Project 1
SOLUTION;
DEAD LOAD
SELF WEIGHT OF SLAB=0.12*25=3 KN/M^2
FLOOR FINISH=3 KN/M^2
TOTAL DEAD LOAD= 6 KN/M^2
6 KN/M^2 *(3*9) M^2=162 KN
162KN/9M =18 KN/M
SELF WEIGHT OF BEAM =0.5*0.75*25=9.375 KN/M
TOTAL DEAD LOAD = 27.375 KN/M
LIVE LOAD =5 KN/M^2*(3*9) M^2=135 KN
135KN/9M=15 KN/M
TOTAL LIVE LOAD = 15KN/M
TOTAL DEAD LOAD +LIVE LOAD =42.375 KN/M
Determine the loading on the beam indicated, analyze the beams and draw bending moment and shear force diagram. Use slope deflection method. Do not consider any load patterning. Also, do not consider any girder deflections for this analysis purpose.
SOLUTION 1;
CASE 1
UDL THROUGHT OUT SPAN OF BEAM
STEP 1; FIXED END MOMENTS
MV@AB= -W*L^2/12 = -286.03125 KNM
MV@BA= W*L^2/12= 286.03125 KNM
MV@BC= - W*L^2/12= -286.03125 KNM
MV@CB= W*L^2/12= 286.03125 KNM
MV@CD = -W*L^2/12=-286.03125 KNM
MV@DC= W*L^2/12= 286.03125 KNM
MV@DE= -W*L^2/12= -286.03125 KNM
MV@ED=W*L^2/12= 286.03125 KNM
STEP 2;SLOPE DEFLECTION EQUATION
SPAN AB
M@AB=MV@AB+(4EI/L)*TA+(2EI/L)*TB+6EI*D/L^2
M@AB=-286.03125+0.4445EI*TA+0.2223EI*TB
M@BA=MV@BA+(4EI/L)*TB+(2EI/L)*TA+6EI*D/L^2
M@BA=286.03125+0.4445EI*TB+0.2223EI*TA
SPAN BC
M@BC=MV@BC+(4EI/L)*TB+(2EI/L)*TC+6EI*D/L^2
M@BC=-286.03125+0.4445EI*TB+0.2223EI*TC
M@CB=MV@CB+(4EI/L)*TC+(2EI/L)*TB+6EI*D/L^2
M@CB=286.03125+0.4445EI*TC+0.2223EI*TB
SPAN CD
M@CD=MV@CD+(4EI/L)*TC+(2EI/L)*TD+6EI*D/L^2
M@CD=-286.03125+0.4445EI*TC+0.2223*EI*TD
M@DC=MV@DC+(4EI/L)*TD+(2EI/L)*TC+6EI*D/L^2
M@DC=286.03125+0.4445EI*TD+0.2223EI*TC
SPAN DE
M@DE=MV@DE+(4EI/L)*TD+(2EI/L)*TE+6EI*D/L^2
M@DE=-286.03125+0.4445EI*TD+0.2223EI*TE
M@ED=MV@ED+(4EI/L)*TE+(2EI/L)*TD+6EI*D/L^2
M@ED=286.03125+0.4445EI*TE+0.2223EI*TD
STEP 3 ;EQUILIBRIUM CONDITIONS
0=-286.03125+0.4445EI*TA+0.2223EI*TB
EI*TA=643.489-0.5001EI*TB
0=286.03125+0.4445EI*TE+0.2223EI*TD
EI*TE=-643.489-0.5001EI*TD
AT JOINT B, SIGMA M@B=0
MBA+MBC=0
0.3334EI*TB+429.078-286.03125+0.4445EI*TB+0.2223EI*TC=0
0.7779EI*TB+0.2223EI*TC=-143.0467
AT JOINT C,SIGMA M@C=0
MCB+MCD=0
286.03125+0.4445EI*TC+0.2223EI*TB-286.03125+0.4445EI*TC+0.2223EI*TD=0
0.2223EI*TB+0.889EI*TC+0.2223EI*TD=0
AT JOINT D, SIGMA M@D=0
MDC+MDE=0
286.03125+0.4445EI*TD+0.2223EI*TC+0.3334EI*TD-429.078=0
0.2223EI*TC+0.7779*EI*TD=143.0467
EI*TB=-183.888 RADIAN
EI*TC= 0 RADIANS
EI*TD= 183.888 RADIAN
EI*TA=735.451 RADIAN
EI*TE=-735.451 RADIAN
M@AB=-286.03125+0.4445*735.451+0.2223*-183.888
M@AB= 0 KNM
M@BA=286.03125+0.4445*-183.888+0.2223*735.451
M@BA=367.783 KNM
M@BC=-286.03125+0.4445*-183.888+0.2223*0
M@BC=-367.769 KNM
M@CB=286.03125+0.4445*0+0.2223*-183.888
M@CB=245.1529 KNM
M@CD=-286.03125+0.4445*0+0.2223*183.888
M@CD=-245.1529 KNM
M@DC=286.03125+0.4445*183.888+0.2223*0
M@DC=367.769 KNM
M@DE=-286.03125+0.4445*183.888+0.2223*-735.451
M@DE=-367.783 KNM
M@ED=286.03125+0.4445*-735.451+0.2223*183.888
M@ED= 0 KNM
STEP 4;SUPPORT REACTION AND BENDNG MOMENT
SPAN AB
V@AB=149.822 KN
V@BA=231.553 KN
MID SPAN B.M@MAX=AB=264.857 KNM
SPAN BC
V@BC*9-367.769+245.1529-42.375*9*9/2=0
V@CB=177.063 KN
MID SPAN B.M@MAX=BC=124.67 KNM
SPAN CD
V@CD*9-245.1529+367.769-42.375*9*9/2=0
V@CD=177.063 KN
MID SPAN B.M@MAX=CD=124.67 KNM
SPAN DE
V@DE=231.552 KN
V@ED=149.822 KN
MID SPAN B.M@MAX=DE=264.857 KNM
For the same beam, consider load patterning to determine maximum positive and negative moments in each span.
SOLUTION 2;
CASE 2
NEGATIVE BENDING MOMENT AT M@B AND SIMILARLY FOR M@D
UDL ON SPAN AB,BC AND DE
STEP 1; FIXED END MOMENTS
MV@AB= -W*L^2/12 = -286.03125 KNM
MV@BA= W*L^2/12= 286.03125 KNM
MV@BC= - W*L^2/12= -286.03125 KNM
MV@CB= W*L^2/12= 286.03125 KNM
MV@CD = -W*L^2/12=-184.78125 KNM
MV@DC= W*L^2/12= 184.78125 KNM
MV@DE= -W*L^2/12= -286.03125 KNM
MV@ED=W*L^2/12= 286.03125 KNM
STEP 2;SLOPE DEFLECTION EQUATION
SPAN AB
M@AB=MV@AB+(4EI/L)*TA+(2EI/L)*TB+6EI*D/L^2
M@AB=-286.03125+0.4445EI*TA+0.2223EI*TB
M@BA=MV@BA+(4EI/L)*TB+(2EI/L)*TA+6EI*D/L^2
M@BA=286.03125+0.4445EI*TB+0.2223EI*TA
SPAN BC
M@BC=MV@BC+(4EI/L)*TB+(2EI/L)*TC+6EI*D/L^2
M@BC=-286.03125+0.4445EI*TB+0.2223EI*TC
M@CB=MV@CB+(4EI/L)*TC+(2EI/L)*TB+6EI*D/L^2
M@CB=286.03125+0.4445EI*TC+0.2223EI*TB
SPAN CD
M@CD=MV@CD+(4EI/L)*TC+(2EI/L)*TD+6EI*D/L^2
M@CD=-184.78125+0.4445EI*TC+0.2223*EI*TD
M@DC=MV@DC+(4EI/L)*TD+(2EI/L)*TC+6EI*D/L^2
M@DC=184.78125+0.4445EI*TD+0.2223EI*TC
SPAN DE
M@DE=MV@DE+(4EI/L)*TD+(2EI/L)*TE+6EI*D/L^2
M@DE=-286.03125+0.4445EI*TD+0.2223EI*TE
M@ED=MV@ED+(4EI/L)*TE+(2EI/L)*TD+6EI*D/L^2
M@ED=286.03125+0.4445EI*TE+0.2223EI*TD
STEP 3 ;EQUILIBRIUM CONDITIONS
0=-286.03125+0.4445EI*TA+0.2223EI*TB
EI*TA=643.489-0.5001EI*TB
0=286.03125+0.4445EI*TE+0.2223EI*TD
EI*TE=-643.489-0.5001EI*TD
AT JOINT B, SIGMA M@B=0
MBA+MBC=0
0.3334EI*TB+429.078-286.03125+0.4445EI*TB+0.2223EI*TC=0
0.7779EI*TB+0.2223EI*TC=-143.0467
AT JOINT C,SIGMA M@C=0
MCB+MCD=0
286.03125+0.4445EI*TC+0.2223EI*TB-286.03125-184.78125+0.4445EI*TC+0.2223EI*TD=0
0.2223EI*TB+0.889EI*TC+0.2223EI*TD=-101.25
AT JOINT D, SIGMA M@D=0
MDC+MDE=0
184.78125+0.4445EI*TD+0.2223EI*TC+0.3334EI*TD-429.078=0
0.2223EI*TC+0.7779*EI*TD=244.2975
EI*TB=-135.0624 RADIAN
EI*TC= -170.8575 RADIANS
EI*TD= 362.8732 RADIAN
EI*TA= 711.0337 RADIAN
EI*TE=-824.9618 RADIAN
M@AB=-286.03125+0.4445*711.0337+0.2223*-135.0624
M@AB= 0 KNM
M@BA=286.03125+0.4445*-135.0624+0.2223*711.0337
M@BA=384.0588 KNM
M@BC=-286.03125+0.4445*-135.0624+0.2223*-170.8575
M@BC=-384.0588 KNM
M@CB=286.03125+0.4445*-170.8575+0.2223*-135.0624
M@CB=180.060 KNM
M@CD=-184.78125+0.4445*-170.8575+0.2223*362.8732
M@CD=-180.060 KNM
M@DC=184.78125+0.4445*362.8732+0.2223*-170.8575
M@DC=308.096 KNM
M@DE=-286.03125+0.4445*362.8732+0.2223*-824.9618
M@DE=-308.096 KNM
M@ED=286.03125+0.4445*-824.9618+0.2223*362.8732
M@ED= 0 KNM
STEP 4;SUPPORT REACTION AND BENDNG MOMENT
SPAN AB
V@AB*9-42.375*9*9/2+384.0588=0
V@AB=148.0143KN
MID SPAN B.M@MAX=AB=258.504 KNM
SPAN BC
V@BC*9-384.0588+180.060-42.375*9*9/2=0
V@BC=213.354 KN
V@CB=168.020 KN
MID SPAN B.M@MAX=BC=153.049 KNM
SPAN CD
V@CD*9-180.060+308.096-27.375*9*9/2=0
V@CD=108.961 KN
MID SPAN B.M@MAX=CD=36.789 KNM
SPAN DE
V@DE=224.92 KN
V@ED=156.454 KN
MID SPAN B.M@MAX=DE=288.823 KNM
CASE 3
NEGATIVE BENDING MOMENT AT MC
LIVE LOAD UDL ON SPAN BC AND CD
STEP 1; FIXED END MOMENTS
MV@AB= -W*L^2/12 = -184.78125 KNM
MV@BA= W*L^2/12= 184.78125 KNM
MV@BC= - W*L^2/12= -286.03125 KNM
MV@CB= W*L^2/12= 286.03125 KNM
MV@CD = -W*L^2/12=-286.03125 KNM
MV@DC= W*L^2/12= 286.03125 KNM
MV@DE= -W*L^2/12=-184.78125 KNM
MV@ED=W*L^2/12= 184.78125 KNM
STEP 2;SLOPE DEFLECTION EQUATION
SPAN AB
M@AB=MV@AB+(4EI/L)*TA+(2EI/L)*TB+6EI*D/L^2
M@AB=-184.78125+0.4445EI*TA+0.2223EI*TB
M@BA=MV@BA+(4EI/L)*TB+(2EI/L)*TA+6EI*D/L^2
M@BA=184.78125+0.4445EI*TB+0.2223EI*TA
SPAN BC
M@BC=MV@BC+(4EI/L)*TB+(2EI/L)*TC+6EI*D/L^2
M@BC=-286.03125+0.4445EI*TB+0.2223EI*TC
M@CB=MV@CB+(4EI/L)*TC+(2EI/L)*TB+6EI*D/L^2
M@CB=286.03125+0.4445EI*TC+0.2223EI*TB
SPAN CD
M@CD=MV@CD+(4EI/L)*TC+(2EI/L)*TD+6EI*D/L^2
M@CD=-286.03125+0.4445EI*TC+0.2223*EI*TD
M@DC=MV@DC+(4EI/L)*TD+(2EI/L)*TC+6EI*D/L^2
M@DC=286.03125+0.4445EI*TD+0.2223EI*TC
SPAN DE
M@DE=MV@DE+(4EI/L)*TD+(2EI/L)*TE+6EI*D/L^2
M@DE=-184.78125 +0.4445EI*TD+0.2223EI*TE
M@ED=MV@ED+(4EI/L)*TE+(2EI/L)*TD+6EI*D/L^2
M@ED=184.78125 +0.4445EI*TE+0.2223EI*TD
STEP 3 ;EQUILIBRIUM CONDITIONS
0=-184.78125+0.4445EI*TA+0.2223EI*TB
EI*TA=415.7058-0.5001EI*TB
0=184.78125+0.4445EI*TE+0.2223EI*TD
EI*TE=-415.7058-0.5001EI*TD
AT JOINT B, SIGMA M@B=0
MBA+MBC=0
0.3334EI*TB+277.1925-286.03125+0.4445EI*TB+0.2223EI*TC=0
0.7779EI*TB+0.2223EI*TC=8.8387
AT JOINT C,SIGMA M@C=0
MCB+MCD=0
286.03125+0.4445EI*TC+0.2223EI*TB-286.03125+0.4445EI*TC+0.2223EI*TD=0
0.2223EI*TB+0.889EI*TC+0.2223EI*TD=0
AT JOINT D, SIGMA M@D=0
MDC+MDE=0
286.03125+0.4445EI*TD+0.2223EI*TC+0.3334EI*TD-277.1925=0
0.2223EI*TC+0.7779*EI*TD=-8.8387
EI*TB=11.3622RADIAN
EI*TC= 0 RADIANS
EI*TD= -11.3622 RADIAN
EI*TA= 410.0235 RADIAN
EI*TE=-410.0235 RADIAN
M@AB=-184.78125+0.4445*410.0235+0.2223*11.3622
M@AB= 0 KNM
M@BA=184.78125+0.4445*11.3622+0.2223*410.0235
M@BA=280.9799 KNM
M@BC=-286.03125+0.4445*11.3622+0.2223*0
M@BC=-280.9799KNM
M@CB=286.03125+0.4445*0+0.2223*11.6322
M@CB=288.5570 KNM
M@CD=-286.03125+0.4445*0+0.2223*-11.6322
M@CD=-288.5570 KNM
M@DC=286.03125+0.4445*-11.3622+0.2223*0
M@DC=280.9807 KNM
M@DE=-184.78125+0.4445*-11.3622+0.2223*-410.0235
M@DE=-280.9807 KNM
M@ED=184.78125+0.4445*-410.0235+0.2223*-11.3622
M@ED= 0 KNM
STEP 4;SUPPORT REACTION AND BENDNG MOMENT
SPAN AB
V@AB=91.9675 KN
V@BA=154.4075 KN
MID SPAN B.M@MAX=AB=154.484 KNM
SPAN BC
V@BC*9-280.980+288.5570-42.375*9*9/2=0
V@BC=189.845 KN
V@CB=191.529 KN
MID SPAN B.M@MAX=BC=144.284 KNM
SPAN CD
V@CD*9-288.5570+280.9807-42.375*9*9/2=0
V@CD=191.529 KN
V@DC=189.845 KN
MID SPAN B.M@MAX=CD=144.284 KNM
SPAN DE
V@DE=154.4075 KN
V@ED=91.9675KN
MID SPAN B.M@MAX=DE=154.484 KNM
CASE 4
POSITIVE BENDING MOMENT AT MB,MC AND MD
LIVE LOAD UDL ON SPAN AB AND CD OR ALTERNATE SPAN
STEP 1; FIXED END MOMENTS
MV@AB= -W*L^2/12 = -286.03125 KNM
MV@BA= W*L^2/12= 286.03125 KNM
MV@BC= - W*L^2/12= -184.78125 KNM
MV@CB= W*L^2/12= 184.78125 KNM
MV@CD = -W*L^2/12=-286.03125 KNM
MV@DC= W*L^2/12= 286.03125 KNM
MV@DE= -W*L^2/12= -184.78125 KNM
MV@ED=W*L^2/12= 184.78125 KNM
STEP 2;SLOPE DEFLECTION EQUATION
SPAN AB
M@AB=MV@AB+(4EI/L)*TA+(2EI/L)*TB+6EI*D/L^2
M@AB=-286.03125+0.4445EI*TA+0.2223EI*TB
M@BA=MV@BA+(4EI/L)*TB+(2EI/L)*TA+6EI*D/L^2
M@BA=286.03125+0.4445EI*TB+0.2223EI*TA
SPAN BC
M@BC=MV@BC+(4EI/L)*TB+(2EI/L)*TC+6EI*D/L^2
M@BC=-184.78125+0.4445EI*TB+0.2223EI*TC
M@CB=MV@CB+(4EI/L)*TC+(2EI/L)*TB+6EI*D/L^2
M@CB=184.78125+0.4445EI*TC+0.2223EI*TB
SPAN CD
M@CD=MV@CD+(4EI/L)*TC+(2EI/L)*TD+6EI*D/L^2
M@CD=-286.03125+0.4445EI*TC+0.2223*EI*TD
M@DC=MV@DC+(4EI/L)*TD+(2EI/L)*TC+6EI*D/L^2
M@DC=286.03125+0.4445EI*TD+0.2223EI*TC
SPAN DE
M@DE=MV@DE+(4EI/L)*TD+(2EI/L)*TE+6EI*D/L^2
M@DE=-184.78125+0.4445EI*TD+0.2223EI*TE
M@ED=MV@ED+(4EI/L)*TE+(2EI/L)*TD+6EI*D/L^2
M@ED=184.78125+0.4445EI*TE+0.2223EI*TD
STEP 3 ;EQUILIBRIUM CONDITIONS
0=-286.03125+0.4445EI*TA+0.2223EI*TB
EI*TA=643.489-0.5001EI*TB
0=184.78125+0.4445EI*TE+0.2223EI*TD
EI*TE=-415.7058-0.5001EI*TD
AT JOINT B, SIGMA M@B=0
MBA+MBC=0
0.3334EI*TB+429.078-184.78125+0.4445EI*TB+0.2223EI*TC=0
0.7779EI*TB+0.2223EI*TC=-244.29675
AT JOINT C,SIGMA M@C=0
MCB+MCD=0
184.78125+0.4445EI*TC+0.2223EI*TB-286.03125+0.4445EI*TC+0.2223EI*TD=0
0.2223EI*TB+0.889EI*TC+0.2223EI*TD=101.25
AT JOINT D, SIGMA M@D=0
MDC+MDE=0
286.03125+0.4445EI*TD+0.2223EI*TC+0.3334EI*TD-277.19255=0
0.2223EI*TC+0.7779*EI*TD= -8.8387
EI*TB=-379.1510 RADIAN
EI*TC= 227.8222 RADIANS
EI*TD= -76.4668 RADIAN
EI*TA=833.1024 RADIAN
EI*TE=-377.4647 RADIAN
M@AB=-286.03125+0.4445*833.1024+0.2223*-379.1510
M@AB= 0 KNM
M@BA=286.03125+0.4445*-379.1510+0.2223*833.1024
M@BA=302.6972 KNM
M@BC=-184.78125+0.4445*-379.1510+0.2223*227.8222
M@BC=-302.6689 KNM
M@CB=184.78125+0.4445*227.8222+0.2223*-379.1510
M@CB=201.7629 KNM
M@CD=-286.03125+0.4445*227.8222+0.2223*-76.4668
M@CD=-201.7629 KNM
M@DC=286.03125+0.4445*-76.4668+0.2223*227.8222
M@DC=302.6866 KNM
M@DE=-184.78125+0.4445*-76.4668+0.2223*-377.4647
M@DE=-302.6866 KNM
M@ED=184.78125+0.4445*-377.4667+0.2223*-76.4668
M@ED= 0 KNM
STEP 4;SUPPORT REACTION AND BENDNG MOMENT
SPAN AB
V@AB*9-42.375*9*9/2+302.6972=0
V@BA=224.3205 KN
MID SPAN B.M@MAX=AB=291.0452 KNM
SPAN BC
V@BC*9-302.6689+201.7629-27.375*9*9/2=0
V@BC=134.399KN
MID SPAN B.M@MAX=BC=27.25KNM
SPAN CD
V@CD*9-201.7629+302.6866-42.375*9*9/2=0
V@DC=201.901 KN
MID SPAN B.M@MAX=CD=178.305 KNm
SPAN DE
V@DE*9-302.6866-27.375*9*9/2=0
V@DE=156.819 KN
V@ED=89.5556 KN
MID SPAN B.M@MAX=DE=146.4859 KNM
