# EX. US-18 Stress Calculation for Plate Elements

This is an example which demonstrates the calculation of principal stresses on a finite element.

This problem is installed with the program by default to C:\Users\Public\Public Documents\STAAD.Pro CONNECT Edition\Samples\Sample Models\US\US-18 Stress Calculation for Plate Elements.STD when you install the program.

### Example Problem No. 18

Where:

• L = 1 ft

Fixed Supports at Joints 1, 2, 3, 4, 5, 9, 13

Load intensity = 1 psi (pound/in2) in negative global Y direction

Actual input is shown in bold lettering followed by explanation.

    STAAD SPACE SAMPLE CALCULATION FOR
* ELEMENT STRESSES


Every input has to start with the term STAAD. The word SPACE signifies that the structure is a space frame (3-D structure).

    UNIT KIP FEET


Defines the input units for the data that follows.

    JOINT COORDINATES
1 0 0 0 4 3 0 0
REPEAT 3 0 0 1


Joint number followed by X, Y and Z coordinates are provided above. The REPEAT command is used to generate coordinates of joints 5 to 16 based on the pattern of joints 1 to 4.

    ELEMENT INCIDENCE
1 1 5 6 2 TO 3
REPEAT 2 3 4


Element connectivities of elements 1 to 3 are defined first, based on which, the connectivities of elements 4 to 9 are generated.

    UNIT INCH
ELEMENT PROPERTIES
1 TO 9 THICK 1.0


Elements 1 to 9 have a thickness of 1.0 in.

    DEFINE MATERIAL START
ISOTROPIC CONCRETE
E 3150
POISSON 0.17
DENSITY 8.7e-005
ALPHA 5e-006
DAMP 0.05
G 1346.15
TYPE CONCRETE
STRENGTH FCU 4
END DEFINE MATERIAL
CONSTANTS
MATERIAL CONCRETE ALL


The DEFINE MATERIAL command is used to specify material properties and the CONSTANT is used to assign the material to all members.

    SUPPORT
1 TO 4 5 9 13 FIXED


Fixed support conditions exist at the above mentioned joints.

    UNIT POUND
1 TO 9 PRESSURE -1.0


A uniform pressure of 1 pound/sq. in is applied on all the elements. In the absence of an explicit direction specification, the load is assumed to act along the local Z axis. The negative value indicates that the load acts opposite to the positive direction of the local Z.

    PERFORM ANALYSIS


The above command instructs the program to proceed with the analysis.

    PRINT SUPPORT REACTION


The above command is self-explanatory.

    PRINT ELEMENT STRESSES LIST 4


Element stresses at the centroid of the element are printed using the above command. The output includes membrane stresses, shear stresses, bending moments per unit width and principal stresses.

    FINISH


## Calculation of principal stresses for element 4

Calculations are presented for the top surface only.

SX = 0.0 lb/in2

SY = 0.0 lb/in2

SXY = 0.0 lb/in2

MX = 16.90 lb·in/in

MY = 85.81 lb·in/in

MXY = 36.43 lb·in/in

S = 1/6t2 = 1/6·12 = 0.1667 in3 (Section Modulus)

σx = SX + MX/S = 0.0 + 16.90/0.1667 = 101.38 psi

σy = SY + MY/S = 0.0 + 85.81/0.1667 = 514.75 psi

τxy = SXY + MXY/S = 0.0 + 36.43/0.1667 = 218.54 psi

$T M a x = ( σ x − σ y ) 2 4 + τ x y 2 = ( 101.38 − 514.75 ) 2 4 + 218.54 2 = 300.80 p s i$

SMax = (σx + σy)/2 + TMax = (101.38 + 514.75)/2 + 300.80 = 608.87 psi

SMin = (σx + σy)/2 - TMax = (101.38 + 514.75)/2 - 300.80 = 7.27 psi

$Angle = 1 2 tan − 1 ( 2 τ x y σ x − σ y ) ⁡ = 1 2 tan − 1 { 2 ( 218.54 ) 101.38 − 514.75 } = − 23.30 ° ⁡$
$VON = 0.707 ( S M a x − S M i n ) 2 + S M a x 2 + S M i n 2 = 0.707 ( 608.87 − 7.27 ) 2 + 608.87 2 + 7.27 2 = 605.18 p s i$

## Input File

STAAD SPACE SAMPLE CALCULATION FOR
* ELEMENT STRESSES
UNIT KIP FEET
JOINT COORDINATES
1 0 0 0 4 3 0 0
REPEAT 3 0 0 1
ELEMENT INCIDENCE
1 1 5 6 2 TO 3
REPEAT 2 3 4
UNIT INCH
ELEMENT PROPERTIES
1 TO 9 THICK 1.0
DEFINE MATERIAL START
ISOTROPIC CONCRETE
E 3150
POISSON 0.17
DENSITY 8.7e-005
ALPHA 5e-006
DAMP 0.05
G 1346.15
TYPE CONCRETE
STRENGTH FCU 4
END DEFINE MATERIAL
CONSTANTS
MATERIAL CONCRETE ALL
SUPPORT
1 TO 4 5 9 13 FIXED
UNIT POUND
1 TO 9 PRESSURE -1.0
PERFORM ANALYSIS
PRINT SUPPORT REACTION
PRINT ELEMENT STRESSES LIST 4
FINISH


                                                                  PAGE NO.    1
****************************************************
*                                                  *
*           Version  22.04.00.**                   *
*           Proprietary Program of                 *
*           Bentley Systems, Inc.                  *
*           Date=    APR 21, 2020                  *
*           Time=    15:41:57                      *
*                                                  *
*  Licensed to: Bentley Systems Inc                *
****************************************************
1. STAAD SPACE SAMPLE CALCULATION FOR
INPUT FILE: US-18 Stress Calculation for Plate Elements.STD
2. * ELEMENT STRESSES
3. UNIT KIP FEET
4. JOINT COORDINATES
5. 1 0 0 0 4 3 0 0
6. REPEAT 3 0 0 1
7. ELEMENT INCIDENCE
8. 1 1 5 6 2 TO 3
9. REPEAT 2 3 4
10. UNIT INCH
11. ELEMENT PROPERTIES
12. 1 TO 9 THICK 1.0
13. DEFINE MATERIAL START
14. ISOTROPIC CONCRETE
15. E 3150
16. POISSON 0.17
17. DENSITY 8.7E-005
18. ALPHA 5E-006
19. DAMP 0.05
20. G 1346.15
21. TYPE CONCRETE
22. STRENGTH FCU 4
23. END DEFINE MATERIAL
24. CONSTANTS
25. MATERIAL CONCRETE ALL
26. SUPPORT
27. 1 TO 4 5 9 13 FIXED
28. UNIT POUND
31. 1 TO 9 PRESSURE -1.0
32. PERFORM ANALYSIS
SAMPLE CALCULATION FOR                                   -- PAGE NO.    2
* ELEMENT STRESSES
P R O B L E M   S T A T I S T I C S
-----------------------------------
NUMBER OF JOINTS         16  NUMBER OF MEMBERS       0
NUMBER OF PLATES          9  NUMBER OF SOLIDS        0
NUMBER OF SURFACES        0  NUMBER OF SUPPORTS      7
Using 64-bit analysis engine.
SOLVER USED IS THE IN-CORE ADVANCED MATH SOLVER
TOTAL      PRIMARY LOAD CASES =     1, TOTAL DEGREES OF FREEDOM =      54
TOTAL LOAD COMBINATION  CASES =     0  SO FAR.
33. PRINT SUPPORT REACTION
SUPPORT  REACTION
SAMPLE CALCULATION FOR                                   -- PAGE NO.    3
* ELEMENT STRESSES
SUPPORT REACTIONS -UNIT POUN INCH    STRUCTURE TYPE = SPACE
-----------------
JOINT  LOAD   FORCE-X   FORCE-Y   FORCE-Z     MOM-X     MOM-Y     MOM Z
1    1      0.00     -9.76      0.00    -12.51      0.00     12.51
2    1      0.00     70.14      0.00   -853.81      0.00    -16.81
3    1      0.00    301.41      0.00  -2821.43      0.00     95.50
4    1      0.00    281.33      0.00  -2127.22      0.00   -769.10
5    1      0.00     70.14      0.00     16.81      0.00    853.81
9    1      0.00    301.41      0.00    -95.50      0.00   2821.43
13    1      0.00    281.33      0.00    769.10      0.00   2127.22
************** END OF LATEST ANALYSIS RESULT **************
34. PRINT ELEMENT STRESSES LIST 4
ELEMENT  STRESSES LIST     4
SAMPLE CALCULATION FOR                                   -- PAGE NO.    4
* ELEMENT STRESSES
ELEMENT STRESSES    FORCE,LENGTH UNITS= POUN INCH
----------------
STRESS = FORCE/UNIT WIDTH/THICK, MOMENT = FORCE-LENGTH/UNIT WIDTH
ELEMENT  LOAD       SQX        SQY          MX          MY          MXY
VONT       VONB         SX          SY          SXY
TRESCAT    TRESCAB
4      1        10.43       -9.94       16.90       85.81       36.43
605.39      605.39        0.00        0.00        0.00
608.99      608.99
TOP : SMAX=     608.99 SMIN=       7.25 TMAX=     300.87 ANGLE= 66.7
BOTT: SMAX=      -7.25 SMIN=    -608.99 TMAX=     300.87 ANGLE=-23.3
**** MAXIMUM STRESSES AMONG SELECTED PLATES AND CASES ****
MAXIMUM       MINIMUM       MAXIMUM       MAXIMUM       MAXIMUM
PRINCIPAL     PRINCIPAL       SHEAR       VONMISES       TRESCA
STRESS        STRESS        STRESS        STRESS        STRESS
6.089873E+02 -6.089873E+02  3.008683E+02  6.053945E+02  6.089873E+02
PLATE NO.       4             4             4             4             4
CASE  NO.       1             1             1             1             1
********************END OF ELEMENT FORCES********************
35. FINISH
SAMPLE CALCULATION FOR                                   -- PAGE NO.    5
* ELEMENT STRESSES
*********** END OF THE STAAD.Pro RUN ***********
**** DATE= APR 21,2020   TIME= 15:41:57 ****
************************************************************