# 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

## Theoretical Solution

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

$Δ max = 5 ( 300 ) ( 100 ) ( 300 ) 3 8 ( 210 ⋅ 10 3 ) ( 100 ⋅ 25 3 12 ) = 405 0 ( 10 ) 9 218.75 ( 10 ) 9 = 18.51 mm$

Maximum Moment:

$M max = W L 2 = 5 ( 300 ) ( 100 ) ( 300 ) 2 = 22.5 ( 10 ) 6 N⋅mm$

## Comparison

Table 1. Comparison of results
Result Type Hand Calculation STAAD.Pro Difference
δmax (mm) 18.51 18.159 1.9%
Mmax (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).

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
1 TO 12 PR 5
PERFORM ANALYSIS
PRINT JOINT DISPLACEMENTS LIST 14
UNIT METER KN
PRINT SUPPORT REACTION
FINISH


   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