Shear force Bending moment of mid point

figure

DRAW SHEAR FORCE BENDING MOMENT DIAGRAM OF GIVEN FIGURE. 

Remember, there are four beams AB,BC,CD & DE & and given point load of 3KN is nodal load not a concentrated load.

SIGN CONVENTION 

sign convention

STEPS FOR FINDING REACTIONS 

Here, In The Above Figure We Have Two Unknown Reaction Force RA And RE. 

1.   Apply First Equilibrium Equation 

∑FY( Summation Of Vertical Forces)= 0 

Here, Two Vertical Reaction Forces Are Acting Upward. They Are +RA  and +RE And Three Downward Forces Acting  On A Beam. They Are UDL Of -4KN/m, -3KN & -6KN/m Respectively. 

Convert The UDL Into Force By Multiplying With The Distance Up to Which The UDL Load Is Acting. 

e.g. UDL Of 4KN/m Acting On A 2m Span Beam BC. 

So, The Force Will Be =(4*2)= 8KN. (Better See The Sign Convention For Positive And Negative Force) 

Take The Summation Of All Vertical Forces Acting On The Beam (Both Upward And Downward). 

2.   Apply Second Equilibrium Equation 

∑M(Summation Of Moments)= 0 

Formula Of Moment= Force*Distance. 

Distance For The Moment Of Udl Load Will Be Half Of Its Span Of The Beam. 

e.g. ∑Md= +(Ra*7)-(4*2*(1+3))-(3*3)-(6*3*1.5) 

Here We Have Taken ∑M From A To D. 

So, Ra is 7m Away From D, UDL Of 4KN/m Is 3m Away From D First We Have Converted It Into Force By Multiplying The UDL With Its Span (4*2) And Again Multiplying With The Distance 1m As Distance For Moment Of UDL Will Be Half Of Its Span (2/2) and added The Remaining Distance Up to D  3m

Nodal Load Of 3KN is 3m Away From D. 

We Have Multiplied Load With The Distance (3*3)

For UDL Of 6KN/m We Have Converted It Into Force By Multiplying The UDL With Its Span (6*3) And Again Multiplying With The Distance  1.5m As Distance For Moment Of UDL Will Be Half Of Its Span (3/2). (Better See The Sign Convention For Positive And Negative Moment Or Clockwise And Anticlockwise Moment) 

By Solving  The Equations We Have Calculated Both The Support Reaction. 

Once We Got The Values Of Reaction We Can Calculate Shear Force And Bending Moment. 

FINDING THE REACTIONS

∑FY=0, 

RA- (4*2)-3-(6*3) +RE =0 

RA+RE=29KN     (Eqn-1) 

LET US USE ∑M=0,  

∑MA=0 

(RA*0)+ (4*2*3) + (3*4) + (6*3*5.5)-9RE=0 

RE=(135/9)= 15KN 

FROM (Eqn-1), 

RA+15=29Kn 

RA= 14KN 

To Calculate The End Shear Force 

 Take summation of forces acting on each node or member. 

e.g. FC= +RA-(4*2)-3  

In the below calculaton L indicates left and R indicates right. 

END SHEAR FORCE  

FA=RA= 14KN 

FB= 14KN 

FCL=14-(4*2)= 6KN 

FCR=14-(4*2)-3= 3KN 

FD=14-(4*2)-3-(6*3)= -15KN 

FE=14-(4*2)-3-(6*3)+15= 0 

To Calculate The Mid Shear Force 

Take summation of forces acting on each node or member. 

e.g. F(C-D)= +Ra-(4*2)-3-(6*1.5) 

Here we have multiplied udl of 6kn/m with 1.5m as we are calculating mid shear force because we want to know the shear force upto mid span of member CD. This will be applied only for udl or uvl load case not for nodal or concentrated load.

MID SHEAR FORCE

F(A-B)= 14KN 

F(B-C)=14-(4*1)= 10KN 

F(C-D)=14-(4*2)-3-(6*1.5)= -6KN 

F(D-E)=14-(4*2)-3-(6*3)= -15KN 

To Calculate The End Bending Moment  

Calculate moments acting on each node or member and add the remaining distance upto which node you want to calculate the moment. 

e.g. ME=(RA*9)-(4*2*(1+5))-(3*5)-(6*3*(1.5+2)) 

END BENDING MOMENTS

MA=(14*0)= 0        (AS, THE DISTANCE IS ZERO) 

MB=(14*2)= 28KNm 

MC=(14*4)-(4*2*1)= 48KNm 

MD=(14*7)-(4*2*4)-(3*3)-(6*3*1.5)= 30KNm 

ME=(14*9)-(4*2*6)-(3*5)-(6*3*3.5)= 0 

To Calculate The Mid Bending Moment  

Calculate moments acting on each node or member and add the remaining distance upto which node you want to calculate the moment. 

e.g. M(D-E)=(RA*8)-(4*2*(1+4))-(3*4)-(6*3*(1.5+1)) 

We are considering distance only upto mid span of member DE. 

MID BENDING MOMENTS

M(A-B)=(14*1)= 14KNm 

M(B-C) = (14*3)-(4*1*0.5)= 40KNm 

M(C-D)=(14*5.5)-(4*2*2.5)-(3*1.5)-(6*1.5*0.75)= 45.75KNm 

M(D-E)=(14*8)-(4*2*5)-(3*4)-(6*3*2.5)= 15KNm 

SFD & BMD MANUAL

FOR BETTER UNDERSTANDING OF SHEAR AND BENDING MOMENTS SEE THE SIGN CONVENTIONS.

STEPS TO ANALYSE THE BEAM IN STAAD PRO 

  • Open the staad pro software. 
  • Click on new project then select any mode space or plane, set the units of force and length according to the figure, give a name to the project and click on next and finish
  • Grid will be open automatically with help of node model the beam AB, BC, CD & DE according to the figure. (you can model in isometric view or click on yellow cube type box located just below the  file menu left top corner of the staad window  view from +Z (front view)). 
  • Go to General tab to define property of the beam, click on define select rectangle define the beam property (depth & witdh) then click on add and close the property tab. 
  • Select the beam size/property which have been defined, click on assign to view, assign and click on yes. 
  • Go to support tab to assign support to the beam, click on create select pinned to create pinned support and then add it and again click on create select Fixed But then release FX and MZ to create Roller support and add it. 
  • Click on Pinned support which you have added and select the first end node(A) and click on assign to assign the pinned support.  
  • Click on Roller support which you have added and select the last end node(B) and click on assign to assign the Roller support. 
  • Goto Load & Definition tab select load case detail and click on add and close it. 
  • Select load case1 and click on add, click on nodal load, node and define the load according to figure. In the given figure 3KN of nodal load is acting downward on the node C as it is downward load so define it as -3KN in FY tab (vertical load) and click on add and close it. 
  • Again select load case1 and click on add, click on member load, uniform force and define the load according to figure. In the given figure 4KN/m of udl load is acting downward on the member BC as it is downward load so define it as -4KN/m in W1 tab and direction should be GY (vertical load) and click on add it.  
  •  Define another 6KN/m of udl load acting on beam CD as it is downward load so define it as  -4KN/m in W1 tab and direction should be GY (vertical load) and click on add and close the load window. 
  • Select the nodal load (-3KN) which have been defined, click on node cursor located near the left of the modeling tab, select the node on which you want to assign (node C), click on assign to selected node, assign and click on yes. 
  • Select the udl load (-4KN/m) which have been defined, click on beam cursor located near the left of the modeling window, select the member on which you want to assign (member BC), click on assign to selected beam, assign and click on yes. 
  • Repeat the same steps for udl of -6KN/m acting on beam BC and Assign it. 
  • Click on Analysis/print  in modeling window click on add and close it. 
  • To analyse the beam click analyze menu located in the same line of file menu at the top portion of staad window click on run analysis and save the file or press CTRL+F5. 
  • Check for the errors, if it is showing zero errors. 
  • Then click on go to post processing mode, click on apply and ok. 
  •  Go to reaction tab to display the support reaction values RA & RE and compare it with the manual reaction value. 
  • Go to Beam tab, it will automatically show the bending moment diagram to see the values.
  • Go to the Results menu located in the same line of file menu at the top portion of staad window, click on view values, then click on Beam results. 
  •  To view end bending moment values click on bending option and then click on Annotate, simultaneously press CTRL and scroll mouse wheel to increase the size of the graph/diagram and compare the Staad values with the manual values. 
  • To view mid bending moment values uncheck the bending option click on mid point option and then click on Annotate, simultaneously press CTRL and scroll mouse wheel to increase the size of the graph/diagram and compare the Staad values with the manual values. 
  • BENDING MOMENT DIAGRAM IN STAAD IS SHOWING DOWNWARD DUE TO NEGATIVE VALUE BECAUSE OF SIGN CONVENTION, WE HAVE TAKEN CLOCK WISE MOMENT AS POSITIVE BUT IN STAAD PRO  CLOCK WISE MOMENT IS TAKEN AS NEGATIVE AND VICE VERSA SO DON’T GET CONFUSE WITH DIAGRAM. 
  • YOU CAN DRAW BMD SAME AS STAAD BY TAKING CLOCKWISE MOMENT AS NEGATIVE AND ANTI CLOCK WISE MOMENT AS POSITIVE.
  • To view the Shear force diagram uncheck the mid point option and turn off the bending moment diagram MZ(bending Z moment) located at the third line of the user interface and turn on the FY(shear force) 
  • To view End shear force values click on shear force End option and then click on Annotate, simultaneously press CTRL and scroll mouse wheel to increase the size of the graph/diagram and compare the Staad values with the manual values. 
  • To view Mid shear force values uncheck the shear end and click on  mid point option and then click on Annotate, simultaneously press CTRL and scroll mouse wheel to increase the size of the graph/diagram and compare the Staad values with the manual values. 

STAAD OUTPUT

reaction staad
end shear force staad
mid shear force staad
end bending moment staad
mid bending moment staad

STAAD RESULT SUMMARY

FX = FORCE IN HORIZONTAL DIRECTION

FY = FORCE IN VERTICAL DIRECTION

MZ = BENDING MOMENT

staad result

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