# V. Forces on a Propped Cantilever 1

To find the deflection and member forces due to an applied load on a propped cantilever beam with a compression only support.

## Reference

Hand calculation using the following reference:

Manual of Steel Construction, Load and Resistance Factor Design, Second Edition, American Institute of Steel Construction, 1998, pp. 4-194, 4-197.

## Problem

A propped cantilever beam with an propped-end support capable of resisting only a compressive force is analyzed for two concentrated loads at 0.6xL:
1. P = +0.5 lbs (up), and
2. P = -0.5 lbs (down)
• E = 10(10)6

• width = 0.6 in

• depth = 0.3 in

• L = 4 ft

Note: A "dummy" member is used to represent a compression only support. The dimensions of this member are arbitrary; just that the member is specified as compression-only.

## Theoretical Solution

General solution equations found on p.2-121 of the reference. In this case, the "propped" end carries no load as the support cannot carry tension. Thus, the member acts like a pure cantilever.

$M ( x < b ) = − P ( b − x ) = − 0.5 l b ( 28.8 − x )$

At the rigid support, M = 14.4 in·lb.

By inspection, fixed end shear is equal to load value = 0.50 lb.

$Δ ( x < b ) = P x 2 6 E I ( 3 x − b )$
$= 0.5 l b ( 28.8 in . ) 2 6 ( 10 , 000 , 000 p s i ) ( 0.00135 in . 4 ) [ 3 ( 28.8 in . ) − 28.8 in . ] = 0.295 in .$

General solution equations found on p.2-118 of the reference. In this case, the "propped" is capable of carrying load and thus the beam acts like a "propped" cantilever case.

$R 1 = P b 2 ( a + 2 ℓ ) 2 ℓ 3 = 0.5 l b ( 48 in . ) 2 2 ( 48 in . ) 3 [ 19.2 in + 2 ( 48 in ) ] = 0.216 l b .$

R2 = P - R1 = 0.50lb - 0.216lb = 0.284lb.

Moment at rigid support:

$M ( x < b ) = R 1 ( ℓ − x ) − P ( ℓ − x − a )$
$= 0.216 l b ( 48 in − 0 ) − 0.5 l b ( 48 in − 0 − 19.2 in ) = − 4.032 in·lb$

$Δ ( x < b ) = − P a x 2 12 E I ℓ 3 ( 3 ℓ 3 − 3 ℓ 2 x − 3 a 2 ℓ + a 2 x )$

## Comparison

Table 1. Comparison of results
LC 1 Moment at fixed end (in·lb) -14.4 -14.4 none
Shear at fixed end (lb) 0.50 0.50 none
Deflection at load (.in) 0.295 0.295 none
LC 2 Moment at fixed end (in·lb) 4.032 4.03 none
Shear at fixed end (lb) 0.284 0.28 1.4%
Deflection at load (.in) -0.040 -.0401 none

The file C:\Users\Public\Public Documents\STAAD.Pro CONNECT Edition\Samples\ Verification Models\01 Beams\Forces on a Propped Cantilever 1.STDis typically installed with the program.

STAAD PLANE :A PROPPED CANTILEVER WITH COMPRESSION ONLY SUPPORT
START JOB INFORMATION
ENGINEER DATE 18-Sep-18
END JOB INFORMATION
*********************************************
* The end support is to be defined as a     *
* compression only support.  A dummy member,*
* #7, is set as compression only to model   *
* this.                                     *
*********************************************
SET NL 2
UNIT INCHES POUND
JOINT COORDINATES
1 0 0 0; 2 9.6 0 0; 3 19.2 0 0; 4 28.8 0 0; 5 38.4 0 0; 6 48 0 0;
7 48 -4 0;
MEMBER INCIDENCES
1 1 2; 2 2 3; 3 3 4; 4 4 5; 5 5 6; 6 6 7;
MEMBER PROPERTY AMERICAN
1 TO 5 PRIS YD 0.3 ZD 0.6
6 PRIS YD 40
MEMBER COMPRESSION
6
DEFINE MATERIAL START
ISOTROPIC MATERIAL1
E 1e+07
POISSON 0.33
END DEFINE MATERIAL
CONSTANTS
MATERIAL MATERIAL1 ALL
SUPPORTS
1 FIXED
7 FIXED
*7 PINNED
4 FY 0.5
PERFORM ANALYSIS
PRINT MEMBER FORCES LIST 1 TO 5
PRINT JOINT DISPLACEMENTS LIST 1 TO 6
PRINT SUPPORT REACTION
CHANGE
4 FY -0.5
PERFORM ANALYSIS
PRINT SUPPORT REACTION
PRINT MEMBER FORCES LIST 1 TO 5
PRINT JOINT DISPLACEMENTS LIST 1 TO 6
FINISH


   ALL UNITS ARE -- POUN INCH     (LOCAL )
MEMBER  LOAD  JT     AXIAL   SHEAR-Y  SHEAR-Z   TORSION     MOM-Y      MOM-Z
1    1     1      0.00     -0.50     0.00      0.00      0.00     -14.40
2      0.00      0.50     0.00      0.00      0.00       9.60
2    1     2      0.00     -0.50     0.00      0.00      0.00      -9.60
3      0.00      0.50     0.00      0.00      0.00       4.80
3    1     3      0.00     -0.50     0.00      0.00      0.00      -4.80
4      0.00      0.50     0.00      0.00      0.00       0.00
4    1     4      0.00     -0.00     0.00      0.00      0.00      -0.00
5      0.00      0.00     0.00      0.00      0.00       0.00
5    1     5      0.00      0.00     0.00      0.00      0.00       0.00
6      0.00     -0.00     0.00      0.00      0.00       0.00
************** END OF LATEST ANALYSIS RESULT **************
39. PRINT JOINT DISPLACEMENTS LIST 1 TO 6
JOINT    DISPLACE LIST     1
:A PROPPED CANTILEVER WITH COMPRESSION ONLY SUPPORT      -- PAGE NO.    4
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   0.00000   0.00000   0.00000   0.00000   0.00000
2    1    0.00000   0.04370   0.00000   0.00000   0.00000   0.00853
3    1    0.00000   0.15293   0.00000   0.00000   0.00000   0.01365
4    1    0.00000   0.29494   0.00000   0.00000   0.00000   0.01536
5    1    0.00000   0.44239   0.00000   0.00000   0.00000   0.01536
6    1    0.00000   0.58985   0.00000   0.00000   0.00000   0.01536
************** END OF LATEST ANALYSIS RESULT **************
40. PRINT SUPPORT REACTION
SUPPORT  REACTION
:A PROPPED CANTILEVER WITH COMPRESSION ONLY SUPPORT      -- 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
1    1      0.00     -0.50      0.00      0.00      0.00    -14.40
7    1      0.00      0.00      0.00      0.00      0.00      0.00
************** END OF LATEST ANALYSIS RESULT **************
41. CHANGE
44. 4 FY -0.5
45. PERFORM ANALYSIS
**NOTE-Tension/Compression converged after  1 iterations, Case=    2
46. PRINT SUPPORT REACTION
SUPPORT  REACTION
:A PROPPED CANTILEVER WITH COMPRESSION ONLY SUPPORT      -- PAGE NO.    6
SUPPORT REACTIONS -UNIT POUN INCH    STRUCTURE TYPE = PLANE
-----------------
JOINT  LOAD   FORCE-X   FORCE-Y   FORCE-Z     MOM-X     MOM-Y     MOM Z
1    2      0.00      0.28      0.00      0.00      0.00      4.03
7    2      0.00      0.22      0.00      0.00      0.00      0.00
************** END OF LATEST ANALYSIS RESULT **************
47. PRINT MEMBER FORCES LIST 1 TO 5
MEMBER   FORCES   LIST     1
:A PROPPED CANTILEVER WITH COMPRESSION ONLY SUPPORT      -- PAGE NO.    7
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    2     1      0.00      0.28     0.00      0.00      0.00       4.03
2      0.00     -0.28     0.00      0.00      0.00      -1.31
2    2     2      0.00      0.28     0.00      0.00      0.00       1.31
3      0.00     -0.28     0.00      0.00      0.00       1.42
3    2     3      0.00      0.28     0.00      0.00      0.00      -1.42
4      0.00     -0.28     0.00      0.00      0.00       4.15
4    2     4      0.00     -0.22     0.00      0.00      0.00      -4.15
5      0.00      0.22     0.00      0.00      0.00       2.07
5    2     5      0.00     -0.22     0.00      0.00      0.00      -2.07
6      0.00      0.22     0.00      0.00      0.00       0.00
************** END OF LATEST ANALYSIS RESULT **************
48. PRINT JOINT DISPLACEMENTS LIST 1 TO 6
JOINT    DISPLACE LIST     1
:A PROPPED CANTILEVER WITH COMPRESSION ONLY SUPPORT      -- PAGE NO.    8
JOINT DISPLACEMENT (INCH RADIANS)    STRUCTURE TYPE = PLANE
------------------
JOINT  LOAD   X-TRANS   Y-TRANS   Z-TRANS   X-ROTAN   Y-ROTAN   Z-ROTAN
1    2    0.00000   0.00000   0.00000   0.00000   0.00000   0.00000
2    2    0.00000  -0.01066   0.00000   0.00000   0.00000  -0.00190
3    2    0.00000  -0.03024   0.00000   0.00000   0.00000  -0.00186
4    2    0.00000  -0.04012   0.00000   0.00000   0.00000   0.00012
5    2    0.00000  -0.02714   0.00000   0.00000   0.00000   0.00233
6    2    0.00000   0.00000   0.00000   0.00000   0.00000   0.00307