shear force and bending moment diagram of trapezoidal load

problem figure

draw shear force and bending moment diagram of given figure. 

SIGN CONVENTION

sign convention
STEPS FOR FINDING REACTIONS 

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

1.   Apply First Equilibrium Equation 

∑FY( Summation Of Vertical Forces)= 0 

Here, TWO Vertical Reaction Forces Are Acting Upward. They Are +RA And +RB And ONE Downward Is Forces Acting On The Beam. i.e., UVL  Of  2KN/m TO 8KN/m 

Convert The UVL into Force by Calculating The Area Of the TRAPEZIUM.  

Divide The Trapezium into Rectangle and Triangle 

(Area of Trapezium = Area of Triangle + Area of Rectangle) 

((0.5*3*6)+(3*2))=15KN

Here, Base of Triangle & Rectanle = 3m 

Height Of  Triangle= 6KN/m

Height Of  Rectangle= 2KN/m

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. 

Force= Area of Triangle= 0.5*3*6=9KN 

if we are taking moment at the perpendicular side of triangle 

= F*(b/3) (Node B

if we are taking moment at the vertex side of triangle=  

F*(2b/3) (Node A) 

bending moment distance

(Better See The Sign Convention For Positive And Negative Moment Or Clockwise And Anticlockwise Moment). 

By Solving The Two Equations We Can Calculate Both The Support Reaction. 

FINDING THE REACTIONS

∑FY=0, 

RA-(0.5*3*6)-(2*3) +RB =0 

RA+RB=15Kn     (Eqn-1) 

LET US USE ∑M=0,  

∑MA=0 

(RA*0)-(2*3*1.5)-(0.5*3*6*(3*(2/3))) +3RB=0 

RB= (27/3) =9KN

FROM (Eqn-1), 

RA+9=15Kn 

RA=6KN 

To Calculate The End Shear Force 

 Take summation of forces acting on node or member. 

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

END SHEAR FORCE

FA=RA= 6KN 

FBL=6-(2*3)-(0.5*3*6) =-9KN 

FBL=6-(2*3)-(0.5*3*6) +9= 0 

To Calculate The Mid Shear Force 

 Take Summation of forces acting on node or member. 

MID SHEAR FORCE

F (A-B) =6-(2*1.5)-(0.5*1.5*(6/2))= 0.75KN 

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. 

END BENDING MOMENTS  

MA= (360) = 0        (AS, THE DISTANCE IS ZERO) 

MB= (6*3)-(2*3*1.5)-(0.5*3*6*(3*(1/3))) = 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. 

MID BENDING MOMENTS

M (A-B) = (6*1.5)-(2*1.5*(1.5/2))-(0.5*1.5*(6/2)*(1.5*(1/3))) = 5.625KNm 

shear force and bending moment diagram manually
shear force and  bending moment diagram manual
STAAD OUTPUT
reaction staad
end shear force staad
mid shear force staad
end moment staad
mid moment staad

STAAD RESULT SUMMARY

FX = FORCE IN HORIZONTAL DIRECTION

FY = FORCE IN VERTICAL DIRECTION

MZ = BENDING MOMENT

result staad

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