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.
References
Hand calculation using the following reference:
Harris, C.O., Introduction to Stress Analysis, The Macmillan Co., 1959. Page 114, Problem 61.
Details
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.
Note: The STAAD.Pro model
is created using 10 beam segments, each with a tapered section. The tapered section is
modeled using the tapered web and (effectively) no flanges, rotated 90°.
Results
| Result | Theory | STAAD.Pro | Difference |
|---|---|---|---|
| Maximum deflection at free end (in.) | -0.04267 | -0.04265 | none |
| Principle stress (psi) | 1,600 | 1,600 | none |
Input
The file C:\Users\Public\Public Documents\STAAD.Pro 2026\Samples \Verification Models\01 Beams\Stresses in a Tapered Cantilever.STDis typically installed with the program.
STAAD
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
FINISHOutput
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