2. STAAD Verification Problems
Release STAAD.Pro 20.07.10.41
Date last updated: 13 December 2013
Quality Assurance Program
STAAD.Pro V8i (SELECTseries 5) is a suite of proprietary computer programs of Bentley
Systems, Inc.. Although every effort has been made to ensure the correctness of these programs,
Bentley Systems, Inc. will not accept responsibility for any mistake, error, or misrepresentation
in or as a result of the usage of these programs.
Notes on Comparisons
In each verification problem, a table is used to summarize the numerical outputs from
STAAD.Pro with a reference or hand calculations. A difference column presents the percent
deviation from the reference, when a significant difference exists.
The deviation is taken as the reference value minus the value determined by STAAD.Pro, divided
by the reference value to express in percent of the reference.
In this context, "significant" is take as larger than a 1% (one percent) difference. When a
difference between 0.5% and 1% exists, then the difference is simply listed as "<1%". Otherwise,
the difference is listed as "none".
This level of significance is intended to reflect that small differences in rounding or calculation
methods do not present a realistic difference in the calculated result.
Verification Manual — 2
9. 1
Static Trusses
Static Truss 1 1
Static Truss 2 4
Static Truss 3 8
Static Truss 4 12
Static Truss 5 16
Static Truss 6 19
Static Truss 7 31
Static Truss 1
Objective
To find member stress due to a joint load in a space truss using static analysis.
Reference
Beer, F.P., and Johnston, Jr., E.R., Vector Mechanics for Engineers, Statics and Dynamics, McGraw -
Hill, Inc., New York, 1962, p.47, Problem 2.70.
Problem
A 50 lb load is supported by three bars which are pinned to a ceiling as shown. Determine the
stress, σ, in each bar.
Area of each bar = 1 in2, E = 30 (10)6 psi
Verification Manual — 1
10. Figure 1-1: Space truss
Comparison
Result Type Theory STAAD.Pro Difference
σAD
31.2 31.2 none
σBD
10.4 10.4 none
σCD
22.9 22.9 none
Table 1-1: Comparison of member stress, in psi, for static truss no. 1
STAAD Input
STAAD TRUSS :A SPACE TRUSS
* FILE TRUSS01.STD
*
* REF: "VECTOR MECHANICS FOR ENGINEERS, STATICS & DYNAMICS",
* MCGRAW HILL BOOK CO., INC., NEW YORK, 1962
* PROBLEM 2.70, PAGE 47.
*
UNIT FT POUND
JOINT COORDINATES
1 0. 0. 0.
2 8. 0. 0.
3 0. 6. 0.
4 4. 2. 6.
MEMBER INCIDENCES
1 1 4
2 2 4
3 3 4
UNIT INCH
MEMBER PROPERTIES
1 TO 3 PRIS AX 1.
CONSTANTS
E 30E6 ALL
SUPPORTS
1 TO 3 PINNED
LOAD 1 WEIGHT
JOINT LOAD
4 FZ 50.
PERFORM ANALYSIS
PRINT MEMBER STRESSES
FINISH
2 — STAAD.Pro
1 Static Trusses
Static Truss 1
11. STAAD Output
PAGE NO. 1
****************************************************
* *
* STAAD.Pro V8i SELECTseries5 *
* Version 20.07.10.41 *
* Proprietary Program of *
* Bentley Systems, Inc. *
* Date= DEC 13, 2013 *
* Time= 12:30: 8 *
* *
* USER ID: Bentley Systems, Inc. *
****************************************************
1. STAAD TRUSS :A SPACE TRUSS
INPUT FILE: TRUSS01.STD
2. * FILE TRUSS01.STD
3. *
4. * REF: "VECTOR MECHANICS FOR ENGINEERS, STATICS & DYNAMICS",
5. * MCGRAW HILL BOOK CO., INC., NEW YORK, 1962
6. * PROBLEM 2.70, PAGE 47.
7. *
8. UNIT FT POUND
9. JOINT COORDINATES
10. 1 0. 0. 0.
11. 2 8. 0. 0.
12. 3 0. 6. 0.
13. 4 4. 2. 6.
14. MEMBER INCIDENCES
15. 1 1 4
16. 2 2 4
17. 3 3 4
18. UNIT INCH
19. MEMBER PROPERTIES
20. 1 TO 3 PRIS AX 1.
21. CONSTANTS
22. E 30E6 ALL
23. SUPPORTS
24. 1 TO 3 PINNED
25. LOAD 1 WEIGHT
26. JOINT LOAD
27. 4 FZ 50.
28. PERFORM ANALYSIS
**WARNING** THE POISSON'S RATIO HAS NOT BEEN SPECIFIED FOR ONE OR MORE
MEMBERS/ELEMENTS/SOLIDS. THE DEFAULT VALUE HAS BEEN SET FOR THE SAME.
:A SPACE TRUSS -- PAGE NO. 2
* FILE TRUSS01.STD
P R O B L E M S T A T I S T I C S
-----------------------------------
NUMBER OF JOINTS 4 NUMBER OF MEMBERS 3
NUMBER OF PLATES 0 NUMBER OF SOLIDS 0
NUMBER OF SURFACES 0 NUMBER OF SUPPORTS 3
SOLVER USED IS THE IN-CORE ADVANCED SOLVER
TOTAL PRIMARY LOAD CASES = 1, TOTAL DEGREES OF FREEDOM = 3
29. PRINT MEMBER STRESSES
MEMBER STRESSES
:A SPACE TRUSS -- PAGE NO. 3
1 Static Trusses
Verification Manual — 3
16. Static Truss 3
Objective
To find member forces due to joint loads in a plane truss.
Reference
Norris C.H., Wilbur J. B., Elementary Structural Analysis, 2nd Edition, McGraw – Hill, Inc., Page
159, Problem 4.3. (Original data is in US Customary Units)
Problem
Compute the bar forces in the bars a, b, c, d, e of the truss due to the loads shown.
Figure 1-3: Plane truss
Comparison
Result Type Theory STAAD.Pro Difference
a -202.13 -202.13 none
b -185.76 -190.28 2.4% (negligible)
c -22.42 -23.42 4.5%
d 266.23 269.52 1.2% (negligible)
e 119.61 117.08 2.1% (negligible)
Table 1-3: Comparison of member force, in kN, for static truss no. 3
8 — STAAD.Pro
1 Static Trusses
Static Truss 3
19. 51. JOINT LOADS
52. 2 9 FY -4.448222E4
53. 4 7 FY -8.896444E4
54. 12 FY -2.224111E4
55. *
56. PERFORM ANALYSIS
**WARNING** THE POISSON'S RATIO HAS NOT BEEN SPECIFIED FOR ONE OR MORE
MEMBERS/ELEMENTS/SOLIDS. THE DEFAULT VALUE HAS BEEN SET FOR THE SAME.
P R O B L E M S T A T I S T I C S
-----------------------------------
NUMBER OF JOINTS 14 NUMBER OF MEMBERS 25
NUMBER OF PLATES 0 NUMBER OF SOLIDS 0
NUMBER OF SURFACES 0 NUMBER OF SUPPORTS 2
SOLVER USED IS THE IN-CORE ADVANCED SOLVER
TOTAL PRIMARY LOAD CASES = 1, TOTAL DEGREES OF FREEDOM = 38
*WARNING- ZERO STIFFNESS IN DIRECTION 6 AT JOINT 1 EQN.NO. 1
LOADS APPLIED OR DISTRIBUTED HERE FROM ELEMENTS WILL BE IGNORED.
THIS MAY BE DUE TO ALL MEMBERS AT THIS JOINT BEING RELEASED OR
EFFECTIVELY RELEASED IN THIS DIRECTION.
*WARNING- ZERO STIFFNESS IN DIRECTION 6 AT JOINT 2 EQN.NO. 4
*WARNING- ZERO STIFFNESS IN DIRECTION 6 AT JOINT 3 EQN.NO. 7
*WARNING- ZERO STIFFNESS IN DIRECTION 6 AT JOINT 4 EQN.NO. 10
*WARNING- ZERO STIFFNESS IN DIRECTION 6 AT JOINT 5 EQN.NO. 13
*WARNING- ZERO STIFFNESS IN DIRECTION 6 AT JOINT 6 EQN.NO. 16
*WARNING- ZERO STIFFNESS IN DIRECTION 6 AT JOINT 7 EQN.NO. 19
*WARNING- ZERO STIFFNESS IN DIRECTION 6 AT JOINT 8 EQN.NO. 22
*WARNING- ZERO STIFFNESS IN DIRECTION 6 AT JOINT 9 EQN.NO. 25
*WARNING- ZERO STIFFNESS IN DIRECTION 6 AT JOINT 10 EQN.NO. 28
*WARNING- ZERO STIFFNESS IN DIRECTION 6 AT JOINT 11 EQN.NO. 31
:A PLANE TRUSS -- PAGE NO. 3
* FILE TRUSS03.STD
*WARNING- ZERO STIFFNESS IN DIRECTION 6 AT JOINT 12 EQN.NO. 34
**WARNING - THERE WERE MORE THAN 12 DOF WITH ZERO STIFFNESS.
THE FIRST 12 ARE LISTED ABOVE.
TOTAL # TRANSLATIONAL= 0 TOTAL # ROTATIONAL= 13
57. UNITS KNS
58. PRINT MEMBER FORCES LIST 2 5 6 10 12
MEMBER FORCES LIST 2
:A PLANE TRUSS -- PAGE NO. 4
* FILE TRUSS03.STD
MEMBER END FORCES STRUCTURE TYPE = PLANE
-----------------
ALL UNITS ARE -- KNS METE (LOCAL )
MEMBER LOAD JT AXIAL SHEAR-Y SHEAR-Z TORSION MOM-Y MOM-Z
2 1 1 190.28 0.00 0.00 0.00 0.00 0.00
5 -190.28 0.00 0.00 0.00 0.00 0.00
5 1 2 -269.52 0.00 0.00 0.00 0.00 0.00
4 269.52 0.00 0.00 0.00 0.00 0.00
6 1 5 -117.08 0.00 0.00 0.00 0.00 0.00
3 117.08 0.00 0.00 0.00 0.00 0.00
10 1 6 202.12 0.00 0.00 0.00 0.00 0.00
8 -202.12 0.00 0.00 0.00 0.00 0.00
12 1 7 23.42 0.00 0.00 0.00 0.00 0.00
10 -23.42 0.00 0.00 0.00 0.00 0.00
************** END OF LATEST ANALYSIS RESULT **************
59. FINISH
*********** END OF THE STAAD.Pro RUN ***********
**** DATE= DEC 13,2013 TIME= 12:30:17 ****
1 Static Trusses
Verification Manual — 11
21. Figure 1-4: Space truss
Comparison
Result Type Theory STAAD.Pro Difference
Bv
0 0 none
Bz
2,700 2,700 none
Cx
1,800 1,800 none
Cv
3,375 3,375 none
Dx
1,800 1,800 none
Dv
3,375 3,375 none
Table 1-4: Comparison of support reactions, in N, for static truss no. 4
Result Type Theory STAAD.Pro Difference
AB 0 0 none
AC 4,275 4,275 none
Table 1-5: Comparison of member forces, in N, for static truss no. 4
1 Static Trusses
Verification Manual — 13
22. Result Type Theory STAAD.Pro Difference
AD 4,275 4,275 none
BC 4,270 4,269 negligible
CD 0 0 none
BD 4,270 4,269 negligible
STAAD Input
STAAD SPACE :A SPACE TRUSS
* FILE: TRUSS04.STD
*
* REFERENCE: VECTOR MECHANICS FOR ENGINEERS - STATICS, BEER AND
* JOHNSTON, 4TH EDITION, MCGRAW-HILL BOOK CO.,
* PAGE 216, PROBLEM 6.20.
*
UNITS NEWTON METER
JOINT COORDINATES
1 1.0 0.0 0.6
2 1.0 0.0 -0.6
3 -0.8 0.0 0.0
4 0.0 1.5 0.0
*
MEMBER INCIDENCES
1 1 4
2 2 4
3 3 4
4 1 2 5
6 3 1
MEMBER PROPERTIES
1 TO 6 PRI AX 0.001
CONSTANTS
E 200.0E9 ALL
MEMBER TRUSS
1 TO 6
*
SUPPORTS
1 FIXED BUT FZ
2 FIXED BUT FZ
3 FIXED BUT FX
*
LOADING 1 HORIZONTAL LOAD
JOINT LOAD
4 FZ 2700.0
PERFORM ANALYSIS
PRINT SUPPORT REACTIONS
PRINT MEMBER FORCES
FINISH
STAAD Output
PAGE NO. 1
****************************************************
* *
* STAAD.Pro V8i SELECTseries5 *
* Version 20.07.10.41 *
* Proprietary Program of *
* Bentley Systems, Inc. *
* Date= DEC 13, 2013 *
* Time= 12:30:18 *
14 — STAAD.Pro
1 Static Trusses
Static Truss 4
23. * *
* USER ID: Bentley Systems, Inc. *
****************************************************
1. STAAD SPACE :A SPACE TRUSS
INPUT FILE: TRUSS04.STD
2. * FILE: TRUSS04.STD
3. *
4. * REFERENCE: VECTOR MECHANICS FOR ENGINEERS - STATICS, BEER AND
5. * JOHNSTON, 4TH EDITION, MCGRAW-HILL BOOK CO.,
6. * PAGE 216, PROBLEM 6.20.
7. *
8. UNITS NEWTON METER
9. JOINT COORDINATES
10. 1 1.0 0.0 0.6
11. 2 1.0 0.0 -0.6
12. 3 -0.8 0.0 0.0
13. 4 0.0 1.5 0.0
14. *
15. MEMBER INCIDENCES
16. 1 1 4
17. 2 2 4
18. 3 3 4
19. 4 1 2 5
20. 6 3 1
21. MEMBER PROPERTIES
22. 1 TO 6 PRI AX 0.001
23. CONSTANTS
24. E 200.0E9 ALL
25. MEMBER TRUSS
26. 1 TO 6
27. *
28. SUPPORTS
29. 1 FIXED BUT FZ
30. 2 FIXED BUT FZ
31. 3 FIXED BUT FX
32. *
33. LOADING 1 HORIZONTAL LOAD
34. JOINT LOAD
35. 4 FZ 2700.0
36. PERFORM ANALYSIS
:A SPACE TRUSS -- PAGE NO. 2
* FILE: TRUSS04.STD
**WARNING** THE POISSON'S RATIO HAS NOT BEEN SPECIFIED FOR ONE OR MORE
MEMBERS/ELEMENTS/SOLIDS. THE DEFAULT VALUE HAS BEEN SET FOR THE SAME.
P R O B L E M S T A T I S T I C S
-----------------------------------
NUMBER OF JOINTS 4 NUMBER OF MEMBERS 6
NUMBER OF PLATES 0 NUMBER OF SOLIDS 0
NUMBER OF SURFACES 0 NUMBER OF SUPPORTS 3
SOLVER USED IS THE IN-CORE ADVANCED SOLVER
TOTAL PRIMARY LOAD CASES = 1, TOTAL DEGREES OF FREEDOM = 9
*WARNING- ZERO STIFFNESS IN DIRECTION 4 AT JOINT 4 EQN.NO. 7
LOADS APPLIED OR DISTRIBUTED HERE FROM ELEMENTS WILL BE IGNORED.
THIS MAY BE DUE TO ALL MEMBERS AT THIS JOINT BEING RELEASED OR
EFFECTIVELY RELEASED IN THIS DIRECTION.
*WARNING- ZERO STIFFNESS IN DIRECTION 5 AT JOINT 4 EQN.NO. 8
*WARNING- ZERO STIFFNESS IN DIRECTION 6 AT JOINT 4 EQN.NO. 9
37. PRINT SUPPORT REACTIONS
SUPPORT REACTION
1 Static Trusses
Verification Manual — 15
37. L = 180 in.
angle = 30°
Figure 1-9: Model of two member truss
Comparison
Result Type Theory
STAAD
Basic Solver
STAAD
Advanced Solver
σA
(psi) 10,000. 10,000. 10,000.
δA
(in) 0.12 0.12 0.12
Table 1-10: Comparison of stress (σ) and Deflection (δ) for verification
problem 12
STAAD Input
STAAD TRUSS VERIFICATION PROBLEM NO 12
*
* THIS EXAMPLE IS TAKEN FROM 'STRENGTH OF MATERIALS'
* (PART 1) BY TIMOSHENKO, PAGE 10 PROB 2.
* THE ANSWER IN THE BOOK , DEFLECTION = 0.12 INCH
* AND STRESS =10000 PSI
*
UNIT INCH POUND
JOINT COORD
1 0. 0. ; 2 155.88457 -90. ; 3 311.76914 0.
MEMB INCI ; 1 1 2 2
MEMB PROP
1 2 PRIS AX 0.5
CONSTANT
E 30E6
POISSON 0.15 ALL
SUPPORT ; 1 3 PINNED
LOAD 1 VERT LOAD
JOINT LOAD ; 2 FY -5000.
PERFORM ANALYSIS
PRINT DISPLACEMENTS
PRINT STRESSES
FINISH
1 Static Trusses
Verification Manual — 29
38. STAAD Output
PAGE NO. 1
****************************************************
* *
* STAAD.Pro V8i SELECTseries5 *
* Version 20.07.10.41 *
* Proprietary Program of *
* Bentley Systems, Inc. *
* Date= DEC 13, 2013 *
* Time= 12:31: 4 *
* *
* USER ID: Bentley Systems, Inc. *
****************************************************
1. STAAD TRUSS VERIFICATION PROBLEM NO 12
INPUT FILE: VER12.STD
2. *
3. * THIS EXAMPLE IS TAKEN FROM 'STRENGTH OF MATERIALS'
4. * (PART 1) BY TIMOSHENKO, PAGE 10 PROB 2.
5. * THE ANSWER IN THE BOOK , DEFLECTION = 0.12 INCH
6. * AND STRESS =10000 PSI
7. *
8. UNIT INCH POUND
9. JOINT COORD
10. 1 0. 0. ; 2 155.88457 -90. ; 3 311.76914 0.
11. MEMB INCI ; 1 1 2 2
12. MEMB PROP
13. 1 2 PRIS AX 0.5
14. CONSTANT
15. E 30E6
16. POISSON 0.15 ALL
17. SUPPORT ; 1 3 PINNED
18. LOAD 1 VERT LOAD
19. JOINT LOAD ; 2 FY -5000.
20. PERFORM ANALYSIS
P R O B L E M S T A T I S T I C S
-----------------------------------
NUMBER OF JOINTS 3 NUMBER OF MEMBERS 2
NUMBER OF PLATES 0 NUMBER OF SOLIDS 0
NUMBER OF SURFACES 0 NUMBER OF SUPPORTS 2
SOLVER USED IS THE IN-CORE ADVANCED SOLVER
TOTAL PRIMARY LOAD CASES = 1, TOTAL DEGREES OF FREEDOM = 2
VERIFICATION PROBLEM NO 12 -- PAGE NO. 2
*
21. PRINT DISPLACEMENTS
DISPLACE
VERIFICATION PROBLEM NO 12 -- PAGE NO. 3
*
JOINT DISPLACEMENT (INCH RADIANS) STRUCTURE TYPE = TRUSS
------------------
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 -0.12000 0.00000 0.00000 0.00000 0.00000
3 1 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000
************** END OF LATEST ANALYSIS RESULT **************
22. PRINT STRESSES
STRESSES
30 — STAAD.Pro
1 Static Trusses
Static Truss 6
40. Figure 1-10: Plane truss subject to differential thermal loading
Comparison
Result Type Theory STAAD.Pro Difference
F1
22,100 22,143 0.2% (negligible)
F2
-31,300 -31,315 0.1% (negligible)
Table 1-11: Comparison of member forces, in N, for static truss no. 7
STAAD Input
STAAD TRUSS :A PLANE TRUSS
* FILE: TRUSS10.STD
* REFERENCE: MECHANICS OF MATERIALS, GERE AND TIMOSHENKO, 2ND EDITION
* PWS ENGINEERING, PAGE 121, PROBLEM 2.6-23
*
UNITS NEWTON METER
JOINT COORDINATES
1 -2.12132 2.12132
2 0.0 2.12132
3 2.12132 2.12132
4 0.0 0.0
*
MEMBER INCIDENCES
1 1 4
2 2 4
3 3 4
UNIT NEWTON MMS
MEMBER PROPERTIES
1 TO 3 PRI AX 900
UNIT NEWTON METER
CONSTANTS
E 200.0E9 ALL
ALPHA 14.0E-6 ALL
32 — STAAD.Pro
1 Static Trusses
Static Truss 7
41. *
SUPPORTS
1 2 3 PINNED
*
LOADING 1 TEMPERATURE LOAD
TEMPERATURE LOAD
1 3 TEMP 20 800
2 TEMP 70
PERFORM ANALYSIS
PRINT MEMBER FORCES
FINISH
STAAD Output
PAGE NO. 1
****************************************************
* *
* STAAD.Pro V8i SELECTseries5 *
* Version 20.07.10.41 *
* Proprietary Program of *
* Bentley Systems, Inc. *
* Date= DEC 13, 2013 *
* Time= 12:30:27 *
* *
* USER ID: Bentley Systems, Inc. *
****************************************************
1. STAAD TRUSS :A PLANE TRUSS
INPUT FILE: TRUSS07.STD
2. * FILE: TRUSS10.STD
3. * REFERENCE: MECHANICS OF MATERIALS, GERE AND TIMOSHENKO, 2ND EDITION
4. * PWS ENGINEERING, PAGE 121, PROBLEM 2.6-23
5. *
6. UNITS NEWTON METER
7. JOINT COORDINATES
8. 1 -2.12132 2.12132
9. 2 0.0 2.12132
10. 3 2.12132 2.12132
11. 4 0.0 0.0
12. *
13. MEMBER INCIDENCES
14. 1 1 4
15. 2 2 4
16. 3 3 4
17. UNIT NEWTON MMS
18. MEMBER PROPERTIES
19. 1 TO 3 PRI AX 900
20. UNIT NEWTON METER
21. CONSTANTS
22. E 200.0E9 ALL
23. ALPHA 14.0E-6 ALL
24. *
25. SUPPORTS
26. 1 2 3 PINNED
27. *
28. LOADING 1 TEMPERATURE LOAD
29. TEMPERATURE LOAD
30. 1 3 TEMP 20 800
31. 2 TEMP 70
32. PERFORM ANALYSIS
:A PLANE TRUSS -- PAGE NO. 2
* FILE: TRUSS10.STD
**WARNING** THE POISSON'S RATIO HAS NOT BEEN SPECIFIED FOR ONE OR MORE
MEMBERS/ELEMENTS/SOLIDS. THE DEFAULT VALUE HAS BEEN SET FOR THE SAME.
1 Static Trusses
Verification Manual — 33
43. 2
Static Beams
Static Beam 1 35
Static Beam 2 39
Static Beam 3 43
Static Beam 4 47
Static Beam 5 53
Static Beam 6 56
Static Beam 7 59
Static Beam 8 62
Static Beam 9 65
Static Beam 11 72
Static Beam 12 75
Static Beam 13 78
Static Beam 15 87
Static Beam 1
Objective
To find the deflection and support reactions due to a trapezoidally varying load applied on part
of the span of a pinned-fixed beam.
Reference
Hand calculation using the following reference:
Roark's Formulas for Stress and Strain, Warren C. Young, 6th edition, McGraw-Hill, Table 3, Case
(2c), p.103
Problem
The beam in the following geometric, load, and section properties: a = 3 m, b =4.5 m, wa
= 4
KN/m, wl
= 7 KN/m, IZ
=5,000 cm4, E = 200 KN/mm2.
Verification Manual — 35
44. Figure 2-1: Simple supported beam with partial, trapezoidal load
Comparison
Result Type Theory STAAD.Pro Difference
OA
(radians) 0.0020 0.0020 none
RA
(kN) 3.2886 3.2886 none
RB
(kN) 21.461 21.461 none
MB
(kN·m) 25.9605 25.9605 none
Table 2-1: Comparison of results for static beam no. 1
Where
RA
= Vertical reaction at A
OA
= Rotation at A
RB
= Vertical reaction at B
MB
= Moment at B
Note: In the STAAD model, two load cases are used. In case 1, the load is applied using the
"MEMBER LOAD - TRAP" option. In case 2, the load is applied using a combination of
"MEMBER LOAD - UNI" and "MEMBER LOAD - LIN" options. Both cases yield identical results.
Theoretical Solution
( ) ( )
R l a l a l a l a
= ( − ) 3 + + ( − ) 4 +
A
W
l
W W
l
8
3 −
40
3
a l a
3 3
( ) ( )
O l a l a l a l a
= ( − ) + 3 − ( − ) 2 + 3
A
W
EIl
W W
EIl
−
48
3 −
240
3
a l a
( )
R l a R
= − −
B
W W
A
−
2
a l
M R l l a l a
= − ( − ) − ( − )
B A
W W W
2
2 −
6
2
a l a
Where
l = a + b
36 — STAAD.Pro
2 Static Beams
Static Beam 1
45. STAAD Input
STAAD PLANE REACTIONS AND DISPLACEMENTS OF A PINNED-FIXED BEAM
*
* FILE : BEAM01.STD
*
* REFERENCE : ROARK'S FORMULAS FOR STRESS & STRAIN
* WARREN C. YOUNG, 6TH EDITION, MCGRAW-HILL
*
* TABLE 3, CASE (2C), LOAD ON PARTIAL SPAN
*
UNIT KNS METER
JOINT COORD
1 0 0 0 ; 2 3 0 0 ; 3 7.5 0 0
*
MEMBER INCI
1 1 2 2
*
UNIT CMS
MEMBER PROP
1 2 PRIS AX 50 IZ 5000
*
UNIT METER
CONSTANT
E 200E6 ALL
POISS STEEL ALL
*
SUPPORT
1 PINNED
3 FIXED
*
LOAD 1
MEMBER LOAD
2 TRAP GY -4.0 -7.0
*
LOAD 2
MEMBER LOAD
2 UNI GY -4
2 LIN GY 0 -3
*
PERFORM ANALYSIS
*
PRINT JOINT DISP
*
UNIT NEWTON
PRINT SUPP REAC
FINISH
STAAD Output
PAGE NO. 1
****************************************************
* *
* STAAD.Pro V8i SELECTseries5 *
* Version 20.07.10.41 *
* Proprietary Program of *
* Bentley Systems, Inc. *
* Date= DEC 13, 2013 *
* Time= 12:27:32 *
* *
* USER ID: Bentley Systems, Inc. *
****************************************************
1. STAAD PLANE REACTIONS AND DISPLACEMENTS OF A PINNED-FIXED BEAM
2 Static Beams
Verification Manual — 37
46. INPUT FILE: BEAM01.STD
2. *
3. * FILE : BEAM01.STD
4. *
5. * REFERENCE : ROARK'S FORMULAS FOR STRESS & STRAIN
6. * WARREN C. YOUNG, 6TH EDITION, MCGRAW-HILL
7. *
8. * TABLE 3, CASE (2C), LOAD ON PARTIAL SPAN
9. *
10. UNIT KNS METER
11. JOINT COORD
12. 1 0 0 0 ; 2 3 0 0 ; 3 7.5 0 0
13. *
14. MEMBER INCI
15. 1 1 2 2
16. *
17. UNIT CMS
18. MEMBER PROP
19. 1 2 PRIS AX 50 IZ 5000
20. *
21. UNIT METER
22. CONSTANT
23. E 200E6 ALL
24. POISS STEEL ALL
25. *
26. SUPPORT
27. 1 PINNED
28. 3 FIXED
29. *
30. LOAD 1
31. MEMBER LOAD
32. 2 TRAP GY -4.0 -7.0
33. *
34. LOAD 2
35. MEMBER LOAD
36. 2 UNI GY -4
37. 2 LIN GY 0 -3
38. *
REACTIONS AND DISPLACEMENTS OF A PINNED-FIXED BEAM -- PAGE NO. 2
*
39. PERFORM ANALYSIS
P R O B L E M S T A T I S T I C S
-----------------------------------
NUMBER OF JOINTS 3 NUMBER OF MEMBERS 2
NUMBER OF PLATES 0 NUMBER OF SOLIDS 0
NUMBER OF SURFACES 0 NUMBER OF SUPPORTS 2
SOLVER USED IS THE IN-CORE ADVANCED SOLVER
TOTAL PRIMARY LOAD CASES = 2, TOTAL DEGREES OF FREEDOM = 4
40. *
41. PRINT JOINT DISP
JOINT DISP
REACTIONS AND DISPLACEMENTS OF A PINNED-FIXED BEAM -- PAGE NO. 3
*
JOINT DISPLACEMENT (CM RADIANS) STRUCTURE TYPE = PLANE
------------------
JOINT LOAD X-TRANS Y-TRANS Z-TRANS X-ROTAN Y-ROTAN Z-ROTAN
1 1 0.0000 0.0000 0.0000 0.0000 0.0000 -0.0020
2 0.0000 0.0000 0.0000 0.0000 0.0000 -0.0020
2 1 0.0000 -0.4626 0.0000 0.0000 0.0000 -0.0006
2 0.0000 -0.4626 0.0000 0.0000 0.0000 -0.0006
38 — STAAD.Pro
2 Static Beams
Static Beam 1
48. Figure 2-2: Fixed support beam with temperature load
Comparison
Result Type Theory STAAD.Pro Difference
Horizontal Reaction at Node A (kN) 468 468 none
Moment at Node A (kN·m) 19.5 19.5 none
Table 2-2: Comparison of results for static beam no. 1
Note: In the STAAD model, two load cases are used. In case 1, the uniform expansion is
applied. In case 2, the temperature change between top and bottom flanges is applied.
Theoretical Solution
Horizontal reactions due to case 1 loads:
RA
= - RB
= EAαΔT = [200(10)6]· [50(10)-4]· [11.7(10)-6]· (40) = 468 kN
Moment reactions due to case 2 loads:
⋅
⋅
⋅
( )
M M kN m
= − = = = 19.5
A B
αEI T
d
∆
11.7(10) 200(10) 5, 000(10) 50
0.3
−6 −6 −8
STAAD Input
STAAD PLANE TEMPERATURE LOAD ON A FIXED-FIXED BEAM
*
* FILE : BEAM02.STD
*
* REFERENCE : MATRIX ANALYSIS OF FRAMED STRUCTURES
* GERE & WEAVER, 3RD EDITION, VAN NOSTRAND REINHOLD
*
* TABLE B-2, APPENDIX B, PAGE 500
*
UNIT KNS METER
40 — STAAD.Pro
2 Static Beams
Static Beam 2
49. JOINT COORD
1 0 0 0 ; 2 3 0 0 ; 3 7.5 0 0
*
MEMBER INCI
1 1 2 2
*
UNIT CMS
MEMBER PROP
1 2 PRIS AX 50 IZ 5000 YD 30
*
UNIT METER
CONSTANT
E 200E6 ALL
POISS STEEL ALL
ALPHA 11.7E-6 ALL
*
SUPPORT
1 FIXED
3 FIXED
*
LOAD 1
TEMPERATURE LOAD
1 2 TEMP 40
*
LOAD 2
TEMPERATURE LOAD
1 2 TEMP 0 50
*
PERFORM ANALYSIS
*
PRINT SUPP REAC
FINISH
STAAD Output
PAGE NO. 1
****************************************************
* *
* STAAD.Pro V8i SELECTseries5 *
* Version 20.07.10.41 *
* Proprietary Program of *
* Bentley Systems, Inc. *
* Date= DEC 13, 2013 *
* Time= 12:27:35 *
* *
* USER ID: Bentley Systems, Inc. *
****************************************************
1. STAAD PLANE TEMPERATURE LOAD ON A FIXED-FIXED BEAM
INPUT FILE: BEAM02.STD
2. *
3. * FILE : BEAM02.STD
4. *
5. * REFERENCE : MATRIX ANALYSIS OF FRAMED STRUCTURES
6. * GERE & WEAVER, 3RD EDITION, VAN NOSTRAND REINHOLD
7. *
8. * TABLE B-2, APPENDIX B, PAGE 500
9. *
10. UNIT KNS METER
11. JOINT COORD
12. 1 0 0 0 ; 2 3 0 0 ; 3 7.5 0 0
13. *
14. MEMBER INCI
15. 1 1 2 2
16. *
2 Static Beams
Verification Manual — 41
51. ************************************************************
Static Beam 3
Objective
To find the twist at the free end of a hollow tapered shaft of uniform thickness.
Reference
Hand calculation using the following reference:
Mechanics of Materials, F.P.Beer & E.R.Johnston, 1981, McGraw-Hill Review Problem 3.120, Page
149.
Problem
The beam in the following figure which has the following geometric, load, and section properties:
L = 2 m, outside diameter at fixed end = 80 mm, outside diameter at free end = 40 mm, uniform
wall thickness of 10 mm, , IZ
= 5,000 cm4, E = 200 KN/mm2, T = 2.0 KNm, and Poisson's ratio =
0.3.
2 Static Beams
Verification Manual — 43
52. Figure 2-3: Cantilevered member with a tapered, hollow shaft cross section
Comparison
Result Type Theory STAAD.Pro Difference
Twist at free end (radians) 0.0751 0.0751 none
Table 2-3: Comparison of results for static beam no. 1
Theoretical Solution
According to the reference, the twist at the free end is
44 — STAAD.Pro
2 Static Beams
Static Beam 3
53.
( )
Φ =
A
TL
πGt
C C
C C
4
+
A B
A B
2 2
where:
CA
= centerline radius at free end = 40 mm/2 - 10 mm/2 = 15 mm
CB
= center line radius at fixed end = 80 mm/2 - 10 mm/2 = 35 mm
G= = = 76.9
E
ν
kN mm
2(1 + )
200 /
2(1 + 0.3)
2
⋅
( )
Φ = = 0.0751 rad.
A
kN m m
π kN mm mm
mm mm
mm mm
2, 000 ×2, 000
4 × 76.9 / × 10
15 + 35
(15 ) (35 )
2 2 2
STAAD Input File
STAAD SPACE TORSION ON CONICAL SHAFT
*
* FILE : BEAM03.STD
*
* REFERENCE : MECHANICS OF MATERIALS, F.P.BEER & E.R.JOHNSTON
* 1981, MCGRAW-HILL
*
* REVIEW PROBLEM 3.120, PAGE 149. TWIST AT FREE END SHOULD BE
* ABOUT 0.0751 RADIAN
*
UNIT KNS METER
JOINT COORD
1 0 0 0 ; 2 0 2 0
MEMBER INCI
1 1 2
UNIT MMS
MEMBER PROP
1 PRIS ROUND START 80 END 40 THICK 10
CONST
E 200 ALL
POISSON 0.3 ALL
SUPPORT
1 FIXED
UNIT KNS METER
LOAD 1
JOINT LOAD
2 MY 2.0
PERFORM ANALYSIS
PRINT JOINT DISP
FINISH
STAAD Output
PAGE NO. 1
****************************************************
* *
* STAAD.Pro V8i SELECTseries5 *
* Version 20.07.10.41 *
* Proprietary Program of *
* Bentley Systems, Inc. *
* Date= DEC 13, 2013 *
* Time= 12:27:38 *
* *
* USER ID: Bentley Systems, Inc. *
****************************************************
1. STAAD SPACE TORSION ON CONICAL SHAFT
2 Static Beams
Verification Manual — 45
54. INPUT FILE: BEAM03.STD
2. *
3. * FILE : BEAM03.STD
4. *
5. * REFERENCE : MECHANICS OF MATERIALS, F.P.BEER & E.R.JOHNSTON
6. * 1981, MCGRAW-HILL
7. *
8. * REVIEW PROBLEM 3.120, PAGE 149. TWIST AT FREE END SHOULD BE
9. * ABOUT 0.0751 RADIAN
10. *
11. UNIT KNS METER
12. JOINT COORD
13. 1 0 0 0 ; 2 0 2 0
14. MEMBER INCI
15. 1 1 2
16. UNIT MMS
17. MEMBER PROP
18. 1 PRIS ROUND START 80 END 40 THICK 10
19. CONST
20. E 200 ALL
21. POISSON 0.3 ALL
22. SUPPORT
23. 1 FIXED
24. UNIT KNS METER
25. LOAD 1
26. JOINT LOAD
27. 2 MY 2.0
28. PERFORM ANALYSIS
TORSION ON CONICAL SHAFT -- PAGE NO. 2
*
P R O B L E M S T A T I S T I C S
-----------------------------------
NUMBER OF JOINTS 2 NUMBER OF MEMBERS 1
NUMBER OF PLATES 0 NUMBER OF SOLIDS 0
NUMBER OF SURFACES 0 NUMBER OF SUPPORTS 1
SOLVER USED IS THE IN-CORE ADVANCED SOLVER
TOTAL PRIMARY LOAD CASES = 1, TOTAL DEGREES OF FREEDOM = 6
29. PRINT JOINT DISP
JOINT DISP
TORSION ON CONICAL SHAFT -- PAGE NO. 3
*
JOINT DISPLACEMENT (CM RADIANS) STRUCTURE TYPE = SPACE
------------------
JOINT LOAD X-TRANS Y-TRANS Z-TRANS X-ROTAN Y-ROTAN Z-ROTAN
1 1 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000
2 1 0.0000 0.0000 0.0000 0.0000 0.0751 0.0000
************** END OF LATEST ANALYSIS RESULT **************
30. FINISH
*********** END OF THE STAAD.Pro RUN ***********
**** DATE= DEC 13,2013 TIME= 12:27:40 ****
TORSION ON CONICAL SHAFT -- PAGE NO. 4
*
************************************************************
* For technical assistance on STAAD.Pro, please visit *
* http://selectservices.bentley.com/en-US/ *
* *
* Details about additional assistance from *
* Bentley and Partners can be found at program menu *
* Help->Technical Support *
* *
46 — STAAD.Pro
2 Static Beams
Static Beam 3
56. Result Type Theory STAAD.Pro Difference
Load Case
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
Theoretical Solution
Load Case 1:
General solution equations found on p.2-121 of the reference.
M x b P b x lb x
( < ) = − ( − ) = −0.5 (28.8 − )
At the rigid support, M = 14.4 in-lb.
By inspection, fixed end shear is equal to load value = 0.50 lb.
x b x b
∆ < = 3 −
Px
EI
6
2
( )( )
in in in
= 3 28.8 . − 28.8 . = 0.295 .
lb in
psi in
0.5 (28.8 .)
6 10, 000, 000 0.00135 .
2
4
Load Case 2:
General solution equations found on p.2-118 of the reference
( )
R in in lb
= = 19.2 + 2 48 = 0.216 .
Pb a
lb in
in
1
+ 2 ℓ
2 ℓ
0.5 (48 .)
2(48 .)
2
3
2
3
R2
= P - R1
= 0.50lb - 0.216lb = 0.284lb.
Moment at rigid support:
M x b R x P x a
( < ) = (ℓ − ) − (ℓ − − )
1 lb in lb in in
= 0.216 (48 − 0) − 0.5 (48 − 0 − 19.2 ) = −4.032 in lb
Deflection at point of load:
x b x a a x
∆ < = 3ℓ − 3ℓ − 3 ℓ +
Pax
EI
−
12 ℓ
3 2 2 2
2
3
( )
( )( )
in in in in in in in in
= 3(48 .) − 3(48 .) 28.8 − 3(19.2 .) 48 + (19.2 .) 28.8 = −0.040 .
lb in in
psi in in
−0.5 19.2 (28.8 )
12 10, 000, 000 0.00135 . (48 .)
3 2 2 2
3
4 3
STAAD Input File
STAAD PLANE :A PROPPED CANTILEVER WITH COMPRESSION ONLY SUPPORT
START JOB INFORMATION
ENGINEER DATE 06-Mar-12
END JOB INFORMATION
* FILE: BEAM04.STD
*
* STARDYNE VERIFICATION PROBLEM #16
*********************************************
* THE END SUPPORT IS TO BE DEFINED AS A *
* COMPRESSION ONLY SUPPORT. A DUMMY MEMBER,*
* #7, IS SET AS COMPRESSION ONLY TO MODEL *
* THIS. *
48 — STAAD.Pro
2 Static Beams
Static Beam 4
57. *********************************************
SET NL 2
***********************************************
* STAAD.Pro Generated Comment *
***********************************************
*1 0. 0. 0. 6 48.0 0. 0.
***********************************************
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;
*7 48. -.01 0.
***********************************************
* STAAD.Pro Generated Comment *
***********************************************
*1 1 2 6
************************************************
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 PRISMATIC YD 2.
6 PRIS YD 40
MEMBER COMPRESSION
6
DEFINE MATERIAL START
ISOTROPIC MATERIAL1
E 1e+007
POISSON 0.33
END DEFINE MATERIAL
CONSTANTS
MATERIAL MATERIAL1 ALL
SUPPORTS
1 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
PAGE NO. 1
****************************************************
* *
* STAAD.Pro V8i SELECTseries5 *
* Version 20.07.10.41 *
* Proprietary Program of *
* Bentley Systems, Inc. *
* Date= DEC 13, 2013 *
* Time= 12:27:41 *
* *
* USER ID: Bentley Systems, Inc. *
2 Static Beams
Verification Manual — 49
58. ****************************************************
1. STAAD PLANE :A PROPPED CANTILEVER WITH COMPRESSION ONLY SUPPORT
INPUT FILE: BEAM04.STD
2. START JOB INFORMATION
3. ENGINEER DATE 06-MAR-12
4. END JOB INFORMATION
5. * FILE: BEAM04.STD
6. *
7. * STARDYNE VERIFICATION PROBLEM #16
8. *********************************************
9. * THE END SUPPORT IS TO BE DEFINED AS A *
10. * COMPRESSION ONLY SUPPORT. A DUMMY MEMBER,*
11. * #7, IS SET AS COMPRESSION ONLY TO MODEL *
12. * THIS. *
13. *********************************************
14. SET NL 2
15. ***********************************************
16. * STAAD.PRO GENERATED COMMENT *
17. ***********************************************
18. *1 0. 0. 0. 6 48.0 0. 0.
19. ***********************************************
20. UNIT INCHES POUND
21. JOINT COORDINATES
22. 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
23. 7 48 -4 0
24. *7 48. -.01 0.
25. ***********************************************
26. * STAAD.PRO GENERATED COMMENT *
27. ***********************************************
28. *1 1 2 6
29. ************************************************
30. MEMBER INCIDENCES
31. 1 1 2; 2 2 3; 3 3 4; 4 4 5; 5 5 6; 6 6 7
32. MEMBER PROPERTY AMERICAN
33. 1 TO 5 PRIS YD 0.3 ZD 0.6
34. *6 PRISMATIC YD 2.
35. 6 PRIS YD 40
36. MEMBER COMPRESSION
37. 6
38. DEFINE MATERIAL START
:A PROPPED CANTILEVER WITH COMPRESSION ONLY SUPPORT -- PAGE NO. 2
39. ISOTROPIC MATERIAL1
40. E 1E+007
41. POISSON 0.33
42. END DEFINE MATERIAL
43. CONSTANTS
44. MATERIAL MATERIAL1 ALL
45. SUPPORTS
46. 1 FIXED
47. 7 PINNED
48. LOAD 1 UPWARD
49. JOINT LOAD
50. 4 FY 0.5
51. PERFORM ANALYSIS
P R O B L E M S T A T I S T I C S
-----------------------------------
NUMBER OF JOINTS 7 NUMBER OF MEMBERS 6
NUMBER OF PLATES 0 NUMBER OF SOLIDS 0
NUMBER OF SURFACES 0 NUMBER OF SUPPORTS 2
SOLVER USED IS THE IN-CORE ADVANCED SOLVER
TOTAL PRIMARY LOAD CASES = 1, TOTAL DEGREES OF FREEDOM = 16
50 — STAAD.Pro
2 Static Beams
Static Beam 4
59. *WARNING- ZERO STIFFNESS IN DIRECTION 6 AT JOINT 7 EQN.NO. 16
LOADS APPLIED OR DISTRIBUTED HERE FROM ELEMENTS WILL BE IGNORED.
THIS MAY BE DUE TO ALL MEMBERS AT THIS JOINT BEING RELEASED OR
EFFECTIVELY RELEASED IN THIS DIRECTION.
**START ITERATION NO. 2
*WARNING- ZERO STIFFNESS IN DIRECTION 6 AT JOINT 7 EQN.NO. 16
LOADS APPLIED OR DISTRIBUTED HERE FROM ELEMENTS WILL BE IGNORED.
THIS MAY BE DUE TO ALL MEMBERS AT THIS JOINT BEING RELEASED OR
EFFECTIVELY RELEASED IN THIS DIRECTION.
**NOTE-Tension/Compression converged after 2 iterations, Case= 1
52. PRINT MEMBER FORCES LIST 1 TO 5
MEMBER FORCES LIST 1
:A PROPPED CANTILEVER WITH COMPRESSION ONLY SUPPORT -- PAGE NO. 3
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 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 **************
53. 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 **************
54. 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 **************
55. CHANGE
56. LOAD 2 DOWNWARD
57. JOINT LOAD
58. 4 FY -0.5
59. PERFORM ANALYSIS
*WARNING- ZERO STIFFNESS IN DIRECTION 6 AT JOINT 7 EQN.NO. 16
LOADS APPLIED OR DISTRIBUTED HERE FROM ELEMENTS WILL BE IGNORED.
THIS MAY BE DUE TO ALL MEMBERS AT THIS JOINT BEING RELEASED OR
EFFECTIVELY RELEASED IN THIS DIRECTION.
**NOTE-Tension/Compression converged after 1 iterations, Case= 2
2 Static Beams
Verification Manual — 51
61. Static Beam 5
Objective
To find end rotation due to torques on a stepped cantilever shaft.
Reference
Hand calculation using the following reference:
Gere J. M., and Timoshenko, S. P., Mechanics of Materials, 2nd Edition, PWS Engineering, Page 171,
Problem 3.3 - 1.
Problem
A stepped shaft is subjected to torques as shown in the figure. The material has shear modulus of
elasticity G = 80 Gpa. Determine the angle of twist θx
in degrees at the free end.
Figure 2-5: Cantilevered member subject to torsional loads
Comparison
Result Theory STAAD.Pro Difference
Angle of twist (rad.) 0.0427 0.0427 none
Table 2-5: Comparison of results for static beam no. 5
Theoretical Solution
Moment of Inertia:
I mm
= = 4.021(10)
p
π mm
1
(80 / 2)
2
6 4
4
I mm
= = 1.272(10)
p
π mm
2
(60 / 2)
2
6 4
4
I mm
= = 0.251(10)
p
π mm
2
(40 / 2)
2
6 4
4
2 Static Beams
Verification Manual — 53
62. Angle of twist is given by:
θ = ∑i
L T
GI
i i
p
= + +
mm
mm GPa
mm
mm GPa
mm
mm GPa
5, 800N-mm(500 )
4.021(10) 80
2, 800N-mm(500 )
1.272(10) 80
, 800N-mm(500 )
0.251(10) 80
6 4 6 4 6 4
= 0.0090 + 0.0138 + 0.0199 = 0.0427
which is equal to 2.446°.
STAAD Input File
STAAD SPACE :A STEPPED CANTILEVER SHAFT
* FILE: BEAM05.STD
*
* REFERENCE: MECHANICS OF MATERIALS, GERE AND TIMOSHENKO, 2ND EDITION
* PROBLEM 3.3-1 PAGE 171
*
UNIT KN METER
JOINT COORDINATES
1 0.0 0.0
2 0.5 0.0
3 1.0 0.0
4 1.5 0.0
MEMBER INCIDENCES
1 1 2 3
UNIT MMS
MEMBER PROPERTIES
1 TA ST PIPE OD 80 ID 0.0
2 TA ST PIPE OD 60 ID 0.0
3 TA ST PIPE OD 40 ID 0.0
UNIT METER
CONSTANTS
E 200.0E6 ALL
POISSONS .25 ALL
*
SUPPORTS
1 FIXED
UNIT MMS
LOADING 1 TORSIONAL MOMENT
JOINT LOADS
2 MX 3000.0
3 MX 2000.0
4 MX 800.0
PERFORM ANALYSIS
PRINT JOINT DISPLACEMENTS LIST 4
FINISH
STAAD Output
PAGE NO. 1
****************************************************
* *
* STAAD.Pro V8i SELECTseries5 *
* Version 20.07.10.41 *
* Proprietary Program of *
* Bentley Systems, Inc. *
* Date= DEC 13, 2013 *
* Time= 12:27:44 *
* *
* USER ID: Bentley Systems, Inc. *
****************************************************
1. STAAD SPACE :A STEPPED CANTILEVER SHAFT
INPUT FILE: BEAM05.STD
2. * FILE: BEAM05.STD
3. *
54 — STAAD.Pro
2 Static Beams
Static Beam 5
63. 4. * REFERENCE: MECHANICS OF MATERIALS, GERE AND TIMOSHENKO, 2ND EDITION
5. * PROBLEM 3.3-1 PAGE 171
6. *
7. UNIT KN METER
8. JOINT COORDINATES
9. 1 0.0 0.0
10. 2 0.5 0.0
11. 3 1.0 0.0
12. 4 1.5 0.0
13. MEMBER INCIDENCES
14. 1 1 2 3
15. UNIT MMS
16. MEMBER PROPERTIES
17. 1 TA ST PIPE OD 80 ID 0.0
18. 2 TA ST PIPE OD 60 ID 0.0
19. 3 TA ST PIPE OD 40 ID 0.0
20. UNIT METER
21. CONSTANTS
22. E 200.0E6 ALL
23. POISSONS .25 ALL
24. *
25. SUPPORTS
26. 1 FIXED
27. UNIT MMS
28. LOADING 1 TORSIONAL MOMENT
29. JOINT LOADS
30. 2 MX 3000.0
31. 3 MX 2000.0
32. 4 MX 800.0
33. PERFORM ANALYSIS
:A STEPPED CANTILEVER SHAFT -- PAGE NO. 2
* FILE: BEAM05.STD
P R O B L E M S T A T I S T I C S
-----------------------------------
NUMBER OF JOINTS 4 NUMBER OF MEMBERS 3
NUMBER OF PLATES 0 NUMBER OF SOLIDS 0
NUMBER OF SURFACES 0 NUMBER OF SUPPORTS 1
SOLVER USED IS THE IN-CORE ADVANCED SOLVER
TOTAL PRIMARY LOAD CASES = 1, TOTAL DEGREES OF FREEDOM = 18
34. PRINT JOINT DISPLACEMENTS LIST 4
JOINT DISPLACE LIST 4
:A STEPPED CANTILEVER SHAFT -- PAGE NO. 3
* FILE: BEAM05.STD
JOINT DISPLACEMENT (CM RADIANS) STRUCTURE TYPE = SPACE
------------------
JOINT LOAD X-TRANS Y-TRANS Z-TRANS X-ROTAN Y-ROTAN Z-ROTAN
4 1 0.0000 0.0000 0.0000 0.0427 0.0000 0.0000
************** END OF LATEST ANALYSIS RESULT **************
35. FINISH
*********** END OF THE STAAD.Pro RUN ***********
**** DATE= DEC 13,2013 TIME= 12:27:47 ****
:A STEPPED CANTILEVER SHAFT -- PAGE NO. 4
* FILE: BEAM05.STD
************************************************************
* For technical assistance on STAAD.Pro, please visit *
* http://selectservices.bentley.com/en-US/ *
* *
* Details about additional assistance from *
* Bentley and Partners can be found at program menu *
* Help->Technical Support *
2 Static Beams
Verification Manual — 55
65. Comparison
Result Theory STAAD.Pro Difference
Reaction at Node 1 (lb) 42.31 42.31 none
Reaction at Node 3 (lb) 70.52 70.52 none
Moment at Node 3 (in-lb) 1,410.4 1410.4 none
Bending Stress, σbend
at Node 2 (psi) 585.9 585.9 none
Rotation at Node 1 (rad.) -.000588 -.00059 none
Deflection at Node 2 (in.) -0.01528 -0.01528 none
Table 2-6: Comparison of results for static beam no. 5
STAAD Input File
STAAD PLANE :A FIXED-ROLLER BEAM
* FILE: BEAM06.STD
*
* REFERENCE: ROARK AND YOUNG, PAGE 109, NO. 23.
*
INPUT WIDTH 72
UNIT INCHES POUND
JOINT COORDINATES
1 0. 0. 0.; 2 42.15 0. 0.; 3 100. 0. 0.
MEMBER INCIDENCES
1 1 2; 2 2 3
MEMBER PROPERTY AMERICAN
1 2 PRI AX 4. IZ 1.3333 YD 2.
CONSTANTS
E 30000000. ALL
DENSITY 0.282072 ALL
SUPPORTS
3 FIXED
1 PINNED
LOAD 1 SELF WEIGHT
SELFWEIGHT Y -1.
PERFORM ANALYSIS
PRINT ANALYSIS RESULTS
PRINT MEMBER STRESSES ALL
FINISH
STAAD Output
PAGE NO. 1
****************************************************
* *
* STAAD.Pro V8i SELECTseries5 *
* Version 20.07.10.41 *
* Proprietary Program of *
* Bentley Systems, Inc. *
* Date= DEC 13, 2013 *
* Time= 12:27:48 *
* *
* USER ID: Bentley Systems, Inc. *
****************************************************
1. STAAD PLANE :A FIXED-ROLLER BEAM
INPUT FILE: BEAM06.STD
2 Static Beams
Verification Manual — 57
66. 2. * FILE: BEAM06.STD
3. *
4. * REFERENCE: ROARK AND YOUNG, PAGE 109, NO. 23.
5. *
6. INPUT WIDTH 72
7. UNIT INCHES POUND
8. JOINT COORDINATES
9. 1 0. 0. 0.; 2 42.15 0. 0.; 3 100. 0. 0.
10. MEMBER INCIDENCES
11. 1 1 2; 2 2 3
12. MEMBER PROPERTY AMERICAN
13. 1 2 PRI AX 4. IZ 1.3333 YD 2.
14. CONSTANTS
15. E 30000000. ALL
16. DENSITY 0.282072 ALL
17. SUPPORTS
18. 3 FIXED
19. 1 PINNED
20. LOAD 1 SELF WEIGHT
21. SELFWEIGHT Y -1.
22. PERFORM ANALYSIS
**WARNING** THE POISSON'S RATIO HAS NOT BEEN SPECIFIED FOR ONE OR MORE
MEMBERS/ELEMENTS/SOLIDS. THE DEFAULT VALUE HAS BEEN SET FOR THE SAME.
:A FIXED-ROLLER BEAM -- PAGE NO. 2
* FILE: BEAM06.STD
P R O B L E M S T A T I S T I C S
-----------------------------------
NUMBER OF JOINTS 3 NUMBER OF MEMBERS 2
NUMBER OF PLATES 0 NUMBER OF SOLIDS 0
NUMBER OF SURFACES 0 NUMBER OF SUPPORTS 2
SOLVER USED IS THE IN-CORE ADVANCED SOLVER
TOTAL PRIMARY LOAD CASES = 1, TOTAL DEGREES OF FREEDOM = 4
23. PRINT ANALYSIS RESULTS
ANALYSIS RESULTS
:A FIXED-ROLLER BEAM -- PAGE NO. 3
* FILE: BEAM06.STD
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 -0.00059
2 1 0.00000 -0.01528 0.00000 0.00000 0.00000 0.00000
3 1 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000
:A FIXED-ROLLER BEAM -- PAGE NO. 4
* FILE: BEAM06.STD
SUPPORT REACTIONS -UNIT POUN INCH STRUCTURE TYPE = PLANE
-----------------
JOINT LOAD FORCE-X FORCE-Y FORCE-Z MOM-X MOM-Y MOM Z
3 1 0.00 70.52 0.00 0.00 0.00 -1410.36
1 1 0.00 42.31 0.00 0.00 0.00 0.00
:A FIXED-ROLLER BEAM -- PAGE NO. 5
* FILE: BEAM06.STD
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
58 — STAAD.Pro
2 Static Beams
Static Beam 6
71. Problem
Find the maximum bending stress in the beam.
E = 30(10)6 psi
b = 1.5 in., h = 8 in., L = 10 in.
M = 1,000,000 in. - lb.
Figure 2-8: Beam A) problem sketch and B) mathematical model
Comparison
Result Theory STAAD.Pro Difference
Bending stress, σ (psi) 700 700 none
Table 2-8: Comparison of results for static beam no. 5
STAAD Input File
STAAD PLANE :A CANTILEVERED BEAM OF INVERTED TEE SECTION
* FILE: BEAM08.STD
*
* REFERENCE: CRANDALL & DAHL, AN INTRODUCTION TO THE MECHANICS
* OF SOLIDS, PAGE294, EX. 7.2
*
INPUT WIDTH 79
UNIT INCHES POUND
JOINT COORDINATES
1 0.000 0.000 0.000
2 10.000 0.000 0.000
MEMBER INCIDENCES
1 1 2
MEMBER PROPERTY AMERICAN
1 PRISMATIC YD 20. ZD 9. YB 16. ZB 1.5
CONSTANTS
E 30000000. ALL
BETA 180. ALL
SUPPORTS
2 Static Beams
Verification Manual — 63
72. 1 FIXED
LOAD 1 CONSTANT MOMENT
JOINT LOAD
2 MZ 100000.0
PERFORM ANALYSIS
PRINT MEMBER PROPERTIES ALL
PRINT MEMBER STRESSES ALL
FINISH
STAAD Output
****************************************************
* *
* STAAD.Pro V8i SELECTseries3 *
* Version 20.07.08.20 *
* Proprietary Program of *
* Bentley Systems, Inc. *
* Date= MAR 6, 2012 *
* Time= 13:37:57 *
* *
* USER ID: Bentley Systems *
****************************************************
1. STAAD PLANE :A CANTILEVERED BEAM OF INVERTED TEE SECTION
INPUT FILE: beam8.STD
2. * FILE: BEAM08.STD
3. *
4. * REFERENCE: CRANDALL & DAHL, AN INTRODUCTION TO THE MECHANICS
5. * OF SOLIDS, PAGE294, EX. 7.2
6. *
7. INPUT WIDTH 79
8. UNIT INCHES POUND
9. JOINT COORDINATES
10. 1 0.000 0.000 0.000
11. 2 10.000 0.000 0.000
12. MEMBER INCIDENCES
13. 1 1 2
14. MEMBER PROPERTY AMERICAN
15. 1 PRISMATIC YD 20. ZD 9. YB 16. ZB 1.5
16. CONSTANTS
17. E 30000000. ALL
18. BETA 180. ALL
19. SUPPORTS
20. 1 FIXED
21. LOAD 1 CONSTANT MOMENT
22. JOINT LOAD
23. 2 MZ 100000.0
24. PERFORM ANALYSIS
**WARNING** THE POISSON'S RATIO HAS NOT BEEN SPECIFIED FOR ONE OR MORE
MEMBERS/ELEMENTS/SOLIDS. THE DEFAULT VALUE HAS BEEN SET FOR THE SAME.
P R O B L E M S T A T I S T I C S
-----------------------------------
NUMBER OF JOINTS/MEMBER+ELEMENTS/SUPPORTS = 2/ 1/ 1
SOLVER USED IS THE IN-CORE ADVANCED SOLVER
TOTAL PRIMARY LOAD CASES = 1, TOTAL DEGREES OF FREEDOM = 3
25. PRINT MEMBER PROPERTIES ALL
MEMBER PROPERTIES. UNIT - INCH
-----------------
MEMB PROFILE AX/ IZ/ IY/ IX/
AY AZ SZ SY
1 PRISMATIC 60.00 2000.00 247.50 154.89
30.00 36.00 142.86 55.00
64 — STAAD.Pro
2 Static Beams
Static Beam 8
73. ************ END OF DATA FROM INTERNAL STORAGE ************
26. PRINT MEMBER STRESSES ALL
MEMBER STRESSES
---------------
ALL UNITS ARE POUN/SQ INCH
MEMB LD SECT AXIAL BEND-Y BEND-Z COMBINED SHEAR-Y SHEAR-Z
1 1 .0 0.0 0.0 700.0 700.0 0.0 0.0
1.00 0.0 0.0 700.0 700.0 0.0 0.0
************** END OF LATEST ANALYSIS RESULT **************
27. FINISH
Static Beam 9
Objective
To find deflection and stress at the center due to a uniform, static load on a simply supported
beam on elastic foundation.
Reference
Hand calculation using the following reference:
Peterson, F.E., Elastic Analysis for Structural Engineering (EASE2), Example Problem Manual,
Engineering Analysis Corporation, Berkeley, CA, 1981.
Problem
Find the vertical deflection and bending stress at the center of the beam.Spacing between
E = 30(10)6 psi
b = 1.0 in., h = 7.114 in., L = 240 in.
wu
= 43.3 lb/in.
Figure 2-9: One-half beam for mathematical model
Comparison
Result Theory STAAD.Pro Difference
Bending stress, σ (psi) 18,052 18,053.29 none
Vertical deflection (in.) 1.0453 1.04549 none
Table 2-9: Comparison of results for static beam no. 5
2 Static Beams
Verification Manual — 65