V. End Moments in a Non Uniform Beam

To find end moments due to a uniform load on a beam with nonuniform sections, fixed at both ends.

Reference

Hand calculation using the following reference:

McCormack, J.C., Structural Analysis, Intext Educational Publishers, 3rd Edition, 1975.

Problem

Find the moment at the supports. Assume for input a unit width for the beam. Depths are as shown.

E = 30×(10)⁶ psi

w = 4 k/ft

d₁ = 10 in., d₂ = 20 in.

L₁ = 12 ft, L₂ = 8 ft

Beam A) problem sketch and B) mathematical model

Comparison

Table 1. Comparison of results
Result	Theory	STAAD.Pro	Difference
Moment at Node 1 (in·lb)	-98.2	-96.93	1.3%
Moment at Node 2 (in·lb)	-217.2	-220.42	1.5%

STAAD Input

The file C:\Users\Public\Public Documents\STAAD.Pro CONNECT Edition\Samples\ Verification Models\01 Beams\End Moments in a Non Uniform Beam.STD is typically installed with the program.

STAAD PLANE :A FIXED-FIXED BEAM OF UNIFORM AND TAPERED SECTIONS
START JOB INFORMATION
ENGINEER DATE 18-Sep-18
END JOB INFORMATION
* 
* REFERENCE: MCCORMAC, J.C., STRUCTURAL ANALYSIS, INTEXT
*            EDUCATIONAL PUBLISHERS, NEW YORK, 3RD EDITION, 1975
*
INPUT WIDTH 72
UNIT INCHES KIP
JOINT COORDINATES
1 0 0 0; 2 144 0 0; 3 240 0 0;
MEMBER INCIDENCES
1 1 2; 2 3 2;
MEMBER PROPERTY AMERICAN
1 PRIS YD 10 ZD 1
2 TAPERED 20 1 10 1 1 1 1
PRINT MEMBER PROPERTIES
UNIT FEET KIP
DEFINE MATERIAL START
ISOTROPIC MATERIAL1
E 4.32e+06
POISSON 0.290909
END DEFINE MATERIAL
UNIT INCHES KIP
CONSTANTS
MATERIAL MATERIAL1 ALL
SUPPORTS
1 3 FIXED
LOAD 1 UNIFORM LOAD OVER ENTIRE BEAM
UNIT FEET KIP
MEMBER LOAD
1 2 UNI GY -4
PERFORM ANALYSIS
PRINT SUPPORT REACTION
FINISH

STAAD Output

   SUPPORT REACTIONS -UNIT KIP  FEET    STRUCTURE TYPE = PLANE
   -----------------
 JOINT  LOAD   FORCE-X   FORCE-Y   FORCE-Z     MOM-X     MOM-Y     MOM Z
      1    1      0.00     33.83      0.00      0.00      0.00     96.93
      3    1      0.00     46.17      0.00      0.00      0.00   -220.42