V. 2D Portal Reactions 1

To find section properties, member forces and support reactions for a 1x1 bay plane frame with members of rectangular section.

Reference

Timoshenko, S., Strength of Materials, Part I, Elementary Theory and Problems, 2nd Edition, Van Nostrand Company, 1940, Pages 188-191,

Problem

The frame supports a concentrated load at middle of the horizontal member. Verify the internally calculated section properties, support reactions and bending moments at the ends of the horizontal member. Columns are square 2" x 2"; beams are rectangular with b = 2" and h = 4".

P = 1,000 lb

h = 100 in.

l = 120 in.

E = 30×(10)⁶ psi

Symmetric portal frame

Calculations

From the reference:

M = \frac{P l}{8} \frac{1}{1 + \frac{2}{3} \frac{h}{l} \frac{I_{b}}{I_{c}}} = \frac{1, 000 (120)}{8} \frac{1}{1 + \frac{2}{3} \frac{100}{120} \frac{10.67}{1.333}} = 2, 754 in-lb

Comparison

Table 1. Comparison of results
Result Type		Theory	STAAD.Pro	Difference
Column Cross Section	A_x (in.²)	4.0	4.0	none
	I_x (in.⁴)	2.25	2.25	none
	I_y (in.⁴)	1.333	1.33	none
	I_z (in.⁴)	1.333	1.33	none
Beam Cross Section	A_x (in.²)	8.0	8.0	none
	I_x (in.⁴)	7.324	7.32	none
	I_y (in.⁴)	2.667	2.67	none
	I_z (in.⁴)	10.667	10.67	none
R_y (lb)		500	500	none
R_x (lb)		27.55	27.54	none
M(in·lb)		2,754.97	2,754.35	none

STAAD Input

The file C:\Users\Public\Public Documents\STAAD.Pro CONNECT Edition\Samples\ Verification Models\03 Frames\2D Portal Reactions 1.STD is typically installed with the program.

STAAD PLANE :A 1X1 BAY PLANE FRAME OF RECTANGULAR SECTION
START JOB INFORMATION
ENGINEER DATE 17-Sep-18
END JOB INFORMATION
*
* REFERENCE: TIMOSHENKO, S., STRENGTH OF MATERIALS, PART 1,
*           2ND EDITION, D. VAN NOSTRAND CO., 1940,
*           PAGES 188 THRU 191.
*
UNIT INCHES POUND
JOINT COORDINATES
1 0 0 0; 2 0 100 0; 3 60 100 0; 4 120 100 0; 5 120 0 0;
MEMBER INCIDENCES
1 1 2; 2 2 3; 3 3 4; 4 4 5;
MEMBER PROPERTY AMERICAN
1 4 PRIS YD 2 ZD 2
2 3 PRIS YD 4 ZD 2
PRINT MEMBER PROPERTIES
DEFINE MATERIAL START
ISOTROPIC MATERIAL1
E 3e+07
POISSON 0.290909
END DEFINE MATERIAL
CONSTANTS
MATERIAL MATERIAL1 ALL
SUPPORTS
1 5 PINNED
LOAD 1 MID SPAN
JOINT LOAD
3 FY 1000
PERFORM ANALYSIS
PRINT SUPPORT REACTION
PRINT MEMBER FORCES LIST 2 3
FINISH

STAAD Output

      :A 1X1 BAY PLANE FRAME OF RECTANGULAR SECTION            -- PAGE NO.    2
   MEMBER PROPERTIES. UNIT - INCH
   -----------------
   MEMB   PROFILE              AX/          IZ/          IY/        IX/  
                               AY           AZ           SZ         SY
     1  PRISMATIC              4.00         1.33         1.33       2.25
                               3.40         3.40         1.33       1.33
     2  PRISMATIC              8.00        10.67         2.67       7.32
                               6.80         6.80         5.33       2.67
     3  PRISMATIC              8.00        10.67         2.67       7.32
                               6.80         6.80         5.33       2.67
     4  PRISMATIC              4.00         1.33         1.33       2.25
                               3.40         3.40         1.33       1.33
   ************ END OF DATA FROM INTERNAL STORAGE ************
    19. DEFINE MATERIAL START
    20. ISOTROPIC MATERIAL1
    21. E 3E+07
    22. POISSON 0.290909
    23. END DEFINE MATERIAL
    24. CONSTANTS
    25. MATERIAL MATERIAL1 ALL
    26. SUPPORTS
    27. 1 5 PINNED
    28. LOAD 1 MID SPAN
    29. JOINT LOAD
    30. 3 FY 1000
    31. PERFORM ANALYSIS
            P R O B L E M   S T A T I S T I C S
            -----------------------------------
     NUMBER OF JOINTS          5  NUMBER OF MEMBERS       4
     NUMBER OF PLATES          0  NUMBER OF SOLIDS        0
     NUMBER OF SURFACES        0  NUMBER OF SUPPORTS      2
           Using 64-bit analysis engine.
           SOLVER USED IS THE IN-CORE ADVANCED MATH SOLVER
   TOTAL      PRIMARY LOAD CASES =     1, TOTAL DEGREES OF FREEDOM =      11
      :A 1X1 BAY PLANE FRAME OF RECTANGULAR SECTION            -- PAGE NO.    3
   TOTAL LOAD COMBINATION  CASES =     0  SO FAR.
    32. PRINT SUPPORT REACTION
  SUPPORT  REACTION                   
      :A 1X1 BAY PLANE FRAME OF RECTANGULAR SECTION            -- PAGE NO.    4
   SUPPORT REACTIONS -UNIT POUN INCH    STRUCTURE TYPE = PLANE
   -----------------
 JOINT  LOAD   FORCE-X   FORCE-Y   FORCE-Z     MOM-X     MOM-Y     MOM Z
      1    1    -27.54   -500.00      0.00      0.00      0.00      0.00
      5    1     27.54   -500.00      0.00      0.00      0.00      0.00
   ************** END OF LATEST ANALYSIS RESULT **************
    33. PRINT MEMBER FORCES LIST 2 3
  MEMBER   FORCES   LIST     2        
      :A 1X1 BAY PLANE FRAME OF RECTANGULAR SECTION            -- PAGE NO.    5
   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
      2    1     2    -27.54   -500.00     0.00      0.00      0.00   -2754.35
                 3     27.54    500.00     0.00      0.00      0.00  -27245.65
      3    1     3    -27.54    500.00     0.00      0.00      0.00   27245.65
                 4     27.54   -500.00     0.00      0.00      0.00    2754.35
   ************** END OF LATEST ANALYSIS RESULT **************