V. 2D Cantilever Beam End Deflection 2

Find the deflection and moments for plate-bending finite element due to a pressure load.

Reference

Simple hand calculation by considering the entire structure as a cantilever beam.

Problem

A simple cantilever plate is divided into 12 4-noded finite elements. A uniform pressure load is applied and the maximum deflection at the tip of the cantilever and the maximum bending at the support are calculated.

Plate thickness = 25 mm

Uniform pressure= 5 N/mm²

Plate length = 6 spaces at 50 mm = 300 mm

Plate width = 2 space at 50 mm = 100 mm

Finite element mesh of cantilevered plate.

Theoretical Solution

Maximum deflection is equal to WL³/8EI, where:

Δ_{max} = \frac{5 (300) (100) {(300)}^{3}}{8 (210 \cdot 10^{3}) (\frac{100 \cdot 25^{3}}{12})} = \frac{405 {0 (10)}^{9}}{218.75 {(10)}^{9}} = 18.51 mm

Maximum Moment:

M_{max} = \frac{W L}{2} = \frac{5 (300) (100) (300)}{2} = 22.5 {(10)}^{6} N⋅mm

Comparison

Table 1. Comparison of results
Result Type	Hand Calculation	STAAD.Pro	Difference
δ_max (mm)	18.51	18.159	1.9%
M_max (kN·m)	22.50	22.50	none

Note: The maximum moment is taken as the sum of the moments at nodes 1, 8, and 15 (i.e., 5.47 + 11.56 + 5.47 = 22.5 kN·m).

STAAD Input

The file C:\Users\Public\Public Documents\STAAD.Pro CONNECT Edition\Samples\ Verification Models\04 Plates Shells\2D Cantilever Beam End Deflection 2.STD is typically installed with the program.

STAAD SPACE : DEFLECTION OF A CANTILEVER PLATE UNDER UNIFORM PRESSURE
START JOB INFORMATION
ENGINEER DATE 14-Sep-18
END JOB INFORMATION
*
* DEFLECTION OF A CANTILEVER PLATE UNDER UNIFORM PRESSURE.
* COMPARISON WITH ESTABLISHED FORMULA (WL^3/8EI)
*
UNIT MMS KN
JOINT COORDINATES
1 0 0 0; 2 50 0 0; 3 100 0 0; 4 150 0 0; 5 200 0 0; 6 250 0 0;
7 300 0 0; 8 0 50 0; 9 50 50 0; 10 100 50 0; 11 150 50 0; 12 200 50 0;
13 250 50 0; 14 300 50 0; 15 0 100 0; 16 50 100 0; 17 100 100 0;
18 150 100 0; 19 200 100 0; 20 250 100 0; 21 300 100 0;
ELEMENT INCIDENCES SHELL
1 1 2 9 8; 2 2 3 10 9; 3 3 4 11 10; 4 4 5 12 11; 5 5 6 13 12;
6 6 7 14 13; 7 8 9 16 15; 8 9 10 17 16; 9 10 11 18 17; 10 11 12 19 18;
11 12 13 20 19; 12 13 14 21 20;
ELEMENT PROPERTY
1 TO 12 THICKNESS 25
UNIT METER KN
DEFINE MATERIAL START
ISOTROPIC MATERIAL1
E 2.1e+08
POISSON 0.3
END DEFINE MATERIAL
UNIT MMS KN
CONSTANTS
MATERIAL MATERIAL1 ALL
SUPPORTS
1 8 15 FIXED
UNIT MMS NEWTON
LOAD 1 5N/SQ.MM. UNIFORM LOAD
ELEMENT LOAD
1 TO 12 PR 5
PERFORM ANALYSIS
PRINT JOINT DISPLACEMENTS LIST 14
UNIT METER KN
PRINT SUPPORT REACTION
FINISH

STAAD Output

   JOINT DISPLACEMENT (CM   RADIANS)    STRUCTURE TYPE = SPACE
   ------------------
 JOINT  LOAD   X-TRANS   Y-TRANS   Z-TRANS   X-ROTAN   Y-ROTAN   Z-ROTAN
     14    1     0.0000    0.0000    1.8159    0.0000   -0.0813    0.0000
   ************** END OF LATEST ANALYSIS RESULT **************
    38. UNIT METER KN
    39. PRINT SUPPORT REACTION
  SUPPORT  REACTION                   
      : DEFLECTION OF A CANTILEVER PLATE UNDER UNIFORM PRESSUR -- PAGE NO.    4
   SUPPORT REACTIONS -UNIT KN   METE    STRUCTURE TYPE = SPACE
   -----------------
 JOINT  LOAD   FORCE-X   FORCE-Y   FORCE-Z     MOM-X     MOM-Y     MOM Z
      1    1      0.00      0.00    -18.91     -1.54      5.47      0.00
      8    1      0.00      0.00   -112.19      0.00     11.56      0.00
     15    1      0.00      0.00    -18.91      1.54      5.47      0.00