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:

P = +0.5 lbs (up), and
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.

Propped cantilever member with a compression only support

Theoretical Solution

Load Case 1

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) = \frac{P x^{2}}{6 E I} (3 x - b)

= \frac{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 .

Load Case 2

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} = \frac{P b^{2} (a + 2 ℓ)}{2 ℓ^{3}} = \frac{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

Deflection at point of load:

Δ (x < b) = \frac{- P a x^{2}}{12 E I ℓ^{3}} (3 ℓ^{3} - 3 ℓ^{2} x - 3 a^{2} ℓ + a^{2} x)

= \frac{- 0.5 (19.2) {(28.8)}^{3}}{12 (10, 000, 000) (0.00135) {(48)}^{3}} [3 {(48)}^{3} - 3 {(48)}^{2} (28.8) - 3 {(19.2)}^{2} (48) + {(19.2)}^{2} (28.8)] = - 0.040 in.

Comparison

Table 1. Comparison of results
Load Case	Result Type	Theory	STAAD.Pro	Difference
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

STAAD Input

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
LOAD 1 UPWARD
JOINT LOAD
4 FY 0.5
PERFORM ANALYSIS
PRINT MEMBER FORCES LIST 1 TO 5
PRINT JOINT DISPLACEMENTS LIST 1 TO 6
PRINT SUPPORT REACTION
CHANGE
LOAD 2 DOWNWARD
JOINT LOAD
4 FY -0.5
PERFORM ANALYSIS
PRINT SUPPORT REACTION
PRINT MEMBER FORCES LIST 1 TO 5
PRINT JOINT DISPLACEMENTS LIST 1 TO 6
FINISH

STAAD Output

   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
    42. LOAD 2 DOWNWARD
    43. JOINT LOAD
    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