# 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

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°.

## Comparison

Table 1. Comparison of results
Maximum deflection at free end (in.) -0.04267 -0.04265 none
Principle stress (psi) 1,600 1,600 none

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
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
``````

```   MEMBER PROPERTIES. UNIT - INCH
-----------------
MEMB   PROFILE              AX/          IZ/          IY/        IX/
AY           AZ           SZ         SY
1      TAP ERED           1.42         0.97         0.03        0.12
1.42         0.01         0.68        0.12
2      TAP ERED           1.28         0.70         0.03        0.11
1.28         0.01         0.54        0.11
3      TAP ERED           1.12         0.48         0.02        0.09
1.12         0.01         0.42        0.09
4      TAP ERED           0.97         0.31         0.02        0.08
0.97         0.01         0.32        0.08
5      TAP ERED           0.82         0.19         0.02        0.07
0.82         0.01         0.23        0.07
6      TAP ERED           0.68         0.11         0.01        0.06
0.68         0.01         0.15        0.06
7      TAP ERED           0.52         0.05         0.01        0.04
0.52         0.01         0.09        0.04
8      TAP ERED           0.38         0.02         0.01        0.03
0.38         0.01         0.05        0.03
9      TAP ERED           0.23         0.01         0.00        0.02
0.23         0.01         0.02        0.02
10      TAP ERED           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
```