.08 0.97 0.01 0.32 0.08 5 TAPERED 0.82 0.19 0.02 0.07 0.82 0.01 0.23 0.07 6 TAPERED 0.68 0.11 0.01 0.06 0.68 0.01 0.15 0.06 7 TAPERED 0.52 0.05 0.01 0.04 0.52 0.01 0.09 0.04 8 TAPERED 0.38 0.02 0.01 0.03 0.38 0.01 0.05 0.03 9 TAPERED 0.23 0.01 0.00 0.02 0.23 0.01 0.02 0.02 10 TAPERED 0.08 0.00 0.00 0.01 0.08 0.01 0.00 0.01 ************ END OF DATA FROM INTERNAL STORAGE ************ 45. PRINT MEMBER FORCES MEMBER FORCES :A CANTILEVER BEAM OF TAPERED SECTION -- PAGE NO. 4 MEMBER END FORCES STRUCTURE TYPE = PLANE ----------------- ALL UNITS ARE -- POUN INCH (LOCAL ) MEMBER LOAD JT AXIAL SHEAR-Y SHEAR-Z TORSION MOM-Y MOM-Z 1 1 11 0.00 0.00 -10.00 0.00 200.00 0.00 10 0.00 0.00 10.00 0.00 -180.00 0.00 2 1 10 0.00 0.00 -10.00 0.00 180.00 0.00 9 0.00 0.00 10.00 0.00 -160.00 0.00 3 1 9 0.00 0.00 -10.00 0.00 160.00 0.00 8 0.00 0.00 10.00 0.00 -140.00 0.00 4 1 8 0.00 0.00 -10.00 0.00 140.00 0.00 7 0.00 0.00 10.00 0.00 -120.00 0.00 5 1 7 0.00 0.00 -10.00 0.00 120.00 0.00 6 0.00 0.00 10.00 0.00 -100.00 0.00 6 1 6 0.00 0.00 -10.00 0.00 100.00 0.00 5 0.00 0.00 10.00 0.00 -80.00 0.00 7 1 5 0.00 0.00 -10.00 0.00 80.00 0.00 4 0.00 0.00 10.00 0.00 -60.00 0.00 8 1 4 0.00 0.00 -10.00 0.00 60.00 0.00 3 0.00 0.00 10.00 0.00 -40.00 0.00 9 1 3 0.00 0.00 -10.00 0.00 40.00 0.00 2 0.00 0.00 10.00 0.00 -20.00 0.00 10 1 2 0.00 0.00 -10.00 0.00 20.00 0.00 1 0.00 0.00 10.00 0.00 -0.00 0.00 ************** END OF LATEST ANALYSIS RESULT ************** 46. PRINT SUPPORT REACTION SUPPORT REACTION :A CANTILEVER BEAM OF TAPERED SECTION -- PAGE NO. 5 SUPPORT REACTIONS -UNIT POUN INCH STRUCTURE TYPE = PLANE ----------------- JOINT LOAD FORCE-X FORCE-Y FORCE-Z MOM-X MOM-Y MOM Z 11 1 0.00 10.00 0.00 0.00 0.00 -200.00 ************** END OF LATEST ANALYSIS RESULT ************** 47. SECTION 0.5 ALL 48. PRINT MEMBER SECTION FORCES ALL MEMBER SECTION FORCES ALL :A CANTILEVER BEAM OF TAPERED SECTION -- PAGE NO. 6 MEMBER FORCES AT INTERMEDIATE SECTIONS -------------------------------------- ALL UNITS ARE -- POUN INCH MEMB LOAD SEC AXIAL SHEAR-Y SHEAR-Z MOM-X MOM-Y MOM-Z 1 1 0.50 0.00 0.00 -10.00 0.00 190.00 0.00 2 1 0.50 0.00 0.00 -10.00 0.00 170.00 0.00 3 1 0.50 0.00 0.00 -10.00 0.00 150.00 0.00 4 1 0.50 0.00 0.00 -10.00 0.00 130.00 0.00 5 1 0.50 0.00 0.00 -10.00 0.00 110.00 0.00 6 1 0.50 0.00 0.00 -10.00 0.00 90.00 0.00 7 1 0.50 0.00 0.00 -10.00 0.00 70.00 0.00 8 1 0.50 0.00 0.00 -10.00 0.00 50.00 0.00 9 1 0.50 0.00 0.00 -10.00 0.00 30.00 0.00 10 1 0.50 0.00 0.00 -10.00 0.00 10.00 0.00 ************** END OF LATEST ANALYSIS RESULT ************** 49. PRINT JOINT DISPLACEMENTS LIST 1 JOINT DISPLACE LIST 1 :A CANTILEVER BEAM OF TAPERED SECTION -- PAGE NO. 7 JOINT DISPLACEMENT (INCH RADIANS) STRUCTURE TYPE = PLANE ------------------ JOINT LOAD X-TRANS Y-TRANS Z-TRANS X-ROTAN Y-ROTAN Z-ROTAN 1 1 0.00000 -0.04265 0.00000 0.00000 0.00000 0.00419 ************** END OF LATEST ANALYSIS RESULT ************** 50. PRINT MEMBER STRESSES ALL MEMBER STRESSES ALL :A CANTILEVER BEAM OF TAPERED SECTION -- PAGE NO. 8 MEMBER STRESSES --------------- ALL UNITS ARE POUN/SQ INCH MEMB LD SECT AXIAL BEND-Y BEND-Z COMBINED SHEAR-Y SHEAR-Z 1 1 .0 0.0 1600.0 0.0 1600.0 0.0 1500.0 0.50 0.0 1600.0 0.0 1600.0 0.0 1500.0 1.0 0.0 C 1600.0 0.0 1600.0 0.0 1500.0 2 1 .0 0.0 1600.0 0.0 1600.0 0.0 1500.0 0.50 0.0 1600.0 0.0 1600.0 0.0 1500.0 1.0 0.0 C 1600.0 0.0 1600.0 0.0 1500.0 3 1 .0 0.0 1600.0 0.0 1600.0 0.0 1500.0 0.50 0.0 1600.0 0.0 1600.0 0.0 1500.0 1.0 0.0 C 1600.0 0.0 1600.0 0.0 1500.0 4 1 .0 0.0 1600.0 0.0 1600.0 0.0 1500.0 0.50 0.0 1600.0 0.0 1600.0 0.0 1500.0 1.0 0.0 C 1600.0 0.0 1600.0 0.0 1500.0 5 1 .0 0.0 1600.0 0.0 1600.0 0.0 1500.0 0.50 0.0 1600.0 0.0 1600.0 0.0 1500.0 1.0 0.0 C 1600.0 0.0 1600.0 0.0 1500.0 6 1 .0 0.0 1600.0 0.0 1600.0 0.0 1500.0 0.50 0.0 1600.0 0.0 1600.0 0.0 1500.0 1.0 0.0 C 1600.0 0.0 1600.0 0.0 1500.0 7 1 .0 0.0 1600.0 0.0 1600.0 0.0 1500.0 0.50 0.0 1600.0 0.0 1600.0 0.0 1500.0 1.0 0.0 C 1600.0 0.0 1600.0 0.0 1500.0 8 1 .0 0.0 1600.0 0.0 1600.0 0.0 1500.0 0.50 0.0 1600.0 0.0 1600.0 0.0 1500.0 1.0 0.0 C 1600.0 0.0 1600.0 0.0 1500.0 9 1 .0 0.0 1600.0 0.0 1600.0 0.0 1500.0 0.50 0.0 1600.0 0.0 1600.0 0.0 1500.0 1.0 0.0 C 1600.0 0.0 1600.0 0.0 1500.0 10 1 .0 0.0 1600.0 0.0 1600.0 0.0 1500.0 :A CANTILEVER BEAM OF TAPERED SECTION -- PAGE NO. 9 MEMBER STRESSES --------------- ALL UNITS ARE POUN/SQ INCH MEMB LD SECT AXIAL BEND-Y BEND-Z COMBINED SHEAR-Y SHEAR-Z 0.50 0.0 1454.5 0.0 1454.5 0.0 1500.0 1.0 0.0 C 0.0 0.0 0.0 0.0 1500.0